advances in difference equationsdownloads.hindawi.com/journals/specialissues/207192.pdfthis is an...

Volume 2010Hindawi Publishing Corporationhttp://www.hindawi.com

Advances inDifferenceEquationsEditor-in-Chief: Ravi P. Agarwal

Special Issue Abstract Differential and Difference Equations

Guest Editors G. M. N'Guérékata, T. Diagana, and A. Pankov

Abstract Differential and DifferenceEquations

Advances in Difference Equations

Abstract Differential and DifferenceEquations

Guest Editors: G. M. N’Guerekata, T. Diagana, and A. Pankov

Copyright q 2010 Hindawi Publishing Corporation. All rights reserved.

This is an issue published in volume 2010 of “Advances in Difference Equations.” All articles are open access articlesdistributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and repro-duction in any medium, provided the original work is properly cited.

Editor-in-ChiefRavi P. Agarwal, Florida Institute of Technology, USA

Associate Editors

Dumitru Baleanu, TurkeyMouffak Benchohra, AlgeriaLeonid Berezansky, IsraelT. Gnana Bhaskar, USAMartin Bohner, USAA. Boucherif, Saudi ArabiaElena Braverman, CanadaAlberto Cabada, SpainM. Cecchi, ItalyM. A. Chaudhry, Saudi ArabiaClaudio Cuevas, Brazil

Toka Diagana, USAJosef Diblık, Czech RepublicOndrej Dosly, Czech RepublicPaul W. Eloe, USAS. R. Grace, EgyptJohn R. Graef, USAIstvan Gyori, HungaryV. Lakshmikantham, USAYongwimon Lenbury, ThailandRigoberto Medina, ChileR. V. Nicholas Melnik, Canada

Kanishka Perera, USAR. L. Pouso, SpainM. Sambandham, USALeonid Shaikhet, UkraineSvatoslav Stanek, Czech RepublicEthiraju Thandapani, IndiaP. J. Y. Wong, SingaporeJianshe Yu, ChinaA. Zafer, TurkeyBinggen Zhang, China

Contents

Abstract Differential and Difference Equations, G. M. N’Guerekata, T. Diagana,and A. PankovVolume 2010, Article ID 857306, 2 pages

Pairs of Function Spaces and Exponential Dichotomy on the Real Line, Adina Luminita SasuVolume 2010, Article ID 347670, 15 pages

On a Max-Type Difference Equation, Ali Gelisken, Cengiz Cinar, and Ibrahim YalcinkayaVolume 2010, Article ID 584890, 6 pages

Inequalities among Eigenvalues of Second-Order Symmetric Equations on Time Scales,Chao Zhang and Shurong SunVolume 2010, Article ID 317416, 24 pages

Mild Solutions for Fractional Differential Equations with Nonlocal Conditions, Fang LiVolume 2010, Article ID 287861, 9 pages

Uniqueness of Periodic Solution for a Class of Lienard p-Laplacian Equations, Fengjuan Cao,Zhenlai Han, Ping Zhao, and Shurong SunVolume 2010, Article ID 235749, 14 pages

Existence of Periodic Solutions for p-Laplacian Equations on Time Scales, Fengjuan Cao,Zhenlai Han, and Shurong SunVolume 2010, Article ID 584375, 13 pages

A Note on a Semilinear Fractional Differential Equation of Neutral Type with Infinite Delay,Gisle M. Mophou and Gaston M. N’GuerekataVolume 2010, Article ID 674630, 8 pages

Almost Automorphic Solutions to Abstract Fractional Differential Equations, Hui-Sheng Ding,Jin Liang, and Ti-Jun XiaoVolume 2010, Article ID 508374, 9 pages

Existence and Uniqueness of Periodic Solutions for a Class of Nonlinear Equations withp-Laplacian-Like Operators, Hui-Sheng Ding, Guo-Rong Ye, and Wei LongVolume 2010, Article ID 197263, 11 pages

Structure of Eigenvalues of Multi-Point Boundary Value Problems, Jie Gao, Dongmei Sun,and Meirong ZhangVolume 2010, Article ID 381932, 24 pages

Asymptotically Almost Periodic Solutions for Abstract Partial Neutral Integro-DifferentialEquation, Jose Paulo C. dos Santos, Sandro M. Guzzo, and Marcos N. RabeloVolume 2010, Article ID 310951, 26 pages

Existence of a Nonautonomous SIR Epidemic Model with Age Structure,Junyuan Yang and Xiaoyan WangVolume 2010, Article ID 212858, 23 pages

Complete Asymptotic Analysis of a Nonlinear Recurrence Relation with Threshold Control,Qi Ge, Chengmin Hou, and Sui Sun ChengVolume 2010, Article ID 143849, 19 pages

Existence and Uniqueness of Positive Solutions for Discrete Fourth-Order Lidstone Problemwith a Parameter, Yanbin Sang, Zhongli Wei, and Wei DongVolume 2010, Article ID 971540, 18 pages

Exponential Decay of Energy for Some Nonlinear Hyperbolic Equations with StrongDissipation, Yaojun YeVolume 2010, Article ID 357404, 12 pages

Oscillation Criteria for Second-Order Quasilinear Neutral Delay Dynamic Equations on TimeScales, Yibing Sun, Zhenlai Han, Tongxing Li, and Guangrong ZhangVolume 2010, Article ID 512437, 14 pages

Existence of 2n Positive Periodic Solutions to n-Species Nonautonomous Food Chains withHarvesting Terms, Yongkun Li and Kaihong ZhaoVolume 2010, Article ID 262461, 17 pages

Notes on the Propagators of Evolution Equations, Yu Lin, Ti-Jun Xiao, and Jin LiangVolume 2010, Article ID 795484, 8 pages

On the Oscillation of Second-Order Neutral Delay Differential Equations, Zhenlai Han,Tongxing Li, Shurong Sun, and Weisong ChenVolume 2010, Article ID 289340, 8 pages

Solutions to Fractional Differential Equations with Nonlocal Initial Condition in BanachSpaces, Zhi-Wei Lv, Jin Liang, and Ti-Jun XiaoVolume 2010, Article ID 340349, 10 pages

Hindawi Publishing CorporationAdvances in Difference EquationsVolume 2010, Article ID 857306, 2 pagesdoi:10.1155/2010/857306

EditorialAbstract Differential and Difference Equations

G. M. N’Guerekata,1 T. Diagana,2 and A. Pankov1

1 Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA2 Department of Mathematics, Howard University, Washington, DC 20005, USA

Correspondence should be addressed to G. M. N’Guerekata, [email protected]

Received 31 Decemeber 2010; Accepted 31 Decemeber 2010

Copyright q 2010 G. M. N’Guerekata et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

This special issue of Advances in Difference Equations is devoted to highlight somerecent developments in abstract differential equations, fractional differential equations,and difference equations and their applications to mathematical physics, engineering, andbiology. It consists of 20 papers carefully selected through a rigorous peer review.

The first category of papers deals with the asymptotic and oscillatory behavior ofsolutions to various abstract differential equations and fractional differential equations.Periodic problems involving the scalar p-Laplacian equation on time scales, or n-speciesnonautonomous food chains with harvesting terms are studied using the Mawhin’scontinuation theorem. Some new oscillation criteria for the second-order quasilinear neutraldelay dynamic equations and nonlinear delay dynamic equations on a time scale T, areestablished, improving some known results for oscillation of second-order nonlinear delaydynamic equations on time scales.

The study of almost automorphic functions in Bochner’s sense have attractedseveral mathematicians since the publication of N’Guerekata’s book in 2001. Recently, theirapplications to fractional differential equations have become an emerging field. A new andgeneral existence and uniqueness theorem of almost automorphic solutions for the semilinearfractional differential equation

Dαtu(t) = Au(t) + D(α−1)tf(t, u(t)), 1 < α < 2, (1)

in complex Banach spaces, with Stepanov-like almost automorphic coefficients is obtained,and applications to fractional relaxation-oscillation equations are presented. The methodused here can be applied successfully to a large class of fractional differential equations.

Another topic encountered in this issue is the existence of asymptotically almostperiodic mild solutions for a class of abstract partial neutral integrodifferential equations with

2 Advances in Difference Equations

unbounded delay. The study of such equations is motivated by different concrete examplesin various technical fields. For instance the equation

d

dt

[

u(t) − λZ

∫ t

−∞C(t − s)u(s)ds

]

= Au(t) + λZ∫ t

−∞B(t − s)u(s)ds − p(t) + q(t) (2)

arises in the study of the dynamics of income, employment, value of capital stock, andcumulative balance of payment.

Abstract partial neutral differential equations also appear in the theory of heatconduction. In the classic theory of heat conduction, it is assumed that the internal energyand the heat flux depend linearly on the temperature and on its gradient.

Under these conditions, the classic heat equation describes sufficiently well theevolution of the temperature in different types of materials. However, this description is notsatisfactory in materials with fading memory. In the theory developed by J. Nunziato, M. E.Gurtin, and A. C. Pipkin, the internal energy and the heat flux are described as functionals of uand ux. An abstract and more general version of neutral system describing such phenomenais considered. The existence and qualitative properties of an exponentially stable resolventoperator for a class of integrodifferential system is studied.

The theory of functional differential equations has emerged as an important branch ofnonlinear analysis. It is worthwhile mentioning that several important problems of the theoryof ordinary and delay differential equations lead to investigations of functional differentialequations of various types (see the books by Hale and Verduyn Lunel, Wu, and articles byLiang, Xiao, Mophou, N’Guerekata, Benchohra, Lizama, Hernandez, etc. and the referencestherein). On the other hand, the theory of fractional differential equations is also intensivelystudied and finds numerous applications in describing real world problems (see e.g., themonographs of Lakshmikantham et al., Vatsala, Podlubny, and the papers of Agarwal et al.,Benchohra et al.). In this issue, the existence of mild solutions to various fractional differentialequations with nonlocal conditions or with infinite delay is studied using classical fixed pointtheorems.

Also, recently, the study of max-type difference equation attracted a considerableattention. Although max-type difference equations are relatively simple in form, it isunfortunately extremely difficult to understand thoroughly the behavior of their solutions.The max operator arises naturally in certain models in automatic control theory. Furthermore,difference equation appear naturally as a discrete analogue and as a numerical solutionof differential and delay differential equations having applications in various scientificbranches, such as ecology, economy, physics, technics, sociology, and biology. Asymptoticbehavior of the positive solutions of a general difference equations is studied in a fine paper,improving recent results by Yang et al.

G. M. N’GuerekataT. DiaganaA. Pankov


Research ArticlePairs of Function Spaces and ExponentialDichotomy on the Real Line

Adina Luminita Sasu

Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timisoara,C. Coposu Boulevard. no. 4, 300223 Timisoara, Romania

Correspondence should be addressed to Adina Luminita Sasu, [email protected]

Received 15 January 2010; Accepted 21 January 2010

Academic Editor: Gaston Mandata N’Guerekata

Copyright q 2010 Adina Luminita Sasu. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We provide a complete diagram of the relation between the admissibility of pairs of Banachfunction spaces and the exponential dichotomy of evolution families on the real line. We provethat if W ∈ H(R) and V ∈ T(R) are two Banach function spaces with the property that eitherW ∈ W(R) or V ∈ V(R), then the admissibility of the pair (W(R, X), V (R, X)) implies the existenceof the exponential dichotomy. We study when the converse implication holds and show that thehypotheses on the underlying function spaces cannot be dropped and that the obtained results arethe most general in this topic. Finally, our results are applied to the study of exponential dichotomyof C0-semigroups.

1. Introduction

In the study of the asymptotic behavior of evolution equations the input-output conditionsare very efficient tools, with wide applicability area, and give a nice connection betweencontrol theory and the qualitative theory of differential equations (see [1–16] and thereference therein). Starting with the pioneering work of Perron (see [8]) these methodswere developed and improved in remarkable books (see [1, 4, 6]). A new and interestingperspective on this framework was proposed in [5], where the authors presented a completestudy of stability, expansiveness, and dichotomy of evolution families on the half-line interms of input-output methods. This paper was the starting point for an entire collectionof studies dedicated to the input-output techniques and their applications to the qualitativetheory of differential and difference equations.

If one analyzes the dichotomous properties of differential equations, then it is easilyseen that there are some main technical differences between the case of evolution families


on the half-line (see [5, 9, 10]) and the case of evolution families on the real line (see [11–16]), which require a distinct analysis for each case. For instance, when one determinessufficient conditions for the existence of exponential dichotomy on the half-line, an importanthypothesis is that the initial stable subspace is closed and complemented (see, e.g., [5,Theorem 4.3] or [9, Theorem 3.3]). This assumption may be dropped when we study theexponential dichotomy on the real line (see, e.g., [11, Theorem 5.1] or [16, Theorem 5.3]).These facts implicitly generate the differences between the admissibility concepts used on thereal line compared with those used on the half-line and also interesting technical approachesin each case.

The aim of the present paper is to provide new and very general conditions for theexistence of exponential dichotomy on the real line. We consider the problem of findingconnections between the solvability of an integral equation and the existence of exponentialdichotomy of evolution families on the real line. The main purpose is to obtain a completediagram and a classification of the classes of function spaces that may be used in the study ofexponential dichotomy via admissibility.

For the beginning we will present the previous results in this topic and the mainobjectives will be clearly specified in the context of the actual state of knowledge. We denoteby T(R) the class of all Banach sequence spaces B which are invariant under translations,contain the continuous functions with compact support, satisfy an integral property and ifB \ L1(R,R)/= ∅, then there is a continuous function ϕ ∈ B \ L1(R,R). We consider H(R)the subclass of T(R) satisfying the ideal property. We associate two subclasses of H(R):W(R)—the class of all Banach function spaces with unbounded fundamental function andV(R)—the class of all Banach function spaces which contain at least a nonintegrable function.A pair of function spaces (W(R, X), V (R, X)) is called admissible for an evolution familyU = {U(t, s)}t≥s on the Banach space X if for every test function in the input space V (R, X)there exists a unique solution function in the output spaceW(R, X) for the associated integralequation given by the variation of constants formula (see Definition 3.5 below).

For the first time, we have proposed in [11] a sufficient condition for exponentialdichotomy, using certain Banach function spaces which are invariant under translations andwe obtained the following theorem.

Theorem 1.1. If V ∈ V(R) and the pair (Cb(R, X), V (R, X)) is admissible for an evolution familyU = {U(t, s)}t≥s, then U is exponentially dichotomic.

Our study has been continued and extended in [16], both for uniform dichotomyand exponential dichotomy. According to the proof of Theorem 4.8 in [16] we may give thefollowing sufficient condition for uniform dichotomy.

Theorem 1.2. If W ∈ H(R), V ∈ T(R), and the pair (W(R, X), V (R, X)) is admissible for anevolution family U = {U(t, s)}t≥s, then U is uniformly dichotomic.

From the proof of Theorem 5.3(i) in [16] we deduce the following sufficient conditionfor exponential dichotomy.

Theorem 1.3. If W ∈ W(R), V ∈ T(R), and the pair (W(R, X), V (R, X)) is admissible for anevolution family U = {U(t, s)}t≥s, then U is exponentially dichotomic.

Taking into account the above results and their consequences, the natural questionarises whether, in the general case, the output space may belong to the class H(R) and

Advances in Difference Equations 3

if so, which is the most general class where the input space should belong to. The aimof the present paper is to answer this question and to provide a complete study of theexponential dichotomy on the real line via integral admissibility. The answer to the abovequestion will establish clearly how should one modify the hypotheses of Theorem 1.2 suchthat the admissibility of the pair (W(R, X), V (R, X)) implies the existence of the exponentialdichotomy.

We will prove that if W ∈ H(R) and V ∈ V(R), then the admissibility of the pair(W(R, X), V (R, X)) is a sufficient condition for exponential dichotomy. Consequently, wewill deduce a complete diagram of the study of exponential dichotomy on the real line interms of the admissibility of function spaces (see Theorem 3.11). Specifically, if W ∈ H(R)and V ∈ T(R) are two Banach function spaces with the property that either W ∈ W(R)or V ∈ V(R), then the admissibility of the pair (W(R, X), V (R, X)) implies the existenceof the exponential dichotomy. Also, in certain conditions, we deduce that the exponentialdichotomy of an evolution family U = {U(t, s)}t≥s is equivalent with the admissibility of thepair (W(R, X), V (R, X)).

By an example we motivate our techniques and show that the hypotheses from ourmain results cannot be removed. Precisely, if W ∈ H(R) and V ∈ T(R) are such thatW /∈W(R) and V /∈V(R), then we prove that the admissibility of the pair (W(R, X), V (R, X))does not imply the exponential dichotomy. Moreover, we show that the obtained results andtheir consequences are the most general in this topic.

Finally, our results are applied at the study of the exponential dichotomy of C0-semigroups. Using function spaces which are invariant under translations, we obtain aclassification of the classes of input and output spaces which may be used in the study ofexponential dichotomy of semigroups in terms of input-output techniques with respect toassociated integral equations.

2. Preliminaries: Banach Function Spaces

In this section, for the sake of clarity, we present some definitions and notations and weintroduce the main classes of function spaces that will be used in our study. Let M(R) bethe linear space of all Lebesgue measurable functions u : R → R, identifying the functionsequal almost everywhere.

Definition 2.1. A linear subspace B ofM(R) is called normed function space if there is a mapping| · |B : B → R+ such that

(i) |u|B = 0 if and only if u = 0 a.e.;

(ii) |αu|B = |α||u|B, for all (α, u) ∈ R × B;

(iii) |u + v|B ≤ |u|B + |v|B, for all u, v ∈ B;

(iv) if u, v ∈ B and |u| ≤ |v| a.e. then |u|B ≤ |v|B;

(v) if u ∈ B, then |u| ∈ B.

If (B, | · |B) is complete, then B is called Banach function space.

Definition 2.2. A Banach function space (B, | · |B) is said to be invariant under translations if forevery (u, s) ∈ B × R, the function us : R → R, us(t) = u(t − s) belongs to B and |us|B = |u|B.


Notations 1. Let Cc(R,R) denote the linear space of all continuous functions v : R → R withcompact support. Throughout this paper, we denote by T(R) the class of all Banach functionspaces B, which are invariant under translations, Cc(R,R) ⊂ B, and satisfy the followingconditions:

(i) for every t > s there is α(t, s) > 0 such that∫ ts|u(τ)|dτ ≤ α(t, s)|u|B, for all u ∈ B;

(ii) if B \ L1(R,R)/= ∅ then there is a continuous function ϕ ∈ B \ L1(R,R).

For examples of Banach function spaces from the class T(R) we refer to [11].LetH(R) be the class of all Banach function spaces B ∈ T(R) with the property that if

|u| ≤ |v| a.e. and v ∈ B, then u ∈ B.

For every A ⊂ R we denote by χA the characteristic function of the set A. Then, ifB ∈ H(R), we have that χ[a,b) ∈ B, for every a, b ∈ R with a < b.

Definition 2.3. Let B ∈ H(R). The mapping FB : (0,∞) → R, FB(t) = |χ[0,t)|B is called thefundamental function of the space B.

For the proof of the next proposition we refer to [16, Proposition 2.8].

Proposition 2.4. Let B ∈ H(R) and ν > 0. If u : R → R+ is a function, which belongs to B andwith the property that qu : R → R+, qu(t) =

∫ t+1t u(s)ds belongs to B, then the functions

fu, gu : R −→ R+, fu(t) =∫ t

−∞e−ν(t−s)u(s)ds, gu(t) =

∫∞

t

e−ν(s−t)u(s)ds (2.1)

belong to B.

Example 2.5. Let M1(R,R) be the linear space of all u ∈ M(R) with the property thatsupt∈R

∫ t+1t |u(s)|ds <∞. With respect to the norm

‖u‖M1 := supt∈R

∫ t+1

t

|u(s)|ds, (2.2)

this is a Banach function space which belongs toH(R).

Lemma 2.6. If B ∈ T(R), then B ⊂M1(R,R).

Proof. Let α > 0 be such that∫1

0|u(τ)|dτ ≤ α|u|B, for all u ∈ B. Then, we have that

∫ t+1

t

|v(τ)|dτ =∫1

0|vt(τ)|dτ ≤ α|vt|B = α|v|B, ∀t ∈ R, ∀v ∈ B. (2.3)


Notations 2. In what follows we denote by

(i) W(R) the class of all Banach function spaces B ∈ H(R) with supt>0FB(t) =∞;

(ii) V(R) the class of all Banach function spaces B ∈ T(R) with the property that B \L1(R,R)/= ∅;

(iii) O(R) the class of all Banach function spaces B ∈ H(R) with the property that forevery u : R → R+ in B, the function qu : R → R+, qu(t) =

∫ t+1t u(s)ds belongs to B.

Remark 2.7. (i) For examples of Banach function spaces from the class W(R) we refer to [16,Proposition 2.9].

(ii) If B ∈ V(R) then there is a continuous function ϕ : R → R+ with ϕ ∈ B \ L1(R,R).

Notation 1. Let C0(R,R) be the space of all continuous functions v : R → R withlimt→±∞v(t) = 0, which is Banach space with respect to the norm |‖v‖| := supt∈R|v(t)|.

Lemma 2.8. Let B be a Banach function space with B ∈ H(R) \ W(R). Then C0(R,R) ⊂ B.

Proof. Let L := supt>0FB(t). Let v ∈ C0(R,R). Then there is an unbounded increasing sequence(tn) ⊂ (0,∞) such that |v(t)| ≤ 1/(n + 1), for all |t| ≥ tn and all n ∈ N. Setting vn = vχ[−tn,tn] wehave that

∣∣vn+p − vn∣∣B≤ 1n + 1

(∣∣∣χ[−tn+p,−tn)

∣∣∣B+∣∣∣χ(tn,tn+p]

∣∣∣B

)≤ 2Ln + 1

, ∀p ∈ N∗, ∀n ∈ N. (2.4)

From the above inequality we deduce that (vn) is fundamental in the Banach space B, so thereis w ∈ B such that vn → w in B. According to [16, Lemma 2.4] there is a subsequence (vkn)such that vkn → w a.e. This implies that v ≡ w a.e., so v = w in B. Thus v ∈ B and the proofis complete.

Notation 2. Let X be a real or complex Banach space. For every B ∈ T(R) we denote byB(R, X) the linear space of all Bochner measurable functions v : R → X with the propertythat the mappingNv : R → R+, Nv(t) = ‖v(t)‖ lies in B. With respect to the norm ‖v‖B(R,X) :=|Nv|B, B(R, X) is a Banach space.

3. Exponential Dichotomy for Evolution Families on the Real Line

Let X be a real or complex Banach space. The norm on X and on B(X), the Banach algebra ofall bounded linear operators on X, will be denoted by ‖ · ‖. Denote by Id the identity operatoron X. First, we remind some basic definitions.

Definition 3.1. A familyU = {U(t, s)}t≥s of bounded linear operators onX is called an evolutionfamily if the following properties hold:

(i) U(t0, t0) = Id and U(t, s)U(s, t0) = U(t, t0), for all t ≥ s ≥ t0;

(ii) for every x ∈ X and every t0 ∈ R the mapping t → U(t, t0)x is continuous on [t0,∞)and the mapping s → U(t0, s)x is continuous on (−∞, t0];

(iii) there are M ≥ 1 and ω > 0 such that ‖U(t, t0)‖ ≤Meω(t−t0), for all t ≥ t0.


Definition 3.2. An evolution family U = {U(t, s)}t≥s is said to be uniformly dichotomic if thereare a family of projections {P(t)}t∈R and a constant K ≥ 1 such that

(i) U(t, t0)P(t0) = P(t)U(t, t0), for all t ≥ t0;

(ii) the restriction U(t, t0)| : ker P(t0) → ker P(t) is an isomorphism, for all t ≥ t0;

(iii) ‖U(t, t0)x‖ ≤ K‖x‖, for all x ∈ Im P(t0) and all t ≥ t0;

(iv) ‖U(t, t0)y‖ ≥ (1/K)‖y‖, for all y ∈ ker P(t0) and all t ≥ t0.

Definition 3.3. An evolution family U = {U(t, s)}t≥s is said to be exponentially dichotomic ifthere exist a family of projections {P(t)}t∈R and two constants K ≥ 1 and ν > 0 such that

(i) U(t, t0)P(t0) = P(t)U(t, t0), for all t ≥ t0;

(ii) the restriction U(t, t0)| : ker P(t0) → ker P(t) is an isomorphism, for all t ≥ t0;

(iii) ‖U(t, t0)x‖ ≤ Ke−ν(t−t0)‖x‖, for all x ∈ Im P(t0) and all t ≥ t0;

(iv) ‖U(t, t0)y‖ ≥ (1/K)eν(t−t0)‖y‖, for all y ∈ ker P(t0) and all t ≥ t0.

Remark 3.4. It is obvious that if an evolution family is exponentially dichotomic, then it isuniformly dichotomic.

One of the most efficient tool in the study of the dichotomic behavior of an evolutionfamily is represented by the so-called input-output techniques. The input-output methodconsidered in this paper is the admissibility of a pair of function spaces. Indeed, let W,V betwo Banach function spaces such that W ∈ H(R) and V ∈ T(R).

Definition 3.5. The pair (W(R, X), V (R, X)) is said to be admissible for U if for every v ∈V (R, X) there exists a unique f ∈W(R, X) such that the pair (f, v) satisfies the equation

f(t) = U(t, s)f(s) +∫ t

s

U(t, τ)v(τ)dτ, ∀t ≥ s. (EU)

Remark 3.6. If the pair (W(R, X), V (R, X)) is admissible for U, then it makes sense to definethe operator Q : V (R, X) → W(R, X), Q(v) = f , where f ∈ W(R, X) is such that the pair(f, v) satisfies (EU). Then Q is a bounded linear operator (see [16, Proposition 4.4]).

Let U = {U(t, s)}t≥s be an evolution family on X and W ∈ H(R). For every t ∈ R, weconsider the stable subspace Xs(t) as the space of all x ∈ X with the property that the function

δx : R −→ X, δx(τ) =

⎧⎨

⎩

U(τ, t)x, τ ≥ t,

0, τ < t(3.1)

belongs toW(R, X) and we define the unstable subspaceXu(t) as the space of all x ∈ X with theproperty that there is a function ϕx ∈ W(R, X) such that ϕx(t) = x and ϕx(τ) = U(τ, s)ϕx(s),for all s ≤ τ ≤ t.

An important information concerning the structure of the projection family associatedwith a uniformly dichotomic evolution family was obtained in [16, Theorem 4.8] and this isgiven by the following.


Theorem 3.7. LetU = {U(t, s)}t≥s be an evolution family onX and letW,V be two Banach functionspaces withW ∈ H(R) and V ∈ T(R). If the pair (W(R, X), V (R, X)) is admissible for the evolutionfamily U, then U is uniformly dichotomic with respect to the family of projections {P(t)}t∈R, where

ImP(t) = Xs(t), kerP(t) = Xu(t), ∀t ∈ R. (3.2)

Taking into account the results obtained in [11, 16], an interesting open question iswhether in the study of exponential dichotomy, the output space may belong to the generalclassH(R). To answer this question, the first purpose of this paper is to prove the followingtheorem.

Theorem 3.8. Let U = {U(t, s)}t≥s be an evolution family on the Banach space X and let W, V betwo Banach function spaces with W ∈ H(R) and V ∈ V(R). If the pair (W(R, X), V (R, X)) isadmissible for U, then U is uniformly exponentially dichotomic.

The proof will be constructive and therefore, we will present several intermediateresults.

Theorem 3.9. Let U = {U(t, s)}t≥s be an evolution family on the Banach space X and let W,V betwo Banach function spaces with W ∈ H(R) and V ∈ V(R). If the pair (W(R, X), V (R, X)) isadmissible for U, then there are K, ν > 0 such that

‖U(t, t0)x‖ ≤ Ke−ν(t−t0)‖x‖, ∀t ≥ t0, ∀x ∈ Xs(t0). (3.3)

Proof. According to Theorem 3.7 and Definition 3.2(iii) we have that there is λ > 0 such that

‖U(t, t0)x‖ ≤ λ‖x‖, ∀t ≥ t0, ∀x ∈ Xs(t0). (3.4)

Since V ∈ V(R+), from Remark 2.7(ii) we have that there is a continuous function ϕ : R → R+

with ϕ ∈ V \L1(R,R). Using the invariance under translations of the space V , we may assumethat there is r > 1 such that

∫ r

0ϕ(s)ds ≥ 2

eλ2‖Q‖∣∣ϕ∣∣V

FW(1). (3.5)

Since∫r−1/n

1/n ϕ(s)ds →n→∞

∫ r0ϕ(s)ds there is n0 ∈ N

∗ such that

∫ r−1/n0

1/n0

ϕ(s)ds ≥ 12

∫ r

0ϕ(s)ds. (3.6)

Let α : R → [0, 1] be a continuous function with supp α ⊂ (0, r) and α(t) = 1, for t ∈[1/n0, r − 1/n0]. Then, the function ψ : R → R+, ψ(t) = α(t)ϕ(t) is continuous and from (3.5)and (3.6) we have that

∫ r

0ψ(t)dt ≥

∫ r−1/n0

1/n0

ϕ(t)dt ≥eλ2‖Q‖

∣∣ϕ∣∣V

FW(1). (3.7)


Let t0 ∈ R and let x ∈ Xs(t0). We consider the functions

v : R → X, v(t) = ψ(t − t0)U(t, t0)x, (3.8)

f : R → X, f(t) =

⎧⎪⎪⎨

⎪⎪⎩

∫ t

t0

ψ(τ − t0)dτU(t, t0)x, t ≥ t0,

0, t < t0.

(3.9)

Since v ∈ Cc(R, X) it follows that v ∈ V (R, X). Setting q =∫ r

0ψ(τ)dτ we observe thatf(t) = qU(t, t0)x, for all t ≥ t0 + r. Since x ∈ Xs(t0) we deduce that f ∈ W(R, X). A simplecomputation shows that the pair (f, v) satisfies (EU), so f = Q(v). This implies that

∥∥f∥∥W(R,X) ≤ ‖Q‖‖v‖V (R,X). (3.10)

According to relation (3.4) we observe that

‖v(t)‖ = ψ(t − t0)‖U(t, t0)x‖ ≤ λ‖x‖ψt0(t), ∀t ∈ R, (3.11)

and using the invariance under translations of the space V we deduce that ‖v‖V (R,X) ≤λ‖x‖|ψ|V . From ψ(t) ≤ ϕ(t), for all t ∈ R, we have that |ψ|V ≤ |ϕ|V . Thus we obtain that

‖v‖V (R,X) ≤ λ‖x‖∣∣ϕ∣∣V . (3.12)

From ‖U(t0 + r + 1, t0)x‖ ≤ λ‖U(τ, t0)x‖ = (λ/q)‖f(τ)‖, for all τ ∈ [t0 + r, t0 + r + 1], we havethat ‖U(t0 + r + 1, t0)x‖χ[t0+r,t0+r+1)(τ) ≤ (λ/q)‖f(τ)‖, for all τ ∈ R, which implies that

FW(1)‖U(t0 + r + 1, t0)x‖ ≤λ

q

∥∥f∥∥W(R,X). (3.13)

Setting l = r + 1, from relations (3.10)–(3.13) it follows that

qFW(1)‖U(t0 + l, t0)x‖ ≤ λ2‖Q‖∣∣ϕ∣∣V ‖x‖. (3.14)

Using relations (3.7) and (3.14) we deduce that ‖U(t0+l, t0)x‖ ≤ (1/e)‖x‖. Taking into accountthat l does not depend on t0 or x, we have that

‖U(t0 + l, t0)x‖ ≤1e‖x‖, ∀t0 ∈ R, ∀x ∈ Xs(t0). (3.15)

Let ν = 1/l and K = λe. Let t ≥ t0 and x ∈ Xs(t0). Then, there are j ∈ N and s ∈ [0, l) such thatt = t0 + jl + s. Using relations (3.4) and (3.15) we obtain that ‖U(t, t0)x‖ ≤ λ‖U(t0 + jl, t0)x‖ ≤λe−j‖x‖ ≤ Ke−ν(t−t0)‖x‖, which completes the proof.


Theorem 3.10. Let U = {U(t, s)}t≥s be an evolution family on the Banach space X and let W,Vbe two Banach function spaces with W ∈ H(R) and V ∈ V(R). If the pair (W(R, X), V (R, X)) isadmissible for U, then there are K, ν > 0 such that

‖U(t, t0)x‖ ≥1Keν(t−t0)‖x‖, ∀t ≥ t0, ∀x ∈ Xu(t0). (3.16)

Proof. Let M ≥ 1 and ω > 0 be given by Definition 3.1. According to Theorem 3.7 andDefinition 3.2(iv) we have that there is λ > 0 such that

‖U(t, t0)x‖ ≥1λ‖x‖, ∀t ≥ t0, ∀x ∈ Xu(t0). (3.17)

Since V ∈ V(R+), from Remark 2.7(ii) we have that there is a continuous function ϕ : R → R+

with ϕ ∈ V \L1(R,R). Using the invariance under translations of the space V we may assumethat there is r > 1 such that

∫ r

0ϕ(s)ds ≥ 2

λMeω+1‖Q‖∣∣ϕ∣∣V

FW(1). (3.18)

Using similar arguments with those in the proof of Theorem 3.9 we obtain that there is acontinuous function ψ : R → R+ with suppψ ⊂ (0, r), ψ(t) ≤ ϕ(t), for all t ∈ R and

∫ r

0ψ(τ)dτ ≥

λMeω+1‖Q‖∣∣ϕ∣∣V

FW(1). (3.19)

Let t0 ∈ R and x ∈ Xu(t0). Then, there is ϕx ∈ W(R, X) such that ϕx(t0) = x and ϕx(τ) =U(τ, s)ϕx(s), for all s ≤ τ ≤ t0. We consider the functions

v : R −→ X, v(t) = −ψ(t − t0)U(t, t0)x, (3.20)

f : R −→ X, f(t) =

⎧⎪⎪⎨

⎪⎪⎩

∫∞

t

ψ(τ − t0)dτU(t, t0)x, t ≥ t0,

qϕx(t), t < t0,

(3.21)

where q =∫r

0ψ(τ)dτ . We have that v ∈ Cc(R, X), so v ∈ V (R, X). Using relation (3.17) wehave that

∥∥f(t)∥∥ ≤ q

∥∥ϕx(t)∥∥ + qλ‖U(t0 + r, t0)x‖χ[t0,t0+r)(t), ∀t ∈ R. (3.22)

From this inequality, since W ∈ H(R) we deduce that f ∈ W(R, X). An easy computationshows that the pair (f, v) satisfies (EU), so f = Q(v). Then, we have that

∥∥f∥∥W(R,X) ≤ ‖Q‖‖v‖V (R,X). (3.23)


Using relation (3.17) we have that

‖v(t)‖ = ψ(t − t0)‖U(t, t0)x‖ ≤ ψ(t − t0)λ‖U(t0 + r, t0)x‖, ∀t ∈ R (3.24)

which implies that

‖v‖V (R,X) ≤∣∣ψ∣∣V λ‖U(t0 + r, t0)x‖ ≤

∣∣ϕ∣∣V λ‖U(t0 + r, t0)x‖. (3.25)

Since x = ϕx(t0) = U(t0, t)ϕx(t), for all t ∈ [t0 − 1, t0], we deduce that

‖x‖χ[t0−1,t0)(t) ≤Meω∥∥ϕx(t)

∥∥χ[t0−1,t0)(t) ≤

Meω

q

∥∥f(t)

∥∥, ∀t ∈ R. (3.26)

This shows that

q‖x‖FW(1) ≤Meω∥∥f∥∥W(R,X). (3.27)

From relations (3.19)–(3.27) it follows that ‖U(t0 + r, t0)x‖ ≥ e‖x‖. Taking into account that rdoes not depend on t0 or x, we have that

‖U(t0 + r, t0)x‖ ≥ e‖x‖, ∀t0 ∈ R, ∀x ∈ Xu(t0). (3.28)

We set ν = 1/r and K = λe. Let t ≥ t0 and x ∈ Xu(t0). Then, there are j ∈ N and s ∈ [0, r) suchthat t = t0 + jr + s. Using relations (3.17) and (3.28) we obtain that ‖U(t, t0)x‖ ≥ (1/λ)‖U(t0 +jr, t0)x‖ ≥ (1/λ)ej‖x‖ ≥ (1/K)eν(t−t0)‖x‖, which completes the proof.

Now, we may give the proof of Theorem 3.8.

Proof of Theorem 3.8. This immediately follows from Theorems 3.7, 3.9, and 3.10.

Now, we may give the main result of the paper, which establishes a completediagram concerning the study of exponential dichotomy on the real line in terms of integraladmissibility.

Theorem 3.11. Let U = {U(t, s)}t≥s be an evolution family on the Banach space X and letW,V betwo Banach function spaces with W ∈ H(R) and V ∈ T(R). If W ∈ W(R) or V ∈ V(R), then thefollowing assertions hold:

(i) if the pair (W(R, X), V (R, X)) is admissible for U, then U is uniformly exponentiallydichotomic;

(ii) if V ⊂ W and one of the spaces V,W belongs to the class O(R), then U is exponentiallydichotomic if and only if the pair (W(R, X), V (R, X)) is admissible for U.

Proof. (i) This follows from Theorems 1.3 and 3.8.(ii) Necessity. Suppose that U is exponentially dichotomic with respect to the family of

projections {P(t)}t∈R and the constants K, ν > 0. Then, we have that L := supt∈R‖P(t)‖ < ∞(see, e.g., [13]).


Let v ∈ V (R, X). We consider the function

f : R −→ X, f(t) =∫ t

−∞U(t, s)P(s)v(s)ds −

∫∞

t

U(s, t)−1| (I − P(s))v(s)ds, (3.29)

where for every s > t, U(s, t)−1| denotes the inverse of the operator U(s, t)| : kerP(t) →

kerP(s).If V ∈ O(R), then using Proposition 2.4 we obtain that f ∈ V (R, X). Since V ⊂ W we

deduce that f ∈W(R, X).If W ∈ O(R) then, since V ⊂ W we have that v ∈ W(R, X). Using Proposition 2.4 it

follows that f ∈W(R, X).An easy computation shows that the pair (f, v) satisfies (EU). The uniqueness of

f is immediate (see, e.g., [16, the Necessity part of Theorem 5.3]). In conclusion, the pair(W(R, X), V (R, X)) is admissible for the evolution family U.

The natural question arises whether the hypotheses from Theorem 3.11 can bedropped and also if the conditions given by this theorem are the most general in this topic.The answers are given by the following example.

Example 3.12. Let W ∈ H(R) and V ∈ T(R) be such that W /∈W(R) and V /∈V(R). ThenV ⊂ L1(R,R) and according to Lemma 2.8 we have that C0(R,R) ⊂W .

We consider the function

ϕ : R −→ R, ϕ(t) =

⎧⎪⎨

⎪⎩

1t + 1

, t ≥ 0,

1 − t, t < 0.(3.30)

Then ϕ is a decreasing function.Let X = R

2 endowed with the norm ‖(x1, x2)‖ := |x1| + |x2|, for all (x1, x2) ∈ R2. For

every t ≥ s we consider the operator

U(t, s) : X −→ X, U(t, s)(x1, x2) =(ϕ(t)ϕ(s)

x1, et−sx2

). (3.31)

Then U = {U(t, s)}t≥s is an evolution family on X.We prove that the pair (W(R, X), V (R, X)) is admissible for U. Let v = (v1, v2) ∈

V (R, X). Then v1, v2 ∈ L1(R,R). We consider the function

f : R −→ X, f(t) =

(∫ t

−∞

ϕ(t)ϕ(τ)

v1(τ)dτ,−∫∞

t

e−(τ−t)v2(τ)dτ

)

. (3.32)


We have that f is correctly defined and an easy computation shows that the pair (f, v) satisfies(EU). We set

f1(t) =∫ t

−∞

ϕ(t)ϕ(s)

v1(s)dτ, f2(t) = −∫∞

t

e−(τ−t)v2(τ)dτ, ∀t ∈ R. (3.33)

We prove that f1 ∈ C0(R,R). Since v1 ∈ L1(R,R), from

∣∣f1(t)

∣∣ ≤∫ t

−∞|v1(τ)|dτ, ∀t ∈ R, (3.34)

we have that limt→−∞f1(t) = 0. Let ε > 0. Then, there is δ > 0 such that∫∞δ |v1(τ)|dτ < ε. It

follows that

∣∣f1(t)∣∣ ≤

ϕ(t)ϕ(δ)

∫δ

−∞

ϕ(δ)ϕ(τ)

|v1(τ)|dτ +∫ t

δ

|v1(τ)|dτ <δ + 1t + 1

‖v1‖1 + ε, ∀t ≥ δ. (3.35)

The above inequality implies that limt→∞ |f1(t)| ≤ ε. Since ε > 0 was arbitrary we obtainthat there exists limt→∞f1(t) = 0. Using similar arguments with those in (3.34) we deducethat limt→∞f2(t) = 0.

Let ε > 0. Then there is γ < 0 such that∫γ−∞|v2(τ)|dτ < ε. It follows that

∣∣f2(t)∣∣ ≤ et−γ

∫∞

γ

eγ−τ |v2(τ)|dτ +∫ γ

t

|v2(τ)|dτ < et−γ‖v2‖1 + ε, ∀t ≤ γ. (3.36)

From this inequality we have that limt→−∞ |f2(t)| ≤ ε. Since ε > 0 was arbitrary, it followsthat there exists limt→−∞f2(t) = 0. Thus, we deduce that f ∈ C0(R, X), so f ∈W(R, X).

To prove the uniqueness of f , let f ∈W(R, X) be such that the pair (f , v) satisfies (EU).Setting g = f − f we have that g ∈W(R, X) and g(t) = U(t, s)g(s), for all t ≥ s. If g = (g1, g2),then we deduce that

g1(t) =ϕ(t)ϕ(s)

g1(s), ∀t ≥ s, (3.37)

g2(t) = et−sg2(s), ∀t ≥ s. (3.38)

Let t ∈ R. Using Lemma 2.6 and integrating in (3.37) we have that

∣∣g1(t)∣∣ =∫s+1

s

ϕ(t)ϕ(τ)

∣∣g1(τ)∣∣dτ ≤

ϕ(t)ϕ(s + 1)

∥∥g∥∥M1(R,X) ∀s ≤ t − 1. (3.39)

For s → −∞ in (3.39) we obtain that g1(t) = 0.


From relation (3.38) we have that

g2(t) = e−(τ−t)g2(τ), ∀τ ≥ t. (3.40)

Integrating in relation (3.40) on [τ, τ + 1] and using Lemma 2.6, we deduce that

∣∣g2(t)

∣∣ =∫ τ+1

τ

e−(ξ−t)∣∣g2(ξ)

∣∣dξ ≤ e−(τ−t)

∥∥g∥∥M1(R,X), ∀τ ≥ t. (3.41)

For τ → ∞ in (3.41) it follows that g2(t) = 0. Since t ∈ R was arbitrary we have that g = 0, sof = f . Thus, the pair (W(R, X), V (R, X)) is admissible for the evolution family U.

Suppose that U is exponentially dichotomic with respect to the family of projections{P(t)}t∈R and the constantsK, ν > 0. According to [13, Proposition 3.1] we have that Im P(t) ={x ∈ X : supτ≥t‖U(τ, t)x‖ < ∞}, which implies that ImP(t) = R × {0}, for all t ∈ R. Then, weobtain that

∣∣∣∣ϕ(t)ϕ(s)

x1

∣∣∣∣ ≤ Ke−ν(t−s)|x1|, ∀t ≥ s, ∀x1 ∈ R (3.42)

or equivalently

ϕ(t)ϕ(s)

≤ Ke−ν(t−s), ∀t ≥ s (3.43)

which is absurd. In conclusion, the pair (W(R, X), V (R, X)) is admissible for U, but, for allthat, the evolution family U is not exponentially dichotomic.

4. Applications to the Case of C0-Semigroups

In this section, by applying the central theorems from the Section 3 we deduce severalconsequences of the main results for the study of exponential dichotomy of C0-semigroups.Let X be a real or complex Banach space.

Definition 4.1. A family T = {T(t)}t≥0 of bounded linear operators on X is said to be a C0-semigroup if the following properties are satisfied:

(i) T(0) = Id, the identity operator on X;

(ii) T(t + s) = T(t)T(s), for all t, s ≥ 0;

(iii) limt↘0T(t)x = x, for every x ∈ X.


Definition 4.2. A C0-semigroup T = {T(t)}t≥0 is said to be exponentially dichotomic if there exista projection P ∈ B(X) and two constants K ≥ 1 and ν > 0 such that

(i) PT(t) = T(t)P , for all t ≥ 0;

(ii) ‖T(t)x‖ ≤ Ke−νt‖x‖, for all x ∈ ImP and all t ≥ 0;

(iii) ‖T(t)x‖ ≥ (1/K)eνt‖x‖, for all x ∈ kerP and all t ≥ 0;

(iv) the restriction T(t)| : kerP → kerP is an isomorphism, for every t ≥ 0.

Remark 4.3. (i) If T = {T(t)}t≥0 is a C0-semigroup, we can associate to T an evolution familyUT = {UT (t, s)}t≥s, by UT (t, s) = T(t − s), for every t ≥ s.

(ii) A C0-semigroup T = {T(t)}t≥0 is exponentially dichotomic if and only ifthe associated evolution family UT = {UT (t, s)}t≥s is exponentially dichotomic (see [12,Proposition 4.4]).

Let W,V be two Banach function spaces such that W ∈ H(R) and V ∈ T(R).

Definition 4.4. The pair (W(R, X), V (R, X)) is said to be admissible for the C0-semigroup T ={T(t)}t≥0 if for every v ∈ V (R, X) there is a unique f ∈W(R, X) such that

f(t) = T(t − s)f(s) +∫ t

s

T(t − τ)v(τ)dτ, ∀t ≥ s. (ET )

Theorem 4.5. Let T = {T(t)}t≥0 be a C0-semigroup on the Banach space X and let W,V be twoBanach function spaces with W ∈ H(R) and V ∈ T(R). If W ∈ W(R) or V ∈ V(R), then thefollowing assertions hold:

(i) if the pair (W(R, X), V (R, X)) is admissible for T, then T is uniformly exponentiallydichotomic;

(ii) if V ⊂ W and one of the spaces V,W belongs to the class O(R), then T is exponentiallydichotomic if and only if the pair (W(R, X), V (R, X)) is admissible for T.

Proof. This follows from Theorem 3.11 and Remark 4.3(ii).

Remark 4.6. According to the example given in the previous section we deduce that thehypothesis W ∈ W(R) or V ∈ V(R) cannot be removed. Moreover, Theorem 4.5 provides acomplete answer concerning the study of exponential dichotomy of semigroups using input-output techniques with respect to the associated integral equation.

Acknowledgment

This work is supported by the Exploratory Research Project PN 2 ID 1081 code 550/2009.


References

[1] J. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, vol. 4 ofTranslations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1974.

[2] J. H. Liu, G. M. N’Guerekata, and N. Van Minh, Topics on Stability and Periodicity in Abstract DifferentialEquations, vol. 6 of Series on Concrete and Applicable Mathematics, World Scientific, Hackensack, NJ,USA, 2008.

[3] Q. Liu, N. Van Minh, G. Nguerekata, and R. Yuan, “Massera type theorems for abstract functionaldifferential equations,” Funkcialaj Ekvacioj, vol. 51, no. 3, pp. 329–350, 2008.

[4] J. L. Massera and J. J. Schaffer, Linear Differential Equations and Function Spaces, vol. 21 of Pure andApplied Mathematics, Academic Press, New York, NY, USA, 1966.

[5] N. Van Minh, F. Rabiger, and R. Schnaubelt, “Exponential stability, exponential expansiveness, andexponential dichotomy of evolution equations on the half-line,” Integral Equations and Operator Theory,vol. 32, no. 3, pp. 332–353, 1998.

[6] N. Van Minh, G. M. N’Guerekata, and R. Yuan, Lectures on the Asymptotic Behavior of Solutions ofDifferential Equations, Nova Science, New York, NY, USA, 2008.

[7] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of AppliedMathematical Sciences, Springer, New York, NY, USA, 1983.

[8] O. Perron, “Die Stabilitatsfrage bei Differentialgleichungen,” Mathematische Zeitschrift, vol. 32, no. 1,pp. 703–728, 1930.

[9] B. Sasu and A. L. Sasu, “Exponential dichotomy and (lp, lq)-admissibility on the half-line,” Journal ofMathematical Analysis and Applications, vol. 316, no. 2, pp. 397–408, 2006.

[10] B. Sasu, “Uniform dichotomy and exponential dichotomy of evolution families on the half-line,”Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1465–1478, 2006.

[11] A. L. Sasu and B. Sasu, “Exponential dichotomy on the real line and admissibility of function spaces,”Integral Equations and Operator Theory, vol. 54, no. 1, pp. 113–130, 2006.

[12] A. L. Sasu, “Exponential dichotomy for evolution families on the real line,” Abstract and AppliedAnalysis, vol. 2006, Article ID 31641, 6 pages, 2006.

[13] A. L. Sasu and B. Sasu, “Exponential dichotomy and admissibility for evolution families on the realline,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 13, no. 1, pp. 1–26, 2006.

[14] A. L. Sasu and B. Sasu, “Discrete admissibility, lp-spaces and exponential dichotomy on the real line,”Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 13, no. 5, pp. 551–561, 2006.

[15] B. Sasu and A. L. Sasu, “Exponential trichotomy and p-admissibility for evolution families on the realline,” Mathematische Zeitschrift, vol. 253, no. 3, pp. 515–536, 2006.

[16] A. L. Sasu, “Integral equations on function spaces and dichotomy on the real line,” Integral Equationsand Operator Theory, vol. 58, no. 1, pp. 133–152, 2007.


Research ArticleOn a Max-Type Difference Equation

Ali Gelisken, Cengiz Cinar, and Ibrahim Yalcinkaya

Mathematics Department, Ahmet Kelesoglu Education Faculty, Selcuk University, Meram Yeni Yol,42090 Konya, Turkey

Correspondence should be addressed to Ali Gelisken, [email protected]

Received 8 December 2009; Revised 20 April 2010; Accepted 23 April 2010

Academic Editor: Gaston M. N’Guerekata

Copyright q 2010 Ali Gelisken et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We prove that every positive solution of the max-type difference equation xn =max{A/xαn−p, B/x

β

n−k}, n = 0, 1, 2, . . . converges to x = max{A1/(1+α), B1/(1+β)} where p, k arepositive integers, 0 < α, β < 1, and 0 < A,B.

1. Introduction

Recently, the study of max-type difference equations attracted a considerable attention.Although max-type difference equations are relatively simple in form, it is unfortunatelyextremely difficult to understand thoroughly the behavior of their solutions; see, for example,[1–20] and the relevant references cited therein. The max operator arises naturally in certainmodels in automatic control theory (see [13, 14]). Furthermore, difference equation appearnaturally as a discrete analogue and as a numerical solution of differential and delaydifferential equations having applications and various scientific branches, such as in ecology,economy, physics, technics, sociology, and biology.

In [20], Yang et al. proved that every positive solution of the difference equation

xn = max

{1

xαn−1,B

xn−2

}

, n = 0, 1, 2, . . . (1.1)

converges to x = 1 or eventually periodic with period 4, where 0 < α < 1 and 0 < A.


In [9], We proved that every positive solution of the difference equation

xn = max

{A

xn−1,

1xαn−3

}

, n = 0, 1, 2, . . . (1.2)

converges to x = 1 or eventually periodic with period 2, where 0 < α < 1 and 0 < A.In [17], Sun proved that every positive solution of the difference equation

xn = max

⎧⎨

⎩A

xαn−1,B

xβ

n−2

⎫⎬

⎭, n = 0, 1, 2, . . . (1.3)

converges to x = max{A1/(1+α), B1/(1+β)}where 0 < α, β < 1, and 0 < A,B.The following difference equation is more general than (1.3):

xn = max

⎧⎨

⎩A

xαn−p,B

xβ

n−k

⎫⎬

⎭, n = 0, 1, 2, . . . , (1.4)

where p, k are positive integers, 0 < α, β < 1, 0 < A,B, and initial conditions are positive realnumbers.

In this paper, we investigate the asymptotic behavior of the positive solutions of(1.4). We prove that every positive solution of (1.4) converges to x = max{A1/(1+α), B1/(1+β)}.Clearly, we can assume that p ≤ k without loss of generality.

2. Main Results

2.1. The Case B1/(1+β) ≤ A1/(1+α)

In this section, we consider the asymptotic behavior of the positive solutions of (1.4) in thecase B1/(1+β) ≤ A1/(1+α).

It is easy to see that by the change

xn = A1/(1+α)Cyn for 0 < C < 1, n ≥ −k. (2.1)

Equation (1.4) is transformed into the difference equation

Cyn = max{C−αyn−p , DC−βyn−k

}, (2.2)

where D = B/A(1+β)/(1+α) and the initial conditions are real numbers. Since B1/(1+β) ≤ A1/(1+α),we have D ≤ 1.

We need the following two lemmas in order to prove the main result of this section.


Lemma 2.1. Let {yn}∞n=−k be a solution of (2.2). If D = 1, then

∣∣yn∣∣ ≤ max

{α∣∣yn−p

∣∣, β∣∣yn−k

∣∣} ∀n ≥ −k. (2.3)

Proof. Clearly, (2.2) implies the following difference equation:

yn = min{−αyn−p,−βyn−k

}∀n ≥ −k. (2.4)

From (2.4), we get the following statements.

(i) If yn−p ≥ 0 and yn−k ≥ 0, then |yn| ≤ max{α|yn−p|, β|yn−k|}.(ii) If yn−p ≤ 0 and yn−k ≤ 0, then |yn| ≤ max{α|yn−p|, β|yn−k|}.(iii) If yn−p ≥ 0 and yn−k ≤ 0, then |yn| = α|yn−p|.(iv) If yn−p ≤ 0 and yn−k ≥ 0, then |yn| = β|yn−k|.

From the above statements, we have |yn| ≤ max{α|yn−p|, β|yn−k|} for all n ≥ −k.Therefore, the proof is complete.

Lemma 2.2. Let {yn}∞n=−k be a solution of (2.2). If D < 1, then


{α∣∣yn−p

∣∣, β∣∣yn−k

∣∣ − 1}∀n ≥ −k. (2.5)

Proof. Assume that C = D. Then (2.2) implies the following difference equation:

yn = min{−αyn−p, 1 − βyn−k

}∀n ≥ −k. (2.6)

From (2.6), we get the following statements.

(i) If yn−p ≥ 0 and yn−k ≥ 0, then |yn| ≤ max{α|yn−p|, β|yn−k| − 1}.(ii) If yn−p ≤ 0 and yn−k ≤ 0, then |yn| ≤ α|yn−p|.(iii) If yn−p ≥ 0 and yn−k ≤ 0, then |yn| = α|yn−p|.(iv) If yn−p ≤ 0 and yn−k ≥ 0, then |yn| ≤ max{α|yn−p|, β|yn−k| − 1}.

From the above statements, we have |yn| ≤ max{α|yn−p|, β|yn−k| − 1} for all n ≥ −k.Therefore, the proof is complete.

Theorem 2.3. Let {xn}∞n=−k be a solution of (1.4) where B1/(1+β) ≤ A1/(1+α). Then {xn}∞n=−kconverges to x = A1/(1+α).

Proof. Assume that D = 1. {yn}∞n=−k is a solution of (2.2). If it is proved that {yn}∞n=−kconverges to zero as n → ∞, then {xn}∞n=−k converges to x = A1/(1+α).

From Lemma 2.1, we have that


{α∣∣yn−p

∣∣, β∣∣yn−k

∣∣} ∀n ≥ −k. (2.7)


Let γ = max{α, β}. Immediately, we have that the following inequality

∣∣yn∣∣ ≤ γ max

{∣∣yn−p∣∣,∣∣yn−k

∣∣} ∀n ≥ −k. (2.8)

From (2.8) and by induction, we get

∣∣yn∣∣ ≤ γ [|n/k|]+1max

1≤j≤k

{y−j}∀n ≥ −k. (2.9)

From (2.9), it is clear that {yn}∞n=−k converges to zero as n → ∞.Now, we assume that D < 1. From Lemma 2.2, we have that


{α∣∣yn−p

∣∣, β∣∣yn−k

∣∣ − 1}≤ max

{α∣∣yn−p

∣∣, β∣∣yn−k

∣∣} ∀n ≥ −k. (2.10)

Then, the rest of proof is similar to the case D = 1 and will be omitted. Therefore, theproof is complete.

2.2. The Case A1/(1+α) < B1/(1+β)

In this section, we consider the asymptotic behavior of the positive solutions of (1.4) in thecase A1/(1+α) < B1/(1+β).

It is easy to see that by the change

xn = B1/(1+β)Cyn for C =A

B(1+α)/(1+β), n ≥ −k. (2.11)

Equation (1.4) is transformed into the difference equation:

yn = min{

1 − αyn−p,−βyn−k}, (2.12)

where initial conditions are real numbers.We need the following lemma in order to prove the main result of this section.

Lemma 2.4. Let {yn}∞n=−k be a solution of (2.12). Then


{α∣∣yn−p

∣∣ − 1, β∣∣yn−k

∣∣} ∀n ≥ −k. (2.13)

Proof. From (2.12), we get the following statements.

(i) If yn−p ≥ 0 and yn−k ≥ 0, then |yn| ≤ max{α|yn−p| − 1, β|yn−k|}.(ii) If yn−p ≤ 0 and yn−k ≤ 0, then |yn| ≤ β|yn−k|.(iii) If yn−p ≥ 0 and yn−k ≤ 0, then |yn| ≤ max{α|yn−p| − 1, β|yn−k|}.(iv) If yn−p ≤ 0 and yn−k ≥ 0, then |yn| = β|yn−k|.


From the above statements, we have |yn| ≤ max{α|yn−p| − 1, β|yn−k|} for all n ≥ −k.Therefore, the proof is complete.

Theorem 2.5. Let {xn}∞n=−k be a solution of (1.4) where A1/(1+α) < B1/(1+β). Then {xn}∞n=−kconverges to x = B1/(1+β).

Proof. Let {yn}∞n=−k be a solution of (2.12). To prove the desired result, it suffices to prove that{yn}∞n=−k converges to zero.

From Lemma 2.4, we have that


{α∣∣yn−p

∣∣ − 1, β

∣∣yn−k

∣∣} ≤ max

{α∣∣yn−p

∣∣, β∣∣yn−k

∣∣} ∀n ≥ −k. (2.14)

From (2.14) and by induction, we get

∣∣yn∣∣ ≤ γ [|n/k|]+1max

1≤j≤k

{y−j}∀n ≥ −k. (2.15)

From (2.15), it is clear that {yn}∞n=−k converges to zero as n → ∞. Therefore, the proofis complete.

Acknowledgment

The authors are grateful to the anonymous referees for their valuable suggestions thatimproved the quality of this study.

References

[1] R. M. Abu-Saris and F. M. Allan, “Periodic and nonperiodic solutions of the difference equationxn+1 = max{x2

n,A}/xnxn−1,” in Advances in Difference Equations (Veszprem, 1995), pp. 9–17, Gordonand Breach, Amsterdam, The Netherlands, 1997.

[2] A. M. Amleh, J. Hoag, and G. Ladas, “A difference equation with eventually periodic solutions,”Computers & Mathematics with Applications, vol. 36, no. 10–12, pp. 401–404, 1998.

[3] K. S. Berenhaut, J. D. Foley, and S. Stevic, “Boundedness character of positive solutions of a maxdifference equation,” Journal of Difference Equations and Applications, vol. 12, no. 12, pp. 1193–1199,2006.

[4] W. J. Briden, E. A. Grove, G. Ladas, and C. M. Kent, “Eventually periodic solutions ofxn+1 =max{1/xn,An/xn−1},” Communications on Applied Nonlinear Analysis, vol. 6, no. 4, pp. 31–43,1999.

[5] W. J. Briden, E. A. Grove, G. Ladas, and L. C. McGrath, “On the nonautonomous equation xn+1 =max{An/xn, Bn, /xn−1},” in New Developments in Difference Equations and Applications (Taipei, 1997),pp. 49–73, Gordon and Breach, Amsterdam, The Netherlands, 1999.

[6] C. Cinar, S. Stevic, and I. Yalcinkaya, “On positive solutions of a reciprocal difference equation withminimum,” Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 307–314, 2005.

[7] A. Gelisken, C. Cinar, and R. Karatas, “A note on the periodicity of the Lyness max equation,”Advances in Difference Equations, vol. 2008, Article ID 651747, 5 pages, 2008.

[8] A. Gelisken, C. Cinar, and I. Yalcinkaya, “On the periodicity of a difference equation with maximum,”Discrete Dynamics in Nature and Society, vol. 2008, Article ID 820629, 11 pages, 2008.

[9] A. Gelisken and C. Cinar, “On the global attractivity of a max-type difference equation,” DiscreteDynamics in Nature and Society, vol. 2009, Article ID 812674, 5 pages, 2009.


[10] E. A. Grove, C. Kent, G. Ladas, and M. A. Radin, “On the xn+1 = max{1/xn,An/xn−1} with a period3 parameter,” in Fields Institute Communications, vol. 29, pp. 161–180, American Mathematical Society,Providence, RI, USA, 2001.

[11] G. Ladas, “On the recursive sequence xn = max{A1/xn−1, A2/xn−2, . . . , Ap/xn−p},” Journal of DifferenceEquations and Applications, vol. 2, no. 3, pp. 339–341, 1996.

[12] D. P. Mishev, W. T. Patula, and H. D. Voulov, “A reciprocal difference equation with maximum,”Computers & Mathematics with Applications, vol. 43, no. 8-9, pp. 1021–1026, 2002.

[13] A. D. Myskis, “Some problems in the theory of differential equations with deviating argument,”Uspekhi Matematicheskikh Nauk, vol. 32, no. 2(194), pp. 173–202, 1977.

[14] E. P. Popov, Automatic Regulation and Control, Nauka, Moscow, Russia, 1966.[15] I. Szalkai, “On the periodicity of the sequence xn+1 = max{A0/x0, A1/xn−1, . . . , Ak/xn−k},” Journal of

Difference Equations and Applications, vol. 5, no. 1, pp. 25–29, 1999.[16] S. Stevic, “On the recursive sequence xn+1 = max{c, xpn/x

p

n−1},” Applied Mathematics Letters, vol. 21, no.8, pp. 791–796, 2008.

[17] F. Sun, “On the asymptotic behavior of a difference equation with maximum,” Discrete Dynamics inNature and Society, vol. 2008, Article ID 243291, 6 pages, 2008.

[18] H. D. Voulov, “On the periodic character of some difference equations,” Journal of Difference Equationsand Applications, vol. 8, no. 9, pp. 799–810, 2002.

[19] I. Yalcinkaya, B. D. Iricanin, and C. Cinar, “On a max-type difference equation,” Discrete Dynamics inNature and Society, vol. 2007, no. 1, Article ID 47264, 10 pages, 2007.

[20] X. Yang, X. Liao, and C. Li, “On a difference equation with maximum,” Applied Mathematics andComputation, vol. 181, no. 1, pp. 1–5, 2006.


Research ArticleInequalities among Eigenvalues of Second-OrderSymmetric Equations on Time Scales

Chao Zhang and Shurong Sun

School of Science, University of Jinan, Jinan, Shandong 250022, China

Correspondence should be addressed to Chao Zhang, ss [email protected]

Received 28 January 2010; Revised 2 May 2010; Accepted 5 May 2010

Academic Editor: A. Pankov

Copyright q 2010 C. Zhang and S. Sun. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We consider coupled boundary value problems for second-order symmetric equations on timescales. Existence of eigenvalues of this boundary value problem is proved, numbers of theireigenvalues are calculated, and their relationships are obtained. These results not only unifythe existing ones of coupled boundary value problems for second-order symmetric differentialequations but also contain more complicated time scales.

1. Introduction

In this paper we consider the following second-order symmetric equation:

−(p(t)yΔ(t)

)Δ+ q(t)yσ(t) = λr(t)yσ(t), t ∈

[ρ(0), ρ(1)

]∩ T, and 0, 1 ∈ T (1.1)

with the coupled boundary conditions:

(y(1)yΔ(1)

)= eiθK

(y(ρ(0)

)

yΔ(ρ(0))

)

, (1.2)

where T is a time scale; pΔ, q, and r are real and continuous functions in [ρ(0), ρ(1)]∩T, p > 0over [ρ(0), 1] ∩ T, r > 0 over [ρ(0), ρ(1)] ∩ T, and p(ρ(0)) = p(1) = 1; σ(t) and ρ(t) are the


forward and backward jump operators in T, yΔ is the delta derivative, and yσ(t) := y(σ(t));θ /= 0, −π < θ < π, is a constant parameter; i =

√−1,

K =(k11 k12

k21 k22

), kij ∈ R, i, j = 1, 2, with detK = 1. (1.3)

The boundary condition (1.2) contains the two special cases: the periodic andantiperiodic conditions. In fact, (1.2) is the periodic boundary condition in the case whereθ = 0 and K = I, the identity matrix, and (1.2) is the antiperiodic condition in the case whereθ = π and K = I. Equation (1.1) with (1.2) is called a coupled boundary value problem.

Hence, according to [1, Theorem 3.1], the periodic and antiperiodic boundary valueproblems have Nd + 1 real eigenvalues and they satisfy the following inequality:

−∞ < λ0(I) < λ0(−I) ≤ λD0 ≤ λ1(−I) < λ1(I) ≤ λD1 ≤ λ2(I) < λ2(−I) ≤ λD2 ≤ λ3(−I) ≤ λD3 ≤ · · · ,(1.4)

where Nd := |[0, 1] ∩ T| − def(μ(ρ(0))) − 1, λDn denote the nth Dirichlet eigenvalues. Denotethe number of point of a set S ⊂ R by |S| and introduce the following notation for α ∈ R:

def α =

⎧⎨

⎩

0, if α/= 0,

1, if α = 0.(1.5)

Furthermore, if Nd <∞, then

λ0(I) < λ0(−I) ≤ λ1(−I) < λ1(I) ≤ λ2(I) < λ2(−I) ≤ λ3(−I) < λ3(I)

≤ · · · ≤ λNd−1(−I) < λNd−1(I) ≤ λNd(I) < λNd(−I), if Nd + 1 is odd,

λ0(I) < λ0(−I) ≤ λ1(−I) < λ1(I) ≤ λ2(I) < λ2(−I) ≤ λ3(−I) < λ3(I)

≤ · · · ≤ λNd−1(I) < λNd−1(−I) ≤ λNd(−I) < λNd(I), if Nd + 1 is even.

(1.6)

In [2], Eastham et al. considered the second-order differential equation:

−(p(t)x′(t)

)′ + q(t)x(t) = λw(t)x(t), t ∈ [a, b], (1.7)

with the coupled boundary condition:

(y(b)

p(b)y′(b)

)= eiαK

(y(a)

p(a)y′(a)

), (1.8)

where i =√−1, −π < α ≤ π , −∞ < a < b <∞, K =

(k11 k12k21 k22

), kij ∈ R, i, j = 1, 2, detK = 1, and

1/p, q,w ∈ L1([a, b],R), p > 0, w > 0 a.e. on [a, b]. Here R denote the set of real number, andL1([a, b],R) the space of real valued Lebesgue integrable functions on [a, b]. They obtained


the following results: the coupled boundary value problem (1.7) with (1.8) has an infinite butcountable number of only real eigenvalues which can be ordered to form a nondecreasingsequence:

−∞ < λ0(K) < λ0

(eiαK

)< λ0(−K) ≤ λ1(−K) < λ1

(eiαK

)< λ1(K)

≤ λ2(K) < λ2

(eiαK

)< λ2(−K) ≤ λ3(−K) < λ3

(eiαK

)< λ3(K) ≤ · · · .

(1.9)

In the present paper, we try to extend these results on time scales. We shall remarkthat Eastham et al. employed continuous eigenvalue branch which studied in [2], in theirproof. Instead, we will make use of some oscillation results that are extended from the resultsobtained by Agarwal et al. [4] to prove the existence of eigenvalues of (1.1) with (1.2) andcompare the eigenvalues as θ varies.

This paper is organized as follows. Section 2 introduces some basic concepts andfundamental theory about time scales and gives some properties of eigenvalues of a kindof separated boundary value problem for (1.1) which will be used in Section 4. Our mainresult has been introduced in Section 3. Section 4 pays attention to prove some propositions,by which one can easily obtain the existence and the comparison result of eigenvalues of thecoupled boundary value problems (1.1) with (1.2). By using these propositions, we give theproof of our main result in Section 5.

2. Preliminaries

In this section, some basic concepts and some fundamental results on time scales areintroduced. Next, the eigenvalues of the kind of separated boundary value problem for (1.1)and the oscillation of their eigenfunction are studied. Finally, the reality of the eigenvalues ofthe coupled boundary value problems for (1.1) is shown.

Let T ⊂ R be a nonempty closed subset. Define the forward and backward jumpoperators σ, ρ : T → T by

σ(t) = inf{s ∈ T : s > t}, ρ(t) = sup{s ∈ T : s < t}, (2.1)

where inf ∅ = sup T, sup ∅ = inf T. A point t ∈ T is called right-scattered, right-dense, left-scattered, and left-dense if σ(t) > t, σ(t) = t, ρ(t) < t, and ρ(t) = t, respectively.

We assume throughout the paper that if 0 is right-scattered, then it is also left-scattered,and if 1 is left-scattered, then it is also right-scattered.

Since T is a nonempty bounded closed subset of R, we put Tk = T\ (ρ(max T),max T].

The graininess μ : T → [0,∞) is defined by

μ(t) = σ(t) − t. (2.2)


Let f be a function defined on T. f is said to be (delta) differentiable at t ∈ Tk provided

there exists a constant a such that for any ε > 0, there is a neighborhood U of t (i.e., U =(t − δ, t + δ) ∩ T for some δ > 0) with

∣∣f(σ(t)) − f(s) − a(σ(t) − s)

∣∣ ≤ ε |σ(t) − s|, ∀s ∈ U. (2.3)

In this case, denote fΔ(t) := a. If f is (delta) differentiable for every t ∈ Tk, then f is said to

be (delta) differentiable on T. If f is differentiable at t ∈ Tk, then

fΔ(t) =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

lims→ ts∈T

f(t) − f(s)t − s if μ(t) = 0

f(σ(t)) − f(t)μ(t)

, if μ(t) > 0.

(2.4)

If FΔ(t) = f(t) for all t ∈ Tk, then F(t) is called an antiderivative of f on T. In this case, define

the delta integral by

∫ t

s

f(τ)Δτ = F(t) − F(s) ∀s, t ∈ T. (2.5)

Moreover, a function f defined on T is said to be rd-continuous if it is continuous at everyright-dense point in T and its left-sided limit exists at every left-dense point in T.

For convenience, we introduce the following results ([5, Chapter 1], [6, Chapter 1],and [7, Lemma 1]), which are useful in the paper.

Lemma 2.1. Let f, g : T → R and t ∈ Tk.

(i) If f is differentiable at t, then f is continuous at t.

(ii) If f and g are differentiable at t, then fg is differentiable at t and

(fg

)Δ(t) = fσ(t)gΔ(t) + fΔ(t)g(t) = fΔ(t)gσ(t) + f(t)gΔ(t). (2.6)

(iii) If f and g are differentiable at t, and f(t)fσ(t)/= 0, then f−1g is differentiable at t and

(gf−1

)Δ(t) =

(gΔ(t)f(t) − g(t)fΔ(t)

)(fσ(t)f(t)

)−1. (2.7)

(iv) If f is rd-continuous on T, then it has an antiderivative on T.

Now, we turn to discuss some properties of solutions of (1.1) and eigenvalues of itsboundary value problems.

Define the Wronskian by

W(x, y

):= xyΔ − yxΔ, x, y ∈ C2

rd(T), (2.8)


whereC2rd(T) is the set of twice differentiable functions with rd-continuous second derivative.

The following result can be derived from the Lagrange Identity [5, Theorem 4.30].

Lemma 2.2. For any two solutions x and y of (1.1), p(t)W(x, y)(t) is a constant on [ρ(0), 1] ∩ T.

In [4], Agarwal et al. studied the following second-order symmetric linear equation:

yΔΔ + q(t)yσ = −λyσ, t ∈[ρ(a), ρ(b)

]∩ T, and a, b ∈ T (2.9)

with the boundary conditions:

Ra

(y)

:= αy(ρ(a)

)+ βyΔ(ρ(a)

)= 0, Rb

(y)

:= γy(b) + δyΔ(b) = 0, (2.10)

where q : [ρ(a), ρ(b)] ∩ T → R is continuous; (α2 + β2)(γ2 + δ2)/= 0; a < b satisfy thatif a is right-scattered, then it is also left-scattered; and if b is left-scattered, then it is alsoright-scattered. A solution y of (2.9) is said to have a node at (t + σ(t))/2 if y(t)y(σ(t)) < 0.A generalized zero of y is defined as its zero or its node. Without loss of generality, theyassumed that α and β in (2.10) satisfy

(H) β > αμ(ρ(a)) if β /=αμ(ρ(a)) and α = −1 if β = αμ(ρ(a))

and obtained the following oscillation result.

Lemma 2.3 (see [4, Theorem 1]). The eigenvalues of (2.9) with (2.10) may be arranged as −∞ <λ0 < λ1 < λ2 < · · · and an eigenfunction corresponding to λk has exactly k-generalized zeros in theopen interval (a, b).

In order to study the kind of separated boundary value problem for (1.1), we nowextend the above oscillation theorem to the more general equation (1.1) with

R0(y)= R1

(y)= 0. (2.11)

By n(λ) denote the number of generalized zeros of the solution y(t, λ) of (1.1) with the initialconditions

y(ρ(0), λ

)= β, yΔ(ρ(0), λ

)= −α (2.12)

in the open interval (0,1), where α and β satisfy (H) with a and b replaced by 0 and 1,respectively. It can be easily verified that

either y(0, λ) > 0 or y(0, λ) = 0 and yΔ(0, λ) = 1, (2.13)

which is independent of λ.

Lemma 2.4 (see [1, Lemma 2.5]). Let y(t, λ) be the solution of (1.1) with (2.12). ThenyΔ(t, λ)/y(t, λ) is strictly decreasing in λ ∈ R for each t ∈ (0, 1] ∩ T whenever y(t, λ)/= 0.


Lemma 2.5 (see [1, Lemma 2.6]). If there exists λ0 ∈ R such that n(λ0) = 0, then n(λ) = 0 for allλ < λ0.

With a similar argument to that used in the proof of [4, Theorem 1], one can show thefollowing result.

Theorem 2.6. All the eigenvalues of (1.1) with (2.11) are simple and can be arranged as −∞ < λ0 <λ1 < λ2 < · · · , and an eigenfunction corresponding to λk has exactly k-generalized zeros in the openinterval (0, 1), where 0, 1 ∈ T satisfy that if 0 is right-scattered, then it is also left-scattered; if 1 isleft-scattered, then it is also right-scattered. Furthermore, the number of its eigenvalues is equal to|[0, 1] ∩ T| − def(β − αμ(ρ(0))) − def δ.

Setting β = 0, γ = k22, and δ = −k12 in (2.11), where k12, k22 are elements of K in (1.2),we get the following separated boundary conditions:

y(ρ(0)

)= 0, k22y(1) − k12y

Δ(1) = 0. (2.14)

The following result is a direct consequence of Theorem 2.6.

Theorem 2.7. All the eigenvalues of (1.1) with (2.14) are simple and can be arranged as

−∞ < μ0 < μ1 < μ2 < · · · , (2.15)

and an eigenfunction corresponding to μk has exactly k-generalized zeros in (0, 1). Furthermore, thenumber of its eigenvalues is equal toNd := |[0, 1] ∩ T| − def(μ(ρ(0))) − def k12.

For convenience, we shall write μk+1 =∞ if Nd = k <∞.

Lemma 2.8. For each λ ∈ (μk, μk+1], n(λ) = k + 1, k ≥ 0.

Proof. The proof is similar to that of [4, Theorem 6]. So the details are omitted.

Lemma 2.9. All the eigenvalues of the coupled boundary value problem (1.1) with (1.2) are real.

Proof. The proof is similar to that of [1, Lemma 2.8]. So the details are omitted.

3. Main Result

In this section we state our main results: general inequalities among eigenvalues of coupledboundary value problem of (1.1) with (1.2).

Theorem 3.1. If k11 > 0 and k12 ≤ 0 or k11 ≥ 0 and k12 < 0, then, for every fixed θ /= 0, −π < θ < π ,coupled boundary value problem (1.1) with (1.2) hasNd +1 eigenvalues and these eigenvalues satisfythe following inequalities:

λ0(K) < λ0

(eiθK

)< λ0(−K) ≤ λ1(−K) < λ1

(eiθK

)< λ1(K)

≤ λ2(K) < λ2

(eiθK

)< λ2(−K) ≤ λ3(−K) < λ3

(eiθK

)< λ3(K) ≤ · · · ,

(3.1)


whereNd := |[0, 1] ∩ T| − def(μ(ρ(0))) − def k12. Furthermore, ifNd <∞ then

λ0(K) < λ0

(eiθK

)< λ0(−K) ≤ λ1(−K) < λ1

(eiθK

)< λ1(K)

≤ λ2(K) < λ2

(eiθK

)< λ2(−K) ≤ λ3(−K) < λ3

(eiθK

)< λ3(K)

≤ · · · ≤ λNd−1(−K) < λNd−1

(eiθK

)< λNd−1(K) ≤ λNd(K)

< λNd

(eiθK

)< λNd(−K), if N is odd,

λ0(K) < λ0

(eiθK

)< λ0(−K) ≤ λ1(−K) < λ1

(eiθK

)< λ1(K)

≤ λ2(K) < λ2

(eiθK

)< λ2(−K) ≤ λ3(−K) < λ3

(eiθK

)< λ3(K)

≤ · · · ≤ λNd−1(K) < λNd−1

(eiθK

)< λNd−1(−K) ≤ λNd(−K)

< λNd

(eiθK

)< λNd(K), if N is even.

(3.2)

Remark 3.2. If k11 ≤ 0 and k12 > 0 or k11 < 0 and k12 ≥ 0, a similar result can beobtained by applying Theorem 3.1 to −K. In fact, eiθK = ei(π+θ)(−K) for θ ∈ (−π, 0) andeiθK = ei(−π+θ)(−K) for θ ∈ (0, π). Hence, the boundary condition (1.2) in the cases ofk11 ≤ 0, k12 > 0 or k11 < 0, k12 ≥ 0 and θ /= 0, −π < θ < π , can be written as condition(1.2), where θ is replaced by π + θ for θ ∈ (−π, 0) and −π + θ for θ ∈ (0, π), and K is replacedby −K.

4. The Characteristic Function D(λ)

Before showing Theorem 3.1, we need to prove the following six propositions.Let ϕ(t, λ) and ψ(t, λ) be the solutions of (1.1) satisfying the following initial

conditions:

ϕ(ρ(0), λ

)= 1, ϕΔ(ρ(0), λ

)= 0; ψ

(ρ(0), λ

)= 0, ψΔ(ρ(0), λ

)= 1, (4.1)

respectively. Obviously, ϕ(t, λ) and ψ(t, λ) are two linearly independent solutions of (1.1). ByLemma 2.2 we have

p(t)[ϕ(t, λ)ψΔ(t, λ) − ϕΔ(t, λ)ψ(t, λ)

]= 1, t ∈

[ρ(0), 1

]∩ T, (4.2)

which, together with the assumption of p(1) = 1, implies

ϕ(1, λ)ψΔ(1, λ) − ϕΔ(1, λ)ψ(1, λ) = 1. (4.3)


For any fixed K ∈ SL(2, R), detK = 1, and all λ ∈ C, we define

D(λ) = k11ψΔ(1, λ) − k21ψ(1, λ) + k22ϕ(1, λ) − k12ϕ

Δ(1, λ), (4.4)

A(λ) = k11ϕΔ(1, λ) − k21ϕ(1, λ), (4.5)

B(λ) = k11ψΔ(1, λ) + k12ϕ

Δ(1, λ) − k22ϕ(1, λ) − k21ψ(1, λ), (4.6)

B1(λ) = k11ψΔ(1, λ) − k21ψ(1, λ), (4.7)

B2(λ) = k22ϕ(1, λ) − k12ϕΔ(1, λ), (4.8)

C(λ) = k22ψ(1, λ) − k12ψΔ(1, λ). (4.9)

Note that

D(λ) = B1(λ) + B2(λ), B(λ) = B1(λ) − B2(λ). (4.10)

Let

φ(t, λ) =(

ϕ(t, λ) ψ(t, λ)p(t)ϕΔ(t, λ) p(t)ψΔ(t, λ)

), t ∈

[ρ(0), 1

]∩ T. (4.11)

Hence, we have

K−1φ(1, λ) =(B2(λ) C(λ)A(λ) B1(λ)

), (4.12)

and by Lemma 2.2, we get

φ−1(t, λ) =(p(t)ψΔ(t, λ) −ψ(t, λ)−p(t)ϕΔ(t, λ) ϕ(t, λ)

), t ∈

[ρ(0), 1

]∩ T. (4.13)

Proposition 4.1. For λ ∈ C, λ is an eigenvalue of (1.1) with (1.2) if and only if

D(λ) = 2 cos θ. (4.14)

Moreover, λ is a multiple eigenvalue of (1.1) with (1.2) if and only if

φ(1, λ) = eiθK. (4.15)

Proof. Since ϕ(t, λ) and ψ(t, λ) are linearly independent solutions of (1.1), then λ is aneigenvalue of the problem (1.1) with (1.2) if and only if there exist two constants C1 andC2, not both zero, such that C1ϕ(t, λ) + C2ψ(t, λ) satisfies (1.2), which yields

(ϕ(1, λ) − eiθk11 ψ(1, λ) − eiθk12

ϕΔ(1, λ) − eiθk21 ψΔ(1, λ) − eiθk22

)(C1

C2

)= 0. (4.16)


It is evident that (4.15) has a nontrivial solution (C1, C2) if and only if

det

(ϕ(1, λ) − eiθk11 ψ(1, λ) − eiθk12

ϕΔ(1, λ) − eiθk21 ψΔ(1, λ) − eiθk22

)

= 0, (4.17)

which together with (4.3), (4.4) and detK = k11k22 − k12k21 = 1 implies that

1 + e2iθ − eiθD(λ) = 0. (4.18)

It follows from the above relation and the fact that e−iθ + eiθ = 2 cos θ that λ is an eigenvalueof (1.1) with (1.2) if and only if λ satisfies

D(λ) = 2 cos θ. (4.19)

On the other hand, (1.1) has two linearly independent solutions satisfying (1.2) if and only ifall the entries of the coefficient matrix of (4.16) are zero. Hence, λ is a multiple eigenvalue of(1.1) with (1.2) if and only if (4.15) holds. This completes the proof.

The following result is a direct consequence of the first result of Proposition 4.1.

Corollary 4.2. For any θ ∈ (−π,π],

λn(eiθK

)= λn

(e−iθK

), n ≥ 0. (4.20)

For K ∈ SL(2,R) and detK = 1, we consider the separated boundary problem (1.1)with (2.14). Let μn, 0 ≤ n ≤ Nd − 1, be all the eigenvalues of (1.1) with (2.14) and orderedas that in Theorem 2.7. Since ϕ(t, λ) and ψ(t, λ) are all entire functions in λ ∈ C for eacht ∈ [ρ(0), σ(1)] ∩ T, D(λ) is an entire functions in C. Denote

d

dλD(λ) := D′(λ),

d2

dλ2D(λ) := D′′(λ). (4.21)

Proposition 4.3. Assume that k11 > 0 and k12 ≤ 0 or k11 ≥ 0 and k12 < 0. For each n, n ≥ 0,D(μn) ≥ 2 if n is odd, and D(μn) ≤ −2 if n is even.

Proof. It is noted that λ is eigenvalue of (1.1) with (2.14) if and only if k22ψ(1, λ)−k12ψΔ(1, λ) =

0. Hence, ψ(t, μn) is an eigenfunction with respect to μn. By Theorem 2.7 and the last tworelations in (4.1), we have that ψ(t, μn) has exactly n generalized zeros in (0, 1) and

sgnψ(1, μn

)= (−1)n. (4.22)

(i) If k12 < 0, then it follows from k22ψ(1, μn) − k12ψΔ(1, μn) = 0 that

ψ(1, μn

)

k12=ψΔ(1, μn

)

k22, k11k22ψ

(1, μn

)= k11k12ψ

Δ(1, μn). (4.23)


By (4.3) and the first relation in (4.23) we have

1 = ϕ(1, μn

)ψΔ(1, μn

)− ϕΔ(1, μn

)ψ(1, μn

)

= ϕ(1, μn

)k22

k12ψ(1, μn

)− ϕΔ(1, μn

)ψ(1, μn

)

=(k22ϕ

(1, μn

)− k12ϕ

Δ(1, μn))ψ

(1, μn

)

k12.

(4.24)

By the definition of D(λ), the second relation in (4.23), and detK = 1, we get

k12D(μn)= k11k12ψ

Δ(1, μn)− k12k21ψ

(1, μn

)+ k12k22ϕ

(1, μn

)− k2

12ϕΔ(1, μn

)

= k11k22ψ(1, μn

)− k12k21ψ

(1, μn

)+ k12k22ϕ

(1, μn

)− k2

12ϕΔ(1, μn

)

= ψ(1, μn

)+ k12k22ϕ

(1, μn

)− k2

12ϕΔ(1, μn

).

(4.25)

Hence,

D(μn)=(k22ϕ

(1, μn

)− k12ϕ

Δ(1, μn))

+ψ(1, μn

)

k12. (4.26)

Noting (k22ϕ(1, μn) − k12ϕΔ(1, μn))(ψ(1, μn)/k12) = 1, k12 < 0, and (4.22), we have that if n is

odd, then

D(μn)=

⎛

⎝

√ψ(1, μn

)

k12−√k22ϕ

(1, μn

)− k12ϕΔ

(1, μn

)⎞

⎠

2

+ 2 ≥ 2, (4.27)

and if n is even, then

D(μn)= −

⎛

⎝

√

−ψ(1, μn

)

k12−√−(k22ϕ

(1, μn

)− k12ϕΔ

(1, μn

))⎞

⎠

2

− 2 ≤ −2. (4.28)

(ii) If k12 = 0, then it is noted that λ is eigenvalue of (1.1) with (2.14) if and only ifψ(1, λ) = 0. Hence, ψ(t, μn) is an eigenfunction with respect to μn. By Theorem 2.7, ψ(t, μn)has exactly n generalized zeros in (0, 1) and

ψ(ρ(0), μn

)= ψ

(1, μn

)= 0, ψΔ(ρ(0), μn

)= 1, n ≥ 0. (4.29)

Hence ψ(t, μ0) > 0 for all t ∈ (0, 1) ∩ T.


Next we will show ψΔ(1, μ0) < 0. In the case that ρ(1) < 1, ψ(ρ(1), μ0) > 0 andψΔ(ρ(1), μ0) < 0. It follows from (1.1) and (2.4) that

(p(ρ(1)

)ψΔ(ρ(1), μ0

))Δ=p(1)ψΔ(1, μ0

)− p

(ρ(1)

)ψΔ(ρ(1), μ0

)

1 − ρ(1) = 0, (4.30)

which implies

ψΔ(1, μ0)= p

(ρ(1)

)ψΔ(ρ(1), μ0

)< 0. (4.31)

In the other case that ρ(1) = 1, then

ψΔ(1, μ0)= lim

t→ 1−

ψ(t, μ0

)− ψ

(1, μ0

)

t − 1= − lim

t→ 1−

ψ(t, μ0

)

1 − t ≤ 0. (4.32)

Further, by the existence and uniqueness theorem of solutions of initial value problems for(1.1) [5, Theorem 4.5], we obtain that ψΔ(1, μ0) < 0.

With a similar argument from above, we get sgnψΔ(1, μn) = (−1)n+1, n ≥ 0.By referring to ψ(1, μn) = 0 and from (4.3), it follows that

ϕ(1, μn

)ψΔ(1, μn

)= 1. (4.33)

Hence, noting detK = k11k22 = 1 and k22 > 0, if n is odd, then

D(μn)=

k22

ψΔ(1, μn

) + k11ψΔ(1, μn

)≥ 2, (4.34)

and if n is even, then

D(μn)≤ −2. (4.35)

This completes the proof.

Proposition 4.4. Assume that k11 > 0 and k12 ≤ 0 or k11 ≥ 0 and k12 < 0. There exists a constant ν0

such that ν0 < μ0 and D(ν0) ≥ 2.

Proof. Since ϕ(t, λ) and ψ(t, λ) are solutions of (1.1), we have

−(p(t)ϕΔ(t, λ)

)Δ+ q(t)ϕσ(t, λ) = λr(t)ϕσ(t, λ),

−(p(t)ψΔ(t, λ)

)Δ+ q(t)ψσ(t, λ) = λr(t)ψσ(t, λ).

(4.36)


By integration, it follows from (4.1) and (4.36) that

ϕΔ(1, λ) =∫1

ρ(0)

(q(s) − λr(s)

)ϕ(σ(s), λ)Δs,

ψΔ(1, λ) = 1 +∫1

ρ(0)

(q(s) − λr(s)

)ψ(σ(s), λ)Δs,

(4.37)

where p(ρ(0)) = p(1) = 1 is used. In addition, from (4.36), we obtain

(ϕ(t, λ)

(p(t)ϕΔ(t, λ)

))Δ= p(t)

(ϕΔ(t, λ)

)2+(q(t) − λr(t)

)(ϕσ(t, λ)

)2,

(ψ(t, λ)

(p(t)ψΔ(t, λ)

))Δ= p(t)

(ψΔ(t, λ)

)2+(q(t) − λr(t)

)(ψσ(t, λ)

)2,

(4.38)

which, similarly together with (4.1) and by integration, imply that

ϕ(1, λ)ϕΔ(1, λ) =∫1

ρ(0)

(p(s)

(ϕΔ(s, λ)

)2+(q(s) − λr(s)

)ϕ2(σ(s), λ)

)Δs

≥∫1

ρ(0)

(q(s) − λr(s)

)ϕ2(σ(s), λ)Δs,

ψ(1, λ)ψΔ(1, λ) =∫1

ρ(0)

(p(s)

(ψΔ(s, λ)

)2+(q(s) − λr(s)

)ψ2(σ(s), λ)

)Δs

≥∫1

ρ(0)

(q(s) − λr(s)

)ψ2(σ(s), λ)Δs.

(4.39)

On the other hand, it follows from Lemma 2.5 and (4.1) that for all sufficiently large −λ,ϕσ(t) > 0, ψσ(t) > 0, for all t ∈ (ρ(0), 1) ∩ T, where α and β in (2.11) are taken as α = 0, β = 1,and α = −1, β = 0, respectively, which satisfy (H). So, from (4.37) and (4.39), we obtain that

limλ→−∞

ψΔ(1, λ) = limλ→−∞

ϕΔ(1, λ) = limλ→−∞

(ψ(1, λ)ψΔ(1, λ)

)= lim

λ→−∞

(ϕ(1, λ)ϕΔ(1, λ)

)=∞,

(4.40)

and by Lemma 2.4, it implies

limλ→−∞

D(λ) = limλ→−∞

(k22ϕ(1, λ) + k11ψ

Δ(1, λ) − k21ψ(1, λ) − k12ϕΔ(1, λ)

)

= limλ→−∞

(

k11ψ(1, λ)

(ψΔ(1, λ)ψ(1, λ)

− k21

k11

)

− k12ϕ(1, λ)

(ϕΔ(1, λ)ϕ(1, λ)

− k22

k12

))

=∞.

(4.41)


By Proposition 4.3, D(μ0) ≤ −2. Therefore, there exists a ν0 < μ0 such that D(ν0) ≥ 2. Thiscompletes the proof.

Lemma 4.5. For any λ ∈ C one has

D′(λ) =∫1

ρ(0)

(A(λ)ψ2(σ(s), λ) − B(λ)ψ(σ(s), λ)ϕ(σ(s), λ) − C(λ)ϕ2(σ(s), λ)

)r(s)Δs,

(4.42)

4C(λ)D′(λ) = −∫1

ρ(0)

(2C(λ)ϕ(σ(s), λ) + B(λ)ψ(σ(s), λ)

)2r(s)Δs

−(4 −D2(λ)

)∫1

ρ(0)ψ2(σ(s), λ)r(s)Δs,

(4.43)

4A(λ)D′(λ) =∫1

ρ(0)

(2A(λ)ψ(σ(s), λ) − B(λ)ϕ(σ(s), λ)

)2r(s)Δs

+(4 −D2(λ)

)∫1

ρ(0)ϕ2(σ(s), λ)r(s)Δs.

(4.44)

Proof. Since ϕ(t, λ) and ψ(t, λ) are solutions of (1.1) with (4.1), then they satisfy (4.36).Differentiating (4.36) with respect to λ, we have

(p(t)ϕΔ

λ (t, λ))Δ

+(q(t) + λr(t)

)ϕσλ(t, λ) = −r(t)ϕ

σ(t, λ)

ϕλ(ρ(0), λ

)= ϕΔ

λ

(ρ(0), λ

)= 0,

(p(t)ψΔ

λ (t, λ))Δ

+(q(t) + λr(t)

)ψσλ (t, λ) = −r(t)ψ

σ(t, λ)

ψλ(ρ(0), λ

)= ψΔ

λ

(ρ(0), λ

)= 0.

(4.45)

By the variation of constants formula [5, Theorem 4.24], we get

ϕλ(t, λ) =∫ t

ρ(0)r(s)

(ψ(σ(s), λ)ϕ(t, λ) − ϕ(σ(s), λ)ψ(t, λ)

)ϕ(σ(s), λ)Δs,

ψλ(t, λ) =∫ t

ρ(0)r(s)

(ψ(σ(s), λ)ϕ(t, λ) − ϕ(σ(s), λ)ψ(t, λ)

)ψ(σ(s), λ)Δs.

(4.46)


Further, it follows from [5, Theorem 1.117] that

ϕΔλ (t, λ) =

∫ t

ρ(0)r(s)

(ψ(σ(s), λ)ϕΔ(t, λ) − ϕ(σ(s), λ)ψΔ(t, λ)

)ϕ(σ(s), λ)Δs,

ψΔλ (t, λ) =

∫ t

ρ(0)r(s)

(ψ(σ(s), λ)ϕΔ(t, λ) − ϕ(σ(s), λ)ψΔ(t, λ)

)ψ(σ(s), λ)Δs.

(4.47)

From (4.46), (4.47), and (4.11), we have

φλ(t, λ) =

(ϕλ(t, λ) ψλ(t, λ)

p(t)ϕΔλ (t, λ) p(t)ψΔ

λ (t, λ)

)

=∫ t

ρ(0)r(s)

(A11(s, t, λ) A12(s, t, λ)

A21(s, t, λ) A22(s, t, λ)

)

Δs, (4.48)

where

A11(s, t, λ) =(ψσ(s, λ)ϕ(t, λ) − ϕσ(s, λ)ψ(t, λ)

)ϕσ(s, λ),

A12(s, t, λ) =(ψσ(s, λ)ϕ(t, λ) − ϕσ(s, λ)ψ(t, λ)

)ψσ(s, λ),

A21(s, t, λ) = p(t)(ψσ(s, λ)ϕΔ(t, λ) − ϕσ(s, λ)ψΔ(t, λ)

)ϕσ(s, λ),

A22(s, t, λ) = p(t)(ψσ(s, λ)ϕΔ(t, λ) − ϕσ(s, λ)ψΔ(t, λ)

)ψσ(s, λ).

(4.49)

It follows from (4.11) and (4.13) that

φ(t, λ)φ−1(σ(s), λ)R(s)φ(σ(s), λ) = −(A11(s, t, λ) A12(s, t, λ)

A21(s, t, λ) A22(s, t, λ)

)

, (4.50)

where R(t) =(

0 0r(t) 0

), t ∈ [ρ(0), ρ(1)] ∩ T.

Hence,

φλ(t, λ) = −∫1

ρ(0)φ(t, λ)φ−1(σ(s), λ)R(s)φ(σ(s), λ)Δs. (4.51)

By (4.10) and (4.12), we have

D(λ) = B1(λ) + B2(λ) = traceK−1φ(1, λ). (4.52)


Differentiating above relation with respect to λ, and with (4.12), we have

D′(λ)

= traceK−1φλ(1, λ)

= −trace∫1

ρ(0)K−1φ(1, λ)φ−1(σ(s), λ)R(s)φ(σ(s), λ)Δs

= −trace∫1

ρ(0)

(B2(λ) C(λ)

A(λ) B1(λ)

)(ψΔ(σ(s), λ) −ψ(σ(s), λ)−ϕΔ(σ(s), λ) ϕ(σ(s), λ)

)

×(

0 0r(s) 0

)( ϕ(σ(s), λ) ψ(σ(s), λ)

ϕΔ(σ(s), λ) ψΔ(σ(s), λ)

)

Δs

=∫1

ρ(0)

(A(λ)ψ2(σ(s), λ) − (B1(λ) − B2(λ))ψ(σ(s), λ)ϕ(σ(s), λ) − C(λ)ϕ2(σ(s), λ)

)r(s)Δs,

(4.53)

which together with (4.10) confirm (4.42).To establish (4.43), from (4.12) and (4.10), we obtain

B1(λ)B2(λ) −A(λ)C(λ) = det(K−1φ(1, λ)

)= 1,

4 −D2(λ) = 4 − (B1(λ) + B2(λ))2 = 4 − (B1(λ) − B2(λ))

2 − 4B1(λ)B2(λ)

= 4(1 − B1(λ)B2(λ)) − B2(λ) = −(4A(λ)C(λ) + B2(λ)

).

(4.54)

Thus

4C(λ)D′(λ)

=∫1

ρ(0)

(4A(λ)C(λ)ψσ

2(s, λ) − 4B(λ)C(λ)ψσ(s, λ)ϕσ(s, λ) − 4C2(λ)ϕσ

2(s, λ)

)r(s)Δs

=∫1

ρ(0)

(−(2C(λ)ϕσ(s, λ) + B(λ)ψσ(s, λ)

)2 +(

4A(λ)C(λ) + B2(λ))ψσ

2(s, λ)

)r(s)Δs

= −∫1

ρ(0)

(2C(λ)ϕσ(s, λ) + B(λ)ψσ(s, λ)

)2r(s)Δs −

(4 −D2(λ)

)∫1

ρ(0)ψσ

2(s, λ)r(s)Δs.

(4.55)

That is, (4.43) holds. The identity (4.44) can be verified similarly. This completes theproof.

Corollary 4.6. If λ ∈ R satisfies |D(λ)| < 2, then A(λ)/= 0, C(λ)/= 0, and D′(λ)/= 0.

Proof. These are direct consequences of (4.43) and (4.44).


Lemma 4.7. C(λ) = 0 if and only if λ = μn for some n ∈ {0, 1, . . . ,Nd − 1} and ψ(·, μn) is aneigenfunction of μn.

Proof. It is directly follows from the definition of C(λ) and the initial conditions (4.1).

Lemma 4.8. Assum that θ = 0 or θ = π and λ is a multiple eigenvalue of (1.1) with (1.2) if andonly if D′(λ) = 0.

Proof. Assume that θ = 0. By (4.15) λ is a multiple eigenvalue if and only if

φ(1, λ) = K; (4.56)

hence, it follows from (4.12) that

(B2(λ) C(λ)A(λ) B1(λ)

)= K−1φ(1, λ) = K−1K = I. (4.57)

Therefore A(λ) = C(λ) = 0 and B1(λ) = B2(λ) = 1.

(i) Suppose that λ is a multiple eigenvalue of (1.1) with (1.2). Then A(λ) = C(λ) = 0and B(λ) = B1(λ) − B2(λ) = 0. By (4.42), D′(λ) = 0.

(ii) Suppose that λ is an eigenvalue of (1.1) with (1.2) and D′(λ) = 0. Then by (4.14),D(λ) = 0. From (4.43) and (4.44) we get

2C(λ)ϕ(σ(s), λ) + B(λ)ψ(σ(s), λ) = 0,

2A(λ)ψ(σ(s), λ) − B(λ)ϕ(σ(s), λ) = 0.(4.58)

Since ϕ(t, λ) and ψ(t, λ) are linearly independence solutions of (1.1), we have

A(λ) = B(λ) = C(λ) = 0. (4.59)

It follows from B(λ) = B1(λ) − B2(λ) = 0 and D(λ) = B1(λ) + B2(λ) = 2 that B1(λ) = B2(λ) = 1.Thus, λ is a multiple eigenvalue of (1.1) with (1.2).

The case θ = π can be established by replacing K by −K in the above argument. Thiscompletes the proof.

Lemma 4.9. Assume θ = 0 or θ = π . If λ is a multiple eigenvalue of (1.1) with (1.2), then thereexists n ∈ {0, 1, . . . ,Nd − 1} such that λ = μn.

Proof. Assume that λ is a multiple eigenvalue of (1.1) with (1.2). From the proof of Lemma 4.8we see that C(λ) = 0. From (4.9) we have

k22ψ(1, λ) − k12ψΔ(1, λ) = 0. (4.60)

This means that λ = μn for some n ∈ {0, 1, . . . ,Nd − 1}. This completes the proof.


Proposition 4.10. Assume that k11 > 0 and k12 ≤ 0 or k11 ≥ 0 and k12 < 0.

(i) Equations D′(λ) = 0 and D(λ) = 2 or −2 hold if and only if λ is a multiple eigenvalue of(1.1) with (1.2) with θ = 0 or θ = π .

(ii) If D(λ) = 2 or −2 and λ is a multiple eigenvalue of (1.1) with (1.2), then λ = μn, n ∈{0, 1, . . . ,Nd − 1}.

(iii) If D(λ) = 2 or −2 for some λ/=μn, n ∈ {0, 1, . . . ,Nd − 1}, then λ is a simple eigenvalue of(1.1) with (1.2) with θ = 0 or θ = π .

(iv) Moreover, for every λ/=μn, n ∈ {0, 1, . . . ,Nd − 1}, with −2 ≤ D(λ) ≤ 2 one has

D′(λ) < 0, ν0 < λ < μ0;(−1)nD′(λ) > 0, μn < λ < μn+1, n ≥ 0,

(4.61)

and in the case ofNd <∞,

(−1)Nd−1D′(λ) > 0, λ > μNd−1. (4.62)

Proof. Parts (i), (ii), and (iii) follow from Lemmas 4.8 and 4.9. It follows from Propositions4.3 and 4.4 and Corollary 4.6 that D(μ0) ≤ −2, D(ν0) ≥ 2 with ν0 < μ0 and D′(λ)/= 0 when|D(λ)| < 2. Hence, D′(λ) < 0 with ν0 < λ < μ0, −2 ≤ D(λ) ≤ 2. Similarly, by Proposition 4.3and Corollary 4.6, we have

D(μn)D(μn+1

)≤ −4, D′(λ)/= 0 when |D(λ)| < 2, (4.63)

which implies

(−1)nD′(λ) > 0 with μn < λ < μn+1, −2 ≤ D(λ) ≤ 2. (4.64)

If Nd <∞, then all the points of [0, 1] ∩ T are isolated. In this case, (1.1) can be rewritten as

p(σ(t))y(σ2(t)

)= (a(t) + λb(t))y(σ(t)) + c(t)y(t), t ∈

[ρ(0), ρ(1)

]∩ T, (4.65)

where

a(t) = p(σ(t)) + p(t)μ(σ(t))μ(t)

− q(t)μ(t)μ(σ(t)),

b(t) = −r(t)μ(t)μ(σ(t)), c(t) = −p(t)μ(σ(t))μ(t)

.

(4.66)

By Theorem 2.7, (1.1) with (2.14) has Nd eigenvalues:

μ0 < μ1 < · · · < μNd−1. (4.67)


It follows from (4.65) that ψΔ(1, λ) = (ψ(σ(1), λ) − ψ(1, λ))/μ(1) and ϕΔ(1, λ) = (ϕ(σ(1), λ) −ϕ(1, λ))/μ(1) are two polynomials of degree Nd + 1 in λ and ψ(1, λ) and ϕ(1, λ) are twopolynomials of degree Nd in λ. Then D(λ) can be written as

D(λ) = k11ψΔ(1, λ) − k21ψ(1, λ) + k22ϕ(1, λ) − k12ϕ

Δ(1, λ) = (−1)Nd+1ANd+1λNd+1 + h(λ),

(4.68)

where ANd+1 > 0 and h(λ) is a polynomial in λ whose order is not larger than Nd. ByProposition 4.3, if Nd + 1 is odd, then D(μNd−1) ≥ 2, and if Nd + 1 is even, then D(μNd−1) ≤ −2.It follows that if Nd + 1 is odd, then D(λ) → −∞ as λ → ∞, and if Nd + 1 is even, thenD(λ) → ∞ as λ → ∞. Hence, if Nd + 1 is odd, then there exists a constant ξ0 > μNd−1

such that D(ξ0) ≤ −2. Similarly, in the other case that Nd + 1 is even, there exists a constantη0 > μNd−1 such that D(η0) ≥ 2, and by using Corollary 4.6, we have

−D′(λ) > 0 with μNd−1 < λ < ξ0 if Nd + 1 is odd,

D′(λ) > 0 with μNd−1 < λ < η0 if Nd + 1 is even.(4.69)

Hence,

(−1)Nd−1D′(λ) > 0, λ > μNd−1. (4.70)

This completes the proof.

Proposition 4.11. For any fixed θ /= 0, −π < θ < π , each eigenvalue of (1.1) with (1.2) is simple.

Proof. It follows from (4.46) and (4.47) that

D′(λ) = k11ψΔλ (1, λ) − k21ψλ(1, λ) + k22ϕλ(1, λ) − k12ϕ

Δλ (1, λ) =

∫1

ρ(0)r(s)δ(s)Δs, (4.71)

where

δ(s) :=(k11ϕ

Δ(1, λ) − k21ϕ(1, λ))ψ2(σ(s), λ)

+(k22ϕ(1, λ) − k11ψ

Δ(1, λ) + k21ψ(1, λ) − k12ϕΔ(1, λ)

)ϕ(σ(s), λ)ψ(σ(s), λ)

−(k22ψ(1, λ) − k12ψ

Δ(1, λ))ϕ2(σ(s), λ)

=(ψ(σ(s), λ), ϕ(σ(s), λ)

)H(λ)

(ψ(σ(s), λ)

ϕ(σ(s), λ)

)

,

H(λ) :=

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

k11ϕΔ(1, λ) − k21ϕ(1, λ)

12(k22ϕ(1, λ) − k11ψ

Δ(1, λ)

+k21ψ(1, λ) − k12ϕΔ(1, λ)

)

12(k22ϕ(1, λ) − k11ψ

Δ(1, λ)

+k21ψ(1, λ) − k12ϕΔ(1, λ)

)k12ψ

Δ(1, λ) − k22ψ(1, λ)

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

.

(4.72)


Then from (4.1), detK = 1, and the definition of D(λ), we have

detH(λ) =(k11ϕ

Δ(1, λ) − k21ϕ(1, λ))(k12ψ

Δ(1, λ) − k22ψ(1, λ))

−(k22ϕ(1, λ) − k11ψ

Δ(1, λ) + k21ψ(1, λ) − k12ϕΔ(1, λ)

)2

4

=(k11ϕ

Δ(1, λ) − k21ϕ(1, λ))(k12ψ

Δ(1, λ) − k22ψ(1, λ))

− 14D2(λ) +

(k22ϕ(1, λ) − k12ϕ

Δ(1, λ))(k11ψ

Δ(1, λ) − k21ψ(1, λ))

= −14D2(λ) + 1.

(4.73)

Thus, if |D(λ)| ≤ 2, then detH(λ) ≥ 0. H(λ) is always positive semidefinite or negativesemidefinite. Consequently, δ(s) is not change sign in [ρ(0), 1] ∩ T. In this case, D′(λ) cannotvanish unless δ(s) ≡ 0. Because ϕ(t, λ) and ψ(t, λ) are linearly independent, δ(s) ≡ 0 if andonly if all the entries of the matrix H(λ) vanish, namely,

k11ϕΔ(1, λ) − k21ϕ(1, λ) = 0,

k12ψΔ(1, λ) − k22ψ(1, λ) = 0,

k22ϕ(1, λ) − k11ψΔ(1, λ) + k21ψ(1, λ) − k12ϕ

Δ(1, λ) = 0.

(4.74)

Suppose that λ is an eigenvalue of the problem (1.1) with (1.2) and fix θ,−π < θ < π withθ /= 0. By Proposition 4.1, we have D2(λ) = 4 cos2θ < 4, then detH(λ) > 0, and the matrixH(λ) is positive definite or negative definite. Hence, δ(s) > 0 or δ(s) < 0 for s ∈ [ρ(0), 1] ∩ T,since ϕ(t, λ) and ψ(t, λ) are linearly independent.

If λ is a multiple eigenvalue of problem (1.1) with (1.2), then (4.15) holds byProposition 4.1. By using (4.15), it can be easily verified that (4.74) holds; that is, all the entriesof the matrix H(λ) are zeros. Then δ(s) = 0, which is contrary to δ(s)/= 0. Hence, λ is a simpleeigenvalue of (1.1) and (1.2). This completes the proof.

Proposition 4.12. If n is odd, D(μn) = 2, and D′(μn) = 0, then D′′(μn) < 0; if n is even, D(μn) =−2, and D′(μn) = 0, then D′′(μn) > 0.

Proof. Assume D(μn) = 2 and D′(μn) = 0 with n being odd. It follows from Proposition 4.1that

φ(1, μn

)= K. (4.75)


As in the proof of Lemma 4.5 and by (4.11) and (4.13),

D′(λ) = −trace∫1

ρ(0)K−1φ(1, λ)φ−1(σ(s), λ)R(s)φ(σ(s), λ)Δs

= −trace∫1

ρ(0)φ−1(1, μn

)φ(1, λ)

(p(s)ψΔ(σ(s), λ) −ψ(σ(s), λ)−p(s)ϕΔ(σ(s), λ) ϕ(σ(s), λ)

)

×(

0 0r(s) 0

)(ϕ(σ(s), λ) ψ(σ(s), λ)

p(s)ϕΔ(σ(s), λ) p(s)ψΔ(σ(s), λ)

)

Δs

= trace

(

φ−1(1, μn)φ(1, λ)

∫1

ρ(0)

(ψ(σ(s), λ)ϕ(σ(s), λ) ψ2(σ(s), λ)

−ϕ2(σ(s), λ) −ψ(σ(s), λ)ϕ(σ(s), λ)

)

r(s)Δs

)

.

(4.76)

Hence,

D′′(λ)

= trace

(

φ−1(1, μn)φλ(1, λ)

∫1

ρ(0)

(ψ(σ(s), λ)ϕ(σ(s), λ) ψ2(σ(s), λ)−ϕ2(σ(s), λ) −ψ(σ(s), λ)ϕ(σ(s), λ)

)

r(s)Δs

)

+ trace

(

φ−1(1, μn)φ(1, λ)

∂

∂λ

∫1

ρ(0)

(ψ(σ(s), λ)ϕ(σ(s), λ) ψ2(σ(s), λ)−ϕ2(σ(s), λ) −ψ(σ(s), λ)ϕ(σ(s), λ)

)

r(s)Δs

)

= trace

⎛

⎝φ−1(1, μn)φ(1, λ)

∫1

ρ(0)

(ψσ(s, λ)ϕσ(s, λ) ψσ

2(s, λ)

−ϕσ2(s, λ) −ψσ(s, λ)ϕσ(s, λ)

)2

r2(s)Δs

⎞

⎠

+ trace

(

φ−1(1, μn)φ(1, λ)

∂

∂λ

∫1

ρ(0)

(ψσ(s, λ)ϕσ(s, λ) ψσ

2(s, λ)

−ϕσ2(s, λ) −ψσ(s, λ)ϕσ(s, λ)

)

r(s)Δs

)

,

(4.77)

where (4.51) is used and

D′′(μn)= trace

∫1

ρ(0)

(ψ(σ(s), μn)ϕ(σ(s), μn) ψ2(σ(s), μn)

−ϕ2(σ(s), μn) −ψ(σ(s), μn)ϕ(σ(s), μn)

)2

r2(s)Δs

= 2

(∫1

ρ(0)ψ(σ(s), μn)ϕ(σ(s), μn)r(s)Δs

)2

− 2∫1

ρ(0)ψ2(σ(s), μn

)r(s)Δs

∫1

ρ(0)ϕ2(σ(s), μn

)r(s)Δs

≤ 0

(4.78)


D(λ)

2

2 cos θ

λ0(−K) μ0 λ1(−K) λ2(−K)

v0 λ0(K) λ0(eiθK) λ1(eiθK) μ1, (λ1,2(K)) λ2(eiθK) λ

−2

Figure 1: The graph of D(λ).

D(λ)

2

2 cos θ

λNd−1(−K) ξ0λNd(−K)

λNd−1(eiθK) λNd−1(K) μNd−1 λNd(K) λNd

(eiθK) λ

−2

Figure 2: The graph of D(λ) in the case that Nd + 1 is odd.

by the Holder inequality [8, Lemma 2.2(iv)]. Therefore D′′(μn) < 0. Since ϕ(t, λ) and ψ(t, λ)are linearly independent, which proves the first conclusion, the second conclusion can beshown similarly. This completes the proof.

5. Proofs of the Main Results

Proof of Theorem 3.1. By Propositions 4.1–4.12 and the intermediate value theorem, theinequalities in (3.1)–(3.2) can been illustrated with the graph of D(λ) (see Figures 1–3). Wenow give the detail proof of Theorem 3.1.

By Lemma 2.9, all the eigenvalues of the coupled boundary value problem (1.1) with(1.2) are real. By Propositions 4.3–4.10, D(μ0) ≤ −2, D′(λ) < 0 for all λ < μ0 with −2 ≤ D(λ) ≤2, and there exists ν0 < μ0 such that D(ν0) ≥ 2. Therefore, by the continuity of D(λ) and theintermediate value theorem, (1.1) and (1.2) with θ = 0 have only one eigenvalue λ0(K) < μ0,


D(λ)

2

2 cos θ

λNd−1(−K) μNd−1 λNd(−K)

λNd−1(K) λNd−1(eiθK) λNd(eiθK) λNd

(K) η0 λ

−2

Figure 3: The graph of D(λ) in the case that Nd + 1 is even.

(1.1) and (1.2) with θ = π hve only one eigenvalue λ0(−K) ≤ μ0, and (1.1) and (1.2) withθ /= 0, −π < θ < π have only one eigenvalue λ0(K) < λ0(eiθK) < λ0(−K), and they satisfy

ν0 ≤ λ0(K) < λ0

(eiθK

)< λ0(−K) ≤ μ0. (5.1)

Similarly, by Propositions 4.1, 4.3, and 4.10, the continuity ofD(λ), and the intermediate valuetheorem,D(λ) reaches −2, 2 cos θ (θ /= 0, −π < θ < π), and 2 exactly one time between any twoconsecutive eigenvalues of the separated boundary value problem (1.1) with (2.14). Hence,(1.1) and (1.2) with θ = 0, θ /= 0, −π < θ < π , and θ = π have only one eigenvalue between anytwo consecutive eigenvalues of (1.1) with (2.14), respectively. In addition, by Propositions4.10 and 4.12, if D(μn) = 2 or −2 and D′(μn) = 0, then μn is not only an eigenvalue of (1.1)with (2.14) but also a multiple eigenvalue of (1.1) and (1.2) with θ = 0 and θ = π .

If Nd = ∞, then it follows from the above discussion that (1.1) and (1.2) with θ /= 0,−π < θ < π have infinitely many eigenvalues, and they are real and satisfy (3.1).

If Nd < ∞, then all points of [0, 1] ∩ T are isolated. In this case (1.1) and D(λ) can berewritten as (4.65) and (4.68). By the same method in the proof of Proposition 4.10, that ifNd + 1 is even, then there exists a constant ξ0 > μNd−1 such that D(ξ0) ≤ −2, which togetherwith (4.62), implies that (1.1) and (1.2) with θ = 0, θ /= 0, −π < θ < π , and θ = π have onlyone eigenvalue λNd(K), λNd(e

iθK), and λNd(−K), satisfying

μNd−1 ≤ λNd(K) < λNd

(eiθK

)< λNd(−K) ≤ ξ0 (5.2)

(see Figure 2). Similarly, in the other case thatNd+1 is even, there exists a constant η0 > μNd−1

such that D(η0) ≥ 2, which, together with (4.62) implies that (1.1) and (1.2) with θ = 0, θ /= 0,−π < θ < π , and θ = π have only one eigenvalue λNd(K), λNd(e

iθK), and λNd(−K), satisfying

μNd−1 ≤ λNd(−K) < λNd

(eiθK

)< λNd(K) ≤ η0 (5.3)


(see Figure 3). Therefore, we get that (1.1) and (1.2) with θ /= 0, −π < θ < π have Nd + 1eigenvalues and they are real and satisfy

ν0 ≤ λ0(K) < λ0

(eiθK

)< λ0(−K) ≤ μ0 ≤ λ1(−K) < λ1

(eiθK

)< λ1(K) ≤ μ1 ≤ λ2(K)

< · · · < λNd−1(K) ≤ μNd−1 ≤ λNd(K) < λNd

(eiθK

)< λNd(−K) ≤ ξ0

(5.4)

if Nd + 1 is odd; and

ν0 ≤ λ0(K) < λ0

(eiθK

)< λ0(−K) ≤ μ0 ≤ λ1(−K) < λ1

(eiθK

)< λ1(K) ≤ μ1 ≤ λ2(K)

< · · · < λNd−1(−K) ≤ μNd−1 ≤ λNd(−K) < λNd

(eiθK

)< λNd(K) ≤ η0

(5.5)

if Nd + 1 is even. This completes the proof.

Remark 5.1. In the continuous case: μ(t) = 0, Nd = ∞, by Theorem 3.1, the coupled boundaryvalue problems (1.1) and (1.2) have infinitely many eigenvalues: {λn(eiθK)}∞n=0 for θ /= 0, −π <θ < π ; {λn(K)}∞n=0 for θ = 0; {λn(−K)}∞n=0 for θ = π , and they satisfy inequality (3.1). Thisresult is the same as that obtained by Eastham et al. for second-order differential equations[2, Theorem 3.2].

Example 5.2. Consider the following three specific cases:

[ρ(0), 1

]∩ T =

[0,

12

]∪[

23, 1]

;

[ρ(0), 1

]∩ T =

[0,

12

]∪{

12(N − 1)

,1

(N − 1),

32(N − 1)

, . . . , 1}, N > 2;

[ρ(0), 1

]∩ T =

{qk | k ≥ 0, k ∈ Z

}∪ {0}, where 0 < q < 1.

(5.6)

It is evident that |[ρ(0), 1]∩T| =∞ and then Nd =∞ in these three cases. By Theorem 3.1, thecoupled boundary value problems (1.1) and (1.2) have infinitely many real eigenvalues andthey satisfy the inequality (3.1). Obviously, the above three cases are not continuous and notdiscrete. So the existing results are not available now.

By Remark 5.1 and Example 5.2, our result in Theorem 3.1 not only extends the resultsin the discrete cases but also contains more complicated time scales.

Acknowledgments

Many thanks to A. Pankov (the Editor) and the anonymous reviewer(s) for helpful commentsand suggestions. This research was supported by the Natural Science Foundation ofShandong Province (Grant Y2008A28) (Grant ZR2009AL003) and the Natural Science FundProject of the University of Jinan (Grant XKY0918).


References

[1] C. Zhang and Y. Shi, “Eigenvalues of second-order symmetric equations on time scales with periodicand antiperiodic boundary conditions,” Applied Mathematics and Computation, vol. 203, no. 1, pp. 284–296, 2008.

[2] M. S. P. Eastham, Q. Kong, H. Wu, and A. Zettl, “Inequalities among eigenvalues of Sturm-Liouvilleproblems,” Journal of Inequalities and Applications, vol. 3, no. 1, pp. 25–43, 1999.

[3] W. N. Everitt, M. Moller, and A. Zettl, “Discontinuous dependence of the n-th Sturm-Liouvilleeigenvalue,” in General Inequalities, 7, vol. 123, pp. 145–150, Birkhauser, Basel, Switzerland, 1997.

[4] R. P. Agarwal, M. Bohner, and P. J. Y. Wong, “Sturm-Liouville eigenvalue problems on time scales,”Applied Mathematics and Computation, vol. 99, no. 2-3, pp. 153–166, 1999.

[5] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,Birkhauser, Boston, Mass, USA, 2001.

[6] V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, Dynamic Systems on Measure Chains, vol.370 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands,1996.

[7] R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” Results inMathematics, vol. 35, no. 1-2, pp. 3–22, 1999.

[8] M. Bohner and D. A. Lutz, “Asymptotic behavior of dynamic equations on time scales,” Journal ofDifference Equations and Applications, vol. 7, no. 1, pp. 21–50, 2001.


Research ArticleMild Solutions for Fractional DifferentialEquations with Nonlocal Conditions

Fang Li

School of Mathematics, Yunnan Normal University, Kunming 650092, China

Correspondence should be addressed to Fang Li, [email protected]

Received 8 January 2010; Accepted 21 January 2010


Copyright q 2010 Fang Li. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

This paper is concerned with the existence and uniqueness of mild solution of the fractionaldifferential equations with nonlocal conditions dqx(t)/dtq = −Ax(t) + f(t, x(t), Gx(t)), t ∈[0, T], and x(0) + g(x) = x0, in a Banach space X, where 0 < q < 1. General existence anduniqueness theorem, which extends many previous results, are given.

1. Introduction

The fractional differential equations can be used to describe many phenomena arising inengineering, physics, economy, and science, so they have been studied extensively (see, e.g.,[1–8] and references therein).

In this paper, we discuss the existence and uniqueness of mild solution for

dqx(t)dtq

= −Ax(t) + f(t, x(t), Gx(t)), t ∈ [0, T],

x(0) + g(x) = x0,

(1.1)

where 0 < q < 1, T > 0, and −A generates an analytic compact semigroup {S(t)}t≥0 ofuniformly bounded linear operators on a Banach space X. The term Gx(t) which may beinterpreted as a control on the system is defined by

Gx(t) :=∫ t

0K(t, s)x(s)ds, (1.2)


where K ∈ C(D,R+)(the set of all positive function continuous on D := {(t, s) ∈ R2 : 0 ≤ s ≤

t ≤ T}) and

G∗ = supt∈[0,T]

∫ t

0K(t, s)ds <∞. (1.3)

The functions f and g are continuous.The nonlocal condition x(0) + g(x) = x0 can be applied in physics with better effect

than that of the classical initial condition x(0) = x0. There have been many significantdevelopments in the study of nonlocal Cauchy problems (see, e.g., [6, 7, 9–14] and referencescited there).

In this paper, motivated by [1–7, 9–15] (especially the estimating approach given byXiao and Liang [14]), we study the semilinear fractional differential equations with nonlocalcondition (1.1) in a Banach space X, assuming that the nonlinear map f is defined on [0, T] ×Xα ×Xα and g is defined on C([0, T], Xα) where Xα = D(Aα), for 0 < α < 1, the domain of thefractional power of A. New and general existence and uniqueness theorem, which extendsmany previous results, are given.

2. Preliminaries

In this paper, we set I = [0, T], a compact interval in R. We denote by X a Banach spacewith norm ‖ · ‖. Let −A : D(A) → X be the infinitesimal generator of a compact analyticsemigroup of uniformly bounded linear operators {S(t)}t≥0, that is, there exists M > 1 suchthat ‖S(t)‖ ≤ M; and without loss of generality, we assume that 0 ∈ ρ(A). So we can definethe fractional power Aα for 0 < α ≤ 1, as a closed linear operator on its domain D(Aα) withinverse A−α, and one has the following known result.

Lemma 2.1 (see [15]). (1) Xα = D(Aα) is a Banach space with the norm ‖x‖α := ‖Aαx‖ forx ∈ D(Aα).

(2) S(t) : X → Xα for each t > 0 and α > 0.(3) For every u ∈ D(Aα) and t ≥ 0, S(t)Aαu = AαS(t)u.(4) For every t > 0, AαS(t) is bounded on X and there existsMα > 0 such that

‖AαS(t)‖ ≤Mαt−α. (2.1)

Definition 2.2. A continuous function x : I → X satisfying the equation

x(t) = S(t)(x0 − g(x)

)+

1Γ(q)∫ t

0(t − s)q−1S(t − s)f(s, x(s), Gx(s))ds (2.2)

for t ∈ I is called a mild solution of (1.1).


In this paper, we use ‖f‖p to denote the Lp norm of f whenever f ∈ Lp(0, T) for somep with 1 ≤ p < ∞. We denote by Cα the Banach space C([0, T], Xα) endowed with the supnorm given by

‖x‖∞ := supt∈I‖x‖α, (2.3)

for x ∈ Cα.The following well-known theorem will be used later.

Theorem 2.3 (Krasnoselkii, see [16]). Let Ω be a closed convex and nonempty subset of a Banachspace X. Let A,B be two operators such that

(i) Ax + By ∈ Ω whenever x, y ∈ Ω.

(ii) A is compact and continuous,

(iii) B is a contraction mapping.

Then there exists z ∈ Ω such that z = Az + Bz.

3. Main Results

We require the following assumptions.

(H1) The function f : [0, T] × Xα × Xα → X is continuous, and there exists a positivefunction μ(·) : [0, T] → R

+ such that

∥∥f(t, x, y

)∥∥ ≤ μ(t), the function s −→μ(s)

(t − s)αbelongs to Lp([0, t],R+),

γ(t) :=

(∫ t

0

(μ(s)

(t − s)α)p

ds

)1/p

≤MT <∞, for t ∈ [0, T],

(3.1)

where p > 1/q > 1.

(H2) The function g : Cα → Xα is continuous and there exists b > 0 such that

∥∥g(x) − g(y)∥∥α ≤ b

∥∥x − y∥∥∞, (3.2)

for any x, y ∈ Cα.

Theorem 3.1. Let −A be the infinitesimal generator of an analytic compact semigroup {S(t)}t≥0 with‖S(t)‖ ≤M, t ≥ 0, and 0 ∈ ρ(A). If the maps f and g satisfy (H1), (H2), respectively, andMb < 1,then (1.1) has a mild solution for every x0 ∈ Xα.

Proof. Set λ = supx∈Cα‖g(x)‖α and choose r such that

r ≥M(‖x0‖α + λ) +MαMT

Γ(q) Mp,q · T (q−1)/p, (3.3)

where Mp,q := ((p − 1)/(pq − 1))(p−1)/p.


Let Br = {x ∈ C([0, T], Xα) | ‖x‖∞ ≤ r}.Define

(Ax)(t) :=1

Γ(q)∫ t

0(t − s)q−1S(t − s)f(s, x(s), Gx(s))ds,

(Bx)(t) := S(t)(x0 − g(x)

).

(3.4)

Let x, y ∈ Br , then for t ∈ [0, T] we have the estimates

∥∥(Ax)(t) + (By)(t)

∥∥α

≤ ‖S(t)‖(‖x0‖α + λ) +1

Γ(q)∫ t

0(t − s)q−1∥∥AαS(t − s)f(s, x(s), Gx(s))

∥∥ds

≤M(‖x0‖α + λ) +Mα

Γ(q)∫ t

0(t − s)q−1 μ(s)

(t − s)αds

≤M(‖x0‖α + λ) +Mα

Γ(q)

(∫ t

0(t − s)(q−1)p/(p−1)ds

)(p−1)/p

·(∫ t

0

(μ(s)

(t − s)α)p

ds

)1/p

≤M(‖x0‖α + λ) +MαMT

Γ(q) Mp,q · Tq−1/p

≤ r.(3.5)

Hence we obtain Ax + By ∈ Br .Now we show that A is continuous. Let {xn} be a sequence of Br such that xn → x in

Br . Then

f(s, xn(s), Gxn(s)) −→ f(s, x(s), Gx(s)), n −→ ∞, (3.6)

since the function f is continuous on I ×Xα ×Xα. For t ∈ [0, T], using (2.1), we have

‖(Axn)(t) − (Ax)(t)‖α

=1

Γ(q)

∥∥∥∥∥

∫ t

0(t − s)q−1S(t − s)

[f(s, xn(s), Gxn(s)) − f(s, x(s), Gx(s))

]ds

∥∥∥∥∥α

≤ 1Γ(q)∫ t

0(t − s)q−1∥∥AαS(t − s)

[f(s, xn(s), Gxn(s)) − f(s, x(s), Gx(s))

]∥∥ds

≤ Mα

Γ(q)∫ t

0(t − s)q−1∥∥f(s, xn(s), Gxn(s)) − f(s, x(s), Gx(s))

∥∥(t − s)−αds.

(3.7)


In view of the fact that

∥∥f(s, xn(s), Gxn(s)) − f(s, x(s), Gx(s))

∥∥ ≤ 2μ(s), s ∈ [0, T], (3.8)

and the function s → 2μ(s)(t − s)−α is integrable on [0, t], then the Lebesgue DominatedConvergence Theorem ensures that

∫ t

0(t − s)q−1∥∥f(s, xn(s), Gxn(s)) − f(s, x(s), Gx(s))

∥∥(t − s)−αds −→ 0 as n −→ ∞. (3.9)

Therefore, we can see that

limn→∞

‖(Axn)(t) − (Ax)(t)‖∞ = 0, (3.10)

which means that A is continuous.Noting that

‖(Ax)(t)‖α =1

Γ(q)

∥∥∥∥∥

∫ t

0(t − s)q−1S(t − s)f(s, x(s), Gx(s))ds

∥∥∥∥∥α

≤ 1Γ(q)∫ t

0(t − s)q−1∥∥AαS(t − s)f(s, x(s), Gx(s))

∥∥ds

≤ Mα

Γ(q)∫ t

0(t − s)q−1 μ(s)

(t − s)αds

≤ Mα MT

Γ(q) Mp,q · Tq−1/p,

(3.11)

we can see that A is uniformly bounded on Br .Next, we prove that (Ax)(t) is equicontinuous. Let 0 < t2 < t1 < T , and let ε > 0 be

small enough, then we have

‖(Ax)(t1) − (Ax)(t2)‖α ≤1

Γ(q)

∥∥∥∥∥

∫ t2

0

[(t1 − s)q−1 − (t2 − s)q−1

]S(t2 − s)f(s, x(s), Gx(s))ds

∥∥∥∥∥α

+1

Γ(q)

∥∥∥∥∥

∫ t1

t2

(t1 − s)q−1S(t1 − s)f(s, x(s), Gx(s))ds∥∥∥∥∥α

+1

Γ(q)

∥∥∥∥∥

∫ t2

0(t1 − s)q−1[S(t1 − s) − S(t2 − s)]f(s, x(s), Gx(s))ds

∥∥∥∥∥α

= I1 + I2 + I3.

(3.12)


Using (2.1) and (H1), we have

I1 =1

Γ(q)

∥∥∥∥∥

∫ t2

0

[(t1 − s)q−1 − (t2 − s)q−1

]S(t2 − s)f(s, x(s), Gx(s))ds

∥∥∥∥∥α

≤ 1Γ(q)∫ t2

0

∣∣∣(t1 − s)q−1 − (t2 − s)q−1

∣∣∣∥∥AαS(t2 − s)f(s, x(s), Gx(s))

∥∥ds

≤ Mα

Γ(q)∫ t2

0

∣∣∣(t1 − s)q−1 − (t2 − s)q−1

∣∣∣

μ(s)(t2 − s)α

ds

≤ Mα

Γ(q)∫ t2−ε

0

[(t2 − s)q−1 − (t1 − s)q−1

] μ(s)(t2 − s)α

ds

+Mα

Γ(q)∫ t2

t2−ε(t2 − s)q−1 μ(s)

(t2 − s)αds

= I ′1 + I′′1 .

(3.13)

It follows from the assumption of μ(s) that I ′1 tends to 0 as t2 → t1. For I ′′1 , using the Holderinequality, we can see that I ′′1 tends to 0 as t2 → t1 and ε → 0.

For I2, using (2.1), (H1), and the Holder inequality, we have

I2 =1

Γ(q)

∥∥∥∥∥

∫ t1

t2

(t1 − s)q−1S(t1 − s)f(s, x(s), Gx(s))ds∥∥∥∥∥α

≤ 1Γ(q)∫ t1

t2

(t1 − s)q−1∥∥AαS(t1 − s)f(s, x(s), Gx(s))∥∥ds

≤ Mα

Γ(q)∫ t1

t2

(t1 − s)q−1 μ(s)(t1 − s)α

ds −→ 0 as t2 −→ t1.

(3.14)

Moreover,

I3 ≤1

Γ(q)

∥∥∥∥∥

∫ t2−ε

0(t1 − s)q−1[S(t1 − s) − S(t2 − s)]f(s, x(s), Gx(s))ds

∥∥∥∥∥α

+1

Γ(q)

∥∥∥∥∥

∫ t2

t2−ε(t1 − s)q−1[S(t1 − s) − S(t2 − s)]f(s, x(s), Gx(s))ds

∥∥∥∥∥α


≤ 1Γ(q)∫ t2−ε

0(t1 − s)q−1

∥∥∥∥S

(t1 − t2

2+t1 − s

2

)− S

(t2 − s

2

)∥∥∥∥

·∥∥∥∥A

αS

(t2 − s

2

)f(s, x(s), Gx(s))

∥∥∥∥ds

+Mα

Γ(q)∫ t2

t2−ε(t1 − s)q−1

[μ(s)

(t1 − s)α+

μ(s)(t2 − s)α

]ds

≤ 2αMα

Γ(q)∫ t2−ε

0(t1 − s)q−1

∥∥∥∥S

(t1 − t2

2+t1 − s

2

)− S

(t2 − s

2

)∥∥∥∥ ·

μ(s)(t2 − s)α

ds

+Mα

Γ(q)∫ t2

t2−ε(t1 − s)q−1

[μ(s)

(t1 − s)α+

μ(s)(t2 − s)α

]ds

= I ′3 + I′′3 . (3.15)

Using the compactness of S(t) in X implies the continuity of t → ‖S(t)‖ for t ∈ [0, T];integrating with s → μ(s)/(t2 − s)α ∈ L1

loc([0, t2],R+), we see that I ′3 tends to 0, as t2 → t1. For

I ′′3 , from the assumption of μ(s) and the Holder inequality, it is easy to see that I ′′3 tends to 0as t2 → t1 and ε → 0.

Thus, ‖(Ax)(t1) − (Ax)(t2)‖α → 0, as t2 → t1, which does not depend on x.So, A(Br) is relatively compact. By the Arzela-Ascoli Theorem, A is compact.Now, let us prove that B is a contraction mapping. For x, y ∈ C ([0, T], Xα) and t ∈

[0, T], we have

∥∥(Bx)(t) − (By)(t)∥∥α ≤ ‖S(t)‖

∥∥g(x) − g(y)∥∥α ≤Mb

∥∥x − y∥∥∞ <

∥∥x − y∥∥∞. (3.16)

So, we obtain

∥∥(Bx)(t) − (By)(t)∥∥∞ <

∥∥x − y∥∥∞. (3.17)

We now conclude the result of the theorem by Krasnoselkii’s theorem.

Now we assume the following.

(H3) There exists a positive function μ1(·) : [0, T] → R+ such that

∥∥f(t, x(t), Gx(t)) − f(t, y(t), Gy(t)

)∥∥ ≤ μ1(t)(∥∥x − y

∥∥α +

∥∥Gx −Gy∥∥α

), (3.18)

the function s → μ1(s)/(t − s)α belongs to L1([0, t],R+) and

γ ′(t) :=

(∫ t

0

(μ1(s)(t − s)α

)p

ds

)1/p

≤M′T <∞, for t ∈ [0, T]. (3.19)


(H4) The function Lα,q : I → R+, 0 < α, q < 1 satisfies

Lα,q(t) =Mb +MαM

′T

Γ(q) Mp,q · tq−1/p(1 +G∗) ≤ ω < 1, t ∈ [0, T]. (3.20)

Theorem 3.2. Let −A be the infinitesimal generator of an analytic semigroup {S(t)}t≥0 with ‖S(t)‖ ≤M, t ≥ 0 and 0 ∈ ρ(A). If x0 ∈ Xα and (H2)–(H4) hold, then (1.1) has a unique mild solution x ∈ Cα.

Proof. Define the mapping F : C([0, T], Xα) → C([0, T], Xα) by

(Fx)(t) = S(t)(x0 − g(x)

)+

1Γ(q)∫ t

0(t − s)q−1S(t − s)f(s, x(s), Gx(s))ds. (3.21)

Obviously, F is well defined on C([0, T], Xα).Now take x, y ∈ C([0, T], Xα), then we have

∥∥(Fx)(t) − (Fy)(t)∥∥α

≤∥∥S(t)(g(x) − g(y))

∥∥α

+1

Γ(q)∫ t

0(t − s)q−1∥∥S(t − s)[f(s, x(s), Gx(s)) − f(s, y(s), Gy(s))]

∥∥αds

≤M∥∥g(x) − g(y)

∥∥α

+1

Γ(q)∫ t

0(t − s)q−1∥∥AαS(t − s)

[f(s, x(s), Gx(s)) − f

(s, y(s), Gy(s)

)]∥∥ds

≤Mb∥∥x − y

∥∥∞ +

Mα

Γ(q)∫ t

0(t − s)q−1 μ1(s)

(t − s)α(∥∥x − y

∥∥α +

∥∥Gx −Gy∥∥α

)ds

≤Mb∥∥x − y

∥∥∞ +

MαM′T

Γ(q) Mp,q · tq−1/p(1 +G∗)

∥∥x − y∥∥α

≤ Lα,q(t)∥∥x − y

∥∥∞.

(3.22)

Therefore, we obtain

∥∥(Fx)(t) − (Fy)(t)∥∥∞ ≤ ω

∥∥x − y∥∥∞ <

∥∥x − y∥∥∞, (3.23)

and the result follows from the contraction mapping principle.

Acknowledgment

This work is supported by the NSF of Yunnan Province (2009ZC054M).


References

[1] R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations andinclusions involving Riemann-Liouville fractional derivative,” Advances in Difference Equations, vol.2009, Article ID 981728, 47 pages, 2009.

[2] M. M. El-Borai and D. Amar, “On some fractional integro-differential equations with analyticsemigroups,” International Journal of Contemporary Mathematical Sciences, vol. 4, no. 25–28, pp. 1361–1371, 2009.

[3] V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis:Theory, Methods & Applications, vol. 69, no. 10, pp. 3337–3343, 2008.

[4] V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” NonlinearAnalysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008.

[5] Z. W. Lv, J. Liang, and T. J. Xiao, “Solutions to fractional differential equations with nonlocal initialcondition in Banach spaces,” reprint, 2009.

[6] H. Liu and J.-C. Chang, “Existence for a class of partial differential equations with nonlocalconditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3076–3083, 2009.

[7] G. M. N’Guerekata, “A Cauchy problem for some fractional abstract differential equation with nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 5, pp. 1873–1876, 2009.

[8] X.-X. Zhu, “A Cauchy problem for abstract fractional differential equations with infinite delay,”Communications in Mathematical Analysis, vol. 6, no. 1, pp. 94–100, 2009.

[9] J. Liang, J. van Casteren, and T.-J. Xiao, “Nonlocal Cauchy problems for semilinear evolutionequations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 50, no. 2, pp. 173–189, 2002.

[10] J. Liang, J. Liu, and T.-J. Xiao, “Nonlocal Cauchy problems governed by compact operator families,”Nonlinear Analysis: Theory, Methods & Applications, vol. 57, no. 2, pp. 183–189, 2004.

[11] J. Liang, J. H. Liu, and T.-J. Xiao, “Nonlocal Cauchy problems for nonautonomous evolutionequations,” Communications on Pure and Applied Analysis, vol. 5, no. 3, pp. 529–535, 2006.

[12] J. Liang, J. H. Liu, and T.-J. Xiao, “Nonlocal impulsive problems for nonlinear differential equationsin Banach spaces,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 798–804, 2009.

[13] J. Liang and T.-J. Xiao, “Semilinear integrodifferential equations with nonlocal initial conditions,”Computers & Mathematics with Applications, vol. 47, no. 6-7, pp. 863–875, 2004.

[14] T.-J. Xiao and J. Liang, “Existence of classical solutions to nonautonomous nonlocal parabolicproblems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5–7, pp. e225–e232, 2005.

[15] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of AppliedMathematical Sciences, Springer, New York, NY, USA, 1983.

[16] D. R. Smart, Fixed Point Theorems, Cambridge University Press, 1980.


Research ArticleUniqueness of Periodic Solution for a Class ofLienard p-Laplacian Equations

Fengjuan Cao,1 Zhenlai Han,1, 2 Ping Zhao,2 and Shurong Sun1, 3

1 School of Science, University of Jinan, Jinan, Shandong 250022, China2 School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China3 Department of Mathematics and Statistics, Missouri University of Science and Technology,Rolla, MO 65409-0020, USA

Correspondence should be addressed to Zhenlai Han, [email protected]

Received 31 December 2009; Accepted 23 February 2010


Copyright q 2010 Fengjuan Cao et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

By topological degree theory and some analysis skills, we consider a class of generalized Lienardtype p-Laplacian equations. Upon some suitable assumptions, the existence and uniqueness ofperiodic solutions for the generalized Lienard type p-Laplacian differential equations are obtained.It is significant that the nonlinear term contains two variables.

1. Introduction

As it is well known, the existence of periodic and almost periodic solutions is themost attracting topics in the qualitative theory of differential equations due to their vastapplications in physics, mathematical biology, control theory, and others. More generalequations and systems involving periodic boundary conditions have also been considered.Especially, the existence of periodic solutions for the Duffing equation, Rayleigh equation,and Lienard type equation, which are derived from many fields, such as fluid mechanics andnonlinear elastic mechanics, has received a lot of attention.

Many experts and scholars, such as Manasevich, Mawhin, Gaines, Cheung, Ren,Ge, Lu, and Yu, have contributed a series of existence results to the periodicity theoryof differential equations. Fixed point theory, Mawhin’s continuation theorem, upper andlower solutions method, and coincidence degree theory are the common tools to studythe periodicity theory of differential equations. Among these approaches, the Mawhin’scontinuation theorem seems to be a very powerful tool to deal with these problems.

Some contributions on periodic solutions to differential equations have been madein [1–13]. Recently, periodic problems involving the scalar p-Laplacian were studied by


many authors. We mention the works by Manasevich and Mawhin [3] and Cheung and Ren[4, 8, 10].

In [3], Manasevich and Mawhin investigated the existence of periodic solutions to theboundary value problem

(φ(u′))′ = f

(t, u, u′

), u(0) = u(T), u′(0) = u′(T), (1.1)

where the function φ : RN → RN is quite general and satisfies some monotonicity conditionswhich ensure that φ is homeomorphism onto RN. Applying Leray-Schauder degree theory,the authors brought us the widely used Manasevich-Mawhin continuation theorem. Whenφ = φp : R → R is the so-called one-dimensional p-Laplacian operator given by φp(s) =|s|p−2s.

Recently, by Mawhin’s continuation theorem, Cheung and Ren studied the existenceof T -periodic solutions for a p-Laplacian Lienard equation with a deviating argument in [4]as follows:

(ϕp(x′(t)

))′ + f(x(t))x′(t) + g(x(t − τ(t))) = e(t), (1.2)

and two results (Theorems 3.1 and 3.2 ) on the existence of periodic solutions were obtained.Ge and Ren [5] promoted Mawhin’s continuation theorem to the case which involved

the quasilinear operator successfully; this also prepared conditions for using Mawhin’scontinuation theorem to solve nonlinear boundary value problem.

Liu [7] has dealt with the existence and uniqueness of T -periodic solutions of theLienard type p-Laplacian differential equation of the form

(ϕp(x′(t)

))′ + f(x(t))x′(t) + g(x(t)) = e(t) (1.3)

by using topological degree theory, and one sufficient condition for the existence anduniqueness of T -periodic solutions of this equation was established.

The aim of this paper is to study the existence of periodic solutions to a class of p-Laplacian Lienard equations as follows:

(ϕp(x′(t)

))′ + f(t, x(t))x′(t) + g(t, x(t)) = e(t), (1.4)

where p > 2, ϕp : R → R is given by ϕp(s) = |s|p−2s for s /= 0, ϕp(0) = 0, f ∈ C2(R2, R),g ∈ C1(R2, R) and T -periodic in the first variable, where T > 0 is a given constant, e ∈ C(R,R),and e(t + T) = e(t).

The paper is organized as follows. In Section 2, we give the definition of norm inBanach space and the main lemma. In Section 3, combining Lemma 2.1 with some analysisskills, two sufficient conditions about the existence of solutions for (1.4) are obtained. Thenonlinear terms f and g contain two variables in this paper, which is seldom considered inthe other papers, and the results are new.


2. Preliminary Results

For convenience, we define

C1T :=

{x ∈ C1(R,R); x is T -periodic

}, (2.1)

and the norm ‖ · ‖ is defined by ‖x‖ = max{|x|∞, |x′|∞}, for all x,

|x|∞ = maxt∈[0,T]

|x(t)|,∣∣x′∣∣∞ = max

t∈[0,T]

∣∣x′(t)

∣∣, |x|k =

(∫T

0|x(t)|kdt

)1/k

. (2.2)

Clearly, C1T is a Banach space endowed with such norm.

For the periodic boundary value problem

(ϕp(x′(t)

))′ = f(t, x, x′

), x(0) = x(T), x′(0) = x′(T), (2.3)

where f is a continuous function and T -periodic in the first variable, we have the followingresult.

Lemma 2.1 (see [3]). Let Ω be an open bounded set in C1T . If the following conditions hold:

(i) for each λ ∈ (0, 1), the problem

(ϕp(x′(t)

))′ = λf(t, x, x′

), x(0) = x(T), x′(0) = x′(T) (2.4)

has no solution on ∂Ω,

(ii) the equation

F(a) :=1T

∫T

0f(t, a, 0)dt = 0 (2.5)

has no solution on ∂Ω ∩ R,

(iii) the Brouwer degree of F is deg{F,Ω ∩ R, 0}/= 0,

then the periodic boundary value problem (2.3) has at least one T -periodic solution on Ω.

Set

Ψ(t, x(t)) =∫x(t)

0f(t, s)ds, y(t) = ϕ

(x′(t)

)+ Ψ(t, x(t)). (2.6)


We can rewrite (1.4) in the following form:

x′(t) =∣∣y(t) −Ψ(t, x(t))

∣∣q−1 sgn

(y(t) −Ψ(t, x(t))

),

y′(t) =∫x(t)

0

∂

∂tf(t, s)ds − g(t, x(t)) + e(t),

(2.7)

where 1 < q < 2 and 1/p + 1/q = 1.

Lemma 2.2. Suppose the following condition holds:

(A1) ∂f/∂t − ∂g/∂s|(t,s) > 0, for all t ∈ R.

Then (1.4) has at most one T -periodic solution.

Proof. Let x1(t) and x2(t) be two T -periodic solutions of (1.4). Then, from (2.7), we obtain

x′i(t) =∣∣yi(t) −Ψ(t, xi(t))

∣∣q−1 sgn(yi(t) −Ψ(t, xi(t))

),

y′i(t) =∫xi(t)

0

∂

∂tf(t, s)ds − g(t, xi(t)) + e(t),

i = 1, 2. (2.8)

Set

v(t) = x1(t) − x2(t), u(t) = y1(t) − y2(t). (2.9)

Then it follows from (2.8) that

v′(t) =∣∣y1(t) −Ψ(t, x1(t))

∣∣q−1 sgn(y1(t) −Ψ(t, x1(t))

)

−∣∣y2(t) −Ψ(t, x2(t))

∣∣q−1 sgn(y2(t) −Ψ(t, x2(t))

),

u′(t) =∫x1(t)

x2(t)

(∂f

∂t−∂g

∂s

)(t, s)ds.

(2.10)

We claim that u(t) ≤ 0 for all t ∈ R. By way of contradiction, in view of u ∈ C2[0, T]and u(t + T) = u(t) for all t ∈ R, we obtain maxt∈Ru(t) > 0. Then there must exist t∗ ∈ R; forconvenience, we can choose t∗ ∈ [0, T] such that

u(t∗) = maxt∈[0,T]

u(t) = maxt∈R

u(t) > 0, (2.11)


which implies that

u′(t∗) =∫x2(t)

x1(t)

(∂f

∂t−∂g

∂s

)(t, s)ds

∣∣∣∣∣t=t∗

= 0,

u′′(t∗) =

(∫x1(t)

x2(t)

∂

∂tf(t, s)ds

)′∣∣∣∣∣t=t∗

−[∂

∂tg(t, x1(t)) +

∂

∂x1g(t, x1(t))x′1(t) −

∂

∂tg(t, x2(t)) −

∂

∂x2g(t, x2(t))x′2(t)

]∣∣∣∣t=t∗≤ 0.

(2.12)

Set

Δ1 = {t : t ∈ [0, T], x1(t)/=x2(t)}, Δ2 = {t : t ∈ [0, T], x1(t) = x2(t)}. (2.13)

Then, [0, T] = Δ1 ∪Δ2. Since t∗ ∈ [0, T], if t∗ ∈ Δ1, from the first equation of (2.12), we have

(∫x2(t)

x1(t)

(∂f

∂t−∂g

∂s

)(t, s)ds

)∣∣∣∣∣t=t∗

= 0, (2.14)

which contradicts assumption (A1), so t∗∈Δ1; it implies that t∗ ∈ Δ2, that is, x1(t∗) = x2(t∗).Hence we have

∂

∂tf(t, x1(t))

∣∣∣∣t=t∗

=∂

∂tf(t, x2(t))

∣∣∣∣t=t∗,

∂

∂x1g(t, x1(t))

∣∣∣∣t=t∗

=∂

∂x2g(t, x2(t))

∣∣∣∣t=t∗,

(∫x2(t)

x1(t)

∂

∂tf(t, s)ds

)′∣∣∣∣∣t=t∗

=∂

∂tf(t, x1(t))

∣∣∣∣t=t∗

[x′1(t

∗) − x′2(t∗)].

(2.15)

Substituting (2.15) into the second equation of (2.12), we have

u′′(t∗) =∂

∂tf(t, x1(t))

∣∣∣∣t=t∗

[x′1(t

∗) − x′2(t∗)]− ∂

∂x1g(t, x1(t))

∣∣∣∣t=t∗

[x′1(t

∗) − x′2(t∗)]

=[∂

∂tf(t, x1(t)) −

∂

∂x1g(t, x1(t))

]∣∣∣∣t=t∗

[x′1(t

∗) − x′2(t∗)]

=[∂f

∂t−∂g

∂x1

]∣∣∣∣(t∗,x1(t∗))

×[∣∣y1(t∗) −Ψ(t∗, x1(t∗))

∣∣q−1 sgn(y1(t∗) −Ψ(t∗, x1(t∗))

)

−∣∣y2(t∗) −Ψ(t∗, x2(t∗))

∣∣q−1 sgn(y2(t∗) −Ψ(t∗, x2(t∗))

)].

(2.16)


Noticing (A1) and that [∂f/∂t − ∂g/∂x1]|(t∗,x1(t∗)) > 0 and u(t∗) = y1(t∗) − y2(t∗), from (2.16),we know that

u′′(t∗) > 0, (2.17)

this contradicts the second equation of (2.12). So we have u(t) = y1(t)−y2(t) ≤ 0, for all t ∈ R.By using a similar argument, we can also show that

y2(t) − y1(t) ≤ 0, ∀t ∈ R. (2.18)

Then, from (2.10) we obtain

∫x2(t)

x1(t)

(∂f

∂t−∂g

∂x

)(t, s)ds ≡ 0. (2.19)

For every t ∈ [0, T], if t ∈ Δ1, then it contradicts (2.19), so Δ1 = ∅; it implies that[0, T] = Δ2, then x1(t) ≡ x2(t), for all t ∈ [0, T].

Hence, (1.4) has at most one T -periodic solution. The proof of Lemma 2.2 is completednow.

3. Main Results

Theorem 3.1. Let (A1) hold. Suppose that there exists a positive constant d such that

(A2) x(g(t, x) − e(t)) < 0, for all |x| > d,

(A3) limu→+∞|f(t, u)/up−1| = r > 0.

Then (1.4) has one unique T -periodic solution if (1/2p−1)(r + ε)λTp−1 < 1.

Proof. Consider the homotopic equation of (1.4) as follows:

(ϕp(x′(t)

))′ + λf(t, x(t))x′(t) + λg(t, x(t)) = λe(t), λ ∈ (0, 1). (3.1)

By Lemma 2.2, combining (A1), it is easy to see that (1.4) has at least one T -periodicsolution. For the remainder, we will apply Lemma 2.1 to study (3.1). Firstly, we will verifythat all the possible T -periodic solutions of (3.1) are bounded.

Let x ∈ C1T be an arbitrary solution of (3.1) with period T . By integrating the two sides

of (3.1) from 0 to T and x′(0) = x′(T), we obtain

∫T

0f(t, x(t))x′(t)dt +

∫T

0

[g(t, x(t)) − e(t)

]dt = 0. (3.2)


Consider x(0) = x(T) and x ∈ C1T , there exists t0 ∈ [0, T] such that x′(t0) = 0, while for

ϕp(0) = 0 we see that

∣∣ϕp(x′(t)

)∣∣ =

∣∣∣∣∣

∫ t

t0

(ϕp(x′(t)

))′ds

∣∣∣∣∣≤ λ∫T

0

∣∣f(t, x(t))

∣∣∣∣x′(t)

∣∣dt + λ

∫T

0

∣∣g(t, x(t))e(t)

∣∣dt, (3.3)

where t ∈ [t0, t0 + T]. Let t be the global maximum point of x(t) on [0, T]. Then as x′(t) = 0,we claim that

(ϕp(x′(t)))′

=(∣∣∣x′(t)∣∣∣p−2x′(t))′≤ 0. (3.4)

Otherwise, we have (ϕp(x′(t)))′ > 0, there must exist a constant ε > 0 such that (ϕp(x′(t)))′ =(|x′(t)|p−2x′(t))′ > 0, for t ∈ (t−ε, t+ε); therefore, ϕp(x′(t)) is strictly increasing for t ∈ (t−ε, t+ε),which implies that x′(t) is strictly increasing for t ∈ (t − ε, t + ε). Thus, (3.4) is true. Then

g(t, x(t))− e(t)≥ 0. (3.5)

In view of (A2), (3.5) implies that x(t) ≤ d; similar to the global minimum point of x(t)on [0, T]. Since x(t) ∈ C1

T , it follows that there exists a constant ξ ∈ [0, T] such that |x(ξ)| ≤ d.Then we have

|x(t)| =∣∣∣∣∣x(ξ) +

∫ t

ξ

x′(s)ds

∣∣∣∣∣≤ d +

∫ t

ξ

∣∣x′(s)∣∣ds, t ∈ [ξ, ξ + T],

|x(t)| = |x(t − T)| =∣∣∣∣∣x(ξ) −

∫ ξ

t−Tx′(s)ds

∣∣∣∣∣≤ d +

∫ ξ

t−T

∣∣x′(s)∣∣ds, t ∈ [ξ, ξ + T].

(3.6)

Combining the above two inequalities, we obtain

|x|∞ = maxt∈[0,T]

|x(t)| = maxt∈[ξ,ξ+T]

|x(t)|

≤ maxt∈[ξ,ξ+T]

{

d +12

(∫ t

ξ

∣∣x′(s)∣∣ds +

∫ ξ

t−T

∣∣x′(s)∣∣ds

)}

≤ d +12

∫T

0

∣∣x′(s)∣∣ds.

(3.7)

Considering (A3), there exist constants d1 and the sufficiently small ε > 0 such that

∣∣f(t, x(t))∣∣ ≤ (r + ε)|x(t)|p−1, when |x(t)| > d1. (3.8)


Set

Δ1 = {t : t ∈ [0, T], |x(t)| > d1}, Δ2 = {t : t ∈ [0, T], |x(t)| ≤ d1},

E1 = {t : t ∈ [0, T], |x(t)| > d}, E2 = {t : t ∈ [0, T], |x(t)| ≤ d}.(3.9)

From (3.7), we have

∫T

0f(t, x(t))x′(t)x(t)dt =

∫

Δ1

f(t, x(t))x′(t)x(t)dt +∫

Δ2

f(t, x(t))x′(t)x(t)dt

≤ fA|x(t)|∞∫T

0

∣∣x′(t)

∣∣dt + (r + ε)|x(t)|p−1

∞

∫T

0

∣∣x′(t)

∣∣dt

≤ (r + ε)

(

d +12

∫T

0

∣∣x′(t)∣∣dt

)p−1 ∫T

0

∣∣x′(t)∣∣dt +

12fA

(∫T

0

∣∣x′(t)∣∣dt

)2

+ dfA

∫T

0

∣∣x′(t)∣∣dt,

(3.10)

where fA = max{|f(t, x(t))| : t ∈ Δ2}. Combining the classical inequality (1+x)p ≤ 1+(p+1)x,when x ∈ [0, h(p)), where h(p) is a constant, since

(

d +12

∫T

0

∣∣x′(t)∣∣dt

)p−1

=

(12

∫T

0

∣∣x′(t)∣∣dt

)p−1⎛

⎝1 +2d

∫T0 |x′(t)|dt

⎞

⎠

p−1

, (3.11)

then we consider the following two cases.

Case 1. If 2d/∫T

0 |x′(t)|dt ≥ h(p), then

∫T0 |x

′(t)|dt ≤ 2d/h(p). Combining (3.7), we know that

|x|∞ ≤ d +12

∫T

0

∣∣x′(s)∣∣ds ≤ d +

2dh(p) ,

(

d +12

∫T

0

∣∣x′(t)∣∣dt

)p

=

(12

∫T

0

∣∣x′(t)∣∣dt

)p⎛

⎝1 +2d

∫T0 |x′(t)|dt

⎞

⎠

p

,

(3.12)

when 2d/∫T

0 |x′(t)|dt ≥ h(p), then we have

∫T0 |x

′(t)|dt ≤ 2d/h(p).

Case 2. When 0 < 2d/∫T

0 |x′(t)|dt < h(p), then from the above classical inequality, we obtain

(

d +12

∫T

0

∣∣x′(t)∣∣dt

)p−1

=

(12

∫T

0

∣∣x′(t)∣∣dt

)p−1⎛

⎝1 +2dp

∫T0 |x′(t)|dt

⎞

⎠. (3.13)


Substituting the above inequality into (3.10), we get

∫T

0f(t, x(t))x′(t)x(t)dt ≤ 1

2p−1(r + ε)

(∫T

0

∣∣x′(t)

∣∣dt

)p

+1

2p−2dp(r + ε)

(∫T

0

∣∣x′(t)

∣∣dt

)p−1

+12fA

(∫T

0

∣∣x′(t)

∣∣dt

)2

+ dfA

∫T

0

∣∣x′(t)

∣∣dt.

(3.14)

Since x(t) is T -periodic, multiplying x(t) by (3.1) and then integrating from 0 to T , inview of (A2), we have

∫T

0

∣∣x′(t)

∣∣pdt = −

∫T

0

∣∣ϕp(x′(t)

)∣∣′x(t)dt

= λ∫T

0f(t, x(t))x′(t)x(t)dt + λ

∫T

0g(t, x(t))x(t)dt − λ

∫T

0e(t)x(t)dt

= λ∫T

0f(t, x(t))x′(t)x(t)dt + λ

∫T

0

[g(t, x(t)) − e(t)

]x(t)dt.

(3.15)

Substituting (3.14) into (3.15) and since

(∫T

0|x(t)|rdt

)1/r

≤ T (s−r)/s(∫T

0x(t)|sdt

)1/s

, 0 < r ≤ s, (3.16)

we obtain

∫T

0

∣∣x′(t)∣∣pdt = −

∫T

0

∣∣ϕp(x′(t)

)∣∣′x(t)dt

≤ r + ε2p−1

λTp−1∫T

0

∣∣x′(t)∣∣pdt +

12p−2

λT (p−1)2/pdp(r + ε)

(∫T

0

∣∣x′(t)∣∣pdt

)(p−1)/p

+12λT2(p−1)/pfA

(∫T

0


)2/p

+ λT (p−1)/pdfA

(∫T

0


)1/p

+ λ

(

d +12

∫T

0

∣∣x′(t)∣∣dt

)∫T

0max

{∣∣g(t, x(t)) − e(t)∣∣, t ∈ E2

}dt

≤ Q1

∫T

0

∣∣x′(t)∣∣pdt +Q2

(∫T

0


)(p−1)/p

+Q3

(∫T

0


)2/p

+Q4

(∫T

0


)1/p

+Q5,

(3.17)


where

Q1 =1

2p−1(r + ε)λTp−1, Q2 =

12p−2

λdp(r + ε)T (p−1)2/p, Q3 =12λfAT

2(p−1)/p,

Q4 = λdfAT (p−1)/p +12λT (p−1)/p

∫T

0max

{∣∣g(t, x(t)) − e(t)∣∣, t ∈ E2

}dt,

Q5 = λd∫T

0max

{∣∣g(t, x(t)) − e(t)∣∣, t ∈ E2

}dt.

(3.18)

Since p > 2 and (1/2p−1)(r + ε)λTp−1 < 1, from (3.17), we know that there exists aconstant M1 such that

∫T0 |x

′(t)|pdt ≤M1. Then,

(∫T

0

∣∣x′(t)∣∣dt

)p

≤ Tp−1∫T

0

∣∣x′(t)∣∣pdt ≤ Tp−1M1 :=N1. (3.19)

So, there exists a constant such that

|x(t)| ≤M2,∣∣x′(t)

∣∣ ≤M2. (3.20)

Set

Ω ={x ∈ C1

T : |x|∞ ≤M2 + 1,∣∣x′∣∣∞ ≤M2 + 1

}. (3.21)

Then (3.1) has no solution on ∂Ω as λ ∈ (0, 1), and when x(t) ∈ ∂Ω ∩ R, x(t) = M2 + 1 orx(t) = −M2 − 1; from (A2), we can see that

1T

∫T

0

{−g(t,M2 + 1) + e(t)

}dt = − 1

T

∫T

0

{g(t,M2 + 1) − e(t)

}dt > 0,

1T

∫T

0

{−g(t,−M2 − 1) + e(t)

}dt = − 1

T

∫T

0

{g(t,−M2 − 1) − e(t)

}dt < 0,

(3.22)

so condition (ii) holds.Set

H(x, μ)= μx +

(1 − μ

) 1T

∫T

0

{−g(t, x) + e(t)

}dt. (3.23)

Then, when x ∈ ∂Ω ∩ R, μ ∈ [0, 1], we have

xH(x, μ)= μx2 −

(1 − μ

)x

1T

∫T

0

{g(t, x) − e(t)

}dt > 0, (3.24)


thus H(x, μ) is a homotopic transformation and

deg{F,Ω ∩ R, 0} = deg

{

− 1T

∫T

0

{g(t, x) − e(t)

}dt,Ω ∩ R, 0

}

= deg{x,Ω ∩ R, 0}/= 0.

(3.25)

So condition (iii) holds. In view of Lemma 2.1, there exists at least one solution with periodT . This completes the proof.

Theorem 3.2. Let (A1) hold. Suppose that there exist positive constants d1 and d2 satisfying thefollowing conditions:

(A4) − sgn(x)g(t, x) > |e|∞, when |x| > d1;

(A5) f(t, x(t)) > 0, t ∈ R;

(A6) |g(t, x)/x| ≤ l, when |x| > d1.

Then for (1.4) there exists one unique T -periodic solution when σ > lT.

Proof. We can rewrite (3.1) in the following from:

x′1(t) = ϕq(x2(t)),

x′2(t) = −λf(t, x1(t))x′1(t) − λg(t, x1(t)) + λe(t).(3.26)

Let (x1(t), x2(t)) ∈ C1T be a T -periodic solution of (3.26), then x1(t) must be a T -periodic

solution of (3.1). First we claim that there is a constant ξ ∈ R such that

|x1(ξ)| ≤ d. (3.27)

Take t0, t1 as the global maximum point and global minimum point of x(t) on [0, T],respectively, then

x′1(t0) = x′1(t1) = 0, x1(t0) = max

t∈Rx1(t) = max

t∈[0,T]x1(t). (3.28)

From the first equation of (3.26) we have x2(t) = ϕp(x′1(t)), so x2(t0) = ϕp(x′1(t0)) = 0 = x2(t1).We claim that

x′2(t0) ≤ 0. (3.29)

By way of contradiction, (3.29) does not hold, then x′2(t0) > 0. So there exists ε > 0such that x′2(t) > 0, for t ∈ (t0 − ε, t0 + ε); therefore, x2(t) > x2(t0) = 0, for t ∈ [t0, t0 + ε), sox′1(t) = ϕq(x2(t)) > 0, t ∈ [t0, t0 + ε), that is, x1(t) > x1(t0), for t ∈ [t0, t0 + ε). This contradictsthe definition of t0, so we have x′2(t0) ≤ 0.


Substituting t0 into the second equation of (3.26), we obtain

g(t0, x1(t0)) − e(t0) ≥ 0. (3.30)

By condition (A4), we have x1(t0) ≤ d. Similarly, we get x1(t1) ≥ −d.

Case 1. If x1(t1) ≤ d, define ξ = t1. Obviously |x1(ξ)| ≤ d.

Case 2. If x1(t1) ≥ d, from the fact that x1 is a continuous function in R, there exists a constantξ between t0 and t1 such that |x1(ξ)| = d.

So we have that (3.27) holds. Next, in view of ξ ∈ R, there are integer k and constantt∗ ∈ [0, T] such that ξ = kT + t∗, hence |x1(ξ)| = |x1(t∗)| ≤ d.

So

|x1(t)| ≤ d +∫T

0+∣∣x′(s)

∣∣ds. (3.31)

We claim that all the periodic solutions x of (3.1) are bounded and |x(t)| ≤ d +∫T

0 |x′(s)|ds.

Let

E1 = {t : t ∈ [0, T], |x(t)| > d1}, E2 = {t : t ∈ [0, T], |x(t)| ≤ d1}. (3.32)

Multiplying both sides of (3.1) by x′(t) and integrating from 0 to T, together with (A5) and(A6), we have

σ

∫T

0

∣∣x′(t)∣∣2dt ≤

∫T

0f(t, x(t))

∣∣x′(t)∣∣2dt

≤∫T

0

∣∣g(t, x(t))∣∣∣∣x′(t)

∣∣dt +∫T

0|e(t)|

∣∣x′(t)∣∣dt

≤∫

E1

∣∣g(t, x(t))∣∣∣∣x′(t)

∣∣dt +∫

E2

∣∣g(t, x(t))∣∣∣∣x′(t)

∣∣dt +∫T

0|e(t)|

∣∣x′(t)∣∣dt

≤ l∫T

0|x(t)|

∣∣x′(t)∣∣dt + |e|∞

∫T

0

∣∣x′(t)∣∣dt + gA

∫T

0

∣∣x′(t)∣∣dt

≤ l(

d +∫T

0

∣∣x′(t)∣∣dt

)∫T

0

∣∣x′(t)∣∣dt +

(gA + |e|∞

)∫T

0

∣∣x′(t)∣∣dt

=(ld + gA + |e|∞

)∫T

0

∣∣x′(t)∣∣dt + lT

∫T

0

∣∣x′(t)∣∣2dt,

(3.33)


where σ = min{f(t, x(t))}, and gA = max{|g(t, x(t))| : |x(t)| ≤ d1}. That is,

(σ − lT)∫T

0

∣∣x′(t)

∣∣2dt ≤

(ld + gA + |e|∞

)∫T

0

∣∣x′(t)

∣∣dt. (3.34)

Using Holder’s inequality and σ > lT, we have

1T

(∫T

0

∣∣x′(t)

∣∣dt

)2

≤∫T

0

∣∣x′(t)

∣∣2dt ≤

ld + |e|∞ + gAσ − lT

∫T

0

∣∣x′(t)

∣∣dt, (3.35)

so

∫T

0

∣∣x′(t)

∣∣dt ≤ T

ld + |e|0 + gAσ − lT :=M; (3.36)

there must be a positive constant M1 > M such that

∫T

0

∣∣x′(t)∣∣dt ≤M1,

∫T

0|x(t)|dt ≤M1; (3.37)

hence together with (3.31), we have |x1|∞ ≤ d +M1 :=M2.This proves the claim and that the rest of the proof of the theorem is identical to that

of Theorem 3.1.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and usefulcomments that have led to the present improved version of the original manuscript. Thisresearch is supported by the Natural Science Foundation of China (60774004, 60904024),China Postdoctoral Science Foundation Funded Project (20080441126, 200902564), ShandongPostdoctoral Funded Project (200802018), and the Natural Scientific Foundation of ShandongProvince (Y2008A28, ZR2009AL003) and is also supported by University of Jinan ResearchFunds for Doctors (XBS0843).

References

[1] S. Lu, “Existence of periodic solutions to a p-Laplacian Lienard differential equation with a deviatingargument,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 6, pp. 1453–1461, 2008.

[2] B. Xiao and B. Liu, “Periodic solutions for Rayleigh type p-Laplacian equation with a deviatingargument,” Nonlinear Analysis: Real World Applications, vol. 10, no. 1, pp. 16–22, 2009.

[3] R. Manasevich and J. Mawhin, “Periodic solutions for nonlinear systems with p-Laplacian-likeoperators,” Journal of Differential Equations, vol. 145, no. 2, pp. 367–393, 1998.

[4] W.-S. Cheung and J. Ren, “On the existence of periodic solutions for p-Laplacian generalized Lienardequation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 1, pp. 65–75, 2005.

[5] W. Ge and J. Ren, “An extension of Mawhin’s continuation theorem and its application to boundaryvalue problems with a p-Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol. 58, no.3-4, pp. 477–488, 2004.


[6] F. Zhang and Y. Li, “Existence and uniqueness of periodic solutions for a kind of Duffing type p-Laplacian equation,” Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 985–989, 2008.

[7] B. Liu, “Existence and uniqueness of periodic solutions for a kind of Lienard type p-Laplacianequation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 2, pp. 724–729, 2008.

[8] W.-S. Cheung and J. Ren, “Periodic solutions for p-Laplacian Lienard equation with a deviatingargument,” Nonlinear Analysis: Theory, Methods & Applications, vol. 59, no. 1-2, pp. 107–120, 2004.

[9] R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, vol. 568 ofLecture Notes in Mathematics, Springer, Berlin, Germany, 1977.

[10] W.-S. Cheung and J. Ren, “Periodic solutions for p-Laplacian Rayleigh equations,” Nonlinear Analysis:Theory, Methods & Applications, vol. 65, no. 10, pp. 2003–2012, 2006.

[11] S. Peng and S. Zhu, “Periodic solutions for p-Laplacian Rayleigh equations with a deviatingargument,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 1, pp. 138–146, 2007.

[12] F. Cao, Z. Han, and S. Sun, “Existence of periodic solutions for p-Laplacian equations on time scales,”Advances in Difference Equations, vol. 2010, Article ID 584375, 13 pages, 2010.

[13] F. Cao and Z. Han, “Existence of periodic solutions for p-Laplacian differential equation with adeviating arguments,” Journal of University of Jinan (Sci. Tech.), vol. 24, no. 1, pp. 95–98, 2010.


Research ArticleExistence of Periodic Solutions forp-Laplacian Equations on Time Scales

Fengjuan Cao,1 Zhenlai Han,1, 2 and Shurong Sun1, 3

1 School of Science, University of Jinan, Jinan, Shandong 250022, China2 School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China3 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla,MO 65409-0020, USA


Received 30 July 2009; Revised 15 October 2009; Accepted 18 November 2009


Copyright q 2010 Fengjuan Cao et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We systematically explore the periodicity of Lienard type p-Laplacian equations on time scales.Sufficient criteria are established for the existence of periodic solutions for such equations, whichgeneralize many known results for differential equations when the time scale is chosen as the setof the real numbers. The main method is based on the Mawhin’s continuation theorem.

1. Introduction

In the past decades, periodic problems involving the scalar p-Laplacian were studied by manyauthors, especially for the second-order and three-order p-Laplacian differential equation,see [1–8] and the references therein. Of the aforementioned works, Lu in [1] investigated theexistence of periodic solutions for a p-Laplacian Lienard differential equation with a deviatingargument

(ϕp(y′(t)

))′ + f(y(t))y′(t) + h

(y(t))+ g(y(t − τ(t))

)= e(t), (1.1)

by Mawhin’s continuation theorem of coincidence degree theory [3]. The author obtained anew result for the existence of periodic solutions and investigated the relation between theexistence of periodic solutions and the deviating argument τ(t). Cheung and Ren [4] studied


the existence of T -periodic solutions for a p-Laplacian Lienard equation with a deviatingargument

(ϕp(x′(t)

))′ + f(x(t))x′(t) + g(x(t − τ(t))) = e(t), (1.2)

by Mawhin’s continuation theorem. Two results for the existence of periodic solutions wereobtained. Such equations are derived from many fields, such as fluid mechanics and elasticmechanics.

The theory of time scales has recently received a lot of attention since it has atremendous potential for applications. For example, it can be used to describe the behaviorof populations with hibernation periods. The theory of time scales was initiated by Hilger[9] in his Ph.D. thesis in 1990 in order to unify continuous and discrete analysis. By choosingthe time scale to be the set of real numbers, the result on dynamic equations yields a resultconcerning a corresponding ordinary differential equation, while choosing the time scale asthe set of integers, the same result leads to a result for a corresponding difference equation.Later, Bohner and Peterson systematically explore the theory of time scales and obtain manyperfect results in [10] and [11]. Many examples are considered by the authors in these books.

But the research of periodic solutions on time scales has not got much attention, see[12–16]. The methods usually used to explore the existence of periodic solutions on timescales are many fixed point theory, upper and lower solutions, Masseras theorem, and so on.For example, Kaufmann and Raffoul in [12] use a fixed point theorem due to Krasnosel’skito show that the nonlinear neutral dynamic system with delay

xΔ(t) = −a(t)xσ(t) + c(t)xΔ(t − k) + q(t, x(t), x(t − k)), t ∈ T, (1.3)

has a periodic solution. Using the contraction mapping principle the authors show that theperiodic solution is unique under a slightly more stringent inequality.

The Mawhin’s continuation theorem has been extensively applied to explore theexistence problem in ordinary differential (difference) equations but rarely applied todynamic equations on general time scales. In [13], Bohner et al. introduce the Mawhin’scontinuation theorem to explore the existence of periodic solutions in predator-prey andcompetition dynamic systems, where the authors established some suitable sufficient criteriaby defining some operators on time scales.

In [14], Li and Zhang have studied the periodic solutions for a periodic mutualismmodel

xΔ(t) = r1(t)

[k1(t) + α1(t) exp

{y(t − τ2

(t, y(t)

))}

1 + exp{y(t − τ2

(t, y(t)

))} − exp{x(t − σ1(t, x(t)))}]

,

yΔ(t) = r2(t)

[k2(t) + α2(t) exp

{x(t − τ1

(t, y(t)

))}

1 + exp{x(t − τ1(t, x(t)))}− exp

{y(t − σ2

(t, y(t)

))}] (1.4)

on a time scale T by employing Mawhin’s continuation theorem, and have obtained threesufficient criteria.


Combining Brouwer’s fixed point theorem with Horn’s fixed point theorem, twoclasses of one-order linear dynamic equations on time scales

xΔ(t) = a(t)x(t) + h(t),

xΔ(t) = f(t, x), with the initial condition x(t0) = x0,(1.5)

are considered in [15] by Liu and Li. The authors presented some interesting properties ofthe exponential function on time scales and obtain a sufficient and necessary condition thatguarantees the existence of the periodic solutions of the equation xΔ(t) = a(t)x(t) + h(t).

In [16], Bohner et al. consider the system

xΔ(t) = G

(

t, exp{x(g1(t)

)}, exp

{x(g2(t)

)}, . . . , exp

{x(gn(t)

)},

∫ t

−∞c(t, s) exp{x(s)}Δs

)

,

(1.6)

easily verifiable sufficient criteria are established for the existence of periodic solutions of thisclass of nonautonomous scalar dynamic equations on time scales, the approach that authorsused in this paper is based on Mawhin’s continuation theorem.

In this paper, we consider the existence of periodic solutions for p-Laplacian equationson a time scales T

(ϕp(xΔ(t)

))Δ+ f(x(t))xΔ(t) + g(x(t)) = e(t), t ∈ T, (1.7)

where p > 2 is a constant, ϕp(s) = |s|p−2s, f, g ∈ C(R,R), e ∈ C(T, R), and e is a function withperiodic ω > 0. T is a periodic time scale which has the subspace topology inherited fromthe standard topology on R. Sufficient criteria are established for the existence of periodicsolutions for such equations, which generalize many known results for differential equationswhen the time scales are chosen as the set of the real numbers. The main method is based onthe Mawhin’s continuation theorem.

If T = R, (1.7) reduces to the differential equation

(ϕp(x′(t)

))′ + f(x(t))x′(t) + g(x(t)) = e(t). (1.8)

We will use Mawhin’s continuation theorem to study (1.7).

2. Preliminaries

In this section, we briefly give some basic definitions and lemmas on time scales which areused in what follows. Let T be a time scale (a nonempty closed subset of R). The forwardand backward jump operators σ, ρ : T → T and the graininess μ : T → R+ are defined,respectively, by

σ(t) = inf{s ∈ T : s > t}, ρ(t) = sup{s ∈ T : s < t}, μ(t) = σ(t) − t. (2.1)


We say that a point t ∈ T is left-dense if t > inf T and ρ(t) = t. If t < supT andσ(t) = t, then t is called right-dense. A point t ∈ T is called left-scattered if ρ(t) < t, whileright-scattered if σ(t) > t. If T has a left-scattered maximum m, then we set T

k = T \ {m},otherwise set T

k = T. If T has a right-scattered minimum m, then set Tk = T \ {m}, otherwiseset Tk = T.

A function f : T → R is right-dense continuous (rd-continuous) provided that it iscontinuous at right-dense point in T and its left side limits exist at left-dense points in T.If f is continuous at each right-dense point and each left-dense point, then f is said to becontinuous function on T.

Definition 2.1 (see [10]). Assume f : T → R is a function and let t ∈ Tk. We define fΔ(t) to be

the number (if it exists) with the property that for a given ε > 0, there exists a neighborhoodU of t such that

∣∣∣[f(σ(t)) − f(s)

]− fΔ(t)[σ(t) − s]

∣∣∣ < ε|σ(t) − s|, for all s ∈ U. (2.2)

We call fΔ(t) the delta derivative of f at t.

If f is continuous, then f is right-dense continuous, and if f is delta differentiable at t,then f is continuous at t.

Let f be right-dense continuous. If FΔ(t) = f(t), for all t ∈ T, then we define the deltaintegral by

∫ t

a

f(s)Δs = F(t) − F(a), for t, a ∈ T. (2.3)

Definition 2.2 (see [12]). We say that a time scale T is periodic if there is p > 0 such that ift ∈ T, then t ± p ∈ T. For T/=R, the smallest positive p is called the period of the time scale.

Definition 2.3 (see [12]). Let T/=R be a periodic time scale with period p. We say that thefunction f : T → R is periodic with period ω if there exists a natural number n such thatω = np, f(t +ω) = f(t) for all t ∈ T, and ω is the smallest number such that f(t +ω) = f(t). IfT = R, we say that f is periodic with period ω > 0 if ω is the smallest positive number suchthat f(t +ω) = f(t) for all t ∈ T.

Lemma 2.4 (see [10]). If a, b ∈ T, α, β ∈ R, and f, g ∈ C(T, R), then

(A1)∫ba[αf(t) + βg(t)]Δt = α

∫baf(t)Δt + β

∫bag(t)Δt;

(A2) if f(t) ≥ 0 for all a ≤ t < b, then∫baf(t)Δt ≥ 0;

(A3) if |f(t)| ≤ g(t) on [a, b) := {t ∈ T : a ≤ t < b}, then |∫baf(t)Δt| ≤

∫bag(t)Δt.


Lemma 2.5 (Holder’s inequality [11]). Let a, b ∈ T. For rd-continuous functions f, g : [a, b] →R, one has

∫b

a

∣∣f(t)g(t)

∣∣Δt ≤

(∫b

a

∣∣f(t)

∣∣pΔt

)1/p(∫b

a

∣∣g(t)

∣∣qΔt

)1/q

, (2.4)

where p > 1 and q = p/(p − 1).

For convenience, we denote

κ = min{[0,∞) ∩ T}, Iω = [κ, κ +ω] ∩ T, g =1ω

∫

Iω

g(s)Δs =1ω

∫κ+ω

κ

g(s)Δs, (2.5)

where g ∈ C(T, R) is an ω-periodic real function, that is, g(t +ω) = g(t) for all t ∈ T.Next, let us recall the continuation theorem in coincidence degree theory. To do so, we

introduce the following notations.Let X,Y be real Banach spaces, L : Dom L ⊂ X → Y a linear mapping, N : X → Y

a continuous mapping. The mapping L will be called a Fredholm mapping of index zero ifdimKerL = codimImL < +∞ and ImL is closed in Y. If L is a Fredholm mapping of index zeroand there exist continuous projections P : X → X, Q : Y → Y such that ImP = KerL, ImL =KerQ = Im(I −Q), then it follows that L|DomL∩KerP : (I −P)X → ImL is invertible. We denotethe inverse of that map by KP. If Ω is an open bounded subset of X, the mapping N will becalled L-compact on Ω if QN(Ω) is bounded and KP (I − Q)N : Ω → X is compact. SinceImQ is isomorphic to KerL, there exists an isomorphism J : ImQ → KerL.

Lemma 2.6 (continuation theorem). Suppose that X and Y are two Banach spaces, and L :DomL ⊂ X → Y is a Fredholm operator of index 0. Furthermore, let Ω ⊂ X be an open bounded setandN : Ω → Y L-compact on Ω. If

(B1) Lx/=λNx, for all x ∈ ∂Ω ∩DomL, λ ∈ (0, 1),

(B2) Nx/∈ ImL, for all x ∈ ∂Ω ∩ KerL,

(B3) deg{JQN,Ω ∩ KerL, 0}/= 0, where J : ImQ → KerL is an isomorphism,

then the equation Lx =Nx has at least one solution in Ω ∩DomL.

Lemma 2.7 (see [13]). Let t1, t2 ∈ Iω and t ∈ T. If g : T → R is ω-periodic, then

g(t) ≤ g(t1) +∫κ+ω

κ

∣∣∣gΔ(s)∣∣∣Δs, g(t) ≥ g(t2) −

∫κ+ω

κ

∣∣∣gΔ(s)∣∣∣Δs. (2.6)

In order to use Mawhin’s continuation theorem to study the existence of ω-periodicsolutions for (1.7), we consider the following system:

xΔ1 (t) = ϕq(x2(t)) = |x2(t)|q−2x2(t),

xΔ2 (t) = −f(x1(t))ϕq(x2(t)) − g(x1(t)) + e(t),

(2.7)


where 1 < q < 2 is a constant with 1/p + 1/q = 1. Clearly, if x(t) = (x1(t), x2(t)) is an ω-

periodic solution to (2.7), then x1(t) must be an ω-periodic solution to (1.7). Thus, in orderto prove that (1.7) has an ω-periodic solution, it suffices to show that (2.7) has an ω-periodicsolution.

Now, we set Ψω = {(u, v) ∈ C(T, R2) : u(t + ω) = u(t), v(t + ω) = v(t), for all t ∈ T}with the norm ‖(u, v)‖ = maxt∈Iω |u(t)| + maxt∈Iω |v(t)|, for (u, v) ∈ Ψω. It is easy to show thatΨω is a Banach space when it is endowed with the above norm ‖ · ‖.

Let

Ψω0 = {(u, v) ∈ Ψω : u = 0, v = 0},

Ψωc ={(u, v) ∈ Ψω : (u(t), v(t)) ≡ (h1, h2) ∈ R2, for t ∈ T

}.

(2.8)

Then it is easy to show that Ψω0 and Ψω

c are both closed linear subspaces of Ψω. Weclaim that Ψω = Ψω

0 ⊕Ψωc , and dimΨω

c = 2. Since for any (u, v) ∈ Ψω0 ∩Ψω

c ,we have (u(t), v(t)) =(h1, h2) ∈ R2, and

u =1ω

∫κ+ω

κ

u(s)Δs = h1 = 0, v =1ω

∫κ+ω

κ

v(s)Δs = h2 = 0, (2.9)

so we obtain (u, v) = (h1, h2) = (0, 0).Take X = Y = Ψω. Define

L : DomL ={x = (x1, x2) ∈ C1

(T, R2

): x(t +ω) = x(t), xΔ(t +ω) = xΔ(t)

}⊂ X → Y,

(2.10)

by

Lx(t) = xΔ(t) =

(xΔ

1 (t)

xΔ2 (t)

)

, (2.11)

and N : X → Y, by

Nx(t) =

(ϕq(x2(t))

−f(x1(t))ϕq(x2(t)) − g(x1(t)) + e(t)

)

. (2.12)

Define the operator P : X → X and Q : Y → Y by

Px = P

(x1

x2

)

=

(x1

x2

)

, Qy = Q

(y1

y2

)

=

(y1

y2

)

, x ∈ X, y ∈ Y. (2.13)

It is easy to see that (2.7) can be converted to the abstract equation Lx =Nx.


Then KerL = Ψωc , ImL = Ψω

0 , and dimKerL = 2 = codim ImL. Since Ψω0 is closed

in Ψω, it follows that L is a Fredholm mapping of index zero. It is not difficult to show thatP and Q are continuous projections such that ImP = KerL and ImL = KerQ = Im(I − Q).Furthermore, the generalized inverse (to LP )KP : ImL → KerP ∩DomL exists and is givenby

KP

(x1

x2

)

=

⎛

⎝X1 −X1

X2 −X2

⎞

⎠, where Xi(t) =∫ t

κ

xi(s)Δs, i = 1, 2. (2.14)

Since for every x ∈ KerP ∩DomL, we have

KPLx(t) = KP

(xΔ

1 (t)

xΔ2 (t)

)

=

⎛

⎜⎜⎜⎜⎜⎝

∫ t

κ

xΔ1 (s)Δs −

1ω

∫κ+ω

κ

∫ t

κ

xΔ1 (s)ΔsΔt

∫ t

κ

xΔ2 (s)Δs −

1ω

∫κ+ω

κ

∫ t

κ

xΔ2 (s)ΔsΔt

⎞

⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎝

x1(t) − x1(κ) −1ω

∫κ+ω

κ

(x1(t) − x1(κ))Δt

x2(t) − x2(κ) −1ω

∫κ+ω

κ

(x2(t) − x2(κ))Δt

⎞

⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎝

x1(t) −1ω

∫κ+ω

κ

x1(t)Δt

x2(t) −1ω

∫κ+ω

κ

x2(t)Δt

⎞

⎟⎟⎟⎟⎟⎠,

(2.15)

from the definition of P and the condition that x ∈ KerP ∩ DomL, then (1/ω)∫κ+ωκ x1(t)Δt =

(1/ω)∫κ+ωκ x2(t)Δt = 0. Thus, we get KPLx(t) = x(t). Similarly, we can prove that LKPx(t) =

x(t), for every x(t) ∈ ImL. So the operator KP is well defined. Thus,

QN

(x1

x2

)

=

⎛

⎜⎜⎝

1ω

∫κ+ω

κ

ϕq(x2(s))Δs

1ω

∫κ+ω

κ

[−f(x1(s))ϕq(x2(s)) − g(x1(s)) + e(s)

]Δs

⎞

⎟⎟⎠. (2.16)


Denote Nx1 =N1, Nx2 =N2. We have

KP (I −Q)N

(x1

x2

)

=

⎛

⎜⎜⎜⎜⎜⎝

∫ t

κ

[N1(s) −

1ω

∫κ+ω

κ

N1(r)Δr]Δs − 1

ω

∫κ+ω

κ

∫ t

κ

[N1(s) −

1ω

∫κ+ω

κ

N1(r)Δr]ΔsΔt

∫ t

κ

[N2(s) −

1ω

∫κ+ω

κ

N2(r)Δr]Δs − 1

ω

∫κ+ω

κ

∫ t

κ

[N2(s) −

1ω

∫κ+ω

κ

N2(r)Δr]ΔsΔt

⎞

⎟⎟⎟⎟⎟⎠.

(2.17)

Clearly, QN and KP (I − Q)N are continuous. Since X is a Banach space, it is easy to

show that KP (I −Q)N(Ω) is a compact for any open bounded set Ω ⊂ X. Moreover, QN(Ω)is bounded. Thus, N is L-compact on Ω.

3. Main Results

In this section, we present our main results.

Theorem 3.1. Suppose that there exist positive constants d1 and d2 such that the following conditionshold:

(i) u(σ(t))uΔ(t)f(u(t)) < 0, |u(σ(t))| > d1, t ∈ T,

(ii) u(σ(t))(g(u(t)) − e(t)) < 0, |u(σ(t))| > d2, t ∈ T,

then (1.7) has at least one ω-periodic solution.

Proof. Consider the equation Lx = λNx, λ ∈ (0, 1), where L and N are defined by the secondsection. Let Ω1 = {x ∈ X : Lx = λNx, λ ∈ (0, 1)}.

If x =(

x1(t)

x2(t)

)∈ Ω1, then we have

xΔ1 (t) = λϕq(x2(t)),

xΔ2 (t) = −f(x1(t))xΔ

1 (t) − λg(x1(t)) + λe(t).(3.1)

From the first equation of (3.1), we obtain x2(t) = ϕp((1/λ)xΔ1 (t)), and then by

substituting it into the second equation of (3.1), we get

[ϕp

(1λxΔ

1 (t))]Δ

= −f(x1(t))xΔ1 (t) − λg(x1(t)) + λe(t). (3.2)


Integrating both sides of (3.2) from κ to κ +ω, noting that x1(κ) = x1(κ +ω), xΔ1 (κ) =

xΔ1 (κ +ω), and applying Lemma 2.4, we have

∫κ+ω

κ

f(x1(t))xΔ1 (t)Δt = −

∫κ+ω

κ

[g(x1(t)) − e(t)

]Δt, (3.3)

that is,

∫κ+ω

κ

[f(x1(t))xΔ

1 (t) + g(x1(t)) − e(t)]Δt = 0. (3.4)

There must exist ξ ∈ Iω such that

f(x1(ξ))xΔ1 (ξ) + g(x1(ξ)) − e(ξ) ≥ 0. (3.5)

From conditions (i) and (ii), when x(σ(ξ)) > max{d1, d2}, we have f(x1(ξ))xΔ1 (ξ) < 0, and

g(x1(ξ))−e(ξ) < 0, which contradicts to (3.5). Consequently x(σ(ξ)) ≤ max{d1, d2}. Similarly,there must exist η ∈ Iω such that

f(x1(η))xΔ

1

(η)+ g(x1(η))− e(η)≤ 0. (3.6)

Then we have x(σ(η)) ≥ −max{d1, d2}. Applying Lemma 2.7, we get

−max{d1, d2} −∫κ+ω

κ

∣∣∣xΔ1 (s)

∣∣∣Δs ≤ x1(t) ≤ max{d1, d2} +∫κ+ω

κ

∣∣∣xΔ1 (s)

∣∣∣Δs. (3.7)

Let d = max{d1, d2}. Then (3.7) equals to the following inequality:

|x1(t)| ≤ d +∫κ+ω

κ

∣∣∣xΔ1 (s)

∣∣∣Δs. (3.8)

Let E1 = {t ∈ Iω : |x1(t)| ≤ d}, E2 = {t ∈ Iω : |x1(t)| > d}.


Consider the second equation of (3.1) and (3.8), then we have

∫κ+ω

κ

xΔ1 (t)x2(t)Δt = −

∫κ+ω

κ

x1(σ(t))xΔ2 (t)Δt

=∫κ+ω

κ

f(x1(t))xΔ1 (t)x1(σ(t))Δt + λ

∫κ+ω

κ

x1(σ(t))[g(x1(t)) − e(t)

]Δt

≤∫κ+ω

κ

∣∣f(x1(t))

∣∣∣∣∣xΔ

1 (t)∣∣∣|x1(σ(t))|Δt + λ

∫

E1

x1(σ(t))[g(x1(t)) − e(t)

]Δt

+ λ∫

E2

x1(σ(t))[g(x1(t)) − e(t)

]Δt

≤ supt∈Iω

∣∣f(x1(t))∣∣(d +∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣Δt)∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣Δt

+ λ∫

E1

x1(σ(t))[g(x1(t)) − e(t)

]Δt

≤ supt∈Iω

∣∣f(x1(t))∣∣(∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣Δt)2

+ d supt∈Iω

∣∣f(x1(t))∣∣∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣Δt

+ λ∫

E1

x1(σ(t))[g(x1(t)) − e(t)

]Δt.

.

(3.9)

Applying Lemma 2.5, we obtain that

1λp−1

∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣pΔt ≤ ω sup

t∈Iω

∣∣f(x1(t))∣∣∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣

2Δt + d sup

t∈Iω

∣∣f(x1(t))∣∣∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣Δt

+ λ(d +∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣Δt)∫κ+ω

κ

∣∣g(x1(t)) − e(t)∣∣Δt

≤ Q1

∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣

2Δt +Q2

∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣Δt

+ λdω supt∈Iω

∣∣g(x1(t)) − e(t)∣∣

≤ Q1

∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣

2Δt +Q2

∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣Δt +Q3,

.

(3.10)


where

Q1 = ω supt∈Iω

∣∣f(x1(t))

∣∣, Q2 = dsup

t∈Iω

∣∣f(x1(t))

∣∣ + λω sup

t∈Iω

∣∣g(x1(t)) − e(t)

∣∣,

Q3 = λdω supt∈Iω

∣∣g(x1(t)) − e(t)

∣∣.

(3.11)

That is,

1λp−1

∫κ+ω

κ

∣∣∣xΔ

1 (t)∣∣∣pΔt ≤ Q1

∫κ+ω

κ

∣∣∣xΔ

1 (t)∣∣∣

2Δt +Q2

∫κ+ω

κ

∣∣∣xΔ

1 (t)∣∣∣Δt +Q3. (3.12)

Thus,

∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣pΔt ≤ λp−1Q1ω

(p−2)/p(∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣pΔt)2/p

+ λp−1Q2ω(p−1)/p

(∫κ+ω

κ

∣∣∣xΔ1 (t)∣∣∣pΔt)1/p

+ λp−1Q3.

(3.13)

Since p > 2, then we obtain that there exists a positive constant M1 such that

∣∣∣xΔ1 (t)∣∣∣ ≤M1. (3.14)

Therefore,

|x1(t)| ≤ d +M1ω :=M2, |x2(t)| ≤M2

p−1

λp−1:=M3. (3.15)

Let Ω2 = {x : x ∈ KerL, QNx = 0}. If x ∈ Ω2, then x ∈ R2 is a constant vector with

|x2(t)|q−2x2(t) = 0,

1ω

∫κ+ω

κ

[f(x1(t))xΔ

1 (t) + g(x1(t)) − e(t)]Δt = 0.

(3.16)

From the second equation of (3.16) we get

∫κ+ω

κ

f(x1(t))xΔ1 (t)Δt = −

∫κ+ω

κ

[g(x1(t)) − e(t)

]Δt, (3.17)

that is,

∫κ+ω

κ

[f(x1(t))xΔ

1 (t) + g(x1(t)) − e(t)]Δt = 0. (3.18)


By assumptions (i) and (ii), we see that |x1(t)| ≤ M2 and x2(t) = 0, which impliesΩ2 ⊂ Ω1.

Now, we set Ω = {x : x = (x1, x2), |x1| < M2 + 1, |x2| < M3 + 1}. Then Ω1 ⊂ Ω. Thus

from (3.8) and (3.14), we see that conditions (B1) and (B2) of Lemma 2.6 are satisfied. Theremainder is verifying condition (B3) of Lemma 2.6. In order to do it, let

J : ImQ → KerL, J(x1, x2) = (x1, x2). (3.19)

Set

Δ0 ={x = (x1, x2) ∈ R2 : |x1| < M2 + 1, x2 = 0

}. (3.20)

It is easy to see that the equation QN(x1, x2)(t) = (0, 0), that is,

ϕq(x2(t)) = 0,

1ω

∫κ+ω

κ

[f(x1(t))ϕq(x2(t)) + g(x1(t)) − e(t)

]Δt = 0,

(3.21)

has no solution in (Ω ∩ KerL) \Δ0. So deg{JQN, Ω ∩ Ker, 0} = deg{JQN,Δ0, 0}.Let

QN0x(t) =

⎛

⎜⎝

0

1ω

∫κ+ω

κ

[g(x1(t)) − e(t)

]Δt

⎞

⎟⎠. (3.22)

If x ∈ ∂Δ0, then we get

‖JQN0x − JQNx‖ = maxx2=0, |x1|=M2+1

∣∣ϕq(x2)∣∣ +

1ω

maxx2=0, |x1|=M2+1

∣∣∣∣

∫κ+ω

κ

f(x1(t))ϕq(x2)Δt∣∣∣∣ = 0,

(3.23)

so we have

deg{JQN,Δ0, 0} = deg{JQN0,Δ0, 0}/= 0. (3.24)

Then we see that

deg{JQN,Ω ∩ KerL, 0} = deg{JQN0,Δ0, 0}/= 0, (3.25)

so the condition (B3) of Lemma 2.6 is satisfied, the proof is complete.

When∫κ+ωκ e(t)Δt = 0, g(x(t)) = β(t)x(t), where β(t) = β(t + T), t ∈ [0, T], we have the

following result.


Corollary 3.2. Suppose that the following conditions hold:

(i) β(t) > 0, for all t ∈ Iω;

(ii) u(t)uΔ(t)f(u(t)) > 0, |u| > d,

then (1.7) has at least one ω-periodic solution.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and usefulcomments that have led to the present improved version of the original manuscript. Thisresearch is supported by the Natural Science Foundation of China (60774004, 60904024),China Postdoctoral Science Foundation Funded Project (20080441126, 200902564), ShandongPostdoctoral Funded Project (200802018) and supported by Shandong Research Funds(Y2008A28), also supported by University of Jinan Research Funds for Doctors (B0621,XBS0843).

References


[2] B. Xiao and B. Liu, “Periodic solutions for Rayleigh type p-Laplacian equation with a deviatingargument,” Nonlinear Analysis: Real World Applications, vol. 10, no. 1, pp. 16–22, 2009.


[4] W. S. Cheung and J. L. Ren, “On the existence of periodic solutions for p-Laplacian generalizedLienard equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 1, pp. 65–75, 2005.

[5] F. X. Zhang and Y. Li, “Existence and uniqueness of periodic solutions for a kind of Duffing typep-Laplacian equation,” Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 985–989, 2008.



[8] F. J. Cao and Z. L. Han, “Existence of periodic solutions for p-Laplacian differential equation withdeviating arguments,” Journal of Uuniversity of Jinan (Science & Technology), vol. 24, pp. 1–4, 2010.

[9] S. Hilger, “Analysis on measure chains-a unified approach to continuous and discrete calculus,”Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.

[10] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,Birkhauser, Boston, Mass, USA, 2001.

[11] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, Mass,USA, 2003.

[12] E. R. Kaufmann and Y. N. Raffoul, “Periodic solutions for a neutral nonlinear dynamical equation ona time scale,” Journal of Mathematical Analysis and Applications, vol. 319, no. 1, pp. 315–325, 2006.

[13] M. Bohner, M. Fan, and J. M. Zhang, “Existence of periodic solutions in predator-prey andcompetition dynamic systems,” Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp. 1193–1204, 2006.

[14] Y. K. Li and H. T. Zhang, “Existence of periodic solutions for a periodic mutualism model on timescales,” Journal of Mathematical Analysis and Applications, vol. 343, no. 2, pp. 818–825, 2008.

[15] X. L. Liu and W. T. Li, “Periodic solutions for dynamic equations on time scales,” Nonlinear Analysis:Theory, Methods & Applications, vol. 67, no. 5, pp. 1457–1463, 2007.

[16] M. Bohner, M. Fan, and J. M. Zhang, “Periodicity of scalar dynamic equations and applications topopulation models,” Journal of Mathematical Analysis and Applications, vol. 330, no. 1, pp. 1–9, 2007.


Research ArticleA Note on a Semilinear Fractional DifferentialEquation of Neutral Type with Infinite Delay

Gisle M. Mophou1 and Gaston M. N’Guerekata2

1 Departement de Mathematiques et Informatique, Universite des Antilles et de La Guyane,Campus Fouillole 97159 Pointe-a-Pitre Guadeloupe (FWI), France

2 Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore,MD 21251, USA

Correspondence should be addressed to Gisle M. Mophou, [email protected]

Received 28 November 2009; Accepted 21 January 2010


Copyright q 2010 G. M. Mophou and G. M. N’Guerekata. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

We deal in this paper with the mild solution for the semilinear fractional differential equation ofneutral type with infinite delay: Dαx(t) +Ax(t) = f(t, xt), t ∈ [0, T], x(t) = φ(t), t ∈] − ∞, 0], withT > 0 and 0 < α < 1. We prove the existence (and uniqueness) of solutions, assuming that −A isa linear closed operator which generates an analytic semigroup (T(t))t≥0 on a Banach space X bymeans of the Banach’s fixed point theorem. This generalizes some recent results.

1. Introduction

We investigate in this paper the existence and uniqueness of the mild solution for thefractional differential equation with infinite delay

Dαx(t) +Ax(t) = f(t, xt), t ∈ I = [0, T],

x(t) = φ(t), t ∈ ]−∞, 0],(1.1)

where T > 0, 0 < α < 1,−A is a generator of an analytic semigroup (T(t))t≥0 on a Banach spaceX such that ‖T(t)‖ ≤ K for all t ≥ 0 and ‖AT(t)x‖ ≤ K/t‖x‖ for every x ∈ X and t > 0. Thefunction f : I × B → X is continuous functions with additional assumptions.


The fractional derivative Dα is understood here in the Caputo sense, that is,

Dαh(t) =1

Γ(1 − α)

∫ t

0(t − s)−αh′(s)ds. (1.2)

φ ∈ B where B is called phase space to be defined in Section 2. For any function x defined on] −∞, T] and any t ∈ I, we denote by xt the element of B defined by

xt(θ) = x(t + θ), θ ∈ ]−∞, 0]. (1.3)

The function xt represents the history of the state from −∞ up to the present time t.The theory of functional differential equations has emerged as an important branch

of nonlinear analysis. It is worthwhile mentioning that several important problems of thetheory of ordinary and delay differential equations lead to investigations of functionaldifferential equations of various types (see the books by Hale and Verduyn Lunel [1], Wu[2], Liang et al. [3], Liang and Xiao [4–9], and the references therein). On the other handthe theory of fractional differential equations is also intensively studied and finds numerousapplications in describing real world problems (see e.g., the monographs of Lakshmikanthamet al. [10], Lakshmikantham [11], Lakshmikantham and Vatsala [12, 13], Podlubny [14], andthe papers of Agarwal et al. [15], Benchohra et al. [16], Anguraj et al. [17], Mophou andN’Guerekata [18], Mophou et al. [19], Mophou and N’Guerekata [20], and the referencestherein).

Recently we studied in our paper [20] the existence of solutions to the fractionalsemilinear differential equation with nonlocal condition and delay-free

Dα x(t) = Ax(t) + tnf(t, x(t), Bx(t)), t ∈ [0, T], n ∈ Z+,

x(0) = x0 + g(x),(1.4)

where T is a positive real, 0 < α < 1, A is the generator of aC0-semigroup (S(t))t≥0 on a Banachspace X, Bx(t) :=

∫ t0K(t, s)x(s)ds, K ∈ C(D,R+) with D defined as above and

B∗ = supt∈[0,T]

∫ t

0K(t, s)ds <∞, (1.5)

f : R × X × X → X is a nonlinear function, g : C([0, T],X) → D(A) is continuous, and0 < q < 1. The derivative Dα is understood here in the Riemann-Liouville sense.

In the present paper we deal with an infinite time delay. Note that in this case, thephase space B plays a crucial role in the study of both qualitative and quantitative aspectsof theory of functional equations. Its choice is determinant as can be seen in the importantpaper by Hale and Kato [21].


Similar works to the present paper include the paper by Benchohra et al. [16],where the authors studied an existence result related to the nonlinear functional differentialequation

Dα x(t) = f(t, xt), t ∈ I = [0, T], 0 < α < 1,

x(t) = φ(t), t ∈ ]−∞, 0],(1.6)

where Dα is the standard Riemann-Liouville fractional derivative, φ in the phase space B,with φ(0) = 0.

2. Preliminaries

From now on, we set I = [0, T]. We denote by X a Banach space with norm ‖ · ‖, C(I; X) thespace of all X-valued continuous functions on I, and L(X) the Banach space of all linear andbounded operators on X.

We assume that the phase space (B, ‖ · ‖B) is a seminormed linear space of functionsmapping ] − ∞, 0] into X, and satisfying the following fundamental axioms due to Hale andKato (see e.g., in [21]).

(A0) If x : ]−∞, T] → X, is continuous on I and x0 ∈ B, then for every t ∈ I the followingconditions hold:

(i) xt is in B,(ii) ‖x(t)‖ ≤ H‖xt‖B,(iii) ‖xt‖B ≤ C1(t)sup 0≤s≤t ‖x(s)‖ + C2(t)‖x0‖B,

where H ≥ 0 is a constant, C1 : [0,+∞[→ [0,+∞[ is continuous, C2 : [0,+∞[→[0,+∞[ is locally bounded, and H, C1, C2 are independent of x(·).

(A1) For the function x(·) in (A0), xt is a B-valued continuous function on I.

(A2) The space B is complete.

Remark 2.1. Condition (ii) in (A0) is equivalent to ‖φ(0)‖ ≤ H‖φ‖B, for all φ ∈ B.

Let us recall some examples of phase spaces.

Example 2.2. (E1) BUC(] − ∞, 0]); X) the Banach space of all bounded and uniformlycontinuous functions φ :] −∞, 0] → X endowed with the supnorm.

(E2) C0(] − ∞, 0]) : X) the Banach space of all bounded and continuous functionsφ :] −∞, 0] → X such that lim θ→−∞φ(θ) = 0 endowed with the norm

∣∣φ∣∣ := sup

θ≤0

∣∣φ(θ)∣∣. (2.1)

(E3) Cγ := {φ ∈ C(]−∞, 0] : X) : lim θ→−∞ eγθφ(θ)exists in X} endowed with the norm

∣∣φ∣∣ = sup

−∞<θ≤0eγθ∣∣φ(θ)

∣∣. (2.2)

Note that the space Cγ is a uniform fading memory for γ > 0.


Throughout this work f will be a continuous function I×B× → X. Let Ω be set definedby:

Ω ={x : ]−∞, T] −→ X such that x|]−∞,0] ∈ B, x|I ∈ C(I; X)

}. (2.3)

Remark 2.3. We recall that the Cauchy Problem

Dαx(t) +Ax(t) = 0, t ∈ [0, T],

x(0) = x0 ∈ D, t ∈ I,(2.4)

where −A is a closed linear operator defined on a dense subset, D ⊂ X is wellposed, and theunique solution is given by

x(t) =∫∞

0ζα(σ)T(tασ)x0dσ, (2.5)

where ζα is a probability density function defined on (0,∞) such that its Laplace transform isgiven by

∫∞

0e−σxζα(σ)dσ =

∞∑

i=0

(−xi)

Γ(1 + αi), x > 0 (2.6)

([22, cf. Theorem 2.1]).

Following [22, 23] we will introduce now the definition of mild solution to (1.1).

Definition 2.4. A function x ∈ Ω is said to be a mild solution of (1.1) if x satisfies

x(t) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

φ(t), t ∈ ]−∞, 0],

Q(t)φ(0) +∫ t

0R(t − s)f(s, xs)ds, t ∈ I,

(2.7)

where

Q(t) =∫∞

0ζα(σ)T(tασ)dσ, R(t) = α

∫∞

0σtα−1ζα(σ)T(tασ)dσ. (2.8)

Remark 2.5. Note that

‖R(t)‖B(X) ≤ αKtα−1, t ≥ 0, (2.9)

since∫∞

0 σζα(σ)dσ = 1 (cf. [23]).


3. Main Results

We present now our result.

Theorem 3.1. Assume the following.

(H1) There exist μ > 0 such that for all t ∈ I, (ψ, ϕ) ∈ B2

∥∥f(t, ϕ)− f(t, ψ)∥∥ ≤ μ

∥∥ϕ − ψ

∥∥B (3.1)

(H2) There exists δ, with 0 < δ < 1 such that the function Λ : I → ]0,+∞] defined by:

Λ(t) = μKC∗1tα (3.2)

satisfies Λ(t) ≤ δ for all t ∈ I. Here

C∗1 = supt∈I

C1(t). (3.3)

Then (1.1) has a unique mild solution on ] −∞, T].

Proof. Consider the operator N : Ω → Ω defined by

N(x)(t) =

⎧⎪⎪⎨

⎪⎪⎩

φ(t), t ∈ ]−∞, 0],

Q(t)φ(0) +∫ t

0R(t − s)f(s, xs)ds, t ∈ I.

(3.4)

Let y(·) : ] −∞, T] → X be the function defined by

y(t) =

⎧⎨

⎩

φ(t), if t ∈ ]−∞, 0],

Q(t)φ(0), if t ∈ I.(3.5)

Then y0 = φ. For each z ∈ C(I,X) with z(0) = 0, we denote by z the function defined by

z(t) =

⎧⎨

⎩

0, t ∈ ]−∞, 0],

z(t), t ∈ I.(3.6)

If x(·) verifies (2.7) then writing x(t) = y(t)+z(t) for t ∈ I, we have xt = yt +zt for t ∈ Iand

z(t) =∫ t

0R(t − s)f(s, ys + zs

)ds. (3.7)

Moreover z0 = 0.


Let

Z0 = {z ∈ Ω such that z0 = 0}. (3.8)

For any z ∈ Z0, we have

‖z‖Z0= sup

t∈I‖z(t)‖ + ‖z0‖B = sup

t∈I‖z(t)‖. (3.9)

Thus (Z0, ‖ · ‖Z0) is a Banach space. We define the operator Π : Z0 → Z0 by

Π(z)(t) =∫ t

0R(t − s)f(s, ys + zs

)ds. (3.10)

It is clear that the operator N has a unique fixed point if and only if Π has a unique fixedpoint. So let us prove that Π has a unique fixed point. Observe first that Π is obviously welldefined. Now, consider z, z∗ ∈ Z0. For any t ∈ I, we have

‖Π(z)(t) −Π(z∗)(t)‖ =∣∣∣∣∣

∫ t

0R(t − s)f

(s, ys + zs

)ds −

∫ t

0R(t − s)f

(s, ys + z∗s

)ds

∣∣∣∣∣

≤∫ t

0‖R(t − s)‖

∥∥∥f(s, ys + zs

)− f(s, ys + z∗s

)∥∥∥ds

≤ μ∫ t

0‖R(t − s)‖

∥∥∥zs − z∗s∥∥∥Bds.

(3.11)

So using (A0)-(iii), (2.9) and (3.3), we obtain for all t ∈ I

‖Π(z)(t) −Π(z∗)(t)‖ ≤ μαK∫ t

0(t − s)α−1C1(s)‖z − z∗‖Z0

≤ μKC∗1tα‖z − z∗‖Z0

(3.12)

which according to (H2) gives

‖Π(z)(t) −Π(z∗)(t)‖ ≤ Λ(t)‖z − z∗‖Z0

≤ δ‖z − z∗‖Z0.

(3.13)

Therefore

‖Πz −Πz∗‖Z0≤ δ‖z − z∗‖Z0

. (3.14)

And since 0 ≤ δ < 1, we conclude by way of the Banach’s contraction mappingprinciple that Π has a unique fixed point z ∈ Z0. This means that N has a unique fixedpoint x ∈ Ω which is obviously a mild solution of (1.1) on ] −∞, T].


4. Application

To illustrate our result, we consider the following Lotka-Volterra model with diffusion:

Dαt u(t, ξ) =

∂2

∂ξ2u(t, ξ) +

∫0

−∞η(σ)u(t + σ, ξ)dσ, 0 ≤ ξ ≤ π,

u(t, 0) = u(t, π) = 0 for t ∈ R,

(4.1)

where 0 < α < 1 and η is a positive function on (−∞, 0] with∫0−∞η(σ)dσ <∞.

Now let X = L2(0, π) and consider the operator A : D(A) ⊂ X → X defined by

D(A) = H2(0, π) ∩H10(0, π) =

{H2(0, π) : z(0) = z(π) = 0

},

Az = z′′.

(4.2)

Clearly D(A) is dense in L2(0, π).Define

f(φ)(ξ) :=

∫∞

0η(σ)φ(σ)(ξ)dσ, ξ ∈ [0, π], φ ∈ B. (4.3)

We choose B as in Example (E3) above. Put

x(t)(ξ) = u(t, ξ), t ∈ (−∞, T], ξ ∈ [0, π]. (4.4)

Then we get

Dαx(t) = Ax(t) + f(t, xt), (4.5)

where f(t, x) is obviously Lipschitzian in x uniformly in t. Thus we can state what follows.

Theorem 4.1. Under the above assumptions (4.1) has a unique mild solution.


References

[1] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99 of AppliedMathematical Sciences, Springer, New York, NY, USA, 1993.

[2] J. Wu, Theory and Applications of Partial Functional-Differential Equations, vol. 119 of AppliedMathematicalSciences, Springer, New York, NY, USA, 1996.

[3] J. Liang, F. Huang, and T. Xiao, “Exponential stability for abstract linear autonomous functional-differential equations with infinite delay,” International Journal of Mathematics and MathematicalSciences, vol. 21, no. 2, pp. 255–259, 1998.

[4] J. Liang and T. J. Xiao, “Functional-differential equations with infinite delay in Banach spaces,”International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 497–508, 1991.

[5] J. Liang and T.-J. Xiao, “The Cauchy problem for nonlinear abstract functional differential equationswith infinite delay,” Computers & Mathematics with Applications, vol. 40, no. 6-7, pp. 693–703, 2000.

[6] J. Liang and T.-J. Xiao, “Solvability of the Cauchy problem for infinite delay equations,” NonlinearAnalysis: Theory, Methods & Applications, vol. 58, no. 3-4, pp. 271–297, 2004.

[7] J. Liang and T.-J. Xiao, “Solutions to nonautonomous abstract functional equations with infinitedelay,” Taiwanese Journal of Mathematics, vol. 10, no. 1, pp. 163–172, 2006.

[8] J. Liang, T.-J. Xiao, and J. van Casteren, “A note on semilinear abstract functional differential andintegrodifferential equations with infinite delay,” Applied Mathematics Letters, vol. 17, no. 4, pp. 473–477, 2004.

[9] T.-J. Xiao and J. Liang, “Blow-up and global existence of solutions to integral equations with infinitedelay in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e1442–e1447, 2009.

[10] V. Lakshmikantham, S. Leela, and J. Vasundhara, Theory of Fractional Dynamic Systems, CambridgeAcademic, Cambridge, UK, 2009.



[13] V. Lakshmikantham and A. S. Vatsala, “Theory of fractional differential inequalities and applica-tions,” Communications in Applied Analysis, vol. 11, no. 3-4, pp. 395–402, 2007.

[14] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering,Academic Press, San Diego, Calif, USA, 1999.

[15] R. P. Agarwal, M. Benchohra, and B. A. Slimani, “Existence results for differential equations withfractional order and impulses,” Memoirs on Differential Equations and Mathematical Physics, vol. 44, pp.1–21, 2008.

[16] M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional orderfunctional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications,vol. 338, no. 2, pp. 1340–1350, 2008.

[17] A. Anguraj, P. Karthikeyan, and G. M. N’Guerekata, “Nonlocal Cauchy problem for some fractionalabstract integro-differential equations in Banach spaces,” Communications in Mathematical Analysis,vol. 6, no. 1, pp. 31–35, 2009.

[18] G. M. Mophou and G. M. N’Guerekata, “Mild solutions for semilinear fractional differentialequations,” Electronic Journal of Differential Equations, vol. 2009, no. 21, pp. 1–9, 2009.

[19] G. M. Mophou, O. Nakoulima, and G. M. N’Guerekata, “Existence results for some fractionaldifferential equations with nonlocal conditions,” Nonlinear Studies, vol. 17, no. 1, pp. 15–22, 2010.

[20] G. M. Mophou and G. M. N’Guerekata, “Existence of the mild solution for some fractional differentialequations with nonlocal conditions,” Semigroup Forum, vol. 79, no. 2, pp. 315–322, 2009.

[21] J. K. Hale and J. Kato, “Phase space for retarded equations with infinite delay,” Funkcialaj Ekvacioj,vol. 21, no. 1, pp. 11–41, 1978.

[22] M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolutionequations,” Chaos, Solitons and Fractals, vol. 14, no. 3, pp. 433–440, 2002.

[23] M. M. El-Borai, “On some stochastic fractional integro-differential equations,” Advances in DynamicalSystems and Applications, vol. 1, no. 1, pp. 49–57, 2006.


Research ArticleAlmost Automorphic Solutions toAbstract Fractional Differential Equations

Hui-Sheng Ding,1 Jin Liang,2 and Ti-Jun Xiao3

1 College of Mathematics and Information Science, Jiangxi Normal University, Nanchang,Jiangxi 330022, China

2 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China3 School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Correspondence should be addressed to Hui-Sheng Ding, [email protected]

Received 24 October 2009; Accepted 12 January 2010


Copyright q 2010 Hui-Sheng Ding et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

A new and general existence and uniqueness theorem of almost automorphic solutions is obtainedfor the semilinear fractional differential equation Dα

t u(t) = Au(t) + Dα−1t f(t, u(t)) (1 < α < 2), in

complex Banach spaces, with Stepanov-like almost automorphic coefficients. Moreover, an applicationto a fractional relaxation-oscillation equation is given.

1. Introduction

In this paper, we investigate the existence and uniqueness of almost automorphic solutionsto the following semilinear abstract fractional differential equation:

Dαt u(t) = Au(t) +D

α−1t f(t, u(t)), t ∈ R, (1.1)

where 1 < α < 2, A : D(A) ⊂ X → X is a sectorial operator of type ω in a Banach spaceX, and f : R × X → X is Stepanov-like almost automorphic in t ∈ R satisfying some kindof Lipschitz conditions in x ∈ X. In addition, the fractional derivative is understood in theRiemann-Liouville’s sense.

Recently, fractional differential equations have attracted more and more attentions(cf. [1–8] and references therein). On the other hand, the Stepanov-like almost automorphicproblems have been studied by many authors (cf., e.g., [9, 10] and references therein).Stimulated by these works, in this paper, we study the almost automorphy of solutions tothe fractional differential equation (1.1) with Stepanov-like almost automorphic coefficients.


A new and general existence and uniqueness theorem of almost automorphic solutions to theequation is established. Moreover, an application to fractional relaxation-oscillation equationis given to illustrate the abstract result.

Throughout this paper, we denote by N the set of positive integers, by R the set of realnumbers, and by X a complex Banach space. In addition, we assume 1 ≤ p < +∞ if there is nospecial statement. Next, let us recall some definitions of almost automorphic functions andStepanov-like almost automorphic functions (for more details, see, e.g., [9–11]).

Definition 1.1. A continuous function f : R → X is called almost automorphic if for everyreal sequence (sm), there exists a subsequence (sn) such that

g(t) := limn→∞

f(t + sn) (1.2)

is well defined for each t ∈ R and

limn→∞

g(t − sn) = f(t) (1.3)

for each t ∈ R. Denote by AA(X) the set of all such functions.

Definition 1.2. The Bochner transform fb(t, s), t ∈ R, s ∈ [0, 1], of a function f(t) on R, withvalues in X, is defined by

fb(t, s) := f(t + s). (1.4)

Definition 1.3. The space BSp(X) of all Stepanov bounded functions, with the exponent p,consists of all measurable functions f on R with values in X such that

∥∥f∥∥Sp := sup

t∈R

(∫ t+1

t

∥∥f(τ)∥∥pdτ

)1/p

< +∞. (1.5)

It is obvious that Lp(R;X) ⊂ BSp(X) ⊂ Lp

loc(R;X) and BSp(X) ⊂ BSq(X) wheneverp ≥ q ≥ 1.

Definition 1.4. The space ASp(X) of Sp-almost automorphic functions (Sp-a.a. for short)consists of all f ∈ BSp(X) such that fb ∈ AA(Lp(0, 1;X)). In other words, a function f ∈Lp

loc(R;X) is said to be Sp-almost automorphic if its Bochner transform fb : R → Lp(0, 1;X)is almost automorphic in the sense that for every sequence of real numbers (s′n), there exist


a subsequence (sn) and a function g ∈ Lploc(R;X) such that

limn→∞

(∫1

0

∥∥f(t + sn + s) − g(t + s)

∥∥pds

)1/p

= 0,

limn→∞

(∫1

0

∥∥g(t − sn + s) − f(t + s)

∥∥pds

)1/p

= 0,

(1.6)

for each t ∈ R.

Remark 1.5. It is clear that if 1 ≤ p < q < ∞ and f ∈ Lq

loc(R;X) is Sq-almost automorphic,then f is Sp-almost automorphic. Also if f ∈ AA(X), then f is Sp-almost automorphic forany 1 ≤ p <∞.

Definition 1.6. A function f : R × X → X, (t, u) → f(t, u) with f(·, u) ∈ Lploc(R, X) for eachu ∈ X is said to be Sp-almost automorphic in t ∈ R uniformly for u ∈ X, if for every sequenceof real numbers (s′n), there exists a subsequence (sn) and a function g : R × X → X withg(·, u) ∈ Lploc(R, X) such that

limn→∞

(∫1

0

∥∥f(t + sn + s, u) − g(t + s, u)∥∥pds

)1/p

= 0,

limn→∞

(∫1

0

∥∥g(t − sn + s, u) − f(t + s, u)∥∥pds

)1/p

= 0,

(1.7)

for each t ∈ R and for each u ∈ X. We denote by ASp(R ×X,X) the set of all such functions.

2. Almost Automorphic Solution

First, let us recall that a closed and densely defined linear operatorA is called sectorial if thereexist 0 < θ < π/2, M > 0, and ω ∈ R such that its resolvent exists outside the sector

ω + Sθ :={ω + λ : λ ∈ C,

∣∣arg(−λ)∣∣ < θ

},

∥∥∥(λI −A)−1∥∥∥ ≤

M

|λ −ω| , λ /∈ω + Sθ.(2.1)

Recently, in [3], Cuesta proved that if A is sectorial operator for some 0 < θ < π(1 − α/2)(1 < α < 2), M > 0, and ω < 0, then there exits C > 0 such that

‖Eα(t)‖ ≤CM

1 + |ω|tα , t ≥ 0, (2.2)


where

Eα(t) :=1

2πi

∫

γ

eλtλα−1(λα −A)−1dλ, (2.3)

where γ is a suitable path lying outside the sector ω + Sθ.In addition, by [2], we have the following definition.

Definition 2.1. A function u : R → X is called a mild solution of (1.1) if s → Eα(t−s)f(s, u(s))is integrable on (−∞, t) for each t ∈ R and

u(t) =∫ t

−∞Eα(t − s)f(s, u(s))ds, t ∈ R. (2.4)

Lemma 2.2. Let {S(t)}t≥0 ⊂ B(X) be a strongly continuous family of bounded and linear operatorssuch that

‖S(t)‖ ≤ φ(t), t ∈ R+, (2.5)

where φ ∈ L1(R+) is nonincreasing. Then, for each f ∈ AS1(X),

∫ t

−∞S(t − s)f(s)ds ∈ AA(X). (2.6)

Proof. For each n ∈ N, let

fn(t) :=∫ t−n+1

t−nS(t − s)f(s)ds =

∫n

n−1S(s)f(t − s)ds, t ∈ R. (2.7)

In addition, for each n ∈ N, by the principle of uniform boundedness,

Mn := supn−1≤s≤n

‖S(s)‖ < +∞. (2.8)

Fix n ∈ N and t ∈ R. We have

∥∥fn(t + h) − fn(t)∥∥ ≤∫n

n−1‖S(s)‖ ·

∥∥f(t + h − s) − f(t − s)∥∥ds

≤Mn ·∫ t−n+1

t−n

∥∥f(s + h) − f(s)∥∥ds.

(2.9)

In view of f ∈ L1loc(R;X), we get

limh→ 0

∫ t−n+1

t−n

∥∥f(s + h) − f(s)∥∥ds = 0, (2.10)


which yields that

limh→ 0

∥∥fn(t + h) − fn(t)

∥∥ = 0. (2.11)

This means that fn(t) is continuous.Fix n ∈ N. By the definition of AS1(X), for every sequence of real numbers (s′m), there

exist a subsequence (sm) and a function g ∈ L1loc(R;X) such that

limm→∞

∫1

0

∥∥f(t + sm + s) − g(t + s)

∥∥ds = lim

m→∞

∫1

0

∥∥g(t − sm + s) − f(t + s)

∥∥ds = 0, (2.12)

for each t ∈ R. Combining this with

∥∥∥∥fn(t + sm) −∫n

n−1S(s)g(t − s)ds

∥∥∥∥ ≤Mn ·∫n

n−1

∥∥f(t + sm − s) − g(t − s)∥∥ds

=Mn ·∫1

0

∥∥f(t − n + sm + s) − g(t − n + s)∥∥ds,

(2.13)

we get

limm→∞

fn(t + sm) =∫n

n−1S(s)g(t − s)ds (2.14)

for each t ∈ R. Similar to the above proof, one can show that

limm→∞

∫n

n−1S(s)g(t − sm − s)ds = fn(t) (2.15)

for each t ∈ R. Therefore, fn ∈ AA(X) for each n ∈ N.Noticing that

∥∥fn(t)∥∥ ≤∫n

n−1φ(s) ·

∥∥f(t − s)∥∥ds ≤ φ(n − 1) ·

∥∥f∥∥S1 ,

∞∑

n=1

φ(n − 1) ·∥∥f∥∥S1 ≤

(

φ(0) +∞∑

n=2

∫n−1

n−2φ(t)dt

)

·∥∥f∥∥S1

≤(φ(0) +

∥∥φ∥∥L1(R+)

)·∥∥f∥∥S1 < +∞,

(2.16)

we know that∑∞

n=1 fn(t) is uniformly convergent on R. Thus

∫ t

−∞S(t − s)f(s)ds =

∞∑

n=1

fn(t) ∈ AA(X). (2.17)


Remark 2.3. For the case of f ∈ AA(X), the conclusion of Lemma 2.2 was given in [1, Lemma3.1].

The following theorem will play a key role in the proof of our existence and uniquenesstheorem.

Theorem 2.4 (see [11]). Assume that

(i) f ∈ ASp(R ×X,X) with p > 1;

(ii) there exists a nonnegative function L ∈ ASr(R) with r ≥ max{p, p/(p − 1)} such that forall u, v ∈ X and t ∈ R,

∥∥f(t, u) − f(t, v)

∥∥ ≤ L(t)‖u − v‖; (2.18)

(iii) x ∈ ASp(X) and K = {x(t) : t ∈ R} is compact in X.

Then there exists q ∈ [1, p) such that f(·, x(·)) ∈ ASq(X).

Now, we are ready to present the existence and uniqueness theorem of almostautomorphic solutions to (1.1).

Theorem 2.5. Assume that A is sectorial operator for some 0 < θ < π(1 − α/2),M > 0 and ω < 0;and the assumptions (i) and (ii) of Theorem 2.4 hold. Then (1.1) has a unique almost automorphic mildsolution provided that

‖L‖S1 <α sin(π/α)

CM(α sin(π/α) + |ω|−1/απ

) . (2.19)

Proof. For each ϕ ∈ AA(X), let

F(ϕ)(t) :=

∫ t

−∞Eα(t − s)f

(s, ϕ(s)

)ds, t ∈ R. (2.20)

In view of {ϕ(t) : t ∈ R} which is compact in X, by Theorem 2.4, there exists q ∈ [1, p) suchthat f(·, ϕ(·)) ∈ ASq(X). On the other hand, by (2.2), we have

‖Eα(t)‖ ≤CM

1 + |ω|tα , t ≥ 0. (2.21)

Since 1 < α < 2, CM/(1 + |ω|tα) ∈ L1(R+) and is nonincreasing. So Lemma 2.2 yields thatF(ϕ) ∈ AA(X). This means that F maps AA(X) into AA(X).


For each ϕ, ψ ∈ AA(X) and t ∈ R, we have

∥∥F(ϕ)(t) − F

(ψ)(t)∥∥ ≤∫ t

−∞‖Eα(t − s)‖ ·

∥∥f(s, ϕ(s)

)− f(s, ψ(s)

)∥∥ds

≤∫ t

−∞

CM

1 + |ω|(t − s)αL(s)ds ·

∥∥ϕ − ψ

∥∥

≤∫+∞

0

CM

1 + |ω|sα L(t − s)ds ·∥∥ϕ − ψ

∥∥

=∞∑

k=0

∫k+1

k

CM

1 + |ω|sα L(t − s)ds ·∥∥ϕ − ψ

∥∥

≤∞∑

k=0

CM

1 + |ω|kα

∫k+1

k

L(t − s)ds ·∥∥ϕ − ψ

∥∥

≤∞∑

k=0

CM

1 + |ω|kα · ‖L‖S1 ·∥∥ϕ − ψ

∥∥

≤(

CM +∞∑

k=1

∫k

k−1

CM

1 + |ω|sα ds)

· ‖L‖S1 ·∥∥ϕ − ψ

∥∥

=(CM +

∫+∞

0

CM

1 + |ω|sα ds)· ‖L‖S1 ·

∥∥ϕ − ψ∥∥

= CM

(

1 +|ω|−1/απ

α sin(π/α)

)

· ‖L‖S1 ·∥∥ϕ − ψ

∥∥,

(2.22)

which gives

∥∥F(ϕ)− F(ψ)∥∥ ≤ CM

(

1 +|ω|−1/απ

α sin(π/α)

)

· ‖L‖S1 ·∥∥ϕ − ψ

∥∥. (2.23)

In view of (2.19), F is a contraction mapping. On the other hand, it is well known that AA(X)is a Banach space under the supremum norm. Thus, F has a unique fixed point u ∈ AA(X),which satisfies

u(t) =∫ t

−∞Eα(t − s)f(s, u(s))ds, (2.24)

for all t ∈ R. Thus (1.1) has a unique almost automorphic mild solution.

In the case of L(t) ≡ L, by following the proof of Theorem 2.5 and using the standardcontraction principle, one can get the following conclusion.


Theorem 2.6. Assume that A is sectorial operator for some 0 < θ < π(1 − α/2),M > 0 and ω < 0;and the assumptions (i) and (ii) of Theorem 2.4 hold with L(t) ≡ L, then (1.1) has a unique almostautomorphic mild solution provided that

L <α sin(π/α)

CM|ω|−1/απ. (2.25)

Remark 2.7. Theorem 2.6 is due to [2, Theroem 3.4] in the case of f(t, u) being almostautomorphic in t. Thus, Theorem 2.6 is a generalization of [2, Theroem 3.4].

At last, we give an application to illustrate the abstract result.

Example 2.8. Let us consider the following fractional relaxation-oscillation equation given by

∂αt u(t, x) = ∂2xu(t, x) − μu(t, x) + ∂α−1

t [a(t) sin(u(t, x))], t ∈ R, x ∈ [0, π], (2.26)

with boundary conditions

u(t, 0) = u(t, π) = 0, t ∈ R, (2.27)

where 1 < α < 2, μ > 0, and

a(t) =

⎧⎨

⎩sin

12 + cosn + cosπn

, t ∈ (n − ε, n + ε), n ∈ Z,

0, otherwise,(2.28)

for some ε ∈ (0, 1/2).Let X = L2[0, π], Au = u

′′ − μu with

D(A) ={u ∈ L2[0, π] : u

′′ ∈ L2[0, π], u(0) = u(π) = 0}, (2.29)

and f(t, ϕ)(s) = a(t) sin(ϕ(s)) for ϕ ∈ X and s ∈ [0, π]. Then (2.26) is transformed into (1.1).It is well known that A is a sectorial operator for some 0 < θ < π/2, M > 0 and ω < 0. By[10, Example 2.3], a(t) ∈ AS2(R). Then f ∈ AS2(R × X,X). In addition, for each t ∈ R andu, v ∈ X,

∥∥f(t, u) − f(t, v)∥∥ =(∫π

0|a(t) sin(u(s)) − a(t) sin(v(s))|2ds

)1/2

≤ |a(t)| · ‖u − v‖. (2.30)

Since

‖|a(·)|‖S1 = supt∈R

∫ t+1

t

|a(s)|ds ≤ 2ε, (2.31)


by Theorem 2.5, there exists a unique almost automorphic mild solution to (2.26) providedthat 1 < α < 2(1 − θ/π) and ε is sufficiently small.

Remark 2.9. In the above example, for any ε > 0, f(t, u) is Lipschitz continuous aboutu uniformly in t with Lipschitz constant L ≡ 1, this means that f(t, u) has a betterLipschitz continuity than (2.30). However, one cannot ensure the unique existence of almostautomorphic mild solution to (2.26) when

α sin(π/α)

CM|ω|−1/απ≤ 1, (2.32)

by using Theorem 2.6. On the other hand, it is interesting to note that one can use Theorem 2.5to obtain the existence in many cases under this restriction.

Acknowledgments

The authors are grateful to the referee for valuable suggestions and comments, whichimproved the quality of this paper. H. Ding acknowledges the support from the NSF ofChina, the NSF of Jiangxi Province of China (2008GQS0057), and the Youth Foundation ofJiangxi Provincial Education Department(GJJ09456). J. Liang and T. Xiao acknowledge thesupport from the NSF of China (10771202), the Research Fund for Shanghai Key Laboratoryof Modern Applied Mathematics (08DZ2271900), and the Specialized Research Fund for theDoctoral Program of Higher Education of China (2007035805).

References

[1] D. Araya and C. Lizama, “Almost automorphic mild solutions to fractional differential equations,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3692–3705, 2008.

[2] C. Cuevas and C. Lizama, “Almost automorphic solutions to a class of semilinear fractionaldifferential equations,” Applied Mathematics Letters, vol. 21, no. 12, pp. 1315–1319, 2008.

[3] E. Cuesta, “Asymptotic behaviour of the solutions of fractional integro-differential equations andsome time discretizations,” Discrete and Continuous Dynamical Systems, pp. 277–285, 2007.




[7] G. M. Mophou, G. M. N’Guerekata, and V. Valmorina, “Pseudo almost automorphic solutions of aneutral functional fractional differential equations,” International Journal of Evolution Equations, vol. 4,pp. 129–139, 2009.


[9] H. Lee, H. Alkahby, and G. N’Guerekata, “Stepanov-like almost automorphic solutions of semilinearevolution equations with deviated argument,” International Journal of Evolution Equations, vol. 3, no. 2,pp. 217–224, 2008.

[10] G. M. N’Guerekata and A. Pankov, “Stepanov-like almost automorphic functions and monotoneevolution equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 9, pp. 2658–2667,2008.

[11] H. S. Ding, J. Liang, and T. J. Xiao, “Almost automorphic solutions to nonautonomous semilinearevolution equations in Banach spaces,” preprint.


Research ArticleExistence and Uniqueness of PeriodicSolutions for a Class of Nonlinear Equations withp-Laplacian-Like Operators

Hui-Sheng Ding, Guo-Rong Ye, and Wei Long

College of Mathematics and Information Science, Jiangxi Normal University, Nanchang,Jiangxi 330022, China

Correspondence should be addressed to Wei Long, [email protected]

Received 1 February 2010; Accepted 19 March 2010


Copyright q 2010 Hui-Sheng Ding et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We investigate the following nonlinear equations with p-Laplacian-like operators (ϕ(x′(t)))′ +f(x(t))x′(t) + g(x(t)) = e(t): some criteria to guarantee the existence and uniqueness of periodicsolutions of the above equation are given by using Mawhin’s continuation theorem. Our resultsare new and extend some recent results due to Liu (B. Liu, Existence and uniqueness of periodicsolutions for a kind of Lienard type p-Laplacian equation, Nonlinear Analysis TMA, 69, 724–729,2008).

1. Introduction

In this paper, we deal with the existence and uniqueness of periodic solutions for thefollowing nonlinear equations with p-Laplacian-like operators:

(ϕ(x′(t)

))′ + f(x(t))x′(t) + g(x(t)) = e(t), (1.1)

where f , g are continuous functions on R, and e is a continuous function on R with periodT > 0; moreover, ϕ : R → R is a continuous function satisfying the following:

(H1) for any x1, x2 ∈ R, x1 /=x2, [ϕ(x1) − ϕ(x2)] · (x1 − x2) > 0 and ϕ(0) = 0;

(H2) there exists a function α : [0,+∞) → [0,+∞) such that lims→+∞α(s) = +∞ and

ϕ(x) · x ≥ α(|x|)|x|, ∀x ∈ R. (1.2)


It is obvious that under these two conditions, ϕ is an homeomorphism from R onto R and isincreasing on R.

Recall that p-Laplacian equations have been of great interest for many mathematicians.Especially, there is a large literature (see, e.g., [1–7] and references therein) about the existenceof periodic solutions to the following p-Laplacian equation:

(ϕp

(x′(t)

))′ + f(x(t))x′(t) + g(x(t)) = e(t), (1.3)

and its variants, where ϕp(s) = |s|p−2s for s /= 0 and ϕp(0) = 0. Obviously, (1.3) is a special caseof (1.1).

However, there are seldom results about the existence of periodic solutions to (1.1).The main difficulty lies in the p-Laplacian-like operator ϕ of (1.1), which is more complicatedthan ϕp in (1.3). Since there is no concrete form for the p-Laplacian-like operator ϕ of (1.1), itis more difficult to prove the existence of periodic solutions to (1.1).

Therefore, in this paper, we will devote ourselves to investigate the existence ofperiodic solutions to (1.1). As one will see, our theorem generalizes some recent results evenfor the case of ϕ(s) = ϕp(s) (see Remark 2.2).

Next, let us recall some notations and basic results. For convenience, we denote

C1T :=

{x ∈ C1(R,R) : x is T -periodic

}, (1.4)

which is a Banach space endowed with the norm ‖x‖ = max{|x|∞, |x′|∞}, where

|x|∞ = maxt∈[0,T]

|x(t)|,∣∣x′

∣∣∞ = max

t∈[0,T]

∣∣x′(t)∣∣. (1.5)

In the proof of our main results, we will need the following classical Mawhin’scontinuation theorem.

Lemma 1.1 ([8]). Let (H1), (H2) hold and f is Caratheodory. Assume thatΩ is an open bounded setin C1

T such that the following conditions hold.

(S1) For each λ ∈ (0, 1), the problem

(ϕ(x′(t)

))′ = λf(t, x, x′

), x(0) = x(T), x′(0) = x′(T) (1.6)

has no solution on ∂Ω.

(S2) The equation

F(a) :=1T

∫T

0f(t, a, 0)dt = 0 (1.7)

has no solution on ∂Ω ∩ R.

(S3) The Brouwer degree

deg(F,Ω ∩ R, 0)/= 0. (1.8)


Then the periodic boundary value problem

(ϕ(x′(t)

))′ = f(t, x, x′

), x(0) = x(T), x′(0) = x′(T) (1.9)

has at least one T -periodic solution on Ω.

2. Main Results

In this section, we prove an existence and uniqueness theorem for (1.1).

Theorem 2.1. Suppose the following assumptions hold:

(A1) g ∈ C1(R,R) and g ′(x) < 0 for all x ∈ R;

(A2) there exist a constant r ≥ 0 and a function ε(t) ∈ C(R,R) such that for all t ∈ R and|x| > r,

x[g(x) − ε(t)

]< 0,

∫T

0[ε(t) − e(t)]dt ≤ 0,

∫T

0|ε(t) − e(t)|dt < 2.

(2.1)

Then (1.1) has a unique T -periodic solution.

Proof

Existence. For the proof of existence, we use Lemma 1.1. First, let us consider the homotopicequation of (1.1):

(ϕ(x′(t)

))′ + λf(x(t))x′(t) + λg(x(t)) = λe(t), λ ∈ (0, 1). (2.2)

Let x(t) ∈ C1T be an arbitrary solution of (2.2). By integrating the two sides of (2.2)

over [0, T], and noticing that x′(0) = x′(T) and x(0) = x(T), we have

∫T

0

[g(x(t)) − e(t)

]dt = 0, (2.3)

that is,

1T

∫T

0g(x(t))dt = e :=

1T

∫T

0e(t)dt. (2.4)

Since g(x(·)) is continuous, there exists t0 ∈ [0, T] such that

g(x(t0)) ≥ e. (2.5)


In view of (A1), we obtain

x(t0) ≤ e, (2.6)

where e = g−1(e). Then, for each t ∈ [t0, t0 + T], we have

2x(t) = x(t) + x(t − T)

= x(t0) +∫ t

t0

x′(s)ds + x(t0) −∫ t0

t−Tx′(s)ds

≤ 2x(t0) +∫ t

t0

∣∣x′(s)

∣∣ds +

∫ t0

t−T

∣∣x′(s)

∣∣ds

≤ 2e +∫T

0

∣∣x′(s)∣∣ds,

(2.7)

which gives that

|x|∞ ≤ e +12

∫T

0

∣∣x′(s)∣∣ds. (2.8)

Thus,

|x|∞ ≤ |e| +12

∫T

0

∣∣x′(s)∣∣ds. (2.9)

Since lims→+∞α(s) = +∞, there is a constant M > 0 such that

α(s) ≥ 1, ∀s ≥M. (2.10)

Set

E1 ={t : t ∈ [0, T],

∣∣x′(t)∣∣ > M

}, E2 =

{t : t ∈ [0, T],

∣∣x′(t)∣∣ ≤M

},

F1 = {t : t ∈ [0, T], |x(t)| > r}, F2 = {t : t ∈ [0, T], |x(t)| ≤ r}.(2.11)

In view of (A2) and

ϕ(x) · x ≥ α(|x|)|x|, ∀x ∈ R, (2.12)


we get

∫T

0

∣∣x′(t)

∣∣dt =

∫

E1

∣∣x′(t)

∣∣dt +

∫

E2

∣∣x′(t)

∣∣dt

≤∫

E1

∣∣x′(t)

∣∣dt +MT

≤∫

E1

ϕ(x′(t))x′(t)α(|x′(t)|) dt +MT

≤∫

E1

ϕ(x′(t)

)x′(t)dt +MT

≤∫T

0ϕ(x′(t)

)x′(t)dt +MT

=∫T

0ϕ(x′(t)

)dx(t) +MT

= −∫T

0

(ϕ(x′(t)

))′x(t)dt +MT

= λ∫T

0

[g(x(t)) − e(t)

]x(t)dt + λ

∫T

0f(x(t))x′(t)x(t)dt +MT

= λ∫T

0

[g(x(t)) − e(t)

]x(t)dt +MT

= λ∫

F1

[g(x(t)) − ε(t)

]x(t)dt + λ

∫

F2

[g(x(t)) − ε(t)

]x(t)dt

+ λ∫T

0[ε(t) − e(t)]x(t)dt +MT

≤∫

F2

∣∣g(x(t)) − ε(t)∣∣ · |x(t)|dt +MT +

∫T

0|ε(t) − e(t)|dt · |x|∞

≤M′T +MT +∫T

0|ε(t) − e(t)|dt · |x|∞,

(2.13)

where

M′ =[

max|x|≤r

∣∣g(x)∣∣ + max

t∈[0,T]|ε(t)|

]· r. (2.14)

By (2.9), we have

|x|∞ ≤ |e| +(M′ +M)T

2+

12

∫T

0|ε(t) − e(t)|dt · |x|∞. (2.15)


Noticing that

∫T

0|ε(t) − e(t)|dt < 2, (2.16)

there exists a constant M′′ > |e| such that

|x|∞ ≤M′′. (2.17)

On the other hand, it follows from

∣∣ϕ(x)

∣∣ · |x| = ϕ(x) · x ≥ α(|x|)|x|, ∀x ∈ R, (2.18)

that α(|x|) ≤ |ϕ(x)| for x /= 0. In addition, since x(0) = x(T), there exists t1 ∈ [0, T] such thatx′(t1) = 0. Thus ϕ(x′(t1)) = 0.

Then, for all t ∈ E := {t ∈ [0, T] : x′(t)/= 0}, we have

α(∣∣x′(t)

∣∣) ≤∣∣ϕ

(x′(t)

)∣∣

=

∣∣∣∣∣

∫ t

t1

(ϕ(x′(s)

))′ds

∣∣∣∣∣

≤∣∣∣∣∣

∫ t

t1

f(x(s)) · x′(s)ds∣∣∣∣∣+∫T

0

∣∣g(x(s))∣∣ds +

∫T

0|e(s)|ds

≤∣∣∣∣∣

∫x(t)

x(t1)f(u)du

∣∣∣∣∣+[

max|x|≤M′′

∣∣g(x)∣∣ + max

t∈[0,T]|e(t)|

]· T

≤∫M′′

−M′′

∣∣f(u)∣∣du +

[max|x|≤M′′

∣∣g(x)∣∣ + max

t∈[0,T]|e(t)|

]· T

≤ max|x|≤M′′

∣∣f(x)∣∣ · 2M′′ +

[max|x|≤M′′ ′

∣∣g(x)∣∣ + max

t∈[0,T]|e(t)|

]· T :=M′′′.

(2.19)

For the above M′′′, it follows from lims→+∞α(s) = +∞ that there exists G > M′′ such that

α(s) > M′′′, s ≥ G. (2.20)

Combining this with α(|x′(t)|) ≤M′′′, we get

∣∣x′(t)∣∣ < G, t ∈ E, (2.21)

which yields that |x′|∞ < G.Now, we have proved that any solution x(t) ∈ C1

T of (2.2) satisfies

|x|∞ < G,∣∣x′

∣∣∞ < G. (2.22)


Since G > |e|, we have

G > e = g−1(e), −G < e = g−1(e). (2.23)

In view of g being strictly decreasing, we get

g(G) < e, g(−G) > e (2.24)

Set

Ω ={x ∈ C1

T : |x|∞ ≤ G,∣∣x′

∣∣∞ ≤ G

}. (2.25)

Then, we know that (2.2) has no solution on ∂Ω for each λ ∈ (0, 1), that is, the assumption(S1) of Lemma 1.1 holds. In addition, it follows from (2.24) that

− 1T

∫T

0

[g(G) − e(t)

]dt = e − g(G) > 0,

− 1T

∫T

0

[g(−G) − e(t)

]dt = e − g(−G) < 0.

(2.26)

So the assumption (S2) of Lemma 1.1 holds. Let

H(x, μ

)= μx −

(1 − μ

) 1T

∫T

0

[g(x) − e(t)

]dt. (2.27)

For x ∈ ∂Ω ∩ R and μ ∈ [0, 1], by (2.24), we have

xH(x, μ

)= μx2 −

(1 − μ

)x

1T

∫T

0

[g(x) − e(t)

]dt

= μx2 +(1 − μ

)x

1T

∫T

0

[e − g(x)

]dt > 0.

(2.28)

Thus, H(x, μ) is a homotopic transformation. So

deg(F,Ω ∩ R, 0) = deg(H(x, 0),Ω ∩ R, 0)

= deg(H(x, 1),Ω ∩ R, 0)

= deg(I,Ω ∩ R, 0)/= 0,

(2.29)

that is, the assumption (S3) of Lemma 1.1 holds. By applying Lemma 1.1, there exists at leastone solution with period T to (1.1).


Uniqueness. Let

ψ(x) =∫x

0f(u)du, y(t) = ϕ

(x′(t)

)+ ψ(x(t)). (2.30)

Then (1.1) is transformed into

x′(t) = ϕ−1[y(t) − ψ(x(t))],

y′(t) = −g(x(t)) + e(t).(2.31)

Let x1(t) and x2(t) being two T -periodic solutions of (1.1); and

yi(t) = ϕ(x′i(t)

)+ ψ(xi(t)), i = 1, 2. (2.32)

Then we obtain

x′i(t) = ϕ−1[yi(t) − ψ(xi(t))],

y′i(t) = −g(xi(t)) + e(t).i = 1, 2. (2.33)

Setting

v(t) = x1(t) − x2(t), u(t) = y1(t) − y2(t), (2.34)

it follows from (2.33) that

v′(t) = ϕ−1[y1(t) − ψ(x1(t))]− ϕ−1[y2(t) − ψ(x2(t))

],

u′(t) = −[g(x1(t)) − g(x2(t))

].

(2.35)

Now, we claim that

u(t) ≤ 0, ∀t ∈ R. (2.36)

If this is not true, we consider the following two cases.

Case 1. There exists t2 ∈ (0, T) such that

u(t2) = maxt∈[0,T]

u(t) = maxt∈R

u(t) > 0, (2.37)

which implies that

u′(t2) = −[g(x1(t2)) − g(x2(t2))

]= 0,

u′′(t2) = −[g(x1(t)) − g(x2(t))

]′|t=t2 = −[g ′(x1(t2))x′1(t2) − g

′(x2(t2))x′2(t2)]≤ 0.

(2.38)


By (A1), g ′(x) < 0. So it follows from g(x1(t2)) − g(x2(t2)) = 0 that x1(t2) = x2(t2). Thus, inview of

−g ′(x1(t2)) > 0, u(t2) = y1(t2) − y2(t2) > 0, (2.39)

and (H1), we obtain

u′′(t2) = −g ′(x1(t2))[x′1(t2) − x′2(t2)

]

= −g ′(x1(t2)){ϕ−1[y1(t2) − ψ(x1(t2))

]− ϕ−1[y2(t2) − ψ(x2(t2))

]}

= −g ′(x1(t2)){ϕ−1[y1(t2) − ψ(x1(t2))

]− ϕ−1[y2(t2) − ψ(x1(t2))

]}> 0,

(2.40)

which contradicts with u′′(t2) ≤ 0.

Case 2.

u(0) = maxt∈[0,T]

u(t) = maxt∈R

u(t) > 0. (2.41)

Also, we have u′(0) = 0 and u′′(0) ≤ 0. Then, similar to the proof of Case 1, one can get acontradiction.

Now, we have proved that

u(t) ≤ 0, ∀t ∈ R. (2.42)

Analogously, one can show that

u(t) ≥ 0, ∀t ∈ R. (2.43)

So we have u(t) ≡ 0. Then, it follows from (2.35) that

g(x1(t)) − g(x2(t)) ≡ 0, ∀t ∈ R, (2.44)

which implies that

x2(t) ≡ x1(t), ∀t ∈ R. (2.45)

Hence, (1.1) has a unique T -periodic solution. The proof of Theorem 2.1 is now completed.

Remark 2.2. In Theorem 2.1, setting ε(t) ≡ e(t), then (A2) becomes as follows:

(A2′) there exists a constant r ≥ 0 such that for all t ∈ R and |x| > r,

x[g(x) − e(t)

]< 0. (2.46)


In the case ϕ(s) = ϕp(s), Liu [7, Theorem 1] proved that (1.1) has a unique T -periodic solutionunder the assumptions (A1) and (A2′). Thus, even for the case of ϕ(s) = ϕp(s), Theorem 2.1is a generalization of [7, Theorem 1].

In addition, we have the following interesting corollary.

Corollary 2.3. Suppose (A1) and

(A2′′) there exist a constant α ≥ 0 such that

∫T

0

[g(α) − e(t)

]dt ≤ 0,

∫T

0

∣∣g(α) − e(t)

∣∣dt < 2 (2.47)

hold. Then (1.1) has a unique T -periodic solution.

Proof. Let ε(t) ≡ g(α). Noticing that

x[g(x) − g(α)

]< 0, |x| > α, (2.48)

we know that (A2) holds with r = α. This completes the proof.

At last, we give two examples to illustrate our results.

Example 2.4. Consider the following nonlinear equation:

(ϕ(x′(t)

))′ + f(x(t))x′(t) + g(x(t)) = e(t), (2.49)

where ϕ(x) = 2xex2 − 2x, f(x) = e−x, g(x) = −(x3 + x), and e(t) = sin t. One can easily check

that ϕ satisfy (H1) and (H2). Obviously, (A1) holds. Moreover, since

limx→+∞

g(x) = −∞, limx→−∞

g(x) = +∞, (2.50)

it is easy to verify that (A2) holds. By Theorem 2.1, (2.49) has a unique 2π-periodic solution.

Example 2.5. Consider the following p-Laplacian equation:

(ϕp

(x′(t)

))′ + g(x(t)) = e(t), (2.51)

where g(x) = −(1/2) arctanx, and e(t) = sin2t. Obviously, (A1) holds. Moreover, we have

∫π

0

[g(0) − e(t)

]dt ≤ 0,

∫π

0

∣∣g(0) − e(t)∣∣dt =

∫π

0sin2t =

π

2< 2. (2.52)

So (A2′′) holds. Then, by Corollary 2.3, (2.51) has a unique π-periodic solution.


Remark 2.6. In Example 2.5, ∀r > 0, we have

x(g(x) − e

(π2

))> r

(1 − π

4

)> 0, ∀x < −r. (2.53)

Thus, (A2′) does not hold. So [7, Theorem 1] cannot be applied to Example 2.5. This meansthat our results generalize [7, Theorem 1] in essence even for the case of ϕ(s) = ϕp(s).

Acknowledgments

The authors are grateful to the referee for valuable suggestions and comments, whichimproved the quality of this paper. The work was supported by the NSF of China, the NSFof Jiangxi Province of China (2008GQS0057), the Youth Foundation of Jiangxi ProvincialEducation Department (GJJ09456), and the Youth Foundation of Jiangxi Normal University.

References

[1] S. Lu, “New results on the existence of periodic solutions to a p-Laplacian differential equation witha deviating argument,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 1107–1123,2007.


[3] B. Liu, “Periodic solutions for Lienard type p-Laplacian equation with a deviating argument,” Journalof Computational and Applied Mathematics, vol. 214, no. 1, pp. 13–18, 2008.

[4] W.-S. Cheung and J. Ren, “Periodic solutions for p-Laplacian Lienard equation with a deviatingargument,” Nonlinear Analysis: Theory, Methods & Applications, vol. 59, no. 1-2, pp. 107–120, 2004.

[5] W.-S. Cheung and J. Ren, “Periodic solutions for p-Laplacian differential equation with multipledeviating arguments,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 4, pp. 727–742,2005.

[6] W.-S. Cheung and J. Ren, “Periodic solutions for p-Laplacian Rayleigh equations,” Nonlinear Analysis:Theory, Methods & Applications, vol. 65, no. 10, pp. 2003–2012, 2006.




Research ArticleStructure of Eigenvalues of Multi-Point BoundaryValue Problems

Jie Gao,1 Dongmei Sun,2 and Meirong Zhang3

1 School of Mathematics and Information Sciences, Weifang University, Weifang, Shandong 261061, China2 College of Applied Science and Technology, Hainan University, Haikou, Hainan 571101, China3 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Correspondence should be addressed to Meirong Zhang, [email protected]

Received 7 January 2010; Revised 19 March 2010; Accepted 29 March 2010

Academic Editor: Gaston M. N’Guerekata

Copyright q 2010 Jie Gao et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The structure of eigenvalues of −y′′+q(x)y = λy, y(0) = 0, and y(1) =∑m

k=1 αky(ηk), will be studied,where q ∈ L1([0, 1],R), α = (αk) ∈ R

m, and 0 < η1 < · · · < ηm < 1. Due to the nonsymmetry of theproblem, this equation may admit complex eigenvalues. In this paper, a complete structure of allcomplex eigenvalues of this equation will be obtained. In particular, it is proved that this equationhas always a sequence of real eigenvalues tending to +∞. Moreover, there exists some constantAq > 0 depending on q, such that when α satisfies ‖α‖ ≤ Aq, all eigenvalues of this equation arenecessarily real.

1. Introduction

In the recent years, multi-point boundary value problems of ordinary differential equationshave received much attention. Some remarkable results have been obtained, especially forthe existence and multiplicity of (positive) solutions for nonlinear second-order ordinarydifferential equations [1–10]. However, as noted in [5, 6], although it is important in manynonlinear problems, the corresponding eigenvalue theory for linear problems is incomplete.The main reason is that the linear operators are no longer symmetric with respect to multi-point boundary conditions.

In this paper, we will establish some fundamental results for eigenvalue theory ofmulti-point boundary value problems. Precisely, for a real potential q ∈ L1

R:= L1([0, 1],R),

we consider the eigenvalue problem

−y′′ + q(x)y = λy, x ∈ [0, 1], (1.1)


associated with the (m + 2)-point boundary condition

y(0) = 0, y(1) −m∑

k=1

αky(ηk)= 0. (1.2)

Here m ∈ N and the boundary data are α = (α1, . . . , αm) ∈ Rm and

η =(η1, . . . , ηm

)∈ Δm :=

{(η1, . . . , ηm

)∈ R

m : 0 < η1 < η2 < · · · < ηm < 1}. (1.3)

As usual, λ ∈ C is called an eigenvalue of (1.1) and (1.2) if (1.1) has a nonzero complex solutiony = y(x) satisfying conditions of (1.2). The set of all eigenvalues of problem (1.1) and (1.2) isdenoted by Σq

α,η ⊂ C, called the spectrum.When α = (αk) = 0, boundary condition (1.2) is reduced to the Dirichlet boundary

condition

y(0) = y(1) = 0. (1.4)

Problem (1.1)–(1.4) is symmetric and has only real eigenvalues [11, 12]. However, in caseα/= 0, problem (1.1) and (1.2) is not symmetric, thus Σq

α,η may contain nonreal eigenvalues. Asimple example is given by Example 2.1.

When q(x) ≡ 0, (1.1) is

−y′′ = λy, x ∈ [0, 1]. (1.5)

Eigenvalues of problem (1.5)–(1.2) can be analyzed using elementary method, because allsolutions of (1.5) can be found explicitly. However, as far as the authors know, even for thissimple eigenvalue problem, the spectrum theory is incomplete in the literature. In [5, 6], Maand O’Regan have constructed all real eigenvalues of problem (1.5)–(1.2) when all ηk arerational, and α = (αk) satisfies certain nondegeneracy condition. In [8, 9], Rynne has obtainedall real eigenvalues for general η ∈ Δm. See [13] for further extension.

The main topic of this paper is the structure of Σqα,η. Much attention will be paid to the

real eigenvalues due to important applications in nonlinear problems.

Theorem 1.1. Given q ∈ L1Rand (α, η) ∈ R

m × Δm, then Σqα,η is composed of a sequence {λn =

λn,α,η(q)}n∈N ⊂ C which satisfies

Re λ1 ≤ Re λ2 ≤ · · · ≤ Re λn ≤ · · · , limn→+∞

Re λn = +∞. (1.6)

Theorem 1.2. Given q ∈ L1Rand (α, η) ∈ R

m ×Δm, then Σqα,η ∩ R = {λn = λn,α,η(q)}n∈N, where

λ1 < λ2 < · · · < λn < · · · , limn→+∞

λn = +∞. (1.7)

For α ∈ Rm, the norm is ‖α‖ = ‖α‖l1 :=

∑mk=1 |αk|. For q ∈ L1

R, the L1 norm is denoted by

‖q‖ := ‖q‖L1[0,1]. With some restrictions on α, we are able to prove that Σqα,η contains only real

eigenvalues.


Theorem 1.3. If α ∈ Rm satisfies ‖α‖ ≤ 1/2, then the spectrum Σq

α,η contains at most finitely manynonreal eigenvalues.

Theorem 1.4. Given q ∈ L1R, there exists some constant A(‖q‖) > 0, depending on the norm ‖q‖

only, such that if α ∈ Rm satisfies ‖α‖ ≤ A(‖q‖), then one has Σq

α,η ⊂ R.

To sketch our proofs, let us denote

(Mq

)= Problem (1.1) (1.2),

(M0) = Problem (1.4) (1.2),

(D0) = Problem (1.4) (1.3).

(1.8)

Basically, eigenvalues of (Mq) are zeros of some entire functions. See (2.24) and (3.3). In orderto study the distributions of eigenvalues, we will consider (Mq) as a perturbation of (M0) orof (D0). To obtain the existence of infinitely many real eigenvalues as in Theorem 1.2, someproperties of almost periodic functions [14, 15] will be used. See Lemmas 2.3 and 3.2. Inorder to pass the results of Σ0

α,η to general potentials q, many techniques like implicit functiontheorem and the Rouche theorem will be exploited. Moreover, some basic estimates in [11] forfundamental solutions of (1.1) play an important role, especially in the proofs of Theorems1.3 and 1.4. Due to the non-symmetry of problem (Mq), the proofs are complicated than thatin [11] where the Dirichlet problem is considered.

The paper is organized as follows. In Section 2, we will give some detailed analysis onproblem (M0). In Section 3, after developing some basic estimates, we will prove Theorems1.1 and 1.2. In Section 4, we will develop some techniques to exclude nonreal eigenvalues andcomplete the proofs of Theorems 1.3 and 1.4. Some open problem on the spectrum of (Mq)will be mentioned.

2. Structure of Eigenvalues of the Zero Potential

In order to motivate our consideration for Σqα,η with non-zero potentials q, in this section we

consider the spectrum Σ0α,η with the zero potential.

2.1. An Example of Nonreal Eigenvalues

Let m = 1. Boundary condition (1.2) is the following three-point boundary condition:

y(0) = 0, y(1) − αy(η)= 0, (2.1)

where α ∈ R and η ∈ Δ1 = (0, 1). We consider the eigenvalue problems (1.5)–(2.1).Let λ ∈ C. Complex solutions y(x) of (1.5) satisfying y(0) = 0 are y(x) = cSλ(x), c ∈ C,

where

Sλ(x) :=sin√λx√λ

=∞∑

k=0

(−1)k

(2k + 1)!λkx2k+1, x ∈ [0, 1]. (2.2)


Notice that Sλ(x) is an entire function of λ ∈ C. Define

T0(λ) := Sλ(1) − αSλ(η)=

sin√λ√

λ−α sinη

√λ

√λ

, λ ∈ C . (2.3)

Obviously, T0(λ) depends on the boundary data (α, η) as well. Then λ ∈ Σ0α,η if and only if λ

satisfies

T0(λ) = 0. (2.4)

Example 2.1. Let η = 1/3. By (2.3) and (2.4), λ = ω2 ∈ Σ0α,1/3 if and only if ω satisfies

T0(λ) = −4sin(ω/3)

ω

(sin2ω

3− (3 − α)

4

)= 0. (2.5)

That is, either

sin(ω/3)ω

= 0 (2.6)

or

sin2ω

3=

3 − α4

(2.7)

Equation (2.6) shows that Σ0α,1/3 always contains positive eigenvalues (3nπ)2, n ∈ N.

Equation (2.7) has real solutions ω if and only if α ∈ [−1, 3]. In this case, Σ0α,1/3 consists

of non-negative eigenvalues. More precisely,

α ∈ [−1, 3) =⇒ Σ0α,1/3 ⊂ (0,∞),

α = 3 =⇒ Σ0α,1/3 ⊂ [0,∞), 0 ∈ Σ0

α,1/3.(2.8)

Equation (2.7) has nonreal solutions ω if and only if α ∈ (−∞,−1)∪ (3,∞). In this case,we have

α ∈ (−∞,−1) ∪ (3,∞) =⇒ Σ0α,1/3 \ R/= ∅. (2.9)

For example, one has

α ∈ (−∞,−1) =⇒ λ =

(3π2

+ 3i log√−1 − α +

√3 − α

2

)2

∈ Σ0α,1/3, (2.10)


α ∈ (3,+∞) =⇒ λ =

(

3π + 3i log√α − 3 +

√α + 1

2

)2

∈ Σ0α,1/3. (2.11)

Notice that all eigenvalues obtained from (2.7) can be constructed explicitly as (2.10)and (2.11). For example, Σ0

α,1/3 contains negative eigenvalues if and only if α ∈ (3,∞).Moreover, in this case, one has the unique negative eigenvalue given by

λ = −9

(

log√α − 3 +

√α + 1

2

)2

. (2.12)

For more details, see [5, 6, 8].

Results (2.10) and (2.11) show that to guarantee that Σ0α,η contains only real

eigenvalues, some restrictions on parameters (α, η) are necessary.

2.2. Real Eigenvalues with General Parameters

In the following we consider general α ∈ Rm, based on properties of almost periodic functions

[14, 15].

Definition 2.2. Suppose that f : R → R is a bounded continuous function. One calls that fis almost periodic, if for any ε > 0, there exists lε > 0 such that for any a ∈ R, there existsb = ba,ε ∈ [a, a + lε] such that

∥∥f(· + b) − f(·)∥∥L∞ := sup

u∈R

∣∣f(u + b) − f(u)∣∣ < ε. (2.13)

Any almost periodic function f admits a well-defined mean value

f := limT→+∞

1T

∫T

0f(u)du ∈ R. (2.14)

To study (M0) and (Mq), let us prove some properties on almost periodic functions.

Lemma 2.3. Let f : R → R be an almost periodic function.

(i) For any A ∈ R, one has

infu∈[A,∞)

f(u) = infu∈R

f(u), (2.15)

supu∈[A,∞)

f(u) = supu∈R

f(u). (2.16)


(ii) Assume that f is non-zero and f = 0. Then f(u) is oscillatory as u → +∞, that is,

∀A ∈ R ∃u1, u2 > A s.t. f(u1)f(u2) < 0. (2.17)

In particular, f(u) has a sequence of positive zeros tending to +∞.

Proof. (i) Let us only prove (2.15) because (2.16) is similar. For any ε > 0, choose a0 ∈ R suchthat

f0 ≤ f(a0) < f0 + ε, f0 := infu∈R

f(u) ∈ R. (2.18)

By (2.13), there exists b0 ∈ [a0, a0 + lε] such that ‖f(· + b0) − f(·)‖L∞ < ε. For any A ∈ R, let ustake

a = max(a0, A) + lε. (2.19)

By (2.13) again, there exists b ∈ [a, a + lε] such that ‖f(· + b) − f(·)‖L∞ < ε. Hence

∥∥f(· + b) − f(· + b0)∥∥L∞ < 2ε. (2.20)

In particular,

∣∣f((a0 − b0) + b) − f((a0 − b0) + b0)∣∣ ≤∥∥f(· + b) − f(· + b0)

∥∥L∞ < 2ε. (2.21)

By the choice of a, one has u0 := a0 − b0 + b ≥ a − lε ≥ A. Hence

f0 ≤ infu∈[A,∞)

f(u) ≤ f(u0) < f(a0) + 2ε < f0 + 3ε. (2.22)

This proves (2.15).(ii) If f /= 0 and f has mean value 0, it is easy to see that

infu∈R

f(u) < 0 < supu∈R

f(u). (2.23)

Now result (2.17) can be deduced simply from (2.15) and (2.16).

Like (2.3) and (2.4), all eigenvalues λ ∈ C of problem (M0) are determined by thefollowing equation:

M0(λ) = 0, (2.24)

where

M0(λ) := Sλ(1) −m∑

k=1

αkSλ(ηk)=

sin√λ√

λ−

m∑

k=1

αk sinηk√λ

√λ

, λ ∈ C. (2.25)


Notice that M0(λ) is an entire function of λ ∈ C. Hence (2.24) has only isolated zeros in C.For u, v ∈ R, we have the following elementary equalities:

sin(u + iv) = sin u cosh v + i cos u sinh v, |sin(u + iv)|2 = sin2u + sinh2v. (2.26)

For real eigenvalues of problem (M0), we have the following result.

Lemma 2.4. Given (α, η) ∈ Rm ×Δm, then Σ0

α,η ∩ R = {λn = λn,α,η}n∈N, where

λ1 < · · · < λn < · · · , limn→∞

λn = +∞. (2.27)

Proof. Let us first consider possible positive eigenvalues λ = u2 of (M0), where u > 0. By thefirst equality of (2.26), equation (2.24) is the same as

F(u) = Fα,η(u) := sinu −m∑

k=1

αk sinηku = 0. (2.28)

The function Fα,η(u) is a non-zero, almost periodic function and has mean value 0. In fact,Fα,η(u) is quasiperiodic. By Lemma 2.3(ii), Fα,η has infinitely many positive zeros tending to+∞. See Figure 1. Hence Σ0

α,η contains a sequence of positive eigenvalues tending to +∞.Next we consider possible negative eigenvalues λ = −u2 of (M0), where u > 0. In this

case, (2.24) is the same as

F(u) = Fα,η(u) := sinh u −m∑

k=1

αk sinh ηku = 0. (2.29)

See the first equality of (2.26). Notice that F(u) is analytic in u. As ηk ∈ (0, 1), one has

limu→+∞

F(u)sinhu

= 1. (2.30)

Thus (2.29) has at most finitely many positive solutions. Hence Σ0α,η contains at most finitely

many negative eigenvalues.As both (2.28) and (2.29) have only isolated solutions, the above two cases show that

all real eigenvalues of (M0) can be listed as in (2.27).

The quasi-periodic function Fα,η(u) is as in Figure 1.

2.3. Nonexistence of Nonreal Eigenvalues

To study real eigenvalues of problem (M0), the authors of [5, 6, 8] have imposed somerestrictions on α = (αk) ∈ R

m. The typical conditions are

αk > 0, ∀k ∈ {1, 2, . . . , m}, ‖α‖ < 1. (2.31)


With some restrictions on α = (αk), we will prove that Σ0α,η consists of only real eigenvalues.

Lemma 2.5. Suppose that α = (αk) ∈ Rm satisfies

‖α‖l2 :=

(m∑

k=1

α2k

)1/2

≤ 1√m. (2.32)

Then Σ0α,η contains only real eigenvalues. Moreover, one has Σ0

α,η ⊂ ((π/2)2,+∞).

Proof. When α = 0, problem (1.5)–(1.2) is the Dirichlet problem and Σ0α,η = {(nπ)

2 : n ∈ N}. Inthe following, assume that α/= 0.

Suppose that λ = ω2 ∈ Σ0α,η, where ω = u + iv, u, v ∈ R. We assert that v = 0 under

assumption (2.32). Otherwise, assume that v /= 0. By (2.26), equation (2.24) is the followingsystem for (u, v) ∈ R

2:

sinh v cos u =m∑

k=1

αk sinh ηkv cos ηku, cosh v sin u =m∑

k=1

αk cosh ηkv sin ηku. (2.33)

It follows from the Holder inequality that

1 = cos2u + sin2u

=

(m∑

k=1

αk sinh ηkv

sinh vcosηku

)2

+

(m∑

k=1

αk coshηkvcoshv

sinηku

)2

≤(

m∑

k=1

α2k

sinh2ηkv

sinh2v

)(m∑

k=1

cos2ηku

)

+

(m∑

k=1

α2k

cosh2ηkv

cosh2v

)(m∑

k=1

sin2ηku

)

< ‖α‖2l2 ·

m∑

k=1

cos2ηku + ‖α‖2l2 ·

m∑

k=1

sin2ηku

= m‖α‖2l2 ,

(2.34)

which is impossible under assumption (2.32). Thus v = 0 and therefore λ = u2 ≥ 0.Next, by (2.2), (2.25), and the Holder inequality, we have

M0(0) = 1 −m∑

k=1

αkηk ≥ 1 − ‖α‖l2∥∥η∥∥l2 > 0, (2.35)

because ‖α‖l2 ≤ 1/√m and ‖η‖l2 <

√m. By (2.24), 0/∈Σ0

α,η. Hence we have∑0

α,η ⊂ (0,∞).


−5

−4

−3

−2

−1

0

1

2

3

4

5

0 50 100 150 200

Fα,η(u)

u

Figure 1: Function Fα,η(u) and its positive zeros, where m = 1, α = 4 and η = 1/(2π).

Finally, by the Holder inequality, assumption (2.32) implies that ‖α‖ ≤√m · ‖α‖l2 ≤ 1.

For any u ∈ (0, π/2], the function Fα,η(u) of (2.28) satisfies

Fα,η(u) ≥ sin u −m∑

k=1

|αk| sin ηku

> sin u −(

m∑

k=1

|αk|)

sin u

≥ 0.

(2.36)

Hence (2.28) shows that Σ0α,η ⊂ ((π/2) 2,∞).

Remark 2.6. Condition (2.32) is sharp. For example, let m = 1 and η = 1/3. Example 2.1 showsthat Σ0

α,η contains nonreal eigenvalues if α < −1. Similarly, by letting m = 1 and η = 1/5, onecan verify that Σ0

α,η contains nonreal eigenvalues when α > 1.

3. Structure of Eigenvalues of Non-Zero Potentials

Given q ∈ L1R

and complex parameter λ ∈ C, the fundamental solutions of (1.1) are denotedby yk(x, λ, q), k = 1, 2. That is, they are solutions of (1.1) satisfying the initial values

y1(0, λ, q

)= y′2

(0, λ, q

)= 1, y′1

(0, λ, q

)= y2

(0, λ, q

)= 0. (3.1)

Notice that yk(x, λ, q) are entire functions of λ ∈ C. See [11]. To study (Mq), let us introduce

Mq(λ) := y2(1, λ, q

)−

m∑

k=1

αky2(ηk, λ, q

), λ ∈ C, (3.2)


which is an entire function of λ ∈ C. See (2.25) for the case q = 0. Notice that Mq(λ) is real forλ ∈ R. Then λ ∈ Σq

α,η if and only if

Mq(λ) = 0. (3.3)

3.1. Basic Estimates

Lemma 3.1. Given β ∈ (0, 1), one has

limv∈R

|v|→+∞

| sin(u + iv)|exp |v| =

12, lim

v∈R|v|→+∞

∣∣sin β(u + iv)

∣∣

exp|v| = 0 (3.4)

uniformly in u ∈ R.

Proof. Suppose that u, v ∈ R. We have from (2.26)

|sin(u + iv)| =√

sin2u + sinh2v ∈ [sinh |v|, coshv],

∣∣sin β(u + iv)∣∣ =√

sin2βu + sinh2βv ≤ cosh βv ≤ exp(β|v|).

(3.5)

The uniform limits in (3.4) are evident.

For the function F(u) = Fα,η(u) of (2.28), one has the following result on its amplitude.

Lemma 3.2. Given (α, η) ∈ Rm × Δm, there exist a constant cα,η > 0 and a sequence {an}n∈N of

increasing positive numbers such that an ↑ +∞ and

(−1)nF(an) ≥ cα,η ∀n ∈ N. (3.6)

Proof. Recall that F(u) is quasi-periodic and has the mean value 0. Denote that

cα,η :=12

max

(

supu∈R

F(u),−infu∈R

F(u)

)

. (3.7)

Then cα,η > 0. The construction for {an}n∈N is as follows. By (2.15), one has some a1 ∈ (0,∞)such that F(a1) ≤ −cα,η. By letting A = a1 + 1 in (2.16), we have some a2 ∈ [a1 + 1,∞) suchthat F2(a2) ≥ cα,η. Then, by letting A = a2 + 1 in (2.15), we have some a3 ∈ [a2 + 1,∞) suchthat F2(a3) ≤ −cα,η. Inductively, we can use (2.15) and (2.16) to find a sequence {an}n∈N suchthat an ≥ a1 + n − 1, and (3.6) is satisfied for all n ∈ N.


Lemma 3.3 (basic estimates, [11, page 13, Theorem 3]). Let q ∈ L1Rand λ ∈ C. There hold the

following estimates for all x ∈ [0, 1]

∣∣y1(x, λ, q

)− Cλ(x)

∣∣ ≤ 1∣∣∣√λ∣∣∣

exp(∣∣∣Im√λ∣∣∣x +

∥∥q∥∥L1[0,x]

), (3.8)

∣∣y2(x, λ, q

)− Sλ(x)

∣∣ ≤ 1|λ| exp

(∣∣∣Im√λ∣∣∣x +

∥∥q∥∥L1[0,x]

), (3.9)

∣∣y′1(x, λ, q

)− C′λ(x)

∣∣ ≤∥∥q∥∥ exp

(∣∣∣Im√λ∣∣∣x +

∥∥q∥∥L1[0,x]

), (3.10)

∣∣y′2(x, λ, q

)− S′λ(x)

∣∣ ≤

∥∥q∥∥

∣∣∣√λ∣∣∣

exp(∣∣∣Im√λ∣∣∣x +

∥∥q∥∥L1[0,x]

). (3.11)

Remark 3.4. For their purpose, the authors of [11] have proved (3.8)–(3.11) for complexpotentials q ∈ L2

C:= L2([0, 1],C). For example, in (3.8)–(3.11), the terms ‖q‖ and ‖q‖L1[0,x]

are replaced by ‖q‖L2[0,1] and ‖q‖L2[0,1] ·√x, respectively in [11]. Inspecting their proofs,

especially the proof of [11, pages 7–9, Theorem 1], one can find that estimates (3.8)–(3.11)are also true for L1 potentials q. Moreover, these estimates can be established even for linearmeasure differential equations with general measures [16]. By the Holder inequality, one has

∥∥q∥∥ =∥∥q∥∥L1[0,1] ≤

∥∥q∥∥L2[0,1],

∥∥q∥∥L1[0,x] ≤

∥∥q∥∥L2[0,1] ·

√x. (3.12)

This is why the authors of [11] have used these terms in (3.8)–(3.11).

Lemma 3.5. There holds the following estimate forMq(λ):

∣∣Mq(λ) −M0(λ)∣∣ ≤

B exp(|Imω|)|ω|2

, ω :=√λ ∈ C, (3.13)

where

B = B(α, q)

:= (1 + ‖α‖) exp(∥∥q∥∥) ∈ [1,∞). (3.14)

Proof. Define ϕ(x) := y2(x, λ, q) − Sλ(x). From (3.9), we have

∣∣ϕ(x)∣∣ ≤ exp

(∥∥q∥∥)|ω|−2 exp(|Imω|) ∀x ∈ [0, 1]. (3.15)


By (2.25) and (3.2), we have

∣∣Mq(λ) −M0(λ)

∣∣ =

∣∣∣∣∣ϕ(1) −

m∑

k=1

αkϕ(ηk)∣∣∣∣∣

≤∣∣ϕ(1)

∣∣ +

m∑

k=1

|αk|∣∣ϕ(ηk)∣∣

≤ (1 + ‖α‖) exp(∥∥q∥∥)|ω|−2 exp(|Imω|).

(3.16)

This gives (3.13).

Lemma 3.6. One hasMq(λ)/≡ 0 on R. Consequently, there exists λ0 ∈ R such that λ0 /∈Σqα,η.

Proof. Otherwise, we have Mq(λ) ≡ 0 on R. Notice that

M0

(u2)≡ F(u)

u, u > 0. (3.17)

Let λ = a2n in (3.13), where {an}n∈N is as in Lemma 3.2. We have

∣∣∣∣F(an)an

∣∣∣∣ =∣∣∣M0

(a2n

)∣∣∣ ≤B

a2n

∀n ∈ N. (3.18)

Hence limn→+∞|F(an)| ≤ B/an → 0, a contradiction with (3.6).

3.2. Eigenvalues with General Parameters

The most general results on spectrum Σqα,η of (Mq) are stated as in Theorem 1.1.

Proof of Theorem 1.1. We argue as in general spectrum theory [12]. By Lemma 3.6, there existsλ0 ∈ R such that λ0 /∈Σ

qα,η. That is, the following equation:

−y′′ + q(x)y − λ0y = 0 (3.19)

has only the trivial solution y = 0 satisfying boundary condition (1.2). Let G0(x, u) be theGreen function associated with problem (3.19)-(1.2). Then λ ∈ Σq

α,η if and only if λ/=λ0 and

−y′′ +(q(x) − λ0

)y = (λ − λ0)y (3.20)

has nontrivial solution y(x) satisfying (1.2). In other words, λ ∈ Σqα,η if and only if the

following equation:

y = (λ − λ0)Lqy (3.21)


has non-trivial solution y, where

Lqy(x) :=∫1

0G0(x, u)

(q(u) − λ0

)y(u)du. (3.22)

Since Lq is a compact linear operator, one sees that this happens when and only when 1/(λ −λ0) ∈ σ(Lq) ⊂ C, where σ(Lq) is the spectrum of Lq. Hence Σq

α,η consists of a sequence ofeigenvalues which can accumulate only at infinity of C.

For λ ∈ C, denote that

λ = ω2, ω =√λ = u + iv, u, v ∈ R. (3.23)

Suppose that λ ∈ Σqα,η and λ/= 0. Then Mq(λ) = 0 and (3.13) implies that

B exp|v||ω|2

≥∣∣Mq(λ) −M0(λ)

∣∣ = |M0(λ)|

=

∣∣∣∣∣sinω −

∑mk=1 αk sinηkωω

∣∣∣∣∣

≥|sin(u + iv)| −

∑mk=1|αk|

∣∣sinηk(u + iv)∣∣

|ω| .

(3.24)

We conclude that all non-zero eigenvalues λ ∈ Σqα,η satisfy

|ω| ·|sin(u + iv)| −

∑mk=1|αk|


exp|v| ≤ B. (3.25)

Let us derive some consequences from estimate (3.25) for λ ∈ Σqα,η.

(i) Since |ω| ≥ |v|, it follows from the uniform limits in (3.4) that

lim|v|=|Im ω|→+∞

|ω| ·|sin(u + iv)| −

∑mk=1|αk|


exp|v| = +∞. (3.26)

Thus there exists some h = hα,η,q > 0 such that

λ ∈ Σqα,η =⇒ ω =

√λ ∈ Hh := {ω ∈ C : |Im ω| < h}. (3.27)

The horizontal strip Hh of (3.27) in the ω-plane is transformed by (3.23) to the followinghalf-plane Pr in the λ-plane:

Σqα,η⊂ Pr := {λ ∈ C : Re λ > r}, where r := −h2. (3.28)


(ii) Let r > −h2. We assert that

Σqα,η ∩ {λ ∈ C : Re λ ≤ r} = Σq

α,η ∩{λ ∈ C : −h2 < Re λ ≤ r

}(3.29)

contains at most finitely many eigenvalues. Otherwise, suppose that

Σqα,η ∩

{λ ∈ C : −h2 < Re λ ≤ r

}(3.30)

contains infinitely many λn, n ∈ N. Since (3.3) has only isolated solutions, we havenecessarily | Im λn| → +∞. By denoting

√λn = un + ivn, one has

−h2 < u2n − v2

n ≤ r, 2|un||vn| −→ +∞. (3.31)

In particular, |vn| → +∞. Now estimate (3.25) reads as

|sin(un + ivn)|exp|vn|

≤∑m

k=1|αk|∣∣sin ηk(un + ivn)

∣∣

exp|vn|+ o(1) asn −→ ∞. (3.32)

This is impossible because we have the uniform limits (3.4).Combining (i) and (ii), we know that Σq

α,η can be listed as in (1.6).

Though problem (Mq) is not symmetric, Σqα,η always contains infinitely many real

eigenvalues, as stated in Theorem 1.2.

Proof of Theorem 1.2. We need to only consider positive eigenvalues of (Mq). Let λ = a2n in

(3.13), where {an}n∈N is as in Lemma 3.2. By using (3.17), we have

∣∣∣∣Mq

(a2n

)− F(an)

an

∣∣∣∣ ≤B

a2n

∀n ∈ N. (3.33)

Since an ↑ +∞, w.l.o.g., we can assume that an ≥ 2B/cα,η for all n ∈ N. Thus

∣∣∣anMq

(a2n

)− F(an)

∣∣∣ ≤B

an≤cα,η

2∀n ∈ N. (3.34)

By using (3.6), we conclude that

(−1)nMq

(a2n

)> 0 ∀n ∈ N. (3.35)

Hence (3.3) has at least one positive solution λn in each interval (a2n, a

2n+1), n ∈ N. Combining

with Theorem 1.1, Σqα,η ∩ R consists of a sequence of real eigenvalues tending to +∞. Hence

Σqα,η ∩ R can be listed as in (1.7).


4. Nonexistence of Nonreal Eigenvalues for Small α

We will apply the Rouche theorem to give further results on Σqα,η when ‖α‖ is small, following

the approach in [11] for the Dirichlet problem (1.1)-(1.4), which corresponds to (Mq) withα = 0. Let us recall the Rouche theorem.

Lemma 4.1 (Rouche theorem). Suppose that f(z), g(z) are entire functions of z ∈ C. If |g(z)| <|f(z)| on a Jordan curveC, then f(z) and f(z)+g(z) have the same number of zeros insideC, countedmultiplicities.

For later use, let us introduce the following elementary function:

G(ω) :=

√(sin2u + sinh2v

)

coshv=

√

1 −(

cos ucosh v

)2

, ω = u + iv ∈ C.(4.1)

Then G(ω) ≡ G(ω + π). Obviously, G(ω) = 0 if and only if ω = nπ , n ∈ Z. Define

G(r) := minω ∈ Cr

G(ω) ∈ [0, 1], r ∈ [0,∞), (4.2)

where Cr is the circle in the ω-plane

Cr := {ω = u + iv ∈ C : |ω| = r}. (4.3)

Then G(nπ) = 0, n ∈ Z+ := {0, 1, . . . , n, . . .}, and 0 < G(r) < 1 for all r ∈ [0,∞) \ πZ

+.Let r0 be the unique solution of the following equation:

tanh r = sin r, r ∈ (0, π). (4.4)

Numerically, r0.= 1.8751 .= 0.5968π . The following facts can be verified by elementary

arguments.

Lemma 4.2. One has

G(r) =

⎧⎨

⎩

tanh r for r ∈ [0, r0],

sin r for r ∈ [r0, π],

limn→+∞

G

((n +

12

)π

)= 1.

(4.5)

For the graph of G(r), see Figure 2.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20

Figure 2: The function G(r), where r ∈ [0, 7π].

r0

nπ (n + 1)π0

h0

Figure 3: Finding zeros of Mq(ω2) in the ω-plane.

4.1. Large Eigenvalues

In the following we apply the Rouche theorem to study the spectrum Σqα,η, that is, the zeros of

the function Mq(λ) in the λ-plane. To this end, we consider problem (Mq) as a perturbationof the Dirichlet problem (D0), whose eigenvalues are zeros of the function

D0(λ) :=sin√λ√λ

, λ ∈ C. (4.6)

Let λ = ω2. Equation (3.3) is the same as

D0

(ω2)+(Mq

(ω2)−D0

(ω2))

= 0, ω ∈ C, (4.7)


which is considered as a perturbation of the following equation:

D0

(ω2)=

sinωω

= 0, ω ∈ C. (4.8)

Due to the form of (4.7) and (4.8), one needs to only consider solutions in the right half-planeC+ of ω. Notice that all solutions of (4.8) are nπ , n ∈ N, which are simple zeros of D0(ω2). Forany α/= 0, we do not know whether all zeros of (4.7) are real. In order to overcome this, theproof is complicated than that in [11].

Let us derive another consequence from estimate (3.25) with some restriction on α =(αk). Suppose that α ∈ R

m satisfies ‖α‖ < 1. Define the positive function

W(h) =W(h;α, q

) def=B exph

sinh h − ‖α‖ cosh h, h ∈ (arctanh ‖α‖,∞), (4.9)

where B = B(α, q) is as in (3.14). Then W(h) is decreasing in h ∈ (arctanh ‖α‖,∞).

Lemma 4.3. Suppose that

‖α‖ < 1, h > arctanh‖α‖. (4.10)

Then for any λ = ω2 ∈ Σqα,η, where η ∈ Δm, one has

either |Im ω| < h, or |ω| ≤W(h). (4.11)

Proof. We keep the notations in (3.23) Let λ = ω2 ∈ Σqα,η. If |v| = | Im ω| ≥ h, it follows from

(2.26) and (3.25) that

B exp|v||ω| ≥ |sin(u + iv)| −

m∑

k=1

|αk|∣∣sin ηk(u + iv)

∣∣

=√

sin2u + sinh2v −m∑

k=1

|αk|√

sin2ηku + sinh2ηkv

≥ sinh|v| −m∑

k=1

|αk|√

1 + sinh2v

= sinh|v| − ‖α‖ coshv.

(4.12)

Using the function W(·) in (4.9), we obtain |ω| ≤W(|v|) ≤W(h). This proves (4.11).

Consider the following circles of the ω-plane:

Cn,r := {ω ∈ C : |ω − nπ | = r} = nπ + Cr, n ∈ Z, r ∈ (0,∞) \ πN. (4.13)


Lemma 4.4. Let (n, r) be as in (4.13). one has

∣∣∣∣∣Mq(λ) −D0(λ)

D0(λ)

∣∣∣∣∣≤ 1G(r)

(‖α‖ + 2B

|ω|

)∀ω =

√λ ∈ Cn,r . (4.14)

Proof. Let ω = nπ + u + iv ∈ Cn,r , where u + iv ∈ Cr . Then sinω/= 0. By (2.26) we have

∣∣∣∣

sinηkωsinω

∣∣∣∣ =

√sin2ηk(nπ + u) + sinh2ηkv√

sin2u + sinh2v

≤√

1 + sinh2v√

sin2u + sinh2v

=1

G(u + iv)≤ 1G(r)

.

(4.15)

See (4.1) and (4.2). Notice that

∣∣∣∣∣Mq(λ) −D0(λ)

D0(λ)

∣∣∣∣∣≤ |M0(λ) −D0(λ)|

|D0(λ)|+

∣∣Mq(λ) −M0(λ)∣∣

|D0(λ)|=: T1 + T2. (4.16)

By (2.25) and (4.15), we have

T1 ≤m∑

k=1

|αk|∣∣∣∣

sinηkωsinω

∣∣∣∣ ≤‖α‖G(r)

. (4.17)

Since

exp(|Im ω|)|sinω| =

exp|v|coshv

coshv√

sin2u + sinh2v=

exp|v|/ coshv

G(u + iv)≤ 2G(r)

, (4.18)

Compared with (4.1) and (4.2), it follows from (3.13) that

T2 ≤exp(|Im ω|)|sinω|

B

|ω| ≤2B

G(r)|ω| .(4.19)

Thus one has (4.14).

Proof of Theorem 1.3. Let

h0def=

√

r20 −(π

2

)2 .= 1.0240, (4.20)


where the constant r0 ∈ (π/2, π) is as in Lemma 4.2. One has sin r0.= 0.9541 and tanh h0

.=0.7714. Denote by Dn,r0 the disc enclosed by the circle Cn,r0 , that is,

Dn,r0 := {ω ∈ C : |ω − nπ | < r0}, n ∈ Z. (4.21)

Since r0 > π/2, Dn,r0 intersects Dn+1,r0 . See Figure 3.In the following, we always assume that α ∈ R

m satisfies

‖α‖ ≤ 12< min(sin r0, tanh h0). (4.22)

Suggested by (3.14), (4.9), and (4.11), we denote

c0def=

(1 + 1/2) exp h0

sinh h0 − (1/2) cosh h0=

3 exp h0

2 sinh h0 − cosh h0

.= 9.7873. (4.23)

Then, for all α as in (4.22), by (4.9) one has

W(h0;α, q

)=

(1 + ‖α‖) exp h0

sinh h0 − ‖α‖ cosh h0exp(∥∥q∥∥) ≤ c0 exp

(∥∥q∥∥). (4.24)

Suppose that

λ0 = ω20 ∈ Σq

α,η, |ω0| ≥ 11 exp(∥∥q∥∥). (4.25)

Let us show that λ0 must be positive. In fact, (4.25) implies that |ω0| ≥ 11 exp(‖q‖) >W(h0;α, q). By result (4.11), we have | Im ω0| < h0. Hence ω0 is a zero of Mq(ω2) insidesome disc Dn, r0 . See Figure 3. W.l.o.g., let us assume that n > 0. Then n satisfies

nπ ≥ |ω0| − |ω0 − nπ | > 11 exp(∥∥q∥∥) − r0. (4.26)

For any ω ∈ Cn, r0 , one has

|ω| ≥ nπ − |ω − nπ | ≥ nπ − r0 > 11 exp(∥∥q∥∥) − 2r0. (4.27)


It follows from (4.5) that G(r0) = sinh r0 = sin r0. By (4.14), we have the estimate

∣∣∣∣∣Mq

(ω2) −D0

(ω2)

D0(ω2)

∣∣∣∣∣≤ 1G(r0)

(

‖α‖ +2(1 + ‖α‖) exp

(∥∥q∥∥)

|ω|

)

<1

sinh r0

(12+

3 exp(∥∥q∥∥)

11 exp(∥∥q∥∥) − 2r0

)

≤ 1sinh r0

(12+

311 − 2r0

).= 0.9578

< 1.

(4.28)

Since (4.8) has the unique, simple zero ω = nπ in Dn,r0 , by the Rouche theorem, we concludefrom estimate (4.28) that (4.7) has the unique, simple zero ω = ωn in Dn,r0 . Furthermore,denote that

ε± :=Mq

((nπ ± r0)2

)−D0

((nπ ± r0)2

)

D0

((nπ ± r0)2

) ∈ (−1, 1). (4.29)

See (4.28). We have

Mq

((nπ ± r0)2

)= (1 + ε±)D0

((nπ ± r0)2

)=±(−1)n(1 + ε±) sin r0

nπ ± r0. (4.30)

Thus

Mq

((nπ − r0)2

)·Mq

((nπ + r0)2

)= − (1 + ε+)(1 + ε−)sin2r0

(nπ)2 − r20

< 0. (4.31)

Hence (4.7) has at least one real solution in the interval (nπ − r0, nπ + r0) ⊂ Dn,r0 . Due to theuniqueness, all eigenvalues λ0 as in (4.25) must be positive.

Finally, it follows from Theorem 1.1 that Σqα,η contains at most finitely many λ which

do not satisfy (4.25). Thus the proof of Theorem 1.3 is completed.

4.2. Small Eigenvalues

In order to prove Theorem 1.4, we need to show that all “small eigenvalues” are also realprovided that ‖α‖ is small. The proof below is a modification of the proof of Theorem 1.3.


Proof of Theorem 1.4. By (4.4), we can fix some n = n‖q‖ ∈ N such that

n ≥11 exp

(∥∥q∥∥)

π− 1

2, (4.32)

1G((n + 1/2)π)

(12+

3 exp(∥∥q∥∥)

(n + 1/2)π

)

< 1. (4.33)

Denote that

r = r‖q‖ :=((

n +12

)π

)2

, D = D‖q‖ := {λ ∈ C : |λ| < r}. (4.34)

In the following we assume that α ∈ Rm satisfies (4.22), that is, ‖α‖ ≤ 1/2. From the

proof of Theorem 1.3, Σqα,η ∩ (C\D) consists of positive eigenvalues. See conditions (4.25) and

(4.32). Moreover, for λ ∈ ∂D, that is, |√λ| = (n + 1/2)π , we obtain from estimate (4.14) and

condition (4.33) that

∣∣∣∣∣Mq(λ) −D0(λ)

D0(λ)

∣∣∣∣∣≤ 1G((n + 1/2)π)

(

‖α‖ +2(1 + ‖α‖) exp

(∥∥q∥∥)

(n + 1/2)π

)

≤ 1G((n + 1/2)π)

(12+

3 exp(∥∥q∥∥)

(n + 1/2)π

)

< 1.

(4.35)

Notice that equation

D0(λ) =sin√λ√

λ= 0, λ ∈ D, (4.36)

has (simple) solutions λ = (nπ)2, n = 1, . . . ,n. By the Rouche theorem, we conclude that, if‖α‖ ≤ 1/2, the following problem:

Mq(λ, α) = 0, λ ∈ D, (4.37)

has precisely n solutions, counted multiplicity. Here Mq(λ) has been written as Mq(λ, α) toemphasize the dependence on α.

Suppose that α = 0. Equation (4.37) corresponds to the Dirichlet eigenvalue problem(1.1)–(1.4), which has only real eigenvalues. Moreover, all solutions of (4.37) are simple inthis case [11]. Hence solutions of problem (4.37) can be denoted by λ = μn, 1 ≤ n ≤ n, where

−r < μ1 < · · · < μn < r. (4.38)

They are the first n eigenvalues of problem (1.1)–(1.4).


In the following, we apply the implicit function theorem to prove that solutions of(4.37) inside D are actually real when ‖α‖ is small. Notice that Mq(λ, α) is a smooth real-valued function of (λ, α) ∈ R

2. By [11, page 21, Theorem 6], the derivative of Mq(λ, α) w.r.t. λis

∂λMq(λ, α) =∫1

0y2(t, λ, q

)(y1(1, λ, q

)y2(t, λ, q

)− y2

(1, λ, q

)y1(t, λ, q

))dt

−m∑

k=1

αk

∫ηk

0y2(t, λ, q

)(y1(ηk, λ, q

)y2(t, λ, q

)− y2

(ηk, λ, q

)y1(t, λ, q

))dt.

(4.39)

In particular,

∂λMq(λ, α)|(λ,α)=(μn,0) = an∫1

0y2

2(t, μn, q

)dt − bn

∫1

0y1(t, μn, q

)y2(t, μn, q

)dt, (4.40)

where

an := y1(1, μn, q

), bn := y2

(1, μn, q

). (4.41)

Since μn is a Dirichlet eigenvalue of problem (1.1), we have bn = 0. Moreover, the Liouvilletheorem for (1.1) implies that

1 = det(y1(1, μn, q

)y2(1, μn, q

)

y′1(1, μn, q

)y′2(1, μn, q

))

= det(y1(1, μn, q

)0

y′1(1, μn, q

)y′2(1, μn, q

)). (4.42)

In particular, an /= 0. Hence

∂λMq(λ, α)|(λ,α)=(μn,0) = an∫1

0y2

2(t, μn, q

)dt /= 0. (4.43)

Now the implicit function theorem is applicable to (4.37). In conclusion, there exist someconstant A = Aq,η > 0 and a continuously differentiable real-valued functions λn(α) of α suchthat

λn(0) = μn, Mq(λn(α), α) ≡ 0, ‖α‖ ≤ Aq,η, 1 ≤ n ≤ n. (4.44)

Due to (4.38)–(4.44) and the continuity of λn(α), one can assume that

−r < λ1(α) < · · · < λn(α) < r ∀ ‖α‖ ≤ Aq,η. (4.45)

Thus {λn(α)}1≤n≤n are different eigenvalues of (Mq) located in the interval (−r, r). Since (4.37)has precisely n solutions inside D, we conclude that all solutions of (4.37) inside D arenecessarily real. Now we have proved that Σq

α,η ⊂ R for all ‖α‖ ≤ Aq,η.


Notice that the constant Aq,η in (4.44) is constructed from the implicit functiontheorem. Generally speaking, Aq,η depends on η ∈ (0, 1) and all information of the potentialq ∈ L1

R. However, during the application of the implicit function theorem to (4.37), the

derivatives of ∂αλn(α) can be well controlled using estimates in [11], like (3.8)–(3.11). It ispossible to choose some Aq,η such that it depends on the norm ‖q‖ only. We will not givethe detailed construction. Note that this has been already observed for large eigenvalues. Forexample, n and r depend only on the norm ‖q‖ of q.

We end the paper with an open problem. Given (α, η) ∈ Rm ×Δm, for any q ∈ L1

R, due

to Theorem 1.2, problem (Mq) has always a sequence of real eigenvalues λn(q) = λn,α,η(q)which tends to +∞. In applications of eigenvalues to nonlinear problems, the smallest (real)eigenvalues λ1(q) are of great importance. The main reason is that solutions of problem (1.1)–(1.2) are oscillatory only when λ > λ1(q). As for the smallest eigenvalue of the Dirichletproblem (1.1)–(1.4), denoted by λ1(q), one has the following variational characterization:

λ1(q)= inf

y∈H10 (0,1), y /= 0

∫10

(y′2 + q(x)y2

)dx

∫10 y

2 dx. (4.46)

An open problem is what is the characterization like (4.46) for the smallest eigenvalue λ1(q)of (Mq). Once this is clear, some results on nonlinear problems in [5, 6, 8] can be extended byusing eigenvalues of (Mq).

Finally, let us remark that the approaches in this paper also can be applied to othermulti-point boundary conditions like

y′(0) = 0, y(1) −m∑

k=1

αky(ηk)= 0 (4.47)

or to more general Stieltjes boundary conditions [17]. In this sense, eigenvalue theory can beestablished for these nonsymmetric problems.

Acknowledgments

The third author is supported by the Major State Basic Research Development Program (973Program) of China (no. 2006CB805903), the Doctoral Fund of Ministry of Education of China(no. 20090002110079), the Program of Introducing Talents of Discipline to Universities (111Program) of Ministry of Education and State Administration of Foreign Experts Affairs ofChina (2007), and the National Natural Science Foundation of China (no. 10531010). Theauthors would like to express their thanks to Ping Yan for her help during the preparation ofthe paper.

References

[1] D. R. Anderson and R. Ma, “Second-order n-point eigenvalue problems on time scales,” Advances inDifference Equations, vol. 2006, Article ID 59572, 17 pages, 2006.


[2] R. P. Agarwal and I. Kiguradze, “On multi-point boundary value problems for linear ordinarydifferential equations with singularities,” Journal of Mathematical Analysis and Applications, vol. 297,no. 1, pp. 131–151, 2004.

[3] C. P. Gupta, S. K. Ntouyas, and P. Ch. Tsamatos, “On an m-point boundary-value problem for second-order ordinary differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 23, no.11, pp. 1427–1436, 1994.

[4] B. Liu and J. Yu, “Solvability of multi-point boundary value problems at resonance. I,” Indian Journalof Pure and Applied Mathematics, vol. 33, no. 4, pp. 475–494, 2002.

[5] R. Ma, “Nodal solutions for a second-order m-point boundary value problem,” CzechoslovakMathematical Journal, vol. 56(131), no. 4, pp. 1243–1263, 2006.

[6] R. Ma and D. O’Regan, “Nodal solutions for second-order m-point boundary value problems withnonlinearities across several eigenvalues,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64,no. 7, pp. 1562–1577, 2006.

[7] F. Meng and Z. Du, “Solvability of a second-order multi-point boundary value problem at resonance,”Applied Mathematics and Computation, vol. 208, no. 1, pp. 23–30, 2009.

[8] B. P. Rynne, “Spectral properties and nodal solutions for second-order, m-point, boundary valueproblems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 12, pp. 3318–3327, 2007.

[9] B. P. Rynne, “Second-order, three-point boundary value problems with jumping non-linearities,”Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 11, pp. 3294–3306, 2008.

[10] M. Zhang and Y. Han, “On the applications of Leray-Schauder continuation theorem to boundaryvalue problems of semilinear differential equations,” Annals of Differential Equations, vol. 13, no. 2, pp.189–207, 1997.

[11] J. Poschel and E. Trubowitz, The Inverse Spectrum Theory, Academic Press, New York, NY, USA, 1987.[12] A. Zettl, Sturm-Liouville Theory, vol. 121 of Mathematical Surveys and Monographs, American

Mathematical Society, Providence, RI, USA, 2005.[13] B. P. Rynne, “Spectral properties of second-order, multi-point, p-Laplacian boundary value

problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 11, pp. 4244–4253, 2010.[14] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377, Springer,

Berlin, Germany, 1974.[15] J. K. Hale, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 2nd edition, 1969.[16] G. Meng, Continuity of solutions and eigenvalues in measures with weak∗ topology, Ph.D. dissertation,

Tsinghua University, Beijing, China, 2009.[17] M. Garcıa-Huidobro, R. Manasevich, P. Yan, and M. Zhang, “A p-Laplacian problem with a multi-

point boundary condition,” Nonlinear Analysis: Theory, Methods & Applications, vol. 59, no. 3, pp. 319–333, 2004.


Research ArticleAsymptotically Almost PeriodicSolutions for Abstract Partial NeutralIntegro-Differential Equation

Jose Paulo C. dos Santos,1 Sandro M. Guzzo,2and Marcos N. Rabelo3

1 Departamento de Ciencias Exatas, Universidade Federal de Alfenas, Rua Gabriel Monteiro da Silva,700. 37130-000 Alfenas-MG, Brazil

2 Colegiado do curso de Matematica-UNIOESTE, Rua Universitaria, 2069, Caixa Postal 711,85819-110 Cascavel-PR, Brazil

3 Departamento de Matematica, Universidade Federal de Pernambuco, Av. Prof. Luiz Freire,Cidade Universitaria, 50740-540 Recife-PE, Brazil

Correspondence should be addressed to Jose Paulo C. dos Santos, [email protected]

Received 24 November 2009; Revised 23 February 2010; Accepted 24 February 2010

Academic Editor: Toka Diagana

Copyright q 2010 Jose Paulo C. dos Santos et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The existence of asymptotically almost periodic mild solutions for a class of abstract partial neutralintegro-differential equations with unbounded delay is studied.

1. Introduction

In this paper, we study the existence of asymptotically almost periodic mild solutions for aclass of abstract partial neutral integro-differential equations modelled in the form

d

dt

(x(t) + f(t, xt)

)= Ax(t) +

∫ t

0B(t − s)x(s)ds + g(t, xt), (1.1)

where A : D(A) ⊂ X → X and B(t) : D(B(t)) ⊂ X → X, t ≥ 0, are closed linear operators;(X, ‖ · ‖) is a Banach space; the history xt : (−∞, 0] → X, xt(θ) = x(t + θ), belongs to someabstract phase space B defined axiomatically f, g : I × B → X are appropriated functions.

The study of abstract neutral equations is motivated by different practical applicationsin different technical fields. The literature related to ordinary neutral functional differential


equations is very extensive and we refer the reader to Chukwu [1], Hale and Lunel [2], Wu[3], and the references therein. As a practical application, we note that the equation

d

dt

[

u(t) − λ∫ t

−∞C(t − s)u(s)ds

]

= Au(t) + λ∫ t

−∞B(t − s)u(s)ds − p(t) + q(t) (1.2)

arises in the study of the dynamics of income, employment, value of capital stock, andcumulative balance of payment; see [1] for details. In the above system, λ is a real number,the state u(t) ∈ R

n, C(·), B(·) are n × n continuous functions matrices, A is a constant n × nmatrix, p(·) represents the government intervention, and q(·) the private initiative. We notethat by assuming the solution u is known on (−∞, 0], we can transform this system into anabstract system with unbounded delay described as (1.1).

Abstract partial neutral differential equations also appear in the theory of heatconduction. In the classic theory of heat conduction, it is assumed that the internal energyand the heat flux depend linearly on the temperature u and on its gradient ∇u. Underthese conditions, the classic heat equation describes sufficiently well the evolution of thetemperature in different types of materials. However, this description is not satisfactory inmaterials with fading memory. In the theory developed in [4, 5], the internal energy and theheat flux are described as functionals of u and ux. The next system, see for instance [6–9], hasbeen frequently used to describe this phenomenon,

d

dt

[

u(t, x) +∫ t

−∞k1(t − s)u(s, x)ds

]

= cΔu(t, x) +∫ t

−∞k2(t − s)Δu(s, x)ds,

u(t, x) = 0, x ∈ ∂Ω.

(1.3)

In this system, Ω ⊂ Rn is open, bounded, and with smooth boundary; (t, x) ∈ [0,∞) × Ω;

u(t, x) represents the temperature in x at the time t; c is a physical constant ki : R → R,i = 1, 2, are the internal energy and the heat flux relaxation, respectively. By assuming thatthe solution u is known on (−∞, 0] and k2 ≡ 0, we can transform this system into an abstractsystem with unbounded delay described in the form (1.1).

Recent contributions on the existence of solutions with some of the previouslyenumerated properties or another type of almost periodicity to neutral functional differentialequations have been made in [10, 11], for the case of neutral ordinary differential equations,and in [12–15] for partial functional differential systems.

The purpose of this work is to study the existence of asymptotically almost periodicmild solutions for the neutral system (1.1). To this end, we study the existence and qualitativeproperties of an exponentially stable resolvent operator for the integro-differential system

dx(t)dt

= Ax(t) +∫ t

0B(t − s)x(s)ds, t ≥ 0,

x(0) = z ∈ X.(1.4)

There exists an extensive literature related to the existence and qualitative properties ofresolvent operator for integro-differential equations. We refer the reader to the book by


Gripenberg et al. [16] which contains an overview of the theory for the case where theunderlying spaceX has finite dimension. For abstract integro-differential equations describedon infinite dimensional spaces, we cite the Pruss book [17] and the papers [18–20], Da Pratoet al. [21, 22], and Lunardi [9, 23]. To finish this short description of the related literature,we cite the papers [24–26] where some of the above topics for the case of abstract neutralintegro-differential equations with unbounded delay are treated.

To the best of our knowledge, the study of the existence of asymptotically almostperiodic solutions of neutral integro-differential equations with unbounded delay describedin the abstract form (1.1) is an untreated topic in the literature and this is the main motivationof this article.

To finish this section, we emphasize some notations used in this paper. Let (Z, ‖ · ‖Z)and (W, ‖·‖W) be Banach spaces. We denote byL(Z,W) the space of bounded linear operatorsfrom Z into W endowed with norm of operators, and we write simplyL(Z) when Z =W . ByR(Q), we denote the range of a map Q, and for a closed linear operator P : D(P) ⊆ Z → W ,the notation [D(P)] represents the domain of P endowed with the graph norm, ‖z‖1 = ‖z‖Z +‖Pz‖W , z ∈ D(P). In the case Z = W , the notation ρ(P) stands for the resolvent set of P, andR(λ, P) = (λI − P)−1 is the resolvent operator of P . Furthermore, for appropriate functionsK : [0,∞) → Z and S : [0,∞) → L(Z,W), the notation K denotes the Laplace transform ofK and S∗K the convolution between S andK, which is defined by S∗K(t) =

∫ t0S(t−s)K(s)ds.

The notation Br(x,Z) stands for the closed ball with center at x and radius r > 0 in Z. Asusual, C0([0,∞), Z) represents the subspace of Cb([0,∞), Z) formed by the functions whichvanish at infinity.

2. Preliminaries

In this work, we will employ an axiomatic definition of the phase space B similar to that in[27]. More precisely, B will denote a vector space of functions defined from (−∞, 0] into Xendowed with a seminorm denoted by ‖ · ‖B and such that the following axioms hold.

(A) If x : (−∞, σ + b) → X, with b > 0, is continuous on [σ, σ + b) and xσ ∈ B, then foreach t ∈ [σ, σ + b) the following conditions hold:

(i) xt is in B,

(ii) ‖x(t)‖ ≤ H‖xt‖B,

(iii) ‖xt‖B ≤ K(t − σ) sup{‖x(s)‖ : σ ≤ s ≤ t} +M(t − σ)‖xσ‖B, where H > 0 is aconstant, and K,M : [0,∞) → [1,∞) are functions such that K(·) and M(·) arerespectively continuous and locally bounded, and H,K,M are independent ofx(·).

(A1) If x(·) is a function as in (A), then xt is a B-valued continuous function on [σ, σ +b).

(B) The space B is complete.

(C2) If (ϕn)n∈N is a sequence in Cb((−∞, 0], X) formed by functions with compactsupport such that ϕn → ϕ uniformly on compact, then ϕ ∈ B and ‖ϕn − ϕ‖B → 0as n → ∞.


Remark 2.1. In the remainder of this paper, L > 0 is such that

∥∥ϕ∥∥B ≤ L sup

θ≤0

∥∥ϕ(θ)

∥∥

(2.1)

for every ϕ : (−∞, 0] → X continuous and bounded; see [27, Proposition 7.1.1] for details.

Definition 2.2. Let S(t) : B → B be theC0-semigroup defined by S(t)ϕ(θ) = ϕ(0) on [−t, 0] andS(t)ϕ(θ) = ϕ(t+θ) on (−∞,−t]. The phase space B is called a fading memory if ‖S(t)ϕ‖B → 0as t → ∞ for each ϕ ∈ B with ϕ(0) = 0.

Remark 2.3. In this work, we suppose that there exists a positive K such that

max{K(t),M(t)} ≤ K (2.2)

for each t ≥ 0. Observe that this condition is verified, for example, if B is a fading memory,see [27, Proposition 7.1.5].

Example 2.4 (The phase space Cr × Lp(ρ,X)). Let r ≥ 0, 1 ≤ p < ∞, and let ρ : (−∞,−r] →R be a nonnegative measurable function which satisfies the conditions (g-5) and (g-6) inthe terminology of [27]. Briefly, this means that ρ is locally integrable, and there exists anonnegative, locally bounded function γ on (−∞, 0] such that ρ(ξ + θ) ≤ γ(ξ)ρ(θ), for allξ ≤ 0 and θ ∈ (−∞,−r) \ Nξ, where Nξ ⊆ (−∞,−r) is a set with Lebesgue measure zero.The space Cr × Lp(ρ,X) consists of all classes of functions ϕ : (−∞, 0] → X such that ϕ iscontinuous on [−r, 0], Lebesgue-measurable, and ρ‖ϕ‖p is Lebesgue integrable on (−∞,−r).The seminorm in Cr × Lp(ρ,X) is defined by

‖ϕ‖B : sup{‖ϕ(θ)‖ : −r ≤ θ ≤ 0

}+(∫−r

−∞ρ(θ)∥∥ϕ(θ)

∥∥pdθ)1/p

. (2.3)

The space B = Cr × Lp(ρ;X) satisfies axioms (A), (A-1), and (B). Moreover, when r = 0 and

p = 2, we can take H = 1, M(t) = γ(−t)1/2, and K(t) = 1 + (∫0−tρ(θ)dθ)

1/2, for t ≥ 0; see [27,

Theorem 1.3.8] for details.Now, we need to introduce some concepts, definitions, and technicalities on almost

periodical functions.

Definition 2.5. A function f ∈ C(R, Z) is almost periodic (a.p.) if for every ε > 0, there exists arelatively dense subset of R, denoted byH(ε, f, Z), such that

∥∥f(t + ξ) − f(t)∥∥Z < ε, t ∈ R, ξ ∈ H

(ε, f, Z

). (2.4)

Definition 2.6. A function f ∈ C([0,∞), Z) is asymptotically almost periodic (a.a.p.) if thereexists an almost periodic function g(·) and w ∈ C0([0,∞), Z) such that f(·) = g(·) +w(·).

The next lemmas are useful characterizations of a.p and a.a.p functions.


Lemma 2.7 (see [28, Theorem 5.5]). A function f ∈ C([0,∞), Z) is asymptotically almost periodicif and only if for every ε > 0 there exist L(ε, f, Z) > 0 and a relatively dense subset of [0,∞), denotedby T(ε, f, Z), such that

∥∥f(t + ξ) − f(t)

∥∥Z < ε, t ≥ L

(ε, f, Z

), ξ ∈ T

(ε, f, Z

). (2.5)

In this paper, AP(Z) and AAP(Z) are the spaces

AP(Z) ={f ∈ C(R, Z) : f is a.p.

},

AAP(Z) ={f ∈ C([0,∞), Z) : f is a.a.p.

} (2.6)

endowed with the norms |‖u‖|Z = sups∈R‖u(s)‖Z and ‖u‖Z = sups≥0‖u(s)‖Z, respectively. Weknow from the result in [28] that AP(Z) and AAP(Z) are Banach spaces.

Next, (Z, ‖ · ‖Z) and (W, ‖ · ‖W) are abstract Banach spaces.

Definition 2.8. Let Ω be an open subset of W.

(a) A continuous function f ∈ C(R × Ω, Z) (resp., f ∈ C([0,∞) × Ω;Z)) is calledpointwise almost periodic (p.a.p.), (resp., pointwise asymptotically almost periodic(p.a.a.p.) if f(·, x) ∈ AP(Z) (resp., f(·, x) ∈ AAP(Z)) for every x ∈ Ω.

(b) A continuous function F ∈ C(R×Ω, Z) is called uniformly almost periodic (u.a.p.),if for every ε > 0 and every compact K ⊂ Ω there exists a relatively dense subset ofR, denoted byH(ε, f,K,Z), such that

∥∥f(t + ξ, y

)− f(t, y)∥∥

Z ≤ ε,(t, ξ, y

)∈ R ×H

(ε, f,K,Z

)×K. (2.7)

(c) A continuous function f : C([0,∞)×Ω, Z) is called uniformly asymptotically almostperiodic (u.a.a.p.), if for every ε > 0 and every compact K ⊂ Ω there exists arelatively dense subset of [0,∞), denoted byT(ε, f,K,Z), and L(ε, f,K,Z) > 0 suchthat

∥∥f(t + ξ, y) − f(t, y)∥∥Z ≤ ε, t ≥ L

(ε, f,K,Z

),(ξ, y)∈ T(ε, f,K,Z

)×K. (2.8)

The next lemma summarizes some properties which are fundamental to obtain ourresults.

Lemma 2.9 (see [29, Theorem 1.2.7]). Let Ω ⊂ W be an open set. Then the following propertieshold.

(a) If f ∈ C(R ×Ω, Z) is p.a.p. and satisfies a local Lipschitz condition at x ∈ Ω, uniformly att, then f is u.a.p.

(b) If f ∈ C([0,∞) × Ω, Z) is p.a.a.p. and satisfies a local Lipschitz condition at x ∈ Ω,uniformly at t, then f is u.a.a.p.

(c) If x ∈ AP(X), then t → xt ∈ AP(B). Moreover, if B is a fading memory space andz ∈ C(R, X) is such that z0 ∈ B and z ∈ AAP(X), then t → zt ∈ AAP(B).


(d) If f ∈ C(R ×Ω, Z) is u.a.p. and y ∈ AP(W) is such that {y(t) : t ∈ R}W⊂ Ω, then t →

f(t, y(t)) ∈ AP(Z).

(e) If f ∈ C([0,∞) × Ω, Z) is u.a.a.p and y ∈ AAP(W) is such that {y(t) : t ∈ R}W⊂ Ω,

then t → f(t, y(t)) ∈ AAP(Z).

3. Resolvent Operators

In this section, we study the existence and qualitative properties of an exponentially resolventoperator for the integro-differential abstract Cauchy problem

dx(t)dt

= Ax(t) +∫ t

0B(t − s)x(s)ds,

x(0) = x ∈ X.(3.1)

The results obtained for the resolvent operator in this section are similar to those thatcan be found, for instance, in the papers [21, 23, 30]. In this paper, we prove the necessaryestimates for the proof of an existence theorem of asymptotically almost periodic solutions for(1.1). For the better comprehension of the subject, we will introduce the following definitions,hypothesis, and results.

We introduce the following concept of resolvent operator for integro-differentialproblem (3.1).

Definition 3.1. A one-parameter family of bounded linear operators (R(t))t≥0 on X is called aresolvent operator of (3.1) if the following conditions are verified.

(a) Function R(·) : [0,∞) → L(X) is strongly continuous and R(0)x = x for all x ∈ X.

(b) For x ∈ D(A), R(·)x ∈ C([0,∞), [D(A)])⋂C1([0,∞), X), and

dR(t)xdt

= AR(t)x +∫ t

0B(t − s)R(s)xds, (3.2)

dR(t)xdt

= R(t)Ax +∫ t

0R(t − s)B(s)xds, (3.3)

for every t ≥ 0,

(c) There exist constants M > 0, β such that ‖R(t)‖ ≤Meβt for every t ≥ 0.

Definition 3.2. A resolvent operator (R(t))t≥0 of (3.1) is called exponentially stable if there existpositive constants M,α such that ‖R(t)‖ ≤Me−αt.

In this work, we always assume that the following conditions are verified.

(H1) The operator A : D(A) ⊆ X → X is the infinitesimal generator of an analyticsemigroup (T(t))t≥0 on X, and there are constants M0 > 0, ω ∈ R, and ϑ ∈ (π/2, π)such that ρ(A) ⊇ Λω,ϑ = {λ ∈ C : λ/=ω, | arg(λ−ω)| < ϑ} and ‖R(λ,A)‖ ≤M0/|λ−ω|for all λ ∈ Λω,ϑ.


(H2) For all t ≥ 0, B(t) : D(B(t)) ⊆ X → X is a closed linear operator, D(A) ⊆ D(B(t)),and B(·)x is strongly measurable on (0,∞) for each x ∈ D(A). There exists b(·) ∈L1

loc(R+) such that b(λ) exists for Re(λ) > 0 and ‖B(t)x‖ ≤ b(t)‖x‖1 for all t > 0 and

x ∈ D(A). Moreover, the operator valued function B : Λω,π/2 → L([D(A)], X) hasan analytical extension (still denoted by B) to Λω,ϑ such that ‖B(λ)x‖ ≤ ‖B(λ)‖ ‖x‖1

for all x ∈ D(A), and ‖B(λ)‖ = O(1/|λ|) as |λ| → ∞.

(H3) There exist a subspaceD ⊆ D(A) dense in [D(A)] and positive constantsCi, i = 1, 2,such that A(D) ⊆ D(A), B(λ)(D) ⊆ D(A), and ‖AB(λ)x‖ ≤ C1‖x‖ for every x ∈ Dand all λ ∈ Λω,ϑ.

In the sequel, for r > 0, θ ∈ (π/2, ϑ), and w ∈ R, set

Λr,ω,θ ={λ ∈ C : λ/=ω, |λ| > r,

∣∣arg(λ −ω)

∣∣ < θ

}, (3.4)

and for ω + Γir,θ

, i = 1, 2, 3, the paths

ω + Γ1r,θ =

{ω + teiθ : t ≥ r

},

ω + Γ2r,θ =

{ω + reiξ : −θ ≤ ξ ≤ θ

},

ω + Γ3r,θ =

{ω + te−iθ : t ≥ r

},

(3.5)

with ω + Γr,θ =⋃3i=1 ω + Γir,θ are oriented counterclockwise. In addition, Ψ(G) is the set

Ψ(G) ={λ ∈ C : G(λ) :=

(λI −A − B(λ)

)−1∈ L(X)

}. (3.6)

We next study some preliminary properties needed to establish the existence of aresolvent operator for the problem (3.1).

Lemma 3.3. There exists r1 > 0 such that Λr1,ω,ϑ ⊆ Ψ(G) and the function G : Λr1,ω,ϑ → L(X) isanalytic. Moreover,

G(λ) = R(λ,A)[I − B(λ)R(λ,A)

]−1, (3.7)

and there exist constantsMi for i = 1, 2, 3 such that

‖G(λ)‖ ≤ M1

|λ −ω| , (3.8)

‖AG(λ)x‖ ≤ M2

|λ −ω| ‖x‖1, x ∈ D(A), (3.9)

‖AG(λ)‖ ≤M3, (3.10)

for every λ ∈ Λr1,ω,ϑ.


Proof. Since

∥∥∥B(λ)R(λ,A)

∥∥∥ ≤∥∥∥B(λ)

∥∥∥‖R(λ,A)‖1

≤

⎛

⎝M0

∥∥∥B(λ)

∥∥∥

|λ −ω| +M0|λ|

∥∥∥B(λ)

∥∥∥

|λ −ω| +∥∥∥B(λ)

∥∥∥

⎞

⎠,

(3.11)

fixed ε < 1, there exists a positive number r1 such that ‖B(λ)R(λ,A)‖ ≤ ε for λ ∈Λr1,ω,ϑ. Consequently, the operator I − B(λ)R(λ,A) has a continuous inverse with ‖(I −B(λ)R(λ,A))−1‖ ≤ 1/(1 − ε). Moreover, for x ∈ X, we have

(λI − B(λ) −A

)R(λ,A)

(I − B(λ)R(λ,A)

)−1x = x, (3.12)

and for x ∈ D(A),

R(λ,A)(I − B(λ)R(λ,A)

)−1(λI − B(λ) −A

)x = x, (3.13)

which shows (3.7) and that Λr1,ω,ϑ ⊆ Ψ(G). Now, from (3.7) we obtain R(G(λ)) ⊆ D(A) and

AG(λ) = (λR(λ,A) − I)(I − B(λ)R(λ,A)

)−1. (3.14)

Consequently,

‖AG(λ)‖ ≤ 11 − ε‖λR(λ,A) − I‖

≤ 11 − ε

(M0 +

M0|ω||λ −ω| + 1

)

≤M3,

(3.15)

the functions G,AG : Λr1,ω,ϑ → L(X) are analytic, and estimates (3.8), and (3.10) are valid.In addition, for x ∈ D(A), we can write

‖AG(λ)x‖ ≤∥∥∥∥AR(λ,A)

(I − B(λ)R(λ,A)

)−1x −AR(λ,A)x

∥∥∥∥ + ‖R(λ,A)Ax‖

=∥∥∥∥[AR(λ,A)

(I −(I − B(λ)R(λ,A)

))] (I − B(λ)R(λ,A)

)−1x

∥∥∥∥ + ‖R(λ,A)Ax‖

=∥∥∥AG(λ)B(λ)R(λ,A)x

∥∥∥ + ‖R(λ,A)Ax‖


≤M3‖B(λ)‖‖R(λ,A)x‖1 + ‖R(λ,A)Ax‖

≤ M2

|λ −ω| ‖x‖1,

(3.16)

for |λ| sufficiently large. This proves (3.9) and completes the proof.

Observation 1. If R(·) is a resolvent operator for (3.1), it follows from (3.3) that R(λ)(λI −A − B(λ))x = x for all x ∈ D(A). Applying Lemma 3.3 and the properties of the Laplacetransform, we conclude that R(·) is the unique resolvent operator for (3.1).

In the remainder of this section, r and θ are numbers such that r > r1 and θ ∈ (π/2, ϑ).Moreover, we denote by C a generic constant that represents any of the constants involvedin the statements of Lemma 3.3 as well as any other constant that arises in the estimate thatfollows. We now define the operator family (R(t))t≥0 by

R(t) =

⎧⎪⎨

⎪⎩

12πi

∫

ω+Γr,θeλtG(λ)dλ, t > 0,

I, t = 0.(3.17)

We will next establish that (R(t))t≥0 is a resolvent operator for (3.1).

Lemma 3.4. The function R(·) is exponentially bounded in L(X).

Proof. If t > 1, from (3.17) and estimate (3.8), we get

‖R(t)‖ ≤ C

π

∫∞

r

et(ω+s cos θ)ds

s+C

2π

∫θ

−θet(ω+r cos ξ)dξ

≤(

C

πr|cos θ| +Cθ

πert)eωt.

(3.18)

On the other hand, using that G(·) is analytic on Λr,ω,θ, for t ∈ (0, 1), we obtain

‖R(t)‖ =∥∥∥∥∥

12πi

∫

ω+Γr/t,θeλtG(λ)dλ

∥∥∥∥∥

≤ C

π

∫∞

r/t

et(ω+s cos θ)ds

s+C

2π

∫θ

−θetω+r cos ξdξ

≤(C

π

∫∞

r

eu cos θ du

u+C

2π

∫θ

−θer cos ξdξ

)

eωt

≤(

C

πr|cos θ| +Cθ

πer)eωt.

(3.19)

This complete the proof.


Lemma 3.5. The operator function R(·) is exponentially bounded in L([D(A)]).

Proof. It follows from (3.9) that the integral in

S(t) =1

2πi

∫

ω+Γr,θeλtAG(λ)dλ, t > 0, (3.20)

is absolutely convergent in L([D(A)], X) and defines a linear operator S(t) ∈ L([D(A)], X).Using that A is closed, we can affirm that S(t) = AR(t). From Lemma 3.3, G : Λr,ω,ϑ →L([D(A)]) is analytic and ‖G(λ)‖1 ≤ C|λ −ω|−1. If t > 1 and x ∈ D(A), we have

‖AR(t)x‖ ≤(C

π

∫∞

r

et(ω+s cos θ)ds

s+C

2π

∫θ

−θet(ω+r cos ξ)dξ

)

‖x‖1

≤(

C

πr|cos θ| +Cθ

πert)eωt‖x‖1.

(3.21)

For t ∈ (0, 1) and x ∈ D(A), we get

‖AR(t)x‖ =∥∥∥∥∥

12πi

∫

ω+Γr/t,θeλtAG(λ)xdλ

∥∥∥∥∥

≤(C

π

∫∞

r/t

et(ω+s cos θ)ds

s+C

2π

∫θ

−θetω+r cos ξdξ

)

‖x‖1

≤(

C

πr|cos θ| +Cθ

πer)eωt‖x‖1.

(3.22)

From before and Lemma 3.4, we infer that R(·) is exponentially bounded in L([D(A)]). Theproof is finished.

Lemma 3.6. The function R : [0,∞) → L(X) is strongly continuous.

Proof. It is clear from (3.17) thatR(·)x is continuous at t > 0 for every x ∈ X. We next establishthe continuity at t = 0. Let ω ≥ 0 and N be sufficiently large, using that

12πi

∫

ω+Γr,θλ−1eλtdλ = lim

N→∞

12πi

∫

{ω+Γr,θ : |r|≤N}∪ ω+CN,θ

λ−1eλtdλ = 1, (3.23)

where ω + CN,θ represent the curve ω +Neiξ for θ ≤ ξ ≤ 2π − θ.For x ∈ D(A) and 0 < t ≤ 1, we get

R(t)x − x =1

2πi

∫

ω+Γr,θ

(eλtG(λ)x − λ−1eλtx

)dλ

=1

2πi

∫

ω+Γr,θeλtλ−1G(λ)

(A + B(λ)

)xdλ.

(3.24)


Furthermore, it follows from (3.8), and assumption (H2) that

∥∥∥eλtλ−1G(λ)

(A + B(λ)

)x∥∥∥ ≤

eω+rC

|λ||λ −ω| = Φ(λ), (3.25)

where Φ(·) is integrable for λ ∈ ω+Γr,θ. From the Lebesgue dominated convergence theorem,we infer that

limt→ 0+

(R(t)x − x) 12πi

∫

ω+Γr,θλ−1G(λ)

(A + B(λ)

)xdλ. (3.26)

Let now ω +CL,θ be the curve ω + Leiξ for θ ≤ ξ ≤ 2π − θ. Turning to apply Cauchy’s theoremcombining with the estimate

∥∥∥∥∥

∫

ω+CL,θ

λ−1G(λ)(A + B(λ)

)xdλ

∥∥∥∥∥≤ CθL

(L −ω)2, (3.27)

we obtain

12πi

∫

ω+Γr,θλ−1G(λ)

(A + B(λ)

)xdλ

= limL→∞

12πi

∫

{ω+Γr,θ : |r|≤L}∪ ω+CL,θ

λ−1G(λ)(A + B(λ)

)xdλ = 0,

(3.28)

we can affirm that limt→ 0+(R(t)x − x) = 0 for all x ∈ D(A), which completes and the proofsince D(A) is dense in X and R(·) is bounded on [0, 1].

Notice that ω < 0, the sectors Λr,0,ϑ ⊆ Λr,ω,ϑ, from Lemma 3.3, G : Λr,ω,ϑ → L(X) isanalytic. Consider the contours

γ1 = {λ = s − iN sin(θ) : cos(θ) ≤ s ≤ ω + cos(θ)},

γ2 = {λ = s + iN sin(θ) : ω + cos(θ) ≤ s ≤ cos(θ)},

ω + ΓNr,θ ={ω + teiθ : r ≤ t ≤N

}∪{ω + reiξ : −θ ≤ ξ ≤ θ

}∪{ω + te−iθ : r ≤ t ≤N

},

0 + ΓNr,θ ={teiθ : r ≤ t ≤N

}∪{reiξ : −θ ≤ ξ ≤ θ

}∪{te−iθ : r ≤ t ≤N

},

(3.29)

andRN = ω+ΓNr,θ∪γ2∪0+ΓN

r,θ∪γ1, oriented counterclockwise. By Cauchy theorem for 0 < t ≤ 1,

we obtain

∫

RN

eλtG(λ)dλ = 0. (3.30)


The following estimate:

∥∥∥∥∥

∫

γ1

eλtG(λ)dλ

∥∥∥∥∥≤∫ω+cos(θ)

cos(θ)eRe(s−iN sin(θ))t C

|s − iN sin(θ) −ω|ds

≤∫ω+cos(θ)

cos(θ)est

C

N|sin(θ)|ds ≤C

N|sin(θ)|

(eωt − 1

t

)ecos(θ)t

≤ C

N|sin(θ)| e

(3.31)

is the one responsible for the fact that the integral∫γ1eλtG(λ)dλ tends to 0 as N tend to +∞,

in a similar way the integral∫γ2eλtG(λ)dλ, tend to 0 as N tend to +∞, so that

12πi

∫

ω+Γr,θeλtG(λ)dλ

12πi

∫

0+Γr,θeλtG(λ)dλ. (3.32)

For x ∈ D(A), we obtain

R(t)x − x =1

2πi

∫

0+Γr,θ

(eλtG(λ)x − λ−1eλtx

)dλ

=1

2πi

∫

0+Γr,θeλtλ−1G(λ)

(A + B(λ)

)xdλ,

(3.33)

and proceeding as before, we obtain limt→ 0+(R(t)x − x) = 0 for all x ∈ X, which ends theproof.

The following result can be proved with an argument similar to that used in the proofof the preceding lemma with changing [D(A)] by D.

Lemma 3.7. The function R : [0,∞) → L([D(A)]) is strongly continuous.

We next set δ = min{ϑ − π/2, π − ϑ}.

Lemma 3.8. The function R : (0,∞) → L(X) has an analytic extension to Λδ,0, and

dR(z)dz

=1

2πi

∫

ω+Γr,θλeλzG(λ)dλ, z ∈ Λδ,0. (3.34)


Proof. For λ ∈ ω + Γr,θ and z ∈ Λδ,0, we can write λz = ω|z|ei arg(z) + s|z|ei(arg(z)+ξ), whereπ/2 < arg(z) + ξ < π , −θ ≤ ξ ≤ θ and s ≥ r. If |z| > 1, from (3.8) and (3.17), we obtain

‖R(z)‖ ≤ 12πi

∫

ω+Γr,θeRe(λz) C

|λ −ω| |dλ|

≤ C

π

∫∞

r

eω|z| cos(arg(z))+s|z| cos(arg(z)+θ)ds

s

+C

2π

∫θ

−θeω|z| cos(arg(z))+r|z| cos(arg(z)+ξ)dξ

≤(

C

πr∣∣cos(arg(z) + θ

)∣∣ +Cθ

πer|z|)

eω|z| cos(arg(z)).

(3.35)

Using that G(·) is analytic on Λr,ω,θ, for z ∈ Λδ,0, 0 < |z| < 1, we get

‖R(z)‖ =∥∥∥∥∥

12πi

∫

ω+Γr/|z|,θeλzG(λ)dλ

∥∥∥∥∥

≤ C

π

∫∞

r/|z|eω|z| cos(arg(z))+s|z| cos(arg(z)+θ)ds

s

+C

2π

∫θ

−θeω|z| cos(arg(z))+r cos(arg(z)+ξ)dξ

≤(C

π

∫∞

r

eu cos(arg(z)+θ)du

u+

C2π

∫θ

−θer cos(arg(z)+ξ)dξ

)

eω|z| cos(arg(z))

≤(

C

πr∣∣cos(arg(z) + θ

)∣∣ +Cθ

πer)

eω|z| cos(arg(z)).

(3.36)

This property allows us to define the extension R(z) by this integral.Similarly, the integral on the right hand side of (3.34) is also absolutely convergent in

L(X) and strong, continuous on X for z ∈ Λδ,0. For λ ∈ ω + Γr,θ,

∥∥∥∥∥eλ(z+h) − eλz

hG(λ) − λeλzG(λ)

∥∥∥∥∥≤∣∣∣∣∣eλ(z+h) − eλz

h− λeλz

∣∣∣∣∣C

r−→ 0 as |h| −→ 0,

∥∥∥∥∥eλ(z+h) − eλz

hG(λ) − λeλzG(λ)

∥∥∥∥∥≤ eRe(λz) C

|λ −ω| = Σ(λ),

(3.37)

where Σ(·) is integrable for λ ∈ ω+Γr,θ. From the Lebesgue dominated convergence theorem,we obtain that R′(z) verifies (3.34). The proof is ended.


Lemma 3.9. For every λ ∈ C with Re(λ) > max{0, ω + r}, R(λ) = G(λ).

Proof. Using that G(·) is analytic on Λr,ω,θ and that the integrals involved in the calculus areabsolutely convergent, we have

R(λ) =∫∞

0e−λtR(t)dt =

∫∞

0

12πi

∫

ω+Γr,θe−(λ−γ)tG

(γ)dγdt

=1

2πi

∫

ω+Γr,θ

(λ − γ

)−1G(γ)dγ

= limL→∞

(1

2πi

∫

{ω+Γr,θ : |r|≤L}∪ ω+CL,θ

(λ − γ

)−1G(γ)dγ

)

= G(λ).

(3.38)

Theorem 3.10. The function R(·) is a resolvent operator for the system (3.1).

Proof. Let x ∈ D(A). From Lemma 3.9, for Re(λ) > max{0, ω + r},

R(λ)[λI −A − B(λ)

]x = x, (3.39)

which implies

R(λ)x =1λx +

1λR(λ)A +

1λR(λ)B(λ)x. (3.40)

Applying [31, Proposition 1.6.4, Corollary 1.6.5], we get

R(t)x = x +∫ t

0R(s)Axds +

∫ t

0

∫ s

0R(s − ξ)B(ξ)xdξds (3.41)

which in turn implies that

dR(t)dt

x = R(t)Ax +∫ t

0R(t − s)B(s)xds. (3.42)

Arguing as above but using the equality [λI −A− B(λ)]R(λ)x = x, we obtain that (3.2)holds.

On the other hand, by Lemma 3.8 we infer thatR(·)x ∈ C1((0,∞), X). Next, we analyzethe differentiability on t = 0. Let a > 0 and x ∈ D(A), for all ε > 0; we can choose δ ∈ [0, a]such that

supt∈(0,δ]

‖R(t)Ax +∫ t

0R(t − s)B(s)ds −Ax‖ < ε. (3.43)


For ζ ∈ X′ and t ∈ (0, δ), there exists cζ,t ∈ (0, t) such that

ζ ◦ R(t)x − ζ ◦ R(0)xt

= ζ(R(cζ,t)Ax +

∫ cζ,t

0R(cϕ,t − s

)B(s)xds

). (3.44)

Consequently, for t ∈ (0, δ) we have that

∥∥∥∥R(t)x − R(0)x

t−Ax

∥∥∥∥ = sup

‖ζ‖≤1

∣∣∣∣ζ ◦ R(t)x − ζ ◦ R(0)x

t− ζ(Ax)

∣∣∣∣

≤ sups∈(0,δ]

∥∥∥∥R(s)Ax +

∫s

0R(s − τ)B(τ)xdτ −Ax

∥∥∥∥,

(3.45)

which proves the existence of the right derivative ofR(·) at zero and that (d/dt)R(t)|t=0 = Ax.This proves that resolvent equation (3.3) is valid for every t ≥ 0 and R(·)x ∈ C1([0,∞);X) forevery x ∈ D(A). This completes the proof.

Corollary 3.11. If ω + r < 0, then the function R(·) is an exponentially stable resolvent operator forthe system (3.1).

In the next result, we denote by (−A)ϑ the fractional power of the operator (−A) (see[32] for details).

Theorem 3.12. Suppose that the conditions (H1)–(H3) are satisfied. Then there exists a positivenumber C such that

∥∥∥(−A)ϑR(t)∥∥∥ ≤

⎧⎨

⎩

Ce(r+ω)t, t ≥ 1,

Ce(r+ω)tt−ϑ, t ∈ (0, 1),(3.46)

for all ϑ ∈ (0, 1).

Proof. Let ϑ ∈ (0, 1). From [32, Theorem 6.10], there exists Cϑ > 0 such that

∥∥∥(−A)ϑx∥∥∥ ≤ Cϑ‖Ax‖ϑ‖x‖1−ϑ, x ∈ D(A). (3.47)

Since G(·) is a D(A) valued function, for all x ∈ X

∥∥∥(−A)ϑG(λ)x∥∥∥ ≤ Cϑ‖AG(λ)x‖ϑ‖G(λ)x‖1−ϑ

≤ CϑMϑ3 ‖x‖

ϑ M1−ϑ1

|λ −ω|1−ϑ‖x‖1−ϑ

≤ C

|λ −ω|1−ϑ‖x‖,

(3.48)


where C is independent of λ. From (3.48), we get for t ≥ 1

∥∥∥(−A)ϑR(t)

∥∥∥ ≤∥∥∥∥∥

12πi

∫

ω+Γr,θeλt(−A)ϑG(λ)dλ

∥∥∥∥∥

≤ C

π

∫∞

r

et(ω+s cos θ) ds

s1−ϑ +C

2π

∫θ

−θet(ω+r cos ξ) rdξ

r1−ϑ

≤(

C

πr1−ϑ|cos θ|+Cθrϑ

πert)

eωt ≤ Ce(r+ω)t.

(3.49)

On the other hand, using that G(·) is analytic on Λr,ω,θ, for t ∈ (0, 1), we get

∥∥∥(−A)ϑR(t)∥∥∥ =

∥∥∥∥∥

12πi

∫

ω+Γr/t,θeλt(−A)ϑG(λ)dλ

∥∥∥∥∥

≤ C

π

∫∞

r/t

et(ω+s cos θ) ds

s1−ϑ +C

2π

∫θ

−θetω+r cos ξ rt

−1dξ

r1−ϑtϑ−1

≤(C

π

∫∞

r

eu cos θ t−1du

u1−ϑtϑ−1+

C

2πr−ϑ

∫θ

−θer cos ξ rt

−1dξ

r1−ϑtϑ−1

)

eωt

≤(

C

πr1−ϑ|cos θ|+Cθrϑ

πer)eωt

tϑ.

(3.50)

From the previous facts, we conclude that

∥∥∥(−A)ϑR(t)∥∥∥ ≤ Ce(r+ω)tt−ϑ, t ∈ (0, 1), (3.51)

which ends the proof.

Corollary 3.13. If ω + r < 0 and ϑ ∈ (0, 1), then there exists φ ∈ L1([0,∞)) such that

∥∥∥(−A)ϑR(t)∥∥∥ ≤ φ(t). (3.52)

In the remainder of this section, we discuss the existence and regularity of solutions of

dx(t)dt

= Ax(t) +∫ t

0B(t − s)x(s)ds + f(t), t ∈ [0, a], (3.53)

x(0) = z ∈ X, (3.54)

where f ∈ L1([0, a], X). In the sequel, R(·) is the operator function defined by (3.17). Webegin by introducing the following concept of classical solution.


Definition 3.14. A function x : [0, b] → X, 0 < b ≤ a, is called a classical solution of (3.53)-(3.54) on [0, b] if x ∈ C([0, b], [D(A)]) ∩ C1((0, b], X), the condition (3.54) holds and (3.53) isverified on [0, a].

The next result has been established in [30].

Theorem 3.15 ([30, Theorem 2]). Let z ∈ X. Assume that f ∈ C([0, a], X) and x(·) is a classicalsolution of (3.53)-(3.54) on [0, a]. Then

x(t) = R(t)z +∫ t

0R(t − s)f(s)ds, t ∈ [0, a]. (3.55)

An immediate consequence of the above theorem is the uniqueness of classicalsolutions.

Corollary 3.16. If u, v are classical solutions of (3.53)-(3.54) on [0, b], then u = v on [0, b].

Motivated by (3.55), we introduce the following concept.

Definition 3.17. A function u ∈ C([0, a], X) is called a mild solution of (3.53)-(3.54) if

u(t) = R(t)z +∫ t

0R(t − s)f(s)ds, t ∈ [0, a]. (3.56)

4. Existence Result of Asymptotically Almost Periodic Solutions

In this section, we study the existence of asymptotically almost periodic mild solutions forthe abstract integro-differential system (1.1). To establish our existence result, motivated bythe previous section we introduce the following assumptions.

(P1) There exists a Banach space (Y, ‖ · ‖Y ) continuously included in X such that thefollowing conditions are verified.

(a) For every t ∈ (0,∞), R(t) ∈ L(X) ∩ L(Y, [D(A)]) and B(t) ∈ L(Y,X). Inaddition, AR(·)x, B(·)x ∈ C((0,∞), X) for every x ∈ Y .

(b) There are positive constants M,β such that

max{‖R(s)‖, ‖B(s)‖L(Y,X)

}≤Me−βt, s ≥ 0. (4.1)

(c) There exists φ ∈ L1([0,∞)) such that ‖AR(t)‖L(Y,X) ≤ φ(t), t ≥ 0.

(P2) The continuous function f : R × B → Y is p.a.a.p, and there exists a continuousfunction Lf : [0,∞) → [0,∞), such that

‖f(t, ψ1)− f(t, ψ2)‖Y ≤ Lf(r)‖ψ1 − ψ2‖B,

(t, ψj)∈ R × Br(0,B). (4.2)


(P3) The continuous function g : R × B → X is p.a.a.p, and there exists a continuousfunction Lg : [0,∞) → [0,∞) such that

‖g(t, ψ1)− g(t, ψ2)‖ ≤ Lg(r)‖ψ1 − ψ2‖B,

(t, ψj)∈ R × Br(0,B). (4.3)

Motivated by the theory of resolvent operator, we introduce the following concept ofmild solution for (1.1).

Definition 4.1. A function u : (−∞, b] → X, 0 < b ≤ a, is called a mild solution of (1.1) on[0, b], if u0 = ϕ ∈ B;u|[0,b] ∈ C([0, b] : X); the functions τ → AR(t − τ)f(τ, uτ) and τ →∫τ

0B(τ − ξ)f(ξ, uξ)dξ are integrable on [0, t) for every t ∈ (0, b] and

u(t) = R(t)(ϕ(0) + f

(0, ϕ))− f(t, ut) −

∫ t

0AR(t − s)f(s, us)ds

−∫ t

0R(t − s)

∫s

0B(s − ξ)f

(ξ, uξ)dξds +

∫ t

0R(t − s)g(s, us)ds, t ∈ [0, b].

(4.4)

Lemma 4.2. Let condition (P1)—(c) hold and let v be a function in AAP(Y ). If u : [0,∞) → X isthe function defined by u(t) =

∫ t0AR(t − s)v(s)ds, then u(·) ∈ AAP(X).

Proof. Let η = max{2‖v‖Y ,∫∞

0 φ(s)ds}. Let T((ε/3)η−1, v, Y ), T = T((ε/3)η−1, v, Y ) be as inLemma 2.7 and T1 > 1 such that

∫∞T1φ(s)ds ≤ (ε/3)η−1. For t ≥ T +T1 and ξ ∈ T((ε/3)η−1, v, Y ),

we get

‖u(t + ξ) − u(t)‖ ≤∫0

−ξ‖AR(t − s)‖L(Y,X)‖v(s + ξ)‖Yds

+∫ t

0‖AR(t − s)‖L(Y,X)‖v(s + ξ) − v(s)‖Yds

≤∫0

−ξφ(t − s)‖v(s + ξ)‖Yds

+∫T

0φ(t − s)‖v(s + ξ) − v(s)‖Yds

+∫ t

T

φ(t − s)‖v(s + ξ) − v(s)‖Yds

≤ ‖v‖Y∫ t+ξ

t

φ(s)ds + 2‖v‖Y∫ t

t−Tφ(s)ds + ε

∫ t−T

0φ(s)ds

≤ ‖v‖Y∫∞

T1

φ(s)ds + 2‖v‖Y∫∞

T1

φ(s)ds +ε

3η−1∫∞

0φ(s)ds

(4.5)


which implies that

‖u(t + ξ) − u(t)‖ ≤ ε, t ≥ T(ε

3η−1, v, Y

)+ T1, ξ ∈ T

(ε

3η−1, v, Y

). (4.6)

Now, from inequality (4.6) and Lemma 2.7, we conclude that u(·) is a.a.p. The proof iscomplete.

Lemma 4.3. Assume that the condition (P1) is fulfilled. Let v ∈ AAP(Y ) and letw(·) : [0,∞) → Xbe the function defined by

w(t) =∫ t

0R(t − s)

∫ s

0B(s − ξ)v(ξ)dξds, t ≥ 0. (4.7)

Then w(·) ∈ AAP(X).

Proof. Let T((ε/3)η−1, v, Y ), T = T((ε/3)η−1, v, Y ) be as in Lemma 2.7 and T1 > 1 such that

∫∞

T1

e−βsds ≤ ε

3η−1, te−β(t−T) ≤ ε

3η−1 (4.8)

for t ≥ T1, where η = max{3‖v‖Y (M2/β), supt≥T1(M2/β)te−β(t−T)}. For t ≥ T + T1 and ξ ∈

T((ε/3)η−1, v, Y ), we get

w(t + ξ) −w(t) =∫ t+ξ

0R(t + ξ − s)

∫ s

0B(s − u)v(u)duds −

∫ t

0R(t − s)

∫ s

0B(s − u)v(u)duds

=∫ t

0R(t − s)

∫ s

0B(s − u)(v(u + ξ) − v(u))duds

+∫ ξ

0R(t + ξ − s)

∫s

0B(s − u)v(u)duds

+∫ t

0R(t − s)

∫ ξ

0B(s + ξ − u)v(u)duds.

(4.9)


We obtain

‖w(t + ξ) −w(t)‖

≤∫ t

0‖R(t − s)‖

∫T

0‖B(s − u)‖L(Y,X)‖v(u + ξ) − v(u)‖Yduds

+∫ t

0‖R(t − s)‖

∫ s

T

‖B(s − u)‖L(Y,X)‖v(u + ξ) − v(u)‖Yduds

+∫ ξ

0‖R(t + ξ − s)‖

∫s

0‖B(s − u)‖L(Y,X)‖v(u)‖Yduds

+∫ t

0‖R(t − s)‖

∫ ξ

0‖B(s + ξ − u)‖L(Y,X)‖v(u)‖Yduds.

≤∫ t

0Me−β(t−s)

∫T

0Me−β(s−u)‖v(u + ξ) − v(u)‖Yduds

+∫ t

0Me−β(t−s)

∫ t

T

Me−β(s−u)‖v(u + ξ) − v(u)‖Yduds

+∫ ξ

0Me−β(t+ξ−s)

∫s

0Me−β(s−u)‖v(u)‖Yduds

+∫ t

0Me−β(t−s)

∫ ξ

0Me−β(s+ξ−u)‖v(u)‖Yduds.

≤ 2‖v‖Y∫ t

0Me−βt

∫T

0Meβududs + ε

∫ t

0Me−βt

∫ t

T

Meβududs

+ ‖v‖Y∫ ξ

0Me−β(t+ξ−s)

∫s

0Me−β(s−u)duds

+ ‖v‖Y∫ t

0Me−β(t−s)

∫ ξ

0Meβ(s+ξ−u)duds

≤ 2‖v‖YM2

βte−β(t−T) + ε

M2

βte−β(t−T)

+ ‖v‖YM2

β

∫ t+ξ

t

e−βsds + ‖v‖YM2

βte−β(t−T)

≤ 3‖v‖YM2

βte−β(t−T) + ε

M2

βte−β(t−T) + ‖v‖Y

M2

β

∫∞

T1

e−βsds,

(4.10)


which implies that

‖w(t + ξ) −w(t)‖ ≤ ε, t ≥ T(ε

3η−1, v, Y

)+ T1, ξ ∈ T

(ε

3η−1, v, Y

). (4.11)

From inequality (4.11) and Lemma 2.7, we conclude that w(·) is a.a.p., which ends the proof.

Now, we can establish our existence result.

Theorem 4.4. Assume that B is a fading memory space and (P1), (P2), and (P3) are held. If Lf(0) =Lg(0) = 0 and f(t, 0) = g(t, 0) = 0 for every t ∈ R, then there exists ε > 0 such that for eachϕ ∈ Bε(0,B), there exists a mild solution, u(·, ϕ), of (1.1) on [0,∞) such that u(·, ϕ) ∈ AAP(X) andu0(·, ϕ) = ϕ.

Proof. Let r > 0 and 0 < λ < 1 be such that

Θ =MHλ +MLf(λr)λ + Lf((λ + 1)Kr)(λ + 1)K

(

‖ic‖L(Y,X) +∥∥φ∥∥L1 +

M2

β2

)

+ Lg((λ + 1)Kr)(λ + 1)KM

β< 1,

(4.12)

where K is the constant introduced in Remark 2.3. We affirm that the assertion holds for ε =λr. Let ϕ ∈ Bε(0,B). On the space

D ={x ∈ AAP(X) : x(0) = ϕ(0), ‖x(t)‖ ≤ r, t ≥ 0

}(4.13)

endowed with the metric d(u, v) = ‖u − v‖, we define the operator Γ : D → C([0,∞);X) by

Γu(t) = R(t)(ϕ(0) + f

(0, ϕ))− f(t, ut) −

∫ t

0AR(t − s)f(s, us)ds

−∫ t

0R(t − s)

∫s

0B(s − ξ)f

(ξ, uξ)dξds +

∫ t

0R(t − s)g(s, us)ds, t ≥ 0,

(4.14)

where u : R → X is the function defined by the relation u0 = ϕ and u = u on [0,∞). From thehypothesis (P1), (P2), and (P3), we obtain that Γu is well defined and that Γu ∈ C([0,∞);X).Moreover, from Lemmas 4.2 and 4.3 it follows that Γu ∈ AAP(X).


Next, we prove that Γ(·) is a contraction from D into D. If u ∈ D and t ≥ 0, we get

‖Γu(t)‖ ≤MHλr +MLf(λr)λr + ‖ic‖L(Y,X)Lf((λ + 1)Kr)(λ + 1)Kr

+∫ t

0φ(t − s)Lf((λ + 1)Kr)(λ + 1)Kr ds

+∫ t

0Me−β(t−s)

∫s

0Me−β(s−ξ)Lf((λ + 1)Kr)(λ + 1)Kr dξds

+∫ t

0Me−β(t−s)Lg((λ + 1)Kr)(λ + 1)Kr ds

≤MHλr +MLf(λr)λr + Lf((λ + 1)Kr)(λ + 1)Kr

+(∫∞

0φ(s)ds

)Lf((λ + 1)Kr)(λ + 1)Kr

+(∫∞

0Me−βsds

)(∫∞

0Me−βξdξ

)Lf((λ + 1)Kr)(λ + 1)Kr

+(∫∞

0e−βsds

)Lg((λ + 1)K)(λ + 1)Kr

≤ Θr,

(4.15)

where the inequality ‖ut‖ ≤ (λ+ 1)Kr has been used and ic : Y → X represent the continuousinclusion of Y on X. Thus, Γ(D) ⊂ D. On the other hand, for u, v ∈ D we see that

‖Γu(t) − Γv(t)‖

≤ ‖ic‖L(Y,X)

∥∥f(t, ut) − f(t, vt)∥∥ +∫ t

0‖AR(t − s)‖L(Y,X)

∥∥f(s, us) − f(s, vs)∥∥Yds

+∫ t

0‖R(t − s)‖

∫s

0‖B(s − ξ)‖L(Y,X)

∥∥f(ξ, uξ) − f(ξ, vξ)∥∥Ydξds

+∫ t

0‖R(t − s)‖

∥∥g(s, us) − g(s, vs)∥∥ds

≤ Lf((λ + 1)Kr)K

(

‖ic‖L(Y,X) +∥∥φ∥∥L1 +

M2

β2

)

‖u − v‖

+ Lg((λ + 1)K)KM

β‖u − v‖ ≤ Θ‖u − v‖,

(4.16)

which shows that Γ(·) is a contraction from D into D. The assertion is now a consequence ofthe contraction mapping principle. The proof is complete.


5. Applications

In this section, we study the existence of asymptotically almost periodic solutions of thepartial neutral integro-differential system

∂

∂t

[

u(t, ξ) +∫ t

−∞

∫π

0b(s − t, η, ξ

)u(s, η)dηds

]

=

(∂2

∂ξ2+ μ

)[

u(t, ξ) +∫ t

0e−γ(t−s)u(s, ξ)ds

]

+∫ t

−∞a0(s − t)u(s, ξ)ds,

u(t, 0) = u(t, π) = 0,

u(θ, ξ) = ϕ(θ, ξ),

(5.1)

for (t, ξ) ∈ [0, a] × [0, π], θ ≤ 0, μ < 0, and γ > 0. Moreover, we have identified ϕ(θ)(ξ) =ϕ(θ, ξ).

To represent this system in the abstract form (1.1), we choose the spaces X = L2([0, π])and B = C0 × L2(ρ,X); see Example 2.4 for details. We also consider the operators A,B(t) :D(A) ⊆ X → X, t ≥ 0, given by Ax = x′′ + μx, B(t)x = e−γtAx for x ∈ D(A) = {x ∈ X :x′′ ∈ X, x(0) = x(π) = 0}. Moreover, A has discrete spectrum, the eigenvalues are −n2 + μ,n ∈ N, with corresponding eigenvectors zn(ξ) = (2/π)1/2 sin(nξ), and the set of functions{zn : n ∈ N} is an orthonormal basis of X and T(t)x =

∑∞n=1 e

−(n2−μ)t < x, zn > zn for x ∈ X.For α ∈ (0, 1), from [32] we can define the fractional power (−A)α : D((−A)α) ⊂ X → X ofA is given by (−A)αx =

∑∞n=1(n

2 − μ)α〈x, zn〉zn, where D((−A)α) = {x ∈ X : (−A)αx ∈ X}. Inthe next theorem, we consider Y = D((−A)1/2). We observe that ρ(A) ⊃ {λ ∈ C : Re(λ) ≥ μ}and ‖λR(λ,A)‖ ≤ M1 for Re(λ) ≥ μ; from [33, Proposition 2.2.11], we obtain that A is asectorial operator satisfying ‖R(λ,A)‖ ≤ M/|λ − μ|,M > 0. Moreover, it is easy to see thatconditions (H2)-(H3) in Section 3 are satisfied with b(t) = e−γt, and D = C∞0 ([0, π]) is thespace of infinitely differentiable functions that vanish at ξ = 0 and ξ = π . Under the aboveconditions, we can represent the system

∂u(t, ξ)∂t

=

(∂2

∂ξ2+ μ

)[

u(t, ξ) +∫ t

0e−γ(t−s)u(s, ξ)ds

]

,

u(t, π) = u(t, 0) = 0,

(5.2)

in the abstract form

dx(t)dt

= Ax(t) +∫ t

0B(t − s)x(s)ds,

x(0) = z ∈ X.(5.3)


We define the functions f, g : B → X by

f(ψ)(ξ) =

∫0

−∞

∫π

0b(s, η, ξ

)ψ(s, η)dηds,

g(ψ)(ξ) =

∫0

−∞a0(s)ψ(s, ξ)ds,

(5.4)

where

(i) the functions a0 : R → R are continuous and Lg := (∫0−∞((a0(s))

2/ρ(s))ds)1/2

<∞.;

(ii) the functions b(·), ∂b(s, η, ξ)/∂ξ are measurable, b(s, η, π) = b(s, η, 0) = 0 for all(s, η) and

Lf := max

⎧⎪⎨

⎪⎩

⎛

⎝∫π

0

∫0

−∞

∫π

0ρ−1(θ)

(∂i

∂ξib(θ, η, ξ)

)2

dηdθdξ

⎞

⎠

1/2

: i = 0, 1

⎫⎪⎬

⎪⎭<∞.

(5.5)

Moreover, f, g are bounded linear operators, ‖f‖L(B,X) ≤ Lf , ‖g‖L(B,X) ≤ Lg , and astraightforward estimation using (ii) shows that f(I × B) ⊂ D((−A)1/2) and

∥∥∥(−A)1/2f(t, ·)∥∥∥L(B,X)

≤ Lf (5.6)

for all t ∈ I. This allows us to rewrite the system (5.1) in the abstract form (1.1) withu0 = ϕ ∈ B.

Theorem 5.1. Assume that the previous conditions are verified. Let 2 < K < γ and μ < 0 such that|μ| > max{M(K + 1 + γ), γ}, then there exists a mild solution u(·) ∈ AAP(X) of (5.1) with u0 = ϕ.

Proof. For λ = μ + seiθ, from |λ + γ | ≥ s − |γ + μ|, we obtain

∥∥∥B(λ)R(λ,A)∥∥∥ ≤

1∣∣λ + γ

∣∣

(

1 +M∣∣λ − μ

∣∣ +M|λ|∣∣λ − μ

∣∣

)

≤ 1∣∣λ + γ

∣∣ +

(1

∣∣λ + γ∣∣ +

|λ|∣∣λ + γ

∣∣

)M∣∣λ − μ

∣∣

≤ 1∣∣λ + γ

∣∣ +

(1

∣∣λ + γ∣∣ + 1 +

∣∣γ∣∣

∣∣λ + γ∣∣

)M∣∣λ − μ

∣∣


≤ 1s −∣∣γ + μ

∣∣ +

(1

s −∣∣γ + μ

∣∣ + 1 +

∣∣γ∣∣

s −∣∣γ + μ

∣∣

)M∣∣λ − μ

∣∣

≤ 1K

+(

1 +K + γK

)M∣∣λ − μ

∣∣

≤ 1K

+1K,

(5.7)

since s ≥ r = max{M(K + 1 + γ), K + |γ + μ|}. By using a similar procedure as in theproofs of Lemma 3.3 and Theorem 3.10, we obtain the existence of resolvent operator for (5.2).From the hypothesis, we obtain μ + r < 0; by the Lemma 3.3, Corollaries 3.11 and 3.13, theassumption (P1) is satisfied. From Theorem 4.4, the proof is complete.

Acknowledgment

Jose Paulo C. dos Santos is partially supported by FAPEMIG/Brazil under Grant CEX-APQ-00476-09.

References

[1] E. N. Chukwu, Differential Models and Neutral Systems for Controlling the Wealth of Nations, vol. 54 ofAdvances in Mathematics for Applied Sciences, World Scientific Publishing, River Edge, NJ, USA, 2001.

[2] J. Hale and S. M. Lunel, Introduction to Functional-Differential Equations. Applied Mathematical Sciences,vol. 99, Springer, New York, NY, USA, 1993.

[3] J. Wu, Theory and Applications of Partial functional-Differential Equations, vol. 119 of Applied MathematicalSciences, Springer, New York, NY, USA, 1996.

[4] M. E. Gurtin and A. C. Pipkin, “A general theory of heat conduction with finite wave speeds,” Archivefor Rational Mechanics and Analysis, vol. 31, no. 2, pp. 113–126, 1968.

[5] J. Nunziato, “On heat conduction in materials with memory,” Quarterly of Applied Mathematics, vol.29, pp. 187–204, 1971.

[6] P. Cannarsa and D. Sforza, “Global solutions of abstract semilinear parabolic equations with memoryterms,” Nonlinear Differential Equations and Applications, vol. 10, no. 4, pp. 399–430, 2003.

[7] Ph. Clement and J. Nohel, “Asymptotic behavior of solutions of nonlinear Volterra equations withcompletely positive kernels,” SIAM Journal on Mathematical Analysis, vol. 12, no. 4, pp. 514–535, 1981.

[8] P. Clement and J. Philippe, “Global existence for a semilinear parabolic Volterra equation,”Mathematische Zeitschrift, vol. 209, no. 1, pp. 17–26, 1992.

[9] A. Lunardi, “On the linear heat equation with fading memory,” SIAM Journal onMathematical Analysis,vol. 21, no. 5, pp. 1213–1224, 1990.

[10] N. M. Man and N. V. Minh, “On the existence of quasi periodic and almost periodic solutions ofneutral functional differential equations,” Communications on Pure and Applied Analysis, vol. 3, no. 2,pp. 291–300, 2004.

[11] R. Yuan, “Existence of almost periodic solutions of neutral functional-differential equations viaLiapunov-Razumikhin function,” Zeitschrift fur Angewandte Mathematik und Physik, vol. 49, no. 1, pp.113–136, 1998.

[12] T. Diagana, H. R. Henrıquez, and E. M. Hernandez, “Almost automorphic mild solutions to somepartial neutral functional-differential equations and applications,” Nonlinear Analysis: Theory, Methods& Applications, vol. 69, no. 5-6, pp. 1485–1493, 2008.

[13] E. Hernandez and H. Henrıquez, “Existence of periodic solutions of partial neutral functional-differential equations with unbounded delay,” Journal of Mathematical Analysis and Applications, vol.221, no. 2, pp. 499–522, 1998.


[14] E. Hernandez and M. Pelicer, “Asymptotically almost periodic and almost periodic solutions forpartial neutral differential equations,” Applied Mathematics Letters, vol. 18, no. 11, pp. 1265–1272, 2005.

[15] E. Hernandez and T. Diagana, “Existence and uniqueness of pseudo almost periodic solutions to someabstract partial neutral functional-differential equations and applications,” Journal of MathematicalAnalysis and Applications, vol. 327, no. 2, pp. 776–791, 2007.

[16] G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra Integral and Functional Equations, vol. 34 ofEncyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1990.

[17] J. Pruss, Evolutionary Integral Equations and Applications, vol. 87 of Monographs in Mathematics,Birkhauser, Basel, Switzerland, 1993.

[18] R. Grimmer, “Resolvent operators for integral equations in a Banach space,” Transactions of theAmerican Mathematical Society, vol. 273, no. 1, pp. 333–349, 1982.

[19] R. C. Grimmer and F. Kappel, “Series expansions for resolvents of Volterra integro-differentialequations in Banach space,” SIAM Journal on Mathematical Analysis, vol. 15, no. 3, pp. 595–604, 1984.

[20] R. C. Grimmer and A. J. Pritchard, “Analytic resolvent operators for integral equations in Banachspace,” Journal of Differential Equations, vol. 50, no. 2, pp. 234–259, 1983.

[21] G. Da Prato and M. Iannelli, “Existence and regularity for a class of integro-differential equations ofparabolic type,” Journal of Mathematical Analysis and Applications, vol. 112, no. 1, pp. 36–55, 1985.

[22] G. Da Prato and A. Lunardi, “Solvability on the real line of a class of linear Volterra integro-differentialequations of parabolic type,” Annali di Matematica Pura ed Applicata, vol. 150, no. 1, pp. 67–117, 1988.

[23] A. Lunardi, “Laplace transform methods in integro-differential equations,” Journal of IntegralEquations, vol. 10, no. 1–3, pp. 185–211, 1985.

[24] H. Henriquez, E. Hernandez, and J. P. C. dos Santos, “Asymptotically almost periodic and almostperiodic solutions for partial neutral integrodifferential equations,” Zeitschrift fur Analysis und ihreAnwendungen, vol. 26, no. 3, pp. 363–375, 2007.

[25] E. Hernandez and J. P. C. dos Santos, “Existence results for partial neutral integro-differential equationwith unbounded delay,” Applicable Analysis, vol. 86, no. 2, pp. 223–237, 2007.

[26] H. R. Henriquez, E. Hernandez, and J. P. C. dos Santos, “Existence results for abstract partialneutral integro-differential equation with unbounded delay,” Electronic Journal of Qualitative Theoryof Differential Equations, vol. 29, pp. 1–23, 2009.

[27] Y. Hino, S. Murakami, and T. Naito, Functional-Differential Equations with Infinite Delay, vol. 1473 ofLecture Notes in Mathematics, Springer, Berlin, Germany, 1991.

[28] S. Zaidman, Almost-Periodic Functions in Abstract Spaces, vol. 126 of Research Notes in Mathematics,Pitman (Advanced Publishing Program), Boston, Mass, USA, 1985.

[29] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, vol.14, Springer, New York, NY, USA, 1975.

[30] R. Grimmer and J. Pruss, “On linear Volterra equations in Banach spaces,” Computers & Mathematicswith Applications, vol. 11, no. 1–3, pp. 189–205, 1985.

[31] W. Arendt, C. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and CauchyProblems, vol. 96 of Monographs in Mathematics, Birkhauser, Basel, Switzerland, 2001.

[32] A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44 of AppliedMathematical Sciences, Springer, New York, NY, USA, 1983.

[33] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, vol. 16 of Progress inNonlinear Differential Equations and Their Applications, Birkhauser, Basel, Switzerland, 1995.


Research ArticleExistence of a Nonautonomous SIR EpidemicModel with Age Structure

Junyuan Yang1, 2 and Xiaoyan Wang2

1 Department of Applied Mathematics, Yuncheng University, Yuncheng Shanxi 044000, China2 Beijing Institute of Information and Control, Beijing 100037, China

Correspondence should be addressed to Junyuan Yang, [email protected]

Received 15 December 2009; Accepted 1 February 2010


Copyright q 2010 J. Yang and X. Wang. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

A nonautonomous SIR epidemic model with age structure is studied. Using integro-differentialequation and a fixed point theorem, we prove the existence and uniqueness of a positive solutionto this model. We conclude our results and discuss some problems to this model in the future. Wesimulate our analyzed results.

1. Introduction

Age structure of a population affects the dynamics of disease transmission. Traditionaltransmission dynamics of certain diseases cannot be correctly described by the traditionalepidemic models with no age-dependence. A simplemodel was first proposed by Lotka andVon Foerster [1, 2], where the birth and the death processes were independent of the totalpopulation size and so the limitation of the resources was not taken into account. To overcomethis deficiency, Gurtin and MacCamy [3], in their pioneering work considered a nonlinearage-dependent model, where birth and death rates were function of the total population.Various age-structured epidemic models have been investigated by many authors, and anumber of papers have been published on finding the threshold conditions for the disease tobecome endemic, describing the stability of steady-state solutions, and analyzing the globalbehavior of these age-structured epidemic models (see [4–7]). We may find that the epidemicmodels that most authors discussed mainly include S-I-R that is, the total population of acountry or a district was subdivided into two or three compartments containing susceptibles,infectives, or immunes; it was assumed that there is no latent class, so a person who catches


the disease becomes infectious instantaneously. The basic SIR age-structured epidemic modelis like the following equations:

∂s(a, t)∂t

+∂s(a, t)∂a

= −λ(a, t, i)s(a, t) − μ(a)s(a, t),

∂i(a, t)∂t

+∂i(a, t)∂a

= λ(a, t, i)s(a, t) − μ(a)i(a, t) − γ(a)i(a, t),

∂r(a, t)∂t

+∂r(a, t)∂a

= γ(a)i(a, t) − μ(a)r(a, t),

s(0, t) =∫∞

0β(a)p(a, t)da, i(0, t) = r(0, t) = 0,

s(a, 0) = s0(a) ∈ L1+(0,+∞), i(a, 0) = i0(a) ∈ L1

+(0,+∞),

r(0, a) = r0(a) ∈ L1+(0,+∞).

(1.1)

The non-autonomous phenomenon is so prevalent and all pervasive in the real lifethat modelling biological proceeding under non-autonomous environment should be morerealistic than autonomous situation. The non-autonomous phenomenon is so prevalent in thereal life that many epidemiological problems can be modeled by non-autonomous systemsof nonlinear differential equations [8–11], which should be more realistic than autonomousdifferential equations. In one case, the incidence of many infectious diseases fluctuates overtime and often exhibits periodic behavior. The basic SIR model is formulated by

dS

dt= B(t) − μ(t)S − α(t)SI,

dI

dt= α(t)SI − μ(t)I − γ(t)I,

dR

dt= γ(t)I − μ(t)R.

(1.2)

These works were mainly concerned with finding threshold conditions for the disease tobecome endemic and describing the stability of steady-state solutions, often under theassumption that the population has reached its steady state and the diseases do not affectthe death rate of the population.

However, all of the models which are not mixed age structure and non-autonomousare only concluding age structure or non-autonomous. Birth rate or input function isdependent on age or dependent on time t in these models cited therein. In fact, birth rateor input function is dependent not only on age a and time t but also on the total populationP(t). We know the resource is limited. As recognized by authors, there was only one paper[3, 12] related them. In [3, 12], their model are two dimensions about epidemic dynamics.The population is increasing year after year. The birth rate is a decrease function until thepopulation attend certain level such as Logistic growth rate. At the same time, the death rateshould be dependent on the total population P(t). We can consider now more realistic andcomplex models in which the epidemic acts in a different way on infected, susceptible andrecovered (immune). We consider a well-known expression for the force of infection which is


justified in the literature. We choose as L1(R+) the natural space for the solution because thetotal population is finite.

This paper is organized as follows: Section 2 introduces a non-autonomous SIR modelwith age structure. In Section 3, existence and uniqueness of a solution for an epidemic modelwith different mortality rates on any finite time-interval is obtained. In Section 4, we concludeour results and discuss the defect of our model.

2. The Model Formulation

This section describes the basic model we are going to analyze in this paper. Thepopulation is divided into three subclasses: susceptible, infected, and recovered. WhereS(a, t), I(a, t), R(a, t) denote the associated density functions with these respective epidemi-ological age-structured classes. Let μi(a, t, P(t)), i = 1, 2, 3, be the age-specific mortalityof the susceptible, the infective and the recovered individuals at time t, respectively. Weassume that the disease affects the death rate, so we have μ2(a, t, P(t)) ≥ μ1(a, t, P(t)), andμ2(a, t, P(t)) ≥ μ3(a, t, P(t)). We assume that all new born are susceptible whose birth processis described by

s(0, t) =∫+∞

0β(a, t, P(t))s(a, t)da, (2.1)

where β is the birth rate. We also suppose that the initial age distributions are given bys0, i0, and r0. And the age-specific recovery rate, γ , is independent of the time. Then the jointdynamics of the age-structured epidemiological model for the transmission of SIR can bewritten as

∂s(a, t)∂t

+∂s(a, t)∂a

= −λ(a, t, i)s(a, t) − μ1(a, t, P(t))s(a, t),

∂i(a, t)∂t

+∂i(a, t)∂a

= λ(a, t, i)s(a, t) − μ2(a, t, P(t))i(a, t) − γ(a)i(a, t),

∂r(a, t)∂t

+∂r(a, t)∂a

= γ(a)i(a, t) − μ3(a, t, P(t))r(a, t),

s(0, t) =∫∞

0β(a, t, P(t))p(a, t)da, i(0, t) = r(0, t) = 0,

s(a, 0) = s0(a) ∈ L1+(0,+∞), i(a, 0) = i0(a) ∈ L1

+(0,+∞),

r(0, a) = r0(a) ∈ L1+(0,+∞).

(2.2)

We supposes s(a, t), i(a, t), and r(a, t) belong to W1,1(0,+∞). So, s(a, t), i(a, t), and r(a, t) →0, as a → +∞. It is logical to satisfy the biological meaning. The horizontal transmission ofthe disease occurs according to the following law:

λ(a, t, i) =∫+∞

0K(a, a′)i(t, a′)da′, (2.3)


where K(a, a′) is the rate at which an infective individual of age a′ comes into a diseasetransmitting contact with a susceptible individual of age a. Summing the equations of (2.2),we obtain the following problem for the population density P(a, t) = S(a, t) +E(a, t) + I(a, t).

P(t) = S(t) + I(t) + R(t) =∫∞

0s(a, t)da +

∫∞

0i(a, t)da +

∫∞

0r(a, t)da. (2.4)

In this paper, we prove the existence and uniqueness of a nonnegative solution ofthe model (2.2) on any finite time-interval. Our results are based on a process of the age-dependent problem for the susceptible the infected and the removed, and then a fixed pointmethod. To study existence and uniqueness of a solution for an epidemic model with differentmortality rates, we need the following hypotheses. Given T > 0, we denote I := [0, T] and wesuppose that

(H1) for i = 1, 2, 3, μi(a, t, P) is a nonnegative measurable function such that the mappingl |→ μi(l, l + u, P) belongs to L1

Loc(R+) for almost all (u, P) ∈ R2. Moreover, thereexists a constant C1(T) > 0 such that for all P, P ′ ∈ R,

∣∣μi(a, t, P) − μi(a, t, P ′

)∣∣ ≤ C1(T)∣∣P − P ′

∣∣, a.e. (a, t) ∈ R+ × I. (2.5)

With the notation μ1 = μ2 − μ1, μ2 = μ2 − μ3, there exists another constant Ci(T) > 0,j = 2, 3, such that

∣∣μi(a, t, P)∣∣ ≤ Cj(T) log(|P | + e), a.e. (a, t) ∈ R+ × I, i = 2, 3. (2.6)

(H2) β(a, t, P) is a nonnegative measurable function which has compact support on thevariable a and such that for all P, P ′ ∈ R,

∣∣β(a, t, P) − β(a, t, P ′

)∣∣ ≤ C4(T)∣∣P − P ′

∣∣, a.e. (a, t) ∈ R+ × I, (2.7)

where C4(T) > 0 is another constant which depends only on T . Moreover, thereexists a constant C5(T) > 0 such that for all P ∈ R,

∣∣β(a, t, P)∣∣ ≤ C5(T) log(|P | + e), a.e. (a, t) ∈ R+ × I. (2.8)

(H3) φ0 := (s0, i0, r0) ∈ (L1(R+))3 has a compact support.

(H4) γ(a) ∈ L∞(R+) has compact support and is a nonnegative function. We set γ∞ =ess supa∈(0,∞)γ(a).

(H5) K(a, a′) ∈ L∞(R+ × R+) has a compact support and is a nonnegative function. Wehave K∞ = ess supa∈(0,∞)K(a, a′).

To simplify the calculation of estimates, we perform the change

i(a, t) = p(a, t) − s(a, t) − r(a, t),

μ1 = μ2 − μ1, μ2 = μ2 − μ3.(2.9)


We obtain that the following system is analogous to (2.2).

∂p(a, t)∂t

+∂p(a, t)∂a

= −μ1s(a, t) + μ2r(a, t) − μ2(a, t, P(t))p(a, t),

∂s(a, t)∂t

+∂s(a, t)∂a

= −λ(a, t, p − s − r

)s(a, t) − μ1(a, t, P(t))i(a, t),

∂r(a, t)∂t

+∂r(a, t)∂a

= γ(a)[p(a, t) − r(a, t)

]−(μ3(a, t, P(t)) + γ(a)

)r(a, t),

p(0, t) = s(0, t) =∫∞

0β(a, t, P(t))p(a, t), r(0, t) = 0,

p(a, 0) = s0(a) ∈ L1+(0,+∞), s(a, 0) = i0(a) ∈ L1

+(0,+∞),

r(0, a) = r0(a) ∈ L1+(0,+∞),

(2.10)

where

ρ0(a) =(p0(a), s0(a), r0(a)

)= (s0(a) + r0(a) + i0(a), s0(a), r0(a)), a.e, a ∈ (0,+∞). (2.11)

For biological reasons, we are interested in nonnegative solutions, so we consider that

p(a, t) ≥ s(a, t), p(a, t) ≥ r(a, t). (2.12)

And we will look for solutions to (2.10) belonging to the following space:

V :={ρ ∈ L∞

(I,(L1(R+)3

))| ρ1(a, t) ≥ ρ2(a, t) ≥ 0,

ρ1(a, t) ≥ ρ2(a, t) ≥ 0, a.e, (a, t) ∈ R+ × I} (2.13)

endowed with the norm

∣∣ρ∣∣

1 = ess supt∈I

e−kt∣∣ρ(·, t)

∣∣1, (2.14)

where k is a positive constant which will be chosen later and | · |1 denotes the usual norm inL1(R+) that is, |ρ(·, t)|1 = ‖ρ1(·, t)‖L1 + ‖ρ2(·, t)‖L1 + ‖ρ3(·, t)‖L1 .

Namely, by a solution to (2.10), we mean a function

ρ(·, ·) =(p(·, ·), s(·, ·), r(·, ·)

)∈ V, (2.15)


such that

Dp = −μ1s(a, t) + μ2r(a, t) − μ2(a, t, P(t))p(a, t),

Ds = −λ(a, t, p − s − r

)s(a, t) − μ1(a, t, P(t))i(a, t),

Dr = γ(a)[p(a, t) − r(a, t)

]−(μ3(a, t, P(t)) + γ(a)

)r(a, t),

limt→ 0+

p(a, t + h) =∫∞

0β(a, t, P(t))p(a, t)da,

limt→ 0+

s(a, t + h) =∫∞

0β(a, t, P(t))p(a, t)da,

limt→ 0+

r(a, t + h) = 0,

p(a, t), s(a, t), r(a, t) −→ 0, when a −→ +∞.

(2.16)

In order to prove the existence of solution of (2.10), adding γ(a) in both sides of (2.16)in technical style, we have

Dp = −μ1s(a, t) + μ2r(a, t) + γ(a)p(a, t) −(μ2(a, t, P(t)) + γ(a)

)p(a, t), (2.17)

where Dp, Ds, and Dr denote the directional derivatives of p, s and r, respectively, that is,

Dp(a, t) = limh→ 0

p(a + h, t + h) − p(a, t)h

. (2.18)

Generally, ρ will not be differentiable everywhere; of course,when this occurs, Dp = ∂p/∂a +∂p/∂t, Ds = ∂s/∂a + ∂s/∂t and Dr = ∂r/∂a + ∂r/∂t.

3. Existence of a Solution to the System

If we assume that ρ = (p, s, r) is smooth along the characteristics a = t + c (except perhaps fora zero-measure set of c), considering

Bp =∫∞

0β(a, t, P(t))p(a, t)da, (3.1)


where P(t) =∫+∞

0 p(a, t)da, and integrating equalities of (2.16) along the line, we obtain thefollowing ODS

p(a, t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

p0(a − t)π1(a, t, t, ρ

)

+∫ t

0π1(a, t, σ, ρ

)[μ1(a − σ, t − σ, P(t − σ))s(a − σ, t − σ)

+ μ2(a − σ, t − σ, P(t − σ))r(a − σ, t − σ)]dσ, a ≥ t,

Bp(t − a)π1(a, t, a, ρ

)

+∫a

0π1(a, t, σ, ρ

)[μ1(a − σ, t − σ, P(t − σ))s(a − σ, t − σ)

+ μ2(a − σ, t − σ, P(t − σ))r(a − σ, t − σ)]dσ, a < t.

(3.2)

Integrating (2.7) along a = t + c, we also get p(a, t) for technical need

p(a, t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

p0(a − t)π11

(a, t, t, ρ

)

+∫ t

0π1

1

(a, t, σ, ρ

)[μ1(a − σ, t − σ, P(t − σ)) × s(a − σ, t − σ)

+μ2(a − σ, t − σ, P(t − σ))r(a − σ, t − σ)

+γ(a − σ)p(a − σ, t − σ)]dσ, a ≥ t,

Bp(t − a)π11

(a, t, a, ρ

)

+∫a

0π1

1

(a, t, σ, ρ

)[μ1(a − σ, t − σ, P(t − σ)) × s(a − σ, t − σ)

+μ2(a − σ, t − σ, P(t − σ))r(a − σ, t − σ)

+γ(a − σ)p(a − σ, t − σ)]dσ, a < t.

(3.3)

Integrating the second equation of (2.16) along a = t + c, we have

s(a, t) =

⎧⎨

⎩

s0(a − t)π2(a, t, t, ρ

), a ≥ t,

Bp(t − a)π2(a, t, a, ρ

), a < t.

(3.4)

Integrating the third equation of (2.16) along a = t + c, we obtain

r(a, t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

r0(a − t)π3(a, t, t, ρ

)

+∫ t

0π3(a, t, σ, ρ

)γ(a − σ)

(p(a − σ, t − σ) − s(a − σ, t − σ)

)dσ, a ≥ t,

Bp(t − a)π11

(a, t, a, ρ

)

+∫a

0π3(a, t, σ, ρ

)γ(a − σ)

(p(a − σ, t − σ) − s(a − σ, t − σ)

)dσ, a < t,

(3.5)


where

π1(a, t, x, ρ

)= exp

∫x

0μ2(a − s, t − s, P(t − s))ds,

π2(a, t, x, ρ

)= exp

∫x

0

(μ1(a − s, t − s, P(t − s)) + λ

(a − σ, t − σ, p − s − r

))ds,

π3(a, t, x, ρ

)= exp

∫x

0

(μ1(a − s, t − s, P(t − s)) + γ(a − σ)

)ds,

π11

(a, t, x, ρ

)= exp

∫x

0

(μ2(a − s, t − s, P(t − s)) + γ(a − σ)

)ds.

(3.6)

We can easily see that solving (2.16) is equivalent to finding a solution to (3.2), (3.4)and (3.5) or (3.3), (3.4), and (3.5) (see [3]). So, in the sequel, we restrict our attention to theseintegral equations.

Let us consider r = log(|ρ0|1 + e) with ρ0, and w > 0 fixed. Consider the set

Cr,w ={ρ ∈ V |

∣∣ρ(·, t)∣∣

1 ≤ exp(rewt

)a.e. t ∈ I

}. (3.7)

The following result provides some useful estimates.

Lemma 3.1. Suppose (H1)–(H5), and let ρ := (p, s, r), ρ′ := (p′, s′, r ′) ∈ Cr,w, a ∈ R+, and t ∈ I.Then for x ≤ min{a, t},

(i)

∣∣πi(a, t, x; ρ

)∣∣ ≤ 1, i = 1, 2, 3. (3.8)

(ii) ∃M(T) > 0 such that

|P(t)|,∣∣Bp(t)

∣∣ ≤M(T) a.e. t ∈ I. (3.9)

(iii) ∃Cj(T) > 0, such that


)− πi(a, t, x; ρ

)∣∣ ≤Cj(T)k

∣∣ρ − ρ′∣∣V e

kt, i = 1, 3, j = 6, 7. (3.10)

(iv) ∃C(T,K∞) > 0, such that

∣∣π2(a, t, x; ρ

)− π2

(a, t, x; ρ′

)∣∣ ≤ C(T,K∞)k

∣∣ρ − ρ′∣∣V e

kt. (3.11)


Proof. Firstly, note that (3.8) and (3.9) are immediate. On the other hand,


)− πi(a, t, x; ρ

)∣∣

≤∫x

0

∣∣μi(a − s, t − s, P(t − s)) − μi

(a − s, t − s, P ′(t − s)

)∣∣ds

≤ Cj(T)∫x

0

∣∣P(t − s) − P ′(t − s)

∣∣ds

= Cj(T)∫ t

t−x

∣∣P(s) − P ′(s)

∣∣ds

= Cj(T)∫ t

t−x

∣∣p(s, ·) − p′(s, ·)∣∣L1ds

≤Cj(T)k

∣∣ρ − ρ′∣∣V e

kt, i = 1, 3, j = 6, 7,

∣∣π2(a, t, x; ρ

)− π2

(a, t, x; ρ

)∣∣

≤∫x

0

∣∣μ1(a − s, t − s, P(t − s)) − μ1(a − s, t − s, P ′(t − s)

)∣∣ds

+∫x

0

∣∣λ(a − s, t − s; p − s − r

)− λ(a − s, t − s, p′ − s′ − r ′

)∣∣ds

≤ C1(T)∫x

0

∣∣P(t − s) − P ′(t − s)∣∣ds + C(T,K∞)

∫x

0

∣∣ρ − ρ′∣∣L1ds

= C1(T)∫ t

t−xekte−kt

∣∣p(s, ·) − p′(s, ·)∣∣L1ds + C(T,K∞)

∫x

0ekte−kt

∣∣ρ − ρ′∣∣L1ds

≤ C1(T)k

∣∣ρ − ρ′∣∣V e

kt +C(T,K∞)

k

∣∣ρ − ρ′∣∣V e

kt.

(3.12)

We set C1K = C1(T) + C(T,K∞), and then

∣∣π2(a, t, x; ρ

)− π2

(a, t, x; ρ

)∣∣ ≤ C1K

k

∣∣ρ − ρ′∣∣V e

kt. (3.13)

Lemma 3.2. Suppose (H1)–(H5), if ρ = (p, s, r) ∈ V satisfies (3.2), (3.4), and (3.5), or (3.3), (3.4),and (3.5), then there exists a constant w > 0, depending only on T and γ∞, such that ρ ∈ Cr,w withCr,w defined in (3.7).


Proof. Suppose that ρ = (p, s, r) ∈ V satisfies the above assumptions. Considering (3.2), (3.4)and (3.5), or (3.3), (3.4), and (3.5), thanks to (3.7) and an obvious change of variables in theintegrals, we have for all t ∈ I,

∣∣ρ(t, ·)

∣∣

1

=∥∥ρ1(t, ·)

∥∥ +∥∥ρ2(t, ·)

∥∥ +∥∥ρ3(t, ·)

∥∥

=∫∞

t

p0(a − t)π1(a, t, t; ρ

)da

+∫∞

t

∫ t

0π1(a, t, σ; ρ

)[μ1(a − σ, t − σ, P(t − σ))s(a − σ, t − σ)

+ μ2(a − σ, t − σ, P(t − σ))r(a − σ, t − σ)]dσ da

+∫ t

0Bp(t − a)π1

(a, t, a; ρ

)da

+∫ t

0

∫a

0π1(a, t, σ; ρ

)[μ1(a − σ, t − σ, P(t − σ))s(a − σ, t − σ)

+ μ2(a − σ, t − σ, P(t − σ))r(a − σ, t − σ)]dσ da

+∫∞

t

s0(a − t)π2(a, t, t; ρ

)da +

∫ t

0Bp(t − a)π2

(a, t, a; ρ

)da +

∫∞

t

r0(a − t)π3(a, t, t; ρ

)da

+∫∞

t

∫ t

0γ(a − σ)π3

(a, t, σ; ρ

)×[p(a − σ, t − σ) − s(a − σ, t − σ)

]dσ da

+∫ t

0

∫a

0γ(a − σ)π3

(a, t, σ; ρ

)×[p(a − σ, t − σ) − s(a − σ, t − σ)

]dσ da

=∫∞

t

ρ0(a − t)π1(a, t, t; ρ

)da

+∫ t

0

∫∞

t−σπ1(a + t − σ, t, t − σ; ρ

)[μ1(a, σ, P(σ))s(a, σ) + μ2(a, σ, P(σ))r(a, σ)

]dadσ

+∫ t

0Bp(t − a)π1

(a, t, a; ρ

)da

+∫ t

0

∫ t−σ

0π1(a + t − σ, t, t − σ; ρ

)[μ1(a, σ, P(σ))s(a, σ) + μ2(a, σ, P(σ))r(a, σ)

]dadσ

+∫ t

0Bp(t − a)π2

(a, t, a; ρ

)da

+∫ t

0

∫∞

t−σγ(a)π3

(a + t − σ, t, t − σ; ρ

)[p(a, σ) − s(a, σ)

]dadσ

+∫ t

0

∫ t−σ

0γ(a)π3

(a + t − σ, t, t − σ; ρ

)×[p(a, σ) − s(a, σ)

]dadσ


≤∫∞

t

∣∣ρ0(a − t)

∣∣da +

∫ t

0

∫∞

0

∣∣μ1(a, σ, P(σ))s(a, σ)

∣∣dadσ

+∫ t

0

∫∞

0

∣∣μ2(a, σ, P(σ))r(a, σ)

∣∣dadσ

+ 2∫ t

0

∣∣Bp(t − a)

∣∣da +

∫ t

0

∫∞

0

∣∣γ(a)p(a, σ) − s(a, σ)

∣∣dadσ

≤∣∣ρ0(a − t)

∣∣

1 + μ1

∫ t

0‖s(·, σ)‖L1dσ + μ1

∫ t

0‖r(a, σ)‖L1dσ

+ 2∫ t

0log(|P | + e)

∥∥p(·, σ)

∥∥L1dσ + γ∞

∫ t

0

∥∥p(·, σ)

∥∥L1 + γ∞

∫ t

0‖s(·, σ)‖L1dσ

≤∣∣ρ0(a − t)

∣∣1 +(2 + γ∞

)∫ t

0log(|P | + e)

∥∥p(·, σ)∥∥L1dσ +

(μ1 + γ∞

)∫ t

0‖s(·, σ)‖L1dσ

+ μ2

∫ t

0‖r(a, σ)‖L1dσ

≤∣∣ρ0(a − t)

∣∣1 +(2 + 2γ∞ + μ1 + μ2

)∫ t

0log(|P | + e)

∥∥ρ(·, σ)∥∥

1dσ.

(3.14)

We use the Gronwall’s inequality, and then

∣∣ρ(t, ·)∣∣

1 ≤∣∣ρ0∣∣

1 expw∫ t

0log(|P | + e)du, (3.15)

where w = 2 + 2γ∞ + μ1 + μ2 and μi(a, t, P(t)) ≤ μi, i = 1, 2.Let us consider the map ρ = (ρ1, ρ2, ρ3) ∈ V → F(ρ) = (F1(ρ), F2(ρ), F3(ρ)) ∈ V , where

F(ρ) is defined by

F1(ρ)(a, t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

p0(a − t)π1(a, t, t; ρ

)

+∫ t

0π1(a, t, σ; ρ

)[μ1(a − σ, t − σ) × F2

(ρ)(a − σ, t − σ)

+ μ2 × F3(ρ)(a − σ, t − σ)

]dσ, t < a,

Bp(t − a)π1(a, t, a; ρ

)

+∫a

0π1(a, t, σ; ρ

)[μ1(a − σ, t − σ) × F2

(ρ)(a − σ, t − σ)

+ μ2 × F3(ρ)(a − σ, t − σ)

]dσ, t ≥ a,

(3.16)


and also F1(ρ) can be equal to

F1(ρ)(a, t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

p0(a − t)π11

(a, t, t; ρ

)

+∫ t

0π1

1

(a, t, σ; ρ

)[μ1(a − σ, t − σ) × F2

(ρ)(a − σ, t − σ)

+ μ2 × F3(ρ)(a − σ, t − σ)

]dσ, t < a,

Bp(t − a)π11

(a, t, a; ρ

)

+∫a

0π1

1

(a, t, σ; ρ

)[μ1(a − σ, t − σ) × F2

(ρ)(a − σ, t − σ)

+ μ2 × F3(ρ)(a − σ, t − σ)

]dσ, t ≥ a,

(3.17)

F2(ρ)(a, t) =

⎧⎨

⎩

s0(a − t)π2(a, t, t; ρ

), t < a,

Bp(t − a)π2(a, t, a; ρ

), t ≥ a,

(3.18)

F3(ρ)(a, t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

r0(a − t)π3(a, t, t; ρ

)

+∫ t

0γ(a − σ)π3

(a, t, σ; ρ

)(ρ1(a − σ, t − σ) − F2

(ρ)(a − σ, t − σ)

)dσ, t < a,

∫a

0γ(a − σ)π3

(a, t, σ; ρ

)

×(ρ1(a − σ, t − σ) − F2

(ρ)(a − σ, t − σ)

)dσ, t ≥ a,

(3.19)

where p(t) =∫∞

0 ρ1(a, t)da.

Lemma 3.3. With the assumptions of Lemma 3.2, we have F : V → V .

Proof. In this proof we denote, for abbreviation,

Γk(x, y)= exp

(−∫y

x

μk(a − s, t − s;P(t − s))ds), k = 1, 2,

Φ1(x, y)= exp

(−∫y

x

λ(a − s, t − s; p − s − r

)ds

),

Φ2(x, y)= exp

(−∫y

x

γ(a − s, t − s; p − s − r

)ds

).

(3.20)

If ρ ∈ V , then P(t) ∈ L∞(I). Then β(a, t, P(t)), μi(a, t, P(t)) ∈ L∞(R+ × I), i = 1, 2, by (2.5) and(2.7). Hence, F is clearly measurable in a and essentially bounded on I.

By (3.18), F2(ρ)(a, t) ≥ 0, a.e. (a, t) ∈ R+ × I. So, we only need to show that F1(ρ)(a, t) ≥F2(ρ)(a, t), F1(ρ)(a, t) ≥ F3(ρ)(a, t), a.e. (a, t) ∈ R+ × I, F3(ρ) ≥ 0, and F1(ρ) ≥ 0, a.e. (a, t) ∈


R+ × I. We assume that a ≥ t (the discussion for a < t is similar). Using (3.11) and (3.19) andsubstituting F2 and F3 into F1 we get

F1(ρ)(a, t) − F2

(ρ)(a, t)

= p0(a − t)π1(a, t, t, ρ

)− s0(a − t)π2

(a, t, t, ρ

)

+∫ t

0π1(a, t, σ, ρ

)(μ1(a − σ, t − σ, P(t − σ))F2

(ρ)(a − σ, t − σ)

+ μ2(a − σ, t − σ, P(t − σ))F3(ρ)(a − σ, t − σ)

)dσ

=: p0(a − t)π1(a, t, t, ρ

)− s0(a − t)π2

(a, t, t, ρ

)+A + B.

(3.21)

Now, we proceed to estimate these quantities to see that F1 ≥ F2. By the mean value theorem,there exists t1 ∈ (0, t), such that

A =∫ t

0π1(a, t, σ, ρ

)μ1(a − σ, t − σ, P(t − σ))F2

(ρ)(a − σ, t − σ)dσ

=∫ t

0π1(a, t, σ, ρ

)μ1(a − σ, t − σ, P(t − σ))s0(a − t)π2

(a − σ, t − σ, t − σ, ρ

)dσ

= s0(a − t)π2(a, t, t, ρ

)(1 − Γ(0, t))Ψ−1(0, t1),

(3.22)

where

B =∫ t

0π1(a, t, σ, ρ

)μ2(a − σ, t − σ, P(t − σ))F3

(ρ)(a − σ, t − σ)dσ

=∫ t

0π1(a, t, σ, ρ

)μ2(a − σ, t − σ, P(t − σ))

×[

r0(a − t)π3(a − σ, t − σ, t − σ, ρ

)dσ

+∫ t

σ

γ(a − s)π3(a − σ, t − σ, s − σ, ρ

)(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)]

dsdσ

= r0(a − t)∫ t

0π1(a, t, σ, ρ

)μ2(a − σ, t − σ, P(t − σ))π3

(a − σ, t − σ, t − σ, ρ

)dσ

+∫ t

0π1(a, t, σ, ρ

)μ2(a − σ, t − σ, P(t − σ))

×∫ t

σ

γ(a − s)π3(a − σ, t − σ, s − σ, ρ

)(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)dsdσ

=: B1 + B2.

(3.23)


By the mean value theorem, there exists t2 ∈ (0, t), such that

B1 = r0(a − t)∫ t

0π1(a, t, σ, ρ

)μ2(a − σ, t − σ, P(t − σ))π3

(a − σ, t − σ, t − σ, ρ

)dσ

= r0(a − t)π3(a, t, t, ρ

)(1 − Γ2(0, t))Ψ−1

2 (0, t2),

(3.24)

and by the mean value theorem, there exists ts ∈ (0, t), such that

B2 =∫ t

0π1(a, t, σ, ρ

)μ2(a − σ, t − σ, P(t − σ))

×∫ t

σ

γ(a − s)π3(a − σ, t − σ, s − σ, ρ

)(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)dsdσ

=∫ t

0γ(a − s)

(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)∫s

0π1(a, t, σ, ρ

)

× μ2(a − σ, t − σ, P(t − σ))π3(a − σ, t − σ, s − σ, ρ

)dσ ds

=∫ t

0γ(a − s)

(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)π3(a, t, s, ρ

)

×∫s

0− d

dσΓ2(0, σ)Ψ−1

2 (0, σ)dσ2ds

=∫ t

0γ(a − s)

(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)π3(a, t, s, ρ

)(1 − Γ2(0, s))Ψ−1

2 (0, ts).

(3.25)

We substitute A,B1, and B2 into the formula of F1(ρ)(a, t) − F2(ρ)(a, t). Thus,

F1(ρ)(a, t) − F2

(ρ)(a, t)

= p0(a − t)π1(a, t, t, ρ

)− s0(a − t)π2

(a, t, t, ρ

)

+ s0(a − t)π2(a, t, t, ρ

)(1 − Γ1(0, t))Ψ−1(0, t1)

+ r0(a − t)π3(a, t, t, ρ

)(1 − Γ2(0, t))Ψ−1

2 (0, t2)

+∫ t

0γ(a − s)

(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)

× π3(a, t, s, ρ

)(1 − Γ2(0, s))Ψ−1

2 (0, ts)ds


≥ p0(a − t)π1(a, t, t, ρ

)− s0(a − t)π2

(a, t, t, ρ

)Γ1(0, t)Ψ−1(0, t1) + B1 + B2

≥ p0(a − t)π1(a, t, t, ρ

)− s0(a − t)π1

(a, t, t, ρ

)Ψ1(t1, t) + B1 + B2

≥ p0(a − t)π1(a, t, t, ρ

)(1 −Ψ1(t1, t)) + B1

+∫ t

0γ(a − s)

(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)

× π3(a, t, s, ρ

)(1 − Γ2(0, s))Ψ−1

2 (0, ts)ds

≥(p0(a − t)π1

(a, t, t, ρ

)(1 −Ψ1(t1, t)) + B1

)e∫ t

0γ(a−s)π3(a,t,s,ρ)(1−Γ2(0,s))Ψ−12 (0,ts)

≥ 0.

(3.26)

By the formula of F3(ρ), we have F3(ρ) ≥ 0. Using (3.17) and (3.18) and substitutingF2 and F3 into F1 we get

F1(ρ)(a, t) − F3

(ρ)(a, t)

= p0(a − t)π1(a, t, t, ρ

)− r0(a − t)π3

(a, t, t, ρ

)

+∫ t

0π1

1

(a, t, σ, ρ

)(μ1(a − σ, t − σ, P(t − σ))F2

(ρ)(a − σ, t − σ)

+ μ2(a − σ, t − σ, P(t − σ))F3(ρ)(a − σ, t − σ)

)dσ

−∫ t

0γ(a − σ)π3

(a, t, σ, ρ

)(ρ1(a − σ, t − σ) − F2

(ρ)(a − σ, t − σ)

)dσ

=: p0(a − t)π1(a, t, t, ρ

)− r0(a − t)π3

(a, t, t, ρ

)+A1 + C1 + C2.

(3.27)

By the mean value theorem, there exists t3, t4 ∈ (0, t), such that

A1 =∫ t

0π1

1

(a, t, σ, ρ

)μ1(a − σ, t − σ, P(t − σ))F2

(ρ)(a − σ, t − σ)dσ

=∫ t

0π1

1

(a, t, σ, ρ

)μ1(a − σ, t − σ, P(t − σ))s0(a − t)π2

(a − σ, t − σ, t − σ, ρ

)dσ

= s0(a − t)π2(a, t, t, ρ

)(1 − Γ1(0, t))Ψ−1

1 (0, t3)Ψ−12 (0, t4).

(3.28)


By the mean value theorem, we have the following:

C1 = r0(a − t)∫ t

0π1

1

(a, t, σ, ρ

)μ2(a − σ, t − σ, P(t − σ))π3

(a − σ, t − σ, t − σ, ρ

)dσ

= r0(a − t)π3(a, t, t, ρ

)(1 − Γ2(0, t)),

C2 =∫ t

0π1

1

(a, t, σ, ρ

)μ2(a − σ, t − σ, P(t − σ))

×∫ t

σ

γ(a − s)π3(a − σ, t − σ, s − σ, ρ

)(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)dsdσ

=∫ t

0γ(a − s)

(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)

×∫ s

0π1

1

(a, t, σ, ρ

)μ2(a − σ, t − σ, P(t − σ))π3

(a − σ, t − σ, s − σ, ρ

)dσ ds

=∫ t

0γ(a − s)

(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)π3(a, t, s, ρ

)∫s

0− d

dσΓ2(0, σ)dσ ds

=∫ t

0γ(a − s)

(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)π3(a, t, s, ρ

)(1 − Γ2(0, s)).

(3.29)

Thus,

F1(ρ)(a, t) − F3

(ρ)(a, t)

= p0(a − t)π11

(a, t, t, ρ

)− r0(a − t)π3

(a, t, t, ρ

)

+∫ t

0π1

1

(a, t, σ, ρ

)μ1(a − σ, t − σ, P(t − σ))F2

(ρ)(a − σ, t − σ)dσ

+ r0(a − t)π3(a, t, t, ρ

)(1 − Γ2(0, t))

−∫ t

0γ(a − s)

(ρ1(a − s, t − s) − F2

(ρ)(a − s, t − s)

)π3(a, t, s, ρ

)(1 − Γ2(0, s))ds

≥(p0(a − t) − r0(a − t)

)π1

1

(a, t, t, ρ

)

+∫ t

0π1

1

(a, t, σ, ρ

)μ1(a − σ, t − σ, P(t − σ))F2

(ρ)(a − σ, t − σ)dσ ≥ 0.

(3.30)

So that F1(ρ)(a, t) ≥ F2(ρ)(a, t) ≥ 0, F1(ρ)(a, t) ≥ F3(ρ)(a, t) ≥ 0, a.e. a ∈ (t,+∞), andwe can conclude that for each ρ ∈ V , F(ρ) ∈ V .

Theorem 3.4. Suppose (H1)–(H5), for each T > 0 and for each ρ0 = (p0, s0, r0) ∈ (L1(R+))3, with

p0 ≥ s0, p0 ≥ r0, there exists a unique ρ = (p, s, r) ∈ V satisfying (3.2), (3.4), and (3.5), or (3.3),(3.4), and (3.5). And so, ρ is the unique solution to problem (2.10).


Proof. In order to prove the theorem, it remains to be shown that F (defined by (3.11) and(3.18)) has a unique point fixed in V .

Let Cr,w be defined by (3.7); then for w being large enough F maps Cr,w into Cr,w.Indeed, by estimate of |ρ(·, ·)|1, we get, for almost all t ∈ 1

∣∣F(ρ)(·, t)

∣∣

1 =∥∥F1(ρ)(·, t)

∥∥L1 +

∥∥F2(ρ)(·, t)

∥∥L1 +

∥∥F3(ρ)(·, t)

∥∥L1

=∫+∞

t

(p0(a − t)π1

(a, t, t, ρ

)+ s0(a − t)π2

(a, t, t, ρ

)+ r0(a − t)π3

(a, t, t, ρ

))da

+∫ t

0

(Bp(t − a)π1

(a, t, a, ρ

)+ Bp(t − a)π2

(a, t, a, ρ

))da

+∫ t

0

∫a

0π1(a, t, σ, ρ

)[μ1(a − σ, t − σ)F2

(ρ)(a − σ, t − σ)

+ μ2(a − σ, t − σ)F3(ρ)(a − σ, t − σ)

]dσ da

+∫+∞

t

∫ t

0π1(a, t, σ, ρ

)[μ1(a − σ, t − σ)F2

(ρ)(a − σ, t − σ)

+ μ2(a − σ, t − σ)F3(ρ)(a − σ, t − σ)

]dσ da

+∫ t

0

∫a

0γ(a − σ)π3

(a, t, σ, ρ

)[ρ1(a − σ, t − σ) − F2

(ρ)(a − σ, t − σ)

]dσ da

+∫+∞

t

∫a

0γ(a − σ)π3

(a, t, σ, ρ

)[ρ1(a − σ, t − σ) − F2

(ρ)(a − σ, t − σ)

]dσ da

≤∫+∞

t

(∣∣p0(a − t)∣∣ + |s0(a − t)| + |r0(a − t)|

)da + 2

∫ t

0Bp(t − a)da

+∫ t

0

∫+∞

0π1(a+t − σ, t, t−σ, ρ

)[μ1(a, σ)F2

(ρ)(a, σ)+μ2(a, σ)F3

(ρ)(a, σ)

]dadσ

+∫ t

0

∫+∞

0γ(a − σ)π3

(a + t − σ, t, t − σ, ρ

)[ρ1(a, σ) − F2

(ρ)(a, σ)

]dadσ

≤∣∣ρ0(·, t)

∣∣1 + μ1

∫ t

0

∥∥F2(ρ)(·, u)∥∥L1du + μ2

∫ t

0

∥∥F3(ρ)(·, u)∥∥L1du + γ∞

∫ t

0

∥∥ρ1(·, u)∥∥du

+ γ∞

∫ t

0

∥∥F2(ρ)(·, u)∥∥L1du

≤∣∣ρ0(·, t)

∣∣1 +(μ1 + 2γ∞ + μ2 + 2C4(T)

)∫ t

0log(|P | + e)

∥∥ρ(·, u)∥∥L1du.

(3.31)

And from Gronwall’s inequality, it follows that

∣∣F(ρ)(·, t)∣∣

1 ≤ exp(rewt

), a.e. t ∈ I, (3.32)


for w > 0 depending on T , μ1, μ2, and γ∞. Hence, we have proved that F maps Cr,w into Cr,w.Let us assume that w is fixed such that F(ρ) remains in Cr,w for ρ in Cr,w. Clearly,

Cr,w is closed in V and to prove that F has a unique fixed point in Cr,w, it suffices to provethat F is a strict contraction, for instance for the norm defined in definition of |ρ|V with ksuitable. For convenience in the following we denote M a certain (which may change) butwhich is independent of a, t and P(t). For ρ := (p, s, r), ρ′ := (p′, s′, r ′) ∈ Cr,w, let us estimate|F(ρ) − F(ρ′)|V .

First, for almost all t ∈ I,

∣∣F(ρ) − F(ρ′)

∣∣

1 =∫+∞

0

∣∣F1(ρ)(a, t) − F1

(ρ′)(a, t)

∣∣da +

∫+∞

0

∣∣F2(ρ)(a, t) − F2

(ρ′)(a, t)

∣∣da

+∫+∞

0

∣∣F3(ρ)(a, t) − F3

(ρ′)(a, t)

∣∣da

= f1(t) + f2(t) + f3(t).(3.33)

Now, substituting the expression of F1 into f1, we get

f1(t) ≤∫ t

0

∣∣Bp(t − a)π1(a, t, a, ρ

)− Bp′(t − a)π1

(a, t, a, ρ′

)∣∣da

+∫+∞

t

∣∣p0(a − t)∣∣∣∣π1

(a, t, t, ρ

)− π1

(a, t, t, ρ′

)∣∣da

+∫ t

0

∫∞

0

∣∣π1(a + t − σ, t, t − σ, ρ

)μ1(a, σ, P(σ))F2

(ρ)(a, σ)

− π1(a + t − σ, t, t − σ, ρ′

)μ1(a, σ, P ′(σ)

)F2(ρ′)(a, σ)

∣∣dadσ

+∫ t

0

∫∞

0

∣∣π1(a + t − σ, t, t − σ, ρ

)μ2(a, σ, P(σ))F3

(ρ)(a, σ)

− π1(a + t − σ, t, t − σ, ρ′

)μ2(a, σ, P ′(σ)

)F3(ρ′)(a, σ)

∣∣dadσ

= f11 (t) + f

21 (t) + f

31 (t) + f

41 (t),

(3.34)

where P(t) :=∫+∞

0 p(a, t)da and P ′(t) :=∫+∞

0 p′(a, t)da. Hence

f21 (t) =

∫∞

t

∣∣p0(a − t)∣∣∣∣π1

(a, t, a, ρ

)− π1

(a, t, a, ρ′

)∣∣da.

≤ C(T)k

ekt∣∣ρ − ρ′

∣∣∫+∞

t

∣∣p0(a − t)∣∣da

≤ C(T)k

∣∣ρ − ρ′∣∣V

∥∥ρ0∥∥L1e

kt.

(3.35)


Estimate of f11 . By (3.10), we have

f11 (t) ≤

∫ t

0

∣∣Bp(t − a)

∣∣∣∣π1(a, t, a, ρ

)− π1

(a, t, a, ρ′

)∣∣da

+∫ t

0

∣∣Bp(t − a) − Bp′(t − a)

∣∣∣∣π1(a, t, a, ρ′

)∣∣da

:= f111 + f12

1 .

(3.36)

Let us now estimate f111 . By (3.9) and (3.10), we get

f111 ≤

MC(T)k

∣∣ρ − ρ′

∣∣V e

kt. (3.37)

Let us estimate f121 (t). Thanks to (2.7), (2.8), and (3.8), we have

f121 (t) ≤

∫ t

0

∣∣Bp(t − a) − Bp′(t − a)∣∣da =

∫ t

0

∣∣Bp(u) − Bp′(u)∣∣du

≤∫ t

0

∫+∞

0

∣∣β(a, u, P(u))∣∣∣∣p(a, u) − p′(a, u)

∣∣dadu

+∫ t

0

∫+∞

0

∣∣β(a, u, P(u)) − β(a, u, P ′(u)

)∣∣∣∣p′(a, u)∣∣dadu

≤∫ t

0C4(T) log(|P | + e)

∥∥p(·, u) − p′(·, u)∥∥L1du +

∫ t

0C3(T)M

∥∥p(·, u) − p′(·, u)∥∥L1du

≤(C4(T) log(|P | + e) + C3(T)M

)∥∥p(·, u) − p′(·, u)∥∥L1du

≤M∫ t

0

∣∣ρ − ρ′∣∣

1du ≤M

k

∣∣ρ − ρ′∣∣V e

kt.

(3.38)

Second, let us estimate f2. Substituting the expression for F2 into f2 and applying (3.11), weobtain

f2(t) ≤∫+∞

t

|s0(a − t)|∣∣π2(a, t, a, ρ

)− π2

(a, t, a, ρ′

)∣∣da

+∫ t

0

∣∣Bp(t − a)∣∣∣∣π2

(a, t, a, ρ

)− π2

(a, t, a, ρ′

)∣∣da

≤C(T,K∞)‖s0‖L1

k

∣∣ρ − ρ′∣∣V e

kt +MC(T,K∞)

k

∣∣ρ − ρ′∣∣V e

kt +M

k

∣∣ρ − ρ′∣∣V e

kt

≤ Mk

∣∣ρ − ρ′∣∣V e

kt.

(3.39)


Estimate of f31 . By (2.6) and (3.8), |μi(a, t, P(t))| ≤ μi, i = 1, 2, then

f31 (t) ≤ μ1

∫ t

0

∫+∞

0

∣∣π1(a + t − σ, t, t − σ, ρ

)∣∣∣∣F2(ρ)(a, σ) − F2

(ρ′)(a, σ)

∣∣dadσ

+ μ1

∫ t

0

∫+∞

0

∣∣π1(a + t − σ, t, t − σ, ρ

)− π1

(a + t − σ, t, t − σ, ρ′

)∣∣∣∣F2(ρ′)(a, σ)

∣∣dadσ

+∫ t

0

∫+∞

0

∣∣π1(a + t − σ, t, t − σ, ρ

)∣∣∣∣F2(ρ′)(a, σ)

∣∣∣∣μ1(a, σ, P(σ)) − μ1

(a, σ, P ′(σ)

)∣∣dadσ.

(3.40)

Since F2(ρ) ∈ Cr,w, we have ‖F2(ρ)‖L1 ≤M, and then

f31 (t) ≤

μ1

k

∣∣ρ − ρ′∣∣V e

kt +MC1(T)

k2

∣∣ρ − ρ′∣∣V e

kt +MC1(T)

k

∣∣ρ − ρ′∣∣V e

kt

=(M

k2+M

k

)∣∣ρ − ρ′∣∣V e

kt.

(3.41)

Finally, let us estimate of f3.

f3(t) ≤∫+∞

t

|r0(a − t)|∣∣π3(a, t, a, ρ

)− π3

(a, t, a, ρ′

)∣∣da

+∫ t

0

∫+∞

0γ(a − σ)

∣∣π3(a + t − σ, t, t − σ, ρ

)(F1(ρ)(a, σ) − F2

(ρ)(a, σ)

)

− π3(a + t − σ, t, t − σ, ρ′

)(F1(ρ′)(a, σ) − F2

(ρ′)(a, σ)

)∣∣dadσ

≤C3(T)‖r0‖L1

k

∣∣ρ − ρ′∣∣V e

kt + γ∞

×∫ t

0

∫+∞

0

∣∣F1(ρ)(a, σ)

∣∣∣∣π3(a + t − σ, t, t − σ, ρ

)− π3

(a + t − σ, t, t − σ, ρ′

)∣∣dadσ

+ γ∞

∫ t

0

∫+∞

0

∣∣π3(a + t − σ, t, t − σ, ρ′

)∣∣∣∣F1(ρ)(a, σ) − F1

(ρ′)(a, σ)

∣∣dadσ

+ γ∞

∫ t

0

∫+∞

0

∣∣F2(ρ)(a, σ)

∣∣∣∣π3(a + t − σ, t, t − σ, ρ

)− π3

(a + t − σ, t, t − σ, ρ′

)∣∣dadσ

+ γ∞

∫ t

0

∫+∞

0

∣∣π3(a + t − σ, t, t − σ, ρ′

)∣∣

×∣∣F2(ρ)(a, σ) − F2

(ρ′)(a, σ)

∣∣dadσ

≤(C3(T)‖r0‖L1

k+γ∞C3(T)M

k2+γ∞k

+γ∞M

k2+γ∞M

k2

)∣∣ρ − ρ′∣∣V e

kt

=(M

k2+M

k

)∣∣ρ − ρ′∣∣V e

kt.

(3.42)


0123456789

10

S

0 100 200 300 400 500 600 700 800 900 1000

t

S

Susceptible population

(a)

0123456789

10

I

0 100 200 300 400 500 600 700 800 900 1000

I

t

Infected population

(b)

00.5

11.5

22.5

33.5

44.5

5

R

0 100 200 300 400 500 600 700 800 900 1000

R

t

Removed population

(c)

Figure 1: The temporal solution found by numerical integration of problem with initial values s0(a) = 10,i0(a) = 10, r0(a) = 5. They show that system (2.2) has a unique positive periodic solution.

Therefore, joining all above estimates, we see that for almost all t ∈ 1, there exist M > 0 andM > 0 depending only on ρ0, T , γ∞, μi, i = 1, 2, and K∞, such that

∣∣F(ρ)(·, t) − F(ρ′)(·, t)∣∣

1 ≤(M

k+M

k2

)∣∣ρ − ρ′

∣∣V e

kt. (3.43)

Dividing both sides of this inequality by ekt, we obtain

∣∣F(ρ)(·, t) − F(ρ′)(·, t)∣∣V ≤(M

k+M

k2

)∣∣ρ − ρ′

∣∣V . (3.44)

And thus for k great enough F is a strict contraction with a unique fixed point in Cr,w, and soin V . This concludes the proof.

4. Discussion

In this paper, existence of positive period solution of a non-autonomous SIR epidemic modelwith age structure is studied. We obtained existence and uniqueness of this model usingintegral differential equation and a fixed theorem. The model is different from the classical


age structure epidemic model and non-autonomous epidemic model. The initial condition isnonlocal and dependent on total population. In addition, incidence law is not Lipschitzianity.The classical methods are not valid. We construct a new norm and prove the existence ofour model under definition of the new norm. We can illustrate this through two simulatesexamples. We set

β(a, t, P(t)) =

⎧⎪⎨

⎪⎩

0, 0 < a < 1,

4(

1 − P(t)13400

), a ≥ 1.

λ(a, t, i) =

⎧⎪⎨

⎪⎩

0, 0 < a < 1,

4.86P(t)

, a ≥ 1.

μ1(a, t, P(t)) = 0.02P 2(t), μ2(a, t, P(t)) = 0.05P(t),

μ3(a, t, P(t)) = 0.01P(t), γ(a) =

⎧⎨

⎩

0, 0 < a < 1,

0.01, a ≥ 1.

(4.1)

System (2.2) with above coefficients has a unique positive periodic solution. We can see itfrom Figure 1.

In the future, there are some problems that will be solved. The existence of steady stateand stability of the steady state are still discussed. If birth rate is impulsive, what results willoccur. The simulation of the age structure still to be resolved. Furthermore, what effect willoccurs, if we introduce the delay in our model.

Acknowledgments

This work is Supported by the National Sciences Foundation of China (10971178), theSciences Foundation of Shanxi (20090110053), and the Sciences Exploited Foundation ofShanxi (20081045).

References

[1] A. J. Lotka, Elements of Mathematical Biology, Dover, New York, NY, USA, 1956.[2] H. Von Foerster, “Some remarks on changing populations,” in The Kinetics of Cellular Proliferation, pp.

382–407, Grune & Stratton, New York, NY, USA, 1959.[3] M. E. Gurtin and R. C. MacCamy, “Non-linear age-dependent population dynamics,” Archive for

Rational Mechanics and Analysis, vol. 54, pp. 281–300, 1974.[4] S. N. Busenberg, M. Iannelli, and H. R. Thieme, “Global behavior of an age-structured epidemic

model,” SIAM Journal on Mathematical Analysis, vol. 22, no. 4, pp. 1065–1080, 1991.[5] M. Iannelli, F. A. Milner, and A. Pugliese, “Analytical and numerical results for the age-structured S-I-

S epidemic model with mixed inter-intracohort transmission,” SIAM Journal on Mathematical Analysis,vol. 23, no. 3, pp. 662–688, 1992.

[6] M. El-Doma, “Analysis for an SIR age-structured epidemic model with vertical transmission andvaccination,” International Journal of Ecology and Development, vol. 3, pp. 1–38, 2005.

[7] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, vol. 89 of Monographs and Textbooksin Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1985.

[8] H. R. Thieme, “Uniform weak implies uniform strong persistence for non-autonomous semiflows,”Proceedings of the American Mathematical Society, vol. 127, no. 8, pp. 2395–2403, 1999.

[9] W. Wang and X.-Q. Zhao, “Threshold dynamics for compartmental epidemic models in periodicenvironments,” Journal of Dynamics and Differential Equations, vol. 20, no. 3, pp. 699–717, 2008.


[10] T. Zhang and Z. Teng, “Permanence and extinction for a nonautonomous SIRS epidemic model withtime delay,” Applied Mathematical Modelling, vol. 33, no. 2, pp. 1058–1071, 2009.

[11] Z. Teng, Y. Liu, and L. Zhang, “Persistence and extinction of disease in non-autonomous SIRSepidemic models with disease-induced mortality,” Nonlinear Analysis: Theory, Methods & Applications,vol. 69, no. 8, pp. 2599–2614, 2008.

[12] M. Delgado, M. Molina-Becerra, and A. Suarez, “Analysis of an age-structured predator-prey modelwith disease in the prey,” Nonlinear Analysis: Real World Applications, vol. 7, no. 4, pp. 853–871, 2006.


Research ArticleComplete Asymptotic Analysis of a NonlinearRecurrence Relation with Threshold Control

Qi Ge,1 Chengmin Hou,1 and Sui Sun Cheng2

1 Department of Mathematics, Yanbian University, Yanji 133002, China2 Department of Mathematics, Tsing Hua University, Taiwan 30043, Taiwan

Correspondence should be addressed to Qi Ge, [email protected]



Copyright q 2010 Qi Ge et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We consider a three-term nonlinear recurrence relation involving a nonlinear filtering functionwith a positive threshold λ. We work out a complete asymptotic analysis for all solutions of thisequation when the threshold varies from 0+ to +∞. It is found that all solutions either tends to 0, alimit 1-cycle, or a limit 2-cycle, depending on whether the parameter λ is smaller than, equal to, orgreater than a critical value. It is hoped that techniques in this paper may be useful in explainingnatural bifurcation phenomena and in the investigation of neural networks in which each neuralunit is inherently governed by our nonlinear relation.

1. Introduction

Let N = {0, 1, 2, . . .}. In [1], Zhu and Huang discussed the periodic solutions of the followingdifference equation:

xn = axn−1 + (1 − a)fλ(xn−k), n ∈ N, (1.1)

where a ∈ (0, 1), k is a positive integer, and f : R → R is a nonlinear signal filtering functionof the form

fλ(x) =

⎧⎨

⎩

1, x ∈ (0, λ],

0, x ∈ (−∞, 0] ∪ (λ,∞),(1.2)

in which the positive number λ can be regarded as a threshold parameter.


In this paper, we consider the following delay difference equation:

xn = axn−2 + bfλ(xn−1), n ∈ N, (1.3)

where a ∈ (0, 1) and b > 0. Besides the obvious and complementary differences between(1.1) and our equation, a good reason for studying (1.3) is that the study of its behavioris preparatory to better understanding of more general (neural) network models. Anotherone is that there are only limited materials on basic asymptotic behavior of discretetime dynamical systems with piecewise smooth nonlinearities! (Besides [1], see [2–6]. Inparticular, in [2], Chen considers the equation

xn = xn−1 + g(xn−k−1), n ∈ N, (1.4)

where k is a nonnegative integer and g : R → R is a McCulloch-Pitts type function

g(ξ) =

⎧⎨

⎩

−1, ξ ∈ (σ,∞),

1, ξ ∈ (−∞, σ],(1.5)

in which σ ∈ R is a constant which acts as a threshold. In [3], convergence and periodicity ofsolutions of a discrete time network model of two neurons with Heaviside type nonlinearityare considered, while “polymodal” discrete systems in [4] are discussed in general settings.)Therefore, a complete asymptotic analysis of our equation is essential to further developmentof polymodal discrete time dynamical systems.

We need to be more precise about the statements to be made later. To this end, we firstnote that given x−2, x−1 ∈ R,we may compute from (1.3) the numbers x0, x1, x2, . . . in a uniquemanner. The corresponding sequence {xn}∞n=−2 is called the solution of (1.3) determined bythe initial vector (x−2, x−1). For better description of latter results, we consider initial vectorsin different regions in the plane. In particular, we set

Ω ={(x, y)∈ R2 | x > 0 or y > 0

}, (1.6)

which is the complement of nonpositive orthant (−∞, 0]2 and contains the positive orthant(0,∞)2. Note that Ω is the union of the disjoint sets

U = (0,∞)2 \ (0, λ]2, (1.7)

V = (0, λ]2 ∪ ((−∞, 0] × (0,+∞)) ∪ ((0,+∞) × (−∞, 0]). (1.8)

Recall also that a positive integer η is a period of the sequence {wn}∞n=α ifwη+n = wn forall n ≥ α and that τ is the least or prime period of {wn}∞n=α if τ is the least among all periodsof {wn}∞n=α. The sequence {wn}∞n=α is said to be τ-periodic if τ is the least period of {wn}∞n=α.The sequence w = {wn}∞n=α is said to be asymptotically periodic if there exist real numbers


w(0), w(1), . . . , w(ω−1), where ω is a positive integer, such that

limn→∞

wωn+i = w(i), i = 0, 1, . . . , ω − 1. (1.9)

In case {w(0), w(1), . . . , w(ω−1), w(0), w(1), . . . , w(ω−1), . . .} is an ω-periodic sequence, we saythat w is an asymptotically ω-periodic sequence tending to the limit ω-cycle (This term isintroduced since the underlying concept is similar to that of the limit cycle in the theoryof ordinary differential equations.) 〈w(0), w(1), . . . , w(ω−1)〉. In particular, an asymptotically1-periodic sequence is a convergent sequence and conversely.

Note that (1.3) is equivalent to the following two-dimensional autonomous dynamicalsystem:

(un+1

vn+1

)

=

(vn

aun + bfλ(vn)

)

, n ∈ N, (1.10)

by means of the identification xn = un+2 for n ∈ {−2,−1, 0, . . .}. Therefore our subsequentresults can be interpreted in terms of the dynamics of plane vector sequences defined by(1.10). For the sake of simplicity, such interpretations will be left in the concluding section ofthis paper.

To obtain complete asymptotic behavior of (1.3), we need to derive results for solutionsof (1.3) determined by vectors in the entire plane. The following easy result can help us toconcentrate on solutions determined by vectors in Ω.

Theorem 1.1. A solution {xn}∞n=−2 of (1.3) with (x−2, x−1) in the nonpositive orthant (−∞, 0]2 isnonpositive and tends to 0.

Proof. Let x−2, x−1 ≤ 0. Then by (1.3),

x0 = ax−2 + bfλ(x−1) = ax−2 ≤ 0,

x1 = ax−1 + bfλ(x0) = ax−1 ≤ 0,

x2 = ax0 + bfλ(x1) = a2x−2 ≤ 0,

x3 = ax1 + bfλ(x2) = a2x−1 ≤ 0,

(1.11)

and by induction, for any k ∈ N, we have

x2k = ak+1x−2 ≤ 0 ,

x2k+1 = ak+1x−1 ≤ 0.(1.12)

Since a ∈ (0, 1), we have

limn→∞

xn = 0. (1.13)

The proof is complete.


Note that if we try to solve for an equilibrium solution {x} of (1.3), then

x =b

1 − afλ(x),(1.14)

which has exactly two solutions x = 0, b/(1 − a) when λ ≥ b/(1 − a) and has the uniquesolution x = 0 when λ ∈ (0, b/(1 − a)). However, since fλ is a discontinuous function, thestandard theories that employ continuous arguments cannot be applied to our equilibriumsolutions x = 0 or b/(1−a) to yield a set of complete asymptotic criteria. Fortunately, we mayresort to elementary arguments as to be seen below.

To this end, we first note that our equation is autonomous (time invariant), and henceif {xn}∞n=−2 is a solution of (1.3), then for any k ∈ N, the sequence {yn}∞n=−2, defined by yn =xn+k for n = −2,−1, 0, . . . , is also a solution. For the sake of convenience, we need to let

A0 =λ − ba

,

aAj+1 + b = Aj, j ∈ N.

(1.15)

Then

Aj =λ − b

(1 + a + · · · + aj

)

aj+1=λ(1 − a) − baj+1(1 − a)

+b

1 − a, j ∈ N, (1.16)

Aj+1 −Aj =λ(1 − a) − b

aj+2, j ∈ N, (1.17)

λ = aA0 + b = a2A1 + ab + b = · · · = aj+1Aj + ajb + aj−1b + · · · + ab + b, j ∈N. (1.18)

We also let

B0 = −ba,

aBj+1 + b = Bj, j ∈ N.

(1.19)

Then

Bj =−b(1 + a + · · · + aj

)

aj+1=−b + aj+1b

aj+1(1 − a), j ∈ N, (1.20)

Bj+1 − Bj = −b

aj+2, j ∈ N, (1.21)

aB0 + b = a2B1 + ab + b = · · · = aj+1Bj + ajb + · · · + ab + b, j ∈ N, (1.22)

limj→∞

Bj = −∞. (1.23)


2. The Case λ > b/(1 − a)

Suppose λ > b/(1 − a). Then

limj→∞

Aj = limj→∞

{λ(1 − a) − baj+1(1 − a)

+b

1 − a

}= +∞. (2.1)

We first show the following.

Lemma 2.1. Let λ > b/(1 − a). If {xn}∞n=−2 is a solution of (1.3) with (x−2, x−1) ∈ Ω, then thereexists an integerm ∈ {−2,−1, 0, . . .} such that 0 < xm, xm+1 ≤ λ.

Proof. From our assumption, we have aλ + b < λ. Let {xn}∞n=−2 be a solution of (1.3) with(x−2, x−1) ∈ Ω. Then there are eight cases.Case 1. If 0 < x−2, x−1 ≤ λ, our assertion is true by taking m = −2.Case 2. Suppose (x−2, x−1) ∈ (0, λ] × (λ,+∞). Then (λ − b)/a > λ. Furthermore, in view of(1.17) and (2.1),

(0, λ] × (λ,+∞) = (0, λ] ×{(

λ,λ − ba

]∪∞⋃

k=1

(Ak−1, Ak]

}

. (2.2)

If x−1 ∈ (λ, (λ − b)/a], then by (1.3),

x0 = ax−2 + bfλ(x−1) = ax−2 ∈ (0, λ),

0 < x1 = ax−1 + bfλ(x0) = ax−1 + b ≤ λ − b + b = λ.(2.3)

This means that our assertion is true by taking m = 0. Next, if x−1 ∈ (A0, A1] = ((λ−b)/a, (λ−b − ab)/a2], then by (1.3) and (1.18),

x0 = ax−2 + bfλ(x−1) = ax−2 ∈ (0, λ),

x1 = ax−1 + bfλ(x0) = ax−1 + b > λ,

x2 = ax0 + bfλ(x1) = a2x−2 ∈ (0, λ),

0 < x3 = ax1 + bfλ(x2) = a2x−1 + ab + b

≤ a2A1 + ab + b = λ.

(2.4)


Thus our assertion holds by taking m = 2. If x−1 ∈ (Ap,Ap+1], where p is an arbitrary positiveinteger, then by (1.3),

x0 = ax−2 + bfλ(x−1) = ax−2 ∈ (0, λ),

x1 = ax−1 + bfλ(x0) = ax−1 + b > aAp + b = Ap−1 > λ,

x2 = ax0 + bfλ(x1) = a2x−2 ∈ (0, λ).

(2.5)

By induction,

x2p = ap+1x−2 ∈ (0, λ),

x2p+1 = ap+1x−1 + apb + · · · + ab + b > ap+1Ap + apb + · · · + ab + b

= apAp−1 + ap−1b + · · · + ab + b = aA0 + b = λ,

x2(p+1) = ax2p + bfλ(x2p+1

)= ap+2x−2 ∈ (0, λ),

x2(p+1)+1 = ax2p+1 + bfλ(x2(p+1)

)= ap+2x−1 + ap+1b + · · · + ab + b

≤ ap+2Ap+1 + ap+1b + · · · + ab + b = λ.

(2.6)

Thus our assertion holds by taking m = 2(p + 1).Case 3. Suppose (x−2, x−1) ∈ (λ,+∞)× (λ,+∞). We assert that there is a nonnegative integer μsuch that xn > λ for n = −2,−1, . . . , μ − 1 and xμ ∈ (0, λ]. Otherwise we have xn ∈ (λ,+∞) forn ∈ N. It follows that

x0 = ax−2 + bfλ(x−1) = ax−2 > λ,

x1 = ax−1 + bfλ(x0) = ax−1 > λ,

x2 = ax0 + bfλ(x1) = a2x−2 > λ,

x3 = ax1 + bfλ(x2) = a2x−1 > λ.

(2.7)

By induction, for any k ∈ N, we have

x2k = ak+1x−2 > λ,

x2k+1 = ak+1x−1 > λ,(2.8)

which implies

limk→∞

x2k = 0 = limk→∞

x2k+1. (2.9)

This is contrary to the fact that xn ∈ (λ,+∞) for n ∈ N.


Now that there exists an integer μ ∈ N such that x−2, x−1, . . . , xμ−1 ∈ (λ,+∞) and xμ ∈(0, λ], it then follows

0 < xμ+1 = axμ−1 + bfλ(xμ)= axμ−1 + b. (2.10)

If xμ+1 ≤ λ, then our assertion holds by taking m = μ. If xμ+1 > λ, then xμ−1 > (λ − b)/a.Thus

0 < xμ+2 = axμ + bfλ(xμ+1

)= axμ ≤ aλ < λ,

0 < xμ+3 = axμ+1 + bfλ(xμ+2

)= axμ+1 + b = a2xμ−1 + ab + b.

(2.11)

If 0 < xμ+3 ≤ λ, then our assertion holds by taking m = μ + 2. If xμ+3 > λ, we have xμ−1 >(λ − b − ab)/a2. Hence

0 < xμ+4 = axμ+2 + bfλ(xμ+3

)= axμ+2 = a2xμ < λ,

0 < xμ+5 = axμ+3 + bfλ(xμ+4

)= axμ+3 + b = a3xμ−1 + a2b + ab + b.

(2.12)

Repeating the procedure, we have

0 < xμ+2k = akxμ < λ,

0 < xμ+(2k+1) = ak+1xμ−1 + akb + · · · + ab + b.(2.13)

If 0 < xμ+(2k+1) ≤ λ, then our assertion holds by taking m = μ + (2k + 1). Otherwise,

xμ−1 >λ − b − ab − · · · − akb

ak+1(2.14)

for all k ∈ N. But this is contrary to (2.1). Thus we conclude that 0 < xμ+(2k+1) ≤ λ for some k.Our assertion then holds by taking m = μ + 2k.Case 4. Suppose (x−2, x−1) ∈ (λ,+∞) × (0, λ]. As in Case 2,

(λ,+∞) × (0, λ] ={(

λ,λ − ba

]∪∞⋃

k=1

(Ak−1, Ak]

}

× (0, λ]. (2.15)

If x−2 ∈ (λ, (λ − b)/a], then by (1.3),

0 < x0 = ax−2 + bfλ(x−1) = ax−2 + b ≤ λ. (2.16)


Thus our assertion holds taking m = −1. If x−2 ∈ (A0, A1], then by (1.3),

x0 = ax−2 + bfλ(x−1) = ax−2 + b > λ,

0 < x1 = ax−1 + bfλ(x0) = ax−1 + b ≤ aλ + b < λ,

0 < x2 = ax0 + bfλ(x1) = a2x−2 + ab + b ≤ a2A1 + ab + b = λ.

(2.17)

Thus our assertion holds by taking m = 1. If x−2 ∈ (Ap,Ap+1], where p is an arbitrary positiveinteger, then by (1.3),

x0 = ax−2 + bfλ(x−1) = ax−2 + b > aAp + b = Ap−1 > λ,

0 < x1 = ax−1 + bfλ(x0) = ax−1 < λ,

x2 = ax0 + bfλ(x1) = a2x−2 + ab + b > Ap−2 > λ,

...

x2p = ap+1x−2 + apb + · · · + ab + b > ap+1Ap + apb + · · · + ab + b = λ,

0 < x2p+1 = ap+1x−1 < λ,

0 < x2(p+1) = ax2p + bfλ(x2p+1

)= ap+2x−2 + ap+1b + · · · + ab + b

≤ ap+2Ap+1 + ap+1b + · · · + ab + b = λ.

(2.18)

Thus our assertion holds by taking m = 2p + 1.Case 5. Suppose (x−2, x−1) ∈ (−∞, 0] × (0, λ]. Then by (1.21) and (1.23),

(−∞, 0] × (0, λ] =

⎧⎨

⎩

⎛

⎝∞⋃

j=1

(Bj, Bj−1

]⎞

⎠ ∪(−ba, 0]⎫⎬

⎭× (0, λ]. (2.19)

If x−2 ∈ (−b/a, 0], then by (1.3),

0 < x0 = ax−2 + bfλ(x−1) = ax−2 + b ≤ b < λ. (2.20)

Thus our assertion holds by m = −1. If x−2 ∈ (B1, B0] = ((−b − ab)/a2,−b/a], then by (1.3),

x0 = ax−2 + bfλ(x−1) = ax−2 + b ≤ 0,

0 < x1 = ax−1 + bfλ(x0) = ax−1 < λ,

0 = −b − ab + ab + b < x2 = ax0 + bfλ(x1)

= a2x−2 + ab + b ≤ a2B0 + ab + b = b < λ.

(2.21)


Thus our assertion holds by taking m = 1. If x−2 ∈ (Bp+1, Bp], where p is an arbitrary positiveinteger, then by (1.3), we have

Bp = aBp+1 + b < x0 = ax−2 + bfλ(x−1) = ax−2 + b ≤ aBp + b = Bp−1,

0 < x1 = ax−1 + bfλ(x0) = ax−1 < λ.(2.22)

That is, (x0, x1) ∈ (Bp, Bp−1] × (0, λ). Therefore we may conclude our assertion by induction.Case 6. Suppose (x−2, x−1) ∈ (−∞, 0] × (λ,+∞). Since

λ

ak<

λ

ak+1, k ∈ N,

limk→∞

λ

ak+1= +∞,

(2.23)

we see that

(−∞, 0] × (λ,+∞) = (−∞, 0] ×∞⋃

k=0

(λ

ak,λ

ak+1

]. (2.24)

If x−1 ∈ (λ, λ/a], then by (1.3),

x0 = ax−2 + bfλ(x−1) = ax−2 ≤ 0,

0 < x1 = ax−1 + bfλ(x0) = ax−1 ≤ λ.(2.25)

That is, (x0, x1) ∈ (−∞, 0] × (0, λ]. We may thus apply the conclusion of Case 5 and the timeinvariance property of (1.3) to deduce our assertion. If x−1 ∈ (λ/ap+1, λ/ap+2], where p is anarbitrary nonnegative integer, then by (1.3), we have

x0 = ax−2 + bfλ(x−1) = ax−2 ≤ 0,

λ

ap< x1 = ax−1 + bfλ(x0) = ax−1 ≤

λ

ap+1.

(2.26)

That is, (x0, x1) ∈ (−∞, 0] × (λ/ap, λ/ap+1]. We may thus use induction to conclude ourassertion.Case 7. Suppose (x−2, x−1) ∈ (0, λ] × (−∞, 0]. As in Case 5,

(0, λ] × (−∞, 0] = (0, λ] ×{(

∞⋃

k=1

(Bk, Bk−1]

)

∪(−ba, 0]}

. (2.27)


If x−1 ∈ (−b/a, 0], then by (1.3),

0 < x0 = ax−2 + bfλ(x−1) = ax−2 < λ,

0 < x1 = ax−1 + bfλ(x0) = ax−1 + b ≤ b < λ.(2.28)

Thus our assertion holds by taking m = 0. If x−1 ∈ (B1, B0] = ((−b − ab)/a2,−b/a], then by(1.3),

0 < x0 = ax−2 + bfλ(x−1) = ax−2 < λ,

−ba=−b − ab

a+ b < x1 = ax−1 + bfλ(x0) = ax−1 + b ≤ 0.

(2.29)

That is, (x0, x1) ∈ (0, λ) × (−b/a, 0]. Thus our assertion holds by taking m = 2.If x−1 ∈ (Bp+1, Bp]. where p is an arbitrary positive integer, then by (1.3), we have

0 < x0 = ax−2 + bfλ(x−1) = ax−2 < λ ,

Bp < x1 = ax−1 + bfλ(x0) = ax−1 + b ≤ Bp−1.(2.30)

That is, (x0, x1) ∈ (0, λ) × (Bp, Bp−1]. Thus our assertion follows from induction.Case 8. Suppose (x−2, x−1) ∈ (λ,+∞) × (−∞, 0]. Then

(λ,+∞) × (−∞, 0] =(∞⋃

k=0

(λ

ak,λ

ak+1

])

× (−∞, 0]. (2.31)

If x−2 ∈ (λ, λ/a], then by (1.3),

0 < x0 = ax−2 + bfλ(x−1) = ax−2 ≤ λ. (2.32)

That is, (x−1, x0) ∈ (−∞, 0]× (0, λ]. We may now apply the assertion in Case 5 to conclude ourproof. If x−2 ∈ (λ/ap+1, λ/ap+2], where p is an arbitrary nonnegative integer, then by (1.3), wehave

λ

ap< x0 = ax−2 + bfλ(x−1) = ax−2 ≤

λ

ap+1,

x1 = ax−1 + bfλ(x0) = ax−1 ≤ 0.

(2.33)

That is, (x0, x1) ∈ (λ/ap, λ/ap+1] × (−∞, 0]. We may thus complete our proof byinduction.

Theorem 2.2. Suppose λ > b/(1 − a), then a solution x = {xn}∞n=−2 of (1.3) with (x−2, x−1) ∈ Ωwill tend to b/(1 − a).


Proof. In view of Lemma 2.1, we may assume without loss of generality that 0 < x−2, x−1 ≤ λ.From our assumption, we have aλ + b < λ. Furthermore, by (1.3),

0 < x0 = ax−2 + bfλ(x−1) = ax−2 + b ≤ aλ + b < λ,

0 < x1 = ax−1 + bfλ(x0) = ax−1 + b ≤ aλ + b < λ,

0 < x2 = ax0 + bfλ(x1) = a2x−2 + ab + b ≤ a2λ + ab + b = a(aλ + b) + b < aλ + b < λ,

0 < x3 = ax1 + bfλ(x2) = a2x−1 + ab + b ≤ a2λ + ab + b = a(aλ + b) + b < aλ + b < λ,

0 < x4 = ax2 + bfλ(x3) = a3x−2 + a2b + ab + b ≤ a3λ + a2b + ab + b

= a2(aλ + b) + ab + b < a2λ + ab + b < λ,


= a2(aλ + b) + ab + b < a2λ + ab + b < λ.

(2.34)


0 < x2k = ak+1x−2 + akb + ak−1b + · · · + ab + b

≤ ak+1λ + akb + ak−1b + · · · + ab + b = ak(aλ + b) + ak−1b + · · · + ab + b

< akλ + ak−1b + · · · + ab + b = ak−1(aλ + b) + ak−2b + · · · + ab + b

< ak−1λ + ak−2b + · · · + ab + b < · · · < a2λ + ab + b < λ,

(2.35)

and similarly

0 < x2k+1 = ak+1x−1 + akb + ak−1b + · · · + ab + b < λ. (2.36)

Thus x2k, x2k+1 ∈ (0, λ] for any k ∈ N and

limk→∞

x2k = limk→∞

{

ak+1x−2 + b ×1 − ak+1

1 − a

}

=b

1 − a,

limk→∞

x2k+1 = limk→∞

{

ak+1x−1 + b ×1 − ak+1

1 − a

}

=b

1 − a.

(2.37)


3. The Case λ ∈ (0, b/(1 − a))

We first show that following result.


Lemma 3.1. Let 0 < λ < b/(1 − a). If x = {xn}∞n=−2 is a solution of (1.3) with (x−2, x−1) ∈ Ω,there exists an integer m ∈ {−2,−1, 0, . . .} such that 0 < xm ≤ λ and xm+1 > λ (or xm > λ and0 < xm+1 ≤ λ).

Proof. From our assumption, we have aλ + b > λ. Let {xn}∞n=−2 be the solution of (1.3)determined by (x−2, x−1) ∈ Ω. Then there are eight cases to show that there exists an integerm ∈ {−2,−1, 0, . . .} such that 0 < xm ≤ λ and xm+1 > λ.Case 1. Suppose (x−2, x−1) ∈ (0, λ] × (λ,+∞). Then our assertion is true by taking m = −2.Case 2. Suppose (x−2, x−1) ∈ (λ,+∞) × (0, λ]. By (1.3)

x0 = ax−2 + bfλ(x−1) = ax−2 + b > aλ + b > λ. (3.1)

This means that our assertion is true by taking m = −1.Case 3. Suppose (x−2, x−1) ∈ (0, λ] × (0, λ]. If xn ∈ (0, λ] for any n ∈ N, then by (1.3),

x0 = ax−2 + bfλ(x−1) = ax−2 + b,

x1 = ax−1 + bfλ(x0) = ax−1 + b,

x2 = ax0 + bfλ(x1) = a2x−2 + ab + b,

x3 = ax1 + bfλ(x2) = a2x−1 + ab + b.

(3.2)


x2k = ak+1x−2 + akb + · · · + ab + b = ak+1x−2 + b ×1 − ak+1

1 − a ,

x2k+1 = ak+1x−1 + akb + · · · + ab + b = ak+1x−1 + b ×1 − ak+1

1 − a .

(3.3)

Hence

limk→∞

x2k =b

1 − a = limk→∞

x2k+1. (3.4)

But this is contrary to our assumption that 0 < λ < b/(1 − a). Hence there exists an integerμ ∈ {−1, 0, 1, . . .} such that x−2, x−1, . . . , xμ ∈ (0, λ] and xμ+1 ∈ (λ,+∞). Thus our assertionholds by taking m = μ.Case 4. Suppose (x−2, x−1) ∈ (λ,+∞)× (λ,+∞). As in Case 3 of Lemma 2.1, we may show thatif xn ∈ (λ,+∞) for all n ∈ N, then it follows that

limn→∞

xn = 0. (3.5)


But this is contrary to the fact that xn ∈ (λ,+∞) for n ∈ N. Hence there exists an integer μ ∈ Nsuch that x−2, x−1, . . . , xμ−1 ∈ (λ,+∞) and xμ ∈ (0, λ], it then follows

xμ+1 = axμ−1 + bfλ(xμ)= axμ−1 + b > aλ + b > λ. (3.6)

This means that our assertion is true by taking m = μ.Case 5. Suppose (x−2, x−1) ∈ (−∞, 0] × (0, λ]. Then by (1.21) and (1.23),

(−∞, 0] × (0, λ] =

⎧⎨

⎩

⎛

⎝∞⋃

j=1

(Bj, Bj−1

]⎞

⎠ ∪(−ba, 0]⎫⎬

⎭× (0, λ]. (3.7)

If x−2 ∈ (−b/a, 0], then by (1.3),

x0 = ax−2 + bfλ(x−1) = ax−2 + b > −b + b = 0. (3.8)

When λ ≥ b, we have

0 < x0 = ax−2 + b ≤ b ≤ λ. (3.9)

That is, (x−1, x0) ∈ (0, λ] × (0, λ]. We may thus apply the conclusion of Case 3 to deduce ourassertion.

Suppose λ < b. If −b/a < x−2 ≤ (λ − b)/a < 0, then we have

0 < x0 = ax−2 + b ≤ λ. (3.10)

We may apply the conclusion of Case 3 to deduce our assertion. If (λ − b)/a < x−2 ≤ 0, wehave

x0 = ax−2 + b > λ. (3.11)

Thus our assertion holds by taking m = −1. If x−2 ∈ (B1, B0] = ((−b − ab)/a2,−b/a], then by(1.3),

B0 = aB1 + b < x0 = ax−2 + bfλ(x−1) = ax−2 + b ≤ 0,

0 < x1 = ax−1 + bfλ(x0) = ax−1 < λ.(3.12)


That is, (x0, x1) ∈ (−b/a, 0] × (0, λ]. In view of the above discussions, our assertion is true. Ifx−2 ∈ (Bp+1, Bp], where p is an arbitrary positive integer, then by (1.3), we have

Bp = aBp+1 + b < x0 = ax−2 + bfλ(x−1) = ax−2 + b ≤ aBp + b = Bp−1,

0 < x1 = ax−1 + bfλ(x0) = ax−1 < λ.(3.13)

That is, (x0, x1) ∈ (Bp, Bp−1] × (0, λ]. Therefore we may conclude our assertion by induction.Case 6. Suppose

(x−2, x−1) ∈ (−∞, 0] × (λ,+∞) = (−∞, 0] ×∞⋃

k=0

(λ

ak,λ

ak+1

]. (3.14)

As in Case 6 of Lemma 2.1, if x−1 ∈ (λ, λ/a], then by (1.3), we have (x0, x1) ∈ (−∞, 0] × (0, λ].We may thus apply the conclusion of Case 5 to deduce our assertion. If x−1 ∈ (λ/ap+1, λ/ap+2],where p is an arbitrary nonnegative integer, then by (1.3), we have (x0, x1) ∈ (−∞, 0] ×(λ/ap, λ/ap+1]. We may thus use induction to conclude our assertion.Case 7. Suppose (x−2, x−1) ∈ (0, λ] × (−∞, 0]. By (1.3), we have

0 < x0 = ax−2 + bfλ(x−1) = ax−2 < λ. (3.15)

That is, (x−1, x0) ∈ (−∞, 0]× (0, λ]. We may thus apply the conclusion of Case 5 to deduce ourassertion.Case 8. Suppose (x−2, x−1) ∈ (λ,+∞) × (−∞, 0]. Then

(λ,+∞) × (−∞, 0] =∞⋃

k=0

(λ

ak,λ

ak+1

]× (−∞, 0]. (3.16)

As in Case 8 of Lemma 2.1, if x−2 ∈ (λ, λ/a], then by (1.3), we have (x−1, x0) ∈ (−∞, 0]× (0, λ].We may now apply the assertion in Case 5 to conclude our proof. If x−2 ∈ (λ/ap+1, λ/ap+2],where p is an arbitrary nonnegative integer, then by (1.3), we have (x0, x1) ∈ (λ/ap, λ/ap+1]×(−∞, 0]. We may thus complete our proof by induction.

Theorem 3.2. Let 0 < λ < b/(1 − a). Then any solution {xn}∞n=−2 of (1.3) with (x−2, x−1) ∈ Ω isasymptotically 2-periodic with limit 2-cycle 〈0, b/(1 − a)〉.


Proof. In view of Lemma 3.1, we may assume without loss of generality that 0 < x−2 ≤ λ andx−1 > λ. Then by (1.3),

0 < x0 = ax−2 + bfλ(x−1) = ax−2 < λ,

x1 = ax−1 + bfλ(x0) = ax−1 + b > aλ + b > λ,

0 < x2 = ax0 + bfλ(x1) = a2x−2 < λ,

x3 = ax1 + bfλ(x2) = a2x−1 + ab + b > a2λ + ab + b

= a(aλ + b) + b > aλ + b > λ,

0 < x4 = ax2 + bfλ(x3) = a3x−2 < λ,

x5 = ax3 + bfλ(x4) = a3x−1 + a2b + ab + b > a3λ + a2b + ab + b

= a2(aλ + b) + ab + b > a2λ + ab + b > λ.

(3.17)


0 < x2k = ak+1x−2 < λ,

x2k+1 = ak+1x−1 + akb + · · · + ab + b > ak+1λ + akb + · · · + ab + b

= ak(aλ + b) + ak−1b + · · · + ab + b > akλ + ak−1b + · · · + ab + b

> · · · > a2λ + ab + b > λ.

(3.18)

Thus x2k ∈ (0, λ] and x2k+1 ∈ (λ,+∞) for any k ∈ N. Then

limk→∞

x2k = limk→∞

ak+1x−2 = 0,

limk→∞

x2k+1 = limk→∞

{

ak+1x−1 + b ×1 − ak+1

1 − a

}

=b

1 − a.(3.19)

4. The Case λ = b/(1 − a)

Suppose λ = b/(1−a). Then λ = aλ+ b > b. We need to consider solutions with initial vectorsin U or V defined by (1.7) and (1.8), respectively.

Lemma 4.1. Let λ = b/(1 − a). If {xn}∞n=−2 is a solution of (1.3) with (x−2, x−1) ∈ V, then thereexists an integerm ∈ N such that 0 < xm, xm+1 ≤ λ.

The proof is the same as the discussions in Cases 5 through Case 8 in the proof ofLemma 2.1, and hence is skipped.

Theorem 4.2. Suppose λ = b/(1 − a), then a solution x = {xn}∞n=−2 of (1.3) with (x−2, x−1) ∈ Vwill tend to b/(1 − a).


Proof. In view of Lemma 4.1, we may assume without loss of generality that 0 < x−2, x−1 ≤ λ.By (1.3),

0 < x0 = ax−2 + bfλ(x−1) = ax−2 + b ≤ aλ + b = λ,

0 < x1 = ax−1 + bfλ(x0) = ax−1 + b ≤ aλ + b = λ,

0 < x2 = ax0 + bfλ(x1) = a2x−2 + ab + b ≤ a2λ + ab + b = a(aλ + b) + b = aλ + b = λ,

0 < x3 = ax1 + bfλ(x2) = a2x−1 + ab + b ≤ a2λ + ab + b = a(aλ + b) + b = aλ + b = λ,


= a2(aλ + b) + ab + b = a2λ + ab + b = λ,


= a2(aλ + b) + ab + b = a2λ + ab + b = λ.

(4.1)


0 < x2k = ak+1x−2 + akb + ak−1b + · · · + ab + b

≤ ak+1λ + akb + ak−1b + · · · + ab + b = ak(aλ + b) + ak−1b + · · · + ab + b

= akλ + ak−1b + · · · + ab + b = ak−1(aλ + b) + ak−2b + · · · + ab + b

= ak−1λ + ak−2b + · · · + ab + b = · · · = a2λ + ab + b = λ,

(4.2)

and similarly

0 < x2k+1 = ak+1x−1 + akb + ak−1b + · · · + ab + b ≤ λ. (4.3)

Thus x2k, x2k+1 ∈ (0, λ] for any k ∈ N. Thus (2.37) hold so that

limn→∞

xn =b

1 − a.(4.4)


Theorem 4.3. Suppose λ = b/(1 − a), then any solution {xn}∞n=−2 of (1.3) with (x−2, x−1) ∈ U isasymptotically 2-periodic with limit 2-cycle 〈0, b/(1 − a)〉.


Proof. We first discuss the case, where (x−2, x−1) ∈ (0, λ] × (λ,+∞). By (1.3),

0 < x0 = ax−2 + bfλ(x−1) = ax−2 < λ,

x1 = ax−1 + bfλ(x0) = ax−1 + b > aλ + b = λ,

0 < x2 = ax0 + bfλ(x1) = a2x−2 < λ,

x3 = ax1 + bfλ(x2) = a2x−1 + ab + b > a2λ + ab + b

= a(aλ + b) + b = aλ + b = λ,

0 < x4 = ax2 + bfλ(x3) = a3x−2 < λ,

x5 = ax3 + bfλ(x4) = a3x−1 + a2b + ab + b > a3λ + a2b + ab + b

= a2(aλ + b) + ab + b = a2λ + ab + b = λ.

(4.5)


0 < x2k = ak+1x−2 < λ,

x2k+1 = ak+1x−1 + akb + · · · + ab + b > ak+1λ + akb + · · · + ab + b = λ.(4.6)

Thus x2k ∈ (0, λ] and x2k+1 ∈ (λ,+∞) for any k ∈ N. Then

limk→∞

x2k = 0,

limk→∞

x2k+1 = limk→∞

{

ak+1x−1 + b ×1 − ak+1

1 − a

}

=b

1 − a.(4.7)

If (x−2, x−1) ∈ (λ,+∞) × (0, λ], then by (1.3),

x0 = ax−2 + bfλ(x−1) = ax−2 + b > aλ + b = λ. (4.8)

That is, (x−1, x0) ∈ (0, λ]× (λ,+∞). We may thus apply the previous conclusion to deduce ourassertion.

If (x−2, x−1) ∈ (λ,+∞)×(λ,+∞), then similar to the discussions of Case 3 of Lemma 2.1,there exists an integer μ ∈ N such that x−2, x−1, . . . , xμ−1 ∈ (λ,+∞) and xμ ∈ (0, λ]. That is,(xμ−1, xμ) ∈ (λ,+∞) × (0, λ]. In view of the previous case, our assertion holds. The proof iscomplete.

5. Concluding Remarks

The results in the previous sections can be stated in terms of the two-dimensional dynamicalsystem (1.10). Indeed, a solution of (1.10) is a vector sequence of the form {(un, vn)†}

∞n=0 that

renders (1.10) into an identity for each n ∈ N. It is uniquely determined by (u0, v0)†.


Let us say that a solution {(un, vn)†}∞n=0 of (1.10) eventually falls into a plane region Ψ

if (un, vn)† ∈ Ψ for all large n; that it is eventually falls into two disjoint plane regions Ψ1 and

Ψ2 alternately if there is somem ∈ N such that (um+2i, vm+2i)† ∈ Ψ1 and (um+2i+1, vm+2i+1)

† ∈ Ψ2

for all i ∈ N; and that it approaches a limit 2-cycle 〈(α1, β1)†, (α2, β2)

†〉 if there is some m ∈ Nsuch that (um+2i, vm+2i)

† → (α1, β1)† and (um+2i+1, vm+2i+1)

† → (α2, β2)† as i → +∞. Then we

may restate the previous theorems as follows.

(i) The vectors (0, 0)†, (0, b/(1−a))†, (b/(1 − a), b/(1 − a))†, and (b/(1−a), 0)† form thecorners of a square in the plane.

(ii) A solution {(un, vn)†}∞n=0 of (1.10) with (u0, v0)

† in the nonpositive orthant (−∞, 0]2

(is nonpositive and) tends to (0, 0)†.

(iii) Suppose λ > b/(1 − a) , then a solution {(un, vn)†}∞n=0 of (1.10) with (u0, v0)

† in Ωwill (eventually falls into (0, λ]2 and) tend to (b/(1 − a), b/(1 − a))†.

(iv) Suppose 0 < λ < b/(1 − a) , then a solution {(un, vn)†}∞n=0 of (1.10) with (u0, v0)

†

in Ω will (eventually falls into (0, λ] × (λ,+∞) and (λ,+∞) × (0, λ] alternately and)approach the limit 2-cycle 〈(0, b/(1 − a))†, (b/(1 − a), 0)†〉.

(v) Suppose λ = b/(1 − a), then a solution {(un, vn)†}∞n=0 of (1.10) with (u0, v0)

† in Vwill (eventually falls into (0, λ]2) tend to (b/(1 − a), b/(1 − a))†.

(vi) Suppose λ = b/(1 − a), Then a solution {(un, vn)†}∞n=0 of (1.10) with (u0, v0)

† in Uwill (eventually falls into (0, λ] × (λ,+∞) and (λ,+∞) × (0, λ] alternately) approachthe limit 2-cycle 〈(0, b/(1 − a))†, (b/(1 − a), 0)†〉.

Since we have obtained a complete set of asymptotic criteria, we may deduce(bifurcation) results such as the following.

If 0 < λ < b/(1−a), then all solutions {(un, vn)†}∞n=0 originated from the positive orthant

approach the limit 2-cycle 〈(0, b/(1 − a))†, (b/(1 − a), 0)†〉; if λ > b/(1 − a), then all solutionsoriginated from the positive orthant tend to (b/(1 − a), b/(1 − a))†; if λ = b/(1 − a), thenall solutions originated from the positive orthant tend to (b/(1 − a), b/(1 − a))† if (u0, v0)

† ∈(0, λ]2 and approach the limit cycle 〈(0, b/(1 − a))†, (b/(1 − a), 0)†〉 otherwise.

Roughly the above statements show that when the threshold parameter λ is a relativelysmall positive parameter, all solutions from the positive orthant tend to a limit 2-cycle; when itreaches the critical value b/(1−a), some of these solutions (those from (0, b/(1−a)]2) switchaway and tend to a limit 1-cycle, and when λ drifts beyond the critical value, all solutionstend to the limit 1-cycle. Such an observation seems to appear in many natural processesand hence our model may be used to explain such phenomena. It is also expected that whena group of neural units interact with each other in a network where each unit is governedby evolutionary laws of the form (1.3), complex but manageable analytical results can beobtained. These will be left to other studies in the future.

Acknowledgment

This project was supported by the National Natural Science Foundation of China (10661011).


References

[1] H. Y. Zhu and L. H. Huang, “Asymptotic behavior of solutions for a class of delay difference equation,”Annals of Differential Equations, vol. 21, no. 1, pp. 99–105, 2005.

[2] Y. Chen, “All solutions of a class of difference equations are truncated periodic,” Applied MathematicsLetters, vol. 15, no. 8, pp. 975–979, 2002.

[3] Z. H. Yuan, L. H. Huang, and Y. M. Chen, “Convergence and periodicity of solutions for a discrete-timenetwork model of two neurons,” Mathematical and Computer Modelling, vol. 35, no. 9-10, pp. 941–950,2002.

[4] H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, vol.15 of Mathematical Modelling: Theory and Applications, Kluwer Academic Publishers, Dordrecht, TheNetherlands, 2003.

[5] M. di Bernardo, C. J. Budd, A. R. Champneys, and P. Kowalczyk, Piecewise Smooth Dynamical Systems,Springer, New York, NY, USA, 2008.

[6] C. M. Hou and S. S. Cheng, “Eventually periodic solutions for difference equations with periodiccoefficients and nonlinear control functions,” Discrete Dynamics in Nature and Society, vol. 2008, ArticleID 179589, 21 pages, 2008.


Research ArticleExistence and Uniqueness of Positive Solutions forDiscrete Fourth-Order Lidstone Problem witha Parameter

Yanbin Sang,1, 2 Zhongli Wei,2, 3 and Wei Dong4

1 Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China2 School of Mathematics, Shandong University, Jinan, Shandong 250100, China3 Department of Mathematics, Shandong Jianzhu University, Jinan, Shandong 250101, China4 Department of Mathematics, Hebei University of Engineering, Handan, Hebei 056021, China

Correspondence should be addressed to Yanbin Sang, [email protected]

Received 9 January 2010; Revised 23 March 2010; Accepted 26 March 2010


Copyright q 2010 Yanbin Sang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This work presents sufficient conditions for the existence and uniqueness of positive solutions fora discrete fourth-order beam equation under Lidstone boundary conditions with a parameter; theiterative sequences yielding approximate solutions are also given. The main tool used is monotoneiterative technique.

1. Introduction

In this paper, we are interested in the existence, uniqueness, and iteration of positive solutionsfor the following nonlinear discrete fourth-order beam equation under Lidstone boundaryconditions with explicit parameter β given by

Δ4y(t − 2) − βΔ2y(t − 1) = h(t)[f1(y(t)

)+ f2

(y(t)

)], t ∈ [a + 1, b − 1]

Z, (1.1)

y(a) = 0 = Δ2y(a − 1), y(b) = 0 = Δ2y(b − 1), (1.2)

where Δ is the usual forward difference operator given by Δy(t) = y(t + 1) − y(t), Δny(t) =Δn−1(Δy(t)), [c, d]

Z:= {c, c + 1, . . . , d − 1, d}, and β > 0 is a real parameter.

In recent years, the theory of nonlinear difference equations has been widely appliedto many fields such as economics, neural network, ecology, and cybernetics, for details, see


[1–7] and references therein. Especially, there was much attention focused on the existenceand multiplicity of positive solutions of fourth-order problem, for example, [8–10], and inparticular the discrete problem with Lidstone boundary conditions [11–17]. However, verylittle work has been done on the uniqueness and iteration of positive solutions of discretefourth-order equation under Lidstone boundary conditions. We would like to mention someresults of Anderson and Minhos [11] and He and Su [12], which motivated us to consider theBVP (1.1) and (1.2).

In [11], Anderson and Minhos studied the following nonlinear discrete fourth-orderequation with explicit parameters β and λ given by

Δ4y(t − 2) − βΔ2y(t − 1) = λf(t, y(t)

), t ∈ [a + 1, b − 1]

Z, (1.3)

with Lidstone boundary conditions (1.2), where β > 0 and λ > 0 are real parameters. Theauthors obtained the following result.

Theorem 1.1 (see [11]). Assume that the following condition is satisfied

(A1) f(t, y) = g(t)w(y), where g : [a + 1, b − 1]Z→ [0,∞) with

∑b−1z=a+1 g(z) > 0, w :

[0,∞) → (0,∞) is continuous and nondecreasing, and there exists θ ∈ (0, 1) such thatw(κy) ≥ κθw(y) for κ ∈ (0, 1) and y ∈ [0,∞),

then, for any λ ∈ (0,+∞), the BVP (1.3) and (1.2) has a unique positive solution yλ. Furthermore,such a solution yλ satisfies the following properties:

(i) limλ→ 0+ ‖yλ‖ = 0 and limλ→∞ ‖yλ‖ =∞;

(ii) yλ is nondecreasing in λ;

(iii) yλ is continuous in λ, that is, if λ → λ0, then ‖yλ − yλ0‖ → 0.

Very recently, in [12], He and Su investigated the existence, multiplicity, andnonexistence of nontrivial solutions to the following discrete nonlinear fourth-orderboundary value problem

Δ4u(t − 2) + ηΔ2u(t − 1) − ξu(t) = λf(t, u(t)), t ∈ Z[a + 1, b + 1],

u(a) = 0 = Δ2u(a − 1), u(b + 2) = 0 = Δ2u(b + 1),(1.4)

where Δ denotes the forward difference operator defined by Δu(t) = u(t + 1) − u(t), Δnu(t) =Δ(Δn−1u(t)), Z[a + 1, b + 1] is the discrete interval given by {a + 1, a + 2, . . . , b + 1}with a andb (a < b) integers, η, ξ, λ are real parameters and satisfy

η < 8 sin2 π

2(b − a + 2), η2+4ξ ≥ 0, ξ+4η sin2 π

2(b − a + 2)< 16 sin4 π

2(b − a + 2), λ > 0.

(1.5)

For the function f , the authors imposed the following assumption:

(B1) f(t, x) = g(t)h(x), where g : Z[a + 1, b + 1] → [0,∞) with∑b+1

t=a+1 g(t) > 0, h :R → (0,∞) is continuous and nondecreasing, and there exists θ ∈ (0, 1) such thath(μx) ≥ μθh(x) for μ ∈ (0, 1) and x ∈ [0,∞).


Their main result is the following theorem.

Theorem 1.2 (see [12]). Assume that (B1) holds. Then for any λ ∈ (0,+∞), the BVP (1.4) has aunique positive solution uλ. Furthermore, such a solution uλ satisfies the properties (i)–(iii) stated inTheorem 1.1.

The aim of this work is to relax the assumptions (A1) and (B1) on the nonlinear term,without demanding the existence of upper and lower solutions, we present conditions for theBVP (1.1) and (1.2) to have a unique solution and then study the convergence of the iterativesequence. The ideas come from Zhai et al. [18, 19] and Liang [20].

Let B denote the Banach space of real-valued functions on [a − 1, b + 1]Z

, with thesupremum norm

∥∥y∥∥ = sup

t∈[a−1,b+1]Z

∣∣y(t)

∣∣. (1.6)

Throughout this paper, we need the following hypotheses:

(H1) fi : [0,+∞) → [0,+∞) are continuous and fi(y) > 0 for y > 0 (i = 1, 2);

(H2) h : [a + 1, b − 1]Z→ [0,+∞) with

∑b−1z=a+1 h(z) > 0;

(H3) f1 : [0,+∞) → [0,+∞) is nondecreasing, f2 : [0,+∞) → [0,+∞) is nonincreasing,and there exist ϕ(τ), ψ(τ) on interval [a+1, b−1]

Zwith ϕ : [a+1, b−1]

Z→ (0, 1), for

all e0 ∈ (0, 1), there exists τ0 ∈ [a+1, b−1]Z

such that ϕ(τ0) = e0, and ψ(τ) > ϕ(τ), forall τ ∈ [a + 1, b − 1]

Zwhich satisfy

f1(ϕ(τ)y

)≥ ψ(τ)f1

(y), f2

(1

ϕ(τ)y

)≥ ψ(τ)f2

(y), ∀τ ∈ [a + 1, b − 1]

Z, y ≥ 0. (1.7)

2. Two Lemmas

To prove the main results in this paper, we will employ two lemmas. These lemmas are basedon the linear discrete fourth-order equation

Δ4y(t − 2) − βΔ2y(t − 1) = u(t), t ∈ [a + 1, b − 1]Z, (2.1)

with Lidstone boundary conditions (1.2).

Lemma 2.1 (see [11]). Let u : [a+1, b−1]Z→ R be a function. Then the nonhomogeneous discrete

fourth-order Lidstone boundary value problem (2.1), (1.2) has solution

y(t) =b∑

s=a

b−1∑

z=a+1

G2(t, s)G1(s, z)u(z), t ∈ [a − 1, b + 1]Z, (2.2)


where G2(t, s) given by

G2(t, s) =1

�(1, 0)�(b, a)

⎧⎨

⎩

�(t, a)�(b, s): t ≤ s,

�(s, a)�(b, t): s ≤ t,(t, s) ∈ [a − 1, b + 1]

Z× [a, b]

Z(2.3)

with �(t, s) = μt−s − μs−t for μ = (β + 2 +√β(β + 4))/2, is the Green’s function for the second-order

discrete boundary value problem

−(Δ2y(t − 1) − βy(t)

)= 0, t ∈ [a, b]

Z,

y(a) = 0 = y(b),(2.4)

and G1(s, z) given by

G1(s, z) =1

b − a

⎧⎨

⎩

(s − a)(b − z): s ≤ z,

(z − a)(b − s): z ≤ s,(s, z) ∈ [a, b]

Z× [a + 1, b − 1]

Z(2.5)

is the Green’s function for the second-order discrete boundary value problem

−Δ2x(s − 1) = 0, s ∈ [a + 1, b − 1]Z,

x(a) = 0 = x(b).(2.6)

Lemma 2.2 (see [11]). Let

m :=�(1, 0)�(b, a + 1)(b − a)�2(b, a)

, M :=(b − a)�2(b/2, a/2)

4�(1, 0)�(b, a). (2.7)

Then, for (t, s, z) ∈ [a + 1, b − 1]3Z, one has

m ≤ G2(t, s)G1(s, z) ≤M. (2.8)

3. Main Results

Theorem 3.1. Assume that (H1)–(H3) hold. Then, the BVP (1.1) and (1.2) has a unique solutiony∗(t) in D, where

D ={y ∈ B | y(a) = 0 = y(b), y(t) > 0, t ∈ [a + 1, b − 1]

Z

}. (3.1)


Moreover, for any x0, y0 ∈ D, constructing successively the sequences

xn+1(t) =b∑

s=a

b−1∑

z=a+1

G2(t, s)G1(s, z)h(z)[f1(xn(z)) + f2

(yn(z)

)],

t ∈ [a − 1, b + 1]Z, n = 0, 1, 2, . . . ,

yn+1(t) =b∑

s=a

b−1∑

z=a+1

G2(t, s)G1(s, z)h(z)[f1(yn(z)

)+ f2(xn(z))

],

t ∈ [a − 1, b + 1]Z, n = 0, 1, 2, . . . ,

(3.2)

One has xn(t), yn(t) converge uniformly to y∗(t) in [a − 1, b + 1]Z.

Proof. First, we show that the BVP (1.1) and (1.2) has a solution.It is easy to see that the BVP (1.1) and (1.2) has a solution y = y(t) if and only if y is a

fixed point of the operator equation

A(y1, y2

)(t) =

b∑

s=a

b−1∑

z=a+1

G2(t, s)G1(s, z)h(z)[f1(y1(z)

)+ f2

(y2(z)

)], t ∈ [a − 1, b + 1]

Z.

(3.3)

In view of (H3) and (3.3),A(y1, y2) is nondecreasing in y1 and nonincreasing in y2. Moreover,for any τ ∈ [a + 1, b − 1]

Z, we have

A

(ϕ(τ)y1,

1ϕ(τ)

y2

)(t) =

b−1∑

s=a+1

b−1∑

z=a+1

G2(t, s)G1(s, z)h(z)[f1(ϕ(τ)y1(z)

)+ f2

(1

ϕ(τ)y2(z)

)]

≥ ψ(τ)b−1∑

s=a+1

b−1∑

z=a+1

G2(t, s)G1(s, z)h(z)[f1(y1(z)

)+ f2

(y2(z)

)]

= ψ(τ)A(y1, y2

)(t)

(3.4)

for t ∈ [a, b]Z

and y1, y2 ∈ D.Let

L = (b − a − 1)b−1∑

z=a+1

h(z), (3.5)


condition (H2) implies L > 0. Since fi(y) > 0 for y > 0 (i = 1, 2), by Lemma 2.2, we have

A(L, L) =b−1∑

s=a+1

b−1∑

z=a+1

G2(t, s)G1(s, z)h(z)[f1(L) + f2(L)

]

≥ m[f1(L) + f2(L)

] b−1∑

s=a+1

b−1∑

z=a+1

h(z)

= m[f1(L) + f2(L)

]L

(3.6)

for m in (2.1) and L in (3.5).Moreover, we obtain

A(L, L) ≤M[f1(L) + f2(L)

]L (3.7)

for M in (2.1).Thus

m[f1(L) + f2(L)

]L ≤ A(L, L) ≤M

[f1(L) + f2(L)

]L. (3.8)

Therefore, we can choose a sufficiently small number e1 ∈ (0, 1) such that

e1L ≤ A(L, L) ≤ L

e1, (3.9)

which together with (H3) implies that there exists τ1 ∈ [a + 1, b − 1]Z

such that ϕ(τ1) = e1, so

ϕ(τ1)L ≤ A(L, L) ≤ L

ϕ(τ1). (3.10)

Since ψ(τ1)/ϕ(τ1) > 1, we can take a sufficiently large positive integer k such that

[ψ(τ1)ϕ(τ1)

]k≥ 1ϕ(τ1)

. (3.11)

It is clear that

[ϕ(τ1)ψ(τ1)

]k≤ ϕ(τ1). (3.12)


We define

u0(t) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

−[ϕ(τ1)

]kL: t = a − 1, b + 1,

0: t = a, b,[ϕ(τ1)

]kL: t ∈ [a + 1, b − 1]

Z,

v0(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

− L[ϕ(τ1)

]k : t = a − 1, b + 1,

0: t = a, b,

L[ϕ(τ1)

]k : t ∈ [a + 1, b − 1]Z.

(3.13)

Evidently, for t ∈ [a, b]Z

, u0 ≤ v0. Take any λ ∈ (0, [ϕ(τ1)]2k], then λ ∈ (0, 1) and u0 ≥ λv0.

By the mixed monotonicity ofA, we haveA(u0, v0) ≤ A(v0, u0). In addition, combining(H3) with (3.10) and (3.11), we get

A(u0, v0) = A

([ϕ(τ1)

]kL,

1[ϕ(τ1)

]k L

)

= A

(

ϕ(τ1)[ϕ(τ1)

]k−1L,

1

ϕ(τ1)[ϕ(τ1)

]k−1L

)

≥ ψ(τ1)A

([ϕ(τ1)

]k−1L,

1[ϕ(τ1)

]k−1L

)

≥ · · ·

≥[ψ(τ1)

]kA(L, L) ≥

[ψ(τ1)

]kϕ(τ1)L

≥[ϕ(τ1)

]kL = u0.

(3.14)

From (H3), we have

A(y1, y2

)= A

(ϕ(s)

y1

ϕ(s),

1ϕ(s)

ϕ(s)y2

)

≥ ψ(s)A(

y1

ϕ(s), ϕ(s)y2

), ∀s ∈ [a + 1, b − 1]

Z, y1, y2 ≥ 0,

(3.15)

and hence

A

(y1

ϕ(s), ϕ(s)y2

)≤ 1ψ(s)

A(y1, y2

), ∀s ∈ [a + 1, b − 1]

Z, y1, y2 ≥ 0. (3.16)


Thus, we have

A(v0, u0) = A

(L

[ϕ(τ1)

]k ,[ϕ(τ1)

]kL

)

= A

(L

ϕ(τ1)[ϕ(τ1)

]k−1, ϕ(τ1)

[ϕ(τ1)

]k−1L

)

≤ 1ψ(τ1)

A

(L

[ϕ(τ1)

]k−1,[ϕ(τ1)

]k−1L

)

≤ · · ·

≤ 1[ψ(τ1)

]k A(L, L) ≤ 1[ψ(τ1)

]kL

ϕ(τ1).

(3.17)

In accordance with (3.12), we can see that

A(v0, u0) ≤L

[ϕ(τ1)

]k = v0. (3.18)

Construct successively the sequences

un = A(un−1, vn−1), vn = A(vn−1, un−1), n = 1, 2, . . . . (3.19)

By the mixed monotonicity of A, we have u1 = A(u0, v0) ≤ A(v0, u0) = v1. By induction, weobtain un ≤ vn, n = 1, 2, . . .. It follows from (3.14), (3.18), and the mixed monotonicity of Athat

u0 ≤ u1 ≤ · · · ≤ un ≤ · · · ≤ vn ≤ · · · ≤ v1 ≤ v0. (3.20)

Note that u0 ≥ λv0, so we can get un(t) ≥ u0(t) ≥ λv0(t) ≥ λvn(t), t ∈ [a, b]Z, n = 1, 2, . . .. Let

λn = sup{λ > 0 | un(t) ≥ λvn(t), t ∈ [a, b]Z}, n = 1, 2, . . . . (3.21)

Thus, we have

un(t) ≥ λnvn(t), t ∈ [a, b]Z, n = 1, 2, . . . , (3.22)

and then

un+1(t) ≥ un(t) ≥ λnvn(t) ≥ λnvn+1(t), t ∈ [a, b]Z, n = 1, 2, . . . . (3.23)

Therefore, λn+1 ≥ λn, that is, {λn} is increasing with {λn} ⊂ (0, 1]. Set λ = limn→∞ λn. We canshow that λ = 1. In fact, if 0 < λ < 1, by (H3), there exists τ2 ∈ [a+1, b−1]

Zsuch that ϕ(τ2) = λ.

Consider the following two cases.


(i) There exists an integer N such that λN = λ. In this case, we have λn = λ for alln ≥N holds. Hence, for n ≥N, it follows from (3.4) and the mixed monotonicity of A that

un+1 = A(un, vn) ≥ A(λvn,

1

λun

)= A

(ϕ(τ2)vn,

1ϕ(τ2)

un

)≥ ψ(τ2)A(vn, un) = ψ(τ2)vn+1.

(3.24)

By the definition of λn, we have

λn+1 = λ ≥ ψ(τ2) > ϕ(τ2) = λ. (3.25)

This is a contradiction.(ii) For all integer n, λn < λ. In this case, we have 0 < λn/λ < 1. In accordance with

(H3), there exists θn ∈ [a + 1, b − 1]Z

such that ϕ(θn) = λn/λ. Hence, combining (3.4) with themixed monotonicity of A, we have

un+1 = A(un, vn) ≥ A(λnvn,

1λnun

)

= A

⎛

⎜⎝λn

λλvn,

un(λn/λ

)λ

⎞

⎟⎠ = A

(ϕ(θn)ϕ(τ2)vn,

unϕ(θn)ϕ(τ2)

)

≥ ψ(θn)A(ϕ(τ2)vn,

unϕ(τ2)

)≥ ψ(θn)ψ(τ2)A(vn, un)

= ψ(θn)ψ(τ2)vn+1.

(3.26)

By the definition of λn, we have

λn+1 ≥ ψ(θn)ψ(τ2) > ϕ(θn)ψ(τ2) =λn

λψ(τ2). (3.27)

Let n → ∞, we have λ ≥ (λ/λ)ψ(τ2) > (λ/λ)ϕ(τ2) = ϕ(τ2) = λ, and this is also a contradiction.Hence, limn→∞ λn = 1.

Thus, combining (3.20) with (3.22), we have

0 ≤ un+l(t) − un(t) ≤ vn(t) − un(t) ≤ vn(t) − λnvn(t) = (1 − λn)vn(t) ≤ (1 − λn)v0(t) (3.28)

for t ∈ [a, b]Z

, where l is a nonnegative integer. Thus,

‖un+l − un‖ ≤ ‖vn − un‖ ≤ (1 − λn)v0. (3.29)

Therefore, there exists a function y∗ ∈ D such that

limn→∞

un(t) = limn→∞

vn(t) = y∗(t) for t ∈ [a − 1, b + 1]Z. (3.30)


By the mixed monotonicity of A and (3.20), we have

un+1(t) = A(un(t), vn(t)) ≤ A(y∗(t), y∗(t)

)≤ A(vn(t), un(t)) = vn+1(t). (3.31)

Let n → ∞ and we get A(y∗(t), y∗(t)) = y∗(t), t ∈ [a − 1, b + 1]Z

. That is, y∗ is a nontrivialsolution of the BVP (1.1) and (1.2).

Next, we show the uniqueness of solutions of the BVP (1.1) and (1.2). Assume, to thecontrary, that there exist two nontrivial solutions y1 and y2 of the BVP (1.1) and (1.2) suchthat A(y1(t), y1(t)) = y1(t) and A(y2(t), y2(t)) = y2(t) for t ∈ [a−1, b+1]

Z. According to (3.9),

we can know that there exists 0 < η ≤ 1 such that ηy2(t) ≤ y1(t) ≤ (1/η)y2(t) for t ∈ [a, b]Z

.Let

η0 = sup{

0 < η ≤ 1 | ηy2 ≤ y1 ≤1ηy2

}. (3.32)

Then 0 < η0 ≤ 1 and η0y2(t) ≤ y1(t) ≤ (1/η0)y2(t) for t ∈ [a, b]Z

.We now show that η0 = 1. In fact, if 0 < η0 < 1, then, in view of (H3), there exists

τ ∈ [a + 1, b − 1]Z

such that ϕ(τ) = η0. Furthermore, we have

y1 = A(y1, y1

)≥ A

(η0y2,

1η0y2

)= A

(ϕ(τ)y2,

1ϕ(τ)

y2

)≥ ψ(τ)A

(y2, y2

)= ψ(τ)y2, (3.33)

y1 = A(y1, y1

)≤ A

(y2

η0, η0y2

)= A

(y2

ϕ(τ), ϕ(τ)y2

)≤ 1ψ(τ)

A(y2, y2

)=

1ψ(τ)

y2. (3.34)

In (3.34), we used the relation formula (3.16). Since ψ(τ) > ϕ(τ) = η0, this contradicts thedefinition of η0. Hence η0 = 1. Therefore, the BVP (1.1) and (1.2) has a unique solution.

Finally, we show that “moreover” part of the theorem. For any initial x0, y0 ∈ D, inaccordance with (3.9), we can choose a sufficiently small number e2 ∈ (0, 1) such that

e2L ≤ x0 ≤1e2L, e2L ≤ y0 ≤

1e2L. (3.35)

It follows from (H3) that there exists τ3 ∈ [a + 1, b − 1]Z

such that ϕ(τ3) = e2, and hence

ϕ(τ3)L ≤ x0 ≤L

ϕ(τ3), ϕ(τ3)L ≤ y0 ≤

L

ϕ(τ3). (3.36)

Thus, we can choose a sufficiently large positive integer k such that

[ψ(τ3)ϕ(τ3)

]k≥ 1ϕ(τ3)

. (3.37)

Define

u0 =[ϕ(τ3)

]kL, v0 =

L[ϕ(τ3)

]k . (3.38)


Obviously, u0 < x0, y0 < v0. Let

un = A(un−1, vn−1), vn = A(vn−1, un−1), n = 1, 2, . . . ,

xn(t) = A(xn−1, yn−1

)(t) =

b∑

s=a

b−1∑

z=a+1

G2(t, s)G1(s, z)h(z)[f1(xn−1(z)) + f2

(yn−1(z)

)],

yn(t) = A(yn−1, xn−1

)(t) =

b∑

s=a

b−1∑

z=a+1

G2(t, s)G1(s, z)h(z)[f1(yn−1(z)

)+ f2(xn−1(z))

]

(3.39)

for t ∈ [a− 1, b + 1]Z, n = 1, 2, . . .. By induction, we get un ≤ xn ≤ vn, un ≤ yn ≤ vn, n = 1, 2, . . ..

Similarly to the above proof, it follows that there exists y ∈ D such that

limn→∞

un = limn→∞

vn = y, A(y, y

)= y. (3.40)

By the uniqueness of fixed points A in D, we get y = y∗. Therefore, we have

limn→∞

xn = limn→∞

yn = y∗. (3.41)

This completes the proof of the theorem.

Remark 3.2. From the proof of Theorem 3.1, we easily know that assume y = A(y, x), x =A(x, y), thus, let y0 = y, x0 = x, we have

yn = y, xn = x, n = 1, 2, . . . . (3.42)

Therefore y = x = y∗.

Theorem 3.3. Assume that (H2) holds, and the following conditions are satisfied:

(C1) f : [0,+∞) → [0,+∞) is continuous and f(y) > 0 for y > 0;

(C2) f : [0,+∞) → [0,+∞) is nondecreasing;

b∑

s=a

b−1∑

z=a+1

G2(t, s)G1(s, z)h(z)f(ϕ(τ)y(z)

)≥ ψ

(τ, y

) b∑

s=a

b−1∑

z=a+1

G2(t, s)G1(s, z)h(z)f(y(z)

),

(3.43)

for all τ ∈ [a+1, b−1]Z, y ∈ [0,+∞), where ϕ : [a+1, b−1]

Z→ (0, 1), for all e0 ∈ (0, 1),

there exists τ0 ∈ [a + 1, b − 1]Zsuch that ϕ(τ0) = e0, and ψ : [a + 1, b − 1]

Z× [0,+∞) →

(0,+∞), with ψ(τ, y) > ϕ(τ), for all τ ∈ [a + 1, b − 1]Z, y ∈ [0,+∞);


(C3) for fixed τ ∈ [a + 1, b − 1]Z, one has

(i) ψ(τ, y) is nonincreasing with respect to y, and there exists τ4 ∈ [a + 1, b − 1]Zsuch

that

mf(L) ≥ ϕ(τ4),ψ(τ4, L/ϕ(τ4)

)

ϕ(τ4)≥Mf(L) (3.44)

or

(ii) ψ(τ, y) is nondecreasing with respect to y, and there exists τ5 ∈ [a + 1, b − 1]Zsuch

that

mf(L) ≥ϕ(τ5)ψ(τ5, L)

,1

ϕ(τ5)≥Mf(L), (3.45)

wherem, M are defined in (2.1), L is defined in (3.5). Then, the BVP

Δ4y(t − 2) − βΔ2y(t − 1) = h(t)f(y(t)

), t ∈ [a + 1, b − 1]

Z,

y(a) = 0 = Δ2y(a − 1), y(b) = 0 = Δ2y(b − 1)(3.46)

has a unique solution y∗.

Proof. For convenience, we still define the operator equation A by

Ay(t) =b∑

s=a

b−1∑

z=a+1

G2(t, s)G1(s, z)h(z)f(y(z)

), t ∈ [a − 1, b + 1]

Z. (3.47)

In the following, we consider the following two cases.(i) For fixed τ ∈ [a + 1, b − 1]

Z, ψ(τ, y) is nonincreasing with respect to y.

According to condition (C3) and Lemma 2.2, we can know that there exists τ4 ∈ [a +1, b − 1]

Zsuch that

ϕ(τ4)L ≤ A(L) ≤ψ(τ4, L/ϕ(τ4)

)

ϕ(τ4)L. (3.48)

Since ψ(τ4, L)/ϕ(τ4) > 1, we can find a sufficiently large positive integer k such that

[ψ(τ4, L)ϕ(τ4)

]k≥ 1ϕ(τ4)

. (3.49)


For t ∈ [a + 1, b − 1]Z

, we still define

u0(t) =[ϕ(τ4)

]kL, v0(t) =

L[ϕ(τ4)

]k ,

un(t) = Aun−1(t), vn(t) = Avn−1(t), n = 1, 2, . . . .

(3.50)

By the proof of Theorem 3.1, it is sufficient to show that

u0 ≤ u1 ≤ v1 ≤ v0. (3.51)

Obviously, u0 ≤ v0 and u1 ≤ v1.In this case, it follows from conditions (C2), (C3), and (3.49) that

u1 = Au0 = A([ϕ(τ4)

]kL)

= A(ϕ(τ4)

[ϕ(τ4)

]k−1L)

≥ ψ(τ4,[ϕ(τ4)

]k−1L)A([ϕ(τ4)

]k−1L)

= ψ(τ4,[ϕ(τ4)

]k−1L)A(ϕ(τ4)

[ϕ(τ4)

]k−2L)

≥ ψ(τ4,[ϕ(τ4)

]k−1L)ψ(τ4,[ϕ(τ4)

]k−2L)A([ϕ(τ4)

]k−2L)

≥ · · ·

≥ ψ(τ4,[ϕ(τ4)

]k−1L)ψ(τ4,[ϕ(τ4)

]k−2L)· · ·ψ(τ4, L)A(L)

≥[ψ(τ4, L)

]kϕ(τ4)L

≥[ϕ(τ4)

]kL = u0.

(3.52)

In accordance with (3.16), we have

A

(y

ϕ(s)

)≤ 1ψ(s, y/ϕ(s)

)Ay, (3.53)


which together with condition (C2) and (3.48) implies that

v1 = Av0 = A

(L

[ϕ(τ4)

]k

)

= A

(L

ϕ(τ4)[ϕ(τ4)

]k−1

)

≤ 1

ψ(τ4, L/

[ϕ(τ4)

]k)A

(L

[ϕ(τ4)

]k−1

)

=1

ψ(τ4, L/

[ϕ(τ4)

]k)A

(L

ϕ(τ4)[ϕ(τ4)

]k−2

)

≤ 1

ψ(τ4, L/

[ϕ(τ4)

]k)1

ψ(τ4, L/

[ϕ(τ4)

]k−1)A

(L

[ϕ(τ4)

]k−2

)

≤ 1

ψ(τ4, L/

[ϕ(τ4)

]k)1

ψ(τ4, L/

[ϕ(τ4)

]k−1) · · · 1

ψ(τ4, L/ϕ(τ4)

)A(L)

≤ 1[ϕ(τ4)

]k−1

1ψ(τ4, L/ϕ(τ4)

)A(L)

≤ L[ϕ(τ4)

]k = v0.

(3.54)

(ii) For fixed τ ∈ [a + 1, b − 1]Z

, ψ(τ, y) is nondecreasing with respect to y.In this case, by condition (C3) and Lemma 2.2, we can know that there exists τ5 ∈

[a + 1, b − 1]Z

such that

ϕ(τ5)Lψ(τ5, L)

≤ A(L) ≤ L

ϕ(τ5). (3.55)

Since 0 < ϕ(τ5)/ψ(τ5, L/ϕ(τ5)) < 1, we can take a sufficiently large positive integer k suchthat

[ϕ(τ5)

ψ(τ5, L/ϕ(τ5)

)

]k≤ ϕ(τ5). (3.56)


For t ∈ [a + 1, b − 1]Z

, we still define

u0(t) =[ϕ(τ5)

]kL, v0(t) =

L[ϕ(τ5)

]k ,

un(t) = Aun−1(t), vn(t) = Avn−1(t), n = 1, 2, . . . .

(3.57)

We continue to prove that

u1 ≥ u0, v1 ≤ v0. (3.58)

By (3.52), combining (3.55) with the monotonicity of ψ, we have

u1 = Au0 = A([ϕ(τ5)

]kL)

≥ ψ(τ5,[ϕ(τ5)

]k−1L)ψ(τ5,[ϕ(τ5)

]k−2L)· · ·ψ(τ5, L)A(L)

≥[ϕ(τ5)

]k−1ψ(τ5, L)A(L)

≥[ϕ(τ5)

]kL = u0.

(3.59)

In accordance with (3.54), combining the monotonicity of ψ and (3.55), we get

v1 = Av0 = A

(L

[ϕ(τ5)

]k

)

≤ 1

ψ(τ5, L/

[ϕ(τ5)

]k)1

ψ(τ5, L/

[ϕ(τ5)

]k−1) · · · 1

ψ(τ5, L/ϕ(τ5)

)A(L)

≤ 1[ψ(τ5, L/ϕ(τ5)

)]kL

ϕ(τ5).

(3.60)

An application of (3.56) yields

v1 ≤1

[ϕ(τ5)

]k L = v0. (3.61)

Therefore, we obtain

u0 ≤ u1 ≤ v1 ≤ v0. (3.62)

For t = a − 1, b + 1, the proof is similar and hence omitted. This completes the proof of thetheorem.


Remark 3.4. In Theorem 3.1, the more general conditions are imposed on the nonlinear termthan Theorem 1.1. In particular, in Theorem 3.3, ψ(τ, y) contains the variable y; therefore, themore comprehensive functions can be incorporated.

4. An Example

Example 4.1. Consider the following discrete fourth-order Lidstone problem:

Δ4y(t − 2) −Δ2y(t − 1) = t

[

1 + y1/4(t) + 2 +1

y1/4(t)

]

, t ∈ [2 + 1, 7 − 1]Z,

y(2) = 0 = Δ2y(1), y(7) = 0 = Δ2y(6).

(4.1)

We claim that the BVP (4.1) and (1.2) has a unique solution y∗(t) in D, where

D ={y ∈ B | y(2) = 0 = y(7), y(t) > 0, t ∈ [3, 6]

Z

}. (4.2)

Moreover, for any x0, y0 ∈ D, constructing successively the sequences

xn+1(t) =7∑

s=2

6∑

z=3

G2(t, s)G1(s, z)z

[

1 + x1/4n (z) + 2 +

1

y1/4n (z)

]

, t ∈ [1, 8]Z, n = 0, 1, 2, . . . ,

yn+1(t) =7∑

s=2

6∑

z=3

G2(t, s)G1(s, z)z

[

1 + y1/4n (z) + 2 +

1

x1/4n (z)

]

, t ∈ [2, 8]Z, n = 0, 1, 2, . . . ,

(4.3)

we have xn(t), yn(t) converge uniformly to y∗(t) in [2, 8]Z

.In fact, we choose f1(y) = 1 + y1/4, f2(y) = 2 + 1/y1/4, h(z) = z, thus fi(y) > 0 for

y > 0 (i = 1, 2),∑6

z=3 h(z) =∑6

z=3 z = 18 > 0. It is easy to check that f1 is nondecreasing on[0,+∞), f2 is nonincreasing on [0,+∞). In addition, we set

τ =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

3, ϕ(τ) ∈(

0,14

],

4, ϕ(τ) ∈(

14,

12

],

5, ϕ(τ) ∈(

12,

34

],

6, ϕ(τ) ∈(

34, 1),

(4.4)


ψ(τ) = [ϕ(τ)]1/2. It is easy to see that

f1(ϕ(τ)y

)= 1 +

(ϕ(τ)y

)1/4 ≥ ψ(τ)(

1 + y1/4)= ψ(τ)f1

(y), ∀τ ∈ [3, 6]

Z, y ≥ 0,

f2

(y

ϕ(τ)

)= 2 +

1(y/ϕ(τ)

)1/4≥ ψ(τ)

(

2 +1

y1/4

)

, ∀τ ∈ [3, 6]Z, y ≥ 0.

(4.5)

The conclusion then follows from Theorem 3.1.

Acknowledgments

The authors were supported financially by the National Natural Science Foundation of China(10971046), the Natural Science Foundation of Shandong Province (ZR2009AM004), and theYouth Science Foundation of Shanxi Province (2009021001-2).

References

[1] R. P. Agarwal, Difference Equations and Inequalities, vol. 155, Marcel Dekker, New York, NY, USA, 1992.[2] R. P. Agarwal, D. O’Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral

Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.[3] R. P. Agarwal, K. Perera, and D. O’Regan, “Multiple positive solutions of singular and nonsingular

discrete problems via variational methods,” Nonlinear Analysis: Theory, Methods & Applications, vol.58, no. 1-2, pp. 69–73, 2004.

[4] V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods andApplications, vol. 181, Academic Press, Boston, Mass, USA, 1988.

[5] W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, AcademicPress, Boston, Mass, USA, 1991.

[6] J. Yu and Z. Guo, “On boundary value problems for a discrete generalized Emden-Fowler equation,”Journal of Differential Equations, vol. 231, no. 1, pp. 18–31, 2006.

[7] D. B. Wang and W. Guan, “Three positive solutions of boundary value problems for p-Laplaciandifference equations,” Computers & Mathematics with Applications, vol. 55, no. 9, pp. 1943–1949, 2008.

[8] B. Zhang, L. Kong, Y. Sun, and X. Deng, “Existence of positive solutions for BVPs of fourth-orderdifference equations,” Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 583–591, 2002.

[9] Z. He and J. Yu, “On the existence of positive solutions of fourth-order difference equations,” AppliedMathematics and Computation, vol. 161, no. 1, pp. 139–148, 2005.

[10] J. V. Manojlovic, “Classification and existence of positive solutions of fourth-order nonlineardifference equations,” Lithuanian Mathematical Journal, vol. 49, no. 1, pp. 71–92, 2009.

[11] D. R. Anderson and F. Minhos, “A discrete fourth-order Lidstone problem with parameters,” AppliedMathematics and Computation, vol. 214, no. 2, pp. 523–533, 2009.

[12] T. He and Y. Su, “On discrete fourth-order boundary value problems with three parameters,” Journalof Computational and Applied Mathematics, vol. 233, no. 10, pp. 2506–2520, 2010.

[13] R. P. Agarwal and D. O’Regan, “Lidstone continuous and discrete boundary value problems,”Memoirs on Differential Equations and Mathematical Physics, vol. 19, pp. 107–125, 2000.

[14] P. J. Y. Wong and R. P. Agarwal, “Multiple solutions of difference and partial difference equationswith Lidstone conditions,” Mathematical and Computer Modelling, vol. 32, no. 5-6, pp. 699–725, 2000.

[15] P. J. Y. Wong and R. P. Agarwal, “Results and estimates on multiple solutions of Lidstone boundaryvalue problems,” Acta Mathematica Hungarica, vol. 86, no. 1-2, pp. 137–168, 2000.

[16] P. J. Y. Wong and R. P. Agarwal, “Characterization of eigenvalues for difference equations subject toLidstone conditions,” Japan Journal of Industrial and Applied Mathematics, vol. 19, no. 1, pp. 1–18, 2002.

[17] P. J. Y. Wong and L. Xie, “Three symmetric solutions of Lidstone boundary value problems fordifference and partial difference equations,” Computers & Mathematics with Applications, vol. 45, no.6–9, pp. 1445–1460, 2003.


[18] C.-B. Zhai and X.-M. Cao, “Fixed point theorems for τ-ϕ-concave operators and applications,”Computers and Mathematics with Applications, vol. 59, no. 1, pp. 532–538, 2010.

[19] C. B. Zhai, W. X. Wang, and L. L. Zhang, “Generalizations for a class of concave and convexoperators,” Acta Mathematica Sinica, vol. 51, no. 3, pp. 529–540, 2008 (Chinese).

[20] Z. D. Liang, “Existence and uniqueness of fixed points for mixed monotone operators,” Journal ofDezhou University, vol. 24, no. 4, pp. 1–6, 2008 (Chinese).


Research ArticleExponential Decay of Energy for Some NonlinearHyperbolic Equations with Strong Dissipation

Yaojun Ye

Department of Mathematics and Information Science, Zhejiang University of Science and Technology,Hangzhou 310023, China

Correspondence should be addressed to Yaojun Ye, [email protected]

Received 14 December 2009; Revised 21 May 2010; Accepted 4 August 2010

Academic Editor: Tocka Diagana

Copyright q 2010 Yaojun Ye. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The initial boundary value problem for a class of hyperbolic equations with strong dissipativeterm utt−

∑ni=1(∂/∂xi)(|∂u/∂xi|

p−2(∂u/∂xi))−aΔut = b|u|r−2u in a bounded domain is studied. Theexistence of global solutions for this problem is proved by constructing a stable set in W1,p

0 (Ω) andshowing the exponential decay of the energy of global solutions through the use of an importantlemma of V. Komornik.

1. Introduction

We are concerned with the global solvability and exponential asymptotic stability for thefollowing hyperbolic equation in a bounded domain:

utt −Δpu − aΔut = b|u|r−2u, x ∈ Ω, t > 0 (1.1)

with initial conditions

u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω (1.2)

and boundary condition

u(x, t) = 0, x ∈ ∂Ω, t ≥ 0, (1.3)


where Ω is a bounded domain in Rn with a smooth boundary ∂Ω, a, b > 0 and r, p > 2 arereal numbers, and Δp =

∑ni=1(∂/∂xi) (|∂/∂xi|p−2(∂/∂xi)) is a divergence operator (degenerate

Laplace operator) with p > 2, which is called a p-Laplace operator.Equations of type (1.1) are used to describe longitudinal motion in viscoelasticity

mechanics and can also be seen as field equations governing the longitudinal motion of aviscoelastic configuration obeying the nonlinear Voight model [1–4].

For b = 0, it is well known that the damping term assures global existence and decayof the solution energy for arbitrary initial data [4–6]. For a = 0, the source term causes finitetime blow up of solutions with negative initial energy if r > p [7].

In [8–10], Yang studied the problem (1.1)–(1.3) and obtained global existence resultsunder the growth assumptions on the nonlinear terms and initial data. These global existenceresults have been improved by Liu and Zhao [11] by using a new method. As for thenonexistence of global solutions, Yang [12] obtained the blow up properties for the problem(1.1)–(1.3) with the following restriction on the initial energy E(0) < min{−(rk1 + pk2/r −p)1/δ,−1}, where r > p and k1, k2, and δ are some positive constants.

Because the p-Laplace operator Δp is nonlinear operator, the reasoning of proof andcomputation are greatly different from the Laplace operator Δ =

∑ni=1(∂

2/∂x2i ). By means

of the Galerkin method and compactness criteria and a difference inequality introduced byNakao [13], Ye [14, 15] has proved the existence and decay estimate of global solutions forthe problem (1.1)–(1.3) with inhomogeneous term f(x, t) and p ≥ r.

In this paper we are going to investigate the global existence for the problem (1.1)–(1.3) by applying the potential well theory introduced by Sattinger [16], and we showthe exponential asymptotic behavior of global solutions through the use of the lemma ofKomornik [17].

We adopt the usual notation and convention. Let Wk,p(Ω) denote the Sobolev spacewith the norm ‖u‖Wk,p(Ω) = (

∑|α|≤k ‖Dαu‖p

Lp(Ω))1/p and W

k,p

0 (Ω) denote the closure in Wk,p(Ω)of C∞0 (Ω). For simplicity of notation, hereafter we denote by ‖ · ‖p the Lebesgue space Lp(Ω)

norm, ‖ · ‖ denotes L2(Ω) norm, and write equivalent norm ‖∇ · ‖p instead of W1,p0 (Ω)

norm ‖ · ‖W

1,p0 (Ω). Moreover, M denotes various positive constants depending on the known

constants, and it may be different at each appearance.

2. The Global Existence and Nonexistence

In order to state and study our main results, we first define the following functionals:

K(u) = ‖∇u‖pp − b‖u‖rr ,

J(u) =1p‖∇u‖pp −

b

r‖u‖rr ,

(2.1)

for u ∈W1,p0 (Ω). Then we define the stable set H by

H ={u ∈W1,p

0 (Ω), K(u) > 0, J(u) < d}∪ {0}, (2.2)


where

d = inf

{

supλ>0

J(λu), u ∈W1,p0 (Ω)/{0}

}

. (2.3)

We denote the total energy associated with (1.1)–(1.3) by

E(t) =12‖ut‖2 +

1p‖∇u‖pp −

b

r‖u‖rr =

12‖ut‖2 + J(u) (2.4)

for u ∈W1,p0 (Ω), t ≥ 0, and E(0) = (1/2)‖u1‖2 + J(u0) is the total energy of the initial data.

Definition 2.1. The solution u(x, t) is called the weak solution of the problem (1.1)–(1.3) onΩ × [0, T), if u ∈ L∞(0, T ;W1,p

0 (Ω)) and ut ∈ L∞(0, T ; L2(Ω)) satisfy

(ut, v) −∫ t

0

(Δpu, v

)dτ + a(∇u,∇v) = b

∫ t

0

(|u|r−2u, v

)dτ + (u1, v) + a(∇u0,∇v) (2.5)

for all v ∈W1,p0 (Ω) and u(x, 0) = u0(x) in W1,p

0 (Ω), ut(x, 0) = u1(x) in L2(Ω).We need the following local existence result, which is known as a standard one (see

[14, 18, 19]).

Theorem 2.2. Suppose that 2 < p < r < np/(n − p) if p < n and 2 < p < r < ∞ if n ≤ p. Ifu0 ∈ W

1,p0 (Ω), u1 ∈ L2(Ω), then there exists T > 0 such that the problem (1.1)–(1.3) has a unique

local solution u(t) in the class

u ∈ L∞([0, T);W1,p

0 (Ω)), ut ∈ L∞

([0, T);L2(Ω)

). (2.6)

For latter applications, we list up some lemmas.

Lemma 2.3 (see [20, 21]). Let u ∈ W1,p0 (Ω), then u ∈ Lq(Ω), and the inequality ‖u‖q ≤

C‖u‖W

1,p0 (Ω) holds with a constant C > 0 depending on Ω, p, and q, provided that, (i) 2 ≤ q < +∞ if

2 ≤ n ≤ p and (ii) 2 ≤ q ≤ np/(n − p), 2 < p < n.

Lemma 2.4. Let u(t, x) be a solution to problem (1.1)–(1.3). Then E(t) is a nonincreasing functionfor t > 0 and

d

dtE(t) = −a‖∇ut(t)‖2. (2.7)

Proof. By multiplying (1.1) by ut and integrating over Ω, we get

12d

dt‖ut‖2 +

1p

d

dt‖∇u‖pp −

b

r

d

dt‖u‖rr = −a‖∇ut(t)‖

2, (2.8)


which implies from (2.4) that

d

dtE(u(t)) = −a‖∇ut(t)‖2 ≤ 0. (2.9)

Therefore, E(t) is a nonincreasing function on t.

Lemma 2.5. Let u ∈W1,p0 (Ω); if the hypotheses in Theorem 2.2 hold, then d > 0.

Proof. Since

J(λu) =λp

p‖∇u‖pp −

bλr

r‖u‖rr , (2.10)

so, we get

d

dλJ(λu) = λp−1‖∇u‖pp − bλr−1‖u‖rr . (2.11)

Let (d/dλ)J(λu) = 0, which implies that

λ1 = b−1/(r−p)(‖u‖rr‖∇u‖pp

)−1/(r−p)

. (2.12)

As λ = λ1, an elementary calculation shows that

d2

dλ2J(λu) < 0. (2.13)

Hence, we have from Lemma 2.3 that

supλ≥0

J(λu) = J(λ1u) =r − prp

b−p/(r−p)(‖u‖r‖∇u‖p

)−rp/(r−p)

≥r − prp

(bCr)−p/(r−p) > 0.

(2.14)

We get from the definition of d that d > 0.

Lemma 2.6. Let u ∈ H, then

r − prp‖∇u‖pp < J(u). (2.15)


Proof. By the definition of K(u) and J(u), we have the following identity:

rJ(u) = K(u) +r − pp‖∇u‖pp. (2.16)

Since u ∈ H, so we have K(u) > 0. Therefore, we obtain from (2.16) that

r − prp‖∇u‖pp ≤ J(u). (2.17)

In order to prove the existence of global solutions for the problem (1.1)-(1.3), we needthe following lemma.

Lemma 2.7. Suppose that 2 < p < r < np/(n − p) if p < n and 2 < p < r < ∞ if n ≤ p. Ifu0 ∈ H, u1 ∈ L2(Ω), and E(0) < d, then u ∈ H, for each t ∈ [0, T).

Proof. Assume that there exists a number t∗ ∈ [0, T) such that u(t) ∈ H on [0, t∗) and u(t∗)/∈H.Then, in virtue of the continuity of u(t), we see that u(t∗) ∈ ∂H. From the definition of H andthe continuity of J(u(t)) and K(u(t)) in t, we have either

J(u(t∗)) = d, (2.18)

or

K(u(t∗)) = 0. (2.19)

It follows from (2.4) that

J(u(t∗)) =1p‖∇u(t∗)‖pp −

b

r‖u(t∗)‖rr ≤ E(t∗) ≤ E(0) < d. (2.20)

So, case (2.18) is impossible.Assume that (2.19) holds, then we get that

d

dλJ(λu(t∗)) = λp−1(1 − λr−p

)‖∇u‖pp. (2.21)

We obtain from (d/dλ)J(λu(t∗)) = 0 that λ = 1.Since

d2

dλ2J(λu(t∗))

∣∣∣∣∣λ=1

= −(r − p

)‖∇u(t∗)‖p < 0, (2.22)


consequently, we get from (2.20) that

supλ≥0

J(λu(t∗)) = J(λu(t∗))|λ=1 = J(u(t∗)) < d, (2.23)

which contradicts the definition of d. Therefore, case (2.19) is impossible as well. Thus, weconclude that u(t) ∈ H on [0, T).

Theorem 2.8. Assume that 2 < p < r < np/(n − p) if p < n and 2 < p < r < ∞ if n ≤ p. u(t)is a local solution of problem (1.1)–(1.3) on [0, T). If u0 ∈ H, u1 ∈ L2(Ω), and E(0) < d, then thesolution u(t) is a global solution of the problem (1.1)–(1.3).

Proof. It suffices to show that ‖ut‖2 + ‖∇u‖pp is bounded independently of t.Under the hypotheses in Theorem 2.8, we get from Lemma 2.7 that u(t) ∈ H on

[0, T). So formula (2.15) in Lemma 2.6 holds on [0, T). Therefore, we have from (2.15) andLemma 2.4 that

12‖ut‖2 +

r − prp‖∇u‖pp ≤

12‖ut‖2 + J(u) = E(t) ≤ E(0) < d. (2.24)

Hence, we get

‖ut‖2 + ‖∇u‖pp ≤ max(

2,rp

r − p

)d < +∞. (2.25)

The above inequality and the continuation principle lead to the global existence of thesolution, that is, T = +∞. Thus, the solution u(t) is a global solution of the problem (1.1)–(1.3).

Now we employ the analysis method to discuss the blow-up solutions of the problem(1.1)–(1.3) in finite time. Our result reads as follows.

Theorem 2.9. Suppose that 2 < p < r < np/(n − p) if p < n and 2 < p < r < ∞ if n ≤ p. Ifu0 ∈ H, u1 ∈ L2(Ω), assume that the initial value is such that

E(0) < Q0, ‖u(0)‖r > S0, (2.26)

where

Q0 =r − prp

Cpr/(p−r), S0 = Cp/(p−r) (2.27)

with C > 0 is a positive Sobolev constant. Then the solution of the problem (1.1)–(1.3) does not existglobally in time.

Proof. On the contrary, under the conditions in Theorem 2.9, let u(x, t) be a global solution ofthe problem (1.1)–(1.3); then by Lemma 2.3, it is well known that there exists a constant C > 0depending only on n, p, and r such that ‖u‖r ≤ C‖∇u‖p for all u ∈W1,p

0 (Ω).


From the above inequality, we conclude that

‖∇u‖pp ≥ C−p‖u‖pr . (2.28)

By using (2.28), it follows from the definition of E(t) that

E(t) =12‖ut‖2 + J(u(t)) =

12‖ut‖2 +

1p‖∇u‖pp −

b

r‖u‖rr

≥ 1p‖∇u‖pp −

b

r‖u‖rr ≥

1pCp‖u‖pr −

b

r‖u‖rr .

(2.29)

Setting

s = s(t) = ‖u(t)‖r ={∫

Ω|u(x, t)|rdx

}1/r

, (2.30)

we denote the right side of (2.29) by Q(s) = Q(‖u(t)‖r), then

Q(s) =1pCp

sp − brsr , s ≥ 0. (2.31)

We have

Q′(s) = C−psp−1 − bsr−1. (2.32)

Letting Q′(t) = 0, we obtain S0 = (bCp)1/(p−r).As s = S0, we have

Q′′(s)∣∣s=S0

=(p − 1Cp

sp−2 − b(r − 1)sr−2)∣∣∣∣

s=S0

=(p − r

)(bp−2C(r−2)p

)1/(p−r)< 0. (2.33)

Consequently, the function Q(s) has a single maximum value Q0 at S0, where

Q0 = Q(S0) =1pCp

(bCp)p/(p−r) − br(bCp)r/(p−r) =

r − prp

(bpCpr)1/(p−r). (2.34)

Since the initial data is such that E(0), s(0) satisfies

E(0) < Q0, ‖u(0)‖r > S0. (2.35)


Therefore, from Lemma 2.4 we get

E(u(t)) ≤ E(0) < Q0, ∀t > 0. (2.36)

At the same time, by (2.29) and (2.31), it is clear that there can be no time t > 0 for which

E(u(t)) < Q0, s(t) = S0. (2.37)

Hence we have also s(t) > S0 for all t > 0 from the continuity of E(u(t)) and s(t).According to the above contradiction, we know that the global solution of the problem

(1.1)–(1.3) does not exist, that is, the solution blows up in some finite time.This completes the proof of Theorem 2.9.

3. The Exponential Asymptotic Behavior

Lemma 3.1 (see [17]). Let y(t) : R+ → R+ be a nonincreasing function, and assume that there is aconstant A > 0 such that

∫+∞

s

y(t)dt ≤ Ay(s), 0 ≤ s < +∞, (3.1)

then y(t) ≤ y(0)e1−(t/A), for all t ≥ 0.

The following theorem shows the exponential asymptotic behavior of global solutionsof problem (1.1)–(1.3).

Theorem 3.2. If the hypotheses in Theorem 2.8 are valid, then the global solutions of problem (1.1)–(1.3) have the following exponential asymptotic behavior:

12‖ut‖2 +

r − prp‖∇u‖pp ≤ E(0)e1−(t/M), ∀t ≥ 0. (3.2)

Proof. Multiplying by u on both sides of (1.1) and integrating over Ω × [S, T] gives

0 =∫T

S

∫

Ωu[utt −Δpu − aΔut − bu|u|r−2

]dx dt, (3.3)

where 0 ≤ S < T < +∞.Since

∫T

S

∫

Ωuuttdx dt =

∫

Ωuutdx

∣∣∣∣

T

S

−∫T

S

∫

Ω|ut|2dx dt, (3.4)


so, substituting the formula (3.4) into the right-hand side of (3.3) gives

0 =∫T

S

∫

Ω

(|ut|2 +

2p|∇u|pp −

2br|u|r

)dx dt

−∫T

S

∫

Ω

[2|ut|2 − a∇ut∇u

]dx dt +

∫

Ωuutdx

∣∣∣∣

T

S

+ b(

2r− 1

)∫T

S

‖u‖rrdt +p − 2p

∫T

S

‖∇u‖ppdt.

(3.5)

By exploiting Lemma 2.3 and (2.24), we easily arrive at

b‖u(t)‖rr ≤ bCr‖∇u(t)‖rp = bCr‖∇u(t)‖r−pp ‖∇u(t)‖pp

< bCr

(rpd

r − p

)(r−p)/p∥∥∥∇u(t)pp∥∥∥.

(3.6)

We obtain from (3.6) and (2.24) that

b

(1 − 2

r

)‖u‖rr ≤ bCr

(rpd

r − p

)(r−p)/p r − 2r‖∇u(t)‖pp

≤ bCr

(rpd

r − p

)(r−p)/p r − 2r·rp

r − pE(t)

=bp(r − 2)Cr

r − p

(rpd

r − p

)(r−p)/pE(t),

p − 2p

∫T

S

‖∇u‖ppdx dt ≤r(p − 2

)

r − p

∫T

S

E(t)dt.

(3.7)

It follows from (3.7) and (3.5) that

[

2 −bp(r − 2)Cr

r − p

(rpd

r − p

)(r−p)/p−r(p − 2

)

r − p

]∫T

S

E(t)dt

≤∫T

S

∫

Ω

[2|ut|2 − a∇ut∇u

]dx dt −

∫

Ωuut dx

∣∣∣∣

T

S

.

(3.8)

We have from Holder inequality, Lemma 2.3 and (2.24) that

∣∣∣∣∣−∫

Ωuut dx

∣∣∣∣

T

S

∣∣∣∣∣≤∣∣∣∣∣

(Cprp

r − p ·r − prp‖∇u‖pp +

12‖ut‖2

)∣∣∣∣

T

S

∣∣∣∣∣

≤ max(Cprp

r − p , 1)∣∣∣ E(t)|TS

∣∣∣ ≤ME(S).

(3.9)


Substituting the estimates of (3.9) into (3.8), we conclude that

[

2 −bp(r − 2)Cr

r − p

(rpd

r − p

)(r−p)/p−r(p − 2

)

r − p

]∫T

S

E(t)dt

≤∫T

S

∫

Ω

[2|ut|2 − a∇ut∇u

]dx dt +ME(S).

(3.10)

We get from Lemma 2.3 and Lemma 2.4 that

2∫T

S

∫

Ω|ut|2dx dt = 2

∫T

S

‖ut‖2dt ≤ 2C2∫T

S

‖∇ut‖2dt

= −2C2

a(E(T) − E(S)) ≤ 2C2

aE(S).

(3.11)

From Young inequality, Lemmas 2.3 and 2.4, and (2.24), it follows that

−a∫T

S

∫

Ω∇u∇utdx dt ≤ a

∫T

S

(εC2‖∇u‖2

p +M(ε)‖∇ut‖2)dt

≤aC2rpε

r − p

∫T

S

E(t)dt +M(ε)(E(S) − E(T))

≤aC2rpε

r − p

∫T

S

E(t)dt +M(ε)E(S).

(3.12)

Choosing ε small enough, such that

12

[bp(r − 2)Cr

r − p

(rpd

r − p

)(r−p)/p+r(p − 2

)

r − p +aC2rpε

r − p

]

< 1, (3.13)

and, substituting (3.11) and (3.12) into (3.10), we get

∫T

S

E(t)dt ≤ME(S). (3.14)

We let T → +∞ in (3.14) to get

∫+∞

S

E(t)dt ≤ME(S). (3.15)

Therefore, we have from (3.15) and Lemma 3.1 that

E(t) ≤ E(0)e1−(t/M), t ∈ [0,+∞). (3.16)


We conclude from u ∈ H, (2.4) and (3.16) that

12‖ut‖2 +

r − prp‖∇u‖pp ≤ E(0)e1−(t/M), ∀t ≥ 0. (3.17)

The proof of Theorem 3.2 is thus finished.

Acknowledgments

This paper was supported by the Natural Science Foundation of Zhejiang Province (no.Y6100016), the Science and Research Project of Zhejiang Province Education Commission (no.Y200803804 and Y200907298). The Research Foundation of Zhejiang University of Scienceand Technology (no. 200803), and the Middle-aged and Young Leader in Zhejiang Universityof Science and Technology (2008–2010).

References

[1] G. Andrews, “On the existence of solutions to the equation utt − uxtx = σ(ux)x,” Journal of DifferentialEquations, vol. 35, no. 2, pp. 200–231, 1980.

[2] G. Andrews and J. M. Ball, “Asymptotic behaviour and changes of phase in one-dimensionalnonlinear viscoelasticity,” Journal of Differential Equations, vol. 44, no. 2, pp. 306–341, 1982.

[3] D. D. Ang and P. N. Dinh, “Strong solutions of quasilinear wave equation with non-linear damping,”SIAM Journal on Mathematical Analysis, vol. 19, pp. 337–347, 1985.

[4] S. Kawashima and Y. Shibata, “Global existence and exponential stability of small solutions tononlinear viscoelasticity,” Communications in Mathematical Physics, vol. 148, no. 1, pp. 189–208, 1992.

[5] A. Haraux and E. Zuazua, “Decay estimates for some semilinear damped hyperbolic problems,”Archive for Rational Mechanics and Analysis, vol. 100, no. 2, pp. 191–206, 1988.

[6] M. Kopackova, “Remarks on bounded solutions of a semilinear dissipative hyperbolic equation,”Commentationes Mathematicae Universitatis Carolinae, vol. 30, no. 4, pp. 713–719, 1989.

[7] J. M. Ball, “Remarks on blow-up and nonexistence theorems for nonlinear evolution equations,” TheQuarterly Journal of Mathematics. Oxford, vol. 28, no. 112, pp. 473–486, 1977.

[8] Z. Yang, “Existence and asymptotic behaviour of solutions for a class of quasi-linear evolutionequations with non-linear damping and source terms,” Mathematical Methods in the Applied Sciences,vol. 25, no. 10, pp. 795–814, 2002.

[9] Z. Yang and G. Chen, “Global existence of solutions for quasi-linear wave equations with viscousdamping,” Journal of Mathematical Analysis and Applications, vol. 285, no. 2, pp. 604–618, 2003.

[10] Y. Zhijian, “Initial boundary value problem for a class of non-linear strongly damped waveequations,” Mathematical Methods in the Applied Sciences, vol. 26, no. 12, pp. 1047–1066, 2003.

[11] L. Yacheng and Z. Junsheng, “Multidimensional viscoelasticity equations with nonlinear dampingand source terms,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 6, pp. 851–865, 2004.

[12] Z. Yang, “Blowup of solutions for a class of non-linear evolution equations with non-linear dampingand source terms,” Mathematical Methods in the Applied Sciences, vol. 25, no. 10, pp. 825–833, 2002.

[13] M. Nakao, “A difference inequality and its application to nonlinear evolution equations,” Journal ofthe Mathematical Society of Japan, vol. 30, no. 4, pp. 747–762, 1978.

[14] Y. Ye, “Existence of global solutions for some nonlinear hyperbolic equation with a nonlineardissipative term,” Journal of Zhengzhou University, vol. 29, no. 3, pp. 18–23, 1997.

[15] Y. Ye, “On the decay of solutions for some nonlinear dissipative hyperbolic equations,” ActaMathematicae Applicatae Sinica. English Series, vol. 20, no. 1, pp. 93–100, 2004.

[16] D. H. Sattinger, “On global solution of nonlinear hyperbolic equations,” Archive for Rational Mechanicsand Analysis, vol. 30, pp. 148–172, 1968.

[17] V. Komornik, Exact Controllability and Stabilization, Research in Applied Mathematics, Masson, Paris,France, 1994.


[18] Y. Ye, “Existence and nonexistence of global solutions of the initial-boundary value problem for somedegenerate hyperbolic equation,” Acta Mathematica Scientia, vol. 25, no. 4, pp. 703–709, 2005.

[19] H. Gao and T. F. Ma, “Global solutions for a nonlinear wave equation with the p-Laplacian operator,”Electronic Journal of Qualitative Theory of Differential Equations, no. 11, pp. 1–13, 1999.

[20] R. A. Adams, Sobolev Spaces, vol. 6 of Pure and Applied Mathematics, Academic Press, New York, NY,USA, 1975.

[21] O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, vol. 49 of AppliedMathematical Sciences, Springer, New York, NY, USA, 1985.


Research ArticleOscillation Criteria for Second-Order QuasilinearNeutral Delay Dynamic Equations on Time Scales

Yibing Sun,1 Zhenlai Han,1, 2 Tongxing Li,1and Guangrong Zhang1

1 School of Science, University of Jinan, Jinan, Shandong 250022, China2 School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China


Received 25 January 2010; Accepted 24 February 2010


Copyright q 2010 Yibing Sun et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We establish some new oscillation criteria for the second-order quasilinear neutral delay dynamicequations [r(t)(zΔ(t))γ ]Δ + q1(t)xα(τ1(t)) + q2(t)xβ(τ2(t)) = 0 on a time scale T, where z(t) = x(t) +p(t)x(τ0(t)), 0 < α < γ < β. Our results generalize and improve some known results for oscillationof second-order nonlinear delay dynamic equations on time scales. Some examples are consideredto illustrate our main results.

1. Introduction

In this paper, we are concerned with oscillation behavior of the second order quasilinearneutral delay dynamic equations

[r(t)(zΔ(t)

)γ]Δ+ q1(t)xα(τ1(t)) + q2(t)xβ(τ2(t)) = 0, (1.1)

on an arbitrary time scale T, where z(t) = x(t) + p(t)x(τ0(t)), γ, α, and β are quotient of oddpositive integers such that 0 < α < γ < β, r, p, q1, and q2 are rd-continuous functions onT, and r, q1, and q2 are positive, −1 < −p0 ≤ p(t) < 1, p0 > 0; the so-called delay functionsτi : T → T satisfy that τi(t) ≤ t for t ∈ T and τi(t) → ∞ as t → ∞, for i = 0, 1, 2, and thereexists a function τ : T → T which satisfies that τ(t) ≤ τ1(t), τ(t) ≤ τ2(t), and τ(t) → ∞ ast → ∞.

Since we are interested in the oscillatory and asymptotic behavior of solutions nearinfinity, we assume that sup T = ∞ and define the time scale interval [t0,∞)

Tby [t0,∞)

T:=

[t0,∞) ∩ T.


We will also consider the two cases

∫∞

t0

Δtr1/γ(t)

=∞, (1.2)

∫∞

t0

Δtr1/γ(t)

<∞. (1.3)

Recently, there has been a large number of papers devoted to the delay dynamicequations on time scales, and we refer the reader to the papers in [1–17].

Agarwal et al. [1], Sahiner [10], Saker [11], Saker et al. [12], and Wu et al. [15] studiedthe second-order nonlinear neutral delay dynamic equations on time scales

(r(t)((y(t) + p(t)y(τ(t))

)Δ)γ)Δ + f(t, y(δ(t))

)= 0, t ∈ T, (1.4)

where 0 ≤ p(t) < 1, and (1.2) holds. By means of Riccati transformation technique, the authorsestablished some sufficient conditions for oscillation of (1.4).

Sun et al. [14] considered (1.1), where rΔ(t) ≥ 0, −1 < −p0 ≤ p(t) ≤ 0, and (1.2) holds.The authors established some oscillation results of (1.1). To the best of our knowledge, thereare no results regarding the oscillation of the solutions of (1.1) when (1.3) holds.

We note that if T = R, (1.1) becomes the second-order Emden-Fowler neutral delaydifferential equation

[r(t)(z′(t)

)γ]′ + q1(t)xα(τ1(t)) + q2(t)xβ(τ2(t)) = 0, t ≥ t0. (1.5)

Chen and Xu [18] as well as Xu and Liu [19] considered (1.5) and obtained some oscillationcriteria for (1.5) when r(t) = 1. Qin et al. [20] found that some results under the case when−1 < p0 ≤ p(t) ≤ 0 in [18, 19] are incorrect.

The paper is organized as follows. In the next section, by developing a Riccatitransformation technique some sufficient conditions for oscillation of all solutions of (1.1)on time scales are established. In Section 3, we give some examples to illustrate our mainresults.

2. Main Results

In this section, by employing the Riccati transformation technique, we establish some newoscillation criteria for (1.1). In order to prove our main results, we will use the formula

(xγ(t))Δ = γ∫1

0[hxσ(t) + (1 − h)x(t)]γ−1xΔ(t)dh, (2.1)

which is a simple consequence of Keller’s chain rule [21, Theorem 1.90]. Also, we need thefollowing lemmas.


It will be convenient to make the following notations:

d+(t) := max{0, d(t)}, θ(a, b;u) :=

∫auΔs/r

1/γ(s)∫buΔs/r

1/γ(s),

α(t, u) := θ(τ(t), σ(t);u), β(t, u) := θ(t, σ(t);u), ν := min{β − αβ − γ ,

β − αγ − α

},

Q1(t) := ν(q1(t)

(1 − p(τ1(t))

)α)(β−γ)/(β−α)(q2(t)

(1 − p(τ2(t))

)β)(γ−α)/(β−α)(α(t, T))γ ,

Q2(t) := ν(q1(t)

)(β−γ)/(β−α)(q2(t)

)(γ−α)/(β−α)(α(t, T))γ ,

Q1∗(t) = Q1(t) − ηΔ(t), Q2∗(t) = Q2(t) − ηΔ(t).

(2.2)

Lemma 2.1 (see [3, Lemma 2.4]). Assume that there exists T ≥ t0, sufficiently large, such that

x(t) > 0, xΔ(t) > 0,(r(t)(xΔ(t)

)γ)Δ< 0, t ≥ T. (2.3)

Then

x(τ(t)) ≥ α(t, T)xσ(t), x(t) ≥ β(t, T)xσ(t), for t ≥ T1 ≥ T. (2.4)

Lemma 2.2. Assume that (1.2) holds; 0 ≤ p(t) < 1. Furthermore, x is an eventually positive solutionof (1.1). Then there exists t1 ≥ t0 such that

z(t) > 0, zΔ(t) > 0,(r(t)(zΔ(t)

)γ)Δ< 0, for t ≥ t1. (2.5)

Proof. Let x be an eventually positive solution of (1.1). Then there exists t1 ≥ t0 such thatx(t) > 0, and x(τi(t)) > 0 for t ≥ t1, i = 0, 1, 2. From (1.1), we have

[r(t)(zΔ(t)

)γ]Δ= −q1(t)xα(τ1(t)) − q2(t)xβ(τ2(t)) < 0 (2.6)

for all t ≥ t1, and so r(t)(zΔ(t))γ is an eventually decreasing function.We first show that r(t)(zΔ(t))γ is eventually positive. Otherwise, there exists t2 ≥ t1

such that r(t2)(zΔ(t2))γ = c < 0; then from (2.6) we have r(t)(zΔ(t))γ ≤ r(t2)(zΔ(t2))γ = c for

t ≥ t2, and so

zΔ(t) ≤ c1/γ(

1r(t)

)1/γ

, (2.7)


which implies by (1.2) that

z(t) ≤ z(t2) + c1/γ∫ t

t2

(1

r(s)

)1/γ

Δs −→ −∞ as t −→ ∞, (2.8)

and this contradicts the fact that z(t) ≥ x(t) > 0 for all t ≥ t1. Hence, we have that (2.5) holdsand completes the proof.

Lemma 2.3. Assume that (1.2) holds, −1 < −p0 ≤ p(t) ≤ 0, and limt→∞p(t) = p > −1. Furthermore,assume that there exists {ck}k≥0 such that limk→∞ck = ∞ and τ0(ck+1) = ck. Then an eventuallypositive solution x of (1.1) satisfies eventually (2.5) or limt→∞x(t) = 0.

Proof. Suppose that x is an eventually positive solution of (1.1). Then there exists t1 ≥ t0 suchthat x(t) > 0, and x(τi(t)) > 0 for t ≥ t1, i = 0, 1, 2. From (1.1), we have that (2.6) holds for allt ≥ t1, and so r(t)(zΔ(t))γ is an eventually decreasing function.

We first show that r(t)(zΔ(t))γ is eventually positive. Otherwise, there exists t2 ≥ t1such that r(t2)(zΔ(t2))

γ = c < 0; then from (2.6) we have r(t)(zΔ(t))γ ≤ r(t2)(zΔ(t2))γ = c fort ≥ t2, and so

zΔ(t) ≤ c1/γ(

1r(t)

)1/γ

, (2.9)

which implies by (1.2) that

z(t) ≤ z(t2) + c1/γ∫ t

t2

(1

r(s)

)1/γ

Δs −→ −∞ as t −→ ∞. (2.10)

Therefore, there exist d > 0 and t3 ≥ t2 such that

x(t) ≤ −d − p(t)x(τ0(t)) ≤ −d + p0x(τ0(t)), t ≥ t3. (2.11)

Thus, we can choose some positive integer k0 such that ck ≥ t3 for k ≥ k0, and

x(ck) ≤ −d + p0x(τ0(ck)) = −d + p0x(ck−1) ≤ −d − p0d + p20x(τ0(ck−1))

= −d − p0d + p20x(ck−2) ≤ · · · ≤ −d − p0d − · · · − pk−k0−1

0 d + pk−k00 x(τ0(ck0+1))

= −d − p0d − · · · − pk−k0−10 d + pk−k0

0 x(ck0).

(2.12)

The above inequality implies that x(ck) < 0 for sufficiently large k, which contradicts the factthat x(t) is eventually positive. Hence zΔ(t) is eventually positive. Consequently, there aretwo possible cases:

(i) z(t) is eventually positive, or

(ii) z(t) is eventually negative.


If there exists a t4 ≥ t1 such that case (ii) holds, then limt→∞z(t) exists, andlimt→∞z(t) = l ≤ 0; we claim that limt→∞z(t) = 0. Otherwise, limt→∞z(t) < 0. We can choosesome positive integer k0 such that ck ≥ t4 for k ≥ k0, and we obtain

x(ck) ≤ p0x(τ0(ck)) = p0x(ck−1) ≤ p20x(τ0(ck−1))

= p20x(ck−2) ≤ · · · ≤ pk−k0

0 x(τ0(ck0+1)) = pk−k00 x(ck0),

(2.13)

which implies that limk→∞x(ck) = 0, and so limk→∞z(ck) = 0, which contradictslimt→∞z(t) = l < 0. Now, we assert that x(t) is bounded. If it is not true, then there exists{tk}with tk → ∞ as k → ∞ such that

x(tk) = maxt0≤s≤tk

x(s), limk→∞

x(tk) =∞. (2.14)

From τ0(t) ≤ t, we obtain

z(tk) = x(tk) + p(tk)x(τ0(tk)) ≥(1 − p0

)x(tk), (2.15)

which implies that limk→∞z(tk) =∞; it contradicts limt→∞z(t) = 0. Therefore, we can assumethat

lim supt→∞

x(t) = x1, lim inft→∞

x(t) = x2. (2.16)

By −1 < p ≤ 0, we get

x1 + px1 ≤ 0 ≤ x2 + px2, (2.17)

which implies that x1 ≤ x2, so x1 = x2. Hence, limt→∞x(t) = 0. The proof is complete.

Theorem 2.4. Assume that (1.2) holds, 0 ≤ p(t) < 1, and γ ≥ 1. Furthermore, assume that thereexist positive rd-continuous Δ-differentiable functions δ and η such that, for all sufficiently large T,for T1 > T

lim supt→∞

∫ t

T1

⎡

⎣δσ(s)Q1∗(s) − δΔ(s)η(s) −r(s)

(γ + 1

)γ+1

((δΔ(s)

)+

)γ+1

(δσ(s))γ(β(s, T)

)−γ2

⎤

⎦Δs =∞. (2.18)

Then every solution of (1.1) is oscillatory.

Proof. Suppose that (1.1) has a nonoscillatory solution x. We may assume without loss ofgenerality that x(τi(t)) > 0, i = 0, 1, 2, for all t ≥ t0. By Lemma 2.2, there exists T ≥ t0 such that(2.5) holds. Define the function ω by

ω(t) = δ(t)

[r(t)(zΔ(t)

)γ

zγ(t)+ η(t)

]

, t ≥ T. (2.19)


Then ω(t) > 0. By the product rule and the quotient rule, noteing (2.19), we have

ωΔ(t) =δΔ(t)δ(t)

ω(t) + δσ(t)

⎡

⎢⎣

(r(t)(zΔ(t)

)γ)Δ

(zσ(t))γ−r(t)(zΔ(t)

)γ(zγ(t))Δ

zγ(t)(zσ(t))γ+ ηΔ(t)

⎤

⎥⎦. (2.20)

By (1.1) and (2.5), we obtain

(r(t)(zΔ(t)

)γ)Δ≤ −q1(t)

((1 − p(τ1(t))

)z(τ1(t))

)α − q2(t)((

1 − p(τ2(t)))z(τ2(t))

)β< 0.

(2.21)

In view of γ ≥ 1, from (2.1), we have (zγ(t))Δ ≥ γ(z(t))γ−1zΔ(t). By (2.20), we obtain

ωΔ(t) ≤ δΔ(t)δ(t)

ω(t) − δσ(t)q1(t)(1 − p(τ1(t))

)α (z(τ1(t)))α

(zσ(t))γ

− δσ(t)q2(t)(1 − p(τ2(t))

)β (z(τ2(t)))β

(zσ(t))γ− γδσ(t)

r(t)(zΔ(t)

)γ+1

z(t)(zσ(t))γ+ δσ(t)ηΔ(t).

(2.22)

By Young’s inequality

|ab| ≤ 1p|a|p + 1

q|b|q, a, b ∈ R, p > 1, q > 1,

1p+

1q= 1, (2.23)

we have

β − γβ − αq1(t)

(1 − p(τ1(t))

)α (z(τ1(t)))α

(zσ(t))γ+γ − αβ − αq2(t)

(1 − p(τ2(t))

)β (z(τ2(t)))β

(zσ(t))γ

≥[q1(t)

(1 − p(τ1(t))

)α (z(τ1(t)))α

(zσ(t))γ

](β−γ)/(β−α)[

q2(t)(1 − p(τ2(t))

)β (z(τ2(t)))β

(zσ(t))γ

](γ−α)/(β−α)

=(q1(t)

(1 − p(τ1(t))

)α)(β−γ)/(β−α)(q2(t)

(1 − p(τ2(t))

)β)(γ−α)/(β−α)((z(τ1(t)))α

(zσ(t))γ

)(β−γ)/(β−α)

×(

(z(τ2(t)))β

(zσ(t))γ

)(γ−α)/(β−α)

≥(q1(t)

(1 − p(τ1(t))

)α)(β−γ)/(β−α)(q2(t)

(1 − p(τ2(t))

)β)(γ−α)/(β−α)(z(τ(t))zσ(t)

)γ.

(2.24)


By Lemma 2.1, we have

z(τ(t))zσ(t)

≥ α(t, T), z(t)zσ(t)

≥ β(t, T). (2.25)

Hence, by (2.19) and (2.22), we obtain


ω(t) − νδσ(t)(q1(t)

(1 − p(τ1(t))

)α)(β−γ)/(β−α)

×(q2(t)

(1 − p(τ2(t))

)β)(γ−α)/(β−α)(α(t, T))γ

− γδσ(t) 1

(r(t))1/γ

(β(t, T)

)γ(ω(t)δ(t)

− η(t))(γ+1)/γ

+ δσ(t)ηΔ(t).

(2.26)

Thus

ωΔ(t) ≤ −δσ(t)[Q1(t) − ηΔ(t)

]+ δΔ(t)η(t) +

(δΔ(t)

)

+

∣∣∣∣ω(t)δ(t)

− η(t)∣∣∣∣

− γδσ(t) 1

(r(t))1/γ

(β(t, T)

)γ(ω(t)δ(t)

− η(t))(γ+1)/γ

.

(2.27)

Set

λ =γ + 1γ

, A = γ1/λ(δσ(t))1/λ 1

(r(t))1/(γ+1)

(β(t, T)

)γ2/(γ+1)∣∣∣∣ω(t)δ(t)

− η(t)∣∣∣∣,

B =((δΔ(t)

)

+

)γ( γ

γ + 1

)γ (r(t))γ/(γ+1)

γγ2/(γ+1)(δσ(t))γ

2/(γ+1)

(1

β(t, T)

)γ3/(γ+1)

.

(2.28)

Using the inequality

λABλ−1 −Aλ ≤ (λ − 1)Bλ, λ ≥ 1, A ≥ 0, B ≥ 0, (2.29)

we obtain

ωΔ(t) ≤ −δσ(t)[Q1(t) − ηΔ(t)

]+ δΔ(t)η(t) +

r(t)(γ + 1

)γ+1

((δΔ(t)

)+

)γ+1

(δσ(t))γ(β(t, T)

)−γ2

. (2.30)


Integrating the last inequality from T1 > T to t > T1, we obtain

−ω(T1) < ω(t) −ω(T1)

≤−∫ t

T1

⎡

⎣δσ(s)(Q1(s)−ηΔ(s)

)−δΔ(s)η(s)− r(s)

(γ + 1

)γ+1

((δΔ(s)

)+

)γ+1


)−γ2

⎤

⎦Δs,

(2.31)

which yields

∫ t

T1

⎡

⎣δσ(s)(Q1(s) − ηΔ(s)

)− δΔ(s)η(s) − r(s)

(γ + 1

)γ+1

((δΔ(s)

)+

)γ+1


)−γ2

⎤

⎦Δs ≤ ω(T1),

(2.32)

which leads to a contradiction to (2.18). The proof is complete.

Theorem 2.5. Assume that (1.2) holds, 0 ≤ p(t) < 1, and γ ≤ 1. Furthermore, assume that thereexist positive rd-continuous Δ-differentiable functions δ and η such that, for all sufficiently large T,for T1 > T

lim supt→∞

∫ t

T1

⎡


(γ + 1

)γ+1

((δΔ(s)

)+

)γ+1


)−γ⎤

⎦Δs =∞.

(2.33)

Then every solution of (1.1) is oscillatory.

Proof. Suppose that (1.1) has a nonoscillatory solution x. We may assume without loss ofgenerality that x(τi(t)) > 0, i = 0, 1, 2, for all t ≥ t0.

By Lemma 2.2, there exists T ≥ t0 such that (2.5) holds. Defining the function ω as(2.19), we proceed as in the proof of Theorem 2.4, and we get (2.20). In view of γ ≤ 1, using(2.1), we have (zγ(t))Δ ≥ γ(zσ(t))γ−1zΔ(t). From (2.20) we obtain


ω(t) − δσ(t)q1(t)(1 − p(τ1(t))

)α (z(τ1(t)))α

(zσ(t))γ

− δσ(t)q2(t)(1 − p(τ2(t))

)β (z(τ2(t)))β

(zσ(t))γ− γδσ(t)

r(t)(zΔ(t)

)γ+1

zγ(t)zσ(t)+ δσ(t)ηΔ(t).

(2.34)

The remainder of the proof is similar to that of Theorem 2.4, and hence it is omitted.


Theorem 2.6. Assume that (1.3) holds, 0 ≤ p(t) < 1, limt→∞p(t) = p1 < 1, and γ ≥ 1. Furthermore,assume that there exist positive rd-continuousΔ-differentiable functions δ, η, and φ such that φΔ(t) ≥0, then for all sufficiently large T, for T1 > T, one has that (2.18) holds, and

∫∞

t0

(1

φ(s)r(s)

∫s

t0

φσ(τ)[q1(τ) + q2(τ)

]Δτ

)1/γ

Δs =∞. (2.35)

Then every solution of (1.1) is either oscillatory or converges to zero.

Proof. We proceed as in Theorem 2.4, and we assume that x(τi(t)) > 0, i = 0, 1, 2, for all t ≥ t0.From the proof of Lemma 2.2, we see that there exist two possible cases for the sign of zΔ(t).

If zΔ(t) is eventually positive, we are then back to the proof of Theorem 2.4 and weobtain a contradiction with (2.18).

If zΔ(t) < 0, t ≥ t1 ≥ t0, then there exist constants c > 0, a > 0 such that z(t) ≤ c,x(t) ≤ z(t) ≤ c, t ≥ t1, and limt→∞z(t) = a ≥ 0. Since x is bounded, we let lim supt→∞x(t) = x1,lim inft→∞x(t) = x2. From definition of z(t), noting 0 ≤ p1 < 1, we have x1 + p1x2 ≤ a ≤x2 + p1x1; hence, we have x1 ≤ x2.

On the other hand, x1 ≥ x2; hence, limt→∞x(t) = a/(1 + p1). Assume that a > 0. Thenthere exist a constant b > 0 and t2 ≥ t1 such that xα(τ1(t)) ≥ b, xβ(τ2(t)) ≥ b for t ≥ t2. Definethe function

u(t) = φ(t)r(t)(zΔ(t)

)γ. (2.36)

Then u(t) < 0 for t ≥ t2. From (1.1) we have

uΔ(t) = φΔ(t)r(t)(zΔ(t)

)γ+ φσ(t)

[r(t)(zΔ(t)

)γ]Δ≤ φσ(t)

[r(t)(zΔ(t)

)γ]Δ

= −φσ(t)[q1(t)xα(τ1(t)) + q2(t)xβ(τ2(t))

]≤ −bφσ(t)

[q1(t) + q2(t)

].

(2.37)

Integrating the above inequality from t2 to t, we obtain

u(t) ≤ u(t2) − b∫ t

t2

φσ(s)[q1(s) + q2(s)

]Δs ≤ −b

∫ t

t2

φσ(s)[q1(s) + q2(s)

]Δs, (2.38)

that is,

zΔ(t) ≤ −b1/γ

(1

φ(t)r(t)

∫ t

t2

φσ(s)[q1(s) + q2(s)

]Δs

)1/γ

. (2.39)


Integrating the last inequality from t2 to t, we get

z(t) ≤ z(t2) − b1/γ∫ t

t2

(1

φ(s)r(s)

∫s

t2

φσ(τ)[q1(τ) + q2(τ)

]Δτ

)1/γ

Δs. (2.40)

We can easily obtain a contradiction with (2.35). Hence, limt→∞x(t) = 0. This completes theproof.

From Theorem 2.6, we have the following result.

Theorem 2.7. Assume that (1.3) holds, 0 ≤ p(t) < 1, limt→∞p(t) = p1 < 1, and γ ≤ 1. Furthermore,assume that there exist positive rd-continuous Δ-differentiable functions δ, η, and φ such that, for allsufficiently large T, for T1 > T, one has that (2.33) and (2.35) hold. Then every solution of (1.1) iseither oscillatory or converges to zero.

The proof is similar to that of the proof of Theorem 2.6; hence, we omit the details.In the following, we give some new oscillation results of (1.1) when p(t) < 0.

Theorem 2.8. Assume that (1.2) holds, −1 < −p0 ≤ p(t) ≤ 0, limt→∞p(t) = p2 > −1, and γ ≥ 1.Furthermore, there exists {ck}k≥0 such that limk→∞ck = ∞ and τ0(ck+1) = ck. If there exist positiverd-continuous Δ-differentiable functions δ and η such that, for all sufficiently large T, for T1 > T ,

lim supt→∞

∫ t

T1

⎡


(γ + 1

)γ+1

((δΔ(s)

)+

)γ+1


)−γ2

⎤

⎦Δs =∞,

(2.41)

then every solution of (1.1) is oscillatory or tends to zero.

Proof. Suppose that (1.1) has a nonoscillatory solution x. We may assume without loss ofgenerality that x(τi(t)) > 0, i = 0, 1, 2, for all t ≥ t0. By Lemma 2.3, there exists T ≥ t0 such that(2.5) holds, or limt→∞x(t) = 0. Assume that (2.5) holds. Define the function ω as (2.19), andthen we get (2.20). By (1.1), we obtain

(r(t)(zΔ(t)

)γ)Δ≤ −q1(t)(z(τ1(t)))α − q2(t)(z(τ2(t)))β < 0. (2.42)

In view of γ ≥ 1, from (2.1), we have (zγ(t))Δ ≥ γ(z(t))γ−1zΔ(t). By (2.20), we obtain


ω(t) − δσ(t)q1(t)(z(τ1(t)))α

(zσ(t))γ− δσ(t)q2(t)

(z(τ2(t)))β

(zσ(t))γ

− γδσ(t)r(t)(zΔ(t)

)γ+1

z(t)(zσ(t))γ+ δσ(t)ηΔ(t).

(2.43)


By Young’s inequality (2.23), we have

β − γβ − αq1(t)

(z(τ1(t)))α

(zσ(t))γ+γ − αβ − αq2(t)

(z(τ2(t)))β

(zσ(t))γ

≥[q1(t)

(z(τ1(t)))α

(zσ(t))γ

](β−γ)/(β−α)[

q2(t)(z(τ2(t)))β

(zσ(t))γ

](γ−α)/(β−α)

=(q1(t)

)(β−γ)/(β−α)(q2(t)

)(γ−α)/(β−α)((z(τ1(t)))α

(zσ(t))γ

)(β−γ)/(β−α)( (z(τ2(t)))β

(zσ(t))γ

)(γ−α)/(β−α)

≥(q1(t)

)(β−γ)/(β−α)(q2(t)

)(γ−α)/(β−α)(z(τ(t))zσ(t)

)γ.

(2.44)

By Lemma 2.1, we have

z(τ(t))zσ(t)

≥ α(t, T), z(t)zσ(t)

≥ β(t, T). (2.45)

Hence, by (2.19) and (2.43), we obtain


ω(t) − νδσ(t)(q1(t)

)(β−γ)/(β−α)(q2(t)

)(γ−α)/(β−α)(α(t, T))γ

− γδσ(t) 1

(r(t))1/γ

(β(t, T)

)γ(ω(t)δ(t)

− η(t))(γ+1)/γ

+ δσ(t)ηΔ(t).

(2.46)

Thus

ωΔ(t) ≤ −δσ(t)[Q2(t) − ηΔ(t)

]+ δΔ(t)η(t) +

(δΔ(t)

)

+

∣∣∣∣ω(t)δ(t)

− η(t)∣∣∣∣

− γδσ(t) 1

(r(t))1/γ

(β(t, T)

)γ(ω(t)δ(t)

− η(t))(γ+1)/γ

.

(2.47)

Set

λ =γ + 1γ

, A = γ1/λ(δσ(t))1/λ 1

(r(t))1/γ+1

(β(t, T)

)γ2/(γ+1)∣∣∣∣ω(t)δ(t)

− η(t)∣∣∣∣,

B =((δΔ(t)

)

+

)γ( γ

γ + 1

)γ (r(t))γ/(γ+1)

γγ2/(γ+1)(δσ(t))γ

2/(γ+1)

(1

β(t, T)

)γ3/(γ+1)

.

(2.48)


Using the inequality (2.29), we obtain

ωΔ(t) ≤ −δσ(t)[Q2(t) − ηΔ(t)

]+ δΔ(t)η(t) +

r(t)(γ + 1

)γ+1

((δΔ(t)

)+

)γ+1

(δσ(t))γ(β(t, T)

)−γ2

. (2.49)

Integrating the last inequality from T1 > T to t > T1, we obtain

−ω(T1) < ω(t) −ω(T1)

≤−∫ t

T1

⎡

⎣δσ(s)(Q2(s)−ηΔ(s)

)−δΔ(s)η(s)− r(s)

(γ + 1

)γ+1

((δΔ(s)

)+

)γ+1


)−γ2

⎤

⎦Δs,

(2.50)

which yields

∫ t

T1

⎡

⎣δσ(s)(Q2(s) − ηΔ(s)

)− δΔ(s)η(s) − r(s)

(γ + 1

)γ+1

((δΔ(s)

)+

)γ+1


)−γ2

⎤

⎦Δs ≤ ω(T1),

(2.51)

which leads to a contradiction with (2.41). The proof is complete.

Theorem 2.9. Assume that (1.2) holds, −1 < −p0 ≤ p(t) ≤ 0, limt→∞p(t) = p2 > −1, and γ ≤ 1.Furthermore, there exists {ck}k≥0 such that limk→∞ck = ∞ and τ0(ck+1) = ck. If there exist positiverd-continuous Δ-differentiable functions δ and η such that, for all sufficiently large T, for T1 > T ,

lim supt→∞

∫ t

T1

⎡


(γ + 1

)γ+1

((δΔ(s)

)+

)γ+1


)−γ⎤

⎦Δs =∞,

(2.52)

then every solution of (1.1) is oscillatory or tends to zero.

Proof. Suppose that (1.1) has a nonoscillatory solution x. We may assume without loss ofgenerality that x(τi(t)) > 0, i = 0, 1, 2, for all t ≥ t0. By Lemma 2.3, there exists T ≥ t0 suchthat (2.5) holds, or limt→∞x(t) = 0. Assume that (2.5) holds.

Define the function ω as (2.19), and then we get (2.20). In view of γ ≤ 1, using (2.1),we have (zγ(t))Δ ≥ γ(zσ(t))γ−1zΔ(t). From (2.20) we obtain


ω(t) − δσ(t)q1(t)(z(τ1(t)))α

(zσ(t))γ− δσ(t)q2(t)

(z(τ2(t)))β

(zσ(t))γ

− γδσ(t)r(t)(zΔ(t)

)γ+1

zγ(t)zσ(t)+ δσ(t)ηΔ(t).

(2.53)

The remainder of the proof is similar to that of Theorem 2.8, and hence it is omitted.


Remark 2.10. One can easily see that the results obtained in [1, 10–12, 15] cannot be appliedin (1.1), so our results are new.

3. Examples

In this section, we will give some examples to illustrate our main results.

Example 3.1. Consider the second-order quasilinear neutral delay dynamic equations on timescales

(

tσ(t)(x(t) +

12x(τ0(t))

)Δ)Δ

+σ(t)τ(t)

x1/3(τ(t)) +σ(t)τ(t)

x5/3(τ(t)) = 0, (3.1)

where t ∈ [t0,∞)T, and we assume that

∫∞t0Δt/tσ(t) =∞.

Let r(t) = tσ(t), p(t) = 1/2, q1(t) = q2(t) = σ(t)/τ(t), γ = 1, α = 1/3, β = 5/3, andτ1(t) = τ2(t) = τ(t). Take δ(t) = η(t) = φ(t) = 1. It is easy to show that (2.18) and (2.35) hold.Hence, by Theorem 2.6, every solution of (3.1) oscillates or tends to zero.

Example 3.2. Consider the second-order quasilinear neutral delay dynamic equations on timescales

((x(t) − 1

2x(τ0(t))

)Δ)Δ

+σ(t)tτ(t)

x1/3(τ(t)) +σ(t)tτ(t)

x5/3(τ(t)) = 0, (3.2)

where t ∈ [t0,∞)T, and we assume there exists {ck}k≥0 such that limk→∞ck =∞ and τ0(ck+1) =

ck.Let r(t) = 1, p(t) = −1/2, q1(t) = q2(t) = σ(t)/tτ(t), γ = 1, α = 1/3, β = 5/3, τ1(t) =

τ2(t) = τ(t). Take δ(t) = η(t) = 1. It is easy to show that (2.41) holds. Hence, by Theorem 2.8,every solution of (3.2) oscillates or tends to zero.

Acknowledgment

This research is supported by the Natural Science Foundation of China (60774004, 60904024),China Postdoctoral Science Foundation funded project (20080441126, 200902564), ShandongPostdoctoral funded project (200802018), the Natural Science Foundation of Shandong(Y2008A28, ZR2009AL003), and also the University of Jinan Research Funds for Doctors(B0621, XBS0843).

References

[1] R. P. Agarwal, D. O’Regan, and S. H. Saker, “Oscillation criteria for second-order nonlinear neutraldelay dynamic equations,” Journal of Mathematical Analysis and Applications, vol. 300, no. 1, pp. 203–217, 2004.

[2] L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for second-order nonlinear delay dynamicequations,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 505–522, 2007.

[3] L. Erbe, T. S. Hassan, and A. Peterson, “Oscillation criteria for nonlinear damped dynamic equationson time scales,” Applied Mathematics and Computation, vol. 203, no. 1, pp. 343–357, 2008.


[4] Z. Han, S. Sun, and B. Shi, “Oscillation criteria for a class of second-order Emden-Fowler delaydynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2,pp. 847–858, 2007.

[5] Z. Han, T. Li, S. Sun, and C. Zhang, “Oscillation for second-order nonlinear delay dynamic equationson time scales,” Advances in Difference Equations, vol. 2009, Article ID 756171, pp. 1–13, 2009.

[6] Z. Han, B. Shi, and S. Sun, “Oscillation criteria for second-order delay dynamic equations on timescales,” Advances in Difference Equations, vol. 2007, Article ID 70730, pp. 1–16, 2007.

[7] Z. Han, T. Li, S. Sun, and C. Zhang, “Oscillation behavior of third order neutral Emden-Fowler delaydynamic equations on time scales,” Advances in Differential Equations, vol. 2010, Article ID 586312, pp.1–23, 2010.

[8] T. Li, Z. Han, S. Sun, and D. Yang, “Existence of nonoscillatory solutions to second-order neutral delaydynamic equations on time scales,” Advances in Difference Equations, vol. 209, Article ID 562329, pp.1–10, 2009.

[9] T. Li, Z. Han, S. Sun, and C. Zhang, “Forced oscillation of second-order nonlinear dynamic equationson time scales,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 60, pp. 1–8, 2009.

[10] Y. Sahiner, “Oscillation of second-order neutral delay and mixed-type dynamic equations on timescales,” Advances in Difference Equations, vol. 2006, Article ID 65626, pp. 1–9, 2006.

[11] S. H. Saker, “Oscillation of second-order nonlinear neutral delay dynamic equations on time scales,”Journal of Computational and Applied Mathematics, vol. 187, no. 2, pp. 123–141, 2006.

[12] S. H. Saker, R. P. Agarwal, and D. O’Regan, “Oscillation results for second-order nonlinear neutraldelay dynamic equations on time scales,” Applicable Analysis, vol. 86, no. 1, pp. 1–17, 2007.

[13] S. Sun, Z. Han, and C. Zhang, “Oscillation of second-order delay dynamic equations on time scales,”Journal of Applied Mathematics and Computing, vol. 30, no. 1-2, pp. 459–468, 2009.

[14] S. Sun, Z. Han, and C. Zhang, “Oscillation criteria of second-order Emden-Fowler neutral delaydynamic equations on time scales,” Journal of Shanghai Jiaotong University, vol. 42, no. 12, pp. 2070–2075, 2008.

[15] H. Wu, R. Zhuang, and R. M. Mathsen, “Oscillation criteria for second-order nonlinear neutralvariable delay dynamic equations,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 321–331,2006.

[16] B. G. Zhang and Z. Shanliang, “Oscillation of second-order nonlinear delay dynamic equations ontime scales,” Computers & Mathematics with Applications, vol. 49, no. 4, pp. 599–609, 2005.

[17] Z. Zhu and Q. Wang, “Existence of nonoscillatory solutions to neutral dynamic equations on timescales,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 751–762, 2007.

[18] M. Chen and Z. Xu, “Interval oscillation of second-order Emden-Fowler neutral delay differentialequations,” Electronic Journal of Differential Equations, vol. 58, pp. 1–9, 2007.

[19] Z. Xu and X. Liu, “Philos-type oscillation criteria for Emden-Fowler neutral delay differentialequations,” Journal of Computational and Applied Mathematics, vol. 206, no. 2, pp. 1116–1126, 2007.

[20] H. Qin, N. Shang, and Y. Lu, “A note on oscillation criteria of second order nonlinear neutral delaydifferential equations,” Computers &Mathematics with Applications, vol. 56, no. 12, pp. 2987–2992, 2008.

[21] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Application,Birkhauser, Boston, Mass, USA, 2001.


Research ArticleExistence of 2n Positive Periodic Solutions ton-Species Nonautonomous Food Chains withHarvesting Terms

Yongkun Li1 and Kaihong Zhao1, 2

1 Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China2 Department of Mathematics, Yuxi Normal University, Yuxi, Yunnan 653100, China

Correspondence should be addressed to Yongkun Li, [email protected]



Copyright q 2010 Y. Li and K. Zhao. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

By using Mawhin’s continuation theorem of coincidence degree theory and some skills ofinequalities, we establish the existence of at least 2n positive periodic solutions for n-speciesnonautonomous Lotka-Volterra type food chains with harvesting terms. An example is given toillustrate the effectiveness of our results.

1. Introduction

The dynamic relationship between predators and their prey has long been and will continueto be one of the dominant themes in both ecology and mathematical ecology due to itsuniversal existence and importance. These problems may appear to be simple mathematicallyat first; sight, they are, in fact, very challenging and complicated. There are many differentkinds of predator-prey models in the literature. For more details, we refer to [1, 2]. Foodchain predator-prey system, as one of the most important predator-prey system, has beenextensively studied by many scholars, many excellent results concerned with the persistentproperty and positive periodic solution of the system; see [3–13] and the references citedtherein. However, to the best of the authors’ knowledge, to this day, still no scholar study then-species nonautonomous case of Food chain predator-prey system with harvesting terms.Indeed, the exploitation of biological resources and the harvest of population species arecommonly practiced in fishery, forestry, and wildlife management; the study of populationdynamics with harvesting is an important subject in mathematical bioeconomics, which isrelated to the optimal management of renewable resources (see [14–16]). This motivates usto consider the following n-species nonautonomous Lotka-Volterra type food chain modelwith harvesting terms:


x1(t) = x1(t)(a1(t) − b1(t)x1(t) − c12(t)x2(t)) − h1(t),

...

xi(t) = xi(t)(−di(t) − bi(t)xi(t) + ci,i−1(t)xi−1(t) − ci,i+1(t)xi+1(t)) − hi(t),

...

xn(t) = xn(t)(−dn(t) − bn(t)xn(t) + cn,n−1(t)xn−1(t)) − hn(t),

(1.1)

where i = 2, 3, . . . , n − 1, xi(t) (i = 1, 2, . . . , n) is the ith species population density, a1(t) is thegrowth rate of the first species that is the only producer in system (1.1), bi(t) (i = 1, 2, . . . , n)and hi(t) (i = 1, 2, . . . , n) stand for the ith species intraspecific competition rate and harvestingrate, respectively, di(t) (i = 2, 3, . . . , n) is the death rate of the ith species, ci,i+1(t) (i =1, 2, . . . , n− 1) represents the (i+ 1)th species predation rate on the ith species, and ci,i−1(t)(i =2, 3, . . . , n) stands for the transformation rate from the (i − 1)th species to the ith species. Inaddition, the effects of a periodically varying environment are important for evolutionarytheory as the selective forces on systems in a fluctuating environment differ from thosein a stable environment. Therefore, the assumptions of periodicity of the parameters are away of incorporating the periodicity of the environment (e.g, seasonal effects of weather,food supplies, mating habits, etc), which leads us to assume that a1(t), bi(t), di(t), cij(t), andhi(t) (i, j = 1, 2, . . . , n) are all positive continuous ω-periodic functions.

Since a very basic and important problem in the study of a population growth modelwith a periodic environment is the global existence and stability of a positive periodicsolution, which plays a similar role as a globally stable equilibrium does in an autonomousmodel, this motivates us to investigate the existence of a positive periodic or multiple positiveperiodic solutions for system (1.1). In fact, it is more likely for some biological species to takeon multiple periodic change regulations and have multiple local stable periodic phenomena.Therefore, it is essential for us to investigate the existence of multiple positive periodicsolutions for population models. Our main purpose of this paper is by using Mawhin’scontinuation theorem of coincidence degree theory [17], to establish the existence of 2n

positive periodic solutions for system (1.1). For the work concerning the multiple existence ofperiodic solutions of periodic population models which was done using coincidence degreetheory, we refer to [18–21].

The organization of the rest of this paper is as follows. In Section 2, by employing thecontinuation theorem of coincidence degree theory and the skills of inequalities, we establishthe existence of at least 2n positive periodic solutions of system (1.1). In Section 3, an exampleis given to illustrate the effectiveness of our results.

2. Existence of at Least 2n Positive Periodic Solutions

In this section, by using Mawhin’s continuation theorem and the skills of inequalities, weshall show the existence of positive periodic solutions of (1.1). To do so, we need to makesome preparations.

Let X and Z be real normed vector spaces. Let L : Dom L ⊂ X → Z be a linearmapping and N : X × [0, 1] → Z be a continuous mapping. The mapping L will be calleda Fredholm mapping of index zero if dim Ker L = codim ImL < ∞ and ImL is closed in Z.


If L is a Fredholm mapping of index zero, then there exist continuous projectors P : X → Xand Q : Z → Z such that ImP = Ker L and Ker Q = ImL = Im (I − Q), and X = Ker L ⊕Ker P, Z = ImL ⊕ ImQ. It follows that L|DomL∩KerP : (I − P)X → ImL is invertible andits inverse is denoted by KP . If Ω is a bounded open subset of X, the mapping N is called L-compact on Ω×[0, 1], ifQN(Ω×[0, 1]) is bounded andKP (I−Q)N : Ω×[0, 1] → X is compact.Because Im Q is isomorphic to Ker L, there exists an isomorphism J : ImQ → Ker L.

The Mawhin’s continuous theorem [17, page 40] is given as follows.

Lemma 2.1 (see [9]). Let L be a Fredholm mapping of index zero and let N be L-compact on Ω ×[0, 1]. Assume that

(a) for each λ ∈ (0, 1), every solution x of Lx = λN(x, λ) is such that x /∈ ∂Ω ∩DomL;

(b) QN(x, 0)x /= 0 for each x ∈ ∂Ω ∩ KerL;

(c) deg(JQN(x, 0),Ω ∩ KerL, 0)/= 0.

Then Lx =N(x, 1) has at least one solution in Ω ∩Dom L.

For the sake of convenience, we denote fl = mint∈[0,ω]f(t), fM = maxt∈[0,ω]f(t), f =(1/ω)

∫ω0 f(t) dt, respectively; here f(t) is a continuous ω-periodic function.For simplicity, we need to introduce some notations as follows:

A±1 =

(al1 − c

M12 l

+2

)±√(

al1 − cM12 l

+2

)2− 4bM1 hM1

2bM1, l±i =

cMi,i−1l+i−1 ±

√(cMi,i−1l

+i−1

)2− 4blih

li

2bli,

l±1 =aM1 ±

√(aM1)2 − 4blnhln

2bln, A±n =

(cln,n−1l

−n−1 − dMn

)±√(

cln,n−1l−n−1 − d

Mn

)2− 4bMn hMn

2bMn,

A±i =

(cli,i−1l

−i−1 − c

Mi,i+1l

+i+1 − d

Mi

)±√(

cli,i−1l−i−1 − c

Mi,i+1l

+i+1 − d

Mi

)2− 4bMi h

Mi

2bMi,

B±1 =al1 ±

√(al1

)2− 4bMn hMn

2bMn, B±i =

cli,i−1l−i−1 ±

√(cli,i−1l

−i−1

)2− 4bMi h

Mi

2bMi,

(2.1)

where i = 2, 3, . . . , n.Throughout this paper, we need the following assumptions:

(H1) al1 − cM12 l

+2 > 2

√bM1 hM1 and cln,n−1l

−n−1 − dMn > 2

√bMn h

Mn ;

(H2) cli,i−1l−i−1 − c

Mi,i+1l

+i+1 − d

Mi > 2

√bMi h

Mi , i = 2, 3, . . . , n − 1.


Lemma 2.2. Let x > 0, y > 0, z > 0 and x > 2√yz, for the functions f(x, y, z) = (x +√x2 − 4yz)/2z and g(x, y, z) = (x −

√x2 − 4yz)/2z, the following assertions hold:

(1) f(x, y, z) and g(x, y, z) are monotonically increasing and monotonically decreasing on thevariable x ∈ (0,∞), respectively.

(2) f(x, y, z) and g(x, y, z) are monotonically decreasing and monotonically increasing on thevariable y ∈ (0,∞), respectively.

(3) f(x, y, z) and g(x, y, z) are monotonically decreasing and monotonically increasing on thevariable z ∈ (0,∞), respectively.

Proof. In fact, for all x > 0, y > 0, z > 0, we have

∂f

∂x=x +√x2 − 4yz

2z√x2 − 4yz

> 0,∂g

∂x=

√x2 − 4yz − x

2z√x2 − 4yz

< 0,∂f

∂y=

−1√x2 − 4yz

< 0,

∂g

∂y=

1√x2 − 4yz

> 0,∂f

∂z=−x(x +√x2 − 4yz

)

2z2√x2 − 4yz

< 0,∂g

∂z=x(x −√x2 − 4yz

)

2z2√x2 − 4yz

> 0.

(2.2)

By the relationship of the derivative and the monotonicity, the above assertions obviouslyhold. The proof of Lemma 2.2 is complete.

Lemma 2.3. Assume that (H1) and (H2) hold, then we have the following inequalities:

ln l−i < lnB−i < lnA−i < lnA

+i < lnB

+i < lnl

+i , i = 1, 2, . . . , n. (2.3)

Proof. Since

aM1 ≥ al1 > a

l1 − c

M12 l

+2 > 2

√bM1 hM1 > 0, 0 < bl1 ≤ b

M1 , 0 < hl1 ≤ h

M1 ,

cMi,i−1l+i−1 > c

li,i−1l

−i > c

li,i−1l

−i−1 − c

Mi,i+1l

+i+1 − d

Mi > 0,

0 < bli ≤ bMi , 0 < hli ≤ h

Mi , i = 2, 3, . . . , n − 1,

cMn,n−1l+n−1 > c

ln,n−1l

−n−1 > c

ln,n−1l

−n−1 − dMn > 2

√bMn h

Mn > 0, 0 < bln ≤ bMn , 0 < hln ≤ hMn .

(2.4)


By assumptions (H1), (H2), Lemma 2.2 and the expressions of A±i , B±i , and l±i , we have

0 < l−1 = g(aM1 , bl1, h

l1

)< g(al1, b

M1 , hM1

)= B−1 < g

(al1 − c

M12 l

+2 , b

M1 , hM1

)= A−1

< A+1 = f

(al1 − c

M12 l

+2 , b

M1 , hM1

)< B+

1 = f(al1, b

M1 , hM1

)< f(aM1 , bl1, h

l1

)= l+1 ,

0 < l−i = g(cMi,i−1l

+i−1, b

li, h

li

)< g(cli,i−1l

−i−1, b

Mi , h

Mi

)= B−i

< g(cli,i−1l

−i−1 − c

Mi,i+1l

+i+1 − d

Mi , b

Mi , h

Mi

)= A−i

< A+i = f

(cli,i−1l

−i−1 − c

Mi,i+1l

+i+1 − d

Mi , b

Mi , h

Mi

)

< B+i = f

(cli,i−1l

−i−1, b

Mi , h

Mi

)< f(cMi,i−1l

+i−1, b

li, h

li

)= l+i ,

0 < l−n = g(cMn,n−1l

+n−1, b

ln, h

ln

)< g(cln,n−1l

−n−1, b

Mn , h

Mn

)= B−n < g

(cln,n−1l

−n−1, b

Mn , h

Mn

)= A−n

< A+n = f

(cln,n−1l

−n−1, b

Mn , h

Mn

)< B+

n = f(cln,n−1l

−n−1, b

Mn , h

Mn

)< f(cMn,n−1l

+n−1, b

ln, h

ln

)= l+n,

(2.5)

where i = 2, 3, . . . , n − 1, that is 0 < l−i < B−i < A−i < A+i < B+

i < l+i , i = 1, 2, . . . , n. Thus, wehave ln l−i < lnB−i < lnA−i < lnA+

i < lnB+i < ln l+i , i = 1, 2, . . . , n. The proof of Lemma 2.3 is

complete.

Theorem 2.4. Assume that (H1) and (H2) hold. Then system (1.1) has at least 2n positiveω-periodicsolutions.

Proof. By making the substitution

xi(t) = exp{ui(t)}, i = 1, 2, . . . , n, (2.6)

system (1.1) can be reformulated as

u1(t) = a1(t) − b1(t)eu1(t) − c12(t)eu2(t) − h1(t)e−u1(t),

...

ui(t) = −di(t) − bi(t)eui(t) + ci,i−1(t)eui−1(t) − ci,i+1(t)eui+1(t) − hi(t)e−ui(t),

...

un(t) = −dn(t) − bn(t)eun(t) + cn,n−1(t)eun−1(t) − hn(t)e−un(t),

(2.7)

where i = 2, 3, . . . , n − 1.


Let

X = Z ={u = (u1, u2, . . . , un)T ∈ C(R,Rn) : u(t +ω) = u(t)

}(2.8)

and define

‖u‖ =n∑

i=1

maxt∈[0,ω]

|ui(t)|, u ∈ X or Z. (2.9)

Equipped with the above norm ‖ · ‖, X and Z are Banach spaces. Let

N(u, λ) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a1(t) − b1(t)eu1(t) − λc12(t)eu2(t) − h1(t)e−u1(t)

...

−λdi(t) − bi(t)eui(t) + ci,i−1(t)eui−1(t) − λci,i+1(t)eui+1(t) − hi(t)e−ui(t)

...

−λdn(t) − bn(t)eun(t) + cn,n−1(t)eun−1(t) − hn(t)e−un(t)

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

n×1

, (2.10)

where i = 2, 3, . . . , n − 1, Lu = u = (du(t))/dt. We put Pu = (1/ω)∫ω

0 u(t) dt, u ∈ X;Qz =(1/ω)

∫ω0 z(t) dt, z ∈ Z. Thus it follows that KerL = Rn, ImL = {z ∈ Z :

∫ω0 z(t) dt = 0} is

closed in Z, dim KerL = n = codim ImL, and P,Q are continuous projectors such that

ImP = KerL, KerQ = ImL = Im (I −Q). (2.11)

Hence, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L)KP : ImL → Ker P

⋂Dom L is given by

KP (z) =∫ t

0z(s) ds − 1

ω

∫ω

0

∫ s

0z(s) ds. (2.12)


Then

QN(u, λ) =

⎛

⎜⎜⎜⎜⎜⎜⎝

1ω

∫ω

0F1(s, λ)ds

...

1ω

∫ω

0Fn(s, λ)ds

⎞

⎟⎟⎟⎟⎟⎟⎠

n×1

, (2.13)

KP (I −Q)N(u, λ) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

∫ t

0F1(s, λ) ds − 1

ω

∫ω

0

∫ t

0F1(s, λ) dsdt +

(12− t

ω

)∫ω

0F1(s, λ)ds

...∫ t

0Fn(s, λ) ds − 1

ω

∫ω

0

∫ t

0Fn(s, λ) dsdt +

(12− t

ω

)∫ω

0Fn(s, λ)ds

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

n×1

,

(2.14)

where

F(u, λ) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a1(s) − b1(s)eu1(s) − λc12(s)eu2(s) − h1(s)e−u1(s)

...

−λdi(s) − bi(s)eui(s) + ci,i−1(s)eui−1(s) − λci,i+1(s)eui+1(s) − hi(s)e−ui(s)

...

−λdn(s) − bn(s)eun(s) + cn,n−1(s)eun−1(s) − hn(s)e−un(s)

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

n×1

. (2.15)

Obviously,QN andKP (I−Q)N are continuous. It is not difficult to show thatKP (I−Q)N(Ω)is compact for any open bounded set Ω ⊂ X by using the Arzela-Ascoli theorem. Moreover,QN(Ω) is clearly bounded. Thus, N is L-compact on Ω with any open bounded set Ω ⊂ X.

In order to use Lemma 2.1, we have to find at least 2n appropriate open boundedsubsets of X. Corresponding to the operator equation Lu = λN(u, λ), λ ∈ (0, 1), we have

u1(t) = λ(a1(t) − b1(t)eu1(t) − λc12(t)eu2(t) − h1(t)e−u1(t)

),

...

ui(t) = λ(−λdi(t) − bi(t)eui(t) + ci,i−1(t)eui−1(t) − λci,i+1(t)eui+1(t) − hi(t)e−ui(t)

),

...

un(t) = λ(−λdn(t) − bn(t)eun(t) + cn,n−1(t)eun−1(t) − hn(t)e−un(t)

),

(2.16)


where i = 2, 3, . . . , n − 1. Assume that u ∈ X is an ω-periodic solution of system (2.16) forsome λ ∈ (0, 1). Then there exist ξi, ηi ∈ [0, ω] such that ui(ξi) = maxt∈[0,ω]ui(t), ui(ηi) =mint∈[0,ω]ui(t), i = 1, 2, . . . , n. It is clear that ui(ξi) = 0, ui(ηi) = 0, i = 1, 2, . . . , n. From this and(2.16), we have

a1(ξ1) − b1(ξ1)eu1(ξ1) − λc12(ξ1)eu2(ξ1) − h1(ξ1)e−u1(ξ1) = 0,

...

−λdi(ξi) − bi(ξi)eui(ξi) + ci,i−1(ξi)eui−1(ξi) − λci,i+1(ξi)eui+1(ξi) − hi(ξi)e−ui(ξi) = 0,

...

−λdn(ξi) − bn(ξn)eun(ξn) + cn,n−1(ξn)eun−1(ξn) − hn(ξn)e−un(ξn) = 0

(2.17)

a1(η1)− b1(η1)eu1(η1) − λc12

(η1)eu2(η1) − h1

(η1)e−u1(η1) = 0,

...

−λdi(ηi)− bi(ηi)eui(ηi) + ci,i−1

(ηi)eui−1(ηi) − λci,i+1

(ηi)eui+1(ηi) − hi

(ηi)e−ui(ηi) = 0,

...

−λdn(ηi)− bn(ηn)eun(ηn) + cn,n−1

(ηn)eun−1(ηn) − hn

(ηn)e−un(ηn) = 0,

(2.18)

where i = 2, 3, . . . , n − 1.On one hand, according to (2.17), we have

bl1e2u1(ξ1) − aM1 eu1(ξ1) + hl1 ≤ b1(ξ1)e2u1(ξ1) − a1(ξ1)eu1(ξ1) + h1(ξ1) = −λc12(ξ1)eu1(ξ1)+u2(ξ1) < 0,

(2.19)namely,

bl1e2u1(ξ1) − aM1 eu1(ξ1) + hl1 < 0, (2.20)

which implies that

ln l−1 < u1(ξ1) < ln l+1 , (2.21)

bl2e2u2(ξ2) + hl2 < b2(ξ2)e2u2(ξ2) + λc23(ξ2)eu2(ξ2)+u3(ξ2) + λd2(ξ2)eu2(ξ2) + h2(ξ2)

= c21(ξ2)eu2(ξ2)+u1(ξ2) < cM21 l+1e

u2(ξ2);(2.22)

that is,

bl2e2u2(ξ2) − cM21 l

+1e

u2(ξ2) + hl2 < 0, (2.23)


which implies that

ln l−2 < u2(ξ2) < ln l+2 . (2.24)

By deducing for i = 3, 4, . . . , n − 1, we obtain

blie2ui(ξi) + hli < bi(ξi)e

2ui(ξi) + λci,i+1(ξi)eui(ξi)+ui+1(ξi) + λdi(ξi)eui(ξi) + hi(ξi)

= ci,i−1(ξi)eui(ξi)+ui−1(ξi) < cMi,i−1l+i−1e

ui(ξi),(2.25)

namely,

blie2ui(ξi) − cMi,i−1l

+i−1e

ui(ξi) + hli < 0, (2.26)

which implies that

ln l−i < ui(ξi) < ln l+i , i = 3, 4, . . . , n − 1, (2.27)

blne2un(ξi) + hln < bn(ξn)e

2un(ξn) + λdn(ξn)eun(ξn) + hn(ξn)

= cn,n−1(ξn)eun(ξn)+un−1(ξn) < cMn,n−1l+n−1e

un(ξn),(2.28)

namely,

blne2un(ξn) − cMn,n−1l

+n−1e

un(ξn) + hln < 0, (2.29)

which implies that

ln l−n < un(ξn) < ln l+n. (2.30)

In view of (2.20), (2.23), (2.26), and (2.29), we have

ln l−i < ui(ξi) < ln l+i , i = 1, 2, . . . , n. (2.31)

From (2.18), one can analogously obtain

ln l−i < ui(ηi)< ln l+i , i = 1, 2, . . . , n. (2.32)

By (2.30) and (2.31), we get

ln l−i < ui(ηi)< ui(ξi) < ln l+i , i = 1, 2, . . . , n. (2.33)


On the other hand, in view of (2.17), we have

−bM1 e2u1(ξ1) + al1eu1(ξ1) − hM1 ≤ −b1(ξ1)e2u1(ξ1) + a1(ξ1)eu1(ξ1) − h1(ξ1)

= λc12(ξ1)eu1(ξ1)+u2(ξ1) < cM12 l+2e

u1(ξ1),(2.34)

namely,

bM1 e2u1(ξ1) −(al1 − c

M12 l

+2

)eu1(ξ1) + hM1 > 0, (2.35)

which implies that

lnA+1 < u1(ξ1) or u1(ξ1) < lnA−1 , (2.36)

cl21l−1 < c21(ξ2)eu1(ξ2) = b2(ξ2)eu2(ξ2) + h2(ξ2)e−u2(ξ2) + λd2(ξ2) + λc23(ξ2)eu3(ξ2)

< bM2 eu2(ξ2) + hM2 e−u2(ξ2) + dM2 + cM23 l+3 ,

(2.37)

that is,

bM2 e2u2(ξ2) −(cl21l

−1 − c

M23 l

+3 − dM2

)eu2(ξ2) + hM2 > 0, (2.38)

which implies that

lnA+2 < u2(ξ2) 2003or u2(ξ2) < lnA−2 . (2.39)

By deducing for i = 3, 4, . . . , n − 1, we obtain

cli,i−1l−i−1 < ci,i−1(ξi)eui−1(ξi) = bi(ξi)eui(ξi) + hi(ξi)e−ui(ξi) + λdi(ξi) + λci,i+1(ξi)eui+1(ξi)

< bMi eui(ξi) + hMi e−ui(ξi) + dMi + cMi,i+1l

+i+1,

(2.40)

that is,

bMi e2ui(ξi) −

(cli,i−1l

−i−1 − c

Mi,i+1l

+i+1 − d

Mi

)eui(ξi) + hMi > 0, (2.41)

which implies that

lnA+i < ui(ξi) or ui(ξi) < lnA−i , i = 3, 4, . . . , n − 1, (2.42)

cln,n−1l−n−1 < cn,n−1(ξn)eun−1(ξn) = bn(ξn)eun(ξn) + hn(ξn)e−un(ξn) + λdn(ξn)

< bMn eun(ξn) + hMn e

−un(ξn) + dMn ,(2.43)


namely,

bMn e2un(ξn) −

(cln,n−1l

−n−1 − d

Mn

)eun(ξn) + hMn > 0, (2.44)

which implies that

lnA+n < un(ξn) or un(ξn) < lnA−n. (2.45)

It follows from (2.35), (2.38), (2.41), and (2.44) that

lnA+i < ui(ξi) or ui(ξi) < lnA−i , i = 1, 2, . . . , n. (2.46)

From (2.18), one can analogously obtain

lnA+i < ui

(ηi)

or ui(ηi)< lnA−i , i = 1, 2, . . . , n. (2.47)

By (2.32), (2.45), (2.46), and Lemma 2.3, we get

lnA+i < ui

(ηi)< ui(ξi) < ln l+i or ln l−i < ui

(ηi)< ui(ξi) < lnA−i , i = 1, 2, . . . , n, (2.48)

which implies that, for all t ∈ R,

lnA+i < ui(t) < ln l+i or ln l−i < ui(t) < lnA−i , i = 1, 2, . . . , n. (2.49)

For convenience, we denote

Gi =(ln l−i , lnA

−i

), Hi =

(lnA+

i , ln l+i

), i = 1, 2, . . . , n. (2.50)

Clearly, l±i (i = 1, 2, . . . , n) and A±i (i = 1, 2, . . . , n) are independent of λ. For each i =1, 2, . . . , n, we choose an interval between two intervals Gi and Hi, and denote it as Δi, thendefine the set

{u = (u1, u2, . . . , un)T ∈ X : ui(t) ∈ Δi, t ∈ R, i = 1, 2, . . . , n

}. (2.51)

Obviously, the number of the above sets is 2n. We denote these sets as Ωk, k =1, 2, . . . , 2n. Ωk, k = 1, 2, . . . , 2n are bounded open subsets of X,Ωi ∩ Ωj = φ, i /= j. ThusΩk(k = 1, 2, . . . , 2n) satisfies the requirement (a) in Lemma 2.1.


Now we show that (b) of Lemma 2.1 holds; that is, we prove when u ∈ ∂Ωk ∩ KerL =∂Ωk ∩ Rn, QN(u, 0)/= (0, 0, . . . , 0)T , k = 1, 2, . . . , 2n. If it is not true, then when u ∈ ∂Ωk ∩KerL = ∂Ωk ∩ Rn, i = 1, 2, . . . , 2n, constant vector u = (u1, u2, . . . , un)

T , with u ∈ ∂Ωk, k =1, 2, . . . , 2n, satisfies

∫ω

0

(a1(t) − b1(t)eu1 − h1(t)e−u1

)dt = 0,

...∫ω

0

(−bi(t)eui + ci,i−1(t)eui−1 − hi(t)e−ui

)ds = 0,

...∫ω

0

(−bn(t)eun + cn,n−1(t)eun−1 − hn(t)e−un

)ds = 0,

(2.52)

where i = 2, 3, . . . , n − 1. In view of the mean value theorem of calculous, there exist n pointsti (i = 1, 2, . . . , n) such that

a1(t1) − b1(t1)eu1 − h1(t1)e−u1 = 0,

...

−bi(ti)eui + ci,i−1(ti)eui−1 − hi(ti)e−ui = 0,

...

−bn(tn)eun + cn,n−1(tn)eun−1 − hn(tn)e−un = 0,

(2.53)

where i = 2, 3, . . . , n − 1. Following the argument of (2.20)–(2.47), from (2.52), we obtain

ln l−i < ui < lnB−i < lnA−i or lnA+i < lnB+

i < ui < lnl+i , i = 1, 2, . . . , n. (2.54)

Then u belongs to one of Ωk ∩ Rn, k = 1, 2, . . . , 2n. This contradicts the fact that u ∈ ∂Ωk ∩Rn, k = 1, 2, . . . , 2n. This proves that (b) in Lemma 2.1 holds.

Finally, in order to show that (c) in Lemma 2.1 holds, we only prove that foru ∈ ∂Ωk ∩ KerL = ∂Ωk ∩ Rn, k = 1, 2, . . . , 2n, then it holds that deg{JQN(u, 0),Ωk ∩KerL, (0, 0, . . . , 0)T}/= 0. To this end, we define the mapping φ : Dom L × [0, 1] → X by

φ(u, μ)= μQN(u, 0) +

(1 − μ

)G(u); (2.55)


here μ ∈ [0, 1] is a parameter and G(u) is defined by

G(u) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∫ω

0

(a1(s) − b1(s)eu1(s) − h1(s)e−u1(s)

)ds

...∫ω

0

(cMi,i−1l

+i−1 − bi(s)eui(s) − hi(s)e−ui(s)

)ds

...∫ω

0

(cMn,n−1l

+n−1 − bn(s)eun(s) − hn(s)e−un(s)

)ds

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

n×1

, (2.56)

where i = 2, 3, . . . , n − 1. We show that for u ∈ ∂Ωk ∩ KerL = ∂Ωk ∩ Rn, k = 1, 2, . . . , 2n, μ ∈[0, 1], then it holds that φ(u, μ)/= (0, 0, . . . , 0)T . Otherwise, parameter μ and constant vectoru = (u1, u2, . . . , un)

T ∈ Rn satisfy φ(u, μ) = (0, 0 . . . , 0)T , that is,

0 = μ∫ω

0

(a1(s) − b1(s)eu1 − h1(s)e−u1

)ds +

(1 − μ

)∫ω

0

(a1(s) − b1(s)eu1 − h1(s)e−u1

)ds,

...

0 = μ∫ω

0

(ci,i−1(s)eui−1 − bi(s)eui − hi(s)e−ui

)ds +

(1 − μ

)∫ω

0

(cMi,i−1l

+i−1 − bi(s)e

ui − hi(s)e−ui)

ds,

...

0 = μ∫ω

0

(cn,n−1(s)eun−1 − bn(s)eun − hn(s)e−un

)ds

+(1 − μ

)∫ω

0

(cMn,n−1l

+n−1 − bn(s)e

un − hn(s)e−un)

ds,

(2.57)

where i = 2, 3, . . . , n − 1. In view of the mean value theorem of calculous, there exist n pointsti ∈ [0, ω] (i = 1, 2, . . . , n) such that

a1

(t1)− b1

(t1)eu1 − h1

(t1)e−u1 = 0,

...

cMi,i−1l+i−1 − bi

(ti)eui − hi

(ti)e−ui = μ

(cMi,i−1l

+i−1 − ci,i−1

(ti)eui−1

),

...

cMn,n−1l+n−1 − bn

(tn)eun − hn

(tn)e−un = μ

(cMn,n−1l

+n−1 − ci,n−1

(tn)eun−1

),

(2.58)


where i = 2, 3, . . . , n − 1. Following the argument of (2.20)–(2.47), from (2.57), we obtain

ln l−i < ui < lnB−i < lnA−i or lnA+i < lnB+

i < ui < ln l+i , i = 1, 2, . . . , n. (2.59)

given that u belongs to one of Ωk ∩ Rn, k = 1, 2, . . . , 2n. This contradicts the fact that u ∈∂Ωk ∩Rn, k = 1, 2, . . . , 2n. This proves φ(u, μ)/= (0, 0, . . . , 0)T holds. Note that the system of thefollowing algebraic equations:

a1

(t1)− b1

(t1)ex1 − h1

(t1)e−x1 = 0,

...

cMi,i−1l+i−1 − bi

(ti)exi − hi

(ti)e−xi = 0,

...

cMn,n−1l+n−1 − bn

(tn)exn − hn

(tn)e−xn = 0

(2.60)

has 2n distinct solutions since (H1) and (H2) hold, (x∗1, x∗2, . . . , x

∗n) = (ln x1, ln x2, . . . , ln xn),

where x±1 = (a1(t1) ±√(a1(t1))

2 − 4b1(t1)h1(t1))/(2b1(t1)), x±k = ((cMk,k−1l+k−1 ±

(cMk,k−1l

+k−1)

2 − 4bk(tk)hk(tk))/(2bk(tk))) (k = 2, 3, . . . , n), xi = x−i or xi = x+i , i = 1, 2, . . . , n.

Similar to the proof of Lemma 2.3, it is easy to verify that

ln l−i < lnx−i < lnB−i < lnA−i < lnA+i < lnB+

i < lnx+i < ln l+i , i = 1, 2, . . . , n. (2.61)

Therefore, (x∗1, x∗2, . . . , x

∗n) uniquely belongs to the corresponding Ωk. Since KerL = ImQ, we

can take J = I. A direct computation gives, for k = 1, 2, . . . , 2n,

deg{JQN(u, 0),Ωk ∩ KerL, (0, 0, . . . , 0)T

}

= deg{φ(u, 1),Ωk ∩ KerL, (0, 0, . . . , 0)T

}

= deg{φ(u, 0),Ωk ∩ KerL, (0, 0, . . . , 0)T

}

= sign

⎡

⎢⎣

n∏

i=1

⎛

⎜⎝−bi

(ti)x∗i +

hi(ti)

x∗i

⎞

⎟⎠

⎤

⎥⎦.

(2.62)


Since a1(t1) − b1(t1)x∗1 − (h1(t1))/(x∗1) = 0, cMi,i−1l+i−1 − bi(ti)x

∗i − (hi(ti))/(x

∗i ) = 0 (i = 2, 3, . . . , n),

then

deg{JQN(u, 0),Ωk ∩ KerL, (0, 0, . . . , 0)T

}

= sign

[n∏

i=2

(a1

(t1 − 2b1

(t1)x∗1

))(cMi,i−1l

+i−1 − 2bMi x

∗i

)]

= ±1, k = 1, 2, . . . , 2n.(2.63)

So far, we have proved that Ωk(k = 1, 2, . . . , 2n) satisfies all the assumptions inLemma 2.1. Hence, system (2.7) has at least 2n different ω-periodic solutions. Thus, by (2.6)system (1.1) has at least 2n different positive ω-periodic solutions. This completes the proofof Theorem 2.4.

In system (1.1), if ci,i−1(t) ≥ 0 (i = 2, 3, . . . , n), cj,j+1(t) ≥ 0 (j = 1, 2, . . . , n − 1), anda1(t) > 0, di(t) > 0, bi(t) > 0, hi(t) > 0 are continuous periodic functions, then similar to theproof of Theorem 2.4, one can prove the following

Theorem 2.5. Assume that (H1) and (H2) hold. Then system (1.1) has at least 2n positiveω-periodicsolutions.

Remark 2.6. In Theorem 2.5, ci−1,i(t) = 0 means that the ith species does not prey the (i − 1)thspecies, thus ci,i−1(t) = 0. That is to say, there is no relationship between the ith species andthe (i − 1)th species.

3. Illustrative Examples

Example 3.1. Consider the following three-species food chain with harvesting terms:

x(t) = x(t)(

3 + sin t − 4 + sin t10

x(t) − c12(t)y(t))− 9 + cos t

20,

y(t) = y(t)(−3 + cos t

10− 5 + cos t

10y(t) + c21(t)x(t) − c23(t)z(t)

)− 2 + cos t

5,

z(t) = z(t)(−3 + sin 2t

10− 8 + sin 2t

10z(t) + c32(t)y(t)

)− 8 + cos 2t

10.

(3.1)

In this case, a1(t) = 3 + sin t, b1(t) = (4 + sin t)/10, h1(t) = (9 + cos t)/20, d2(t) = (3 +cos t)/10, b2(t) = (5 + cos t)/10, h2(t) = (2 + cos t)/5, d3(t) = (3 + sin 2t)/10, b3(t) =(8 + sin 2t)/10, and h3(t) = (8 + cos 2t)/10. Since

l±1 =aM1 ±

√(aM1)2 − 4bl1h

l1

2bl1=

20 ± 2√

973

, (3.2)


taking c21(t) ≡ 14/(5l−1 ), then we have

l±2 =cM21 l

+1 ±√(

cM21 l+1

)2 − 4bl2hl2

2bl2=

7l+1 ±√

49(l+1)2 − 2

(l−1)2

2l−1. (3.3)

Take c32(t) ≡ 12/(5l−2 ), then

l±3 =cM32 l

+2 ±√(

cM32 l+2

)2 − 4bl3hl3

2bl3=

12l+2 ±√(

12l+2)2 −

(7l−2)2

7l−2. (3.4)

Take c12(t) ≡ 1/(2l+2 ), c23(t) ≡ 1/(l+3 ), then

al1 − cM12 l

+2 =

32> 1 = 2

√bM1 hM1 , cl32l

−2 − d

M3 = 2 >

95= 2√bM3 hM3 ,

cl21l−1 − c

M23 l

+3 − d

M2 =

75>

65= 2√bM2 hM2 .

(3.5)

Therefore, all conditions of Theorem 2.4 are satisfied. By Theorem 2.4, system (3.1) has at leasteight positive 2π-periodic solutions.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of People’s Republic ofChina under Grant no. 10971183.

References

[1] A. A. Berryman, “The origins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530–1535, 1992.

[2] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57 of Monographs andTextbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.

[3] S.-R. Zhou, W.-T. Li, and G. Wang, “Persistence and global stability of positive periodic solutions ofthree species food chains with omnivory,” Journal of Mathematical Analysis and Applications, vol. 324,no. 1, pp. 397–408, 2006.

[4] Y. G. Sun and S. H. Saker, “Positive periodic solutions of discrete three-level food-chain model ofHolling type II,” Applied Mathematics and Computation, vol. 180, no. 1, pp. 353–365, 2006.

[5] Y. Li, L. Lu, and X. Zhu, “Existence of periodic solutions in n-species food-chain system withimpulsive,” Nonlinear Analysis: Real World Applications, vol. 7, no. 3, pp. 414–431, 2006.

[6] S. Zhang, D. Tan, and L. Chen, “Dynamic complexities of a food chain model with impulsiveperturbations and Beddington-DeAngelis functional response,” Chaos, Solitons & Fractals, vol. 27, no.3, pp. 768–777, 2006.

[7] R. Xu, L. Chen, and F. Hao, “Periodic solutions of a discrete time Lotka-Volterra type food-chainmodel with delays,” Applied Mathematics and Computation, vol. 171, no. 1, pp. 91–103, 2005.

[8] S. Zhang and L. Chen, “A Holling II functional response food chain model with impulsiveperturbations,” Chaos, Solitons & Fractals, vol. 24, no. 5, pp. 1269–1278, 2005.

[9] R. Xu, M. A. J. Chaplain, and F. A. Davidson, “Periodic solution for a three-species Lotka-Volterrafood-chain model with time delays,” Mathematical and Computer Modelling, vol. 40, no. 7-8, pp. 823–837, 2004.


[10] F. Wang, C. Hao, and L. Chen, “Bifurcation and chaos in a Monod-Haldene type food chain chemostatwith pulsed input and washout,” Chaos, Solitons & Fractals, vol. 32, no. 1, pp. 181–194, 2007.

[11] W. Wang, H. Wang, and Z. Li, “The dynamic complexity of a three-species Beddington-type foodchain with impulsive control strategy,” Chaos, Solitons & Fractals, vol. 32, no. 5, pp. 1772–1785, 2007.

[12] Y. Li and L. Lu, “Positive periodic solutions of discrete n-species food-chain systems,” AppliedMathematics and Computation, vol. 167, no. 1, pp. 324–344, 2005.

[13] H.-F. Huo and W.-T. Li, “Existence and global stability of periodic solutions of a discrete ratio-dependent food chain model with delay,” Applied Mathematics and Computation, vol. 162, no. 3, pp.1333–1349, 2005.

[14] C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Pure andApplied Mathematics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1990.

[15] J. L. Troutman, Variational Calculus and Optimal Control, Undergraduate Texts in Mathematics,Springer, New York, NY, USA, 2nd edition, 1996.

[16] A. W. Leung, “Optimal harvesting-coefficient control of steady-state prey-predator diffusive Volterra-Lotka systems,” Applied Mathematics and Optimization, vol. 31, no. 2, pp. 219–241, 1995.


[18] Y. Chen, “Multiple periodic solutions of delayed predator-prey systems with type IV functionalresponses,” Nonlinear Analysis: Real World Applications, vol. 5, no. 1, pp. 45–53, 2004.

[19] Q. Wang, B. Dai, and Y. Chen, “Multiple periodic solutions of an impulsive predator-prey modelwith Holling-type IV functional response,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp.1829–1836, 2009.

[20] D. Hu and Z. Zhang, “Four positive periodic solutions to a Lotka-Volterra cooperative system withharvesting terms,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 1115–1121, 2010.

[21] K. Zhao and Y. Ye, “Four positive periodic solutions to a periodic Lotka-Volterra predatory-preysystem with harvesting terms,” Nonlinear Analysis: Real World Applications. In press.


Research ArticleNotes on the Propagators of Evolution Equations

Yu Lin,1 Ti-Jun Xiao,2 and Jin Liang3

1 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China2 Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences,Fudan University, Shanghai 200433, China

3 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Correspondence should be addressed to Ti-Jun Xiao, [email protected]



Copyright q 2010 Yu Lin et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We consider the propagator of an evolution equation, which is a semigroup of linear operators.Questions related to its operator norm function and its behavior at the critical point for normcontinuity or compactness or differentiability are studied.

1. Introduction

As it is well known, each well-posed Cauchy problem for first-order evolution equation inBanach spaces

u′(t) = Au(t), t ≥ 0,

u(0) = u0

(1.1)

gives rise to a well-defined propagator, which is a semigroup of linear operators, and thetheory of semigroups of linear operators on Banach spaces has developed quite rapidly sincethe discovery of the generation theorem by Hille and Yosida in 1948. By now, it is a rich theorywith substantial applications to many fields (cf., e.g., [1–6]).

In this paper, we pay attention to some basic problems on the semigroups of linearoperators and reveal some essential properties of theirs.

Let X be a Banach space.

Definition 1.1 (see [1–6]). A one-parameter family (T(t))t≥0 of bounded linear operators on Xis called a strongly continuous semigroup (or simply C0-semigroup) if it satisfies the followingconditions:


(i) T(0) = I, with (I being the identity operator on X),

(ii) T(t + s) = T(t)T(s) for t, s ≥ 0,

(iii) the map t → T(t)x is continuous on [0,∞) for every x ∈ X.

The infinitesimal generator A of (T(t))t≥0 is defined as

Ax = limt→ 0+

T(t)x − xt

(1.2)

with domain

D(A) ={x ∈ X; lim

t→ 0+

T(t)x − xt

exists}. (1.3)

For a comprehensive theory of C0-semigroups we refer to [2].

2. Properties of the Function t �→ ‖T(t)‖

Let (T(t))t≥0 be a C0-semigroup on X and define g(t) := ‖T(t)‖ for t ≥ 0. Clearly, fromDefinition 1.1 we see that

(I) g(0) = 1, g(t) ≥ 0 for t ≥ 0;(II) g(t + s) ≤ g(t)g(s) for t, s ≥ 0.

Furthermore, we can infer from the strong continuity of (T(t))t≥0 that(III) g(t) is lower-semicontinuous, that is,

g(t) ≤ lim infs→ t

g(s). (2.1)

In fact,

‖T(t)x‖ = lim infs→ t

‖T(s)x‖ ≤ lim infs→ t

sup‖x‖=1‖T(s)x‖ = lim inf

s→ tg(s) (2.2)

holds for all x ∈ X with ‖x‖ = 1. Thus, taking the supremum for all x ∈ X with ‖x‖ = 1 on theleft-hand side leads to (2.1).

We ask the following question

For every function g(·) satisfying (I), (II), and (III), does there exist a C0 semigroup(T(t))t≥0 on some Banach space X such that ‖T(t)‖ = g(t) for all t ≥ 0?

We show that this is not true even if X is a finite-dimensional space.

Theorem 2.1. Let X be an n-dimensional Banach space with 2 ≤ n <∞. Let

g(t) =

⎧⎪⎨

⎪⎩

1 + t if 0 ≤ t ≤ 2n − 1,(

22n − 1

)ntn if t > 2n − 1.

(2.3)


Then g(·) satisfies (I), (II), and (III), and there exists no C0 semigroup (T(t))t≥0 on X such that‖T(t)‖ = g(t) for all t ≥ 0.

Proof. First, we show that g(·) satisfies (I), (II), and (III).(I) is clearly satisfied.To show (III) and (II), we write

γ := 2n − 1. (2.4)

Then

limt→ γ−

g(t) = g(γ)= 1 + γ = 2n = lim

t→ γ+g(t), (2.5)

hence g(·) satisfies (III).For (II), suppose t, s � 0, and consider the following four cases.

Case 1 (t � γ and s � γ). In this case

g(t + s)g(t)g(s)

=(γ

2

)n(1t+

1s

)n�(γ

2

)n(2γ

)n= 1, (2.6)

that is,

g(t + s) � g(t)g(s). (2.7)

Case 2 (t � γ and s � γ). Let

h(s) =(

1 +s

γ

)n− 1 − s. (2.8)

Then

h′′(s) =n(n − 1)

γ2

(1 +

s

γ

)n−2

� 0, (2.9)

and h(·) is a convex function on [0, γ]. So by Jensen’s inequality, we have

h(s) � s

γh(0) +

γ − sγ

h(γ)=γ − sγ

(2n − 1 − γ

)= 0, (2.10)

that is,

(1 +

s

γ

)n� 1 + s for 0 � s � γ. (2.11)


Therefore

g(t + s)g(t)

=(

1 +s

t

)n�(

1 +s

γ

)n� 1 + s = g(s), (2.12)

that is, g(t + s) � g(t)g(s).

Case 3 (t + s > γ , but t � γ and s � γ). It follows from Case 2 that

g(t + s) � g(γ)g(t + s − γ

)

=(1 + γ

)(1 + t + s − γ

)

= (1 + t)(1 + s) −(γ − s

)(γ − t

)

� (1 + t)(1 + s)

= g(t)g(s).

(2.13)

Case 4 (t + s � γ). Again we have

g(t + s) = 1 + t + s � 1 + t + s + st = (1 + t)(1 + s) = g(t)g(s). (2.14)

Next, we prove that there does not exist any C0 semigroup (T(t))t≥0 on X such that‖T(t)‖ = g(t). Suppose g(t) = ‖T(t)‖ (t ≥ 0) for some C0 semigroup (T(t))t≥0 on X and let Abe its infinitesimal generator.

First we note from (2.3) that

limt→+∞

∥∥e−εtT(t)∥∥ = lim

t→+∞e−εtg(t) = 0 (2.15)

for every ε > 0, while

‖T(t)‖ = O(tn) (t −→ +∞). (2.16)

By the well-known Lyapunov theorem [2, Chapter I, Theorem 2.10], all eigenvalues of A − εI(the infinitesimal generator of e−εtT(t)) have negative parts for every ε > 0. Letting λ1, . . . , λrbe the eigenvalues of A, we then have

Reλj − ε < 0 for every ε > 0, j = 1, . . . , r, (2.17)


and this implies that

Reλj ≤ 0, j = 1, . . . , r. (2.18)

It is known that there is an isomorphism P of Cn onto X such that

A = P

⎛

⎜⎜⎜⎝

J1 · · · 0

......

...

0 · · · Jr

⎞

⎟⎟⎟⎠P−1, (2.19)

where Jj is the Jordan block corresponding to λj . Therefore

T(t) = P

⎛

⎜⎜⎜⎝

etJ1 · · · 0

......

...

0 · · · etJr

⎞

⎟⎟⎟⎠P−1. (2.20)

Set

Nj = Jj − λjI(kj ), 1 ≤ j ≤ r, (2.21)

where kj is the order of Jj . Then Nj is a kj th nilpotent matrix with kj ≤ n for each 1 ≤ j ≤ r.According to (2.20) and (2.18), we have

‖T(t)‖ ≤ ‖P‖ ·∥∥∥P−1

∥∥∥max

1≤j≤r

∥∥∥etJj

∥∥∥

= ‖P‖ ·∥∥∥P−1

∥∥∥max1≤j≤r

∥∥∥etλj etNj

∥∥∥

≤ ‖P‖ ·∥∥∥P−1

∥∥∥max1≤j≤r

∥∥∥etNj

∥∥∥.

(2.22)

Observing

∥∥∥etNi

∥∥∥ =

∥∥∥∥∥I(ki) + tNi + · · · +

tki−1

(ki − 1)!Nki−1

i

∥∥∥∥∥

= O(tki−1)

(t −→ +∞),

(2.23)


we see that

∥∥∥etNi

∥∥∥ = O

(tn−1)

(t −→ +∞). (2.24)

Thus,

g(t) = ‖T(t)‖ = O(tn−1)

(t −→ +∞), (2.25)

which is a contradiction to (2.16).

Open Problem 1. Is it possible that there exists an X with dimX = ∞ and a C0 semigroup(T(t))t≥0 on X such that ‖T(t)‖ = g(t) for all t ≥ 0?

3. The Critical Point of Norm-Continuous (Compact, Differentiable)Semigroups

The following definitions are basic [1–6].

Definition 3.1. A C0-semigroup (T(t))t≥0 is called norm-continuous for t > t0 if t �→ T(t) iscontinuous in the uniform operator topology for t > t0.

Definition 3.2. AC0-semigroup (T(t))t≥0 is called compact for t > t0 if (T(t))t≥0 is a compact operatorfor t > t0.

Definition 3.3. A C0-semigroup (T(t))t≥0 is called differentiable for t > t0 if for every x ∈ X, t �→T(t)x is differentiable for t > t0.

It is known that if a C0-semigroup (T(t))t≥0 is norm continuous (compact, differen-tiable) at t = t0, then it remains so for all t > t0. For instance, the following holds.

Proposition 3.4. If the map t → T(t)x is right differentiable at t = t0, then it is also differentiablefor t > t0.

Therefore, if we write

Jn := {t0 ≥ 0; T(t)is norm-continuousfor t > t0},

Jc :={t0 ≥ 0; T(t) is compact for t > t0

},

Jd := {t0 ≥ 0; T(t) is differentiable for t > t0},

(3.1)

and suppose Jn /= ∅ (Jc /= ∅, Jd /= ∅), then Jn (Jc, Jd) takes the form of [τ0,∞) for a nonnegativereal number τ0. In other words, if Jn /= ∅ (Jc /= ∅, Jd /= ∅), then (T(t))t≥0 is norm continuous(compact, differentiable) on the interval (τ0,∞) but not at any point in [0, τ0). We call τ0 thecritical point of the norm continuity (compactness, differentiability) of operator semigroup(T(t))t≥0.


A natural question is the following

Suppose that τ0 is the critical point of the norm continuity (compactness, differentiability )of the operator semigroup (T(t))t≥0. Is (T(t))t≥0 also norm continuous (compact,differentiable) at τ0? Of course, concerning norm continuity or differentiability at τ0

we only mean right continuity or right differentiability.

We show that the answer is “yes” in some cases and “no” for other cases.

Example 3.5. Let X = L2[0, 1] and

T(t)f(s) =

⎧⎨

⎩

f(s + t), s + t ≤ 1,

0, s + t > 1,t ≥ 0. (3.2)

Then clearly T(t) = 0 for t > 1. Moreover, (T(t))t≥0 is not norm continuous (not compact, notdifferentiable) for any 0 ≤ t < 1 since

‖T(t + h) − T(t)‖ ≥∥∥(T(t + h) − T(t))fh

∥∥ = 2 (3.3)

for sufficiently small h > 0, where

fh(s) =

⎧⎨

⎩

h−1/2, t + h ≤ s ≤ t + 2h,

0, otherwise.(3.4)

Therefore, in this case we have τ0 = 1. Since T(1) = 0, we see that T(1) is compact and (T(t))t≥0is differentiable at t = 1 from the right.

Example 3.6. Let

X = c0 ={(x1, . . . , xn, . . .); xn ∈ C, lim

n→∞xn exists

}(3.5)

with supremum norm. For any x = (x1, . . . , xn, . . .) ∈ X set

T(t)x =(e−t(cos(te) + i sin(te))x1, . . . , e

−nt(cos(ten) + i sin(ten))xn, . . .), t ≥ 0. (3.6)

Then, (T(t))t≥0 is compact (hence norm-continuous) for t > 0 since T(t) is the operator-normlimit of a sequence (Tk(t))k∈N of finite-rank operators:

Tk(t)x =(y1(t), . . . , yn(t), . . .

), for each x = (x1, . . . , xn, . . .) ∈ X, k ∈ N, (3.7)

where

yn(t) =

⎧⎨

⎩

e−nt(cos(ten) + i sin(ten))xn, n ≤ k,

0, n > k.(3.8)


So the critical point for compactness and norm continuity is τ0 = 0. However, the infinitesimalgenerator of (T(t))t≥0 is given by

Ax = ((−1 + ie)x1, . . . , (−n + ien)xn, . . .) for x = (x1, . . . , xn, . . .) ∈ D(A) (3.9)

with

D(A) = {x ∈ X; Ax ∈ X}. (3.10)

In view of that A is unbounded, we know that (T(t))t≥0 is not norm continuous at t = 0.For differentiability, we note that T(t)x is differentiable at t = t0 if and only if T(t0)x ∈

D(A) for each x ∈ X. From

AT(t)x =((e(1−t)(− sin(te) + i cos(te)) − e−t(cos(te) + i sin(te))

)x1, . . . ,

(en(1−t)(− sin(ten) + i cos(ten)) − ne−nt(cos(ten) + i sin(ten))

)xn, . . .

),

for each x = (x1, . . . , xn, . . .) ∈ X, t ≥ 0,

(3.11)

it follows that when t > 1, T(t)x ∈ D(A) for every x ∈ X. On the other hand, when 0 ≤t ≤ 1 and x is any nonzero constant sequence, T(t)x /∈D(A). Therefore the critical point fordifferentiability is τ0 = 1. But (T(t))t≥0 is not differentiable at t = 1.

Acknowledgments

The authors acknowledge the support from the NSF of China (10771202), the ResearchFund for Shanghai Key Laboratory of Modern Applied Mathematics (08DZ2271900), andthe Specialized Research Fund for the Doctoral Program of Higher Education of China(2007035805).

References

[1] E. B. Davies, One-Parameter Semigroups, vol. 15 of London Mathematical Society Monographs, AcademicPress, London, UK, 1980.

[2] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of GraduateTexts in Mathematics, Springer, New York, NY, USA, 2000.

[3] H. O. Fattorini, The Cauchy Problem, vol. 18 of Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, Mass, USA, 1983.

[4] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, TheClarendon Press/Oxford University Press, New York, NY, USA, 1985.

[5] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical SocietyColloquium Publications, vol. 31, American Mathematical Society, Providence, RI, USA, 1957.

[6] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of AppliedMathematical Sciences, Springer, Berlin, Germany, 1983.


Research ArticleOn the Oscillation of Second-Order Neutral DelayDifferential Equations

Zhenlai Han,1, 2 Tongxing Li,1 Shurong Sun,1, 3

and Weisong Chen1

1 School of Science, University of Jinan, Jinan, Shandong 250022, China2 School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China3 Department of Mathematics and Statistics, Missouri University of Science and Technology,Rolla, MO 65409-0020, China


Received 8 October 2009; Accepted 10 January 2010


Copyright q 2010 Zhenlai Han et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Some new oscillation criteria for the second-order neutral delay differential equation (r(t)z′(t))′ +q(t)x(σ(t)) = 0, t ≥ t0 are established, where

∫∞t0(1/r(t))dt = ∞, z(t) = x(t) + p(t)x(τ(t)), 0 ≤ p(t) ≤

p0 <∞, q(t) > 0. These oscillation criteria extend and improve some known results. An example isconsidered to illustrate the main results.

1. Introduction

Neutral differential equations find numerous applications in natural science and technology.For instance, they are frequently used for the study of distributed networks containinglossless transmission lines; see Hale [1]. In recent years, many studies have been made onthe oscillatory behavior of solutions of neutral delay differential equations, and we refer tothe recent papers [2–23] and the references cited therein.

This paper is concerned with the oscillatory behavior of the second-order neutral delaydifferential equation

(r(t)z′(t)

)′ + q(t)x(σ(t)) = 0, t ≥ t0, (1.1)

where z(t) = x(t) + p(t)x(τ(t)).


In what follows we assume that

(I1) p, q ∈ C([t0,∞), R), 0 ≤ p(t) ≤ p0 <∞, q(t) > 0,

(I2) r ∈ C([t0,∞), R), r(t) > 0,∫∞t0(1/r(t))dt =∞,

(I3) τ, σ ∈ C([t0,∞), R), τ(t) ≤ t, σ(t) ≤ t, τ ′(t) = τ0 > 0, σ ′(t) > 0, limt→∞τ(t) =limt→∞σ(t) =∞, τ(σ(t)) = σ(τ(t)), where τ0 is a constant.

Some known results are established for (1.1) under the condition 0 ≤ p(t) < 1.Grammatikopoulos et al. [6] obtained that if 0 ≤ p(t) ≤ 1, q(t) ≥ 0 and,

∫∞t0q(s)[1−p(s−σ)]ds =

∞, then the second-order neutral delay differential equation

[y(t) + p(t)y(t − τ)

]′′ + q(t)y(t − σ) = 0 (1.2)

oscillates. In [13], by employing Riccati technique and averaging functions method, Ruanestablished some general oscillation criteria for second-order neutral delay differentialequation

[a(t)

(x(t) + p(t)x(t − τ)

)′]′ + q(t)f(x(t − σ)) = 0. (1.3)

Xu and Meng [18] as well as Zhuang and Li [23] studied the oscillation of the second-orderneutral delay differential equation

[r(t)

(y(t) + p(t)y(τ(t))

)′]′ +n∑

i=1

qi(t)fi(y(σi(t))

)= 0. (1.4)

Motivated by [11], we will further the investigation and offer some more general newoscillation criteria for (1.1), by employing a class of function Y, operator T, and the Riccatitechnique and averaging technique.

Following [11], we say that a function φ = φ(t, s, l) belongs to the function class Y,denoted by φ ∈ Y if φ ∈ C(E,R), where E = {(t, s, l) : t0 ≤ l ≤ s ≤ t < ∞}, which satisfiesφ(t, t, l) = 0, φ(t, l, l) = 0, and φ(t, s, l) > 0, for l < s < t, and has the partial derivative ∂φ/∂son E such that ∂φ/∂s is locally integrable with respect to s in E. By choosing the specialfunction φ, it is possible to derive several oscillation criteria for a wide range of differentialequations.

Define the operator T[·; l, t] by

T[g; l, t

]=∫ t

l

φ(t, s, l)g(s)ds, (1.5)

for t ≥ s ≥ l ≥ t0 and g ∈ C1[t0,∞). The function ϕ = ϕ(t, s, l) is defined by

∂φ(t, s, l)∂s

= ϕ(t, s, l)φ(t, s, l). (1.6)


It is easy to see that T[·; l, t] is a linear operator and that it satisfies

T[g ′; l, t

]= −T

[gϕ; l, t

], for g(s) ∈ C1[t0,∞). (1.7)

2. Main Results

In this section, we give some new oscillation criteria for (1.1). We start with the followingoscillation criteria.

Theorem 2.1. If

∫∞

t0

Q(t)dt =∞, (2.1)

where Q(t) := min{q(t), q(τ(t))}, then (1.1) oscillates.

Proof. Let x be a nonoscillatory solution of (1.1). Then there exists t1 ≥ t0 such that x(t)/= 0, forall t ≥ t1. Without loss of generality, we assume that x(t) > 0, x(τ(t)) > 0, and x(σ(t)) > 0,for all t ≥ t1. From (1.1), we have

(r(t)z′(t)

)′ = −q(t)x(σ(t)) < 0, t ≥ t1. (2.2)

Therefore r(t)z′(t) is a decreasing function. We claim that z′(t) > 0 for t ≥ t1. Otherwise, thereexists t2 ≥ t1 such that z′(t2) < 0. Then from (2.2) we obtain

r(t)z′(t) ≤ r(t2)z′(t2), t ≥ t2, (2.3)

and hence,

z(t) ≤ z(t2) −[−r(t2)z′(t2)

]∫ t

t2

dsr(s)

. (2.4)

Taking t → ∞, we get z(t) → −∞, t → ∞. This contradiction proves that z′(t) > 0 for t ≥ t1.Using definition of z(t) and applying (1.1), we get for sufficiently large t

(r(t)z′(t)

)′ + q(t)x(σ(t)) + p0q(τ(t))x(σ(τ(t))) +p0

τ ′(t)(r(τ(t))z′(τ(t))

)′ = 0, (2.5)

and thus,

(r(t)z′(t)

)′ +Q(t)z(σ(t)) +p0

τ ′(t)(r(τ(t))z′(τ(t))

)′ ≤ 0. (2.6)


Integrating (2.6) from t3 (≥ t1) to t, we obtain

∫ t

t3

(r(s)z′(s)

)′ds +∫ t

t3

Q(s)z(σ(s))ds + p0

∫ t

t3

1τ ′(s)

(r(τ(s))z′(τ(s))

)′ds ≤ 0. (2.7)

Noting that τ ′(t) = τ0 > 0, we have

∫ t

t3

Q(s)z(σ(s))ds ≤ −∫ t

t3

(r(s)z′(s)

)′ds − p0

∫ t

t3

1

(τ ′(s))2

(r(τ(s))z′(τ(s))

)′d(τ(s))

= −∫ t

t3

(r(s)z′(s)

)′ds −p0

τ20

∫ τ(t)

τ(t3)

(r(u)z′(u)

)′du

= r(t3)z′(t3) − r(t)z′(t) +p0

τ20

r(τ(t3))z′(τ(t3)) −p0

τ20

r(τ(t))z′(τ(t)).

(2.8)

Since z′(t) > 0 for t ≥ t1, we can find a constant c > 0 such that z(σ(t)) ≥ c for t ≥ t3 ≥ t1. Thenfrom (2.8) and the fact that r(t)z′(t) is eventually decreasing, we have

∫∞

t3

Q(t)dt <∞, (2.9)

which is a contradiction to (2.1). This completes the proof.

Theorem 2.2. Assume that σ(t) ≤ τ(t), and there exist functions φ ∈ Y and k ∈ C1([t0,∞), R+)such that

lim supt→∞

T

[

k(s)Q(s) −(1 +

(p0/τ0

))(ϕ + (k′(s)/k(s))

)2

4r(σ(s))k(s)

σ ′(s); l, t

]

> 0, (2.10)

whereQ(t) is defined as in Theorem 2.1, the operator T is defined by (1.5), and ϕ = ϕ(t, s, l) is definedby (1.6). Then every solution x of (1.1) is oscillatory.

Proof. Let x be a nonoscillatory solution of (1.1). Then there exists t1 ≥ t0 such that x(t)/= 0 forall t ≥ t1. Without loss of generality, we assume that x(t) > 0, x(τ(t)) > 0, and x(σ(t)) > 0, forall t ≥ t1. Define

ω(t) = k(t)r(t)z′(t)z(σ(t))

, t ≥ t1. (2.11)

Then w(t) > 0 and

ω′(t) = k′(t)r(t)z′(t)z(σ(t))

+ k(t)(r(t)z′(t))′z(σ(t)) − r(t)z′(t)z′(σ(t))σ ′(t)

z2(σ(t)). (2.12)


By (2.2) and the fact z′(t) > 0, we get

z′(σ(t))z′(t)

≥ r(t)r(σ(t))

. (2.13)

From (2.11), (2.12), and (2.13), we have

ω′(t) ≤ k(t)(r(t)z′(t))′

z(σ(t))+k′(t)k(t)

ω(t) − σ ′(t)r(σ(t))k(t)

ω2(t). (2.14)

Similarly, define

ν(t) = k(t)r(τ(t))z′(τ(t))

z(σ(t)), t ≥ t1. (2.15)

Then ν(t) > 0 and

ν′(t) = k′(t)r(τ(t))z′(τ(t))

z(σ(t))+ k(t)

(r(τ(t))z′(τ(t)))′z(σ(t)) − r(τ(t))z′(τ(t))z′(σ(t))σ ′(t)z2(σ(t))

.

(2.16)

By (2.2) and the facting z′(t) > 0, noting that σ(t) ≤ τ(t), we get

z′(σ(t))z′(τ(t))

≥ r(τ(t))r(σ(t))

. (2.17)

From (2.15), (2.16), and (2.17), we have

ν′(t) ≤ k(t) (r(τ(t))z′(τ(t)))′

z(σ(t))+k′(t)k(t)

ν(t) − σ ′(t)r(σ(t))k(t)

ν2(t). (2.18)

Therefore, from (2.14) and (2.18), we get

ω′(t) +p0

τ0ν′(t) ≤ k(t) (r(t)z

′(t))′

z(σ(t))+p0

τ0k(t)

(r(τ(t))z′(τ(t)))′

z(σ(t))

+k′(t)k(t)

ω(t) − σ ′(t)r(σ(t))k(t)

ω2(t) +p0

τ0

k′(t)k(t)

ν(t) −p0

τ0

σ ′(t)r(σ(t))k(t)

ν2(t).

(2.19)

From (2.6), we obtain

ω′(t) +p0

τ0ν′(t) ≤ −k(t)Q(t) +

k′(t)k(t)

ω(t) − σ ′(t)r(σ(t))k(t)

ω2(t)

+p0

τ0

k′(t)k(t)

ν(t) −p0

τ0

σ ′(t)r(σ(t))k(t)

ν2(t).

(2.20)


Applying T[·; l, t] to (2.20), we get

T

[ω′(s) +

p0

τ0ν′(s); l, t

]

≤ T[−k(s)Q(s) +

k′(s)k(s)

ω(s) − σ ′(s)r(σ(s))k(s)

ω2(s) +p0

τ0

k′(s)k(s)

ν(s) −p0

τ0

σ ′(s)r(σ(s))k(s)

ν2(s); l, t].

(2.21)

By (1.7) and the above inequality, we obtain

T[k(s)Q(s); l, t]

≤ T[(

ϕ +k′(s)k(s)

)ω(s) − σ ′(s)

r(σ(s))k(s)ω2(s) +

p0

τ0

(ϕ +

k′(s)k(s)

)ν(s) −

p0

τ0

σ ′(s)r(σ(s))k(s)

ν2(s); l, t].

(2.22)

Hence, from (2.22) we have

T[k(s)Q(s); l, t] ≤ T[((

ϕ + (k′(s)/k(s)))2

4+

(p0/τ0

)(ϕ + (k′(s)/k(s))

)2

4

)r(σ(s))k(s)

σ ′(s); l, t

]

,

(2.23)

that is,

T

[

k(s)Q(s) −(1 +

(p0/τ0

))(ϕ + (k′(s)/k(s))

)2

4r(σ(s))k(s)

σ ′(s); l, t

]

≤ 0. (2.24)

Taking the super limit in the above inequality, we get

lim supt→∞

T

[

k(s)Q(s) −(1 +

(p0/τ0

))(ϕ + (k′(s)/k(s))

)2

4r(σ(s))k(s)

σ ′(s); l, t

]

≤ 0, (2.25)

which contradicts (2.10). This completes the proof.

Remark 2.3. With the different choice of k and φ, Theorem 2.2 can be stated with differentconditions for oscillation of (1.1). For example, if we choose φ(t, s, l) = ρ(s)(t − s)σ(s − l)μ forσ > 1/2, μ > 1/2, ρ ∈ C1([t0,∞), (0,∞)), then

ϕ(t, s, l) =ρ′(s)ρ(s)

+μt −

(σ + μ

)s + σl

(t − s)(s − l) . (2.26)

By Theorem 2.2 we can obtain the oscillation criterion for (1.1), the details are left to thereader.


For an application, we give the following example to illustrate the main results.

Example 2.4. Consider the following equation:

(x(t) + 2x(t − π))′′ + x(t − π) = 0, t ≥ t0. (2.27)

Let r(t) = 1, p(t) = 2, q(t) = 1, and τ(t) = σ(t) = t − π, then by Theorem 2.1 every solution of(2.27) oscillates; for example, x(t) = sin t is an oscillatory solution of (2.27).

Remark 2.5. The recent results cannot be applied in (2.27) since p(t) = 2 > 1; so our results arenew ones.

Acknowledgments

This research is supported by the Natural Science Foundation of China (60774004, 60904024),China Postdoctoral Science Foundation Funded Project (20080441126, 200902564), ShandongPostdoctoral Funded Project (200802018) and the Natural Scientific Foundation of ShandongProvince (Y2008A28, ZR2009AL003), also supported by University of Jinan Research Fundsfor Doctors (XBS0843).

References

[1] J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977,Applied Mathematical Sciences.

[2] R. P. Agarwal and S. R. Grace, “Oscillation theorems for certain neutral functional-differentialequations,” Computers & Mathematics with Applications, vol. 38, no. 11-12, pp. 1–11, 1999.

[3] L. Berezansky, J. Diblik, and Z. Smarda, “On connection between second-order delay differentialequations and integrodifferential equations with delay,” Advances in Difference Equations, vol. 2010,Article ID 143298, 8 pages, 2010.

[4] J. Dzurina and I. P. Stavroulakis, “Oscillation criteria for second-order delay differential equations,”Applied Mathematics and Computation, vol. 140, no. 2-3, pp. 445–453, 2003.

[5] S. R. Grace, “Oscillation theorems for nonlinear differential equations of second order,” Journal ofMathematical Analysis and Applications, vol. 171, no. 1, pp. 220–241, 1992.

[6] M. K. Grammatikopoulos, G. Ladas, and A. Meimaridou, “Oscillations of second order neutral delaydifferential equations,” Radovi Matematicki, vol. 1, no. 2, pp. 267–274, 1985.

[7] Z. Han, T. Li, S. Sun, and Y. Sun, “Remarks on the paper [Appl. Math. Comput. 207 (2009) 388–396],”Applied Mathematics and Computation, vol. 215, no. 11, pp. 3998–4007, 2010.

[8] B. Karpuz, J. V. Manojlovic, O. Ocalan, and Y. Shoukaku, “Oscillation criteria for a class of second-order neutral delay differential equations,” Applied Mathematics and Computation, vol. 210, no. 2, pp.303–312, 2009.

[9] H.-J. Li and C.-C. Yeh, “Oscillation criteria for second-order neutral delay difference equations,”Computers & Mathematics with Applications, vol. 36, no. 10-12, pp. 123–132, 1998.

[10] X. Lin and X. H. Tang, “Oscillation of solutions of neutral differential equations with a superlinearneutral term,” Applied Mathematics Letters, vol. 20, no. 9, pp. 1016–1022, 2007.

[11] L. Liu and Y. Bai, “New oscillation criteria for second-order nonlinear neutral delay differentialequations,” Journal of Computational and Applied Mathematics, vol. 231, no. 2, pp. 657–663, 2009.

[12] R. N. Rath, N. Misra, and L. N. Padhy, “Oscillatory and asymptotic behaviour of a nonlinear secondorder neutral differential equation,” Mathematica Slovaca, vol. 57, no. 2, pp. 157–170, 2007.

[13] S. G. Ruan, “Oscillations of second order neutral differential equations,” Canadian MathematicalBulletin, vol. 36, no. 4, pp. 485–496, 1993.

[14] Y. G. Sun and F. Meng, “Note on the paper of Dourina and Stavroulakis,” Applied Mathematics andComputation, vol. 174, no. 2, pp. 1634–1641, 2006.


[15] Y. Sahiner, “On oscillation of second order neutral type delay differential equations,” AppliedMathematics and Computation, vol. 150, no. 3, pp. 697–706, 2004.

[16] R. Xu and F. Meng, “Some new oscillation criteria for second order quasi-linear neutral delaydifferential equations,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 797–803, 2006.

[17] R. Xu and F. Meng, “Oscillation criteria for second order quasi-linear neutral delay differentialequations,” Applied Mathematics and Computation, vol. 192, no. 1, pp. 216–222, 2007.

[18] R. Xu and F. Meng, “New Kamenev-type oscillation criteria for second order neutral nonlineardifferential equations,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1364–1370, 2007.

[19] Z. Xu and X. Liu, “Philos-type oscillation criteria for Emden-Fowler neutral delay differentialequations,” Journal of Computational and Applied Mathematics, vol. 206, no. 2, pp. 1116–1126, 2007.

[20] L. Ye and Z. Xu, “Oscillation criteria for second order quasilinear neutral delay differential equations,”Applied Mathematics and Computation, vol. 207, no. 2, pp. 388–396, 2009.

[21] A. Zafer, “Oscillation criteria for even order neutral differential equations,” Applied MathematicsLetters, vol. 11, no. 3, pp. 21–25, 1998.

[22] Q. Zhang, J. Yan, and L. Gao, “Oscillation behavior of even-order nonlinear neutral differentialequations with variable coefficients,” Computers and Mathematics with Applications, vol. 59, no. 1, pp.426–430, 2010.

[23] R.-K. Zhuang and W.-T. Li, “Interval oscillation criteria for second order neutral nonlinear differentialequations,” Applied Mathematics and Computation, vol. 157, no. 1, pp. 39–51, 2004.


Research ArticleSolutions to Fractional Differential Equations withNonlocal Initial Condition in Banach Spaces

Zhi-Wei Lv,1 Jin Liang,2 and Ti-Jun Xiao3

1 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China2 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China3 Shanghai Key Laboratory for Contemporary AppliedMathematics, School of Mathematical Sciences, FudanUniversity, Shanghai 200433, China

Correspondence should be addressed to Jin Liang, [email protected]



Copyright q 2010 Zhi-Wei Lv et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

A new existence and uniqueness theorem is given for solutions to differential equations involvingthe Caputo fractional derivative with nonlocal initial condition in Banach spaces. An applicationis also given.

1. Introduction

Fractional differential equations have played a significant role in physics, mechanics,chemistry, engineering, and so forth. In recent years, there are many papers dealing withthe existence of solutions to various fractional differential equations; see, for example, [1–6].

In this paper, we discuss the existence of solutions to the nonlocal Cauchy problem forthe following fractional differential equations in a Banach space E:

cDαx(t) = f(t, x(t)), 0 ≤ t ≤ 1,

x(0) =∫1

0g(s)x(s)ds,

(1.1)

where cDα is the standard Caputo’s derivative of order 0 < α < 1, g ∈ L1([0, 1], R+), g(t) ∈[0, 1), and f is a given E-valued function.


2. Basic Lemmas

Let E be a real Banach space, and θ the zero element of E. Denote by C([0, 1], E) theBanach space of all continuous functions x : [0, 1] → E with norm ‖x‖c = supt∈[0,1]‖x(t)‖.Let L1([0, 1], E) be the Banach space of measurable functions x : [0, 1] → E which areLebesgue integrable, equipped with the norm ‖x‖L1 =

∫10‖x(s)‖ds. Let R+ = [0,+∞), R+ =

(0,+∞), and μ =∫1

0g(s)ds.A function x ∈ C([0, 1], E) is called a solution of (1.1) if it satisfies(1.1).

Recall the following defenition

Definition 2.1. Let B be a bounded subset of a Banach space X. The Kuratowski measure ofnoncompactness of B is defined as

α(B) := inf{γ > 0; B admits a finite cover by sets of diameter ≤ γ

}. (2.1)

Clearly, 0 ≤ α(B) < ∞. For details on properties of the measure, the reader is referredto [2].

Definition 2.2 (see [7, 8]). The fractional integral of order q with the lower limit t0 for afunction f is defined as

Iqf(t) =1

Γ(q)∫ t

t0

(t − s)q−1f(s)ds, t > t0, q > 0, (2.2)

where Γ is the gamma function.

Definition 2.3 (see [7, 8]). Caputo’s derivative of order q with the lower limit t0 for a functionf can be written as

cDqf(t) =1

Γ(n − q

)∫ t

t0

(t − s)n−q−1f (n)(s)ds, t > t0, q > 0, n =[q]+ 1. (2.3)

Remark 2.4. Caputo’s derivative of a constant is equal to θ.

Lemma 2.5 (see [7]). Let α > 0. Then we have

cDq(Iqf(t))= f(t). (2.4)

Lemma 2.6 (see [7]). Let α > 0 and n = [α] + 1. Then

Iα(cDαf(t)

)= f(t) −

n−1∑

k=0

f (k)(0)k!

tk. (2.5)


Lemma 2.7 (see [9]). IfH ⊂ C([0, 1], E) is bounded and equicontinuous, then

(a) αC(H) = α(H([0, 1]));

(b) α(H([0, 1])) = maxt∈[0,1]α(H(t)), whereH([0, 1]) = {x(t) : x ∈ H, t ∈ [0, 1]}.

Lemma 2.8.

Q(τ)Γ(α)

< e,

∫ t0(t − s)

α−1ds

Γ(α)< e, (2.6)

where Q(τ) =∫1τg(s)(s − τ)

α−1ds, t, τ ∈ [0, 1].

Proof. A direct computation shows

Q(τ)Γ(α)

=

∫1τg(s)(s − τ)

α−1ds∫∞

0 sα−1e−sds

<

∫1τ(s − τ)

α−1ds∫∞

0 sα−1e−sds

=

∫1−τ0 sα−1ds

∫∞0 s

α−1e−sds

≤e∫1−τ

0 sα−1e−sds∫∞

0 sα−1e−sds

< e

(2.7)

and

∫ t0(t − s)

α−1ds

Γ(α)=

∫ t0s

α−1ds∫∞

0 sα−1e−sds

≤e∫ t

0sα−1e−sds

∫∞0 s

α−1e−sds< e. (2.8)

3. Main Results

(H1) f ∈ ([0, 1] × E, E), and there exist M > 0, pf(t) ≤ M for t ∈ [0, 1], pf ∈L1([0, 1], R+) such that ‖f(t, x)‖ ≤ pf(t)‖x‖ for t ∈ [0, 1] and each x ∈ E.

(H2) For any t ∈ [0, 1] and R > 0, f(t, BR) = {f(t, x) : x ∈ BR} is relatively compact in E,where BR = {x ∈ C([0, 1], E), ‖x‖C ≤ R}and

Λ1 =

(2 − μ

)e

1 − μ M < 1. (3.1)


Lemma 3.1. If (H1) holds, then the problem (1.1) is equivalent to the following equation:

x(t) =1

(1 − μ

)Γ(α)

∫1

0Q(τ)f(τ, x(τ))dτ +

1Γ(α)

∫ t

0(t − s)α−1f(s, x(s))ds. (3.2)

Proof. By Lemma 2.6 and (1.1), we have

x(t) = x(0) +1

Γ(α)

∫ t

0(t − s)α−1f(s, x(s))ds. (3.3)

Therefore,

x(0) =∫1

0g(s)x(s)ds

=∫1

0g(s)

[x(0) +

1Γ(α)

∫s

0(s − τ)α−1f(τ, x(τ))dτ

]ds

=∫1

0g(s)dsx(0) +

1Γ(α)

∫1

0g(s)

∫s

0(s − τ)α−1f(τ, x(τ))dτds.

(3.4)

So,

x(0) =1

(1 −

∫10g(s)ds

)Γ(α)

∫1

0g(s)

∫ s

0(s − τ)α−1f(τ, x(τ))dτds

=1

(1 − μ

)Γ(α)

∫1

0f(τ, x(τ))

[∫1

τ

(s − τ)α−1g(s)ds

]

dτ

=1

(1 − μ

)Γ(α)

∫1

0Q(τ)f(τ, x(τ))dτ,

(3.5)

and then

x(t) =1

(1 − μ

)Γ(α)

∫1


1Γ(α)

∫ t

0(t − s)α−1f(s, x(s))ds. (3.6)


Conversely, if x is a solution of (3.2), then for every t ∈ [0, 1], according to Remark 2.4and Lemma 2.5, we have

cDαx(t)=cDα

[1

(1 − μ

)Γ(α)

∫1


1Γ(α)

∫ t

0(t − s)α−1f(s, x(s))ds

]

=cDα

[1

(1 − μ

)Γ(α)

∫1

0Q(τ)f(τ, x(τ))dτ

]

+cDα

[1

Γ(α)

∫ t

0(t − s)α−1f(s, x(s))ds

]

= θ+cDα(Iαf(t, x(t)))

= f(t, x(t)).

(3.7)

It is obvious that x(0) =∫1

0g(s)x(s)ds. This completes the proof.

Theorem 3.2. If (H1) and (H2) hold, then the initial value problem (1.1) has at least one solution.

Proof. Define operator A : C([0, 1], E) → C([0, 1], E), by

(Ax)(t) =1

(1 − μ

)Γ(α)

∫1


1Γ(α)

∫ t

0(t − s)α−1f(s, x(s))ds. (3.8)

Clearly, the fixed points of the operator A are solutions of problem (1.1).It is obvious that BR is closed, bounded, and convex.

Step 1. We prove that A is continuous.Let

xn, x ∈ C([0, 1], E), ‖xn − x‖c −→ 0 (n −→ ∞). (3.9)

Then r = supn‖xn‖C <∞ and ‖x‖C ≤ r. For each t ∈ [0, 1],

‖(Axn)(t) − (Ax)(t)‖ ≤e

1 − μ

∫1

0

∥∥f(τ, xn(τ)) − f(τ, x(τ))∥∥dτ

+1

Γ(α)

∫ t

0(t − s)(α−1)∥∥f(s, xn(s)) − f(s, x(s))

∥∥ds.

(3.10)

It is clear that

f(t, xn(t)) −→ f(t, x(t)), as n −→ ∞, t ∈ [0, 1],∥∥f(t, xn(t)) − f(t, x(t))

∥∥ ≤ 2Mr.(3.11)


It follows from (3.11) and the dominated convergence theorem that

‖(Axn) − (Ax)‖C −→ 0, as n −→ ∞. (3.12)

Step 2. We prove that A(BR) ⊂ BR.Let x ∈ BR. Then for each t ∈ [0, 1], we have

‖(Ax)(t)‖ ≤ 11 − μ

∫1

0

Q(τ)Γ(α)

∥∥f(τ, x(τ))

∥∥dτ +

1Γ(α)

∫ t

0(t − s)α−1∥∥f(s, x(s))

∥∥ds

≤ 11 − μ

∫1

0

Q(τ)Γ(α)

pf(τ)‖x(τ)‖dτ +1

Γ(α)

∫ t

0(t − s)α−1pf(s)‖x(s)‖ds

≤(

e

1 − μM + eM)‖x‖C

< R.

(3.13)

Step 3. We prove that A(BR) is equicontinuous.Let t1, t2 ∈ [0, 1], t1 < t2, and x ∈ BR. We deduce that

‖(Ax)(t2) − (Ax)(t1)‖

=1

Γ(α)

∥∥∥∥∥

∫ t2

0(t2 − s)α−1f(s, x(s))ds −

∫ t1

0(t1 − s)α−1f(s, x(s))ds

∥∥∥∥∥

≤ 1Γ(α)

∫ t1

0

∣∣∣(t2 − s)α−1 − (t1 − s)α−1∣∣∣∥∥f(s, x(s))

∥∥ds

+1

Γ(α)

∫ t2

t1

(t2 − s)α−1∥∥f(s, x(s))∥∥ds

≤[∫ t1

0

∣∣∣(t2 − s)α−1 − (t1 − s)α−1∣∣∣ds +

∫ t2

t1

(t2 − s)α−1ds

]MR

Γ(α)

≤[2(t2 − t1)α +

(tα2 − t

α1

)] MR

Γ(α + 1).

(3.14)

As t1 → t2, the right-hand side of the above inequality tends to zero.

Step 4. We prove that A(BR) is relatively compact.Let 5 ⊂ BR be arbitrarily given. Using the formula

∫b

a

y(t)dt ∈ (b − a)co{y(t) : t ∈ [0, 1]

}(3.15)


for y ∈ C([a, b], E) and (H2), we obtain

α((AV )(t)) ≤ α(

co{

Q(s)(1 − u)Γ(α)f(s, x(s)) : s ∈ [0, 1], x ∈ V

})

+ α

(

co

{(t − s)α−1

Γ(α)f(s, x(s)) : s ∈ [0, t], t ∈ [0, 1], x ∈ V

})

≤{

Q(s)(1 − u)Γ(α)α

(f(s, V (s))

): s ∈ [0, 1]

}

+

{(t − s)α−1

Γ(α)α(f(s, V (s))

): s ∈ [0, t], t ∈ [0, 1]

}

= 0.

(3.16)

It follows from (3.16) that α((AV )(t)) = 0 for t ∈ [0, 1]. This, together with Lemma 2.7,yields that

αC(AV ) = 0. (3.17)

From (3.17), we see that A(BR) is relatively compact. Hence, A : BR → BR is completelycontinuous. Finally, the Schauder fixed point theorem guarantees that A has a fixed point inBR.

Theorem 3.3. Besides the hypotheses of Theorem 3.2, we suppose that there exists a constant L suchthat

0 < L < Λ2, (3.18)∥∥f(t, u) − f(t,w)

∥∥ ≤ L‖u −w‖, for every u,w ∈ BR, (3.19)

where

Λ2 =1 − μ

(2 − μ

)e. (3.20)

Then, the solution x(t) of (1.1) is unique in BR.


Proof. From Theorem 3.2, we know that there exists at least one solution x(t) in BR. Wesuppose to the contrary that there exist two different solutions u(t) and w(t) in BR. It followsfrom (3.8) that

‖u(t) −w(t)‖ ≤ e

1 − μ

∫1

0

∥∥f(τ, u(τ)) − f(τ,w(τ))

∥∥dτ

+1

Γ(α)

∫ t

0(t − s)α−1∥∥f(s, u(s)) − f(s,w(s))

∥∥ds

≤ e

1 − μ

∫1

0L‖u(τ) −w(τ)‖dτ

+1

Γ(α)

∫ t

0(t − s)α−1L‖u(s) −w(s)‖ds.

(3.21)

Therefore, we get

‖u −w‖C ≤2 − μ1 − μeL‖u −w‖C. (3.22)

By (3.18), we obtain ‖u −w‖C = 0. So, the two solutions are identical in BR.

4. Example

Let

E = c0 = {x = (x1, . . . , xn, . . .) : xn −→ 0} (4.1)

with the norm ‖x‖ = supn|xn|. Consider the following nonlocal Cauchy problem for thefollowing fractional differential equation in E:

c

Dαxn(t) =1 + t

100n2xn(t), t ∈ [0, 1], 0 < α < 1,

xn(0) =∫1

0

12xn(s)ds.

(4.2)

Conclusion. Problem (4.2) has only one solution on [0, 1].

Proof. Write

fn(t, x) =1 + t

100n2xn, f =

(f1, . . . , fn, . . .

),

g(s) =12, pf(t) =

1 + t100n

.

(4.3)


Then it is clear that

f ∈ C([0, 1] × E, E), pf(t) ≤150

=M,

pf ∈ L([0, 1], R+),∥∥f(t, x)

∥∥ ≤ pf‖x‖.

(4.4)

So, (H1) is satisfied.In the same way as in Example 3.2.1 in [9], we can prove that f(t, BR) is relatively

compact in c0.By a direct computation, we get

Λ1 =

(2 − μ

)e

1 − μ M ≤(2 − μ

)e

1 − μ1

50=

3e50

< 1. (4.5)

Hence, condition (H2) is also satisfied.Moreover, we have

∣∣fn(t, u) − fn(t,w)∣∣ =

∣∣∣∣1 + t

100n2un −

1 + t100n2

wn

∣∣∣∣ ≤1

50|un −wn|, (4.6)

so

∥∥f(t, u) − f(t,w)∥∥ ≤ 1

50‖u −w‖. (4.7)

Clearly,

Λ2 =1 − μ

(2 − μ

)e=

1 − 1/23e/2

=1

3e. (4.8)

Therefore, L = 1/50 < 1/3e. Thus, our conclusion follows from Theorem 3.3.

Acknowledgments

This work was supported partially by the NSF of China (10771202), the Research Fundfor Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900) andthe Specialized Research Fund for the Doctoral Program of Higher Education of China(2007035805).

References

[1] S. Abbas and M. Benchohra, “Darboux problem for perturbed partial differential equations offractional order with finite delay,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 597–604, 2009.

[2] J. Henderson and A. Ouahab, “Fractional functional differential inclusions with finite delay,” NonlinearAnalysis: Theory, Methods & Applications, vol. 70, no. 5, pp. 2091–2105, 2009.



[4] V. Lakshmikantham and S. Leela, “Nagumo-type uniqueness result for fractional differentialequations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2886–2889, 2009.



[7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujjllo, Theory and Applications of Fractional Differential Equations,North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006.

[8] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1993.[9] D. J. Guo, V. Lakshmikantham, and X. Z. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer

Academic Publishers, Dordrecht, The Netherlands, 1996.

advances in difference equationsdownloads.hindawi.com/journals/specialissues/207192.pdfthis is an...

Documents