almost automorphy profile of solutions for difference equations of volterra type

JAMCJ Appl Math Comput (2013) 42:1–18DOI 10.1007/s12190-012-0615-3

A P P L I E D M AT H E M AT I C S

Almost automorphy profile of solutions for differenceequations of Volterra type

Ravi P. Agarwal · Claudio Cuevas · Filipe Dantas

Received: 26 July 2012 / Published online: 10 November 2012© Korean Society for Computational and Applied Mathematics 2012

Abstract This work deals with the almost automorphic profile of solutions ofthe nonlinear Volterra difference equation u(n + 1) = λ

∑nj=−∞ a(n − j)u(j) +

f (n,u(n)), n ∈ Z, for λ in a distinguished subset of the complex plane, where a(n) isa complex summable sequence and the perturbation f is a non-Lipschitz nonlinearity.Many illustrating remarks and examples are considered.

Keywords Almost automorphic functions · Volterra difference equation ·Perturbation theory

Mathematics Subject Classification (2010) 39A60 · 39A12 · 43A60 · 45D05

1 Introduction

In this work, we study the existence of discrete almost automorphic solutions to non-linear Volterra difference equation of convolution type

u(n + 1) = λ

n∑

j=−∞a(n − j)u(j) + f

(n,u(n)

), n ∈ Z, (1.1)

R.P. AgarwalDepartment of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363-8202, USAe-mail: [email protected]

C. Cuevas (�) · F. DantasDepartamento de Matemática, Universidade Federal de Pernambuco, Recife, PE, 50540-740, Brazile-mail: [email protected]

F. Dantase-mail: [email protected]

mailto:[email protected]



2 R.P. Agarwal et al.

where λ is a complex number and∑∞

n=0 |a(n)| < +∞. Volterra difference equa-tions can be considered as natural generalization of difference equations. During thelast few years Volterra difference equations have emerged vigorously in several ap-plied fields and nowadays there is a wide interest in developing the qualitative theoryfor such equations. Volterra difference equations mainly arise to model many realphenomena, like the study of competitive species in population dynamics and thestudy of motions of interacting bodies, or by applying numerical methods for solv-ing Volterra integral or integro-differential equations. It would be noted that Volterrasystems can describe process whose current state is determined by their entire pre-history. These processes are encountered in models of propagation of perturbation inmaterials with memory, various models to describe the evolution of epidemics, thetheory of viscoelasticity and the study of optimal control problems (cf. [20, 28–31]and the references therein).

Discrete almost automorphic functions, a class of functions which are more gen-eral than discrete almost periodic ones, was considered by Minh et al. [34] in con-nection with the study of (continuous) almost automorphic bounded mild solutions ofdifferential equations. The subject has also been systematically studied in the recentpaper by Araya et al. [5], where the authors review their main properties and dis-cuss the existence of discrete almost automorphic solutions of first order differenceequations in linear and nonlinear cases.

The study of almost automorphy for Volterra difference equations does not existat this time and should be developed, so that to produce a progress in the qualita-tive theory of Volterra difference equations. To the best of the author’s knowledge,the only paper in such direction is due to Cuevas et al. [17], where the authors haveconsidered the existence of discrete almost automorphic solutions of linear and non-linear Volterra difference equation of type (1.1) in the case when the perturbation f

satisfies a global Lipschitz condition. We note that the results in [17] are not sharpenough to include more general perturbations. Indeed, the authors did not study thecase when the perturbation f is a non-Lipschitz nonlinearity, even the locally Lips-chitz case was not considered. Anticipating a wide interest in the subject, this papercontributes in filling this important gap. Investigations in such directions are techni-cally more complicated, because it is necessary to apply efficient composition resultsas well as using powerful fixed point argument. The latter has been a important toolin the study of nonlinear phenomena. Specifically, we have used the Leray-Schauderalternative theorem. We remark that to use those result, some compactness assump-tions must be imposed in the perturbations. We emphasize that the implementationof this approach is a priori not trivial as the reader will find out from this work. Froman applied perspective, this work provides a set of results and techniques which areinteresting when applied to study regularity of abstract difference equations (see [9,10, 12, 13, 15]).

We will now present a summary of this work. We have tried to make the presenta-tion almost self-contained. Section 2 provides the definitions and preliminary resultsto be used in the theorems stated and proved in this article. In particular, to facilitateaccess to the individual topics, we review in Sect. 2.1 some of the standard propertiesof discrete almost automorphic (d.a.a.) functions. In Sect. 2.2, we have established acomposition result for d.a.a. functions, which is of central importance in Sect. 3. In

Almost automorphy profile of solutions for difference equations 3

Sect. 2.3, we present the notion of fundamental solution for Eq. (1.1) and we establishthe existence of discrete almost automorphic solutions for linear Volterra differenceequations, while in Sect. 2.4 we give an useful compactness criterion in C0

h(Z,X)

(see Lemma 2.1). Actually the compactness arguments are very important for deepinvestigations on perturbation theory. In Sect. 3, we obtain very general results onthe existence of discrete almost automorphic solutions for nonlinear Volterra differ-ence equations. In Sect. 3.1 we treat the locally Lipschitzian case, while in Sect. 3.2we study the existence of d.a.a. solutions to (1.1) when the perturbation f does notsatisfy a Lipschitz condition.

2 Preliminaries and basic results

In this section, we introduce notations, definitions and preliminary facts which areused throughout this work. Fix once and all a complex Banach space (X,‖ · ‖). Wedenote Bρ(X) the closed ball in X with center 0 and radius ρ. We denote by R,C,Z

and Z+ the set of all real numbers, the set of complex numbers, the set of all integers

and the set of all non-negative integers, respectively.

2.1 Discrete almost automorphic functions

From the applied point of view, the dynamic behavior of the discrete systems arerather complex and richer than those of the continuous one. Therefore, it is morerealistic to consider dynamic systems governed by difference equations. A very im-portant aspect of the qualitative study of the solutions of difference equations is theirperiodicity and in general their asymptotic periodicity. Results in such direction can-not be deduced directly from the theory on the continuous case (e.g. [40]). Almostperiodicity of a discrete function was first introduced by Walther [43, 44]. There ismuch interest in developing the periodicity study for difference equations. For de-tails, including some applications and recent developments, see the monographs ofAgarwal [1], Elaydi [20] and Corduneanu [11], and the paper by Halanay [25]. Re-cently several works (see [37–39, 41]) have been devoted to study the existence ofalmost periodic solutions of discrete systems with delay. The main method employedin these papers is to assume certain stability properties of a bounded solution. In re-cent paper Agarwal et al. [3] have studied the existence of asymptotically periodicsolutions for a class of semilinear difference equations with infinite delay.

The range of applications of almost automorphic functions include at present lin-ear and nonlinear evolution equations, integro-differential and functional-differentialequations, dynamical systems. Almost automorphic dynamics have been given a no-table amount of attention in recent years with respect to the study of almost period-ically forced monotone systems. We observe that there have been extensive studieson periodically forced second order oscillations. In engineering applications, peri-odic oscillations are referred as harmonic oscillations and almost periodic ones areinterpreted as harmonic oscillations covered with small “noise”. Accordingly, almostautomorphic oscillations can be regarded as harmonic ones covered with big “noise”(see [45]). The theory of continuous almost automorphic functions was introduced


by S. Bochner in relation to some aspects of differential geometry [7, 8]. A unifiedexposition of the theory and its applications to differential equations was first givenby N’Guérékata in his book [35]. Important contributions to the theory of almost au-tomorphic functions have been obtained, for example, in the papers [4, 14, 18, 19, 22,23, 26, 33, 46–48] and in the books [36, 49].

We use the same setting as in [5, 17]. We begin by recalling the concept of discretealmost automorphic function.

Definition 2.1 ([5]) A function f : Z → X is said to be discrete almost auto-morphic if for every integer sequence (k′

n), there is a subsequence (kn) such thatlimn→∞ f (k + kn) =: f (k) is well defined for each k ∈ Z and limn→∞ f (k − kn) =f (k) for each k ∈ Z. Denote by AAd(X) the set of all such functions.

Instead of given detailed statements, we made several remark.

Remark 2.1

(i) Note that if the convergence in the above definition is uniform in Z, then we getdiscrete almost periodicity.

(ii) If {an}n∈Z is an almost periodic sequence, then there is an almost periodic func-tion f (t) such that f (n) = an for all n ∈ Z. If f is an almost periodic function,then {f (n)}n∈Z is an almost periodic sequence (see [21] for details).

(iii) If f is a continuous almost automorphic function in R, then f |Z is discretealmost automorphic. In [17, Theorem 2.1], the authors have proved that ifx : Z → X is a discrete almost automorphic function, then there is an almostautomorphic function f : R → X such that x(n) = f (n), for all n ∈ Z.

(iv) We have that discrete almost automorphy is a more general concept than discretealmost periodicity, that is APd(X) ⊆ AAd(X). Examples of discrete almost au-tomorphic functions which are not discrete almost periodic were first constructedby Veech [42]. Notice that the function

f (k) = 1

2 + cosk + cos√

2k, k ∈ Z

is discrete almost automorphic, but not discrete almost periodic (see [5] for de-tails).

(v) Let u be a discrete almost automorphic, then for each l ∈ Z, the function ul(k) :=u(k + l), k ∈ Z is discrete almost automorphic.

(vi) Note that each discrete almost automorphic function is bounded. AAd(X) en-dowed with the norm ‖u‖d = supk∈Z ‖u(k)‖ is a Banach space. A standard ref-erence containing this result is [5, Theorem 2.4].

For applications to nonlinear difference equations, the following concept of dis-crete almost automorphic function depending on parameters will be useful.

Definition 2.2 ([5]) A function u : Z×X → X is said to be discrete almost automor-phic in k ∈ Z for each x ∈ X, if for every sequence of integers numbers (k′

n), there


is a subsequence (kn) such that limn→∞ u(k + kn, x) =: u(k, x) is well defined foreach k ∈ Z, x ∈ X and limn→∞ u(k − kn, x) =: u(k, x) for each k ∈ Z and x ∈ X.We denote by AAd(Z × X) the space of all discrete almost automorphic functions ink ∈ Z for each x ∈ X.

Let us agree to employ the following notation ‖u(·, x)‖d := supk∈Z ‖u(k, x)‖.

Remark 2.2 ([5]) If u,v : Z × X → X are discrete almost automorphic functionsin k ∈ Z for each x ∈ X and c is an arbitrary scalar, the following assertions arefulfilled:

(i) u + cv is discrete almost automorphic in k ∈ Z for each x ∈ X.(ii) ‖u(·, x)‖d = ‖u(·, x)‖d < +∞, for each x ∈ X, where u is a function involved

in Definition 2.2.

The following result is the essential property to study the existence of discretealmost automorphic solutions of linear and nonlinear Volterra difference equations ofconvolution type.

Theorem 2.1 ([5]) Let b : Z+ → C be a summable function, that is

∑∞k=0 |b(k)| <

+∞. Then for any discrete almost automorphic function u : Z → X, the functionb ∗ u defined by b ∗ u(k) = ∑k

j=−∞ b(k − j)u(j), k ∈ Z is also discrete almostautomorphic.

2.2 A composition theorem

It is well known that the study of composition of two functions with special proper-ties is important and basic for deep investigations. Our composition theorem (Theo-rem 2.2) improves the known one by making use of a uniform continuity conditioninstead of the Lipschitz condition used in [5].1 Hence, Theorem 2.2 became more ef-ficient in the applied theory associated with difference equations than Theorem 2.10of [5] (the reader can compare and realize that we have gained a much better result).

We need the following definition.

Definition 2.3 ([2]) A function f : Z × X → X is said to be uniformly continuous ineach bounded subsets of X uniformly in k ∈ Z if for every ε > 0 and every boundedsubset K of X, there is δε,K > 0 such that ‖f (k, x) − f (k, y)‖ ≤ ε for all k ∈ Z andall x, y ∈ K with ‖x − y‖ ≤ δε,K .

We are now ready to give the composition theorem of discrete almost automorphicfunctions.

1This seems reasonable and necessary since the uniform continuity condition is the main condition neededfor the composition theorems of almost periodic functions, asymptotically almost periodic functions andpseudo-almost periodic functions (see [2, 32]).


Theorem 2.2 Let f : Z×X → X be a discrete almost automorphic function in k ∈ Z

for each x ∈ X such that f (k, x) is uniformly continuous in each bounded subset ofX uniformly in k ∈ Z. Suppose u : Z → X is a discrete almost automorphic function.Then, the Niemytski map U : Z → X defined by U(k) = f (k,u(k)) is discrete almostautomorphic.

Proof Let (s′n) be a sequence of integers numbers. From Definitions 2.1, 2.2 and 2.3,

there are a subsequence (sn) of (s′n), a bounded subset B of X such that, for k ∈ Z be

given and ε > 0 fixed, there are n0 = n0(ε, k) ∈ N and δ = δ(ε,B) > 0 (here, δ is theconstant given in Definition 2.3) so that the following conditions hold:

(i) u(t), u(t) ∈ B , for all t ∈ Z and ‖u(k + sn) − u(k)‖ < δ, for all n ≥ n0;(ii) ‖f (k + sn, u(k)) − f (k,u(k))‖ < ε

2 , for all n ≥ n0;(iii) ‖f (t, u(k + sn)) − f (t, u(k))‖ < ε

2 , for all t ∈ Z and n ≥ n0,

where u and f are the functions involved in Definitions 2.1 and 2.2, respectively.We define U : Z → X by U(k) = f (k,u(k)), k ∈ Z. For all n ≥ n0, we have the

following estimate:∥∥U(k + sn) − U(k)

∥∥ = ∥

∥f(k + sn, u(k + sn)

) − f(k,u(k)

)∥∥

≤ ∥∥f

(k + sn, u(k + sn)

) − f(k + sn, u(k)

)∥∥

+ ∥∥f

(k + sn, u(k)

) − f(k,u(k)

)∥∥ <

ε

2+ ε

2= ε.

This in turn implies limn→∞ U(k + sn) = U(k) is well defined. Since f : Z×X → X

is uniformly continuous in each bounded subset of X uniformly in k ∈ Z. We canverify, without essential modification in the previous argument, that limn→∞ U(k −sn) = U(k) for each k ∈ Z, which completes the proof. �

We obtain Araya et al.’s result [5, Theorem 2.10] as an immediate consequence ofprevious theorem.

Corollary 2.1 Let f : Z×X → X be a discrete almost automorphic function in k ∈ Z

for each x ∈ X that satisfies a Lipschitz condition in x ∈ X uniformly in k ∈ Z, that is,there is a constant L ≥ 0 such that ‖f (k, x)−f (k, y)‖ ≤ L‖x −y‖, for all x, y ∈ X,k ∈ Z. Then, the conclusion of the previous theorem is true.

Corollary 2.1 admits a new version with local conditions on the function f . In theprocess of obtaining such version, we require the following assumption:

(H1) Assume that f : Z × X → X is locally Lipschitz with respect to the secondvariable, that is, for each positive number σ , for all k ∈ Z and for all x, y ∈ X

with ‖x‖ ≤ σ and ‖y‖ ≤ σ , we have ‖f (k, x) − f (k, y)‖ ≤ Lf (σ )‖x − y‖,where Lf : R

+ → R+ is a nondecreasing function.

Corollary 2.2 Let f : Z×X → X be a discrete almost automorphic function in k ∈ Z

for each x ∈ X that satisfies condition (H1). Then, the conclusion of the previoustheorem is true.


Remark 2.3 In the proof of Corollary 2.2 is not necessary to assume that Lf (·) is anondecreasing function.

2.3 Linear Volterra difference equations

In this section we establish the existence of discrete almost automorphic solutions forlinear Volterra difference equations in a Banach space X and described by

u(n + 1) = λ

n∑

j=−∞a(n − j)u(j) + f (n), n ∈ Z, (2.1)

where λ is a complex number, a : N → C is a summable function and f is in AAd(X).For a given λ ∈ C, let s(λ, k) ∈ C be the solution of the difference equation

s(λ, k + 1) = λ

k∑

j=0

a(k − j)s(λ, j), k = 0,1,2, . . . , s(λ,0) = 1. (2.2)

In this case, s(λ, k) is called the fundamental solution to Eq. (2.1) generated bya(·). We define the set Ωs := {λ ∈ C : ‖s(λ, ·)‖1 := ∑∞

k=0 |s(λ, k)| < +∞}.

Example 2.1 For a(k) = pk , where |p| < 1, we obtain, after a calculation usingin (2.2) the unilateral Z-transform, that s(λ, k) = λ(λ + p)k−1, k ≥ 1 and henceD(−p,1) := {z ∈ C : |z + p| < 1} ⊆ Ωs .

Another result that will come handy is the following:

Theorem 2.3 ([17]) Let λ be in Ωs . If f : Z → X is a discrete almost automorphicfunction, then there is a discrete almost automorphic solution of (2.1) given by

u(n + 1) =n∑

j=−∞s(λ,n − j)f (j). (2.3)

Remark 2.4 In view of Theorem 2.3 one may wonder whether the solution u(n) givenby (2.3) is the unique solution of (2.1). The answer is the following: note that if f

is bounded, then s(λ, ·) ∗ f is well defined, that is s(λ, ·) ∗ f ∈ l∞(Z,X). Hence allbounded solutions of (2.1) is given by (2.3). In fact, let u(n) be a bounded functionsatisfying (2.1), we obtain

n∑

j=−∞s(λ,n − j)f (j)

=n∑

j=−∞s(λ,n − j)

[

u(j + 1) − λ

j∑

l=−∞a(j − l)u(l)

]


=n∑

j=−∞s(λ,n − j)u(j + 1) − λ

n∑

j=−∞

j∑

l=−∞a(j − l)s(λ,n − j)u(l)

=n+1∑

l=−∞s(λ,n + 1 − l)u(l) − λ

n∑

l=−∞

n∑

j=l

a(j − l)s(λ,n − j)u(l)

=n+1∑

l=−∞s(λ,n + 1 − l)u(l) −

n∑

l=−∞

(

λ

n−l∑

j=0

a(n − l − j)s(λ, j)

)

u(l)

=n+1∑

l=−∞s(λ,n + 1 − l)u(l) −

n∑

l=−∞s(λ,n + 1 − l)u(l) = u(n + 1).

Hence, u(n) satisfies (2.3). This finishes the discussion of Remark 2.4.

2.4 Compactness criterion

The perturbations problem is justified no only by aspiration for mathematical gener-ality, but also by the obvious fact that the overwhelming majority of real problems arenonlinear. The compactness criterions provide us with a sensible tool to pursue newresults on perturbations. Because any concrete nonlinear situation requires a compactoperator and hence a very detailed knowledge on the relatively compact sets con-tained in the state space involved in the evolution problem studied and consequentlywe need a good compactness criterion. Let us recall that in general developing thismachinery is technically not trivial.

Let h : Z → R+ be a function such that h(n) ≥ 1 for all n ∈ Z, and h(n) → ∞ as

|n| → ∞. We consider the space

C0h(Z,X) =

{

ξ : Z → X : lim|n|→∞‖ξ(n)‖h(n)

= 0

}

,

endowed with the norm ‖ξ‖h = supn∈Z

‖ξ(n)‖h(n)

. It is clear that C0h(Z,X) is a Banach

space isometrically isomorphic with the space C0(Z,X) consisting of all sequencesx : Z → X that vanish at ±∞. According to a compactness criterion due to Cuevasand Pinto (see [16, Lemma 2.3]), we have the following result.

Lemma 2.1 Let S be a subset of C0h(Z,X). Suppose the following conditions are

satisfied:

(H2) The set Hn(S) = {u(n)h(n)

: u ∈ S} is relatively compact in X, for all n ∈ Z;(H3) S is weighted equiconvergent at ±∞, that is for every ε > 0, there is a T > 0

such that ‖u(n)‖ < εh(n), for each |n| ≥ T , for all u ∈ S.

Then, S is relatively compact in C0h(Z,X).

Proof To reader’s convenience, we give the proof. Let (um)m be a sequence in S. Itfollows from (H2) that there is a subsequence (umj

)j of (um)m such that the limit


a(n) = limj→∞ umj(n)h(n)−1 exists. A simple computation shows that (umj

)j is aCauchy sequence in C0

h(Z,X). To see this, let T be a number ensured in condition(H3). Then,

maxn∈[−T ,T ]∩Z

∥∥umj

(n) − umk(n)

∥∥h(n)−1 ≤ max

n∈[−T ,T ]∩Z

∥∥umj

(n)h(n)−1 − a(n)∥∥

+ maxn∈[−T ,T ]∩Z

∥∥umk

(n)h(n)−1 − a(n)∥∥,

whereas

sup|n|>T

∥∥umj

(n) − umk(n)

∥∥h(n)−1 ≤ sup

|n|>T

‖umj(n)‖

h(n)+ sup

|n|>T

‖umk(n)‖

h(n).

This concludes the proof of the lemma. �

Before proceeding further, we shall give the following remark.

Remark 2.5 If S is relatively compact on C0h(Z,X), then (H2) and (H3) hold. In

fact, we consider the continuous map Φn : C0h(Z,X) → X defined by Φn(u) = u(n)

h(n),

then Hn(S) = Φn(S). Whence condition (H2) holds. To prove (H3), we considerε > 0 arbitrary since S is relatively compact there are fi, i = 1, . . . ,m such thatS ⊂ ⋃m

i=1 B(fi, ε). If f ∈ S, f ∈ B(fi, ε), for some i, then ‖f (n)h(n)−1‖ ≤‖f − fi‖h + ‖fi(n)h(n)−1‖. Taking n0 ∈ N such that ‖fi(n)h(n)−1‖ < ε for all|n| > n0 and each i = 1, . . . ,m we have ‖f (n)h(n)−1‖ < 2ε for |n| > n0 and allf ∈ S.

Let H : R → R be a continuous function such that H(t) ≥ 1 for all t ∈ R, andH(t) → ∞ as |t | → ∞. We consider the following Banach space CH (R,X) = {u ∈C(R,X) : lim|t |→∞ u(t)

H(t)= 0}, endowed with the norm ‖u‖H = supt∈R

‖u(t)‖H(t)

.

Remark 2.6 If v ∈ CH (R,X), then u := v|Z ∈ C0h(Z,X), where h = H |Z.

We have the following result.

Theorem 2.4 Let u be in C0h(Z,X). Then, there are two functions Hh and vu satis-

fying the following conditions:

(i) Hh : R → [1,∞) such that h(n) = Hh(n) for all n ∈ Z and lim|t |→∞ Hh(t)

= ∞;(ii) vu ∈ CHh

(R,X) and u = vu|Z.

Proof We define Hh : R → R by Hh(t) = (1 + [t] − t)h([t]) + (t − [t])h([t] + 1),here [s] denotes the biggest integer ≤ s. It is clear that Hh(t) ≥ 1, for all t ∈ R. Wenow show that Hh is a continuous function. Set t0 = [t0] + r0 ∈ ([t0], [t0] + 1), thenif t = [t0] + r ∈ ([t0], [t0] + 1), t → t0 ⇔ r → r0. Then, Hh(t) = (1 − r)h([t0]) +rh([t0] + 1) −→ (1 − r0)h([t0]) + r0h([t0] + 1) = Hh(t0), as t → t0. Similarly, we


have the continuity in t0 ∈ Z. We can assert that lim|t |→∞ Hh(t) = ∞. In fact, fix anumber ε > 0, there is N ∈ N such that if |m| ≥ N , we get h(m) > ε. Then, if t ∈ R

with |t | ≥ N , we have that Hh(t) > (1+[t]− t)ε+(t −[t])ε = ε, thus our assertion isproved. Now, we define vu : R → X by vu(t) = (1+[t]− t)u([t])+(t −[t])u([t]+1).Since vu is continuous and ‖vu(t)‖Hh(t)

−1 ≤ ‖u([t])‖h([t])−1 +‖u([t]+1)‖h([t]+1)−1, this shows that vu ∈ CHh

(R,X). �

Remark 2.7 An alternative approach to show that C0h(Z,X) is a Banach space could

be to use Theorem 2.4. Indeed, Cauchy sequences in C0h(Z,X) can be extended to

Cauchy sequences in CHh(R,X). Hence, the completeness of (CHh

(R,X),‖ · ‖Hh)

implies that (C0h(Z,X),‖ · ‖h) is a Banach space.

3 Almost automorphy

In this section, we examine the existence of discrete almost automorphic solutionsfor the semilinear Volterra difference equation (1.1).

3.1 Locally Lipschitz case

We begin with the following result.

Theorem 3.1 Let λ be in Ωs and let f : Z × X → X be a discrete almost auto-morphic function in k ∈ Z for each x ∈ X that satisfies the condition (H1). If there isr > 0 such that ‖s(λ, ·)‖1(Lf (r)+ ‖f (·,0)‖d

r) < 1, then Eq. (1.1) has a discrete almost

automorphic solution u(n) with ‖u‖d ≤ r satisfying

u(n + 1) =n∑

j=−∞s(λ,n − j)f

(j,u(j)

). (3.1)

Remark 3.1 In view of Theorem 3.1, notice that if u(n) is a bounded solution of (1.1),then it also satisfies (3.1) (see Remark 2.4).

Proof of Theorem 3.1 Let F : Br(AAd(X)) → Br(AAd(X)) be the map defined by

Fu(n) =n−1∑

j=−∞s(λ,n − 1 − j)f

(j,u(j)

). (3.2)

Since u ∈ AAd(X) and f (k, x) satisfies (H1), we obtain by Corollary 2.2 thatf (·, u(·)) is in AAd(X). Recalling that AAd(X) is translation invariant (see [5]), itfollows from Theorem 2.1 that Fu is in AAd(X). Let u be in Br(AAd(X)), we havethe following estimates:

∥∥Fu(n)

∥∥ ≤

n−1∑

j=−∞

∣∣s(λ,n − 1 − j)

∣∣∥∥f

(j,u(j)

) − f (j,0)∥∥


+n−1∑

j=−∞

∣∣s(λ,n − 1 − j)

∣∣∥∥f (j,0)

∥∥

≤ Lf (r)

n−1∑

j=−∞

∣∣s(λ,n − 1 − j)

∣∣∥∥u(j)

∥∥ + ∥

∥s(λ, ·)∥∥1

∥∥f (·,0)

∥∥

d

≤ ∥∥s(λ, ·)∥∥1

(

Lf (r) + ‖f (·,0)‖d

r

)

r ≤ r.

Hence Fu ∈ Br(AAd(X)) and then F is well defined. On the other hand, for u,v ∈Br(AAd(X)), we have that

∥∥Fu(n) − Fv(n)

∥∥ ≤

n−1∑

j=−∞

∣∣s(λ,n − 1 − j)

∣∣∥∥f

(j,u(j)

) − f(j, v(j)

)∥∥

≤ Lf (r)

n−1∑

j=−∞

∣∣s(λ,n − 1 − j)

∣∣∥∥u(j) − v(j)

∥∥

≤ ∥∥s(λ, ·)∥∥1Lf (r)‖u − v‖d .

Recalling that ‖s(λ, ·)‖1Lf (r) < 1, it follows that F is a contraction on Br(AAd(X)).Then, there is a unique function u ∈ Br(AAd(X)) such that

u(n + 1) =n∑

j=−∞s(λ,n − j)f

(j,u(j)

)

and, applying Theorem 2.3, we obtain that u is a discrete almost automorphic solutionof Eq. (1.1). �

We obtain [5, Theorem 4.1] as an immediate consequence of the previous result.

Corollary 3.1 Let λ ∈ Ωs and assume that f is a function that satisfies the conditionsin Corollary 2.1. If L‖s(λ, ·)‖1 < 1, where L is the Lipschitz constant of f , thenEq. (1.1) has a unique discrete almost automorphic solution.

Example 3.1 Let p be in C such that |p| < 1 and take λ ∈ D(−p,1) (see Exam-ple 2.1). Let g : Z → C be a discrete almost automorphic function. Let ν and r betwo real numbers such that

|ν| < 1 − |λ + p|2(1 − |λ + p| + |p|)‖g‖d

, 0 < r <1

2. (3.3)

We consider the following difference equation in the Banach space X:

u(n + 1) = λ

n∑

j=−∞pn−j u(j) + νg(n)

1 + ‖u(n)‖u(n), n ∈ Z. (3.4)


Let f (k, x) be the perturbation associated to (3.4). We first note that the functionf (k, x) is discrete almost automorphic in k ∈ Z for each x ∈ X. We observe thatcondition (H1) follows as an immediate consequence of the following estimate:

∥∥f (k, x) − f (k, y)

∥∥ ≤ |ν|‖g‖d

(1 + 2‖y‖)‖x − y‖, (3.5)

for all x, y ∈ X and k ∈ Z. Looking at the right hand side of (3.5), we can choose thefunction Lf (·) in condition (H1) as

Lf (σ ) = |ν|‖g‖d(1 + 2σ). (3.6)

Combining (3.3) and (3.6), we have that ‖s(λ, ·)‖1Lf (r) < 1. By Theorem 3.1 thereis a unique discrete almost automorphic solution u(n) of (3.4) given by u(n + 1) =λν

∑nj=−∞

(λ+p)n−j g(j)1+‖u(j)‖ u(j). This finishes the discussion of Example 3.1.

The next example is a modification of the first one.

Example 3.2 Let p, λ and g be as in the previous example. We consider the followingdifference equation

u(n + 1) = λ

n∑

j=−∞pn−j u(j) + g(n)

∥∥u(n)

∥∥β

u(n), n ∈ Z, (3.7)

where β > 1. It is important to note that∣∣|a|β − |b|β ∣

∣ ≤ β|a − b|(|a|β−1 + |b|β−1), a, b ∈ R. (3.8)

Let r be a real number such that 0 < rβ < [1−|λ+p|][2β(1−|λ+p|+|λ|)‖g‖d ]−1.As we will see below, its choice justifies the application of the previous result. LetF(k, x) be the perturbation associated to (3.7). From (3.8) we derive the followingestimate:

∥∥F(k, x) − F(k, y)

∥∥ ≤ 2β‖g‖d

(‖x‖β + ‖y‖β)‖x − y‖,

for all x, y ∈ X and k ∈ Z. We can choose the function LF (·) in condition (H1) asLF (σ) = 2β‖g‖dσβ . To complete the discussion of this example again we invokeTheorem 3.1 and hence we conclude that there is a discrete almost automorphic so-lution u(n) of (3.7).

Remark 3.2 We call the attention of the reader that in partial differential equa-tions (see [6]) is quite often to find locally Lipschitz perturbations f (t, x) satisfying‖f (t, x) − f (t, y)‖ ≤ C(1 + ‖x‖β + ‖y‖β)‖x − y‖, for all x, y ∈ X, t ∈ R, whereβ > 1. Of course, one gets the same results like Theorem 3.1 for this class of pertur-bations.

Remark 3.3 Let λ be in Ωs and let f : Z × X → X be a function such thatf (·,0) is bounded and that satisfies the condition (H1). If there is r > 0 such that‖s(λ, ·)‖1(Lf (r) + ‖f (·,0)‖dr−1) < 1, then Eq. (1.1) has a bounded solution. Thestraightforward change in the details may safely be left to the reader.


3.2 Non-Lipschitz case

We next study the existence of discrete almost automorphic solutions to (1.1) whenthe perturbation f does not satisfy a Lipschitz condition. To understand more generalperturbations, we need to deal with more general fixed point theorems than contrac-tion principle. In this subsection, we will use the Leray-Schauder alternative and ageneralization of Banach’s contraction principle [24, Theorem 5.2]. To establish ourresult, we consider functions f : Z×X → X that satisfies the following boundednesscondition:

(H4) There are a nondecreasing function W : R+ → R

+ and a function M : Z → R+

such that ‖f (k, x)‖ ≤ M(k)W(‖x‖), for all k ∈ Z and x ∈ X.

We can formulate our result.

Theorem 3.2 Let λ be in Ωs and let f : Z×X → X be a discrete almost automorphicfunction in k ∈ Z for each x ∈ X that satisfies assumption (H4). Suppose, in addition,that the following conditions are satisfied:

(H5) The function f is uniformly continuous in each bounded subset of X uniformlyin k ∈ Z;

(H6) For each ν > 0, lim|n|→∞ 1h(n+1)

∑nj=−∞ |s(λ,n − j)|M(j)W(νh(j)) = 0,

where h is given by Lemma 2.1;(H7) For each ε > 0, there is a δ > 0 such that for every u,v ∈ C0

h(Z,X),‖u−v‖h ≤ δ implies that

∑nj=−∞ |s(λ,n− j)|‖f (j,u(j))−f (j, v(j))‖ ≤ ε,

for all n ∈ Z;(H8) For all a, b ∈ Z, a ≤ b, and σ > 0, the set {f (k, x) : a ≤ k ≤ b,‖x‖ ≤ σ } is

relatively compact in X;(H9) lim infr→∞ r

β(r)> 1, where β(r) := supn∈Z( 1

h(n+1)

∑nj=−∞ |s(λ,n − j)| ×

M(j)W(rh(j))).

Then (1.1) has a discrete almost automorphic solution.

The proof of Theorem 3.2 uses two basic ingredients:

(i) A compactness criterion Lemma 2.1 due to Cuevas and Pinto [16];(ii) The Leray-Schauder alternative theorem [24].

Proof of Theorem 3.2 We define the operator G : C0h(Z,X) → C0

h(Z,X) by (3.2).We want to show that G has a fixed point in AAd(X). We must show that we have allthe necessary conditions to apply Leray-Schauder alternative theorem. We divide theproof into several steps.

Step 1 The first step in our analysis will be to prove that G is well defined. Let u bein C0

h(Z,X), by (H4) we have that

∥∥Gu(n)

∥∥ ≤

n−1∑

j=−∞

∣∣s(λ,n − 1 − j)

∣∣M(j)W

(∥∥u(j)

∥∥)


≤n−1∑

j=−∞

∣∣s(λ,n − 1 − j)

∣∣M(j)W

(‖u‖hh(j)),

whence

‖Gu(n)‖h(n)

≤ 1

h(n)

n−1∑

j=−∞

∣∣s(λ,n − 1 − j)

∣∣M(j)W

(‖u‖hh(j)).

It follows from the condition (H6) that G is C0h(Z,X)-valued.

Step 2 The map G is continuous from C0h(Z,X) into C0

h(Z,X). In fact, take anyε > 0, let δ > 0 be the constant involved in condition (H7). For u,v ∈C0

h(Z,X) and ‖u − v‖h ≤ δ, we have

∥∥Gu(n) − Gv(n)

∥∥ ≤

n−1∑

j=−∞

∣∣s(λ,n − 1 − j)

∣∣∥∥f

(j,u(j)

) − f(j, v(j)

)∥∥.

Keeping in mind that h(n) ≥ 1 for all n ∈ Z, we obtain ‖Gu(n) − Gv(n)‖ ×h(n)−1 ≤ ε, for all n ∈ Z, which implies that ‖Gu − Gv‖h ≤ ε. Since ε > 0is arbitrary, this shows the assertion.

Step 3 We next show that G is completely continuous. We set V = G(BR(C0h(Z,X)))

and v = Gu for u ∈ BR(C0h(Z,X)). Initially, we prove that

Hn(V ) :={

v(n)

h(n): v ∈ V

}

is relatively compact in X for each n ∈ Z. To do this, we look at the condition(H6). Hence, for ε > 0, we can choose l ∈ Z

+ such that∑∞

j=l |s(λ, j)|M(n−1 − j)W(Rh(n − 1 − j)) ≤ ε. Let v be in V , then

v(n) =l−1∑

j=0

s(λ, j)f(n − 1 − j,u(n − 1 − j)

)

+∞∑

j=l

s(λ, j)f(n − 1 − j,u(n − 1 − j)

). (3.9)

Thus

v(n)

h(n)= l

h(n)

(1

l

l−1∑

j=0

s(λ, j)f(n − 1 − j,u(n − 1 − j)

))

+ 1

h(n)

∞∑

j=l

s(λ, j)f(n − 1 − j,u(n − 1 − j)

). (3.10)


We shall check that the second term on right hand side above is dominatedby ε.

1

h(n)

∣∣∣∣∣

∣∣∣∣∣

∞∑

j=l

s(λ, j)f(n − 1 − j,u(n − 1 − j)

)∣∣∣∣∣

∣∣∣∣∣

≤ 1

h(n)

∞∑

j=l

∣∣s(λ, j)

∣∣M(n − 1 − j)W

(‖u‖hh(n − 1 − j))

≤ 1

h(n)

∞∑

j=l

∣∣s(λ, j)

∣∣M(n − 1 − j)W

(Rh(n − 1 − j)

)

≤ ε. (3.11)

Summarizing from (3.10) and (3.11) we have v(n)h(n)

∈ lh(n)

co(K) + Bε(X),

where co(K) denotes the convex hull of K2 and

K =l−1⋃

j=0

{s(λ, j)f (ξ, x) : ξ ∈ [n − l, n − 1] ∩ Z, ‖x‖ ≤ R#},

where R# = R maxξ∈[n−l,n−1]∩Z h(ξ). Using the property (H8), we infer thatK is relatively compact, and since Hn(V ) ⊆ l

h(n)co(K) + Bε(X), we have

that Hn(V ) is relatively compact in X for all n ∈ Z, which establishes ourassertion. We observe that V is weighted equiconvergent at ±∞. In fact, weget

‖v(n)‖h(n)

≤ 1

h(n)

n−1∑

j=−∞

∣∣s(λ,n − 1 − j)

∣∣M(j)W

(Rh(j)

).

This last estimate guarantees the fact that v(n)h(n)

→ 0 as |n| → ∞ and

this convergence is independent of u ∈ BR(C0h(Z,X)). Then according to

Lemma 2.1, we have that V is a relatively compact set in C0h(Z,X).

Step 4 We now analyze the boundedness of the set {u ∈ C0h(Z,X) : u = γGu, γ ∈

(0,1)}. Let u be in C0h(Z,X) so that is a solution of equation u = γGu,

γ ∈ (0,1). Next we compute ‖u‖h as

∥∥u(n)

∥∥ ≤

n−1∑

j=−∞

∣∣s(λ,n − 1 − j)

∣∣M(j)W

(‖u‖hh(j)) ≤ h(n)β

(‖u‖h

).

2The convex hull of a set K is the set of all convex combinations of point in K : co(K) := {θ1x1 + · · · +θkxk : xi ∈ K, θi ≥ 0, i = 1, . . . , k; θ1 + · · · + θk = 1}. As the name suggests, the convex hull co(K) isalways convex. It is the smallest convex set that contain K .


Therefore, ‖u(n)‖h(n)

≤ β(‖u‖h), for each n ∈ Z. Thus, we have proved the fol-lowing estimate:

‖u‖h

β(‖u‖h)≤ 1. (3.12)

From the estimate (3.12) and the condition (H9), we conclude our assertion.Step 5 It follows, from condition (H5) and Theorem 2.2, that the sequence k �→

f (k,u(k)) belongs to AAd(X), whenever u ∈ AAd(X). Hence using The-orem 2.1, we get that G(AAd(X)) ⊆ AAd(X), and consequently, we can

consider G : AAd(X)C0

h(Z,X) → AAd(X)C0

h(Z,X), where B

C0h(Z,X)

denotes theclosure of a set B in the space C0

h(Z,X). The argument used in the first partof this proof yields immediately that this map is completely continuous. Ap-plying (iv) and the Leray-Schauder alternative theorem [24, Theorem 6.5.4]

we deduce that the map G has a fixed point u ∈ AAd(X)C0

h(Z,X).

Step 6 Let (un)n be a sequence in AAd(X) such that un → u, as n → ∞ in thenorm of C0

h(Z,X). For ε > 0, let δ > 0 be the constant in (H7), one can selectn0 ∈ Z

+ large enough so that ‖un − u‖h ≤ δ for all n ≥ n0. We observe thatfor n ≥ n0,

‖Gun − Gu‖d ≤ supm∈Z

m−1∑

k=−∞

∣∣s(λ,m − 1 − k)

∣∣∥∥f

(k,un(k)

) − f(k,u(k)

)∥∥

≤ ε.

This means that (Gun)n converges to Gu = u uniformly in Z. This impliesthat u ∈ AAd(X), and this completes the proof. �

Remark 3.4

(i) The conclusion of Theorem 3.2 is true when M(·) ≡ 1 in (H4);(ii) If dim X < ∞, we can remove condition (H8);

(iii) If W(·) ≡ 1 and M∞ := supk∈Z M(k) < +∞ in condition (H4), then we canremove in Theorem 3.2 conditions (H6) and (H9).

4 Conclusions

We have studied almost automorphy of Volterra difference equations in the case whenthe perturbation is not necessarily Lipschitz. Anticipating a wide interest in the sub-ject, this paper contributes in filling this important gap. Investigations in such di-rections are technically more complicated, because it is necessary to apply efficientcomposition results as well as using powerful fixed point arguments. The usefulnessof the fixed point methods for applications have increased enormously by the devel-opment of accurate and efficient techniques for computing fixed point, making fixedpoint arguments a powerful weapon in the arsenal of applied mathematics (see [27]).This work provides a set of results and techniques which can be applied to study


regularity and qualitative behavior of abstract functional difference equations. Theinvestigations of these and other similar problems form the scope for further researchwork in theory of discrete evolution equations.

Acknowledgements The results in this work were partially obtained during a visit (July-September2012) of the second author to the Department of Mathematics and Statistics of Universidad de La Fronteraunder Programa Atracción e Inserción (PAI-MEC) Grant 80112008 (CONICYT-CHILE). He is grateful toprofessor Herme Soto and the Department of Mathematics and Statistics, for its generous hospitality andproviding a stimulating atmosphere to work.

References

1. Agarwal, R.P.: Difference Equations and Inequalities. Monographs and Textbooks in Pure and AppliedMathematics, vol. 155. Dekker, New York (1992)

2. Agarwal, R.P., de Andrade, B., Cuevas, C.: Weighted pseudo-almost periodic solutions of a classof semilinear fractional differential equations. Nonlinear Anal., Real World Appl. 11(5), 3532–3554(2010)

3. Agarwal, R.P., Cuevas, C., Frasson, M.: Semilinear functional difference equations with infinite delay.Math. Comput. Model. 55(3–4), 1083–1105 (2012)

4. Allouche, J.-P., Bacher, R.: Toeplitz sequences, paperfolding, towers of Hanoi and progression-freesequences of integers. Enseign. Math. (2) 38(3–4), 315–327 (1992)

5. Araya, D., Castro, R., Lizama, C.: Almost automorphic solutions of difference equations. Adv. Differ.Equ. 2009, 591380 (2009)

6. Arrieta, J.M., Carvalho, A.N.: Abstract parabolic problems with critical nonlinearities and applica-tions to Navier-Stokes and heat equations. Trans. Am. Math. Soc. 352(1), 285–310 (1999)

7. Bochner, S.: Uniform convergence of monotone sequences of functions. Proc. Natl. Acad. Sci. USA47, 582–585 (1961)

8. Bochner, S., Von Neumann, J.: On compact solutions of operational-differential equations. I. Ann.Math. (2) 36(1), 255–291 (1935)

9. Cardoso, F., Cuevas, C.: Exponential dichotomy and boundedness for retarded functional differenceequations. J. Differ. Equ. Appl. 15(3), 261–290 (2009)

10. Castro, A., Cuevas, C.: Perturbation theory, stability, boundedness and asymptotic behavior for secondorder evolution in discrete time. J. Differ. Equ. Appl. 17(3), 327–358 (2011)

11. Corduneanu, C.: Almost periodicity in functional equations, differential equations, chaos and varia-tional problems. Prog. Nonlinear Differ. Equ. Appl. 75, 157–163 (2008)

12. Cuevas, C., de Souza, J.C.: A perturbation theory for the discrete harmonic oscillator equation. J. Dif-fer. Equ. Appl. 16(12), 1413–1428 (2010)

13. Cuevas, C., Lizama, C.: Semilinear evolution equations of second order via maximal regularity. Adv.Differ. Equ. 2008, 316207 (2008)

14. Cuevas, C., Lizama, C.: Almost automorphic solutions to integral equations on the line. SemigroupForum 79(3), 461–472 (2009)

15. Cuevas, C., Lizama, C.: Semilinear evolution equations on discrete time and maximal regularity.J. Math. Anal. Appl. 361(1), 234–245 (2010)

16. Cuevas, C., Pinto, M.: Convergent solutions of linear functional difference equations in phase space.J. Math. Anal. Appl. 277(1), 324–341 (2003)

17. Cuevas, C., Henríquez, H.R., Lizama, C.: On the existence of almost automorphic solutions ofVolterra difference equations. J. Differ. Equ. Appl. (2011). doi:10.1080/10236198.2011.603311

18. De Andrade, B., Cuevas, C., Henríquez, E.: Asymptotic periodicity and almost automorphy for a classof Volterra integro-differential equations. Math. Methods Appl. Sci. (2012). doi:10.1002/mma.1607

19. Diagana, T., N’Guérékata, G., Minh, N.V.: Almost automorphic solutions of evolution equations.Proc. Am. Math. Soc. 132(11), 3289–3298 (2004)

20. Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Undergraduate Texts in Mathematics.Springer, New York (2005)

21. Fink, A.M.: Almost Differential Equations. Lecture Notes in Mathematics, vol. 377. Springer, Berlin(1974)

http://dx.doi.org/10.1080/10236198.2011.603311

http://dx.doi.org/10.1002/mma.1607


22. Gjerde, R., Johansen, O.: Bratteli-Vershik models for Cantor minimal systems: applications toToeplitz flows. Ergod. Theory Dyn. Syst. 20(6), 1687–1710 (2000)

23. Goldstein, J.A., N’Guérékata, G.M.: Almost automorphic solutions of semilinear evolution equations.Proc. Am. Math. Soc. 133(8), 2401–2408 (2005)

24. Granas, A., Dugundji, J.: Fixed Point Theory. Springer Monographs in Mathematics. Springer, NewYork (2003)

25. Halanay, A.: Solutions périodiques et presque-périodiques des systèmes d’équations aux différencesfinies. Arch. Ration. Mech. Anal. 12, 134–149 (1963)

26. Henríquez, H., Cuevas, C.: Almost automorphy for abstract neutral differential equation via controltheory. Ann. Mat. Pura Appl. (2011). doi:10.1007/s10231-011-0229-7

27. Istratescu, V.I.: Fixed Point Theory, An Introduction. Mathematics and Its Applications, vol. 7. Reidel,Dordrecht (1981)

28. Kolmanovskii, V., Shaikhet, L.: Some conditions for boundedness of solutions of difference Volterraequations. Appl. Math. Lett. 16(6), 857–862 (2003)

29. Kolmanovskii, V.B., Myshkis, A.D.: Stability in the first approximation of some Volterra differenceequations. J. Differ. Equ. Appl. 3(5–6), 563–569 (1998)

30. Kolmanovskii, V.B., Myshkis, A.D., Richard, J.-P.: Estimate of solutions for some Volterra differenceequations. Nonlinear Anal. 40(1–8), 345–363 (2000)

31. Kolmanovskii, V.B., Castellanos-Velasco, E., Torres-Muñoz, J.A.: A survey: stability and bounded-ness of Volterra difference equations. Nonlinear Anal. 53(7–8), 861–928 (2003)

32. Li, H.-X., Huang, F.-L., Li, J.-Y.: Composition of pseudo almost-periodic functions and semilineardifferential equations. J. Math. Anal. Appl. 255(2), 436–446 (2001)

33. Liu, J.-h., Song, X.-q.: Almost automorphic and weighted pseudo almost automorphic solutions ofsemilinear evolution equations. J. Funct. Anal. 258(1), 196–207 (2010)

34. Minh, N.V., Naito, T., N’Guérékata, G.: A spectral countability condition for almost automorphy ofsolutions of differential equations. Proc. Am. Math. Soc. 134(11), 3257–3266 (2006)

35. N’Guérékata, G.M.: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. KluwerAcademic/Plenum, New York (2001)

36. N’Guérékata, G.M.: Topics in Almost Automorphy. Springer, New York (2005)37. Song, Y.: Almost periodic solutions of discrete Volterra equations. J. Math. Anal. Appl. 314(1), 174–

194 (2006)38. Song, Y.: Periodic and almost periodic solutions of functional difference equations with finite delay.

Adv. Differ. Equ. 2007, 68023 (2007)39. Song, Y.: Asymptotically almost periodic solutions of nonlinear Volterra difference equations with

unbounded delay. J. Differ. Equ. Appl. 14(9), 971–986 (2008)40. Song, Y.: Positive almost periodic solutions of nonlinear discrete systems with finite delay. Comput.

Math. Appl. 58(1), 128–134 (2009)41. Song, Y., Tian, H.: Periodic and almost periodic solutions of nonlinear discrete Volterra equations

with unbounded delay. J. Comput. Appl. Math. 205(2), 859–870 (2007)42. Veech, W.A.: Almost automorphic functions. Proc. Natl. Acad. Sci. USA 49, 462–464 (1963)43. Walther, A.: Fastperiodische Folgen und ihre Fouriersche Analysis. Alti Congresso Inst. Mat. Bologna

2, 289–298 (1928)44. Walther, A.: Fastperiodische Folgen und Potenzreihen mit fastperiodischen Koeffizienten. Abh. Math.

Sem. Hamburg Univ. 6, 217–234 (1928)45. Yi, Y.: On almost automorphic oscillations. In: Differences and Differential Equations. Fields Institute

Communications, vol. 42, pp. 75–99. Am. Math. Soc., Providence (2004)46. Zaidman, S.: Almost automorphic solutions of some abstract evolution equations. Rend. - Ist. Lomb.,

Accad. Sci. Lett. A 110(2), 578–588 (1976)47. Zaidman, S.: Behavior of trajectories of C0-semigroups. Rend. - Ist. Lomb., Accad. Sci. Lett. A 114,

205–208 (1980–1982).48. Zaidman, S.: Topics in abstract differential equations. Nonlinear Anal. 23(7), 849–870 (1994)49. Zaidman, S.: Topics in Abstract Differential Equations. II. Pitman Research Notes in Mathematics,

vol. 321. Longman Scientific & Technical, Harlow (1995)

http://dx.doi.org/10.1007/s10231-011-0229-7

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