s-asymptotically ω-periodic solutions for semilinear volterra equations

Research Article

Received 20 August 2009 Published online 18 February 2010 in Wiley InterScience

(www.interscience.wiley.com) DOI: 10.1002/mma.1284MOS subject classification: primary: 47 A 60; secondary: 34 G 20; 26 A 33

S-asymptotically x-periodic solutions forsemilinear Volterra equations

Claudio Cuevasa∗† and Carlos Lizamab

We study S-asymptotically x-periodic mild solutions of the semilinear Volterra equation u′(t)= (a∗Au)(t)+f (t, u(t)),considered in a Banach space X, where A is the generator of an (exponentially) stable resolvent family. In partic-ular, we extend the recent results for semilinear fractional integro-differential equations considered in (Appl. Math.Lett. 2009; 22:865–870) and for semilinear Cauchy problems of first order given in (J. Math. Anal. Appl. 2008; 343(2):1119–1130). Applications to integral equations arising in viscoelasticity theory are shown. Copyright © 2010 John Wiley& Sons, Ltd.

Keywords: S-asymptotically �-periodic function; stable resolvent families; semilinear Volterra equations

1. Introduction

The study of the existence of almost periodic, asymptotically almost periodic, almost automorphic, asymptotically almost automor-phic, compact almost automorphic and pseudo almost periodic solutions is one of the most interesting topics in the qualitativetheory of differential equations both due to its mathematical interest as well as due to their applications in physics and mathematicalbiology, among other areas. Some recent contributions on the existence of these types of solutions for abstract differential equationshave been made. Related with this subject, we refer the reader to the extensive bibliography in [1--15].

A vector-valued function f ∈Cb([0,∞), X) is called S-asymptotically �-periodic (see Henríquez, et al. [16]) if there exists �>0 suchthat

limt→∞(f (t+�)−f (t))=0. (1)

In [16] is shown the surprising fact that the property (1) does not characterize asymptotically �-periodic functions, that is, boundedand continuous functions, which admits the decomposition f =g+�, where g is �-periodic and limt→∞ �(t)=0.

On the other hand, the literature concerning the qualitative behavior (1) for evolution equations is incipient and limited essentiallyto the study of the existence of solutions of ordinary differential equations described on finite-dimensional spaces (see [17--21]).Only recently, has been developed a theory of S-asymptotically �-periodic functions with values in Banach spaces, and stated theexistence of S-asymptotically �-periodic functions for the first-order semilinear Cauchy problem [16] (see also [22]). These resultswere now used in [23, 24] (respectively, [25]) to establish S-asymptotically �-solutions of semilinear fractional (respectively, abstractpartial neutral) integro-differential equations.

In this work we study the existence and uniqueness of S-asymptotically �-periodic solutions of the semilinear Volterra equation

v′(t) =∫ t

0a(t−s)Av(s) ds+f (t, v(t)), t�0, (2)

v(0) = u0 ∈X, (3)

where A : D(A)⊂X →X is the generator of a bounded analytic semigroup or, more generally, the generator of a resolvent family ona complex Banach space X, a∈L1

loc(R+) and f : [0,∞)×X →X is a continuous function satisfying suitable conditions.

aUniversidade Federal de Pernambuco, Departamento de Matemática, Av. Prof. Luiz Freire, S/N, Recife-PE, CEP 50540-740, BrazilbUniversidad de Santiago de Chile, Departamento de Matemática, Facultad de Ciencias, Casilla 307-Correo 2, Santiago, Chile∗Correspondence to: Claudio Cuevas, Universidade Federal de Pernambuco, Departamento de Matemática, Av. Prof. Luiz Freire, S/N, Recife-PE, CEP 50540-740,

Brazil.†E-mail: [email protected]

Contract/grant sponsor: CNPQ/Brazil; contract/grant number: 300365/2008-0Contract/grant sponsor: Laboratorio de Analisis Estocástico; contract/grant number: PBCT ACT-13

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Copyright © 2010 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2010, 33 1628–1636

C. CUEVAS AND C. LIZAMA

Owing to their applications in several fields of science (see [26--28]), type (2) equations as well as their numerical treatment areattracting increasing interest. The properties of the solutions of (2) have been extensively studied in the previous years. In theinfinite-dimensional setting, we refer to the classical monograph [29] and references therein.

The results in the present paper are, on one hand, an extension of the results in [16, 23] and, on the other hand, a contributionto the study of the qualitative properties for the Volterra Equation (2), which are new even in the scalar case.

In particular, when we consider viscoelastic material behavior modeled by semilinear Volterra-type Equations (2), it describes themotion of a viscoelastic body with fading memory [30]. The memory term, represented by the convolution term in the equation,expresses the fact that the stress at any instant t depends on the past history of strains which the material has undergone fromtime 0 to t. Our results show that the solutions are asymptotically �-periodic as time goes to infinity. This is established under mildconditions on the relaxation kernel a(t) and the operator A. The reader will perceive throughout this work that our abstract resultsand techniques suggest that it is possible to develop a nice connection with other applications.

This work is organized as follows: In Section 2, we state and review the main definitions and results of other sources to be used inthe paper. In Section 3 we study the linear case by means of an integrated version of (2). As a consequence of the theory of linearevolution equations for Volterra equations [29] we derive our main result (Theorem 3.4) which states maximal regularity under theconditions that a(t) is 1-regular, A is the generator of a strongly integrable resolvent, and the initial condition belongs to the domainof the operator A (generally unbounded). In Section 4, we study the existence and uniqueness of S-asymptotically �-periodic mildsolutions of the semilinear problem (2)–(3). To achieve our results, we require an f (t, x) Lipschitz-type conditions (Theorems 4.2, 4.6and 4.7) or compactness (Theorem 4.9). In passing, we give easy-to-check conditions solely in terms of a(t), f , and A (being now thegenerator of a bounded analytic C0-semigroup) to guarantee that problem (2)–(3) has a unique S-asymptotically �-periodic mildsolution (Corollary 4.4). To illustrate our main results, at the end of this paper we examine sufficient conditions for the existenceand uniqueness of S-asymptotically �-periodic mild solution to a specific integral equation arising in viscoelasticity theory.

2. Preliminaries

We recall that the Laplace transform of a function f ∈L1loc(R+, X) is given by

L(f )(�) := f (�) :=∫ ∞

0e−�tf (t) dt, Re�>�,

where the integral is absolutely convergent for Re�>�. Furthermore, we denote by B(X) the space of bounded linear operatorsfrom X into X endowed with the norm of operators, and the notation �(A) stands for the resolvent set of A.

In order to give an operator theoretical approach to Equation (2) we recall the following definition (cf. [29, 31]).

Definition 2.1Let A be a closed and linear operator with domain D(A) defined on a Banach space X . We call A the generator of a solution operator(or resolvent family) if there exists �∈R and a strongly continuous function S :R+ →B(X), such that {1 / a(�) : Re�>�}⊂�(A) and

1

�a(�)

(1

a(�)−A

)−1x =

∫ ∞

0e−�tS(t)x dt, Re�>�, x ∈X.

In this case, S(t) is called the solution operator generated by A.

In the scalar case there is a large bibliography that studies the concept of resolvent, we refer to the monograph by Gripenberget al. [32], and references therein. We emphasize the fact that because of the uniqueness of the Laplace transform, in the casea(t)≡1 the family S(t) corresponds to a C0-semigroup, whereas in case a(t)= t a solution operator corresponds to the concept ofcosine family, see e.g. [33, 34]. We note that solution operators, as well as resolvent families, are a particular case of (a, k)-regularizedfamilies introduced in [31]. According to [31] a solution operator S(t) corresponds to a (1, a)-regularized family.

In this work Cb([0,∞), X) denotes the space consisting of the continuous and bounded functions from [0,∞) into X , endowed withthe norm of the uniform convergence, which is denoted by ||•||∞, and the notation SAP�(X) stands for the subspace of Cb([0,∞), X)consisting of the S-asymptotically �-periodic functions. We note that SAP�(X) is a Banach space (see [16], Proposition 3.5).

Definition 2.2 (Henríquez et al. [16])A continuous function f : [0,∞)×X →X is said to be uniformly S-asymptotically �-periodic on bounded sets if for every boundedsubset K of X , the set {f (t, x) : t�0, x ∈K} is bounded and limt→∞(f (t, x)−f (t+�, x))=0 uniformly in x ∈K .

Definition 2.3 (Henríquez et al. [16])A continuous function f : [0,∞)×X →X is said to be asymptotically uniformly continuous on bounded sets if for every �>0 and everybounded subset K of X , there exist L�,K�0 and ��,K>0 such that ||f (t, x)−f (t, y)||��, for all t�L�,K and all x, y ∈K with ||x−y||��,K .

Lemma 2.4 (Henríquez et al. [16])Let f : [0,∞)×X →X be a uniformly S-asymptotically �-periodic on bounded sets and asymptotically uniformly continuouson bounded sets function and, let u : [0,∞)→X be an S-asymptotically �-periodic function. Then the function v(t)= f (t, u(t)) isS-asymptotically �-periodic.


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Definition 2.5 (Prüss [29])A strongly measurable family of operators {T(t)}t�0 ⊂B(X) is called uniformly integrable (or strongly integrable) if

∫ ∞0 ||T(t)||dt<∞.

We also recall the following concept, studied for resolvent families in [35]:

Definition 2.6A strongly continuous family of operators {T(t)}t�0 ⊂B(X) is called uniformly stable if ||T(t)||→0 as t →∞.

Definition 2.7 (Henríquez et al. [16])A strongly continuous family of operators {T(t)}t�0 ⊂B(X) is said to be strongly S-asymptotically �-periodic if there is �>0 suchthat T(·)x is S-asymptotically �-periodic for all x ∈X .

Definition 2.8 (Prüss [29])Let a∈L1

loc(R+) be of subexponential growth and k ∈N. The function a(t) is called k-regular if there is a constant C>0 such that

|�nan(�)|�C|a(�)| for all Re(�)>0, 0�n�k.

3. The linear case

In this section we consider the linear version for Equation (2), that is

v′(t) =∫ t

0b(t−s)Av(s) ds+f (t), t�0, (4)

v(0) = u0 ∈X. (5)

or, equivalently, the integrated form

v(t)=∫ t

0a(t−s)Av(s) ds+

∫ t

0f (s) ds+u0, t�0, (6)

where u0 ∈X and a(t)=∫ t0 b(s) ds. Recall that a function v ∈C(R+; X) is called a strong solution of (6) on R+ if v ∈C(R+; D(A)) and (6)

holds on R+. If A generates a resolvent family S(t), the variation of parameters formula allows us to write the solution of problem (6)as (cf. [29, Proposition 1.2])

v(t)=S(t)u0 +∫ t

0S(t−s)f (s) ds, t�0.

Moreover, v(t) is a strong solution of (6) if u0 ∈D(A), see [29, Proposition 1.2].

Lemma 3.1Suppose that A generates a uniformly integrable resolvent family S(t) and let f ∈SAP�(X). Then

∫ t

0S(t−s)f (s) ds∈SAP�(X).

ProofLet v(t) :=∫ t

0 S(t−s)f (s) ds. We have

v(t+�)−v(t) =∫ t+�

0S(t+�−s)f (s) ds−

∫ t

0S(t−s)f (s) ds

=∫ �

0S(t+�−s)f (s) ds+

∫ t+�

�S(t+�−s)f (s) ds−

∫ t

0S(t−s)f (s) ds

=∫ �

0S(t+�−s)f (s) ds+

∫ t

0S(t−s)f (s+�) ds−

∫ t

0S(t−s)f (s) ds

For each �>0, there is a positive constant L� such that ||f (t+�)−f (t)||��, for every t�L�. Under these conditions, for t�L�, we canestimate

||v(t+�)−v(t)|| �∫ �

0||S(t+�−s)f (s)||ds+

∫ L�

0||S(t−s)[f (s+�)−f (s)]||ds+

∫ t

L�

||S(t−s)[f (s+�)−f (s)]||ds

� ||f ||∞∫ �

0||S(t+�−s)||ds+2||f ||∞

∫ L�

0||S(t−s)||ds+�

∫ t

L�

||S(t−s)||ds

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= ||f ||∞∫ t+�

t||S(s)||ds+2||f ||∞

∫ t

t−L�

||S(s)||ds+�∫ t−L�

0||S(s)||ds

� ||f ||∞∫ ∞

t||S(s)||ds+2||f ||∞

∫ ∞

t−L�

||S(s)||ds+�∫ ∞

0||S(s)||ds

which permit to infer that v(t+�)−v(t)→0 as t →∞. �

A direct consequence of the previous lemma is the following theorem.

Theorem 3.2Suppose that A generates a uniformly integrable resolvent S(t) which is S-asymptotically �-periodic, and let f ∈SAP�(X). Then foreach u0 ∈D(A) Equation (6) admits a unique S-asymptotically �-periodic strong solution.

We recall the following definition:

Definition 3.3We say that A generates a parabolic resolvent family if the following conditions are satisfied.

(P1) a(�) =0, 1 / a(�)∈�(A) for all Re�>0.(P2) There exists a constant M�1 such that

‖(�−�a(�)A)−1‖� M

|� | for all Re�>0. (7)

If A generates an analytic resolvent which is bounded on some sector �(0,) then A generates a parabolic resolvent family. Theconverse is not true. A standard situation leading with generators of parabolic resolvents is the following: Let a(t) of subexponentialgrowth of positive type, and let A generate a bounded analytic C0-semigroup in X , then A generate a parabolic resolvent (cf. [29,Proposition 3.1]). Based on the above definitions, we are ready to state the following result.

Theorem 3.4Suppose that a(t) is 1-regular and A generates a parabolic and uniformly integrable resolvent family {S(t)}t�0, and let f ∈SAP�(X).Then for each u0 ∈D(A) Equation (6) admits a unique S-asymptotically �-periodic strong solution.

ProofAs a(t) is 1-regular and A generates a parabolic and uniformly integrable resolvent family, we obtain by the main result in [35] that{S(t)}t�0 is uniformly stable. In particular, {S(t)}t�0 is S-asymptotically �-periodic for any �>0. The result is now a consequence ofTheorem 3.2. �

4. The semilinear case

In this section we consider the existence and uniqueness of S-asymptotically �-periodic mild solutions of the problem (2)–(3). Theconsiderations in the linear case motivates the following definition.

Definition 4.1A function u∈Cb([0,∞), X) is said to be S-asymptotically �-periodic mild solution of problem (2)–(3) if u(·) is S-asymptotically�-periodic and

u(t)=S(t)u0 +∫ t

0S(t−s)f (s, u(s)) ds for all t�0. (8)

In the case that the resolvent family is differentiable, we can give an alternative definition of mild solution of problem (2) (withoutinitial condition) as follows:

u(t)= f (t, u(t))+∫ t

0S(t−s)f (s, u(s)) ds for all t�0. (9)

Note that in case of a C0-semigroup T(t), the last definition is generally used when the semigroup is, in addition, analytic whereT(t)=AT(t). Moreover, we observe that if A generates a parabolic resolvent family S(t) (see Definition 3.3) and the kernel a(t) is2-regular (see Definition 2.8), then S(t) is differentiable (cf. [29, Theorem 3.1]).

Theorem 4.2Suppose A generates a uniformly integrable solution operator S(t), which is in addition strongly S-asymptotically �-periodic. Letf : [0,∞)×X →X be a continuous function such that f (·, 0) is integrable in [0,∞) and there exists a continuous integrable functionL : [0,∞)→R such that

‖f (t, x)−f (t, y)‖�L(t)‖x−y‖ for all x, y ∈X, t�0. (10)

Then the problem (2)–(3) has a unique S-asymptotically �-periodic mild solution.


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ProofWe define the operator � on the space SAP�(X) by

(�u)(t) :=S(t)u0 +∫ t

0S(t−s)f (s, u(s)) ds=S(t)u0 +v(t). (11)

We show initially that �u∈SAP�(X). In fact, we observe that by hypothesis S(·)u0 ∈SAP�(X). It follows from the inequality||f (s, u(s))||�L(s)||u(s)||+||f (s, 0)||, that the function s→ f (s, u(s)) is integrable in [0,∞). Hence, we obtain that v(t)∈Cb([0,∞), X) and∫ t

a S(t−s)f (s, u(s)) ds→0, as a→∞, uniformly for t�a. In addition, for fixed a, the set {f (s, u(s)) : 0�s�a} is compact, which impliesthat S(t+�)f (s, u(s))−S(t)f (s, u(s))→0, as t →∞, uniformly in s∈ [0, a]. Combining these properties with the decomposition

v(t+�)−v(t)=∫ a

0[S(t+�−s)−S(t−s)]f (s, u(s)) ds+

∫ t+�

aS(t+�−s)f (s, u(s)) ds−

∫ t

aS(t−s)f (s, u(s)) ds.

Hence v(t+�)−v(t)→0 as t →∞. Furthermore, for u1, u2 ∈SAP�(X) the inequality ||(�u1)(t)−(�u2)(t)||�C∫ t

0 L(s)||u1(s)−u2(s)||dsshows that � : SAP�(X)→SAP�(X) is a continuous map. On the other hand, we define the linear map B : Cb([0,∞))→Cb([0,∞)) by(Bg)(t)=C

∫ t0 L(s)g(s) ds, for t�0. It is clear that B is continuous. Moreover, B is completely continuous. To establish this assertion, for

each �>0, we take a�0 such that C∫ ∞

a L(s) ds�� and, for each g∈Cb([0,∞)) with ||g||∞�1, we define the functions

�1(g)(t) :=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

C

∫ t

0L(s)g(s) ds, 0�t�a,

C

∫ a

0L(s)g(s) ds, t�a,

�2(g)(t) :=

⎧⎪⎨⎪⎩

0, 0�t�a,

C

∫ t

0L(s)g(s) ds, t�a.

It follows from the Ascoli–Arzela Theorem that the set R0 :={�1(g) : ||g||∞�1} is relatively compact. As Bg(t)=�1(g)(t)+�2(g)(t) forall t�0, we can infer that {Bg : ||g||∞�1}⊂R0 +{ :∈Cb([0,∞)), ||||∞��}, which shows that the set {Bg : ||g||∞�1} is relativelycompact and, in turn that B is completely continuous. Moreover, since the point spectrum �p(B)={0}, the spectral radius of B isequal to zero. Let m : Cb([0,∞), X)→Cb([0,∞)) be the map defined by m(u)(t)=sup0�s�t ||u(s)||. It is not difficult to verify that themaps �, B, and m satisfy all the conditions of Theorem 1 in [36] which implies that � has a unique fixed point u. �

Theorem 4.2 together with the argument used in the proof of Theorem 3.4 permits to infer the following consequence.

Corollary 4.3Suppose a(t) is 1-regular and A generates a parabolic and uniformly integrable resolvent family S(t). Let f : [0,∞)×X →X be acontinuous function such that f (·, 0) is integrable in [0,∞) and there exists a continuous integrable function L : [0,∞)→R such that



The following result will be of more practical use.

Corollary 4.4Suppose a(t) is 1-regular, of subexponential growth, of positive type, completely monotonic and satisfies a(∞)= limt→∞ a(t)>0.Assume that A generates a bounded analytic C0-semigroup and 0∈�(A). Let f : [0,∞)×X →X be a continuous function such thatf (·, 0) is integrable in [0,∞) and there exists a continuous integrable function L : [0,∞)→R such that



ProofUnder the stated hypothesis, it follows from [29, Corollary 10.1] that A generates an uniformly integrable analytic resolvent. HenceA generates a uniformly integrable parabolic resolvent, which is then also uniformly stable. The result follows. �

Example 4.5Let

a(t)= t−1

�(), t>0.

Then a(t) satisfies the hypotheses in the previous corollary if and only if �1. Note that we essentially recover [23, Theorem 3.2],and [16, Theorem 4.3] in case =1. Also observe that in the above cited works, the examples given are always corresponding togenerators of analytic semigroups with 0∈�(A).

The case where L in (10) is a constant, is analyzed in the following two results.

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Theorem 4.6Suppose A generates a uniformly integrable resolvent family S(t) which is strongly S-asymptotically �-periodic. Let f : [0,∞)×X →Xbe a function uniformly S-asymptotically �-periodic on bounded sets and satisfies the Lipschitz condition

‖f (t, x)−f (t, y)‖�L‖x−y‖ for all x, y ∈X, t�0. (14)

If L<||S||−11 , then the problem (2)–(3) has a unique S-asymptotically �-periodic mild solution.

ProofProceeding as in the proof of Theorem 4.2, we define the map � on the space SAP�(X) by the expression (11). We next prove that� is a contraction from SAP�(X) into SAP�(X). We show initially that � is SAP�(X)-valued. Let u be in SAP�(X). By hypothesis thefunction S(·)u0 ∈SAP�(X) and the problem is reduced to show that the function v given by (11) belongs to SAP�(X). In view of thefact that f is asymptotically uniformly continuous on bounded sets and applying Lemma 2.4 and 3.1 we get that �u∈SAP�(X). Onthe other hand, for u1, u2 ∈SAP�(X), we have the inequality

||(�u1)(t)−(�u2)(t)||�L||S||1||u1 −u2||∞,

which proves that � is a contraction. Now, the assertion is consequence of the contraction mapping principle. The proof is complete.�

Theorem 4.7Suppose A generates a uniformly bounded and integrable resolvent family S(t) such that limt→∞(S(t)x−S(t+n�)x)=0 uniformly inn∈N, for all x ∈X . Let Condition (14) hold and assume that f (·, 0) is a bounded function and limt→∞(f (t, x)−f (t+n�, x))=0 uniformlyin x ∈K and n∈N, for every bounded subset K of X . If L<||S||−1

1 , then the problem (2)–(3) has a unique asymptotically �-periodicmild solution.

ProofLet S(X) be the space consisting of functions u∈Cb([0,∞), X) such that limt→∞ (u(t)−u(t+n�))=0 uniformly in n∈N. It is easy tosee that S(X) is a closed subspace of Cb([0,∞), X) (see [16]). Let u be in S(X). It follows from our assumptions that for each �>0, thereis a positive constant L� such that ||f (t+n�, u(t+n�))−f (t, u(t))||�� for every t�L� and every n∈N. We consider the map � on S(X)by the expression (11). We have the following estimate: ||�u||∞�||S||∞||u0||+[L||u||∞+||f (·, 0)||∞]||S||1. Let v(t) :=∫ t

0 S(t−s)f (s) ds,proceeding as the proof of Lemma 3.1, we have for t�L�

||v(t+n�)−v(t)||�[L||u||∞+||f (·, 0)||∞]

(∫ ∞

t||S(s)||ds+2

∫ ∞

t−L�

||S(s)||ds

)+�||S||1.

We get that � is S(X)-valued. Therefore, the fixed point of � belongs to S(X) and the assertion is a consequence of Corollary 3.1in [16]. The proof is complete. �

We next study the existence of S-asymptotically �-periodic mild solutions of Equation (2) when the function f is not Lipschitzcontinuous. We will consider function f that satisfies the following boundedness condition.

(B) There exists a continuous nondecreasing function W :R+ := [0,∞)→R+ such that ‖f (t, x)‖�W(‖x‖) for all t ∈R+ and x ∈X .Let h :R+ →R be a continuous function such that h(t)�1 for all t ∈R+, and h(t)→∞ as t →∞. We consider the space

Ch(X)={

u∈C(R+, X) : limt→∞

u(t)

h(t)=0

}

endowed with the norm

‖u‖h =supt�0

‖u(t)‖h(t)

.

We will use the following result (cf. [Lemma 2.2, (Cuevas and Henríquez, submitted)]).

Lemma 4.8A subset K ⊆Ch(X) is a relatively compact set if it verifies the following conditions:

(c-1) The set Kb ={u|[0,b] : u∈K} is relatively compact in C([0, b], X) for all b�0.(c-2) limt→∞ ||u(t)|| / h(t)=0 uniformly for all u∈K .

Theorem 4.9Assume that A generates a uniformly bounded and integrable resolvent family {S(t)}t�0 which is in addition strongly S-asymptotically�-periodic. Let f :R+×X →X be a uniformly S-asymptotically �-periodic on bounded sets and asymptotically uniformly continuouson bounded sets function and that satisfies assumption (B), and the following conditions:

(a) For each C�0, the function t �→∫ t0 ||S(t−s)||W(Ch(s)) ds is included in BC(R+). We set

(C)=∥∥∥∥||S(·)||||u0||+

∫ ·

0||S(·−s)||W(Ch(s)) ds

∥∥∥∥h

.


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(b) For each �>0 there is �>0 such that for every u, v ∈Ch(X), ‖v−u‖h�� implies that∫ t

0 ||S(t−s)||‖f (s, v(s))−f (s, u(s))‖ds�� for allt�0.

(c) lim inf�→∞ � / (�)>1.(d) For all a, b∈R, a<b, and r>0, the set {f (s, h(s)x) : a�s�b, x ∈X,‖x‖�r} is relatively compact in X .

Then the problem (2)–(3) has an S-asymptotically �-periodic mild solution.

ProofWe define the operator � on Ch(X) as in (11). We show that � has a fixed point in SAP�(X). We divide the proof into several steps.

(i) For u∈Ch(X), we have that

‖�u(t)‖ � ||S(t)||||u0||+∫ t

0||S(t−s)||W(‖u(s)‖) ds

� ||S||∞||u0||+∫ t

0||S(t−s)||W(‖u‖hh(s)) ds. (15)

It follows from condition (a) that � : Ch(X)→Ch(X).(ii) The map � is continuous. In fact, for �>0, we take � involved in condition (b). If u, v ∈Ch(X) and ‖u−v‖h��, then

‖�u(t)−�v(t)‖�∫ t

0||S(t−s)||‖f (s, u(s))−f (s, v(s))‖ds��,

which shows the assertion.(iii) We show that � is completely continuous. To abbreviate the text, we set Br(Z) for the closed ball with center at 0 and radius

r in a space Z. Let V =�(Br(Ch(X))) and v =�(u) for u∈Br(Ch(X)).Initially, we will prove that Vb(t) is a relatively compact subset of X for each t ∈ [0, b]. We get

v(t)=S(t)u0 +∫ t

0S(s)f (t−s, u(t−s)) ds∈S(t)u0 +tc(K),

where c(K) denotes the convex hull of K and K ={S(s)f (�, h(�)x) : 0�s�t, 0��t,‖x‖�r}. Using that S(·) is strongly continuous and theproperty (d) of f , we infer that K is a relatively compact set, and V(t)⊆S(t)u0 +tc(K), which establishes our assertion.

We next show that the set Vb is equicontinuous. In fact, we can decompose

v(t+s)−v(t)= (S(t+s)−S(t))u0 +∫ s

0S(�)f (t+s−�, u(t+s−�))d�+

∫ t

0(S(�+s)−S(�))f (t−�, u(t−�)) d�.

For each �>0, we can choose �1>0 such that∥∥∥∥∫ s

0S(�)f (t+s−�, u(t+s−�)) d�

∥∥∥∥�∫ s

0||S(�)||W(rh(t+s−�)) d�� / 2,

for s��1. Moreover, since {f (t−�, u(t−�)) : 0��t, u∈Br(Ch(X))} is a relatively compact set and S(·) is strongly continuous, wecan choose �2>0 and �3>0 such that ||(S(t+s)−S(t))u0||<� / 4, for s��2 and ‖(S(�+s)−S(�))f (t−�, u(t−�))‖�� / 2(t+1) for s��3.Combining these estimates, we get ‖v(t+s)−v(t)‖�� for s enough small and independent of u∈Br(Ch(X)).

Finally, applying condition (a), we can show that

v(t)

h(t)� 1

h(t)

[||S(t)||||u0||+

∫ t

0||S(t−s)||W(rh(s)) ds

]→0, t →∞,

and this convergence is independent of u∈Br(Ch(X)).Hence V satisfies conditions (c-1), (c-2) of Lemma 4.8, which completes the proof that V is a relatively compact set in Ch(X).(iv) If u�(·) is a solution of equation u� =��(u�) for some 0<�<1, from the estimate

‖u�(t)‖ = �

∥∥∥∥S(t)u0 +∫ t

0S(t−s)f (s, u�(s)) ds

∥∥∥∥

� ||S(t)||||u0||+∫ t

0||S(t−s)||W(‖u�‖hh(s)) ds

� (‖u�‖h)h(t),

we get

‖u�‖h

(‖u�‖h)�1

and, combining with condition (c), we conclude that the set {u� : u� =��(u�),�∈ (0, 1)} is bounded.

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(v) It follows from Lemmas 2.4 and 3.1 that �(SAP�(X))⊆SAP�(X) and, consequently, we can consider � : SAP�(X)→SAP�(X). Usingproperties (i)–(iii) we have that this map is completely continuous. Applying Leray–Schauder alternative theorem [37, Theorem6.5.4], we infer that � has a fixed point u∈SAP�(X). Let (un)n be a sequence in SAP�(X) that converges to u. We see that (�un)nconverges to �u=u uniformly in [0,∞). This implies that u∈SAP�(X), and completes the proof. �

Corollary 4.10Assume that A generates a uniformly bounded and integrable resolvent family {S(t)}t�0 which is in addition strongly S-asymptotically�-periodic. Let f :R+×X →X be a uniformly S-asymptotically �-periodic on bounded sets that satisfy the Hölder-type condition

‖f (t, y)−f (t, x)‖�C1‖y−x‖ , 0< <1, (16)

for all x, y ∈X , t�0, where C1>0 is a constant. Moreover, assume the following conditions:

(i) f (t, 0)=q.(ii) supt�0

∫ t0 ||S(t−s)||h(s) ds<+∞.

(iii) For all a, b�0, a<b, and r>0, the set {f (s, h(s)x) : a�s�b, x ∈X,‖x‖�r} is relatively compact in X .

Then the problem (2)–(3) has an S-asymptotically �-periodic mild solution.

To finish this work, we examine the existence and uniqueness of S-asymptotically �-periodic mild solution to the integro-differentialequation

ut(t, x) =∫ t

0da(s)uxx(t−s, x)+f (t, u(t, x)), t�0, x ∈ [0, 1],

u(t, 0) = u(t, 1)=0,

u(0, x) = u0(x), x ∈ [0, 1].

(17)

Here a :R→R+ is a function of bounded variation on each compact interval J= [0, T](T>0), with a(0)=0. The above initial-boundaryproblem is a typical example of one-dimensional problems in viscoelasticity, such as simple shearing motions, simple tension, torsionof a rod; see [29, Section 5.4].

To obtain a formulation as an abstract evolutionary integral equation like (2)–(3), we choose X =L2[0, 1] and define an operatorA by means of Au(x)=uxx(x) with domain D(A)={u∈X : uxx ∈X, u(0)=u(1)=0}. It is well known that A generates a bounded analyticsemigroup with 0∈�(A). Then Corollary 4.4 implies the following result.

Corollary 4.11Suppose a(t) is 1-regular, of subexponential growth, of positive type, completely monotonic and satisfies a(∞)= limt→∞ a(t)>0. Letf : [0,∞)×X →X be a continuous function such that f (·, 0) is integrable in [0,∞) and there exists a continuous integrable functionL : [0,∞)→R such that


Then the problem (17) has a unique S-asymptotically �-periodic mild solution.

Acknowledgements

The first author is partially supported by CNPQ/Brazil under Grant 300365/2008-0; the second author is partially supported byLaboratorio de Analisis Estocástico, project PBCT ACT-13.

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s-asymptotically ω-periodic solutions for semilinear volterra equations

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