International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 537890 | https://doi.org/10.5402/2011/537890

Alejandro Caicedo, Claudio Cuevas, Hernán R. Henríquez, "Asymptotic Periodicity for a Class of Partial Integrodifferential Equations", International Scholarly Research Notices, vol. 2011, Article ID 537890, 18 pages, 2011. https://doi.org/10.5402/2011/537890

Asymptotic Periodicity for a Class of Partial Integrodifferential Equations

Academic Editor: M. S. Moslehian
Received01 Mar 2011
Accepted27 Apr 2011
Published29 Jun 2011

Abstract

We study the existence of 𝑆-asymptotically 𝜔-periodic solutions for a class of abstract partial integro-differential equations and for a class of abstract partial integrodifferential equations with delay. Applications to integral equations arising in the study of heat conduction in materials with memory are shown.

1. Introduction

In this paper we study the existence of 𝑆-asymptotically 𝜔-periodic solutions for a class of abstract integrodifferential equations of the form𝑢(𝑡)=𝐴𝑢(𝑡)+𝑡0𝐵(𝑡𝑠)𝑢(𝑠)𝑑𝑠+𝑔(𝑡,𝑢(𝑡)),𝑡0,(1.1)𝑢(0)=𝑥0,(1.2) where 𝐴𝐷(𝐴)𝑋𝑋 and 𝐵(𝑡)𝐷(𝐵(𝑡))𝑋𝑋 for 𝑡0 are densely defined closed linear operators in a Banach space (𝑋,). We assume that 𝐷(𝐴)𝐷(𝐵(𝑡)) for every 𝑡0 and that 𝑔[0,)×𝑋𝑋 is a suitable function.

Due to its numerous applications in several branches of science, abstract integrodifferential equations of type (1.1) have received much attention in recent years. Properties of the solutions of (1.1) have been studied from different point of view. We refer the reader to ([13]) for well posedness to ([46] and references therein) for the existence of mild solutions; to [7] for the existence of asymptotically almost periodic and almost periodic solutions, and to [8] for the existence of asymptotically almost automorphic solutions.

The literature concerning 𝑆-asymptotically 𝜔-periodic functions with values in Banach spaces is very recent (see [916]). To the best of our knowledge, the study of the existence of 𝑆-asymptotically 𝜔-periodic solutions for equations of type (1.1) is a topic not yet considered in the literature. To fill this gap is the main motivation of this paper. To obtain our results, we use the theory of resolvent operators (see [13] for details). This theory is related to abstract integrodifferential equations in a similar manner as the semigroup theory is related to first-order linear abstract partial differential equations.

This paper has five sections. In the next section, we consider some definitions, technical aspects and basic properties related with 𝑆-asymptotically 𝜔-periodic functions and resolvent operators. In the third section, we establish very general results about the existence of 𝑆-asymptotically 𝜔-periodic mild solutions to the problem (1.1)-(1.2). In the fourth section, we present similar results for abstract partial integrodifferential equations with delay. Finally, as an application of our abstract results, in the fourth section, we establish conditions for the existence of 𝑆-asymptotically 𝜔-periodic mild solutions of a specific integral equation arising in the study of heat conduction in materials with memory.

2. Preliminaries

In this section, we introduce some notations and results to be used in this paper. Let (𝑍,𝑍) and (𝑊,𝑊) be Banach spaces. In this work 𝐶𝑏([0,);𝑍) denotes the Banach space consisting of all continuous and bounded functions from [0,) into 𝑍 endowed with the norm of the uniform convergence which is denoted by . As usual, 𝐶0([0,);𝑍) is the vector space of all functions 𝑧𝐶𝑏([0,);𝑍) such that lim𝑡𝑧(𝑡)=0. Also, we denote by 𝐶0([0,)×𝑊;𝑍) the vector space of all continuous functions 𝐻[0,)×𝑊𝑍 such that lim𝑡𝐻(𝑡,𝑤)=0 uniformly for 𝑤 in compact subsets of 𝑊. The notation (𝑍) stands for the Banach space of bounded linear operators from 𝑍 into 𝑍. Besides, we denote by 𝐵𝑟(𝑍) the closed ball with center at 0 and radius 𝑟. We begin by recalling the concept of 𝑆-asymptotically 𝜔-periodic functions. In the rest of this paper, 𝜔>0 is a fixed real number.

Definition 2.1 (see [14]). A function 𝑓𝐶𝑏([0,);𝑋) is called 𝑆-asymptotically 𝜔-periodic if lim𝑡(𝑓(𝑡+𝜔)𝑓(𝑡))=0.(2.1) In this case, we say that 𝜔 is an asymptotic period of 𝑓.

In this work, 𝑆𝐴𝑃𝜔(𝑋) represents the subspace of 𝐶𝑏([0,);𝑋) consisting of all 𝑆-asymptotically 𝜔-periodic functions. It is easy to see that (𝑆𝐴𝑃𝜔(𝑋),) is a Banach space.

Definition 2.2 (see [14]). A function 𝑓𝐶([0,)×𝑊;𝑍) is said to be uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets if for every bounded set 𝐾𝑊, the set {𝑓(𝑡,𝑥)𝑡0,𝑥𝐾} is bounded and lim𝑡(𝑓(𝑡+𝜔,𝑥)𝑓(𝑡,𝑥))=0, uniformly for 𝑥𝐾.

Definition 2.3 (see [14]). A function 𝑓𝐶([0,)×𝑊;𝑍) is said to be asymptotically uniformly continuous on bounded sets if for every 𝜖>0 and every bounded set 𝐾𝑊, there are constants 𝑇𝜖,𝐾0 and 𝛿𝜖,𝐾>0 such that 𝑓(𝑡,𝑥)𝑓(𝑡,𝑦)𝑍𝜖 for all 𝑡𝑇𝜖,𝐾 and every 𝑥,𝑦𝐾 with 𝑥𝑦𝑊𝛿𝜖,𝐾.

Lemma 2.4 (see [14]). Assume that 𝑓𝐶([0,)×𝑊;𝑍) is uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets and asymptotically uniformly continuous on bounded sets. If 𝑢𝑆𝐴𝑃𝜔(𝑊), then the function 𝑡𝑓(𝑡,𝑢(𝑡)) belongs to 𝑆𝐴𝑃𝜔(𝑍).

Let [0,)[1,) be a continuous nondecreasing function such that (𝑡) as 𝑡. Next, the notation 𝐶(𝑍) stands for the space 𝐶(𝑍)={𝑢𝐶([0,),𝑍)lim𝑡(𝑢(𝑡)/(𝑡))=0} endowed with the norm 𝑢=sup𝑡0(𝑢(𝑡)/(𝑡)).

Lemma 2.5 (see [17]). A set 𝐾𝐶(𝑍) is relatively compact in 𝐶(𝑍) if it verifies the following conditions. (c-1)For all 𝑏>0, the set 𝐾𝑏={𝑢|[0,𝑏]𝑢𝐾} is relatively compact in 𝐶([0,𝑏];𝑍).(c-2)lim𝑡(𝑢(𝑡)/(𝑡))=0 uniformly for 𝑢𝐾.

Now, we include some preliminaries concerning resolvent operators. In the following definition, [𝐷(𝐴)] represents the space 𝐷(𝐴) endowed with the graph norm given by 𝑥𝐴=𝑥+𝐴𝑥.

Definition 2.6 (see [18]). A family {𝑅(𝑡)𝑡0} of continuous linear operators on 𝑋 is called a resolvent operator for (1.1) if the following conditions are fulfilled. (R1)For each 𝑥𝑋, 𝑅(0)𝑥=𝑥 and 𝑅()𝑥𝐶([0,);𝑋). (R2)The map 𝑅[0,)([𝐷(𝐴)]) is strongly continuous. (R3)For each 𝑦𝐷(𝐴), the function 𝑡𝑅(𝑡)𝑦 is continuously differentiable and 𝑑𝑑𝑡𝑅(𝑡)𝑦=𝐴𝑅(𝑡)𝑦+𝑡0𝐵(𝑡𝑠)𝑅(𝑠)𝑦𝑑𝑠=𝑅(𝑡)𝐴𝑦+𝑡0𝑅(𝑡𝑠)𝐵(𝑠)𝑦𝑑𝑠,𝑡0.(2.2)

In what follows, we assume that there exists a resolvent operator for (1.1). The existence of solutions of the problem 𝑢(𝑡)=𝐴𝑢(𝑡)+𝑡0𝐵(𝑡𝑠)𝑢(𝑠)𝑑𝑠+𝑓(𝑡),𝑡0,𝑢(0)=𝑥0,(2.3) has been studied for many authors. Assuming that 𝑓[0,)𝑋 is locally integrable and following [2] we affirm that𝑢(𝑡)=𝑅(𝑡)𝑥0+𝑡0𝑅(𝑡𝑠)𝑓(𝑠)𝑑𝑠,𝑡0,(2.4) is the mild solution of the problem (2.3).

Motivated by this result, we adopt the following concept of solution.

Definition 2.7 (see [3]). A function 𝑢𝐶([0,);𝑋) is called a mild solution of (1.1)-(1.2) if 𝑢(𝑡)=𝑅(𝑡)𝑥0+𝑡0𝑅(𝑡𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠,𝑡0.(2.5)

To establish our results, we introduce the following condition.(H-1)There are positive constants 𝑀,𝜇 such that 𝑅(𝑡)𝑀𝑒𝜇𝑡 for all 𝑡0.

Remark 2.8. For additional details on resolvent operators and applications to partial integrodifferential equations we refer the reader to [2].

3. Existence Results

In this section, we consider the existence and uniqueness of 𝑆-asymptotically 𝜔-periodic mild solutions for the problem (1.1)-(1.2). We will assume that there exists a resolvent operator 𝑅() which satisfies the condition (H-1). Initially we establish a basic property.

Lemma 3.1. Let 𝑢𝑆𝐴𝑃𝜔(𝑋). Then 𝑣(𝑡)=𝑡0𝑅(𝑡𝑠)𝑢(𝑠)𝑑𝑠𝑆𝐴𝑃𝜔(𝑋).(3.1)

Proof. The estimate 𝑣(𝑀/𝜇)𝑢 shows that 𝑣𝐶𝑏([0,);𝑋). For 𝜖>0, we select 𝑇>0 such that 𝑢(𝑡+𝜔)𝑢(𝑡)𝜖 for all 𝑡𝑇 and 𝑇𝑒𝜇𝑠𝑑𝑠𝜖. We have the following decomposition: 𝑣(𝑡+𝜔)𝑣(𝑡)=𝜔0𝑅(𝑡+𝜔𝑠)𝑢(𝑠)𝑑𝑠+𝜔𝑡+𝜔𝑅(𝑡+𝜔𝑠)𝑢(𝑠)𝑑𝑠𝑡0=𝑅(𝑡𝑠)𝑢(𝑠)𝑑𝑠𝑡𝑡+𝜔𝑅(𝑠)𝑢(𝑡+𝜔𝑠)𝑑𝑠+𝑡0[]=𝑅(𝑡𝑠)𝑢(𝑠+𝜔)𝑢(𝑠)𝑑𝑠𝑡𝑡+𝜔𝑅(𝑠)𝑢(𝑡+𝜔𝑠)𝑑𝑠+𝑇0[]+𝑅(𝑡𝑠)𝑢(𝑠+𝜔)𝑢(𝑠)𝑑𝑠𝑡𝑇[]𝑅(𝑡𝑠)𝑢(𝑠+𝜔)𝑢(𝑠)𝑑𝑠.(3.2) Hence, for 𝑡2𝑇, we obtain 𝑣(𝑡+𝜔)𝑣(𝑡)𝑀𝑢𝑡𝑡+𝜔𝑒𝜇𝑠𝑑𝑠+2𝑀𝑢𝑡𝑡𝑇𝑒𝜇𝑠𝑑𝑠+𝜖𝑀0𝑡𝑇𝑒𝜇𝑠𝑑𝑠𝑀𝑢𝑡𝑡+𝜔𝑒𝜇𝑠𝑑𝑠+2𝑀𝑢𝑡𝑇𝑒𝜇𝑠𝑑𝑠+𝜖𝑀𝑡0𝑒𝜇𝑠𝑑𝑠3𝑀𝑢𝑇𝑒𝜇𝑠𝑀𝑑𝑠+𝜇𝜖=𝑀3𝑢+1𝜇𝜖,(3.3) which completes the proof.

Theorem 3.2. Assume that 𝑔[0,)×𝑋𝑋 is a uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets function that verifies the Lipschitz condition 𝑔(𝑡,𝑥)𝑔(𝑡,𝑦)𝐿𝑥𝑦,(3.4) for all 𝑥,𝑦𝑋 and every 𝑡0. If 𝐿𝑀/𝜇<1, then the problem (1.1)-(1.2) has a unique 𝑆-asymptotically 𝜔-periodic mild solution.

Proof. We define the map Γ on the space 𝑆𝐴𝑃𝜔(𝑋) by the expression Γ𝑢(𝑡)=𝑅(𝑡)𝑥0+𝑡0𝑅(𝑡𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠,𝑡0.(3.5) We next prove that Γ is a contraction from 𝑆𝐴𝑃𝜔(𝑋) into 𝑆𝐴𝑃𝜔(𝑋). Initially we show that Γ is a map 𝑆𝐴𝑃𝜔(𝑋)-valued. Let 𝑢𝑆𝐴𝑃𝜔(𝑋). We abbreviate the notation by writing 𝑣(𝑡)=𝑡0𝑅(𝑡𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠(3.6) Since 𝑅()𝑥0𝑆𝐴𝑃𝜔(𝑋), it remains to show that the function 𝑣() given by (3.6) belongs to 𝑆𝐴𝑃𝜔(𝑋). Considering that 𝑔 is asymptotically uniformly continuous on bounded sets and applying the Lemma 2.4, 𝑔(,𝑢())𝑆𝐴𝑃𝜔(𝑋). By Lemma 3.1, 𝑣𝑆𝐴𝑃𝜔(𝑋). On the other hand, if 𝑢1,𝑢2𝑆𝐴𝑃𝜔(𝑋) we have the estimate Γ𝑢1(𝑡)Γ𝑢2(𝑡)𝐿𝑀𝑡0𝑒𝜇(𝑡𝑠)𝑢1(𝑠)𝑢2(𝑠)𝑑𝑠𝐿𝑀𝜇𝑢1𝑢2.(3.7) The fixed point of Γ is the unique mild solution of (1.1)-(1.2). The proof is complete.

A similar result can be established when 𝑔 satisfies a local Lipschitz condition.

Theorem 3.3. Assume that 𝑔[0,)×𝑋𝑋 is a function uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets that satisfies the local Lipschitz condition 𝑔(𝑡,𝑥)𝑔(𝑡,𝑦)𝐿(𝑟)𝑥𝑦,(3.8) for all 𝑡0 and for all 𝑥,𝑦𝑋 with 𝑥𝑟 and 𝑦𝑟, where 𝐿[0,)[0,) is a nondecreasing function. Let 𝐶=sup𝑡0𝑔(𝑡,𝑅(𝑡)𝑥0). If there is 𝑟>0 such that 𝑀𝜇𝐿𝑥𝑟+𝑀0+𝐶𝑟<1,(3.9) then there is a unique 𝑆-asymptotically 𝜔-periodic mild solution of (1.1)-(1.2).

Proof. Let 𝑆𝐴𝑃0𝜔(𝑋)={𝑣𝑆𝐴𝑃𝜔(𝑋)𝑣(0)=0}. It is clear that 𝑆𝐴𝑃0𝜔(𝑋) is a closed vector subspace of 𝑆𝐴𝑃𝜔(𝑋). Let 𝐹𝑆𝐴𝑃0𝜔(𝑋)𝑆𝐴𝑃0𝜔(𝑋) be the map defined by 𝐹𝑣(𝑡)=𝑡0𝑅(𝑡𝑠)𝑔𝑠,𝑣(𝑠)+𝑅(𝑠)𝑥0𝑑𝑠.(3.10) Since 𝑔 satisfies (3.8) and we have that it is asymptotically uniformly continuous on bounded sets, we can argue as in the proof of the Theorem 3.2 to conclude that 𝐹 is well defined. For 𝑣1,𝑣2𝑆𝐴𝑃0𝜔(𝑋) with 𝑣1𝑟 and 𝑣2𝑟, we obtain that 𝐹𝑣1(𝑡)𝐹𝑣2(𝑡)𝑀𝑡0𝑒𝜇(𝑡𝑠)𝐿𝑥𝑟+𝑀0𝑣1(𝑠)𝑣2(𝑀𝑠)𝑑𝑠𝜇𝐿𝑥𝑟+𝑀0𝑣1𝑣2.(3.11) Hence 𝐹𝑣1𝐹𝑣2𝑀𝜇𝐿𝑥𝑟+𝑀0𝑣1𝑣2.(3.12) On the other hand, for 𝑣𝑆𝐴𝑃0𝜔(𝑋) with 𝑣𝑟, we get 𝐹𝑣𝐹𝑣𝐹(0)+𝐹(0)𝑀𝜇𝐿𝑥𝑟+𝑀0𝑣+𝐶𝑀𝜇.(3.13) Let 𝑟>0 be such that 𝑀𝜇𝐿𝑥𝑟+𝑀0𝑟+𝐶<𝑟.(3.14) From the above remarks it follows that 𝐹 is a contraction on 𝐵𝑟(𝑆𝐴𝑃0𝜔(𝑋)). Thus there is a unique fixed point 𝑣𝐵𝑟(𝑆𝐴𝑃0𝜔(𝑋)) of 𝐹. To finish the proof we note that 𝑢(𝑡)=𝑣(𝑡)+𝑅(𝑡)𝑥0 is the 𝑆-asymptotically 𝜔-periodic mild solution of (1.1)-(1.2).

We can also avoid the uniform Lipschitz conditions such as (3.4) or (3.8).

Theorem 3.4. Assume that 𝑔[0,)×𝑋𝑋 is a function uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets that verifies the Lipschitz condition 𝑔(𝑡,𝑥)𝑔(𝑡,𝑦)𝐿(𝑡)𝑥𝑦,(3.15) for all 𝑥,𝑦𝑋 and every 𝑡0, where the function 𝐿() is locally integrable on [0,). If Θ=𝑀sup𝑡0𝑡0𝑒𝜇(𝑡𝑠)𝐿(𝑠)𝑑𝑠<1,(3.16) then the problem (1.1)-(1.2) has a unique 𝑆-asymptotically 𝜔-periodic mild solution.

Proof. We define the map Γ on the space 𝑆𝐴𝑃𝜔(𝑋) by the expression (3.5). For 𝑢𝑆𝐴𝑃𝜔(𝑋), let 𝑣 be the function given by (3.6). Since the function 𝑢() is bounded, it follows from the Definition 2.2 that 𝐶=sup𝑠0𝑔(𝑠,𝑢(𝑠))<. Consequently, (𝑣𝑡)𝑡0𝑀𝑒𝜇(𝑡𝑠)(𝑔𝑠,𝑢(𝑠))𝑑𝑠𝐶𝑀𝜇,𝑡0,(3.17) which shows that Γ𝑢 is a bounded continuous function on [0,).
We next prove that Γ is a Θ-contraction from 𝑆𝐴𝑃𝜔(𝑋) into 𝑆𝐴𝑃𝜔(𝑋). Let 𝑢𝑆𝐴𝑃𝜔(𝑋). Next we set 𝐵={𝑢(𝑡)𝑡0}. We can write𝑣(𝑡+𝜔)𝑣(𝑡)=𝜔0𝑅(𝑡+𝜔𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠+𝜔𝑡+𝜔𝑅(𝑡+𝜔𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠𝑡0=𝑅(𝑡𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠𝑡0[]+𝑅(𝑡𝑠)𝑔(𝑠+𝜔,𝑢(𝑠+𝜔))𝑔(𝑠,𝑢(𝑠))𝑑𝑠𝑡𝑡+𝜔=𝑅(𝑠)𝑔(𝑡+𝜔𝑠,𝑢(𝑡+𝜔𝑠))𝑑𝑠𝑡0[]+𝑅(𝑠)𝑔(𝑡+𝜔𝑠,𝑢(𝑡+𝜔𝑠))𝑔(𝑡𝑠,𝑢(𝑡𝑠))𝑑𝑠𝑡𝑡+𝜔=𝑅(𝑠)𝑔(𝑡+𝜔𝑠,𝑢(𝑡+𝜔𝑠))𝑑𝑠𝑇0[]+𝑅(𝑠)𝑔(𝑡+𝜔𝑠,𝑢(𝑡+𝜔𝑠))𝑔(𝑡𝑠,𝑢(𝑡+𝜔𝑠))𝑑𝑠𝑇0[]+𝑅(𝑠)𝑔(𝑡𝑠,𝑢(𝑡+𝜔𝑠))𝑔(𝑡𝑠,𝑢(𝑡𝑠))𝑑𝑠𝑡𝑇[]+𝑅(𝑠)𝑔(𝑡+𝜔𝑠,𝑢(𝑡+𝜔𝑠))𝑔(𝑡𝑠,𝑢(𝑡𝑠))𝑑𝑠𝑡𝑡+𝜔𝑅(𝑠)𝑔(𝑡+𝜔𝑠,𝑢(𝑡+𝜔𝑠))𝑑𝑠=𝐼1+𝐼2+𝐼3+𝐼4.(3.18) Below we will estimate each one of the terms 𝐼𝑖, 1𝑖4, of the above expression separately. For 𝜀>0, let 𝜀=min{𝜇/𝑀,1/Θ}(𝜀/3). We choose 𝑇>0 such that the following conditions hold:(i)𝑒𝜇𝑇𝜀𝜇/9𝐶𝑀,(ii)𝑢(𝑡+𝜔)𝑢(𝑡)𝜀,(iii)𝑔(𝑡+𝜔,𝑥)𝑔(𝑡,𝑥)𝜀,for all 𝑡𝑇 and 𝑥𝐵. Let 𝑡2𝑇. Since 𝑡𝑠𝑡𝑇𝑇 for 0𝑠𝑇, we get 𝐼1𝑀𝜀𝑇0𝑒𝜇𝑠𝜀𝑑𝑠3,𝐼2𝜀𝑀𝑇0𝑒𝜇𝑠𝐿(𝑡𝑠)𝑑𝑠𝜀𝑀𝑡0𝑒𝜇𝑠𝐿(𝑡𝑠)𝑑𝑠𝜀𝐼Θ,3𝑀𝑡𝑇𝑒𝜇𝑠2𝐶𝑑𝑠2𝐶𝑀𝜇𝑒𝜇𝑇,𝐼4𝑀𝑡𝑡+𝜔𝑒𝜇𝑠𝐶𝑑𝑠𝐶𝑀𝜇𝑒𝜇𝑇.(3.19) Combining these estimates, we find 𝑣(𝑡+𝜔)𝑣(𝑡)𝜀,(3.20) for 𝑡2𝑇. Hence 𝑣𝑆𝐴𝑃𝜔(𝑋).
On the other hand, if 𝑢1,𝑢2𝑆𝐴𝑃𝜔(𝑋), and 𝑡0, we have Γ𝑢1(𝑡)Γ𝑢2(𝑡)𝑀𝑡0𝑒𝜇(𝑡𝑠)𝑢𝐿(𝑠)1(𝑠)𝑢2(𝑢𝑠)𝑑𝑠Θ1𝑢2.(3.21) The proof is complete.

As a consequence of the Lipchitz conditions (3.4), (3.8), or (3.15), our previous results show the existence of solutions of the problem (1.1)-(1.2) for functions 𝑔 such that 𝑔(𝑡,𝑥)/𝑥 is bounded as 𝑥. In what follows, we will show that using properly the stability of the resolvent operator we can establish existence results for functions 𝑔 with another type of asymptotic behavior at infinity. To establish our result, we consider functions 𝑔[0,)×𝑋𝑋 that satisfies the following boundedness condition.(H-2)There is a continuous nondecreasing function 𝑊[0,)[0,) such that 𝑔(𝑡,𝑥)𝑊(𝑥) for all 𝑡[0,) and 𝑥𝑋.

Theorem 3.5. Assume that 𝑔[0,)×𝑋𝑋 satisfies the hypotheses in the statement of Lemma 2.4 and the assumption (H-2). Suppose, in addition, that the following conditions are fulfilled. (a)For each 𝜈0, lim𝑡(1/(𝑡))𝑡0𝑒𝜇(𝑡𝑠)𝑊(𝜈(𝑠))𝑑𝑠=0, where is the function in Lemma 2.5. We set 𝜎𝜈(𝑡)=𝑅()𝑥0+𝑀𝑡0𝑒𝜇(𝑡𝑠)𝑊(𝜈(𝑠))𝑑𝑠,𝑡0,(3.22) and 𝜌(𝜈)=𝜎𝜈.(b)For each 𝜖>0, there is 𝛿>0 such that for every 𝑢,𝑣𝐶(𝑋), 𝑢𝑣𝛿 implies that 𝑡0𝑒𝜇(𝑡𝑠)(𝑔𝑠,𝑢(𝑠))𝑔(𝑠,𝑣(𝑠))𝑑𝑠𝜖,(3.23) for all 𝑡0.(c)For all 𝑎0 and 𝑟>0, the set {𝑔(𝑠,𝑥)0𝑠𝑎,𝑥𝑋,𝑥𝑟} is relatively compact in 𝑋.(d)liminf𝜉(𝜉/𝜌(𝜉))>1.Then the problem (1.1)-(1.2) has an 𝑆-asymptotically 𝜔-periodic mild solution.

Proof. Let Γ𝐶(𝑋)𝐶([0,);𝑋) be the map defined by the expression (3.5). Next, we prove that Γ has a fixed point in 𝑆𝐴𝑃𝜔(𝑋). We divide the proof in several steps.(i)For 𝑢𝐶(𝑋), we have that Γ𝑢(𝑡)𝑀(𝑡)𝑥(𝑡)0+𝑀(𝑡)𝑡0𝑒𝜇(𝑡𝑠)𝑊𝑢(𝑠)𝑑𝑠.(3.24) It follows from the condition (a) that Γ𝐶(𝑋)𝐶(𝑋).(ii)The map Γ is continuous from 𝐶(𝑋) into 𝐶(𝑋). In fact, for 𝜖>0, let 𝛿>0 be the constant involved in the condition (b). For 𝑢,𝑣𝐶(𝑋), 𝑢𝑣𝛿, taking into account that (𝑡)1, we get Γ𝑢(𝑡)Γ𝑣(𝑡)𝑀(𝑡)(𝑡)𝑡0𝑒𝜇(𝑡𝑠)(𝑔𝑠,𝑢(𝑠))𝑔(𝑠,𝑣(𝑠))𝑑𝑠𝑀𝜖,(3.25) which implies that Γ𝑢Γ𝑣𝑀𝜖. Since 𝜖>0 is arbitrary, this shows the assertion.(iii)We next show that Γ is completely continuous. Let 𝑉=Γ(𝐵𝑟(𝐶(𝑋))). We set 𝑣=Γ(𝑢) for 𝑢𝐵𝑟(𝐶(𝑋)). Initially, we prove that 𝑉(𝑡)={𝑣(𝑡)𝑣𝑉} is a relatively compact subset of 𝑋 for each 𝑡0. From the mean value theorem, 𝑣(𝑡)=𝑅(𝑡)𝑥0+𝑡0𝑅(𝑠)𝑔(𝑡𝑠,𝑢(𝑡𝑠))𝑑𝑠𝑅(𝑡)𝑥0+𝑡𝑐(𝐾),(3.26) where 𝑐(𝐾) denotes the convex hull of 𝐾 and 𝐾={𝑅(𝑠)𝑔(𝜉,𝑥)0𝑠𝑡,0𝜉𝑡,𝑥𝑟}. Combining the fact that 𝑅() is strongly continuous with the property (c), we infer that 𝐾 is a relatively compact set, and 𝑉(𝑡)𝑅(𝑡)𝑥0+𝑡𝑐(𝐾) is also a relatively compact set. Let 𝑏>0. We next show that the set 𝑉𝑏={𝑣|[0,𝑏]𝑣𝑉} is equicontinuous. In fact, for 𝑡0 fixed we can decompose 𝑣(𝑡+𝑠)𝑣(𝑡) as 𝑣(𝑡+𝑠)𝑣(𝑡)=(𝑅(𝑡+𝑠)𝑅(𝑡))𝑥0+𝑡𝑡+𝑠+𝑅(𝑡+𝑠𝜉)𝑔(𝜉,𝑢(𝜉))𝑑𝜉𝑡0(𝑅(𝜉+𝑠)𝑅(𝜉))𝑔(𝑡𝜉,𝑢(𝑡𝜉))𝑑𝜉.(3.27) For each 𝜖>0, we can choose 𝛿1>0 such that 𝑡𝑡+𝑠𝑅(𝑡+𝑠𝜉)𝑔(𝜉,𝑢(𝜉))𝑑𝜉𝑀𝑡𝑡+𝑠𝑒𝜇(𝑡+𝑠𝜉)𝜖𝑊(𝑟(𝜉))𝑑𝜉3,(3.28) for 𝑠𝛿1. Moreover, since {𝑔(𝑡𝜉,𝑢(𝑡𝜉))0𝜉𝑡,𝑢𝐵𝑟(𝐶(𝑋))} is a relatively compact set and 𝑅() is strongly continuous, we can choose 𝛿2>0 and 𝛿3>0 such that (𝑅(𝑡+𝑠)𝑅(𝑡))𝑥0𝜖/3, for 𝑠𝛿2 and (𝑅(𝜉+𝑠)𝑅(𝜉))𝑔(𝑡𝜉,𝑢(𝑡𝜉))𝜖/3(𝑏+1), for 𝑠𝛿3. Combining these estimates, we get 𝑣(𝑡+𝑠)𝑣(𝑡)𝜖 for |𝑠|min{𝛿1,𝛿2,𝛿3} with 𝑡+𝑠0 and for all 𝑢𝐵𝑟(𝐶(𝑋)).
Finally, applying condition (a), we can show that𝑣(𝑡)𝑀𝑥(𝑡)0+𝑀(𝑡)(𝑡)𝑡0𝑒𝜇(𝑡𝑠)𝑊(𝑟(𝑠))𝑑𝑠0,𝑡,(3.29) and this convergence is independent of 𝑢𝐵𝑟(𝐶(𝑋)). Hence 𝑉 satisfies the conditions (c-1) and (c-2) of Lemma 2.5, which completes the proof that 𝑉 is a relatively compact set in 𝐶(𝑋).(iv)If 𝑢𝜆() is a solution of the equation 𝑢𝜆=𝜆Γ(𝑢𝜆) for some 0<𝜆<1, we have the estimate 𝑢𝜆/𝜌(𝑢𝜆)1 and, combining this estimate with the condition (d), we conclude that the set 𝐾={𝑢𝜆𝑢𝜆=𝜆Γ(𝑢𝜆),𝜆(0,1)} is bounded.(v) Since 𝑆𝐴𝑃𝜔(𝑋)𝐶(𝑋), it follows from Lemmas 2.4 and 3.1 that Γ(𝑆𝐴𝑃𝜔(𝑋))𝑆𝐴𝑃𝜔(𝑋) and, consequently, we can consider Γ𝑆𝐴𝑃𝜔(𝑋)𝐶(𝑋)𝑆𝐴𝑃𝜔(𝑋)𝐶(𝑋), where 𝐵𝐶(𝑋) denotes the closure of a set 𝐵 in the space 𝐶(𝑋). We have that this map is completely continuous. Applying (iv) and the Leray-Schauder alternative theorem ([19, Theorem 6.5.4]), we deduce that the map Γ has a fixed point 𝑢𝑆𝐴𝑃𝜔(𝑋)𝐶(𝑋). Let (𝑢𝑛)𝑛 be a sequence in 𝑆𝐴𝑃𝜔(𝑋) such that 𝑢𝑛𝑢 in the norm of 𝐶(𝑋). For 𝜀>0, let 𝛿>0 be the constant in (b), there is 𝑛0 so that 𝑢𝑛𝑢𝛿, for all 𝑛𝑛0. We observe that for 𝑛𝑛0Γ𝑢𝑛Γ𝑢=sup𝑡0𝑡0𝑔𝑅(𝑡𝑠)𝑠,𝑢𝑛(𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠𝑀sup𝑡0𝑡0𝑒𝜇(𝑡𝑠)𝑔𝑠,𝑢𝑛(𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠𝜖.(3.30) Hence (Γ𝑢𝑛)𝑛 converges to Γ𝑢=𝑢 uniformly in [0,). This implies that 𝑢𝑆𝐴𝑃𝜔(𝑋) and completes the proof.

4. Existence Results for Functional Equations

In this section, we apply the results established in the Section 3 to study the existence of 𝑆-asymptotically 𝜔-periodic mild solutions for abstract functional integrodifferential equations. We keep the notations and the standing hypotheses considered in the Section 3. Initially we are concerned with the initial value problem 𝑢(𝑡)=𝐴𝑢(𝑡)+𝑡0𝐵(𝑡𝑠)𝑢(𝑠)𝑑𝑠+𝑔𝑡,𝑢𝑡𝑢,𝑡0,0=𝜑,(4.1) here 𝜑𝐶([𝑟,0];𝑋), the history of the function 𝑢() is given by 𝑢𝑡[𝑟,0]𝑋, 𝑢𝑡(𝜃)=𝑢(𝑡+𝜃), and 𝑔[0,)×𝐶([𝑟,0];𝑋)𝑋 is a continuous function. The following property is an immediate consequence of our definitions.

Lemma 4.1. Let 𝑢[𝑟,)𝑋 be a continuous function. If 𝑢|[0,)𝑆𝐴𝑃𝜔(𝑋). Then the function [0,)𝐶([𝑟,0];𝑋), 𝑡𝑢𝑡, is 𝑆-asymptotically 𝜔-periodic.

Definition 4.2. A function 𝑢𝐶([𝑟,);𝑋) is said to be a mild solution of (4.1) if 𝑢0=𝜑 and the integral equation 𝑢(𝑡)=𝑅(𝑡)𝜑(0)+𝑡0𝑅(𝑡𝑠)𝑔𝑠,𝑢𝑠𝑑𝑠,𝑡0,(4.2) is verified.

The following result is an immediate consequence of the Lemma 4.1 and the Theorem 3.2.

Corollary 4.3. Assume that 𝑔[0,)×𝐶([𝑟,0];𝑋)𝑋 is a uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets function that verifies the Lipschitz condition 𝑔𝑡,𝜓1𝑔𝑡,𝜓2𝜓𝐿1𝜓2,(4.3) for all 𝜓1,𝜓2𝐶([𝑟,0];𝑋) and every 𝑡0. If 𝐿𝑀/𝜇<1, then the problem (4.1) has a unique 𝑆-asymptotically 𝜔-periodic mild solution.

For this type of problems we can also establish existence results similar to Theorems 3.3 and 3.4. For the sake of brevity we omit the details. On the other hand, proceeding in a similar way, we can study existence of solutions for equations with infinite delay. Specifically, in what follows we will be concerned with the problem (4.1) when the history 𝑢𝑡 is given by 𝑢𝑡(,0]𝑋, 𝑢𝑡(𝜃)=𝑢(𝑡+𝜃). To model this problem we assume that 𝑢𝑡 belongs to some phase space which satisfies appropriate conditions. We will employ the axiomatic definition of the phase space introduced in [20]. Specifically, will be a linear space of functions mapping (,0] into 𝑋 endowed with a seminorm and verifying the following axioms. (A)If 𝑥(,𝜎+𝑎)𝑋, 𝑎>0,𝜎, is continuous on [𝜎,𝜎+𝑎) and 𝑥𝜎, then for every 𝑡[𝜎,𝜎+𝑎) the following hold: (i)𝑥𝑡 is in . (ii)𝑥(𝑡)𝐻𝑥𝑡. (iii)𝑥𝑡𝐾(𝑡𝜎)sup{𝑥(𝑠)𝜎𝑠𝑡}+𝑀(𝑡𝜎)𝑥𝜎,

where 𝐻>0 is a constant; 𝐾,𝑀[0,)[1,), 𝐾 is continuous, 𝑀 is locally bounded and 𝐻,𝐾,𝑀 are independent of 𝑥(). (A1)For the function 𝑥() in (A), the function 𝑡𝑥𝑡 is continuous from [𝜎,𝜎+𝑎) into . (B)The space is complete. (C-2)If (𝜓𝑛)𝑛 is a uniformly bounded sequence of continuous functions with compact support and 𝜓𝑛𝜓, 𝑛, in the compact-open topology, then 𝜓 and 𝜓𝑛𝜓0 as 𝑛.

We introduce the space 0={𝜓𝜓(0)=0} and the operator 𝑆(𝑡) given by []𝑆(𝑡)𝜓(𝜃)=𝜓(0),𝑡𝜃0,𝜓(𝑡+𝜃),<𝜃<𝑡.(4.4) It is well known that (𝑆(𝑡))𝑡0 is a 𝐶0-semigroup ([20]).

Definition 4.4. The phase space is said to be a fading memory space if 𝑆(𝑡)𝜓0 as 𝑡 for every 𝜓0.

Remark 4.5. Since verifies axiom (C-2), the space 𝐶𝑏((,0],𝑋) consisting of continuous and bounded functions 𝜓(,0]𝑋 is continuously included in . Thus, there exists a constant 𝑄0 such that 𝜓𝑄𝜓, for every 𝜓𝐶𝑏((,0],𝑋) ([20, Proposition 7.1.1]).
Moreover, if is a fading memory space, then 𝐾,𝑀 are bounded functions and we can choose 𝐾=𝑄 (see [20, Proposition 7.1.5]).

Example 4.6. The phase space 𝐶𝑟×𝐿𝑝(𝜌,𝑋)

Let 𝑟0,1𝑝< and let ̃𝜌(,𝑟] be a nonnegative measurable function which satisfies the conditions (g-5)-(g-6) in the terminology of [20]. Briefly, this means that ̃𝜌 is locally integrable and there exists a nonnegative locally bounded function 𝛾 on (,0] such that ̃𝜌(𝜉+𝜃)𝛾(𝜉)̃𝜌(𝜃), for all 𝜉0 and 𝜃(,𝑟)𝑁𝜉, where 𝑁𝜉(,𝑟) is a set whose Lebesgue measure zero.

The space =𝐶𝑟×𝐿𝑝(̃𝜌,𝑋) consists of all classes of functions 𝜑(,0]𝑋 such that 𝜑 is continuous on [𝑟,0], Lebesgue-measurable, and ̃𝜌𝜑𝑝 is Lebesgue integrable on (,𝑟). The seminorm in 𝐶𝑟×𝐿𝑝(̃𝜌,𝑋) is defined as follows:𝜑=sup{𝜑(𝜃)𝑟𝜃0}+𝑟̃𝜌(𝜃)𝜑(𝜃)𝑝𝑑𝜃1/𝑝.(4.5) The space =𝐶𝑟×𝐿𝑝(̃𝜌,𝑋) satisfies axioms (A), (A-1), and (B). Moreover, when 𝑟=0 and 𝑝=2, it is possible to choose 𝐻=1, 𝑀(𝑡)=𝛾(𝑡)1/2 and 𝐾(𝑡)=1+(0𝑡̃𝜌(𝜃)𝑑𝜃)1/2 for 𝑡0 (see [20, Theorem 1.3.8] for details). Note that if conditions (g-6)-(g-7) of [20] hold, then is a fading memory space ([20, Example 7.1.8]).

For fading memory spaces the following property holds ([15, Lemma 2.10]).

Lemma 4.7. Assume that is a fading memory space. Let 𝑢𝑋 be a function with 𝑢0 and 𝑢|[0,)𝑆𝐴𝑃𝜔(𝑋). Then the function 𝑡𝑢𝑡 belongs to 𝑆𝐴𝑃𝜔().

Next we assume that 𝜑 and that 𝑔[0,)×𝑋 is a continuous function.

Definition 4.8. A function 𝑢𝐶(;𝑋) is said to be a mild solution of (4.1) if 𝑢0=𝜑 and the integral equation 𝑢(𝑡)=𝑅(𝑡)𝜑(0)+𝑡0𝑅(𝑡𝑠)𝑔𝑠,𝑢𝑠𝑑𝑠,𝑡0,(4.6) is verified.

The following result is an immediate consequence of the Lemma 4.7 and the Theorem 3.2.

Corollary 4.9. Assume that 𝑔[0,)×𝑋 is a uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets function that verifies the Lipschitz condition 𝑔𝑡,𝜓1𝑔𝑡,𝜓2𝜓𝐿1𝜓2,(4.7) for all 𝜓1,𝜓2 and every 𝑡0. If 𝐿𝑀𝑄/𝜇<1, then the problem (4.1) has a unique 𝑆-asymptotically 𝜔-periodic mild solution.

As was mentioned for the problem (4.1) with finite delay, in this case we can also establish results similar to Theorems 3.3 and 3.4.

5. Applications to the Heat Conduction

Let Ω be a bounded open connected subset of 3 with 𝐶 boundary; let 𝛼 and 𝛽 be in 𝐶2([0,),) with 𝛼(0) and 𝛽(0) positive, and let a[0,) and 𝑏𝐻10(Ω)𝐿2(Ω) be functions.

Let us consider the following equation that arises in the study of heat conduction in materials with memory (see [2, 2123])𝜃(𝑡)+𝛽(0)𝜃(𝑡)=𝛼(0)Δ𝜃(𝑡)𝑡0𝛽(𝑡𝑠)𝜃(𝑠)𝑑𝑠+𝑡0𝛼(𝑡𝑠)Δ𝜃(𝑠)𝑑𝑠+𝑎(𝑡)𝑏(𝜃(𝑡)),(5.1) for 𝑡0, where Δ is the Laplacian on Ω.

We consider (5.1) with initial condition𝜃(0)=𝜃0,𝜃(0)=𝜂0.(5.2) To model this problem, we consider the space 𝑋=𝐻10(Ω)×𝐿2(Ω) and the linear operators 𝐴=0𝐼𝛼(0)Δ𝛽(0)𝐼(5.3) on the domain 𝐷(𝐴)=(𝐻2(Ω)𝐻10(Ω))×𝐻10(Ω), and 𝐵(𝑡)=𝐹(𝑡)𝐴, where 𝐹(𝑡)=[𝐹𝑖𝑗(𝑡)]𝑋𝑋, 𝑡0, is defined by 𝐹11(𝑡)=𝐹12(𝑡)=0, 𝐹21(𝑡)=𝛽(𝑡)𝐼+𝛽(0)(𝛼(𝑡)/𝛼(0))𝐼, 𝐹22(𝑡)=(𝛼(𝑡)/𝛼(0))𝐼.

Introducing the variable𝜃𝑢(𝑡)=𝜃(𝑡)(𝑡)𝑋,(5.4) and defining0𝑔(𝑡,𝑢(𝑡))=𝑎(𝑡)𝑏(𝜃(𝑡)),(5.5) the equation (5.1) with initial condition𝜃𝑢(0)=0𝜂0(5.6) takes the abstract form (1.1)-(1.2).

It follows from [24] that 𝐴 generates a 𝐶0-semigroup (𝑇(𝑡))𝑡0 such that 𝑇(𝑡)𝑀𝑒𝛾𝑡 for all 𝑡0 and some constants 𝑀,𝛾>0. Assume that 𝛼(𝑡)𝑒𝛾𝑡, 𝛼(𝑡)𝑒𝛾𝑡, 𝛽(𝑡)𝑒𝛾𝑡, and 𝛽(𝑡)𝑒𝛾𝑡 are bounded and uniformly continuous functions on [0,), and that for all 𝑡0, ||𝛽||||𝛼(𝑡)+max{𝛽(0),1}||(𝑡)𝛼(0)𝛾𝑒𝛾𝑡2𝑀,||𝛽||||𝛼(𝑡)+max{𝛽(0),1}||(𝑡)𝛼𝛾(0)2𝑒𝛾𝑡4𝑀2.(5.7)

Then, by Grimmer [2, Theorem 4.1], there is a resolvent operator 𝑅(𝑡) associated to the operators 𝐴 and 𝐵() and satisfying(𝑅𝑡)𝑀𝑒(𝛾/2)𝑡,𝑡0.(5.8) In addition, suppose that 𝑎𝑆𝐴𝑃𝜔() and 𝑏𝐻10(Ω)𝐿2(Ω) satisfies𝑏(𝜃1)𝑏(𝜃2)𝐿2(Ω)𝐿𝑏𝜃1𝜃2𝐻10(Ω),(5.9) for all 𝜃1,𝜃2𝐻10(Ω).

We claim that for each 𝜃0𝐻10(Ω) and 𝜂0𝐿2(Ω) the problem (5.1)-(5.2) satisfies the assumptions of the Theorem 3.2. In fact, the assumption (H-1) follows from (5.8). It is immediate also that 𝑔 satisfies the Lipschitz condition (3.4) with 𝐿=𝑎𝐿𝑏. In addition, estimates 𝑔(𝑡,𝑢)𝐻10(Ω)×𝐿2(Ω)𝑎𝐿𝑏𝑢𝐻10(Ω)×𝐿2(Ω)+𝑏(0)𝐿2(Ω),𝑔(𝑡+𝜔,𝑢)𝑔(𝑡,𝑢)𝐻10(Ω)×𝐿2(Ω)||||𝐿𝑎(𝑡+𝜔)𝑎(𝑡)𝑏𝑢𝐻10(Ω)×𝐿2(Ω)+𝑏(0)𝐿2(Ω),(5.10) which are verified for 𝑡0 and 𝑢𝐻10(Ω)×𝐿2(Ω) show that 𝑔[0,)×𝐻10(Ω)×𝐿2(Ω)𝐻10(Ω)×𝐿2(Ω) is a function uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets. As a consequence of Theorem 3.2, we obtain the following result.

Proposition 5.1. Under the above conditions, if (2𝑀/𝛾)𝑎𝐿𝑏<1, then the problem (5.1)-(5.2) has a unique 𝑆-asymptotically 𝜔-periodic mild solution.

To establish our next result, we assume the following conditions.(H-3)Let 𝑎𝑆𝐴𝑃𝜔(), 𝜏(0,1) and let 𝑏𝐻10(Ω)𝐿2(Ω) be a function that satisfies the Hölder type condition 𝑏(𝜃1)𝑏(𝜃2)𝐿2(Ω)𝐿𝑏𝜃1𝜃2𝜏𝐻10(Ω),(5.11) for all 𝜃1,𝜃2𝐻10(Ω).(H-4)There is a continuous nondecreasing function [0,)[1,) such that (𝑡) as 𝑡 and (1)lim𝑡(1/(𝑡))𝑡0𝑒(𝛾/2)(𝑡𝑠)(𝑠)𝜏𝑑𝑠=0, (2)sup𝑡0𝑡0𝑒(𝛾/2)(𝑡𝑠)|𝑎(𝑠)|(𝑠)𝜏𝑑𝑠<.

For a concrete example, take 0<𝜆<𝛾/2(1𝜏), 𝑎(𝑡)=𝑒𝜆𝑡, and (𝑡)=𝑒𝜆𝑡.

From Theorem 3.5, we can deduce the following result.

Proposition 5.2. Suppose that assumptions (H-3), (H-4) hold. Then the problem (5.1)-(5.2) has an 𝑆-asymptotically 𝜔-periodic mild solution.

Proof. We have the following estimates: 𝑔(𝑡,𝑢)𝐻10(Ω)×𝐿2(Ω)𝑎𝐿𝑏𝑢𝜏𝐻10(Ω)×𝐿2(Ω)+𝑏(0)𝐿2(Ω),(5.12)𝑔(𝑡+𝜔,𝑢)𝑔(𝑡,𝑢)𝐻10(Ω)×𝐿2(Ω)||||𝐿𝑎(𝑡+𝜔)𝑎(𝑡)𝑏𝑢𝜏𝐻10(Ω)×𝐿2(Ω)+𝑏(0)𝐿2(Ω),(5.13) for all 𝑡0 and 𝑢𝐻10(Ω)×𝐿2(Ω). It follows from (5.12)-(5.13) that the function 𝑔[0,)×𝐻10(Ω)×𝐿2(Ω)𝐻10(Ω)×𝐿2(Ω) is uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets. In addition, from the estimate 𝑔(𝑡,𝑢)𝑔(𝑡,𝑣)𝐻10(Ω)×𝐿2(Ω)||||𝐿𝑎(𝑡)𝑏𝑢𝑣𝜏𝐻10(Ω)×𝐿2(Ω),(5.14) for all 𝑡0 and 𝑢,𝑣𝐻10(Ω)×𝐿2(Ω), we obtain that 𝑔 is asymptotically uniformly continuous on bounded sets. By (5.12) we are led to define 𝑊(𝜉)=𝑎(𝐿𝑏𝜉𝜏+𝑏(0)𝐿2(Ω)). Consequently, the function 𝑔 satisfies the assumption (H-2). From (H-4), for 𝑢,𝑣𝐶(𝐻10(Ω)×𝐿2(Ω)) we can infer that 1(𝑡)𝑡0𝑒(𝛾/2)(𝑡𝑠)𝑊(𝜈(𝑠))𝑑𝑠𝑎𝐿𝑏𝜈𝜏1(𝑡)𝑡0𝑒(𝛾/2)(𝑡𝑠)(𝑠)𝜏𝑑𝑠+2𝑏(0)𝐿2(Ω)𝛾(𝑡)0,𝑡,sup𝑡0𝑡0𝑒(𝛾/2)(𝑡𝑠)𝑔(𝑠,𝑢(𝑠))𝑔(𝑠,𝑣(𝑠))𝐻10(Ω)×𝐿2(Ω)𝑑𝑠sup𝑡0𝑡0𝑒(𝛾/2)(𝑡𝑠)||𝑎||(𝑠)(𝑠)𝜏𝑑𝑠𝑢𝑣𝜏.(5.15) Therefore, conditions (a) and (b) of Theorem 3.5 are satisfied. A straightforward computation shows that liminf𝜉(𝜉/𝜌(𝜉))>1, where 𝜌(𝜉)=sup𝑡01𝜃(𝑡)𝑅()0𝜂0+𝑀𝑡0𝑒(𝛾/2)(𝑡𝑠)𝑊(𝜉(𝑠))𝑑𝑠.(5.16) Finally, since Ω is bounded set with 𝐶 boundary, Rellich-Kondrachov's Theorem [25, Theorem IX.16] leads to the conclusion that {𝑎(𝑠)𝑏(𝜃)0𝑠𝑠0,𝜃𝐻10(Ω),𝜃𝐻10(Ω)𝑟}, 𝑠0>0, is relatively compact in 𝐿2(Ω). Hence, condition (c) holds. Using the Theorem 3.5, we conclude that the problem (5.1)-(5.2) has an 𝑆-asymptotically 𝜔-periodic mild solution.

To complete these applications we consider (5.1) with a heat source depending on the past of the temperature. This is a usual situation in control systems. To simplify our exposition, we consider only a system which presents a finite transmission delay time 𝑟>0. In this case the equation is 𝜃(𝑡)+𝛽(0)𝜃(𝑡)=𝛼(0)Δ𝜃(𝑡)𝑡0𝛽(𝑡𝑠)𝜃(𝑠)𝑑𝑠+𝑡0𝛼(𝑡𝑠)Δ𝜃(𝑠)𝑑𝑠+𝑎(𝑡)𝑏(𝜃(𝑡𝑟)),(5.17) for 𝑡0. Using the previous development, we model this problem in the space 𝑋=𝐻10(Ω)×𝐿2(Ω), and we consider 𝑢𝑡𝐶([𝑟,0];𝑋) for 𝑡0. To be consistent with our model, we study the equation (5.17) with initial condition𝑢0=𝜑𝜓,(5.18) where 𝜑𝐶([𝑟,0];𝐻10(Ω)) and 𝜓𝐶([𝑟,0];𝐿2(Ω)). The function 𝑔[0,)×𝐶([𝑟,0];𝑋)𝑋 is given by𝑔𝜑𝑡,1,𝜓1=0𝜑𝑎(𝑡)𝑏1(𝑟).(5.19) Applying now the Corollary 4.3, we obtain the following result.

Proposition 5.3. Under the above conditions, if (2𝑀/𝛾)𝑎𝐿𝑏<1, then the problem (5.17)-(5.18) has a unique 𝑆-asymptotically 𝜔-periodic mild solution.

Acknowledgments

Claudio Cuevas is partially supported by CNPQ/Brazil under Grant 300365/2008-0. Hernán R. Henríquez is supported in part by CONICYT under Grant FONDECYT no. 1090009.

References

  1. W. Desch, R. Grimmer, and W. Schappacher, “Well-posedness and wave propagation for a class of integrodifferential equations in Banach space,” Journal of Differential Equations, vol. 74, no. 2, pp. 391–411, 1988. View at: Publisher Site | Google Scholar
  2. R. C. Grimmer, “Resolvent operators for integral equations in a Banach space,” Transactions of the American Mathematical Society, vol. 273, no. 1, pp. 333–349, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. R. Grimmer and J. Prüss, “On linear Volterra equations in Banach spaces,” Computers & Mathematics with Applications, vol. 11, no. 1–3, pp. 189–205, 1985. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. S. Aizicovici and M. McKibben, “Existence results for a class of abstract nonlocal Cauchy problems,” Nonlinear Analysis, vol. 39, no. 5, pp. 649–668, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. S. Aizicovici and H. Lee, “Nonlinear nonlocal Cauchy problems in Banach spaces,” Applied Mathematics Letters, vol. 18, no. 4, pp. 401–407, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. J. Liang, J. van Casteren, and T. J. Xiao, “Nonlocal Cauchy problems for semilinear evolution equations,” Nonlinear Analysis, vol. 50, no. 2, pp. 173–189, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. E. Hernández and J. P. C. dos Santos, “Asymptotically almost periodic and almost periodic solutions for a class of partial integro-differential equations,” Electronic Journal of Differential Equations, vol. 38, pp. 1–8, 2006. View at: Google Scholar | Zentralblatt MATH
  8. H. S. Ding, T. J. Xiao, and J. Liang, “Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 141–151, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. B. de Andrade and C. Cuevas, “S-asymptotically ω-periodic and asymptotically ω-periodic solutions to semilinear Cauchy problems with non dense domain,” Nonlinear Analysis, vol. 72, no. 6, pp. 3190–3208, 2010. View at: Publisher Site | Google Scholar
  10. C. Cuevas, M. Pierri, and A. Sepulveda, “Weighted S-asymptotically ω-periodic solutions of a class of fractional differential equations,” Advances in Difference Equations, vol. 2011, Article ID 584874, 13 pages, 2011. View at: Publisher Site | Google Scholar
  11. C. Cuevas and C. Lizama, “S-asymptotically ω-periodic solutions for semilinear Volterra equations,” Mathematical Methods in the Applied Sciences, vol. 33, no. 13, pp. 1628–1636, 2010. View at: Google Scholar
  12. C. Cuevas and J. C. de Souza, “Existence of S-asymptotically ω-periodic solutions for fractional order functional integro-differential equations with infinite delay,” Nonlinear Analysis, vol. 72, no. 3-4, pp. 1683–1689, 2010. View at: Publisher Site | Google Scholar
  13. C. Cuevas and J. C. de Souza, “S-asymptotically ω-periodic solutions of semilinear fractional integrodifferential equations,” Applied Mathematics Letters, vol. 22, no. 6, pp. 865–870, 2009. View at: Publisher Site | Google Scholar
  14. H. R. Henríquez, M. Pierri, and P. Táboas, “On S-asymptotically ω-periodic functions on Banach spaces and applications,” Journal of Mathematical Analysis and Applications, vol. 343, no. 2, pp. 1119–1130, 2008. View at: Publisher Site | Google Scholar
  15. H. R. Henríquez, M. Pierri, and P. Táboas, “Existence of S-asymptotically ω-periodic for abstract neutral equations,” Bulletin of the Australian Mathematical Society, vol. 78, no. 3, pp. 365–382, 2008. View at: Publisher Site | Google Scholar
  16. S. Nicola and M. Pierri, “A note on S-asymptotically ω-periodic functions,” Nonlinear Analysis, vol. 10, no. 5, pp. 2937–2938, 2009. View at: Publisher Site | Google Scholar
  17. C. Cuevas and H. R. Henríquez, “Solutions of second order abstract retarded functional differential equations on the line,” Journal of Nonlinear and Convex Analysis. In press. View at: Google Scholar
  18. W. Desch, R. Grimmer, and W. Schappacher, “Some considerations for linear integro-differential equations,” Journal of Mathematical Analysis and Applications, vol. 104, no. 1, pp. 219–234, 1984. View at: Publisher Site | Google Scholar
  19. A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, NY, USA, 2003.
  20. Y. Hino, S. Murakami, and T. Naito, Functional-Differential Equations with Infinite Delay, vol. 1473 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1991.
  21. B. D. Coleman and M. E. Gurtin, “Equipresence and constitutive equations for rigid heat conductors,” Zeitschrift fur Angewandte Mathematik und Physik, vol. 18, pp. 199–208, 1967. View at: Google Scholar
  22. 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  23. R. K. Miller, “An integro-differential equation for rigid heat conductors with memory,” Journal of Mathematical Analysis and Applications, vol. 66, no. 2, pp. 313–332, 1978. View at: Publisher Site | Google Scholar
  24. C. Chen, “Control and stabilization for the wave equation in a bounded domain,” SIAM Journal on Control and Optimization, vol. 17, no. 1, pp. 66–81, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  25. H. Brézis, Analyse Fonctionnelle, Masson, Paris, Farnce, 1993.

Copyright © 2011 Alejandro Caicedo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views802
Downloads402
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.