Table of Contents 
ISRN Mathematical Analysis
VolumeΒ 2011, Article IDΒ 537890, 18 pages
http://dx.doi.org/10.5402/2011/537890
Research Article

Asymptotic Periodicity for a Class of Partial Integrodifferential Equations

Alejandro Caicedo,1Β Claudio Cuevas,1Β and HernΓ‘n R. HenrΓ­quez2

1Departamento de MatemΓ‘tica, Universidade Federal de Pernambuco, Recife 50540-740, PE, Brazil
2Departamento de MatemΓ‘tica, Universidad de Santiago, USACH, Casilla 307, Correo 2, Santiago, Chile

Received 1 March 2011; Accepted 27 April 2011

Academic Editor: M. S.Β Moslehian

Copyright Β© 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.

Abstract

We study the existence of 𝑆-asymptotically πœ”-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 πœ”-periodic solutions for a class of abstract integrodifferential equations of the formπ‘’ξ…ž(ξ€œπ‘‘)=𝐴𝑒(𝑑)+𝑑0𝐡(π‘‘βˆ’π‘ )𝑒(𝑠)𝑑𝑠+𝑔(𝑑,𝑒(𝑑)),𝑑β‰₯0,(1.1)𝑒(0)=π‘₯0,(1.2) where 𝐴∢𝐷(𝐴)βŠ†π‘‹β†’π‘‹ and 𝐡(𝑑)∢𝐷(𝐡(𝑑))βŠ†π‘‹β†’π‘‹ for 𝑑β‰₯0 are densely defined closed linear operators in a Banach space (𝑋,β€–β‹…β€–). We assume that 𝐷(𝐴)βŠ‚π·(𝐡(𝑑)) for every 𝑑β‰₯0 and that π‘”βˆΆ[0,∞)×𝑋→𝑋 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 ([1–3]) for well posedness to ([4–6] 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 πœ”-periodic functions with values in Banach spaces is very recent (see [9–16]). To the best of our knowledge, the study of the existence of 𝑆-asymptotically πœ”-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 [1–3] 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 πœ”-periodic functions and resolvent operators. In the third section, we establish very general results about the existence of 𝑆-asymptotically πœ”-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 πœ”-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 (π‘Š,β€–β‹…β€–π‘Š) be Banach spaces. In this work 𝐢𝑏([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 β€–β‹…β€–βˆž. As usual, 𝐢0([0,∞);𝑍) is the vector space of all functions π‘§βˆˆπΆπ‘([0,∞);𝑍) such that limπ‘‘β†’βˆžπ‘§(𝑑)=0. Also, we denote by 𝐢0([0,∞)Γ—π‘Š;𝑍) the vector space of all continuous functions 𝐻∢[0,∞)Γ—π‘Šβ†’π‘ such that limπ‘‘β†’βˆžπ»(𝑑,𝑀)=0 uniformly for 𝑀 in compact subsets of π‘Š. The notation β„’(𝑍) stands for the Banach space of bounded linear operators from 𝑍 into 𝑍. Besides, we denote by π΅π‘Ÿ(𝑍) the closed ball with center at 0 and radius π‘Ÿ. We begin by recalling the concept of 𝑆-asymptotically πœ”-periodic functions. In the rest of this paper, πœ”>0 is a fixed real number.

Definition 2.1 (see [14]). A function π‘“βˆˆπΆπ‘([0,∞);𝑋) is called 𝑆-asymptotically πœ”-periodic if limπ‘‘β†’βˆž(𝑓(𝑑+πœ”)βˆ’π‘“(𝑑))=0.(2.1) In this case, we say that πœ” is an asymptotic period of 𝑓.

In this work, π‘†π΄π‘ƒπœ”(𝑋) represents the subspace of 𝐢𝑏([0,∞);𝑋) consisting of all 𝑆-asymptotically πœ”-periodic functions. It is easy to see that (π‘†π΄π‘ƒπœ”(𝑋),β€–β‹…β€–βˆž) is a Banach space.

Definition 2.2 (see [14]). A function π‘“βˆˆπΆ([0,∞)Γ—π‘Š;𝑍) is said to be uniformly 𝑆-asymptotically πœ”-periodic on bounded sets if for every bounded set πΎβŠ‚π‘Š, the set {𝑓(𝑑,π‘₯)βˆΆπ‘‘β‰₯0,π‘₯∈𝐾} is bounded and limπ‘‘β†’βˆž(𝑓(𝑑+πœ”,π‘₯)βˆ’π‘“(𝑑,π‘₯))=0, uniformly for π‘₯∈𝐾.

Definition 2.3 (see [14]). A function π‘“βˆˆπΆ([0,∞)Γ—π‘Š;𝑍) is said to be asymptotically uniformly continuous on bounded sets if for every πœ–>0 and every bounded set πΎβŠ‚π‘Š, there are constants π‘‡πœ–,𝐾β‰₯0 and π›Ώπœ–,𝐾>0 such that ‖𝑓(𝑑,π‘₯)βˆ’π‘“(𝑑,𝑦)β€–π‘β‰€πœ– for all 𝑑β‰₯π‘‡πœ–,𝐾 and every π‘₯,π‘¦βˆˆπΎ with β€–π‘₯βˆ’π‘¦β€–π‘Šβ‰€π›Ώπœ–,𝐾.

Lemma 2.4 (see [14]). Assume that π‘“βˆˆπΆ([0,∞)Γ—π‘Š;𝑍) is uniformly 𝑆-asymptotically πœ”-periodic on bounded sets and asymptotically uniformly continuous on bounded sets. If π‘’βˆˆπ‘†π΄π‘ƒπœ”(π‘Š), then the function 𝑑↦𝑓(𝑑,𝑒(𝑑)) belongs to π‘†π΄π‘ƒπœ”(𝑍).

Let β„ŽβˆΆ[0,∞)β†’[1,∞) be a continuous nondecreasing function such that β„Ž(𝑑)β†’βˆž as π‘‘β†’βˆž. Next, the notation πΆβ„Ž(𝑍) stands for the space πΆβ„Ž(𝑍)={π‘’βˆˆπΆ([0,∞),𝑍)∢limπ‘‘β†’βˆž(𝑒(𝑑)/β„Ž(𝑑))=0} endowed with the norm β€–π‘’β€–β„Ž=sup𝑑β‰₯0(‖𝑒(𝑑)β€–/β„Ž(𝑑)).

Lemma 2.5 (see [17]). A set πΎβŠ†πΆβ„Ž(𝑍) is relatively compact in πΆβ„Ž(𝑍) if it verifies the following conditions. (c-1)For all 𝑏>0, the set 𝐾𝑏={𝑒|[0,𝑏]βˆΆπ‘’βˆˆπΎ} is relatively compact in 𝐢([0,𝑏];𝑍).(c-2)limπ‘‘β†’βˆž(‖𝑒(𝑑)β€–/β„Ž(𝑑))=0 uniformly for π‘’βˆˆπΎ.

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 β€–π‘₯‖𝐴=β€–π‘₯β€–+‖𝐴π‘₯β€–.

Definition 2.6 (see [18]). A family {𝑅(𝑑)βˆΆπ‘‘β‰₯0} of continuous linear operators on 𝑋 is called a resolvent operator for (1.1) if the following conditions are fulfilled. (R1)For each π‘₯βˆˆπ‘‹, 𝑅(0)π‘₯=π‘₯ and 𝑅(β‹…)π‘₯∈𝐢([0,∞);𝑋). (R2)The map π‘…βˆΆ[0,∞)β†’β„’([𝐷(𝐴)]) is strongly continuous. (R3)For each π‘¦βˆˆπ·(𝐴), the function 𝑑↦𝑅(𝑑)𝑦 is continuously differentiable and π‘‘ξ€œπ‘‘π‘‘π‘…(𝑑)𝑦=𝐴𝑅(𝑑)𝑦+𝑑0ξ€œπ΅(π‘‘βˆ’π‘ )𝑅(𝑠)𝑦𝑑𝑠=𝑅(𝑑)𝐴𝑦+𝑑0𝑅(π‘‘βˆ’π‘ )𝐡(𝑠)𝑦𝑑𝑠,𝑑β‰₯0.(2.2)

In what follows, we assume that there exists a resolvent operator for (1.1). The existence of solutions of the problem π‘’ξ…ž(ξ€œπ‘‘)=𝐴𝑒(𝑑)+𝑑0𝐡(π‘‘βˆ’π‘ )𝑒(𝑠)𝑑𝑠+𝑓(𝑑),𝑑β‰₯0,𝑒(0)=π‘₯0,(2.3) has been studied for many authors. Assuming that π‘“βˆΆ[0,∞)→𝑋 is locally integrable and following [2] we affirm that𝑒(𝑑)=𝑅(𝑑)π‘₯0+ξ€œπ‘‘0𝑅(π‘‘βˆ’π‘ )𝑓(𝑠)𝑑𝑠,𝑑β‰₯0,(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 π‘’βˆˆπΆ([0,∞);𝑋) is called a mild solution of (1.1)-(1.2) if 𝑒(𝑑)=𝑅(𝑑)π‘₯0+ξ€œπ‘‘0𝑅(π‘‘βˆ’π‘ )𝑔(𝑠,𝑒(𝑠))𝑑𝑠,𝑑β‰₯0.(2.5)

To establish our results, we introduce the following condition.(H-1)There are positive constants 𝑀,πœ‡ such that ‖𝑅(𝑑)β€–β‰€π‘€π‘’βˆ’πœ‡π‘‘ for all 𝑑β‰₯0.

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 πœ”-periodic mild solutions for the problem (1.1)-(1.2). We will assume that there exists a resolvent operator 𝑅(β‹…) which satisfies the condition (H-1). Initially we establish a basic property.

Lemma 3.1. Let π‘’βˆˆπ‘†π΄π‘ƒπœ”(𝑋). Then ξ€œπ‘£(𝑑)=𝑑0𝑅(π‘‘βˆ’π‘ )𝑒(𝑠)π‘‘π‘ βˆˆπ‘†π΄π‘ƒπœ”(𝑋).(3.1)

Proof. The estimate β€–π‘£β€–βˆžβ‰€(𝑀/πœ‡)β€–π‘’β€–βˆž shows that π‘£βˆˆπΆπ‘([0,∞);𝑋). For πœ–>0, we select 𝑇>0 such that ‖𝑒(𝑑+πœ”)βˆ’π‘’(𝑑)β€–β‰€πœ– for all 𝑑β‰₯𝑇 and βˆ«βˆžπ‘‡π‘’βˆ’πœ‡π‘ π‘‘π‘ β‰€πœ–. We have the following decomposition: ξ€œπ‘£(𝑑+πœ”)βˆ’π‘£(𝑑)=πœ”0ξ€œπ‘…(𝑑+πœ”βˆ’π‘ )𝑒(𝑠)𝑑𝑠+πœ”π‘‘+πœ”ξ€œπ‘…(𝑑+πœ”βˆ’π‘ )𝑒(𝑠)π‘‘π‘ βˆ’π‘‘0=ξ€œπ‘…(π‘‘βˆ’π‘ )𝑒(𝑠)𝑑𝑠𝑑𝑑+πœ”ξ€œπ‘…(𝑠)𝑒(𝑑+πœ”βˆ’π‘ )𝑑𝑠+𝑑0[]=ξ€œπ‘…(π‘‘βˆ’π‘ )𝑒(𝑠+πœ”)βˆ’π‘’(𝑠)𝑑𝑠𝑑𝑑+πœ”ξ€œπ‘…(𝑠)𝑒(𝑑+πœ”βˆ’π‘ )𝑑𝑠+𝑇0[]+ξ€œπ‘…(π‘‘βˆ’π‘ )𝑒(𝑠+πœ”)βˆ’π‘’(𝑠)𝑑𝑠𝑑𝑇[]𝑅(π‘‘βˆ’π‘ )𝑒(𝑠+πœ”)βˆ’π‘’(𝑠)𝑑𝑠.(3.2) Hence, for 𝑑β‰₯2𝑇, we obtain ‖𝑣(𝑑+πœ”)βˆ’π‘£(𝑑)β€–β‰€π‘€β€–π‘’β€–βˆžξ€œπ‘‘π‘‘+πœ”π‘’βˆ’πœ‡π‘ π‘‘π‘ +2π‘€β€–π‘’β€–βˆžξ€œπ‘‘π‘‘βˆ’π‘‡π‘’βˆ’πœ‡π‘ ξ€œπ‘‘π‘ +πœ–π‘€0π‘‘βˆ’π‘‡π‘’βˆ’πœ‡π‘ π‘‘π‘ β‰€π‘€β€–π‘’β€–βˆžξ€œπ‘‘π‘‘+πœ”π‘’βˆ’πœ‡π‘ π‘‘π‘ +2π‘€β€–π‘’β€–βˆžξ€œπ‘‘π‘‡π‘’βˆ’πœ‡π‘ ξ€œπ‘‘π‘ +πœ–π‘€π‘‘0π‘’βˆ’πœ‡π‘ π‘‘π‘ β‰€3π‘€β€–π‘’β€–βˆžξ€œβˆžπ‘‡π‘’βˆ’πœ‡π‘ π‘€π‘‘π‘ +πœ‡ξ‚΅πœ–=𝑀3β€–π‘’β€–βˆž+1πœ‡ξ‚Άπœ–,(3.3) which completes the proof.

Theorem 3.2. Assume that π‘”βˆΆ[0,∞)×𝑋→𝑋 is a uniformly 𝑆-asymptotically πœ”-periodic on bounded sets function that verifies the Lipschitz condition ‖𝑔(𝑑,π‘₯)βˆ’π‘”(𝑑,𝑦)‖≀𝐿‖π‘₯βˆ’π‘¦β€–,(3.4) for all π‘₯,π‘¦βˆˆπ‘‹ and every 𝑑β‰₯0. If 𝐿𝑀/πœ‡<1, then the problem (1.1)-(1.2) has a unique 𝑆-asymptotically πœ”-periodic mild solution.

Proof. We define the map Ξ“ on the space π‘†π΄π‘ƒπœ”(𝑋) by the expression Γ𝑒(𝑑)=𝑅(𝑑)π‘₯0+ξ€œπ‘‘0𝑅(π‘‘βˆ’π‘ )𝑔(𝑠,𝑒(𝑠))𝑑𝑠,𝑑β‰₯0.(3.5) We next prove that Ξ“ is a contraction from π‘†π΄π‘ƒπœ”(𝑋) into π‘†π΄π‘ƒπœ”(𝑋). Initially we show that Ξ“ is a map π‘†π΄π‘ƒπœ”(𝑋)-valued. Let π‘’βˆˆπ‘†π΄π‘ƒπœ”(𝑋). We abbreviate the notation by writing ξ€œπ‘£(𝑑)=𝑑0𝑅(π‘‘βˆ’π‘ )𝑔(𝑠,𝑒(𝑠))𝑑𝑠(3.6) Since 𝑅(β‹…)π‘₯0βˆˆπ‘†π΄π‘ƒπœ”(𝑋), it remains to show that the function 𝑣(β‹…) given by (3.6) belongs to π‘†π΄π‘ƒπœ”(𝑋). Considering that 𝑔 is asymptotically uniformly continuous on bounded sets and applying the Lemma 2.4, 𝑔(β‹…,𝑒(β‹…))βˆˆπ‘†π΄π‘ƒπœ”(𝑋). By Lemma 3.1, π‘£βˆˆπ‘†π΄π‘ƒπœ”(𝑋). On the other hand, if 𝑒1,𝑒2βˆˆπ‘†π΄π‘ƒπœ”(𝑋) we have the estimate ‖‖Γ𝑒1ξ€Έ(𝑑)βˆ’Ξ“π‘’2ξ€Έ(β€–β€–ξ€œπ‘‘)≀𝐿𝑀𝑑0π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )‖‖𝑒1(𝑠)βˆ’π‘’2(‖‖𝑠)π‘‘π‘ β‰€πΏπ‘€πœ‡β€–β€–π‘’1βˆ’π‘’2β€–β€–βˆž.(3.7) The fixed point of Ξ“ 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 π‘”βˆΆ[0,∞)×𝑋→𝑋 is a function uniformly 𝑆-asymptotically πœ”-periodic on bounded sets that satisfies the local Lipschitz condition ‖𝑔(𝑑,π‘₯)βˆ’π‘”(𝑑,𝑦)‖≀𝐿(π‘Ÿ)β€–π‘₯βˆ’π‘¦β€–,(3.8) for all 𝑑β‰₯0 and for all π‘₯,π‘¦βˆˆπ‘‹ with β€–π‘₯β€–β‰€π‘Ÿ and β€–π‘¦β€–β‰€π‘Ÿ, where 𝐿∢[0,∞)β†’[0,∞) is a nondecreasing function. Let 𝐢=sup𝑑β‰₯0‖𝑔(𝑑,𝑅(𝑑)π‘₯0)β€–. If there is π‘Ÿ>0 such that π‘€πœ‡ξ‚€πΏξ€·β€–β€–π‘₯π‘Ÿ+𝑀0β€–β€–ξ€Έ+πΆπ‘Ÿξ‚<1,(3.9) then there is a unique 𝑆-asymptotically πœ”-periodic mild solution of (1.1)-(1.2).

Proof. Let 𝑆𝐴𝑃0πœ”(𝑋)={π‘£βˆˆπ‘†π΄π‘ƒπœ”(𝑋)βˆΆπ‘£(0)=0}. It is clear that 𝑆𝐴𝑃0πœ”(𝑋) is a closed vector subspace of π‘†π΄π‘ƒπœ”(𝑋). Let πΉβˆΆπ‘†π΄π‘ƒ0πœ”(𝑋)→𝑆𝐴𝑃0πœ”(𝑋) be the map defined by ξ€œπΉπ‘£(𝑑)=𝑑0𝑅(π‘‘βˆ’π‘ )𝑔𝑠,𝑣(𝑠)+𝑅(𝑠)π‘₯0𝑑𝑠.(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,𝑣2βˆˆπ‘†π΄π‘ƒ0πœ”(𝑋) with ‖𝑣1β€–βˆžβ‰€π‘Ÿ and ‖𝑣2β€–βˆžβ‰€π‘Ÿ, we obtain that ‖‖𝐹𝑣1(𝑑)βˆ’πΉπ‘£2(β€–β€–ξ€œπ‘‘)≀𝑀𝑑0π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )𝐿‖‖π‘₯π‘Ÿ+𝑀0‖‖‖‖𝑣1(𝑠)βˆ’π‘£2(‖‖≀𝑀𝑠)π‘‘π‘ πœ‡πΏξ€·β€–β€–π‘₯π‘Ÿ+𝑀0‖‖‖‖𝑣1βˆ’π‘£2β€–β€–βˆž.(3.11) Hence ‖‖𝐹𝑣1βˆ’πΉπ‘£2β€–β€–βˆžβ‰€π‘€πœ‡πΏξ€·β€–β€–π‘₯π‘Ÿ+𝑀0‖‖‖‖𝑣1βˆ’π‘£2β€–β€–βˆž.(3.12) On the other hand, for π‘£βˆˆπ‘†π΄π‘ƒ0πœ”(𝑋) with β€–π‘£β€–βˆžβ‰€π‘Ÿ, we get β€–πΉπ‘£β€–βˆžβ‰€β€–πΉπ‘£βˆ’πΉ(0)β€–βˆž+‖𝐹(0)β€–βˆžβ‰€π‘€πœ‡πΏξ€·β€–β€–π‘₯π‘Ÿ+𝑀0β€–β€–ξ€Έβ€–π‘£β€–βˆž+πΆπ‘€πœ‡.(3.13) Let π‘Ÿ>0 be such that π‘€πœ‡ξ€·πΏξ€·β€–β€–π‘₯π‘Ÿ+𝑀0β€–β€–ξ€Έπ‘Ÿξ€Έ+𝐢<π‘Ÿ.(3.14) From the above remarks it follows that 𝐹 is a contraction on π΅π‘Ÿ(𝑆𝐴𝑃0πœ”(𝑋)). Thus there is a unique fixed point π‘£βˆˆπ΅π‘Ÿ(𝑆𝐴𝑃0πœ”(𝑋)) of 𝐹. To finish the proof we note that 𝑒(𝑑)=𝑣(𝑑)+𝑅(𝑑)π‘₯0 is the 𝑆-asymptotically πœ”-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 π‘”βˆΆ[0,∞)×𝑋→𝑋 is a function uniformly 𝑆-asymptotically πœ”-periodic on bounded sets that verifies the Lipschitz condition ‖𝑔(𝑑,π‘₯)βˆ’π‘”(𝑑,𝑦)‖≀𝐿(𝑑)β€–π‘₯βˆ’π‘¦β€–,(3.15) for all π‘₯,π‘¦βˆˆπ‘‹ and every 𝑑β‰₯0, where the function 𝐿(β‹…) is locally integrable on [0,∞). If Θ=𝑀sup𝑑β‰₯0ξ€œπ‘‘0π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )𝐿(𝑠)𝑑𝑠<1,(3.16) then the problem (1.1)-(1.2) has a unique 𝑆-asymptotically πœ”-periodic mild solution.

Proof. We define the map Ξ“ on the space π‘†π΄π‘ƒπœ”(𝑋) by the expression (3.5). For π‘’βˆˆπ‘†π΄π‘ƒπœ”(𝑋), let 𝑣 be the function given by (3.6). Since the function 𝑒(β‹…) is bounded, it follows from the Definition 2.2 that 𝐢=sup𝑠β‰₯0‖𝑔(𝑠,𝑒(𝑠))β€–<∞. Consequently, (ξ€œβ€–π‘£π‘‘)‖≀𝑑0π‘€π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )(‖𝑔𝑠,𝑒(𝑠))β€–π‘‘π‘ β‰€πΆπ‘€πœ‡,𝑑β‰₯0,(3.17) which shows that Γ𝑒 is a bounded continuous function on [0,∞).
We next prove that Ξ“ is a Θ-contraction from π‘†π΄π‘ƒπœ”(𝑋) into π‘†π΄π‘ƒπœ”(𝑋). Let π‘’βˆˆπ‘†π΄π‘ƒπœ”(𝑋). Next we set 𝐡={𝑒(𝑑)βˆΆπ‘‘β‰₯0}. We can writeξ€œπ‘£(𝑑+πœ”)βˆ’π‘£(𝑑)=πœ”0ξ€œπ‘…(𝑑+πœ”βˆ’π‘ )𝑔(𝑠,𝑒(𝑠))𝑑𝑠+πœ”π‘‘+πœ”βˆ’ξ€œπ‘…(𝑑+πœ”βˆ’π‘ )𝑔(𝑠,𝑒(𝑠))𝑑𝑠𝑑0=ξ€œπ‘…(π‘‘βˆ’π‘ )𝑔(𝑠,𝑒(𝑠))𝑑𝑠𝑑0[]+ξ€œπ‘…(π‘‘βˆ’π‘ )𝑔(𝑠+πœ”,𝑒(𝑠+πœ”))βˆ’π‘”(𝑠,𝑒(𝑠))𝑑𝑠𝑑𝑑+πœ”=ξ€œπ‘…(𝑠)𝑔(𝑑+πœ”βˆ’π‘ ,𝑒(𝑑+πœ”βˆ’π‘ ))𝑑𝑠𝑑0[]+ξ€œπ‘…(𝑠)𝑔(𝑑+πœ”βˆ’π‘ ,𝑒(𝑑+πœ”βˆ’π‘ ))βˆ’π‘”(π‘‘βˆ’π‘ ,𝑒(π‘‘βˆ’π‘ ))𝑑𝑠𝑑𝑑+πœ”=ξ€œπ‘…(𝑠)𝑔(𝑑+πœ”βˆ’π‘ ,𝑒(𝑑+πœ”βˆ’π‘ ))𝑑𝑠𝑇0[]+ξ€œπ‘…(𝑠)𝑔(𝑑+πœ”βˆ’π‘ ,𝑒(𝑑+πœ”βˆ’π‘ ))βˆ’π‘”(π‘‘βˆ’π‘ ,𝑒(𝑑+πœ”βˆ’π‘ ))𝑑𝑠𝑇0[]+ξ€œπ‘…(𝑠)𝑔(π‘‘βˆ’π‘ ,𝑒(𝑑+πœ”βˆ’π‘ ))βˆ’π‘”(π‘‘βˆ’π‘ ,𝑒(π‘‘βˆ’π‘ ))𝑑𝑠𝑑𝑇[]+ξ€œπ‘…(𝑠)𝑔(𝑑+πœ”βˆ’π‘ ,𝑒(𝑑+πœ”βˆ’π‘ ))βˆ’π‘”(π‘‘βˆ’π‘ ,𝑒(π‘‘βˆ’π‘ ))𝑑𝑠𝑑𝑑+πœ”π‘…(𝑠)𝑔(𝑑+πœ”βˆ’π‘ ,𝑒(𝑑+πœ”βˆ’π‘ ))𝑑𝑠=𝐼1+𝐼2+𝐼3+𝐼4.(3.18) Below we will estimate each one of the terms 𝐼𝑖, 1≀𝑖≀4, of the above expression separately. For πœ€>0, let πœ€ξ…ž=min{πœ‡/𝑀,1/Θ}(πœ€/3). We choose 𝑇>0 such that the following conditions hold:(i)π‘’βˆ’πœ‡π‘‡β‰€πœ€πœ‡/9𝐢𝑀,(ii)‖𝑒(𝑑+πœ”)βˆ’π‘’(𝑑)β€–β‰€πœ€ξ…ž,(iii)‖𝑔(𝑑+πœ”,π‘₯)βˆ’π‘”(𝑑,π‘₯)β€–β‰€πœ€ξ…ž,for all 𝑑β‰₯𝑇 and π‘₯∈𝐡. Let 𝑑β‰₯2𝑇. Since π‘‘βˆ’π‘ β‰₯π‘‘βˆ’π‘‡β‰₯𝑇 for 0≀𝑠≀𝑇, we get ‖‖𝐼1β€–β€–β‰€π‘€πœ€ξ…žξ€œπ‘‡0π‘’βˆ’πœ‡π‘ πœ€π‘‘π‘ β‰€3,‖‖𝐼2β€–β€–β‰€πœ€ξ…žπ‘€ξ€œπ‘‡0π‘’βˆ’πœ‡π‘ πΏ(π‘‘βˆ’π‘ )π‘‘π‘ β‰€πœ€ξ…žπ‘€ξ€œπ‘‘0π‘’βˆ’πœ‡π‘ πΏ(π‘‘βˆ’π‘ )π‘‘π‘ β‰€πœ€ξ…žβ€–β€–πΌΞ˜,3β€–β€–ξ€œβ‰€π‘€π‘‘π‘‡π‘’βˆ’πœ‡π‘ 2𝐢𝑑𝑠≀2πΆπ‘€πœ‡π‘’βˆ’πœ‡π‘‡,‖‖𝐼4β€–β€–ξ€œβ‰€π‘€π‘‘π‘‘+πœ”π‘’βˆ’πœ‡π‘ πΆπ‘‘π‘ β‰€πΆπ‘€πœ‡π‘’βˆ’πœ‡π‘‡.(3.19) Combining these estimates, we find ‖𝑣(𝑑+πœ”)βˆ’π‘£(𝑑)β€–β‰€πœ€,(3.20) for 𝑑β‰₯2𝑇. Hence π‘£βˆˆπ‘†π΄π‘ƒπœ”(𝑋).
On the other hand, if 𝑒1,𝑒2βˆˆπ‘†π΄π‘ƒπœ”(𝑋), and 𝑑β‰₯0, we have ‖‖Γ𝑒1ξ€Έ(𝑑)βˆ’Ξ“π‘’2ξ€Έ(β€–β€–ξ€œπ‘‘)≀𝑀𝑑0π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )‖‖𝑒𝐿(𝑠)1(𝑠)βˆ’π‘’2(‖‖‖‖𝑒𝑠)π‘‘π‘ β‰€Ξ˜1βˆ’π‘’2β€–β€–βˆž.(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(𝑑,π‘₯)β€–/β€–π‘₯β€– is bounded as β€–π‘₯β€–β†’βˆž. 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 π‘”βˆΆ[0,∞)×𝑋→𝑋 that satisfies the following boundedness condition.(H-2)There is a continuous nondecreasing function π‘ŠβˆΆ[0,∞)β†’[0,∞) such that ‖𝑔(𝑑,π‘₯)β€–β‰€π‘Š(β€–π‘₯β€–) for all π‘‘βˆˆ[0,∞) and π‘₯βˆˆπ‘‹.

Theorem 3.5. Assume that π‘”βˆΆ[0,∞)×𝑋→𝑋 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 𝜈β‰₯0, limπ‘‘β†’βˆžβˆ«(1/β„Ž(𝑑))𝑑0π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )π‘Š(πœˆβ„Ž(𝑠))𝑑𝑠=0, where β„Ž is the function in Lemma 2.5. We set 𝜎𝜈(‖‖𝑑)=𝑅(β‹…)π‘₯0β€–β€–βˆžξ€œ+𝑀𝑑0π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )π‘Š(πœˆβ„Ž(𝑠))𝑑𝑠,𝑑β‰₯0,(3.22) and 𝜌(𝜈)=β€–πœŽπœˆβ€–β„Ž.(b)For each πœ–>0, there is 𝛿>0 such that for every 𝑒,π‘£βˆˆπΆβ„Ž(𝑋), β€–π‘’βˆ’π‘£β€–β„Žβ‰€π›Ώ implies that ξ€œπ‘‘0π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )(‖𝑔𝑠,𝑒(𝑠))βˆ’π‘”(𝑠,𝑣(𝑠))β€–π‘‘π‘ β‰€πœ–,(3.23) for all 𝑑β‰₯0.(c)For all π‘Žβ‰₯0 and π‘Ÿ>0, the set {𝑔(𝑠,π‘₯)∢0β‰€π‘ β‰€π‘Ž,π‘₯βˆˆπ‘‹,β€–π‘₯β€–β‰€π‘Ÿ} is relatively compact in 𝑋.(d)liminfπœ‰β†’βˆž(πœ‰/𝜌(πœ‰))>1.Then the problem (1.1)-(1.2) has an 𝑆-asymptotically πœ”-periodic mild solution.

Proof. Let Ξ“βˆΆπΆβ„Ž(𝑋)→𝐢([0,∞);𝑋) be the map defined by the expression (3.5). Next, we prove that Ξ“ has a fixed point in π‘†π΄π‘ƒπœ”(𝑋). We divide the proof in several steps.(i)For π‘’βˆˆπΆβ„Ž(𝑋), we have that ‖Γ𝑒(𝑑)β€–β‰€π‘€β„Ž(𝑑)β€–β€–π‘₯β„Ž(𝑑)0β€–β€–+π‘€ξ€œβ„Ž(𝑑)𝑑0π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )π‘Šξ€·β€–π‘’β€–β„Žξ€Έβ„Ž(𝑠)𝑑𝑠.(3.24) It follows from the condition (a) that Ξ“βˆΆπΆβ„Ž(𝑋)β†’πΆβ„Ž(𝑋).(ii)The map Ξ“ is continuous from πΆβ„Ž(𝑋) into πΆβ„Ž(𝑋). In fact, for πœ–>0, let 𝛿>0 be the constant involved in the condition (b). For 𝑒,π‘£βˆˆπΆβ„Ž(𝑋), β€–π‘’βˆ’π‘£β€–β„Žβ‰€π›Ώ, taking into account that β„Ž(𝑑)β‰₯1, we get ‖Γ𝑒(𝑑)βˆ’Ξ“π‘£(𝑑)β€–β‰€π‘€β„Ž(𝑑)ξ€œβ„Ž(𝑑)𝑑0π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )(‖𝑔𝑠,𝑒(𝑠))βˆ’π‘”(𝑠,𝑣(𝑠))β€–π‘‘π‘ β‰€π‘€πœ–,(3.25) which implies that β€–Ξ“π‘’βˆ’Ξ“π‘£β€–β„Žβ‰€π‘€πœ–. Since πœ–>0 is arbitrary, this shows the assertion.(iii)We next show that Ξ“ is completely continuous. Let 𝑉=Ξ“(π΅π‘Ÿ(πΆβ„Ž(𝑋))). We set 𝑣=Ξ“(𝑒) for π‘’βˆˆπ΅π‘Ÿ(πΆβ„Ž(𝑋)). Initially, we prove that 𝑉(𝑑)={𝑣(𝑑)βˆΆπ‘£βˆˆπ‘‰} is a relatively compact subset of 𝑋 for each 𝑑β‰₯0. From the mean value theorem, 𝑣(𝑑)=𝑅(𝑑)π‘₯0+ξ€œπ‘‘0𝑅(𝑠)𝑔(π‘‘βˆ’π‘ ,𝑒(π‘‘βˆ’π‘ ))π‘‘π‘ βˆˆπ‘…(𝑑)π‘₯0+𝑑𝑐(𝐾),(3.26) where 𝑐(𝐾) denotes the convex hull of 𝐾 and 𝐾={𝑅(𝑠)𝑔(πœ‰,π‘₯)∢0≀𝑠≀𝑑,0β‰€πœ‰β‰€π‘‘,β€–π‘₯β€–β‰€π‘Ÿ}. Combining the fact that 𝑅(β‹…) is strongly continuous with the property (c), we infer that 𝐾 is a relatively compact set, and 𝑉(𝑑)βŠ†π‘…(𝑑)π‘₯0+𝑑𝑐(𝐾) is also a relatively compact set. Let 𝑏>0. We next show that the set 𝑉𝑏={𝑣|[0,𝑏]βˆΆπ‘£βˆˆπ‘‰} is equicontinuous. In fact, for 𝑑β‰₯0 fixed we can decompose 𝑣(𝑑+𝑠)βˆ’π‘£(𝑑) as 𝑣(𝑑+𝑠)βˆ’π‘£(𝑑)=(𝑅(𝑑+𝑠)βˆ’π‘…(𝑑))π‘₯0+ξ€œπ‘‘π‘‘+𝑠+ξ€œπ‘…(𝑑+π‘ βˆ’πœ‰)𝑔(πœ‰,𝑒(πœ‰))π‘‘πœ‰π‘‘0(𝑅(πœ‰+𝑠)βˆ’π‘…(πœ‰))𝑔(π‘‘βˆ’πœ‰,𝑒(π‘‘βˆ’πœ‰))π‘‘πœ‰.(3.27) For each πœ–>0, we can choose 𝛿1>0 such that β€–β€–β€–ξ€œπ‘‘π‘‘+π‘ β€–β€–β€–ξ€œπ‘…(𝑑+π‘ βˆ’πœ‰)𝑔(πœ‰,𝑒(πœ‰))π‘‘πœ‰β‰€π‘€π‘‘π‘‘+π‘ π‘’βˆ’πœ‡(𝑑+π‘ βˆ’πœ‰)πœ–π‘Š(π‘Ÿβ„Ž(πœ‰))π‘‘πœ‰β‰€3,(3.28) for 𝑠≀𝛿1. Moreover, since {𝑔(π‘‘βˆ’πœ‰,𝑒(π‘‘βˆ’πœ‰))∢0β‰€πœ‰β‰€π‘‘,π‘’βˆˆπ΅π‘Ÿ(πΆβ„Ž(𝑋))} is a relatively compact set and 𝑅(β‹…) is strongly continuous, we can choose 𝛿2>0 and 𝛿3>0 such that β€–(𝑅(𝑑+𝑠)βˆ’π‘…(𝑑))π‘₯0β€–β‰€πœ–/3, for 𝑠≀𝛿2 and β€–(𝑅(πœ‰+𝑠)βˆ’π‘…(πœ‰))𝑔(π‘‘βˆ’πœ‰,𝑒(π‘‘βˆ’πœ‰))β€–β‰€πœ–/3(𝑏+1), for 𝑠≀𝛿3. Combining these estimates, we get ‖𝑣(𝑑+𝑠)βˆ’π‘£(𝑑)β€–β‰€πœ– for |𝑠|≀min{𝛿1,𝛿2,𝛿3} with 𝑑+𝑠β‰₯0 and for all π‘’βˆˆπ΅π‘Ÿ(πΆβ„Ž(𝑋)).
Finally, applying condition (a), we can show that‖𝑣(𝑑)‖≀𝑀‖‖π‘₯β„Ž(𝑑)0β€–β€–+π‘€β„Ž(𝑑)ξ€œβ„Ž(𝑑)𝑑0π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )π‘Š(π‘Ÿβ„Ž(𝑠))π‘‘π‘ βŸΆ0,π‘‘βŸΆβˆž,(3.29) and this convergence is independent of π‘’βˆˆπ΅π‘Ÿ(πΆβ„Ž(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 πΆβ„Ž(𝑋).(iv)If π‘’πœ†(β‹…) is a solution of the equation π‘’πœ†=πœ†Ξ“(π‘’πœ†) for some 0<πœ†<1, we have the estimate β€–π‘’πœ†β€–β„Ž/𝜌(β€–π‘’πœ†β€–β„Ž)≀1 and, combining this estimate with the condition (d), we conclude that the set 𝐾={π‘’πœ†βˆΆπ‘’πœ†=πœ†Ξ“(π‘’πœ†),πœ†βˆˆ(0,1)} is bounded.(v) Since π‘†π΄π‘ƒπœ”(𝑋)βŠ‚πΆβ„Ž(𝑋), it follows from Lemmas 2.4 and 3.1 that Ξ“(π‘†π΄π‘ƒπœ”(𝑋))βŠ†π‘†π΄π‘ƒπœ”(𝑋) and, consequently, we can consider Ξ“βˆΆπ‘†π΄π‘ƒπœ”(𝑋)πΆβ„Ž(𝑋)β†’π‘†π΄π‘ƒπœ”(𝑋)πΆβ„Ž(𝑋), where π΅πΆβ„Ž(𝑋) denotes the closure of a set 𝐡 in the space πΆβ„Ž(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 Ξ“ has a fixed point π‘’βˆˆπ‘†π΄π‘ƒπœ”(𝑋)πΆβ„Ž(𝑋). Let (𝑒𝑛)𝑛 be a sequence in π‘†π΄π‘ƒπœ”(𝑋) such that 𝑒𝑛→𝑒 in the norm of πΆβ„Ž(𝑋). For πœ€>0, let 𝛿>0 be the constant in (b), there is 𝑛0βˆˆβ„• so that β€–π‘’π‘›βˆ’π‘’β€–β„Žβ‰€π›Ώ, for all 𝑛β‰₯𝑛0. We observe that for 𝑛β‰₯𝑛0β€–β€–Ξ“π‘’π‘›β€–β€–βˆ’Ξ“π‘’βˆž=sup𝑑β‰₯0β€–β€–β€–ξ€œπ‘‘0𝑔𝑅(π‘‘βˆ’π‘ )𝑠,𝑒𝑛(‖‖‖𝑠)βˆ’π‘”(𝑠,𝑒(𝑠))𝑑𝑠≀𝑀sup𝑑β‰₯0ξ€œπ‘‘0π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )‖‖𝑔𝑠,𝑒𝑛(‖‖𝑠)βˆ’π‘”(𝑠,𝑒(𝑠))π‘‘π‘ β‰€πœ–.(3.30) Hence (Γ𝑒𝑛)𝑛 converges to Γ𝑒=𝑒 uniformly in [0,∞). This implies that π‘’βˆˆπ‘†π΄π‘ƒπœ”(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 πœ”-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 π‘’ξ…ž(ξ€œπ‘‘)=𝐴𝑒(𝑑)+𝑑0𝐡(π‘‘βˆ’π‘ )𝑒(𝑠)𝑑𝑠+𝑔𝑑,𝑒𝑑𝑒,𝑑β‰₯0,0=πœ‘,(4.1) here πœ‘βˆˆπΆ([βˆ’π‘Ÿ,0];𝑋), the history of the function 𝑒(β‹…) is given by π‘’π‘‘βˆΆ[βˆ’π‘Ÿ,0]→𝑋, 𝑒𝑑(πœƒ)=𝑒(𝑑+πœƒ), and π‘”βˆΆ[0,∞)×𝐢([βˆ’π‘Ÿ,0];𝑋)→𝑋 is a continuous function. The following property is an immediate consequence of our definitions.

Lemma 4.1. Let π‘’βˆΆ[βˆ’π‘Ÿ,∞)→𝑋 be a continuous function. If 𝑒|[0,∞)βˆˆπ‘†π΄π‘ƒπœ”(𝑋). Then the function [0,∞)→𝐢([βˆ’π‘Ÿ,0];𝑋), 𝑑↦𝑒𝑑, is 𝑆-asymptotically πœ”-periodic.

Definition 4.2. A function π‘’βˆˆπΆ([βˆ’π‘Ÿ,∞);𝑋) is said to be a mild solution of (4.1) if 𝑒0=πœ‘ and the integral equation ξ€œπ‘’(𝑑)=𝑅(𝑑)πœ‘(0)+𝑑0𝑅(π‘‘βˆ’π‘ )𝑔𝑠,𝑒𝑠𝑑𝑠,𝑑β‰₯0,(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 π‘”βˆΆ[0,∞)×𝐢([βˆ’π‘Ÿ,0];𝑋)→𝑋 is a uniformly 𝑆-asymptotically πœ”-periodic on bounded sets function that verifies the Lipschitz condition ‖‖𝑔𝑑,πœ“1ξ€Έξ€·βˆ’π‘”π‘‘,πœ“2ξ€Έβ€–β€–β€–β€–πœ“β‰€πΏ1βˆ’πœ“2β€–β€–βˆž,(4.3) for all πœ“1,πœ“2∈𝐢([βˆ’π‘Ÿ,0];𝑋) and every 𝑑β‰₯0. If 𝐿𝑀/πœ‡<1, then the problem (4.1) has a unique 𝑆-asymptotically πœ”-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 𝑒𝑑 is given by π‘’π‘‘βˆΆ(βˆ’βˆž,0]→𝑋, 𝑒𝑑(πœƒ)=𝑒(𝑑+πœƒ). To model this problem we assume that 𝑒𝑑 belongs to some phase space ℬ which satisfies appropriate conditions. We will employ the axiomatic definition of the phase space ℬ introduced in [20]. Specifically, ℬ will be a linear space of functions mapping (βˆ’βˆž,0] into 𝑋 endowed with a seminorm ‖⋅‖ℬ and verifying the following axioms. (A)If π‘₯∢(βˆ’βˆž,𝜎+π‘Ž)→𝑋, π‘Ž>0,πœŽβˆˆβ„, is continuous on [𝜎,𝜎+π‘Ž) and π‘₯πœŽβˆˆβ„¬, then for every π‘‘βˆˆ[𝜎,𝜎+π‘Ž) the following hold: (i)π‘₯𝑑 is in ℬ. (ii)β€–π‘₯(𝑑)‖≀𝐻‖π‘₯𝑑‖ℬ. (iii)β€–π‘₯𝑑‖ℬ≀𝐾(π‘‘βˆ’πœŽ)sup{β€–π‘₯(𝑠)β€–βˆΆπœŽβ‰€π‘ β‰€π‘‘}+𝑀(π‘‘βˆ’πœŽ)β€–π‘₯πœŽβ€–β„¬,

where 𝐻>0 is a constant; 𝐾,π‘€βˆΆ[0,∞)β†’[1,∞), 𝐾 is continuous, 𝑀 is locally bounded and 𝐻,𝐾,𝑀 are independent of π‘₯(β‹…). (A1)For the function π‘₯(β‹…) in (A), the function 𝑑→π‘₯𝑑 is continuous from [𝜎,𝜎+π‘Ž) into ℬ. (B)The space ℬ is complete. (C-2)If (πœ“π‘›)π‘›βˆˆβ„• is a uniformly bounded sequence of continuous functions with compact support and πœ“π‘›β†’πœ“, π‘›β†’βˆž, in the compact-open topology, then πœ“βˆˆβ„¬ and β€–πœ“π‘›βˆ’πœ“β€–β„¬β†’0 as π‘›β†’βˆž.

We introduce the space ℬ0={πœ“βˆˆβ„¬βˆΆπœ“(0)=0} and the operator 𝑆(𝑑)βˆΆβ„¬β†’β„¬ given by []𝑆(𝑑)πœ“(πœƒ)=πœ“(0),βˆ’π‘‘β‰€πœƒβ‰€0,πœ“(𝑑+πœƒ),βˆ’βˆž<πœƒ<βˆ’π‘‘.(4.4) It is well known that (𝑆(𝑑))𝑑β‰₯0 is a 𝐢0-semigroup ([20]).

Definition 4.4. The phase space ℬ is said to be a fading memory space if ‖𝑆(𝑑)πœ“β€–β„¬β†’0 as π‘‘β†’βˆž for every πœ“βˆˆβ„¬0.

Remark 4.5. Since ℬ verifies axiom (C-2), the space 𝐢𝑏((βˆ’βˆž,0],𝑋) consisting of continuous and bounded functions πœ“βˆΆ(βˆ’βˆž,0]→𝑋 is continuously included in ℬ. Thus, there exists a constant 𝑄β‰₯0 such that β€–πœ“β€–β„¬β‰€π‘„β€–πœ“β€–βˆž, for every πœ“βˆˆπΆπ‘((βˆ’βˆž,0],𝑋) ([20, Proposition 7.1.1]).
Moreover, if ℬ 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 πΆπ‘ŸΓ—πΏπ‘(𝜌,𝑋)

Let π‘Ÿβ‰₯0,1≀𝑝<∞ and let ΜƒπœŒβˆΆ(βˆ’βˆž,βˆ’π‘Ÿ]→ℝ be a nonnegative measurable function which satisfies the conditions (g-5)-(g-6) in the terminology of [20]. Briefly, this means that ΜƒπœŒ is locally integrable and there exists a nonnegative locally bounded function 𝛾 on (βˆ’βˆž,0] such that ΜƒπœŒ(πœ‰+πœƒ)≀𝛾(πœ‰)ΜƒπœŒ(πœƒ), for all πœ‰β‰€0 and πœƒβˆˆ(βˆ’βˆž,βˆ’π‘Ÿ)β§΅π‘πœ‰, where π‘πœ‰βŠ†(βˆ’βˆž,βˆ’π‘Ÿ) is a set whose Lebesgue measure zero.

The space ℬ=πΆπ‘ŸΓ—πΏπ‘(ΜƒπœŒ,𝑋) consists of all classes of functions πœ‘βˆΆ(βˆ’βˆž,0]↦𝑋 such that πœ‘ is continuous on [βˆ’π‘Ÿ,0], Lebesgue-measurable, and ΜƒπœŒβ€–πœ‘β€–π‘ is Lebesgue integrable on (βˆ’βˆž,βˆ’π‘Ÿ). The seminorm in πΆπ‘ŸΓ—πΏπ‘(ΜƒπœŒ,𝑋) is defined as follows:β€–πœ‘β€–β„¬ξ‚΅ξ€œ=sup{β€–πœ‘(πœƒ)β€–βˆΆβˆ’π‘Ÿβ‰€πœƒβ‰€0}+βˆ’π‘Ÿβˆ’βˆžΜƒπœŒ(πœƒ)β€–πœ‘(πœƒ)β€–π‘ξ‚Άπ‘‘πœƒ1/𝑝.(4.5) The space ℬ=πΆπ‘ŸΓ—πΏπ‘(ΜƒπœŒ,𝑋) satisfies axioms (A), (A-1), and (B). Moreover, when π‘Ÿ=0 and 𝑝=2, it is possible to choose 𝐻=1, 𝑀(𝑑)=𝛾(βˆ’π‘‘)1/2 and ∫𝐾(𝑑)=1+(0βˆ’π‘‘ΜƒπœŒ(πœƒ)π‘‘πœƒ)1/2 for 𝑑β‰₯0 (see [20, Theorem 1.3.8] for details). Note that if conditions (g-6)-(g-7) of [20] hold, then ℬ 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 ℬ is a fading memory space. Let π‘’βˆΆβ„β†’π‘‹ be a function with 𝑒0βˆˆβ„¬ and 𝑒|[0,∞)βˆˆπ‘†π΄π‘ƒπœ”(𝑋). Then the function 𝑑→𝑒𝑑 belongs to π‘†π΄π‘ƒπœ”(ℬ).

Next we assume that πœ‘βˆˆβ„¬ and that π‘”βˆΆ[0,∞)×ℬ→𝑋 is a continuous function.

Definition 4.8. A function π‘’βˆˆπΆ(ℝ;𝑋) is said to be a mild solution of (4.1) if 𝑒0=πœ‘ and the integral equation ξ€œπ‘’(𝑑)=𝑅(𝑑)πœ‘(0)+𝑑0𝑅(π‘‘βˆ’π‘ )𝑔𝑠,𝑒𝑠𝑑𝑠,𝑑β‰₯0,(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 π‘”βˆΆ[0,∞)×ℬ→𝑋 is a uniformly 𝑆-asymptotically πœ”-periodic on bounded sets function that verifies the Lipschitz condition ‖‖𝑔𝑑,πœ“1ξ€Έξ€·βˆ’π‘”π‘‘,πœ“2ξ€Έβ€–β€–β€–β€–πœ“β‰€πΏ1βˆ’πœ“2‖‖ℬ,(4.7) for all πœ“1,πœ“2βˆˆβ„¬ and every 𝑑β‰₯0. If 𝐿𝑀𝑄/πœ‡<1, then the problem (4.1) has a unique 𝑆-asymptotically πœ”-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 Ξ© be a bounded open connected subset of ℝ3 with 𝐢∞ boundary; let 𝛼 and 𝛽 be in 𝐢2([0,∞),ℝ) with 𝛼(0) and 𝛽(0) positive, and let a∢[0,∞)→ℝ and π‘βˆΆπ»10(Ξ©)→𝐿2(Ξ©) be functions.

Let us consider the following equation that arises in the study of heat conduction in materials with memory (see [2, 21–23])πœƒξ…žξ…ž(𝑑)+𝛽(0)πœƒξ…ž(ξ€œπ‘‘)=𝛼(0)Ξ”πœƒ(𝑑)βˆ’π‘‘0π›½ξ…ž(π‘‘βˆ’π‘ )πœƒξ…ž(ξ€œπ‘ )𝑑𝑠+𝑑0π›Όξ…ž(π‘‘βˆ’π‘ )Ξ”πœƒ(𝑠)𝑑𝑠+π‘Ž(𝑑)𝑏(πœƒ(𝑑)),(5.1) for 𝑑β‰₯0, where Ξ” is the Laplacian on Ξ©.

We consider (5.1) with initial conditionπœƒ(0)=πœƒ0,πœƒξ…ž(0)=πœ‚0.(5.2) To model this problem, we consider the space 𝑋=𝐻10(Ξ©)×𝐿2(Ξ©) and the linear operators ξƒͺ𝐴=0𝐼𝛼(0)Ξ”βˆ’π›½(0)𝐼(5.3) on the domain 𝐷(𝐴)=(𝐻2(Ξ©)∩𝐻10(Ξ©))×𝐻10(Ξ©), and 𝐡(𝑑)=𝐹(𝑑)𝐴, where 𝐹(𝑑)=[𝐹𝑖𝑗(𝑑)]βˆΆπ‘‹β†’π‘‹, 𝑑β‰₯0, is defined by 𝐹11(𝑑)=𝐹12(𝑑)=0, 𝐹21(𝑑)=βˆ’π›½ξ…ž(𝑑)𝐼+𝛽(0)(π›Όξ…ž(𝑑)/𝛼(0))𝐼, 𝐹22(𝑑)=(π›Όξ…ž(𝑑)/𝛼(0))𝐼.

Introducing the variableξƒ©πœƒπ‘’(𝑑)=πœƒ(𝑑)ξ…žξƒͺ(𝑑)βˆˆπ‘‹,(5.4) and defining0ξƒͺ𝑔(𝑑,𝑒(𝑑))=π‘Ž(𝑑)𝑏(πœƒ(𝑑)),(5.5) the equation (5.1) with initial conditionξƒ©πœƒπ‘’(0)=0πœ‚0ξƒͺ(5.6) takes the abstract form (1.1)-(1.2).

It follows from [24] that 𝐴 generates a 𝐢0-semigroup (𝑇(𝑑))𝑑β‰₯0 such that ‖𝑇(𝑑)β€–β‰€π‘€π‘’βˆ’π›Ύπ‘‘ for all 𝑑β‰₯0 and some constants 𝑀,𝛾>0. Assume that π›Όξ…ž(𝑑)𝑒𝛾𝑑, π›Όξ…žξ…ž(𝑑)𝑒𝛾𝑑, π›½ξ…ž(𝑑)𝑒𝛾𝑑, and π›½ξ…žξ…ž(𝑑)𝑒𝛾𝑑 are bounded and uniformly continuous functions on [0,∞), and that for all 𝑑β‰₯0, ||π›½ξ…ž||||𝛼(𝑑)+max{𝛽(0),1}ξ…ž||(𝑑)≀𝛼(0)π›Ύπ‘’βˆ’π›Ύπ‘‘2𝑀,||π›½ξ…žξ…ž||||𝛼(𝑑)+max{𝛽(0),1}ξ…žξ…ž||(𝑑)𝛼≀𝛾(0)2π‘’βˆ’π›Ύπ‘‘4𝑀2.(5.7)

Then, by Grimmer [2, Theorem 4.1], there is a resolvent operator 𝑅(𝑑) associated to the operators 𝐴 and 𝐡(β‹…) and satisfying(‖𝑅𝑑)‖≀𝑀𝑒(βˆ’π›Ύ/2)𝑑,𝑑β‰₯0.(5.8) In addition, suppose that π‘Žβˆˆπ‘†π΄π‘ƒπœ”(ℝ) and π‘βˆΆπ»10(Ξ©)→𝐿2(Ξ©) satisfies‖‖𝑏(πœƒ1)βˆ’π‘(πœƒ2)‖‖𝐿2(Ξ©)β‰€πΏπ‘β€–β€–πœƒ1βˆ’πœƒ2‖‖𝐻10(Ξ©),(5.9) for all πœƒ1,πœƒ2∈𝐻10(Ξ©).

We claim that for each πœƒ0∈𝐻10(Ξ©) and πœ‚0∈𝐿2(Ξ©) 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=β€–π‘Žβ€–βˆžπΏπ‘. In addition, estimates ‖𝑔(𝑑,𝑒)‖𝐻10(Ξ©)×𝐿2(Ξ©)β‰€β€–π‘Žβ€–βˆžξ‚€πΏπ‘β€–π‘’β€–π»10(Ξ©)×𝐿2(Ξ©)+‖𝑏(0)‖𝐿2(Ξ©),‖𝑔(𝑑+πœ”,𝑒)βˆ’π‘”(𝑑,𝑒)‖𝐻10(Ξ©)×𝐿2(Ξ©)≀||||ξ‚€πΏπ‘Ž(𝑑+πœ”)βˆ’π‘Ž(𝑑)𝑏‖𝑒‖𝐻10(Ξ©)×𝐿2(Ξ©)+‖𝑏(0)‖𝐿2(Ξ©),(5.10) which are verified for 𝑑β‰₯0 and π‘’βˆˆπ»10(Ξ©)×𝐿2(Ξ©) show that π‘”βˆΆ[0,∞)×𝐻10(Ξ©)×𝐿2(Ξ©)→𝐻10(Ξ©)×𝐿2(Ξ©) is a function uniformly 𝑆-asymptotically πœ”-periodic on bounded sets. As a consequence of Theorem 3.2, we obtain the following result.

Proposition 5.1. Under the above conditions, if ξ‚‹(2𝑀/𝛾)β€–π‘Žβ€–βˆžπΏπ‘<1, then the problem (5.1)-(5.2) has a unique 𝑆-asymptotically πœ”-periodic mild solution.

To establish our next result, we assume the following conditions.(H-3)Let π‘Žβˆˆπ‘†π΄π‘ƒπœ”(ℝ), 𝜏∈(0,1) and let π‘βˆΆπ»10(Ξ©)→𝐿2(Ξ©) be a function that satisfies the HΓΆlder type condition ‖‖𝑏(πœƒ1)βˆ’π‘(πœƒ2)‖‖𝐿2(Ξ©)β‰€πΏπ‘β€–β€–πœƒ1βˆ’πœƒ2β€–β€–πœπ»10(Ξ©),(5.11) for all πœƒ1,πœƒ2∈𝐻10(Ξ©).(H-4)There is a continuous nondecreasing function β„ŽβˆΆ[0,∞)β†’[1,∞) such that β„Ž(𝑑)β†’βˆž as π‘‘β†’βˆž and (1)limπ‘‘β†’βˆžβˆ«(1/β„Ž(𝑑))𝑑0𝑒(βˆ’π›Ύ/2)(π‘‘βˆ’π‘ )β„Ž(𝑠)πœπ‘‘π‘ =0, (2)sup𝑑β‰₯0βˆ«π‘‘0𝑒(βˆ’π›Ύ/2)(π‘‘βˆ’π‘ )|π‘Ž(𝑠)|β„Ž(𝑠)πœπ‘‘π‘ <∞.

For a concrete example, take 0<πœ†<𝛾/2(1βˆ’πœ), π‘Ž(𝑑)=π‘’βˆ’πœ†π‘‘, and β„Ž(𝑑)=π‘’πœ†π‘‘.

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 πœ”-periodic mild solution.

Proof. We have the following estimates: ‖𝑔(𝑑,𝑒)‖𝐻10(Ξ©)×𝐿2(Ξ©)β‰€β€–π‘Žβ€–βˆžξ‚€πΏπ‘β€–π‘’β€–πœπ»10(Ξ©)×𝐿2(Ξ©)+‖𝑏(0)‖𝐿2(Ξ©),(5.12)‖𝑔(𝑑+πœ”,𝑒)βˆ’π‘”(𝑑,𝑒)‖𝐻10(Ξ©)×𝐿2(Ξ©)≀||||ξ‚€πΏπ‘Ž(𝑑+πœ”)βˆ’π‘Ž(𝑑)π‘β€–π‘’β€–πœπ»10(Ξ©)×𝐿2(Ξ©)+‖𝑏(0)‖𝐿2(Ξ©),(5.13) for all 𝑑β‰₯0 and π‘’βˆˆπ»10(Ξ©)×𝐿2(Ξ©). It follows from (5.12)-(5.13) that the function π‘”βˆΆ[0,∞)×𝐻10(Ξ©)×𝐿2(Ξ©)→𝐻10(Ξ©)×𝐿2(Ξ©) is uniformly 𝑆-asymptotically πœ”-periodic on bounded sets. In addition, from the estimate ‖𝑔(𝑑,𝑒)βˆ’π‘”(𝑑,𝑣)‖𝐻10(Ξ©)×𝐿2(Ξ©)≀||||πΏπ‘Ž(𝑑)π‘β€–π‘’βˆ’π‘£β€–πœπ»10(Ξ©)×𝐿2(Ξ©),(5.14) for all 𝑑β‰₯0 and 𝑒,π‘£βˆˆπ»10(Ξ©)×𝐿2(Ξ©), we obtain that 𝑔 is asymptotically uniformly continuous on bounded sets. By (5.12) we are led to define π‘Š(πœ‰)=β€–π‘Žβ€–βˆž(πΏπ‘πœ‰πœ+‖𝑏(0)‖𝐿2(Ξ©)). Consequently, the function 𝑔 satisfies the assumption (H-2). From (H-4), for 𝑒,π‘£βˆˆπΆβ„Ž(𝐻10(Ξ©)×𝐿2(Ξ©)) we can infer that 1ξ€œβ„Ž(𝑑)𝑑0π‘’βˆ’(𝛾/2)(π‘‘βˆ’π‘ )π‘Š(πœˆβ„Ž(𝑠))π‘‘π‘ β‰€β€–π‘Žβ€–βˆžπΏπ‘πœˆπœξ‚΅1ξ€œβ„Ž(𝑑)𝑑0𝑒(βˆ’π›Ύ/2)(π‘‘βˆ’π‘ )β„Ž(𝑠)πœξ‚Άπ‘‘π‘ +2‖𝑏(0)‖𝐿2(Ξ©)π›Ύβ„Ž(𝑑)⟢0,π‘‘βŸΆβˆž,sup𝑑β‰₯0ξ€œπ‘‘0π‘’βˆ’(𝛾/2)(π‘‘βˆ’π‘ )‖𝑔(𝑠,𝑒(𝑠))βˆ’π‘”(𝑠,𝑣(𝑠))‖𝐻10(Ξ©)×𝐿2(Ξ©)≀𝑑𝑠sup𝑑β‰₯0ξ€œπ‘‘0𝑒(βˆ’π›Ύ/2)(π‘‘βˆ’π‘ )||π‘Ž||β„Ž(𝑠)(𝑠)πœξ‚Άπ‘‘π‘ β€–π‘’βˆ’π‘£β€–πœβ„Ž.(5.15) Therefore, conditions (a) and (b) of Theorem 3.5 are satisfied. A straightforward computation shows that liminfπœ‰β†’βˆž(πœ‰/𝜌(πœ‰))>1, where 𝜌(πœ‰)=sup𝑑β‰₯01ξƒ©β€–β€–β€–β€–ξƒ©πœƒβ„Ž(𝑑)𝑅(β‹…)0πœ‚0ξƒͺβ€–β€–β€–β€–+ξ‚‹π‘€ξ€œπ‘‘0𝑒(βˆ’π›Ύ/2)(π‘‘βˆ’π‘ )ξƒͺπ‘Š(πœ‰β„Ž(𝑠))𝑑𝑠.(5.16) Finally, since Ξ© is bounded set with 𝐢∞ boundary, Rellich-Kondrachov's Theorem [25, Theorem IX.16] leads to the conclusion that {π‘Ž(𝑠)𝑏(πœƒ)∢0≀𝑠≀𝑠0,πœƒβˆˆπ»10(Ξ©),β€–πœƒβ€–π»10(Ξ©)β‰€π‘Ÿ}, 𝑠0>0, is relatively compact in 𝐿2(Ξ©). Hence, condition (c) holds. Using the Theorem 3.5, we conclude that the problem (5.1)-(5.2) has an 𝑆-asymptotically πœ”-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 π‘Ÿ>0. In this case the equation is πœƒξ…žξ…ž(𝑑)+𝛽(0)πœƒξ…ž(ξ€œπ‘‘)=𝛼(0)Ξ”πœƒ(𝑑)βˆ’π‘‘0π›½ξ…ž(π‘‘βˆ’π‘ )πœƒξ…ž(ξ€œπ‘ )𝑑𝑠+𝑑0π›Όξ…ž(π‘‘βˆ’π‘ )Ξ”πœƒ(𝑠)𝑑𝑠+π‘Ž(𝑑)𝑏(πœƒ(π‘‘βˆ’π‘Ÿ)),(5.17) for 𝑑β‰₯0. Using the previous development, we model this problem in the space 𝑋=𝐻10(Ξ©)×𝐿2(Ξ©), and we consider π‘’π‘‘βˆˆπΆ([βˆ’π‘Ÿ,0];𝑋) for 𝑑β‰₯0. To be consistent with our model, we study the equation (5.17) with initial condition𝑒0=ξƒ©πœ‘πœ“ξƒͺ,(5.18) where πœ‘βˆˆπΆ([βˆ’π‘Ÿ,0];𝐻10(Ξ©)) and πœ“βˆˆπΆ([βˆ’π‘Ÿ,0];𝐿2(Ξ©)). The function π‘”βˆΆ[0,∞)×𝐢([βˆ’π‘Ÿ,0];𝑋)→𝑋 is given byπ‘”ξ€·ξ€·πœ‘π‘‘,1,πœ“1=0ξ€·πœ‘ξ€Έξ€Έπ‘Ž(𝑑)𝑏1ξ€Έξƒͺ(βˆ’π‘Ÿ).(5.19) Applying now the Corollary 4.3, we obtain the following result.

Proposition 5.3. Under the above conditions, if ξ‚‹(2𝑀/𝛾)β€–π‘Žβ€–βˆžπΏπ‘<1, then the problem (5.17)-(5.18) has a unique 𝑆-asymptotically πœ”-periodic mild solution.

Acknowledgments

Claudio Cuevas is partially supported by CNPQ/Brazil under Grant 300365/2008-0. HernΓ‘n R. HenrΓ­quez is supported in part by CONICYT under Grant FONDECYT no. 1090009.

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