Asymptotic Periodicity for a Class of Partial Integrodifferential Equations

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(𝑡𝑢−(0𝑠))𝑢=(𝑠𝑥)₀𝑑𝑠+𝑔(𝑡,𝑢(𝑡)),𝑡≥0,(1.1),(1.2) where 𝐴∶𝐷(𝐴)⊆𝑋→𝑋 and 𝐵(𝑡)∶𝐷(𝐵(𝑡))⊆𝑋→𝑋 for 𝑡≥0 are densely defined closed linear operators in a Banach space (𝑋,‖⋅‖). We assume that 𝐷(𝐴)⊂𝐷(𝐵(𝑡)) for every 𝑡≥0 and that 𝑔∶[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 𝜔-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 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 [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 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 𝐶_𝑏([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 𝑧∈𝐶_𝑏([0,∞);𝑍) such that lim_𝑡→∞𝑧(𝑡)=0. Also, we denote by 𝐶₀([0,∞)×𝑊;𝑍) the vector space of all continuous functions 𝐻∶[0,∞)×𝑊→𝑍 such that lim_𝑡→∞𝐻(𝑡,𝑤)=0 uniformly for 𝑤 in compact subsets of 𝑊. The notation ℒ(𝑍) stands for the Banach space of bounded linear operators from 𝑍 into 𝑍. Besides, we denote by 𝐵_𝑟(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 𝑓∈𝐶_𝑏([0,∞);𝑋) is called 𝑆-asymptotically 𝜔-periodic if lim_𝑡→∞(𝑓(𝑡+𝜔)−𝑓(𝑡))=0.(2.1) In this case, we say that 𝜔 is an asymptotic period of 𝑓.

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

Definition 2.2 (see [14]). A function 𝑓∈𝐶([0,∞)×𝑊;𝑍) is said to be uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets if for every bounded set 𝐾⊂𝑊, the set {𝑓(𝑡,𝑥)∶𝑡≥0,𝑥∈𝐾} is bounded and lim_𝑡→∞(𝑓(𝑡+𝜔,𝑥)−𝑓(𝑡,𝑥))=0, uniformly for 𝑥∈𝐾.

Definition 2.3 (see [14]). A function 𝑓∈𝐶([0,∞)×𝑊;𝑍) is said to be asymptotically uniformly continuous on bounded sets if for every 𝜖>0 and every bounded set 𝐾⊂𝑊, there are constants 𝑇_𝜖,𝐾≥0 and 𝛿_𝜖,𝐾>0 such that ‖𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)‖_𝑍≤𝜖 for all 𝑡≥𝑇_𝜖,𝐾 and every 𝑥,𝑦∈𝐾 with ‖𝑥−𝑦‖_𝑊≤𝛿_𝜖,𝐾.

Lemma 2.4 (see [14]). Assume that 𝑓∈𝐶([0,∞)×𝑊;𝑍) is uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets and asymptotically uniformly continuous on bounded sets. If 𝑢∈𝑆𝐴𝑃_𝜔(𝑊), then the function 𝑡↦𝑓(𝑡,𝑢(𝑡)) belongs to 𝑆𝐴𝑃_𝜔(𝑍).

Let ℎ∶[0,∞)→[1,∞) be a continuous nondecreasing function such that ℎ(𝑡)→∞ as 𝑡→∞. Next, the notation 𝐶_ℎ(𝑍) stands for the space 𝐶_ℎ(𝑍)={𝑢∈𝐶([0,∞),𝑍)∶lim_𝑡→∞(𝑢(𝑡)/ℎ(𝑡))=0} endowed with the norm ‖𝑢‖_ℎ=sup_𝑡≥0(‖𝑢(𝑡)‖/ℎ(𝑡)).

Lemma 2.5 (see [17]). A set 𝐾⊆𝐶_ℎ(𝑍) is relatively compact in 𝐶_ℎ(𝑍) if it verifies the following conditions. (c-1)For all 𝑏>0, the set 𝐾_𝑏={𝑢|_[0,𝑏]∶𝑢∈𝐾} is relatively compact in 𝐶([0,𝑏];𝑍).(c-2)lim_𝑡→∞(‖𝑢(𝑡)‖/ℎ(𝑡))=0 uniformly for 𝑢∈𝐾.

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 ‖𝑥‖_𝐴=‖𝑥‖+‖𝐴𝑥‖.

Definition 2.6 (see [18]). A family {𝑅(𝑡)∶𝑡≥0} of continuous linear operators on 𝑋 is called a resolvent operator for (1.1) if the following conditions are fulfilled. (R1)For each 𝑥∈𝑋, 𝑅(0)𝑥=𝑥 and 𝑅(⋅)𝑥∈𝐶([0,∞);𝑋). (R2)The map 𝑅∶[0,∞)→ℒ([𝐷(𝐴)]) is strongly continuous. (R3)For each 𝑦∈𝐷(𝐴), the function 𝑡↦𝑅(𝑡)𝑦 is continuously differentiable and 𝑑𝑑𝑡𝑅(𝑡)𝑦=𝐴𝑅(𝑡)𝑦+^𝑡0𝐵(𝑡−𝑠)𝑅(𝑠)𝑦𝑑𝑠=𝑅(𝑡)𝐴𝑦+^𝑡0𝑅(𝑡−𝑠)𝐵(𝑠)𝑦𝑑𝑠,𝑡≥0.(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)+^𝑡0𝐵𝑢(𝑡(0−)𝑠=)𝑢𝑥₀(𝑠)𝑑𝑠+𝑓(𝑡),𝑡≥0,,(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(𝑡−𝑠)𝑓(𝑠)𝑑𝑠,𝑡≥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 𝑢∈𝐶([0,∞);𝑋) is called a mild solution of (1.1)-(1.2) if 𝑢(𝑡)=𝑅(𝑡)𝑥₀+^𝑡0𝑅(𝑡−𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠,𝑡≥0.(2.5)

To establish our results, we introduce the following condition.(H-1)There are positive constants 𝑀,𝜇 such that ‖𝑅(𝑡)‖≤𝑀𝑒^−𝜇𝑡 for all 𝑡≥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 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 𝑢∈𝑆𝐴𝑃_𝜔(𝑋). Then 𝑣(𝑡)=^𝑡0𝑅(𝑡−𝑠)𝑢(𝑠)𝑑𝑠∈𝑆𝐴𝑃_𝜔(𝑋).(3.1)

Proof. The estimate ‖𝑣‖_∞≤(𝑀/𝜇)‖𝑢‖_∞ shows that 𝑣∈𝐶_𝑏([0,∞);𝑋). For 𝜖>0, we select 𝑇>0 such that ‖𝑢(𝑡+𝜔)−𝑢(𝑡)‖≤𝜖 for all 𝑡≥𝑇 and ∫^∞𝑇𝑒^−𝜇𝑠𝑑𝑠≤𝜖. We have the following decomposition: 𝑣(𝑡+𝜔)−𝑣(𝑡)==^𝜔0𝑅(𝑡+𝜔−𝑠)𝑢(𝑠)𝑑𝑠+^{𝜔𝑡+𝜔}𝑅(𝑡+𝜔−𝑠)𝑢(𝑠)𝑑𝑠−^𝑡0𝑅(𝑡−𝑠)𝑢(𝑠)𝑑𝑠_{𝑡𝑡+𝜔}=𝑅(𝑠)𝑢(𝑡+𝜔−𝑠)𝑑𝑠+^𝑡0𝑅(𝑡−𝑠)[𝑢(𝑠+𝜔)−𝑢(𝑠)]𝑑𝑠_{𝑡𝑡+𝜔}+𝑅(𝑠)𝑢(𝑡+𝜔−𝑠)𝑑𝑠+^𝑇0𝑅(𝑡−𝑠)[𝑢(𝑠+𝜔)−𝑢(𝑠)]𝑑𝑠_𝑡𝑇𝑅(𝑡−𝑠)[𝑢(𝑠+𝜔)−𝑢(𝑠)]𝑑𝑠.(3.2) Hence, for 𝑡≥2𝑇, we obtain ‖𝑣(𝑡+𝜔)−𝑣(𝑡)‖≤𝑀‖𝑢‖_∞^{𝑡𝑡+𝜔}𝑒^−𝜇𝑠𝑑𝑠+2𝑀‖𝑢‖_∞^{𝑡𝑡−𝑇}≤𝑀‖𝑢‖_∞𝑒^−𝜇𝑠𝑑𝑠+𝜖𝑀^0𝑡−𝑇𝑒^−𝜇𝑠𝑑𝑠^{𝑡𝑡+𝜔}𝑒^−𝜇𝑠𝑑𝑠+2𝑀‖𝑢‖_∞^𝑡𝑇≤3𝑀‖𝑢‖_∞𝑒^−𝜇𝑠𝑑𝑠+𝜖𝑀^𝑡0𝑒^−𝜇𝑠𝑑𝑠^∞𝑇𝑒^−𝜇𝑠𝑑𝑠+𝑀𝜇𝜖=𝑀3‖𝑢‖^∞+1𝜇𝜖,(3.3) which completes the proof.

Theorem 3.2. Assume that 𝑔∶[0,∞)×𝑋→𝑋 is a uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets function that verifies the Lipschitz condition ‖𝑔(𝑡,𝑥)−𝑔(𝑡,𝑦)‖≤𝐿‖𝑥−𝑦‖,(3.4) for all 𝑥,𝑦∈𝑋 and every 𝑡≥0. 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 𝑆𝐴𝑃_𝜔(𝑋) by the expression Γ𝑢(𝑡)=𝑅(𝑡)𝑥₀+^𝑡0𝑅(𝑡−𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠,𝑡≥0.(3.5) We next prove that Γ is a contraction from 𝑆𝐴𝑃_𝜔(𝑋) into 𝑆𝐴𝑃_𝜔(𝑋). Initially we show that Γ is a map 𝑆𝐴𝑃_𝜔(𝑋)-valued. Let 𝑢∈𝑆𝐴𝑃_𝜔(𝑋). We abbreviate the notation by writing 𝑣(𝑡)=^𝑡0𝑅(𝑡−𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠(3.6) Since 𝑅(⋅)𝑥₀∈𝑆𝐴𝑃_𝜔(𝑋), it remains to show that the function 𝑣(⋅) given by (3.6) belongs to 𝑆𝐴𝑃_𝜔(𝑋). Considering that 𝑔 is asymptotically uniformly continuous on bounded sets and applying the Lemma 2.4, 𝑔(⋅,𝑢(⋅))∈𝑆𝐴𝑃_𝜔(𝑋). By Lemma 3.1, 𝑣∈𝑆𝐴𝑃_𝜔(𝑋). On the other hand, if 𝑢₁,𝑢₂∈𝑆𝐴𝑃_𝜔(𝑋) we have the estimate ‖‖Γ𝑢₁(𝑡)−Γ𝑢₂(𝑡)‖‖≤𝐿𝑀^𝑡0𝑒^{−𝜇(𝑡−𝑠)}‖‖𝑢₁(𝑠)−𝑢₂(𝑠)‖‖𝑑𝑠≤𝐿𝑀𝜇‖‖𝑢¹−𝑢₂‖‖_∞.(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,∞)×𝑋→𝑋 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)‖𝑥−𝑦‖,(3.8) for all 𝑡≥0 and for all 𝑥,𝑦∈𝑋 with ‖𝑥‖≤𝑟 and ‖𝑦‖≤𝑟, where 𝐿∶[0,∞)→[0,∞) is a nondecreasing function. Let 𝐶=sup_𝑡≥0‖𝑔(𝑡,𝑅(𝑡)𝑥₀)‖. If there is 𝑟>0 such that 𝑀𝜇𝐿𝑟+𝑀‖‖𝑥⁰‖‖+𝐶𝑟<1,(3.9) then there is a unique 𝑆-asymptotically 𝜔-periodic mild solution of (1.1)-(1.2).

Proof. Let 𝑆𝐴𝑃^0𝜔(𝑋)={𝑣∈𝑆𝐴𝑃_𝜔(𝑋)∶𝑣(0)=0}. It is clear that 𝑆𝐴𝑃^0𝜔(𝑋) is a closed vector subspace of 𝑆𝐴𝑃_𝜔(𝑋). Let 𝐹∶𝑆𝐴𝑃^0𝜔(𝑋)→𝑆𝐴𝑃^0𝜔(𝑋) be the map defined by 𝐹𝑣(𝑡)=^𝑡0𝑅(𝑡−𝑠)𝑔𝑠,𝑣(𝑠)+𝑅(𝑠)𝑥₀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,𝑣₂∈𝑆𝐴𝑃^0𝜔(𝑋) with ‖𝑣₁‖_∞≤𝑟 and ‖𝑣₂‖_∞≤𝑟, we obtain that ‖‖𝐹𝑣₁(𝑡)−𝐹𝑣₂(𝑡)‖‖≤≤𝑀𝑀^𝑡0𝑒^{−𝜇(𝑡−𝑠)}𝐿𝑟+𝑀‖‖𝑥₀‖‖‖‖𝑣₁(𝑠)−𝑣₂(𝑠)‖‖𝑑𝑠𝜇𝐿𝑟+𝑀‖‖𝑥⁰‖‖‖‖𝑣₁−𝑣₂‖‖_∞.(3.11) Hence ‖‖𝐹𝑣₁−𝐹𝑣₂‖‖_∞≤𝑀𝜇𝐿𝑟+𝑀‖‖𝑥⁰‖‖‖‖𝑣₁−𝑣₂‖‖_∞.(3.12) On the other hand, for 𝑣∈𝑆𝐴𝑃^0𝜔(𝑋) with ‖𝑣‖_∞≤𝑟, we get ‖𝐹𝑣‖_∞≤‖𝐹𝑣−𝐹(0)‖_∞+‖𝐹(0)‖_∞≤𝑀𝜇𝐿𝑟+𝑀‖‖𝑥⁰‖‖‖𝑣‖_∞+𝐶𝑀𝜇.(3.13) Let 𝑟>0 be such that 𝑀𝜇𝐿𝑟+𝑀‖‖𝑥⁰‖‖𝑟+𝐶<𝑟.(3.14) From the above remarks it follows that 𝐹 is a contraction on 𝐵_𝑟(𝑆𝐴𝑃^0𝜔(𝑋)). Thus there is a unique fixed point 𝑣∈𝐵_𝑟(𝑆𝐴𝑃^0𝜔(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,∞)×𝑋→𝑋 is a function uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets that verifies the Lipschitz condition ‖𝑔(𝑡,𝑥)−𝑔(𝑡,𝑦)‖≤𝐿(𝑡)‖𝑥−𝑦‖,(3.15) for all 𝑥,𝑦∈𝑋 and every 𝑡≥0, where the function 𝐿(⋅) is locally integrable on [0,∞). If Θ=𝑀sup_𝑡≥0^𝑡0𝑒^{−𝜇(𝑡−𝑠)}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 𝑆𝐴𝑃_𝜔(𝑋) by the expression (3.5). For 𝑢∈𝑆𝐴𝑃_𝜔(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=sup_𝑠≥0‖𝑔(𝑠,𝑢(𝑠))‖<∞. Consequently, ‖𝑣(𝑡)‖≤^𝑡0𝑀𝑒^{−𝜇(𝑡−𝑠)}‖𝑔(𝑠,𝑢(𝑠))‖𝑑𝑠≤𝐶𝑀𝜇,𝑡≥0,(3.17) which shows that Γ𝑢 is a bounded continuous function on [0,∞).
We next prove that Γ is a Θ-contraction from 𝑆𝐴𝑃_𝜔(𝑋) into 𝑆𝐴𝑃_𝜔(𝑋). Let 𝑢∈𝑆𝐴𝑃_𝜔(𝑋). Next we set 𝐵={𝑢(𝑡)∶𝑡≥0}. We can write𝑣(𝑡+𝜔)−𝑣(𝑡)=−^𝜔0𝑅(𝑡+𝜔−𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠+^{𝜔𝑡+𝜔}𝑅(𝑡+𝜔−𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠_𝑡0=𝑅(𝑡−𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠_𝑡0+𝑅(𝑡−𝑠)[𝑔(𝑠+𝜔,𝑢(𝑠+𝜔))−𝑔(𝑠,𝑢(𝑠))]𝑑𝑠_{𝑡𝑡+𝜔}=𝑅(𝑠)𝑔(𝑡+𝜔−𝑠,𝑢(𝑡+𝜔−𝑠))𝑑𝑠_𝑡0+𝑅(𝑠)[𝑔(𝑡+𝜔−𝑠,𝑢(𝑡+𝜔−𝑠))−𝑔(𝑡−𝑠,𝑢(𝑡−𝑠))]𝑑𝑠_{𝑡𝑡+𝜔}=𝑅(𝑠)𝑔(𝑡+𝜔−𝑠,𝑢(𝑡+𝜔−𝑠))𝑑𝑠_𝑇0+𝑅(𝑠)[𝑔(𝑡+𝜔−𝑠,𝑢(𝑡+𝜔−𝑠))−𝑔(𝑡−𝑠,𝑢(𝑡+𝜔−𝑠))]𝑑𝑠_𝑇0+𝑅(𝑠)[𝑔(𝑡−𝑠,𝑢(𝑡+𝜔−𝑠))−𝑔(𝑡−𝑠,𝑢(𝑡−𝑠))]𝑑𝑠_𝑡𝑇+𝑅(𝑠)[𝑔(𝑡+𝜔−𝑠,𝑢(𝑡+𝜔−𝑠))−𝑔(𝑡−𝑠,𝑢(𝑡−𝑠))]𝑑𝑠_{𝑡𝑡+𝜔}=𝐼₁𝑅(𝑠)𝑔(𝑡+𝜔−𝑠,𝑢(𝑡+𝜔−𝑠))𝑑𝑠+𝐼₂+𝐼₃+𝐼₄.(3.18) Below we will estimate each one of the terms 𝐼_𝑖, 1≤𝑖≤4, 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)𝑒^−𝜇𝑇≤𝜀𝜇/9𝐶𝑀,(ii)‖𝑢(𝑡+𝜔)−𝑢(𝑡)‖≤𝜀^,(iii)‖𝑔(𝑡+𝜔,𝑥)−𝑔(𝑡,𝑥)‖≤𝜀^,for all 𝑡≥𝑇 and 𝑥∈𝐵. Let 𝑡≥2𝑇. Since 𝑡−𝑠≥𝑡−𝑇≥𝑇 for 0≤𝑠≤𝑇, we get ‖‖𝐼₁‖‖≤𝑀𝜀^^𝑇0𝑒^−𝜇𝑠𝑑𝑠≤𝜀‖‖𝐼₂3,‖‖≤𝜀^𝑀‖‖^𝑇0𝐼𝑒^−𝜇𝑠𝐿(𝑡−𝑠)𝑑𝑠≤𝜀^𝑀^𝑡0𝑒^−𝜇𝑠𝐿(𝑡−𝑠)𝑑𝑠≤𝜀^Θ,₃‖‖≤𝑀^𝑡𝑇𝑒^−𝜇𝑠2𝐶𝑑𝑠≤2𝐶𝑀𝜇𝑒^−𝜇𝑇‖‖𝐼₄,‖‖≤𝑀^{𝑡𝑡+𝜔}𝑒^−𝜇𝑠𝐶𝑑𝑠≤𝐶𝑀𝜇𝑒^−𝜇𝑇.(3.19) Combining these estimates, we find ‖𝑣(𝑡+𝜔)−𝑣(𝑡)‖≤𝜀,(3.20) for 𝑡≥2𝑇. Hence 𝑣∈𝑆𝐴𝑃_𝜔(𝑋).
On the other hand, if 𝑢₁,𝑢₂∈𝑆𝐴𝑃_𝜔(𝑋), and 𝑡≥0, we have ‖‖Γ𝑢₁(𝑡)−Γ𝑢₂(𝑡)‖‖≤𝑀^𝑡0𝑒^{−𝜇(𝑡−𝑠)}𝐿(𝑠)‖‖𝑢₁(𝑠)−𝑢₂(𝑠)‖‖𝑑𝑠≤Θ‖‖𝑢₁−𝑢₂‖‖_∞.(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,∞)×𝑋→𝑋 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 𝑥∈𝑋.

Theorem 3.5. Assume that 𝑔∶[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 𝜈≥0, lim_𝑡→∞(1/ℎ(𝑡))∫^𝑡0𝑒^{−𝜇(𝑡−𝑠)}𝑊(𝜈ℎ(𝑠))𝑑𝑠=0, where ℎ is the function in Lemma 2.5. We set 𝜎_𝜈(𝑡)=‖‖𝑅(⋅)𝑥₀‖‖_∞+𝑀^𝑡0𝑒^{−𝜇(𝑡−𝑠)}𝑊(𝜈ℎ(𝑠))𝑑𝑠,𝑡≥0,(3.22) and 𝜌(𝜈)=‖𝜎_𝜈‖_ℎ.(b)For each 𝜖>0, there is 𝛿>0 such that for every 𝑢,𝑣∈𝐶_ℎ(𝑋), ‖𝑢−𝑣‖_ℎ≤𝛿 implies that ^𝑡0𝑒^{−𝜇(𝑡−𝑠)}‖𝑔(𝑠,𝑢(𝑠))−𝑔(𝑠,𝑣(𝑠))‖𝑑𝑠≤𝜖,(3.23) for all 𝑡≥0.(c)For all 𝑎≥0 and 𝑟>0, the set {𝑔(𝑠,𝑥)∶0≤𝑠≤𝑎,𝑥∈𝑋,‖𝑥‖≤𝑟} is relatively compact in 𝑋.(d)liminf_𝜉→∞(𝜉/𝜌(𝜉))>1.Then the problem (1.1)-(1.2) has an 𝑆-asymptotically 𝜔-periodic mild solution.

Proof. Let Γ∶𝐶_ℎ(𝑋)→𝐶([0,∞);𝑋) be the map defined by the expression (3.5). Next, we prove that Γ has a fixed point in 𝑆𝐴𝑃_𝜔(𝑋). We divide the proof in several steps.(i)For 𝑢∈𝐶_ℎ(𝑋), we have that ‖Γ𝑢(𝑡)‖ℎ(𝑡)≤ℎ𝑀(𝑡)‖‖𝑥⁰‖‖+𝑀ℎ(𝑡)^𝑡0𝑒^{−𝜇(𝑡−𝑠)}𝑊‖𝑢‖_ℎℎ(𝑠)𝑑𝑠.(3.24) It follows from the condition (a) that Γ∶𝐶_ℎ(𝑋)→𝐶_ℎ(𝑋).(ii)The map Γ is continuous from 𝐶_ℎ(𝑋) into 𝐶_ℎ(𝑋). In fact, for 𝜖>0, let 𝛿>0 be the constant involved in the condition (b). For 𝑢,𝑣∈𝐶_ℎ(𝑋), ‖𝑢−𝑣‖_ℎ≤𝛿, taking into account that ℎ(𝑡)≥1, we get ‖Γ𝑢(𝑡)−Γ𝑣(𝑡)‖ℎ(𝑡)≤ℎ𝑀(𝑡)^𝑡0𝑒^{−𝜇(𝑡−𝑠)}‖𝑔(𝑠,𝑢(𝑠))−𝑔(𝑠,𝑣(𝑠))‖𝑑𝑠≤𝑀𝜖,(3.25) which implies that ‖Γ𝑢−Γ𝑣‖_ℎ≤𝑀𝜖. Since 𝜖>0 is arbitrary, this shows the assertion.(iii)We next show that Γ is completely continuous. Let 𝑉=Γ(𝐵_𝑟(𝐶_ℎ(𝑋))). We set 𝑣=Γ(𝑢) for 𝑢∈𝐵_𝑟(𝐶_ℎ(𝑋)). Initially, we prove that 𝑉(𝑡)={𝑣(𝑡)∶𝑣∈𝑉} is a relatively compact subset of 𝑋 for each 𝑡≥0. From the mean value theorem, 𝑣(𝑡)=𝑅(𝑡)𝑥₀+^𝑡0𝑅(𝑠)𝑔(𝑡−𝑠,𝑢(𝑡−𝑠))𝑑𝑠∈𝑅(𝑡)𝑥₀+𝑡𝑐(𝐾),(3.26) where 𝑐(𝐾) denotes the convex hull of 𝐾 and 𝐾={𝑅(𝑠)𝑔(𝜉,𝑥)∶0≤𝑠≤𝑡,0≤𝜉≤𝑡,‖𝑥‖≤𝑟}. 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 𝑉_𝑏={𝑣|_[0,𝑏]∶𝑣∈𝑉} is equicontinuous. In fact, for 𝑡≥0 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+𝑠−𝜉)𝑔(𝜉,𝑢(𝜉))𝑑𝜉_𝑡0(𝑅(𝜉+𝑠)−𝑅(𝜉))𝑔(𝑡−𝜉,𝑢(𝑡−𝜉))𝑑𝜉.(3.27) For each 𝜖>0, we can choose 𝛿₁>0 such that ‖‖‖^{𝑡𝑡+𝑠}𝑅(𝑡+𝑠−𝜉)𝑔(𝜉,𝑢(𝜉))𝑑𝜉‖‖‖≤𝑀^{𝑡𝑡+𝑠}𝑒^{−𝜇(𝑡+𝑠−𝜉)}𝑊(𝑟ℎ(𝜉))𝑑𝜉≤𝜖3,(3.28) for 𝑠≤𝛿₁. Moreover, since {𝑔(𝑡−𝜉,𝑢(𝑡−𝜉))∶0≤𝜉≤𝑡,𝑢∈𝐵_𝑟(𝐶_ℎ(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))𝑥₀‖≤𝜖/3, for 𝑠≤𝛿₂ and ‖(𝑅(𝜉+𝑠)−𝑅(𝜉))𝑔(𝑡−𝜉,𝑢(𝑡−𝜉))‖≤𝜖/3(𝑏+1), for 𝑠≤𝛿₃. Combining these estimates, we get ‖𝑣(𝑡+𝑠)−𝑣(𝑡)‖≤𝜖 for |𝑠|≤min{𝛿₁,𝛿₂,𝛿₃} with 𝑡+𝑠≥0 and for all 𝑢∈𝐵_𝑟(𝐶_ℎ(𝑋)).
Finally, applying condition (a), we can show that‖𝑣(𝑡)‖ℎ(𝑡)≤𝑀‖‖𝑥⁰‖‖ℎ(𝑡)+ℎ𝑀(𝑡)^𝑡0𝑒^{−𝜇(𝑡−𝑠)}𝑊(𝑟ℎ(𝑠))𝑑𝑠⟶0,𝑡⟶∞,(3.29) and this convergence is independent of 𝑢∈𝐵_𝑟(𝐶_ℎ(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_ℎ(𝑋).(iv)If 𝑢^𝜆(⋅) is a solution of the equation 𝑢^𝜆=𝜆Γ(𝑢^𝜆) for some 0<𝜆<1, we have the estimate ‖𝑢^𝜆‖_ℎ/𝜌(‖𝑢^𝜆‖_ℎ)≤1 and, combining this estimate with the condition (d), we conclude that the set 𝐾={𝑢^𝜆∶𝑢^𝜆=𝜆Γ(𝑢^𝜆),𝜆∈(0,1)} is bounded.(v) Since 𝑆𝐴𝑃_𝜔(𝑋)⊂𝐶_ℎ(𝑋), it follows from Lemmas 2.4 and 3.1 that Γ(𝑆𝐴𝑃_𝜔(𝑋))⊆𝑆𝐴𝑃_𝜔(𝑋) and, consequently, we can consider Γ∶𝑆𝐴𝑃_𝜔(𝑋)^{𝐶ℎ(𝑋)}→𝑆𝐴𝑃_𝜔(𝑋)^{𝐶ℎ(𝑋)}, where 𝐵^{𝐶ℎ(𝑋)} denotes the closure of a set 𝐵 in the space 𝐶_ℎ(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 𝑢∈𝑆𝐴𝑃_𝜔(𝑋)^{𝐶ℎ(𝑋)}. Let (𝑢_𝑛)_𝑛 be a sequence in 𝑆𝐴𝑃_𝜔(𝑋) such that 𝑢_𝑛→𝑢 in the norm of 𝐶_ℎ(𝑋). For 𝜀>0, let 𝛿>0 be the constant in (b), there is 𝑛₀∈ℕ so that ‖𝑢_𝑛−𝑢‖_ℎ≤𝛿, for all 𝑛≥𝑛₀. We observe that for 𝑛≥𝑛₀‖‖Γ𝑢_𝑛−Γ𝑢‖‖_∞=sup_𝑡≥0‖‖‖^𝑡0≤𝑀sup_𝑡≥0𝑅(𝑡−𝑠)𝑔𝑠,𝑢_𝑛(𝑠)−𝑔(𝑠,𝑢(𝑠))𝑑𝑠‖‖‖^𝑡0𝑒^{−𝜇(𝑡−𝑠)}‖‖𝑔𝑠,𝑢_𝑛(𝑠)−𝑔(𝑠,𝑢(𝑠))‖‖𝑑𝑠≤𝜖.(3.30) Hence (Γ𝑢_𝑛)_𝑛 converges to Γ𝑢=𝑢 uniformly in [0,∞). This implies that 𝑢∈𝑆𝐴𝑃_𝜔(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(𝑡−𝑢𝑠)𝑢(𝑠)𝑑𝑠+𝑔𝑡,𝑢_𝑡,𝑡≥0,₀=𝜑,(4.1) here 𝜑∈𝐶([−𝑟,0];𝑋), the history of the function 𝑢(⋅) is given by 𝑢_𝑡∶[−𝑟,0]→𝑋, 𝑢_𝑡(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,∞)∈𝑆𝐴𝑃_𝜔(𝑋). Then the function [0,∞)→𝐶([−𝑟,0];𝑋), 𝑡↦𝑢_𝑡, is 𝑆-asymptotically 𝜔-periodic.

Definition 4.2. A function 𝑢∈𝐶([−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(𝑡−𝑠)𝑔𝑠,𝑢_𝑠𝑑𝑠,𝑡≥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 𝑔∶[0,∞)×𝐶([−𝑟,0];𝑋)→𝑋 is a uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets function that verifies the Lipschitz condition ‖‖𝑔𝑡,𝜓₁−𝑔𝑡,𝜓₂‖‖≤𝐿‖‖𝜓₁−𝜓₂‖‖_∞,(4.3) for all 𝜓₁,𝜓₂∈𝐶([−𝑟,0];𝑋) and every 𝑡≥0. 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 𝑢_𝑡 is given by 𝑢_𝑡∶(−∞,0]→𝑋, 𝑢_𝑡(𝜃)=𝑢(𝑡+𝜃). To model this problem we assume that 𝑢_𝑡 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 (−∞,0] into 𝑋 endowed with a seminorm ‖⋅‖_ℬ and verifying the following axioms. (A)If 𝑥∶(−∞,𝜎+𝑎)→𝑋, 𝑎>0,𝜎∈ℝ, is continuous on [𝜎,𝜎+𝑎) and 𝑥_𝜎∈ℬ, then for every 𝑡∈[𝜎,𝜎+𝑎) the following hold: (i)𝑥_𝑡 is in ℬ. (ii)‖𝑥(𝑡)‖≤𝐻‖𝑥_𝑡‖_ℬ. (iii)‖𝑥_𝑡‖_ℬ≤𝐾(𝑡−𝜎)sup{‖𝑥(𝑠)‖∶𝜎≤𝑠≤𝑡}+𝑀(𝑡−𝜎)‖𝑥_𝜎‖_ℬ,

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 𝑡→𝑥_𝑡 is continuous from [𝜎,𝜎+𝑎) into ℬ. (B)The space ℬ is complete. (C-2)If (𝜓^𝑛)_𝑛∈ℕ is a uniformly bounded sequence of continuous functions with compact support and 𝜓^𝑛→𝜓, 𝑛→∞, in the compact-open topology, then 𝜓∈ℬ and ‖𝜓^𝑛−𝜓‖_ℬ→0 as 𝑛→∞.

We introduce the space ℬ₀={𝜓∈ℬ∶𝜓(0)=0} and the operator 𝑆(𝑡)∶ℬ→ℬ given by [𝑆(𝑡)𝜓](𝜃)=𝜓𝜓((0𝑡)+,𝜃),−−𝑡∞≤<𝜃𝜃≤<0,−𝑡.(4.4) It is well known that (𝑆(𝑡))_𝑡≥0 is a 𝐶₀-semigroup ([20]).

Definition 4.4. The phase space ℬ is said to be a fading memory space if ‖𝑆(𝑡)𝜓‖_ℬ→0 as 𝑡→∞ for every 𝜓∈ℬ₀.

Remark 4.5. Since ℬ verifies axiom (C-2), the space 𝐶_𝑏((−∞,0],𝑋) consisting of continuous and bounded functions 𝜓∶(−∞,0]→𝑋 is continuously included in ℬ. Thus, there exists a constant 𝑄≥0 such that ‖𝜓‖_ℬ≤𝑄‖𝜓‖_∞, for every 𝜓∈𝐶_𝑏((−∞,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 𝐶_𝑟×𝐿^𝑝(𝜌,𝑋)

Let 𝑟≥0,1≤𝑝<∞ and let 𝜌̃∶(−∞,−𝑟]→ℝ be a nonnegative measurable function which satisfies the conditions (g-5)-(g-6) in the terminology of [20]. Briefly, this means that 𝜌̃ is locally integrable and there exists a nonnegative locally bounded function 𝛾 on (−∞,0] such that 𝜌̃(𝜉+𝜃)≤𝛾(𝜉)𝜌̃(𝜃), for all 𝜉≤0 and 𝜃∈(−∞,−𝑟)⧵𝑁_𝜉, where 𝑁_𝜉⊆(−∞,−𝑟) is a set whose Lebesgue measure zero.

The space ℬ=𝐶_𝑟×𝐿^𝑝(𝜌̃,𝑋) consists of all classes of functions 𝜑∶(−∞,0]↦𝑋 such that 𝜑 is continuous on [−𝑟,0], Lebesgue-measurable, and 𝜌̃‖𝜑‖^𝑝 is Lebesgue integrable on (−∞,−𝑟). The seminorm in 𝐶_𝑟×𝐿^𝑝(𝜌̃,𝑋) is defined as follows:‖𝜑‖_ℬ=sup{‖𝜑(𝜃)‖∶−𝑟≤𝜃≤0}+^{−𝑟−∞}𝜌̃(𝜃)‖𝜑(𝜃)‖^𝑝𝑑𝜃^1/𝑝.(4.5) The space ℬ=𝐶_𝑟×𝐿^𝑝(𝜌̃,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+(∫^0−𝑡𝜌̃(𝜃)𝑑𝜃)^1/2 for 𝑡≥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 𝑢∶ℝ→𝑋 be a function with 𝑢₀∈ℬ and 𝑢_|[0,∞)∈𝑆𝐴𝑃_𝜔(𝑋). Then the function 𝑡→𝑢_𝑡 belongs to 𝑆𝐴𝑃_𝜔(ℬ).

Next we assume that 𝜑∈ℬ and that 𝑔∶[0,∞)×ℬ→𝑋 is a continuous function.

Definition 4.8. A function 𝑢∈𝐶(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(𝑡−𝑠)𝑔𝑠,𝑢_𝑠𝑑𝑠,𝑡≥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 𝑔∶[0,∞)×ℬ→𝑋 is a uniformly 𝑆-asymptotically 𝜔-periodic on bounded sets function that verifies the Lipschitz condition ‖‖𝑔𝑡,𝜓₁−𝑔𝑡,𝜓₂‖‖≤𝐿‖‖𝜓₁−𝜓₂‖‖_ℬ,(4.7) for all 𝜓₁,𝜓₂∈ℬ and every 𝑡≥0. 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 ℝ³ with 𝐶^∞ boundary; let 𝛼 and 𝛽 be in 𝐶²([0,∞),ℝ) with 𝛼(0) and 𝛽(0) positive, and let a∶[0,∞)→ℝ and 𝑏∶𝐻¹⁰(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, 21–23])𝜃^(𝑡)+𝛽(0)𝜃^(𝑡)=𝛼(0)Δ𝜃(𝑡)−^𝑡0𝛽^(𝑡−𝑠)𝜃^(𝑠)𝑑𝑠+^𝑡0𝛼^(𝑡−𝑠)Δ𝜃(𝑠)𝑑𝑠+𝑎(𝑡)𝑏(𝜃(𝑡)),(5.1) for 𝑡≥0, 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=𝛼(00)Δ−𝛽𝐼(0)𝐼(5.3) on the domain 𝐷(𝐴)=(𝐻²(Ω)∩𝐻¹⁰(Ω))×𝐻¹⁰(Ω), and 𝐵(𝑡)=𝐹(𝑡)𝐴, where 𝐹(𝑡)=[𝐹_𝑖𝑗(𝑡)]∶𝑋→𝑋, 𝑡≥0, 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 variable𝑢(𝑡)=𝜃𝜃(𝑡)_(𝑡)∈𝑋,(5.4) and defining𝑔(𝑡,𝑢(𝑡))=𝑎(𝑡)𝑏0(𝜃(𝑡)),(5.5) the equation (5.1) with initial condition𝑢(0)=𝜃⁰𝜂₀(5.6) takes the abstract form (1.1)-(1.2).

It follows from [24] that 𝐴 generates a 𝐶₀-semigroup (𝑇(𝑡))_𝑡≥0 such that ‖𝑇(𝑡)‖≤𝑀𝑒^−𝛾𝑡 for all 𝑡≥0 and some constants 𝑀,𝛾>0. Assume that 𝛼^(𝑡)𝑒^𝛾𝑡, 𝛼^(𝑡)𝑒^𝛾𝑡, 𝛽^(𝑡)𝑒^𝛾𝑡, and 𝛽^(𝑡)𝑒^𝛾𝑡 are bounded and uniformly continuous functions on [0,∞), and that for all 𝑡≥0, ||𝛽^(𝑡)||+max{𝛽(0),1}||𝛼^(𝑡)||||𝛽_𝛼(0)≤𝛾2𝑒𝑀^−𝛾𝑡,(𝑡)||+max{𝛽(0),1}||𝛼^(𝑡)||𝛼(0)≤𝛾²𝑒^−𝛾𝑡4𝑀².(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‖𝑅(𝑡)‖≤𝑀𝑒^{(−𝛾/2)𝑡},𝑡≥0.(5.8) In addition, suppose that 𝑎∈𝑆𝐴𝑃_𝜔(ℝ) and 𝑏∶𝐻¹⁰(Ω)→𝐿²(Ω) satisfies‖‖𝑏(𝜃₁)−𝑏(𝜃₂)‖‖_𝐿2(Ω)≤𝐿_𝑏‖‖𝜃₁−𝜃₂‖‖_𝐻10(Ω),(5.9) for all 𝜃₁,𝜃₂∈𝐻¹⁰(Ω).

We claim that for each 𝜃₀∈𝐻¹⁰(Ω) and 𝜂₀∈𝐿²(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=‖𝑎‖_∞𝐿_𝑏. In addition, estimates ‖𝑔(𝑡+𝜔,𝑢)‖−𝑔𝑔(𝑡(,𝑡𝑢,𝑢)‖)‖^{𝐻10(Ω)×𝐿2(Ω)}_{𝐻10(Ω)×𝐿2(Ω)}≤‖𝑎‖_∞𝐿^𝑏‖𝑢‖_𝐻¹⁰_(Ω)×𝐿²_(Ω)+‖𝑏(0)‖_𝐿²_(Ω),≤||𝑎(𝑡+𝜔)−𝑎(𝑡)||𝐿_𝑏‖𝑢‖_𝐻¹⁰_(Ω)×𝐿²_(Ω)+‖𝑏(0)‖_𝐿²_(Ω),(5.10) which are verified for 𝑡≥0 and 𝑢∈𝐻¹⁰(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)‖𝑎‖_∞𝐿_𝑏<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 𝑎∈𝑆𝐴𝑃_𝜔(ℝ), 𝜏∈(0,1) and let 𝑏∶𝐻¹⁰(Ω)→𝐿²(Ω) be a function that satisfies the Hölder type condition ‖‖𝑏(𝜃₁)−𝑏(𝜃₂)‖‖_𝐿2(Ω)≤𝐿_𝑏‖‖𝜃₁−𝜃₂‖‖^𝜏𝐻_10(Ω),(5.11) for all 𝜃₁,𝜃₂∈𝐻¹⁰(Ω).(H-4)There is a continuous nondecreasing function ℎ∶[0,∞)→[1,∞) such that ℎ(𝑡)→∞ as 𝑡→∞ and (1)lim_𝑡→∞(1/ℎ(𝑡))∫^𝑡0𝑒^{(−𝛾/2)(𝑡−𝑠)}ℎ(𝑠)^𝜏𝑑𝑠=0, (2)sup_𝑡≥0∫^𝑡0𝑒^{(−𝛾/2)(𝑡−𝑠)}|𝑎(𝑠)|ℎ(𝑠)^𝜏𝑑𝑠<∞.

For a concrete example, take 0<𝜆<𝛾/2(1−𝜏), 𝑎(𝑡)=𝑒^−𝜆𝑡, and ℎ(𝑡)=𝑒^𝜆𝑡.

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+𝜔,𝑢)−𝑔‖(𝑔𝑡,(𝑢𝑡,)𝑢‖)‖_𝐻¹⁰_(Ω)×𝐿²_{(Ω)𝐻10(Ω)×𝐿2(Ω)}≤‖𝑎‖_∞𝐿^𝑏‖𝑢‖^𝜏𝐻_{10(Ω)×𝐿2(Ω)}+‖𝑏(0)‖_𝐿²_(Ω),(5.12)≤||𝑎(𝑡+𝜔)−𝑎(𝑡)||𝐿_𝑏‖𝑢‖^𝜏𝐻_{10(Ω)×𝐿2(Ω)}+‖𝑏(0)‖_𝐿²_(Ω),(5.13) for all 𝑡≥0 and 𝑢∈𝐻¹⁰(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)≤||𝑎(𝑡)||𝐿_𝑏‖𝑢−𝑣‖^𝜏𝐻_{10(Ω)×𝐿2(Ω)},(5.14) for all 𝑡≥0 and 𝑢,𝑣∈𝐻¹⁰(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‖_∞(𝐿_𝑏𝜉^𝜏+‖𝑏(0)‖_𝐿²_(Ω)). Consequently, the function 𝑔 satisfies the assumption (H-2). From (H-4), for 𝑢,𝑣∈𝐶_ℎ(𝐻¹⁰(Ω)×𝐿²(Ω)) we can infer that 1ℎ(𝑡)^𝑡0≤‖𝑒𝑎‖^{−(𝛾/2)(𝑡−𝑠)}_∞𝑊(𝜈ℎ(𝑠))𝑑𝑠𝐿_𝑏𝜈^𝜏ℎ1(𝑡)^𝑡0𝑒^{(−𝛾/2)(𝑡−𝑠)}ℎ(𝑠)^𝜏𝑑𝑠+2‖𝑏(0)‖^𝐿2(Ω)sup_𝑡≥0𝛾ℎ(𝑡)⟶0,𝑡⟶∞,^𝑡0≤𝑒sup^{−(𝛾/2)(𝑡−𝑠)}_𝑡≥0‖𝑔(𝑠,𝑢(𝑠))−𝑔(𝑠,𝑣(𝑠))‖_𝐻¹⁰_(Ω)×𝐿²_(Ω)𝑑𝑠^𝑡0𝑒^{(−𝛾/2)(𝑡−𝑠)}||𝑎(𝑠)||ℎ(𝑠)^𝜏𝑑𝑠‖𝑢−𝑣‖^𝜏ℎ.(5.15) Therefore, conditions (a) and (b) of Theorem 3.5 are satisfied. A straightforward computation shows that liminf_𝜉→∞(𝜉/𝜌(𝜉))>1, where 𝜌(𝜉)=sup_𝑡≥01ℎ(𝑡)‖‖‖‖𝑅(⋅)𝜃𝜂⁰₀‖‖‖‖+𝑀^𝑡0𝑒^{(−𝛾/2)(𝑡−𝑠)}𝑊(𝜉ℎ(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)∶0≤𝑠≤𝑠₀,𝜃∈𝐻¹⁰(Ω),‖𝜃‖_𝐻¹⁰_(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^(𝑡−𝑠)𝜃^(𝑠)𝑑𝑠+^𝑡0𝛼^(𝑡−𝑠)Δ𝜃(𝑠)𝑑𝑠+𝑎(𝑡)𝑏(𝜃(𝑡−𝑟)),(5.17) for 𝑡≥0. Using the previous development, we model this problem in the space 𝑋=𝐻¹⁰(Ω)×𝐿²(Ω), and we consider 𝑢_𝑡∈𝐶([−𝑟,0];𝑋) for 𝑡≥0. To be consistent with our model, we study the equation (5.17) with initial condition𝑢₀=𝜓𝜑,(5.18) where 𝜑∈𝐶([−𝑟,0];𝐻¹⁰(Ω)) and 𝜓∈𝐶([−𝑟,0];𝐿²(Ω)). The function 𝑔∶[0,∞)×𝐶([−𝑟,0];𝑋)→𝑋 is given by𝑔𝑡,𝜑¹,𝜓¹=𝑎(𝑡)𝑏𝜑0₁(−𝑟).(5.19) Applying now the Corollary 4.3, we obtain the following result.

Proposition 5.3. Under the above conditions, if (2𝑀/𝛾)‖𝑎‖_∞𝐿_𝑏<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.