#### Abstract

We study the existence of -asymptotically -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 -asymptotically -periodic solutions for a class of abstract integrodifferential equations of the form where and for are densely defined closed linear operators in a Banach space . We assume that for every and that 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 ([1β3]) for well posedness to ([4β6] 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 -asymptotically -periodic functions with values in Banach spaces is very recent (see [9β16]). To the best of our knowledge, the study of the existence of -asymptotically -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 [1β3] 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 -asymptotically -periodic functions and resolvent operators. In the third section, we establish very general results about the existence of -asymptotically -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 -asymptotically -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 and be Banach spaces. In this work denotes the Banach space consisting of all continuous and bounded functions from into endowed with the norm of the uniform convergence which is denoted by . As usual, is the vector space of all functions such that . Also, we denote by the vector space of all continuous functions such that uniformly for in compact subsets of . The notation stands for the Banach space of bounded linear operators from into . Besides, we denote by the closed ball with center at 0 and radius . We begin by recalling the concept of -asymptotically -periodic functions. In the rest of this paper, is a fixed real number.

*Definition 2.1 (see [14]). *A function is called -asymptotically -periodic if
In this case, we say that is an asymptotic period of .

In this work, represents the subspace of consisting of all -asymptotically -periodic functions. It is easy to see that is a Banach space.

*Definition 2.2 (see [14]). *A function is said to be uniformly -asymptotically -periodic on bounded sets if for every bounded set , the set is bounded and , uniformly for .

*Definition 2.3 (see [14]). *A function is said to be asymptotically uniformly continuous on bounded sets if for every and every bounded set , there are constants and such that for all and every with .

Lemma 2.4 (see [14]). *Assume that is uniformly -asymptotically -periodic on bounded sets and asymptotically uniformly continuous on bounded sets. If , then the function belongs to .*

Let be a continuous nondecreasing function such that as . Next, the notation stands for the space endowed with the norm .

Lemma 2.5 (see [17]). *A set is relatively compact in if it verifies the following conditions. *(c-1)*For all , the set is relatively compact in .*(c-2)* uniformly for .*

Now, we include some preliminaries concerning resolvent operators. In the following definition, represents the space endowed with the graph norm given by .

*Definition 2.6 (see [18]). *A family of continuous linear operators on is called a resolvent operator for (1.1) if the following conditions are fulfilled. (R1)For each , and . (R2)The map is strongly continuous. (R3)For each , the function is continuously differentiable and

In what follows, we assume that there exists a resolvent operator for (1.1). The existence of solutions of the problem has been studied for many authors. Assuming that is locally integrable and following [2] we affirm that 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 is called a mild solution of (1.1)-(1.2) if

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

*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 -asymptotically -periodic mild solutions for the problem (1.1)-(1.2). We will assume that there exists a resolvent operator which satisfies the condition (H-1). Initially we establish a basic property.

Lemma 3.1. *Let . Then
*

* Proof. *The estimate shows that . For , we select such that for all and . We have the following decomposition:
Hence, for , we obtain
which completes the proof.

Theorem 3.2. *Assume that is a uniformly -asymptotically -periodic on bounded sets function that verifies the Lipschitz condition
**
for all and every . If , then the problem (1.1)-(1.2) has a unique -asymptotically -periodic mild solution.*

* Proof. *We define the map on the space by the expression
We next prove that is a contraction from into . Initially we show that is a map -valued. Let . We abbreviate the notation by writing
Since , it remains to show that the function given by (3.6) belongs to . Considering that is asymptotically uniformly continuous on bounded sets and applying the Lemma 2.4, . By Lemma 3.1, . On the other hand, if we have the estimate
The fixed point of is the unique mild solution of (1.1)-(1.2). The proof is complete.

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

Theorem 3.3. *Assume that is a function uniformly -asymptotically -periodic on bounded sets that satisfies the local Lipschitz condition
**
for all and for all with and , where is a nondecreasing function. Let . If there is such that
**
then there is a unique -asymptotically -periodic mild solution of (1.1)-(1.2).*

* Proof. *Let . It is clear that is a closed vector subspace of . Let be the map defined by
Since 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 is well defined. For with and , we obtain that
Hence
On the other hand, for with , we get
Let be such that
From the above remarks it follows that is a contraction on . Thus there is a unique fixed point of . To finish the proof we note that is the -asymptotically -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 is a function uniformly -asymptotically -periodic on bounded sets that verifies the Lipschitz condition
**
for all and every , where the function is locally integrable on . If
**
then the problem (1.1)-(1.2) has a unique -asymptotically -periodic mild solution.*

* Proof. *We define the map on the space by the expression (3.5). For , let be the function given by (3.6). Since the function is bounded, it follows from the Definition 2.2 that . Consequently,
which shows that is a bounded continuous function on .

We next prove that is a -contraction from into . Let . Next we set . We can write
Below we will estimate each one of the terms , , of the above expression separately. For , let . We choose such that the following conditions hold:(i),(ii),(iii),for all and . Let . Since for , we get
Combining these estimates, we find
for . Hence .

On the other hand, if , and , we have
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 such that is bounded as . In what follows, we will show that using properly the stability of the resolvent operator we can establish existence results for functions with another type of asymptotic behavior at infinity. To establish our result, we consider functions that satisfies the following boundedness condition.(H-2)There is a continuous nondecreasing function such that for all and .

Theorem 3.5. *Assume that 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 , , where is the function in Lemma 2.5. We set
and .*(b)*For each , there is such that for every , implies that
for all .*(c)*For all and , the set is relatively compact in .*(d)*.**Then the problem (1.1)-(1.2) has an -asymptotically -periodic mild solution.*

* Proof. *Let be the map defined by the expression (3.5). Next, we prove that has a fixed point in . We divide the proof in several steps.(i)For , we have that
It follows from the condition (a) that .(ii)The map is continuous from into . In fact, for , let be the constant involved in the condition (b). For , , taking into account that , we get
which implies that . Since is arbitrary, this shows the assertion.(iii)We next show that is completely continuous. Let . We set for . Initially, we prove that is a relatively compact subset of for each . From the mean value theorem,
where denotes the convex hull of and . Combining the fact that is strongly continuous with the property (c), we infer that is a relatively compact set, and is also a relatively compact set. Let . We next show that the set is equicontinuous. In fact, for fixed we can decompose as
For each , we can choose such that
for . Moreover, since is a relatively compact set and is strongly continuous, we can choose and such that , for and , for . Combining these estimates, we get for with and for all .

Finally, applying condition (a), we can show that
and this convergence is independent of . Hence satisfies the conditions (c-1) and (c-2) of Lemma 2.5, which completes the proof that is a relatively compact set in .(iv)If is a solution of the equation for some , we have the estimate and, combining this estimate with the condition (d), we conclude that the set is bounded.(v) Since , it follows from Lemmas 2.4 and 3.1 that and, consequently, we can consider , where denotes the closure of a set in the space . 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 has a fixed point . Let be a sequence in such that in the norm of . For , let be the constant in (b), there is so that , for all . We observe that for
Hence converges to uniformly in . This implies that 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 -asymptotically -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 here , the history of the function is given by , , and is a continuous function. The following property is an immediate consequence of our definitions.

Lemma 4.1. *Let be a continuous function. If . Then the function , , is -asymptotically -periodic.*

*Definition 4.2. *A function is said to be a mild solution of (4.1) if and the integral equation
is verified.

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

Corollary 4.3. *Assume that is a uniformly -asymptotically -periodic on bounded sets function that verifies the Lipschitz condition
**
for all and every . If , then the problem (4.1) has a unique -asymptotically -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 is given by , . To model this problem we assume that 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 into endowed with a seminorm and verifying the following axioms. (A)If , , is continuous on and , then for every the following hold: (i) is in . (ii). (iii),

where is a constant; , is continuous, is locally bounded and are independent of . (A1)For the function in (A), the function is continuous from into . (B)The space is complete. (C-2)If is a uniformly bounded sequence of continuous functions with compact support and , , in the compact-open topology, then and as .

We introduce the space and the operator given by It is well known that is a -semigroup ([20]).

*Definition 4.4. *The phase space is said to be a fading memory space if as for every .

*Remark 4.5. * Since verifies axiom (C-2), the space consisting of continuous and bounded functions is continuously included in . Thus, there exists a constant such that , for every ([20, Proposition 7.1.1]).

Moreover, if is a fading memory space, then are bounded functions and we can choose (see [20, Proposition 7.1.5]).

*Example 4.6. *The phase space

Let and let be a nonnegative measurable function which satisfies the conditions (g-5)-(g-6) in the terminology of [20]. Briefly, this means that is locally integrable and there exists a nonnegative locally bounded function on such that , for all and , where is a set whose Lebesgue measure zero.

The space consists of all classes of functions such that is continuous on , Lebesgue-measurable, and is Lebesgue integrable on . The seminorm in is defined as follows: The space satisfies axioms (A), (A-1), and (B). Moreover, when and , it is possible to choose , and for (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 be a function with and . Then the function belongs to .*

Next we assume that and that is a continuous function.

*Definition 4.8. *A function is said to be a mild solution of (4.1) if and the integral equation
is verified.

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

Corollary 4.9. *Assume that is a uniformly -asymptotically -periodic on bounded sets function that verifies the Lipschitz condition
**
for all and every . If , then the problem (4.1) has a unique -asymptotically -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 be a bounded open connected subset of with boundary; let and be in with and positive, and let and be functions.

Let us consider the following equation that arises in the study of heat conduction in materials with memory (see [2, 21β23]) for , where is the Laplacian on .

We consider (5.1) with initial condition To model this problem, we consider the space and the linear operators on the domain , and , where , , is defined by , , .

Introducing the variable and defining the equation (5.1) with initial condition takes the abstract form (1.1)-(1.2).

It follows from [24] that generates a -semigroup such that for all and some constants . Assume that , , , and are bounded and uniformly continuous functions on , and that for all ,

Then, by Grimmer [2, Theorem 4.1], there is a resolvent operator associated to the operators and and satisfying In addition, suppose that and satisfies for all .

We claim that for each and 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 satisfies the Lipschitz condition (3.4) with . In addition, estimates which are verified for and show that is a function uniformly -asymptotically -periodic on bounded sets. As a consequence of Theorem 3.2, we obtain the following result.

Proposition 5.1. *Under the above conditions, if , then the problem (5.1)-(5.2) has a unique -asymptotically -periodic mild solution.*

To establish our next result, we assume the following conditions.(H-3)Let , and let be a function that satisfies the HΓΆlder type condition for all .(H-4)There is a continuous nondecreasing function such that as and (1), (2).

For a concrete example, take , , and .

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 -asymptotically -periodic mild solution.*

* Proof. *We have the following estimates:
for all and . It follows from (5.12)-(5.13) that the function is uniformly -asymptotically -periodic on bounded sets. In addition, from the estimate
for all and , we obtain that is asymptotically uniformly continuous on bounded sets. By (5.12) we are led to define . Consequently, the function satisfies the assumption (H-2). From (H-4), for we can infer that
Therefore, conditions (a) and (b) of Theorem 3.5 are satisfied. A straightforward computation shows that , where
Finally, since is bounded set with boundary, Rellich-Kondrachov's Theorem [25, Theorem IX.16] leads to the conclusion that , , is relatively compact in . Hence, condition (c) holds. Using the Theorem 3.5, we conclude that the problem (5.1)-(5.2) has an -asymptotically -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 . In this case the equation is for . Using the previous development, we model this problem in the space , and we consider for . To be consistent with our model, we study the equation (5.17) with initial condition where and . The function is given by Applying now the Corollary 4.3, we obtain the following result.

Proposition 5.3. *Under the above conditions, if , then the problem (5.17)-(5.18) has a unique -asymptotically -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.