Abstract and Applied Analysis

Volume 2012 (2012), Article ID 647103, 15 pages

doi:10.1155/2012/647103

Research Article

Existence of Mild Solutions for a Semilinear Integrodifferential Equation with Nonlocal Initial Conditions

Carlos Lizama¹ and Juan C. Pozo²

¹Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago 9160000, Chile

²Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Santiago 7810000, Chile

Received 26 October 2012; Accepted 14 December 2012

Academic Editor: Józef Banaś

Copyright © 2012 Carlos Lizama and Juan C. Pozo. 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.

Using Hausdorff measure of noncompactness and a fixed-point argument we prove the existence of mild solutions for the semilinear integrodifferential equation subject to nonlocal initial conditions , , , where , and for every the maps are linear closed operators defined in a Banach space . We assume further that for every , and the functions and are -valued functions which satisfy appropriate conditions.

1. Introduction

The concept of nonlocal initial condition has been introduced to extend the study of classical initial value problems. This notion is more precise for describing nature phenomena than the classical notion because additional information is taken into account. For the importance of nonlocal conditions in different fields, the reader is referred to [1–3] and the references cited therein.

The earliest works related with problems submitted to nonlocal initial conditions were made by Byszewski [4–7]. In these works, using methods of semigroup theory and the Banach fixed point theorem the author has proved the existence of mild and strong solutions for the first order Cauchy problem

where is an operator defined in a Banach space which generates a semigroup , and the maps and are suitable -valued functions.

Henceforth, (1.1) has been extensively studied by many authors. We just mention a few of these works. Byszewski and Lakshmikantham [8] have studied the existence and uniqueness of mild solutions whenever and satisfy Lipschitz-type conditions. Ntouyas and Tsamatos [9, 10] have studied this problem under conditions of compactness for the semigroup generated by and the function . Recently, Zhu et al. [11], have investigated this problem without conditions of compactness on the semigroup generated by , or the function .

On the other hand, the study of abstract integrodifferential equations has been an active topic of research in recent years because it has many applications in different areas. In consequence, there exists an extensive literature about integrodifferential equations with nonlocal initial conditions, (cf., e.g., [12–25]). Our work is a contribution to this theory. Indeed, this paper is devoted to study the existence of mild solutions for the following semilinear integrodifferential evolution equation:

where and for every the mappings are linear closed operators defined in a Banach space . We assume further that for every , and the functions and are -valued functions that satisfy appropriate conditions which we will describe later. In order to abbreviate the text of this paper, henceforth we will denote by the interval , and is the space of all continuous functions from to endowed with the uniform convergence norm.

The classical initial value version of (1.2), that is, for some , has been extensively studied by many researchers because it has many important applications in different fields of natural sciences such as thermodynamics, electrodynamics, heat conduction in materials with memory, continuum mechanics and population biology, among others. For more information, see [26–28]. For this reason the study of existence and other properties of the solutions for (1.2) is a very important problem. However, to the best of our knowledge, the existence of mild solutions for the nonlocal initial value problem (1.2) has not been addressed in the existing literature. Most of the authors obtain the existence of solutions and well-posedness for (1.2) by establishing the existence of a resolvent operator and a variation of parameters formula (see, [29, 30]). Using adaptation of the methods described in [11], we are able to prove the existence of mild solutions of (1.2) under conditions of compactness of the function and continuity of the function for . Furthermore, in the particular case for all , where the operator is the infinitesimal generator of a -semigroup defined in a Hilbert space , and the kernel is a scalar map which satisfies appropriate hypotheses, we are able to give sufficient conditions for the existence of mild solutions only in terms of spectral properties of the operator and regularity properties of the kernel . We show that our abstract results can be applied to concrete situations. Indeed, we consider an example with a particular choice of the function and the operator is defined by

where the given coefficients , , satisfy the usual uniform ellipticity conditions.

2. Preliminaries

Most of the notations used throughout this paper are standard. So, , , , and denote the set of natural integers and real and complex numbers, respectively, , and .

In this work and always are complex Banach spaces with norms and ; the subscript will be dropped when there is no danger of confusion. We denote the space of all bounded linear operators from to by . In the case , we will write briefly . Let be an operator defined in . We will denote its domain by , its domain endowed with the graph norm by , its resolvent set by , and its spectrum by .

As we have already mentioned is the vector space of all continuous functions . This space is a Banach space endowed with the norm

In the same manner, for we write for denoting the space of all functions from to which are -times differentiable. Further, represents the space of all infinitely differentiable functions from to .

We denote by the space of all (equivalent classes of) Bochner-measurable functions such that is integrable for . It is well known that this space is a Banach space with the norm

We next include some preliminaries concerning the theory of resolvent operator for (1.2).

Definition 2.1. Let be a complex Banach space. A family of bounded linear operators defined in is called a resolvent operator for (1.2) if the following conditions are fulfilled.

 For each , and .  The map is strongly continuous.   For each , the function is continuously differentiable and

In what follows we assume that there exists a resolvent operator for (1.2) satisfying the following property.

  The function is continuous from to endowed with the uniform operator norm .

Note that property is also named in different ways in the existing literature on the subject, mainly the theory of -semigroups, namely, norm continuity for , eventually norm continuity, or equicontinuity.

The existence of solutions of the linear problem

has been studied by many authors. Assuming that is locally integrable, it follows from [29] that the function given by

is a mild solution of the problem (2.4). Motivated by this result, we adopt the following concept of solution.

Definition 2.2. A continuous function is called a mild solution of (1.2) if the equation

is verified.

The main results of this paper are based on the concept of measure of noncompactness. For general information the reader can see [31]. In this paper, we use the notion of Hausdorff measure of noncompactness. For this reason we recall a few properties related with this concept.

Definition 2.3. Let be a bounded subset of a normed space . The Hausdorff measure of noncompactness of is defined by

Remark 2.4. Let , be bounded sets of a normed space . The Hausdorff measure of noncompactness has the following properties.

(i)If , then . (ii), where denotes the closure of . (iii) if and only if is totally bounded. (iv) with .(v). (vi), where . (vii), where is the closed convex hull of .

We next collect some specific properties of the Hausdorff measure of noncompactness which are needed to establish our results. Henceforth, when we need to compare the measures of noncompactness in and , we will use to denote the Hausdorff measure of noncompactness defined in and to denote the Hausdorff measure of noncompactness on . Moreover, we will use for the Hausdorff measure of noncompactness for general Banach spaces .

Lemma 2.5. Let be a subset of continuous functions. If is bounded and equicontinuous, then the set is also bounded and equicontinuous.

For the rest of the paper we will use the following notation. Let be a set of functions from to and fixed, and we denote . The proof of Lemma 2.6 can be found in [31].

Lemma 2.6. Let be a bounded set. Then for all . Furthermore, if is equicontinuous on , then is continuous on , and

A set of functions is said to be uniformly integrable if there exists a positive function such that a.e. for all .

The next property has been studied by several authors; the reader can see [32] for more details.

Lemma 2.7. If is uniformly integrable, then for each the function is measurable and

The next result is crucial for our work, the reader can see its proof in [33, Theorem 2].

Lemma 2.8. Let be a Banach space. If is a bounded subset, then for each , there exists a sequence such that

The following lemma is essential for the proof of Theorem 3.2, which is the main result of this paper. For more details of its proof, see [34, Theorem 3.1].

Lemma 2.9. For all , denote . If and and let

then .

Clearly, a manner for proving the existence of mild solutions for (1.2) is using fixed-point arguments. The fixed-point theorem which we will apply has been established in [34, Lemma 2.4].

Lemma 2.10. Let be a closed and convex subset of a complex Banach space , and let be a continuous operator such that is a bounded set. Define

If there exist a constant and such that

then has a fixed point in the set .

3. Main Results

In this section we will present our main results. Henceforth, we assume that the following assertions hold.

  There exists a resolvent operator for (1.2) having the property .   The function is a compact map.   The function satisfies the Carathéodory type conditions; that is, is measurable for all and is continuous for almost all .   There exist a function and a nondecreasing continuous function such that

for all and almost all .   There exists a function such that for any bounded

for almost all .

Remark 3.1. Assuming that the function satisfies the hypothesis , it is clear that takes bounded set into bounded sets. For this reason, for each we will denote by the number .

The following theorem is the main result of this paper.

Theorem 3.2. If the hypotheses – are satisfied and there exists a constant such that

where , then the problem (1.2) has at least one mild solution.

Proof. Define by

for all .

We begin showing that is a continuous map. Let such that as (in the norm of ). Note that

by hypotheses and and by the dominated convergence theorem we have that when .

Let and denote and note that for any we have

Therefore and is a bounded set. Moreover, by continuity of the function on , we have that the set is an equicontinuous set of functions.

Define . It follows from Lemma 2.5 that the set is equicontinuous. In addition, the operator is continuous and is a bounded set of functions.

Let . Since the function is a compact map, by Lemma 2.8 there exists a sequence such that

By the hypothesis , for each we have . Therefore, by the condition we have

Since the function , for there exists satisfying . Hence,

where . Since is arbitrary, we have

Let . Since the function is a compact map and applying the Lemma 2.8 there exists a sequence such that

Using the inequality (3.10) we have that

Since is arbitrary, we have

By an inductive process, for all , it holds

where, for , the symbol denotes the binomial coefficient .

In addition, for all the set is an equicontinuous set of functions. Therefore, using the Lemma 2.6 we conclude that

Since and , it follows from Lemma 2.7 that there exists such that

Consequently, . It follows from Lemma 2.9 that has a fixed point in , and this fixed point is a mild solution of (1.2).

Our next result is related with a particular case of (1.2). Consider the following Volterra equation of convolution type:

where is a closed linear operator defined on a Hilbert space , the kernel , and the function is an appropriate -valued map.

Since (3.17) is a convolution type equation, it is natural to employ the Laplace transform for its study.

Let be a Banach space and . We say that the function is Laplace transformable if there is such that . In addition, we denote by , for , the Laplace transform of the function .

We need the following definitions for proving the existence of a resolvent operator for (3.17). These concepts have been introduced by Prüss in [28].

Definition 3.3. Let be Laplace transformable and . We say that the function is -regular if there exists a constant such that

for all and .

Convolutions of -regular functions are again -regular. Moreover, integration and differentiation are operations which preserve -regularity as well. See [28, page 70].

Definition 3.4. Let . We say that is a completely monotone function if and only if for all and .

Definition 3.5. Let such that is Laplace transformable. We say that is completely positive function if and only if

are completely monotone functions.

Finally, we recall that a one-parameter family of bounded and linear operators is said to be exponentially bounded of type if there are constants and such that

The next proposition guarantees the existence of a resolvent operator for (3.17) satisfying the property . With this purpose we will introduce the conditions and .

  The kernel defined by , for all , is -regular and completely positive.   The operator is the generator of a semigroup of type and there exists such that

Proposition 3.6. Suppose that is the generator of a -semigroup of type in a Hilbert space . If the conditions -are satisfied, then there exists a resolvent operator for (3.17) having the property .

Proof. Integrating in time (3.17) we get

Since the scalar kernel is completely positive and generates a -semigroup, it follows from [28, Theorem 4.2] that there exists a family of operators strongly continuous, exponentially bounded which commutes with , satisfying

On the other hand, using the condition and since the scalar kernel is -regular, it follows from [35, Theorem 2.2] that the function is continuous for . Further, since , it follows from (3.23) that for all the map is differentiable for all and satisfies

From the quality (3.24), we conclude that is a resolvent operator for (3.17) having the property .

Corollary 3.7. Suppose that generates a -semigroup of type in a Hilbert space . Assume further that the conditions - are fulfilled. If the hypotheses – are satisfied and there exists such that

then (3.17) has at least one mild solution.

Proof. It follows from Proposition 3.6 that there exists a resolvent operator for the equation and this resolvent operator has the property . Since the hypotheses – are satisfied, we apply Theorem 3.2 and conclude that (3.17) has at least one mild solution.

4. Applications

In this section we apply the abstract results which we have obtained in the preceding section to study the existence of solutions for a partial differential equation submitted to nonlocal initial conditions. This type of equations arises in the study of heat conduction in materials with memory (see [26, 27]). Specifically, we will study the following problem:

where is a continuous function such that for all , the constant and the constants satisfy the relation . The operator is defined by

where the coefficients , , satisfy the usual uniformly ellipticity conditions, and . The functions and satisfy appropriate conditions which will be specified later.

Identifying we model this problem in the space , where the group is defined as the quotient . We will use the identification between functions on and -periodic functions on . Specifically, in what follows we denote by the space of -periodic and square integrable functions from into . Consequently, (4.1) is rewritten as

where the function is defined by , and , where is integrable on , and is a bounded function satisfying a Lipschitz type condition with Lipschitz constant .

We will prove that there exists sufficiently small such that (4.3) has a mild solution on .

With this purpose, we begin noting that . Moreover, it is a well-known fact that the is a compact map.

Further, the function satisfies , with and . Thus, the conditions – are fulfilled.

Define , for all . Since the kernel defined by is -regular, it follows that is -regular. Furthermore, we claim that is completely positive. In fact, we have

Define the functions and by and , respectively. In other words

A direct calculation shows that

Since , we have that and are completely monotone. Thus, the kernel is completely positive.

On the other hand, it follows from [36] that generates an analytic, noncompact semigroup on . In addition, there exists a constant such that

It follows from the preceding fact and the Hille-Yosida theorem that for all such that . Let . By direct computation we have

Hence, for all , such that . This implies that

Since the semigroup generated by is an analytic semigroup we have

Therefore,

It follows from Proposition 3.6 that (4.3) admits a resolvent operator satisfying property .

Let and .

A direct computation shows that for each the number is equal to .

Therefore the expression is equivalent to .

Since there exists such that , then there exists such that

From Corollary 3.7 we conclude that there exists a mild solution of (4.1).

Acknowledgments

C. Lizama and J. C. Pozo are partially supported by FONDECYT Grant no. 1110090 and Ring Project ACT-1112. J. C. Pozo is also partially financed by MECESUP PUC 0711.

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