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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 503656, 12 pages

http://dx.doi.org/10.1155/2013/503656
Research Article

Existence of Some Semilinear Nonlocal Functional Differential Equations of Neutral Type

Department of Information Management, Yuan Ze University, 135 Yuan-Tung Road, Taoyuan 320, Taiwan

Received 13 August 2013; Accepted 22 September 2013

Academic Editor: Ngai-Ching Wong

Copyright © 2013 Hsiang Liu 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

This paper is concerned with the existence of mild and strong solutions on the interval for some neutral partial differential equations with nonlocal conditions. The linear part of the equations is assumed to generate a compact analytic semigroup of bounded linear operators, whereas the nonlinear part satisfies the Carathëodory condition and is bounded by some suitable functions. We first employ the Schauder fixed-point theorem to prove the existence of solution on the interval for that is small enough, and, then, by letting and using a diagonal argument, we have the existence results on the interval . This approach allows one to drop the compactness assumption on a nonlocal condition, which generalizes recent conclusions on this topic. The obtained results will be applied to a class of functional partial differential equations with nonlocal conditions.

1. Introduction

The purpose of this paper is to study the existence of mild and strong solutions for the following neutral evolution problem with nonlocal initial conditions: in a Banach space , where and generates an analytic compact semigroup on . The functions , , and will be specified later. The Cauchy problem with the nonlocal condition was first considered by Byszewski [1], and since it reflects physical phenomena more precisely than the classical initial condition does, this issue has gained enormous attention in the past several years. For more detailed information about the importance of nonlocal initial conditions in applications, we refer to the works of Byszewski [2], Byszewski and Lakshmikantham [3], and to many other authors [47] and the references therein.

Equation (1) has been studied by many authors under various assumptions on the linear part , the nonlinear terms , , and the nonlocal condition see, for example, [814]. A basic approach to this problem is to define the solution operator by and to use various fixed-point theorems, including Schauder fixed-point theorem, Banach contraction principle, Leray-Schauder alternative, and Sadovskii fixed-point theorem, to show that has a fixed point, which is the mild solution of (1). When using fixed-point theorems, it is necessary that the semigroup generated by the linear part of (1) be compact; that is, is a compact operator, for all , so that the norm continuity of , for , becomes a key point in the study of the existence of mild solutions. Thus, because of the absence of compactness of the solution operator at , most of the papers on the relevant topics (e.g., [810, 14]) assume complete continuity on the nonlocal term . However, it is too restrictive in terms of applications.

Recently, Liang et al. [15] observed the nonlocal Cauchy problem [1, 3, 4, 6] that the nonlocal condition is completely determined on , for some ; that is, such ignores the fact that ; for instance, in [4, 6], the function is given by where ’s are given constants, and in this case, we have measurements at rather than just at . Thus, by assuming that there is a such that the authors utilize fixed-point theorem twice to deduce the existence results. More recently, Liu and Yuan [16] gave existence results using Schauder fixed-point theorem and a limiting process under the following hypothesis.

There is a such that and , for all , , with and with the nonlinear term being bounded by an integrable function.

Motivated by the works in [15, 16], we drop the compactness assumption on the nonlocal condition and discuss the existence of solutions for (1). The obtained results generalize recent conclusions on this topic.

The present work is organized as follows. Section 1 is devoted the introduction of the problem we studied. Section 2, we explain some known notations and results we will use. The basic hypotheses on (1) are also given in this section. In Section 3, we study the existence of mild solutions to (1) and in Section 4, we investigate some conditions for (1) to come up with strong solutions. In Section 5, an example is given to illustrate the existence results.

2. Preliminaries

Throughout this paper, will be a fixed real number, will be a Banach space with norm , and is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators such that . Then, there exists a constant such that , for and it is possible to define the fractional power , for , as a closed linear operator on its domain with inverse (see [17]). The followings are the basic properties of .

Theorem 1 (see [17], pages 69–75). The following assertions hold:(i) is a Banach space with the norm , for .(ii) , for each .(iii) for each and .(iv)For every , is bounded on , and there exist and such that (v) is a bounded linear operator in with .(vi)If , then .

Let be the Banach space endowed with the norm . Then, we denote by the operator norm of , that is, , and let be the Banach space endowed with the supnorm given by and, for any , set . Moreover, let be the Banach space endowed with the supnorm given by The following hypotheses are the basic assumptions of this paper.

(H1) There exist and such that the function satisfies for all and .

(H2) The function satisfies the following conditions.(i)For each , the function is continuous, and, for each , the function is strongly measurable.(ii)For each and , there exists a positive function such that and there is a such that

(H3) The function is continuous, and there exist constants such that for .

3. Mild Solutions

Definition 2. A continuous function is called a mild solution of (1) on if, for each , the function is integrable on , and the following equation is satisfied: for all .

To see the existence of mild solution of nonlocal problem (1), we will, in view of (12), locate the fixed point of a mapping defined on by For this, we first observe the following result, where for all , we let .

Lemma 3. Assume that hypotheses (H1)–(H3) are satisfied, and, in addition, there holds the following inequality:

Then, , for some .

Proof. Suppose, on the contrary, that, for each , there exist and such that . Then, we have Dividing the two sides by and taking the lower limit as , we have which is a contradiction. This completes the proof.

By Lemma 3, we see that the mapping defined by (13) maps into itself. We will show that has a fixed point in . To see this, note first that is continuous by the continuity of , and . We decompose as , where We show that is a contraction in and is a compact operator in .

Lemma 4. Assume that hypotheses (H1)–(H4) are satisfied. If , then is a contraction in .

Proof. Observe that, for and , we have the assumption (H1) as follows: Hence, where which is, by (H4), less than 1. Thus, is a contraction.

Lemma 5. Assume that hypotheses (H1)–(H4) are satisfied, and, in addition, the following is given.

There exists a such that , and , for any , with and .

Then the problem (1) has at least one mild solution in for some .

Proof. Let be given by (H5), and let For any , let be defined by Now, we define on by Then, by Lemma 3, we see that . Consider as the sum , where and is defined on by With a similar argument as in the proof of Lemma 4, one sees that is a contraction on .

For the compactness of , note first that is continuous by the continuity of and . Now, to show that the set is relatively compact in , we will prove that, for each , the two sets are relatively compact in and that are equicontinuous families of functions on . In fact, it follows from (H3) and the compactness of , for that, for each , Moreover, since for each , and , the set is relatively compact, then, in view of we see by (H2) that there are relative compact sets arbitrarily close to , and, hence, is also relatively compact in . Now, by the norm continuity of , for , we see that independently of , and, hence, is an equicontinuous family of functions on . Finally, let be arbitrarily small, and we see that which is, by the norm continuity of , for , arbitrarily small and independent of as . Therefore, is an equicontinuous family of functions on and so is . It follows from Arzela-Ascoli’s theorem that is relatively compact on . Thus, the mapping defined by (24) is compact.

By the fixed-point theorem of Sadovskiĭ [18], this shows that has a fixed point in ; that is, there is a such that Now, define a function on by Then, on , and . Consequently, (H5) guarantees that That is, is a mild solution of (1).

For the main results in this section, we introduce a family of nonlocal neutral problems as follows. Firstly, we define, for each , an operator on by for all . It is clear that is bounded on and , and, hence, . Now, for each , we define by by and by Consider the following nonlocal neutral problem: In view of (35)–(38), the following result is an immediate corollary of Lemmas 4 and 5.

Lemma 6. Suppose that (H1)–(H4) are satisfied. Then, for any , the problem has at least one mild solution in .

Theorem 7. Suppose that, hypotheses (H1)–(H4) are satisfied. Then, problem (1) has at least one mild solution in for some .

Proof. Choose a decreasing sequence so that , and, then, by Lemma 6, we see that for each , there is an such that

Now, for each , we define on by Then, (39) implies that has a fixed point in which is a mild solution for the nonlocal Cauchy problem . Decompose as , where With the same argument as in the proof of Lemma 4, we have that is a contraction. Furthermore, since the sequence lies in , then a similar argument as in the proof of Lemma 5 (see (27)–(31)) shows that, for each , the sets are both relatively compact in and that the sequence of functions is equicontinuous on . Hence, it follows from Ascoli-Arzela theorem that Now, let be a decreasing sequence such that , and let be a subsequence of . Then, a similar argument as in the proof of Lemma 5 insures that is an equicontinuous sequence of functions on . Thus, Ascoli-Arzela theorem guarantees that the sequence Thus, by (44) and (45), we see that is relatively compact in , and, hence, we can select a subsequence of denoted by , which is a Cauchy sequence in . By a similar process, we can select a subsequence of denoted by , which is a Cauchy sequence in . Repeat the above argument, and use a diagonal argument to obtain a subsequence of denoted by . Then, for every , is a Cauchy sequence in , and thus, we can define the function by It is clear that is strongly measurable, , and It therefore follows from Lebesgue’s dominated convergence theorem that there is a subsequence of such that This shows that the sequence is relatively compact on , and, hence, by the continuity of , it follows that By (44) and (49), we see the relative compactness of on . Thus, there is a subsequence of denoted by and a function such that It is clear that . Since then (50) and the uniform continuity of imply that . By taking limits in (39), we see that is a mild solution of (1) and this completes the proof.

We will consider the case more generally; that is, the nonlocal condition is defined on rather than .

Theorem 8. Suppose that, hypotheses (H1) and (H2) are satisfied, and, in addition, there hold the following hypotheses.

The function is continuous, and inequality (11) also holds.

   , where If inequality (H4) holds, then, problem (1) has at least one mild solution in , for some .

Proof. Let and be the sequences defined as in the proof of Theorem 7. With the same arguments as in the proof of Lemma 4, we see that , for some . Moreover, it follows from the same arguments as in the proof of Theorem 7 that (44), and (45) also hold, and, for every subsequence of , there exist a subsequence and a function such that is continuous on and, for every , Let be given. It follows from (H7) and (53) that there is a such that and that, for every , there is an such that implies that Choose that is large enough so that , and define by Thus, (H7), (54), and (55) insure that And, hence, by the continuity of and the compactness of , for , (49) is also valid in this case. Therefore, a similar argument as in the last paragraph of the proof of Theorem 7 shows the existence of a mild solution for (1).

4. Strong Solutions

Definition 9. A mild solution is called a strong solution if is continuously differentiable on with and satisfies (1).

In the following, we establish a result of a strong solution for (1).

Theorem 10. Let be a reflexive Banach space. Suppose that there hold the following hypotheses.

The function is a continuous function and there exists such that for all and .

   is Lipschitz continuous; that is, there exists a constant such that for all .

The function is continuous, , for all , and for some .

There holds the following inequality: If and inequality (H4) also holds with ; is replaced by ; and , respectively, then (1) has a strong solution on .

Proof. Let be the operator defined by (13). By (H8), (H9) and (H10), one can use a similar argument as in the proof of Lemma 3 to deduce that there is a such that . For this , consider the set for some and that are large enough. It is clear that is nonempty, convex, and closed. We will prove that has a fixed point on . Obviously, from the proofs of Lemmas 4 and 5 and Theorem 7, it is sufficient to show that, for any , We first fix an element and observe that, for any , where and . Now, Thus, from (H8), (H9), and (H10), it follows that where is a constant independent of . Since (H11) implies that then whenever Therefore, has a fixed point which is a mild solution of (1). By the above calculation, we see that, for this , all of the functions are Lipschitz continuous, respectively. Since is Lipschitz continuous on and the space is reflexive, then a result of [19] asserts that is a.e. differentiable on and . A similar argument shows that , , , and also have this property. Furthermore, with a standard argument as in [17] (Theorem ), we have So the following holds, for almost all : This shows that is also a strong solution to the nonlocal Cauchy problem (1), and the proof is completed.

The following result is an immediate corollary of Theorems 8 and 10.

Corollary 11. Suppose that the hypotheses (H7)–(H9), and (H11) are satisfied, and in addition, there holds the following hypotheses.

The function is continuous, , for all , and inequality (60) sustains.

If and inequality (H4) also holds with ; and is replaced by ; , respectively, then (1) has a strong solution on .

5. An Example

In the last section, our existence results will be applied to solve the following system: where and equipped with norm .

The operator defined by Then, generates a compact, analytic semigroup of uniformly bounded linear operators. It is well known that , and, thus, the fractional powers of are well-defined where the eigenvalues of are and the corresponding normalized eigenvectors are , . Moreover, with , and the operator is given by with domain .

We need the following assumptions to solve (73) with our results.

(A1) The function satisfies the following conditions.(a) is welldefined and measurable with (b) , for each .

(A2) The function satisfies the following conditions.(a)For each , the function is measurable.(b)For each , the function is continuous.(c)There is an such that

(A3) The functions and satisfy the following conditions, respectively:(a) ,(b) and there is an such that Let be the Banach space equipped with supnorm, let be defined by and let be defined by Moreover, if we define by

Assumptions ( ) and ( ) imply the following conclusions.

Theorem 12. The functions and have the following inequalities. (a) satisfies hypothesis (H1) with and that is,