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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 451252, 11 pages
http://dx.doi.org/10.1155/2013/451252
Research Article

Existence of Solutions in Some Interpolation Spaces for a Class of Semilinear Evolution Equations with Nonlocal Initial Conditions

1Department of Applied Mathematics, I-Shou University, Ta-Hsu, Kaohsiung 84008, Taiwan
2Department of Information Management, Yang Ze University, Jhongli, Taoyuan 32097, Taiwan

Received 29 May 2013; Accepted 7 July 2013

Academic Editor: J. Liang

Copyright © 2013 Jung-Chan Chang and Hsiang Liu. 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 for a class of semilinear evolution equations with nonlocal initial conditions. The linear part is assumed to be a (not necessarily densely defined) sectorial operator in a Banach space . Considering the equations in the norm of some interpolation spaces between and the domain of the linear part, we generalize the recent conclusions on this topic. The obtained results will be applied to a class of semilinear functional partial differential equations with nonlocal conditions.

1. Introduction

The purpose of this paper is to study the existence of mild and strong solutions to the following semilinear evolution problem with nonlocal initial conditions: in a Banach space , where , and are given -valued functions which will be specified later, and is a linear closed operator in . By letting , we see that (1) can be considered as a special case of the following equation: Since the nonlocal condition reflects physical phenomena more precisely than the classical initial condition does, this issue has caught an increasing interest in the past several years. For the importance of nonlocal Cauchy problems in applications, we refer to the papers [1, 2] and the references therein.

The existence and uniqueness of mild and classical solutions of (1) have been investigated by many authors, see for example [114]. A basic approach of this problem is to define the solution operator by and use various fixed-point theorems, including Schauder fixed-point theorem, Banach contraction principle, Leray-Schauder alternative, Sadovskii fixed-point theorem, and some others, to show that has a fixed point, which is the mild solution of (1). To use fixed-point theorems, it is necessary that the semigroup generated by the linear part of (1) is 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 solution. In [13], the authors studied the existence of mild solution for (1) by using the Leray-Schauder alternative. However, as pointed out by Liu [11], the proof of the main result in [13] is invalid because of the absence of compactness of the solution operator at . This gap can be filled by simply assuming complete continuity on the nonlocal term , but it is too restrictive in applications. To take away this unsatisfactory condition, Liang et al. [7] observed that in many studies of nonlocal Cauchy problems, for example [1, 6, 9], the nonlocal condition is completely determined on , for some ; that is, such a ignores ; for instance, in [9] the function is given by where the are given constants, and in this case, we have measurements at rather than just at . Thus, by assuming that there is a such that with the authors use their method of utilizing fixed-point theorem twice to deduce the existence of mild solution of (1) with the nonlinear term being Lipschitz continuous and the nonlocal term being continuous and mapping into bounded sets. Afterward, Xiao and Liang [14] initiated the study of the nonautonomous case of (1) with nondensely defined sectorial linear part. More recently, Liu and Yuan [12] gave existence results by Schauder fixed-point theorem and a limiting process, under the following hypothesis: there is a such that with with the nonlinear term being bounded by an integrable function.

In the present paper, the linear part of (1) and (2) is assumed to be a (not necessarily densely defined) sectorial operator in , which means that there exist constants , , and such that Existence results of solutions to (1) and (2) are discussed in some interpolation spaces , where and (see [15], Chapter 2). We establish some existence results of (1) and (2) without assuming Lipschitz condition on the nonlinear term and complete continuity on the nonlocal condition. Thus, the hypotheses in this note are less restricted than relevant works and hence the obtained results are more applicable than recent conclusions on this topic.

This work is organized as follows. In Section 2, we collect some known notions and results on the interpolation spaces between and and give the basic hypotheses on (2). In Section 3, we study the existence of mild solutions to (2) and (1). Section 4 is devoted to the existence of strong solutions to (2) and (1), and in Section 5, we give an example to illustrate our abstract results.

2. Preliminaries

As usual, for a Banach space , is the space of linear operators on with operator norm denoted by . If satisfies (9), then it is known that the family of Dunford integrals where is the curve , is an analytic semigroup satisfying the following properties (see [15], Proposition 2.1.1, p. 35). (A1) for each , and . Moreover, for all and . (A2) for all . (A3) There are positive constants such that  (A4) The function belongs to and Moreover, it has an analytic extension in the sector We see by (A3), that the family is uniformly bounded on the interval ; that is, For , , and , we denote by the interpolation space with norm where is given by .Let be the Banach space endowed with the supnorm given by and for any , set .By (2.2.1) of [15] (p.44), we have And hence the analytic semigroup also satisfies the following property. (A5) For every Moreover, by (2.2.20) of [15] (see the last line of p.51), there is a constant such that  (A6) .

The following conditions are basic assumptions of this paper. (H1) The operator family is compact; that is, maps bounded subsets of into relatively compact subsets of for all . (H2) The function is continuous and for each , there exists a positive function such that (1) is integrable on for each ,(2) for almost all ,(3) , (4) .  (H3) The function is continuous and there exist constants and such that for . Moreover, let be the number defined by that is, if and if .

We will argue that the hypothesis (H2) is reasonable. In fact, consider a function satisfying that there exist and such that and define . It is easy to see that satisfies (H2) by letting Moreover, one can choose the functions to be in and find easily that there is a lot more functions satisfying (H2).

3. Mild Solutions

Definition 1. Let be a continuous function.(i) is called a mild solution of (1) on if satisfies for . (ii) is called a mild solution of (2) on if satisfies for .

The part of in is defined by By Proposition 2.2.7 of [15], we see that is sectorial in . We denote by the analytic semigroup generated by on . It is clear that for all and for all and .

Remark 2. It is known that is contained in ([15], Corollary 2.2.3 (ii), p.47) and that is continuous at if ([15], Proposition 2.1.4 (i), p.38).

Before stating and proving the main results, we need some lemmas.

Lemma 3. If (H1) is satisfied, then the following conclusions hold. (i) is norm-continuous for .(ii)If , then is compact.(iii)If is a bounded sequence in and , then for any , has a convergent subsequence in .

Proof. (i) Since the operator family is, by (H1), compact for , then is norm-continuous for . Note that which shows the norm continuity of for . A similar argument also shows that for all .
(ii) For the compactness of for , it is sufficient to show that the sequence has a convergent subsequence in for each sequence in the unit ball of and . Since then there is, by the compactness of , a subsequence of and an so that Thus, the boundedness of implies that If and , then (30), (31), and the Lebesgue dominated convergence theorem insure that as . On the other hand, using (14) and (31), we have as . Since , the assertion follows.
(iii) This follows immediately from the proof of (ii).

In the sequel, we assume that and satisfy . To see the existence of mild solution of nonlocal problem (2), we define the mapping on by and for each , let We have the following results.

Lemma 4. Assume that the conditions (H1)–(H3) are satisfied. If and, in addition,(H4) also holds, then for some .

Proof. If this were not the case, then for each , there would exist and such that and hence we see by (14), (28), (H2), (H3), and (A6) that Dividing on both sides by and taking the lower limit as we obtain which contradicts (H4). Hence for some .

The following lemma is useful in the proof of our main result in this section.

Lemma 5. Assume that the conditions (H1)–(H4) are satisfied. If and, in addition, (H5) there exists a , such that , for any with , ,
then the nonlocal problem (2) has at least one mild solution in .

Proof. Let where is the number defined in (H5). For , find such that for all and define and by for and , respectively. By (H2), (H3), and (H5), we see that and are well-defined, continuous and satisfy Now let us define a mapping on as follows: We will show that has a fixed point in . To see this, note first that is continuous by the continuity of and . Then, in view of Schauder’s fixed point theorem, it suffices to show that the set is relatively compact in . Thus, we will prove that for each , both sets are relatively compact in , and that both are equicontinuous families of functions on .
In fact, from the compactness of for it follows that is relatively compact in for . Moreover, for each , , and , consider the mapping defined by Since is relatively compact in by the compactness of and since then we see by (H2) that there are relative compact sets arbitrarily close to and hence is also relatively compact in . Therefore, the set is relatively compact in for each .
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 by (A6) with that which is, by the norm continuity of for and dominated convergence theorem, arbitrarily small independently of as . Therefore, is an equicontinuous family of functions on .
Consequently, by Arzela-Ascoli’s theorem, it follows that is relatively compact in . Thus there is a function such that ; that is, or
Now, define a function on by Then on and we have . By the definition of , , and (H5), we see that , and consequently, that is, is a mild solution of (2), and the proof is completed.

For our main result in this section, we introduce in the following a family of nonlocal Cauchy problems. Firstly, we define, for each , an operator on by It is clear that is bounded on and , and hence . Now, for each , we define by and by and consider the nonlocal Cauchy problems: In view of (52), (53), and (54), the following result is an immediate corollary of Lemma 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 the hypotheses (H1)–(H4) are satisfied. If , then the problem (2) 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 , or is a mild solution for the nonlocal Cauchy problem ( ).
Since the sequence lies in , then a similar argument as in the proof of Lemma 5 (see (44)–(46) shows that that and that Hence it follows by 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 (60) and (61), 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 so, we can define the function by It is clear that is strongly measurable, and It therefore follows by Lebesgue’s dominated convergence theorem that there is a subsequence of such that as . This shows that the sequence
By (60) and (65), we see the relative compactness of in . Thus, there is a subsequence of , denoted by and a function such that It is clear that . Since then (66) and the uniform continuity of imply that . By taking limits in (56), we see that is a mild solution of (2) and this completes the proof.

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

Theorem 8. Suppose that the hypotheses (H1) and (H2) are satisfied and, in addition, that there hold the following conditions. (H6) The function is continuous and there exist constants and such that for . Moreover, the real number is defined by  (H7) , where
If the inequality (H4) holds and , then the problem (2) has at least one mild solution in for some .

Proof. Let be the sequence defined as in the proof of Theorem 7. By the same arguments as in the proofs of Lemma 4 and Theorem 7, we see that for some , (60) and (61) also hold and for every subsequence of , and there exist a subsequence and a function such that is continuous on and for every Let be given. It follows by (H7) and (70) that there is a such that and that for every there is an such that implies that Choose large enough so that and define by Thus, (H7), (71), and (72) insures that And hence by the continuity of and compactness of for , (65) is also valid in this case. Therefore, the same argument as in the last paragraph of the proof of Theorem 7 shows the existence of mild solution for (2).

Now, we consider the nonlocal Cauchy problem (1). For this, we introduce the following hypothesis.(H8) The function satisfies the Carathéodory condition; that is, is measurable for each and is continuous for any . Moreover, for each , there exists a function such that (1) is integrable on for each ,(2) for almost all ,(3) ,(4) . The following existence results are immediate corollaries of Theorems 7 and 8.

Theorem 9. (i) Suppose that (H1), (H3), and (H8) are satisfied. If the inequality (H4) also holds and , then (1) has at least one mild solution in for some .
(ii) Suppose that (H1) and (H6)–(H8) are satisfied. If the inequality (H4) also holds and , then (1) has at least one mild solution in for some .

Proof. By letting one sees that (H8) implies (H2), and hence we obtain the existence results of (1) by Theorems 7 and 8.

4. The Existence of Strong Solutions

Definition 10. A mild solution of (1) is called a strong solution if is continuously differentiable on and satisfies (1). The strong solution of (2) is defined in a similar way.

In the following, we establish existence of strong solutions for (2).

Theorem 11. (i) Suppose that the hypotheses (H1) and (H3) are satisfied, and in addition, that there holds the following conditions.(H9) The function is continuous and for each , there exists a such that (1) is locally bounded function on , (2) is locally Hölder continuous on , (3) for almost all , (4) ,(5) . (H10) If is locally Hölder continuous on , then is locally Hölder continuous on .
If the inequality (H4) also holds and , then (2) has a strong solution on .
(ii) Suppose that the hypotheses (H1), (H6), (H7), (H9), and (H10) are satisfied. If the inequality (H4) also holds and , then (2) has a strong solution on .

Proof. It is clear that (H9) implies (H2). Thus, the conditions (H1)–(H4) hold by assumption and hence Theorem 7 shows that if , then (2) has a mild solution on ; that is, there is a such that for . We will show that this mild solution is a strong solution. In view of [16] (p.113, Corollary ), it suffices to prove that is locally Hölder continuous on .
From (76), for , one derives the following: By (18), (A5), and (A6) we have that where we use (A6) with and that Since is locally bounded and is locally Hölder continuous on by (H9), then for some . This shows that is locally Hölder continuous on and so, by (H10), is locally Hölder continuous on . This completes the proof of (i) and the assertion (ii) follows in a similar argument.

Now, we establish existence results for strong solutions of the nonlocal Cauchy problem (1) and for this purpose, we introduce the following hypotheses.(H11) The function satisfies the Carathéodory condition; that is, is measurable for each and is continuous for any . Moreover, for each , there exists a function such that (1) is locally bounded function on , (2) is locally Hölder continuous on , (3) for almost all , (4) , and (5) . (H12) If is locally Hölder continuous on , then is locally Hölder continuous on .

Theorem 12. (i) Suppose that (H1), (H3), (H11), and (H12) are satisfied. If the inequality (H4) also holds and , then (1) has at least one strong solution in for some .
(ii) Suppose that (H1), (H6), (H7), (H11), and (H12) are satisfied. If the inequality (H4) also holds and , then (1) has at least one strong solution in for some .

Proof. By letting one sees that (H11) implies (H2), and hence we obtain the existence results of (1) by using a similar argument as the proof of Theorem 11.

5. An Example

In this section, we consider the following system: where , , , and . This system will be solved by transforming it into (2). To do this, let and let . Define by . By [15], it is well known that is a sectorial operator on and generates a semigroup on . The nonautonomous cases have been studied by Xiao and Liang [14] and they have shown there that is a compact operator for each . Choose and again by [15] (P.107, (3.1.76)) one sees that . Suppose that , , and satisfy the following conditions.

(1) is continuous and there is a function such that for and , .

(2) is continuous and for each , the mapping belongs to .

(3) is continuous and uniformly bounded. Moreover, there is a constant such that for and .

Let denote the space equipped the supnorm and for each or , let be given by , for all and . Consider the following two operators.(i) is defined by where denotes the partial derivative with respect to the second variable.(ii) is defined by