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International Journal of Differential Equations
Volume 2013 (2013), Article ID 490673, 9 pages
http://dx.doi.org/10.1155/2013/490673
Research Article

Nonlocal Problems for Fractional Differential Equations via Resolvent Operators

1Department of Mathematics, Changshu Institute of Technology, Suzhou, Jiangsu 215500, China
2Laboratoire CEREGMIA, Université des Antilles et de la Guyane, Campus Fouillole, Guadeloupe (FWI) 97159 Pointe-à-Pitre, France

Received 7 April 2013; Accepted 15 May 2013

Academic Editor: Sotiris Ntouyas

Copyright © 2013 Zhenbin Fan and Gisèle Mophou. 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

We discuss the continuity of analytic resolvent in the uniform operator topology and then obtain the compactness of Cauchy operator by means of the analytic resolvent method. Based on this result, we derive the existence of mild solutions for nonlocal fractional differential equations when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively. An example is also given to illustrate our theory.

1. Introduction

In this paper, we are concerned with the existence of mild solutions for a fractional differential equation with nonlocal conditions of the form where is the Caputo fractional derivative of order with , is the infinitesimal generator of a resolvent , , is a real Banach space endowed with the norm , and and are appropriate continuous functions to be specified later.

The theory of fractional differential equations has received much attention over the past twenty years, since they are important in describing the natural models such as diffusion processes, stochastic processes, finance, and hydrology. Many notions associated with resolvent are developed such as integral resolvent, solution operators, -resolvent operator functions, -regularized resolvent, and -order fractional semigroups. All of these notions play a central role in the study of Volterra equations, especially the fractional differential equations. Concerning the literature, we refer the reader to the books [1, 2], the recent papers [320], and the references therein.

On the other hand, abstract differential equations with nonlocal conditions have also been studied extensively in the literature, since it is demonstrated that the nonlocal problems have better effects in applications than the classical ones. It was Byszewski and Lakshmikantham [21] who first studied the existence and uniqueness of mild solutions for nonlocal differential equations. And the main difficulty in dealing with the nonlocal problem is how to get the compactness of solution operator at zero, especially when the nonlocal item is only assumed to be Lipschitz continuous or continuous. Many authors developed different techniques and methods to solve this problem. For more details on this topic, we refer to [10, 11, 2233] and references therein.

In this paper, we combine the above two directions and study the nonlocal fractional differential equation (1) governed by operator generating an analytic resolvent. A standard approach in deriving the mild solution of (1) is to define the solution operator . Then, conditions are given such that some fixed point theorems such as Browder’s and Schauder’s fixed point theorems can be applied to get a fixed point for solution operator , which gives rise to a mild solution of (1). The key step of using this approach is to prove the compactness of Cauchy operator associated with solution operator . When the operator generates a compact semigroup, it is well known that the Cauchy operator is also compact. However, to the best of our knowledge, it is unknown when generates a compact resolvent. The main difficulty of this problem lies in the fact that there is no property of semigroups for resolvent.

To this end, we will first discuss the continuity of resolvent in the uniform operator topology in this paper. In fact, we prove that the compact analytic resolvent is continuous in the uniform operator topology. Based on this result, we can prove the compactness of Cauchy operator. As a consequence, we obtain the existence of mild solutions for (1) when the nonlocal item is Lipschitz continuous. At the same time, we also derive the existence of mild solutions for (1) without the Lipschitz or compact assumption on the nonlocal item by using the techniques developed in [24, 30]. Actually, we only assume that is continuous on and is completely determined on for some small or is continuous on with topology (see Corollaries 1719).

This paper has four sections. In Section 2, we recall some definitions on Caputo fractional derivatives, analytic resolvent, and mild solutions to (1). In Section 3, we prove the compactness of Cauchy operator. Finally, in Section 4 we establish the existence of mild solutions of (1) when the nonlocal item satisfies different conditions. An example is also given in this section.

2. Preliminaries

Throughout this paper, let be fixed, and let , , and be the set of positive integers, real numbers, and nonnegative real numbers, respectively. We denote by the Banach space with the norm , the space of all -valued continuous functions on with the norm , and the space of -valued Bochner integrable functions on with the norm , where . Also, we denote by the space of bounded linear operators from into endowed with the norm of operators.

Now, let us recall some basic definitions and results on fractional derivative and fractional differential equations.

Definition 1 (see [1]). The fractional order integral of the function of order is defined by where is the Gamma function.

Definition 2 (see [1]). The Riemann-Liouville fractional order derivative of order of a function given on the interval is defined by where , .

Definition 3 (see [1]). The Caputo fractional order derivative of order of a function given on the interval is defined by where , .

In the remainder of this paper, we always suppose that and is a closed and densely defined linear operator on .

Definition 4. A family of bounded linear operators in is called a resolvent (or solution operator) generating by if the following conditions are satisfied:(S1) is strong continuous on and ;(S2) and for all and ;(S3) the resolvent equation holds

For , , let

Definition 5. A resolvent is called analytic, if the function admits analytic extension to a sector for some . An analytic resolvent is said to be of analyticity type if for each and there is such that for , where denotes the real part of .

Definition 6. A resolvent is called compact for if for every , is a compact operator.

According to Proposition  1.2 in [2], we can give the following definition of mild solutions for (1).

Definition 7. A function is called a mild solution of fractional evolution equation (1) if it satisfies for every .

Next, we introduce the Hausdorff measure of noncompactness defined on each bounded subset of Banach space by

Some basic properties of are given in the following lemma.

Lemma 8 (see [34]). Let be a real Banach space and let be bounded; then the following properties are satisfied:(1) is precompact if and only if ;(2), where and mean the closure and convex hull of , respectively;(3) when ;(4), where ;(5);(6) for any ;(7)if the map is Lipschitz continuous with constant , then for any bounded subset , where is a Banach space.

The map is said to be a -contraction if there exists a positive constant such that for any bounded closed subset , where is a Banach space.

Lemma 9 ([34], Darbo-Sadovskii). If is bounded closed and convex, the continuous map is a -contraction, then the map has at least one fixed point in .

3. Compactness of Cauchy Operators

Let Cauchy operator be defined by If is a compact -semigroup, it is well known that is compact. However, it is unknown in case of compact resolvent. The main difficulty is that the resolvent does not have the property of semigroups. Thus, it seems to be more complicated to prove the compactness of Cauchy operator. Here, we will first discuss the continuity of resolvent in the uniform operator topology. Then, we can give the positive answer to the above problem.

In the remainder of this paper, we always assume that .

Lemma 10. Suppose is a compact analytic resolvent of analyticity type . Then the following are hold:(i) for ;(ii) for ;(iii) for .

Proof. (i) Let be an analytic resolvent of analyticity type , and let be given. Then, by means of Cauchy integral formula, we have Thus, for any , , there exists a constant such that Now, let and . It follows from (11) that there exists a constant such that which implies that is continuous in the uniform operator topology for , that is,
(ii) Let , , and be given. Since is compact, the set is also compact. Thus, there exists a finite family such that for any with , there exists () such that From the strong continuity of , , there exists such that On the other hand, from (i), there exists such that Thus, for and , it follows from (14)–(16) that which implies that for all .
(iii) Let and . Then, there exists such that It follows from (i) and (ii) that This completes the proof.

Lemma 11. Suppose is a compact analytic resolvent of analyticity type . Then Cauchy operator defined by (9) is a compact operator.

Proof. We will show is compact by using the Arzela-Ascoli theorem. Let be any bounded subset of .
First, we claim that the set is equicontinuous on . In fact, let and ; then we have If , it is easy to see that If , for , we have Note that from Lemma 10(i), we know is operator norm continuous uniformly for . Combining this and the arbitrariness of with the above estimation on , we can conclude that Thus, is equicontinuous on .
Next, we will show that the set is precompact in for every . It is easy to see that the set is precompact in . Now, let be given and . Then is precompact since is compact. Moreover, for arbitrary , we have From Lemma 10(iii), we know Then, it follows from the Lebesgue dominated convergence theorem and the arbitrariness of that On the other hand, Thus, which implies that is precompact in by using the total boundedness, and therefore is compact in view of Arzela-Ascoli theorem.

4. Nonlocal Problems

In this section, we always assume that and that the operator generates a compact analytic resolvent of analyticity type , and we will prove the existence of mild solutions of (1) when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively.

Let be a fixed positive real number and Clearly, is a bounded closed and convex set. We make the following assumptions.(H1) is continuous.(H2) is Lipschitz continuous with Lipschitz constant such that .

Under these assumptions, we can prove the first main result in this paper.

Theorem 12. Assume that conditions (H1) and (H2) are satisfied. Then the nonlocal problem (1) has at least one mild solution provided that

Proof. We consider the solution operator defined by It is easy to see that the fixed point of is the mild solution of nonlocal Cauchy problem (1). Subsequently, we will prove that has a fixed point by using Lemma 9 (Darbo-Sadovskii’s fixed point theorem).
Firstly, we prove that the mapping is continuous on . For this purpose, let be a sequence in with in . Then By the continuity of and , we deduce that is continuous on .
Secondly, we claim that , where is defined by (30). In fact, for any , by (31), we have which implies that maps into itself.
Now, according to Lemma 9, it remains to prove that is a -contraction in . From condition (H2), we get that is Lipschitz continuous with constant . In fact, for , by (H2), we have Thus, it follows from Lemma 8-(30) that .
For operator , we have , where is continuous on and is the Cauchy operator defined by (9). Thus, in view of Lemma 11, we know that is compact on , and hence . Consequently, Since , the mapping is a -contraction on . By Darbo-Sadovskii’s fixed point theorem, the operator has a fixed point in , which is just the mild solution of nonlocal Cauchy problem (1).

Now, we give the following technical condition on function .(H) is continuous, and the set is precompact, where denotes the convex closed hull of set , and is given by (32).

Remark 13. It is easy to see that condition (H) is weaker than the compactness and convexity of . The same hypothesis can be seen from [24, 30], where the authors considered the existence of mild solutions for semilinear nonlocal problems of integer order when is a linear, densely defined operator on which generates a -semigroup. After the proof of our main results, we will give some special types of nonlocal item which is neither Lipschitz nor compact, but satisfies the condition (H) in the next Corollaries.

Theorem 14. Assume that conditions (H1) and (H) are satisfied. Then the nonlocal problem (1) has at least one mild solution provided that (31) holds.

Proof. We will prove that has a fixed point by using Schauder’s fixed point theorem. According to the proof of Theorem 12, we have proven that is continuous. Next, we will prove that there exists a set such that is compact.
For this purpose, let and . It is easy to see that the set is precompact since is compact. On the other hand, by Lemma 10(iii), we obtain which implies that is precompact in by using the total boundedness. Next, we claim is equicontinuous on for any small positive number . In fact, for and , we have By Lemma 10(i), is operator norm continuous for . Thus is operator norm continuous uniformly for , and hence that is, is equicontinuous on . Note that is compact by Lemma 11. Therefore, we have proven that is precompact for every and is equicontinuous on for any small positive number .
Now, let , we get that is a bounded closed and convex subset of and . It is easy to see that is precompact in for every and is equicontinuous on for any small positive number . Moreover, we have that is precompact due to condition (H).
Thus, we can now claim that is compact. In fact, it is easy to see that is precompact since is precompact. It remains to prove that is equicontinuous on . To this end, let and ; then we have In view of the compactness of and the strong continuity of on , we obtain the equicontinuous of on . Thus, is compact by Arzela-Ascoli theorem, and hence is also compact. Now, Schauder’s fixed point theorem implies that has a fixed point on , which gives rise to a mild solution of nonlocal problem (1).

The following theorem is a direct consequence of Theorem 14.

Theorem 15. Assume that conditions (H1) and (H) are satisfied for each . If uniformly for , then the nonlocal problem (1) has at least one mild solution.

Remark 16. It is easy to see that if there exist constants , and , such that for , then conditions (42) are satisfied.

Next, we will give special types of nonlocal item which is neither Lipschitz nor compact, but satisfies condition (H).

We give the following assumptions.(H3) is a continuous mapping which maps into a bounded set, and there is a such that for any with , .(H4) is continuous.

Corollary 17. Assume that conditions (H1) and (H3) are satisfied. Then the nonlocal problem (1) has at least one mild solution on provided that (31) holds.

Proof. Let From the proof of Theorem 14, we know that is precompact in . Moreover, by condition (H3), is also precompact in . Thus, all the hypotheses in Theorem 14 are satisfied. Therefore, there is at least one mild solution of nonlocal problem (1).

Corollary 18. Let condition (H1) be satisfied. Suppose that , where are given positive constants, and . Then the nonlocal problem (1) has at least one mild solution on provided that

Proof. It is easy to see that the mapping with satisfies condition (H3). And all the conditions in Corollary 17 are satisfied. So the conclusion holds.

Corollary 19. Assume that conditions (H1) and (H4) are satisfied. Then the nonlocal problem (1) has at least one mild solution on provided that (31) holds.

Proof. According to Theorem 14, it is sufficient to prove that the hypothesis (H) is satisfied. For arbitrary , there exists such that for all . Let From the proof of Theorem 14, we know that is precompact in , which implies that is precompact in . Thus, is precompact in as it has an -net . By condition is continuous and , it follows that condition (H) is satisfied, where denotes the convex and closed hull of in . Therefore, the nonlocal problem (1) has at least one mild solution on .

Finally, we give a simple example to illustrate our theory.

Example 20. Consider the following fractional partial differential heat equation in : where and denotes the Caputo fractional derivative.
We introduce the abstract frame as follows. Let . Let be the linear operator in defined by and for , . Then, it is well known that generates a compact semigroup for on . From the subordination principle [6, Theorems  3.1 and 3.3], also generates a compact resolvent for .
Assume that is a continuous function defined by , and is also a continuous function defined by , , , where .
Under these assumptions, the fractional partial differential heat equation (47) can be reformulated as the abstract problem (1). If the inequality holds for some constant , there exists at least one mild solution for fractional equation (47) in view of Corollary 18.

Acknowledgments

This work was completed when the first author visited the Pennsylvania State University. The author is grateful to Professor Alberto Bressan and the Department of Mathematics for their hospitality and providing good working conditions. The work was supported by the NSF of China (11001034) and Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers and Presidents.

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