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Discrete Dynamics in Nature and Society
Volume 2012, Article ID 363562, 14 pages
http://dx.doi.org/10.1155/2012/363562
Research Article

Existence of Solutions for Fractional-Order Neutral Differential Inclusions with Impulsive and Nonlocal Conditions

1Department of Basic Courses, Nanjing Institute of Technology, Nanjing 211167, China
2Research Center for Complex Systems and Network Sciences and Department of Mathematics, Southeast University, Nanjing 210096, China

Received 21 March 2012; Accepted 19 May 2012

Academic Editor: Zengji Du

Copyright © 2012 Chao Song 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 investigates the existence of solutions for fractional-order neutral impulsive differential inclusions with nonlocal conditions. Utilizing the fractional calculus and fixed point theorem for multivalued maps, new sufficient conditions are derived for ensuring the existence of solutions. The obtained results improve and generalize some existed results. Finally, an illustrative example is given to show the effectiveness of theoretical results.

1. Introduction

This paper deals with the existence of solutions for the following fractional-order differential inclusions with impulsive nonlocal conditions: where and is the Caputo fractional derivative. and are multivalued maps ( is the family of all nonempty subsets of ). , represents the right and left limits of at , respectively. are given functions to be specified later.

The nonlocal problem was more general and has better effect than the classical Cauchy problems. So it has been studied extensively under various conditions in the literature [17] and the references contained therein. In [2], Byszewski and Lakshmikantham considered the existence and uniqueness of mild solutions when and satisfied Lipschitz conditions. In [3], Ntouyas and Tsamatos studied the case of compactness conditions of and . Liang et al. in [5] discussed the nonlinear nonlocal Cauchy problems when was compact and was Lipschitz and was not Lipschitz and not compact, respectively. Xue in [6] established existence results of mild solutions for semilinear differential equations under various conditions on , and . Subsequently in [7], the author examined the semilinear differential equations when was compact and failed to be compact.

The theory of impulsive differential equations and differential inclusions has received much attention for the past decades because of its wide applicability in control, electrical engineering, mechanics, biology, and so on. For more details on this theory and applications, we refer to the monograph of Lakshmikantham et al. [8], Samoilenko and Perestyuk [9] and the references therein. In [10], Abada et al. discussed the impulsive differential inclusions with delays when was the infinitesimal generator of a strongly continuous semigroup . In the following work [11], they obtained some existence and controllability results when was a nondensely defined closed linear operator.

Since fractional-order differential equations have proved to be valuable tools in the modeling of many phenomena in physics and technical sciences, differential equations involving Riemann-Liouville as well as Caputo derivatives have been investigated extensively in the last decades (see [1228] and the references therein). In [13], Mophou obtained the existence and uniqueness of mild solutions for the semilinear impulsive fractional differential equations by means of the Schauder fixed point theorem in Banach spaces. In [18], Benchohra and Slimani established sufficient conditions for the existence of solutions for a class of initial value problem for impulsive fractional differential equations. In [21], Chang and Nieto studied the existence of solutions for a class of fractional differential inclusions with boundary conditions by means of Bohnenblust-Karlin fixed point theorem. Balachandran et al. in [25] investigated fractional-order impulsive integrodifferential equations in Banach spaces. In [27], Agarwal and Ahmad studied the existence of solutions for nonlinear fractional differential equations and inclusions of order with antiperiodic boundary conditions. The existence results were established for convex as well as nonconvex multivalued maps by some fixed point theorems, such as Leray-Schauder degree theory, and nonlinear alternative of Leray-Schauder type.

However, there is little information in the literature on neutral fractional-order impulsive differential inclusions with nonlocal conditions. Motivated by works mentioned above, we consider the existence results of (1.1) by using fixed point theorem for multivalued maps.

This paper is organized as follows. In Section 2, we will recall briefly some basic definitions and preliminary results which will be used throughout the paper. In Section 3, based on the fractional calculus and fixed point theorem for multivalued maps, we will prove the existence of solution to the problem (1.1). In Section 4, An example is given to show the effectiveness of our abstract results.

2. Preliminaries

In this section, we introduce some definitions, notes, and preliminary facts which will be used in this paper.

Let be a Banach space, with the norm . Let denote the space of Bochner integrable functions on with the norm .

Let be a Banach space. is the family of all nonempty subsets of . , and denote, respectively, the family of all nonempty bounded, closed, bounded-closed, compact-convex, and bounded-closed-convex subsets of . A multivalued map is convex (closed) valued if is convex (closed) for all . We say that is bounded on bounded sets if is bounded in for all .

A mapping is called upper semicontinuous (u.s.c.) on if, for each , the set is a nonempty closed subset of and if, for each open set of containing , there exists an open neighborhood of such that .   is completely continuous if is relatively compact for every . If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph (i.e., imply ).

A mapping has a fixed point if there exists such that . A multivalued map is said to be measurable if, for each , the function is measurable.

Definition 2.1. A multivalued map is -Carathéodory if (i) is measurable for each ; (ii) is upper semicontinuous for almost all ; (iii)for each , there exists such that for all and .

Definition 2.2. A multivalued operator is called (a)-Lipschitz if there exists such that (b)a contraction if it is -Lipschitz with .
The key tool in our approach is the following fixed point theorem.

Lemma 2.3 2.3(see [29]). Let and denote, respectively, the open and closed balls in a Banach space centered at origin and of radius , and let and be two multivalued operators satisfying the following:(a) is a contraction,(b) is completely continuous. Then either(i)the operator inclusion has a solution in or(ii)there exists an with such that , for some .

Lemma 2.4 (see [30]). Let be a compact real interval and a Banach space. Let be an -Caratheodory multivalued mapping with compact convex values with , where and let be a linear continuous mapping. Then the operator is a closed graph operator in .

For more details on multivalued map, see the books of Aubin and Cellina [31], Deimling [32].

Now, we recall some definitions and facts about fractional derivatives and fractional integrals of arbitrary order, see [33].

Definition 2.5. The Riemann-Liouville fractional integral operator of order of a function is defined by where is the gamma function.

Definition 2.6. The Caputo fractional-order derivative of order of a function is defined by

Lemma 2.7 2.7(see [34]). Let ; then the differential equation has solutions , where .

Lemma 2.8 2.8( see [34]). Let ; then for some .

3. Main results

In this section, main results are presented.

Let , and there exist with . This set is a Banach space with the norm .

As a consequence of Lemmas 2.7 and 2.8, we can define the solution of problem (1.1).

Definition 3.1. The function is said to be a solution of the problem (1.1) if , and there exist such that integral equation is satisfied.

For the study of the system (1.1), the following hypotheses are given:(H1) is a Carathéodory map with compact convex values, and for each fixed is nonempty;(H2) there exist a function , where and a nondecreasing continuous function such that  for and each ;(H3) is Lipschtiz continuous with Lipschtiz constant , and there exists such that ;(H4) is Lipschtiz continuous with Lipschtiz constant ;(H5) are multivalued maps with bounded, closed, and convex values. And there exist , such that  for all ;(H6),

The existence of solutions is now presented.

Theorem 3.2. Assume that (H1)–(H6) are satisfied; then the problem (1.1) has at least one solution on provided that there exists a real number such that

Proof. We transform the problem (1.1) into a fixed point problem by considering the multivalued operator : where .
Consider the multivalued operators : It is clear that . This problem of finding a solution of (1.1) is reduced to find a solution of the operator inclusions . In what follows, we aim to show that the operator has a fixed point, which is a solution of the system (1.1). For this purpose, We will show that satisfies all the conditions of Lemma 2.3. For better readability, the proof will be given in several steps.
Step 1. A priori bounded on solution: we show that the second assertion of Lemma 2.3 is not true.
Define an open ball , where the real number satisfies the inequality given in condition (3.5). Let be a possible solution of , for some real number with . Then we have Hence by (H2)–(H5), we have Taking the supremum over , we get Substituting in the above inequality yields which is a contradiction to (3.5).
Step 2. is a contraction mapping.
Indeed, let , and from (H5), we have Hence by (3.4), is a contraction mapping.
Step 3. has compact, convex values, and it is completely continuous.

Claim 1. is convex for each .
In fact, if and belong to , then there exist , such that Let . Then, for each , we have Since is convex (because has convex values), we have

Claim 2. maps bounded sets into bounded sets in . Let be a bounded set in . For , then there exists such that Therefore, Thus

Claim 3. maps bounded sets into equicontinuous sets. Let , , be a bounded set of as in Claim 2 and . Then there exists such that
By elementary computation, we have As , the right-hand side of the above inequality tends to zero. As a consequence of Claims 1 to 3 together with the Arzelá-Ascoli theorem, we conclude that is completely continuous.

Claim 4. has a closed graph. Let , and ; we prove that . Now means that there exists such that
We will prove that there exists such that

Consider the continuous operator

From Lemma 2.4, it follows that is a closed graph operator. Since , we have . That is, there exists such that Therefore, has a closed graph.

As a consequence of Lemma 2.3, the operator inclusion has a solution which implies that the problem (1.1) has a solution on .

Remark 3.3. If we take , where is a continuous operator, then the result corresponds to a single-value problem. In this case, our result is also new in the present configuration.

Corollary 3.4. Assume that (H1)–(H6) are satisfied; then the problem (1.1) has at least one solution on provided that

Proof. From (3.25), we know that there exists a constant , such that
Since the proof of Corollary 3.4 follows the very same lines as the proof of Theorem 3.2, we do not give details.

Corollary 3.5. Assume that (H1), (H3)–(H6), and the following condition are satisfied,
(H2') There exists a function , such that for and each .
Then the problem (1.1) has at least one solution on provided that

Proof. In this case, we take in Corollary 3.4.
At last we would like to discuss the impulsive condition.() are single continuous operator, and there exists , such that , for all .

Corollary 3.6. Assume that (H1)–(H4), , (H 6) are satisfied; then the problem (1.1) has at least one solution on provided that

4. An Illustrative Example

In this section, an example is given to show the effectiveness of our theoretical result. Consider the following fractional-order differential inclusions with impulsive and nonlocal conditions: where , and is a multivalued map given by

Thus, where .

By computation, we have

Suppose that ; we have , , , , , .

Clearly, for suitable , all the conditions of Corollary 3.6 are satisfied. So there exists at least one solution of problem (4.1) on .

Acknowledgments

The authors would like to thank the Associate Editor and all the anonymous reviewers for their valuable comments and constructive suggestions, which lead to the improvement of the presentation of this paper. This work was funded by Scientific Research Foundation of Nanjing Institute of Technology under Grant Nos. QKJB2010028, QKJA2011009 and also jointly supported by the National Natural Science Foundation of China under Grant Nos. 60874088 and 11072059 and the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20110092110017, the JSPS Innovation Program under Grant CXZZ12_0080.

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