ISRN Mathematical Analysis

VolumeΒ 2011, Article IDΒ 436045, 20 pages

http://dx.doi.org/10.5402/2011/436045

## Some Results for Nonlinear ()-Term Fractional Integrodifferential Inclusions with Multipoint Boundary Conditions

Department of Applied Mathematics, Donghua University, Shanghai 201620, China

Received 18 April 2011; Accepted 24 May 2011

Academic Editors: B.Β Djafari-Rouhani, G. L.Β Karakostas, and W.Β Kryszewski

Copyright Β© 2011 Huacheng Zhou 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

We are concerned with the nonlinear -term fractional integrodifferential inclusions , , where , , are constant coefficients, and , , subject to the nonlocal conditions , The existence results are obtained by using two fixed-point theorems due to Bohnenblust-Karlin and Covitz-Nadler, respectively. Our results partly generalize and improve the known ones.

#### 1. Introduction

Fractional differential equations (FDEs) have received increasing interest for the last three decades. It is benefited by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science, engineering economics, and other fields, see for instance [1β12] and references therein.

Babakhani and Gejji [7] considered the existence of positive solutions for the nonlinear fractional differential equations where , , , , and is the standard Riemann-Liouville fractional derivative. Some existence results of positive solutions are obtained by using some fixed-point theorems on a cone.

StojanoviΔ [11] considered the existenceuniqueness of solutions for a nonlinear -term fractional differential equation where , with initial data , .

On the other hand, realistic problems arising from economics, optimal control, and so on can be modeled as differential inclusions. Recently, El-Sayed and Ibrahim [13] initiated the study of fractional differential equations inclusions. Differential inclusions have been widely investigated by many authors, see [14β30] and references therein.

Very recently, in the survey paper [16], Agarwal et al. establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problems for fractional differential equations and inclusions involving the Caputo fractional derivative.

Ouahab [26] studied the following boundary value problem of fractional differential inclusions: where is the standard Riemann-Liouville fractional derivative, is a multivalued map with compact values.

Chang and Nieto [27] studied boundary value problem of fractional differential inclusions where is the Caputo's derivative, .

However, to the best of our knowledge, the existence of solutions for fractional integro-differential inclusions with multipoint boundary conditions has not been paid much attention. Our goal is to fill this gap in literature.

In the present work, we consider more general fractional integro-differential inclusions with multipoint boundary conditions where , , , , is the standard Riemann-Liouville fractional derivative, , , , , are two continuous functions on , and is a given multivalued function ( is the family of all nonempty subsets of ). We shall consider both the cases of convex and nonconvex valued right hand side and establish some sufficient conditions which admit that the integro-differential inclusions problem has at least one solution. These results obtained by applying two fixed point theorems due to Bohnenblust-Karlin and Covitz-Nadler are complement of previously known results.

The organization of this paper is as follows. In Section 2, we present some necessary definitions and preliminaries which are used throughout this paper. Main results and their proofs are given in Section 3.

#### 2. Preliminaries and Several Lemmas

In this section, we recall some basic definitions and notations and give several lemmas which are useful in our discussion.

*Definition 2.1. *The Riemann-Liouville fractional integral of order of a function is given by
provided the right side is pointwise defined on .

*Definition 2.2. *The Riemann-Liouville fractional derivative of function is given by
where , , provided the right side is pointwise defined on .

Lemma 2.3. *Let , then
**
for some , , .*

*Remark 2.4. *If the fractional derivative is integrable, then
If is continuous on , then , and (2.4) reduces to

The reader is referred to [4, 8, 9] for more details on fractional integrals and fractional derivatives.

Lemma 2.5. *Let , , and . Then is a solution of the BVP
**
if and only if satisfies the integral equation
*

*Proof. *In view of Lemma 2.3 and Remark 2.4, (2.6) is equivalent to the integral equation
for some , that is,
The boundary condition implies . Thus,
In view of the boundary condition , we conclude that
Therefore, the solution of (2.6) and (2.7) satisfies (2.8).

Conversely, if is a solution of (2.8), it is easy to verify that satisfies (2.6) and (2.7). The proof is complete.

Now, we recall some facts from multivalued analysis.

Let be a metric space, and , , , , , , and so forth.

Consider , given by where , . Then is a metric space and is a generalized metric space (see [31]).

A multivalued map is said to be measurable if for each , the function , defined by is measurable.

A multivalued map is convex (closed) valued if is convex (closed) for all . is bounded on bounded sets if is bounded in for all . That is, . is called upper semicontinuous (u.s.c for short) on if for each , the set is nonempty closed subset of , and if for each open set of containing , there exists an open neighborhood of such that . is said to be completely continuous if is relatively compact for every . If the multivalued map is completely continuous with nonempty compact valued, then is u.s.c. if and only if has closed graph, that is, , , imply .

More details on multivalued maps can be found in the books of Deimling [3], GΓ³rniewicz [32], Hu and Papageorgiou [33], and Tolstonogov [34].

*Definition 2.6. *The multivalued map is -CarathΓ©odory if (i) is measurable for each ; (ii) is upper semicontinuous for almost all ; (iii)for each , there exists such that for all and for almost all ,

For any , we define the set which is known as the set of selection functions.

Lemma 2.7 (see [35]). *Let be a Banach space. Let be an -CarathΓ©odory multivalued map with , and let be a linear continuous mapping from into . Then the operator
**
is a closed graph operator in .*

The following Bohnenblust-Karlin fixed-point lemma and the fixed-point theorem for contractive multivalued operators given by Covitz and Nadler are of great importance in the proofs of our main results. The proofs of these results can be found in Bohnenblust and Karlin [30] and in Covitz and Nadler [36].

Lemma 2.8 (see [30]). *Let be a Banach space, be a nonempty subset of , which is bounded, closed, and convex. Suppose is u.s.c. with closed, convex values, and such that and is relatively compact. Then has a fixed point.*

Lemma 2.9 (see [36]). *Let be a complete metric space. If is a contraction operator, then . **For convenience, let us list some conditions. *(H1)*;*(H2)* is CarethΓ©odory multivalued map.*

Lemma 2.10. *Assume that hypothesis (H1) is satisfied. For any , the integral equation
**
has a unique solution in .*

*Proof. *We define the operator as follows:
Obviously, is a map from into . Also, we have
Therefore, , where . By (H1) and Banach contraction principle, the conclusion of lemma is true.

*Definition 2.11. *A function is said to be a solution of the BVP (1.5) and (1.6), if there exists a function such that a.e. on and

#### 3. Main Results

In this section, we present our main results and prove them. Firstly, under convexity condition on the multivalued right-hand side, we are to establish the existence theorem of solutions for fractional differential inclusions (1.5) and (1.6), by employing the Bohnenblust-Karlin fixed-point theorem. Then, under nonconvexity condition on the multivalued right-hand side, the existence theorem of solutions are gotten, by employing the Covitz-Nadler fixed-point theorem.

Theorem 3.1. *Assume that hypotheses (H1) and (H2) are satisfied. Then BVP (1.5) and (1.6) has at least one solution in , provided that *(H3)*,
**where
**
and is defined in the proof of Lemma 2.10.*

*Proof. *To transform the problem into a fixed-point problem, we consider the multivalued operator, , where for any is defined by

For any , we have , , . In view of Lemma 2.10, is well defined. Moreover, it follows from the convexity of (because has convex values) that is convex for each . Clearly, the fixed points of are solutions of (1.5) and (1.6).

We shall show that has a fixed point in three steps. *Step 1. *we claim that there exists a , such that , where .

In fact, if it is not true, then for any , there exists a function , but , that is and for some ,

On the other hand, from (H2), we obtain
Thus,

Dividing both sides by and taking the lower limit as , we get

This is a contraction to (H3). Hence there exists a such that .*Step 2. * is equicontinuous.

Let , , , and . Then there exists such that for each , we have
Therefore,
where

The right-hand side of the above inequality tends to zeros independently of , as . This shows that is equicontinuous. *Step 3. * has a closed graph.

Let , and . We will prove that . Now implies that there exists such that for each ,

We need to show that there exists such that for each ,
In (2.18), taking
In view of Lemma 2.10, we know that for any , the integral equation (2.18) has a unique solution, which we denote by . Because of this, we can define the operator
It is easy to verify that the operator is linear. On the other hand, we can get
where
that is, , which implies is continuous. From Lemma 2.7, it follows that is a closed graph operator. Moreover, we know . Since , it follows from Lemma 2.7 that (3.11) holds for some .

Therefore, is a compact multivalued map, u.s.c. with convex closed values. As a consequence of Lemma 2.8, we immediately conclude that has a fixed-point which is a solution of the problem (1.5) and (1.6). The proof is complete.

As a direct corollary of Theorem 3.1, we can immediately obtain the following result when the nonlinearity has sublinear growth in the state variable.

Corollary 3.2. *Suppose that (H1) and (H2) are satisfied. Then the problem (1.5) and (1.6) has at least one solution in , provided that *(H4)*there exist functions , and such that**
for each . *

*Proof. *We only need to verify that (H4) implies (H3) in Theorem 3.1. Taking , it is easy to prove
because of . Then (H3) is satisfied. The proof is finished.

In the next part, we are concerned with the BVP (1.5) and (1.6) with nonconvex valued right-hand side. By using Covitz and Nadler fixed-point theorem, we obtain the following result.

Theorem 3.3. *Assume that the following hypotheses hold: *(A1) *; is measurable for each ;*(A2)*There exist three functions such that for a.e. and ,**
Then the BVP (1.5) and (1.6) has at least one solution in provided that
**
where
**
β and defined as in Lemma 2.10, defined by (3.1) in Theorem 3.1.*

*Proof. *Transform the problem into a fixed-point problem. Let be defined as in Theorem 3.1. We shall show that satisfies the assumptions of Lemma 2.9. The proof will be given in two steps.*Step 1. * for each .

Indeed, let such that in . Then there exists such that for each ,

For every , from (A2), we have
Then,
that is, , where

It is clear that is a multivalued integrable bounded map. Since , we may pass to a subsequence if necessary to get that converges weakly to in . From Mauz's lemma [37], there exists , then there exists a subsequence in , such that converges strongly to in , which implies . Then for each ,
So, . *Step 2. *There exists , such that

Let and . Then there exists such that for each
It follows from (A2) that
Hence, there is such that

Consider , given by

Since the multivalued operator is measurable (see in [38, PropositionββIII.4]), there exists a function , which is a measurable selection for . So, and

For each , define