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ISRN Mathematical Analysis
Volume 2013 (2013), Article ID 967192, 8 pages
http://dx.doi.org/10.1155/2013/967192
Research Article

Solutions to Dirichlet-Type Boundary Value Problems of Fractional Order in Banach Spaces

College of Applied Sciences, Beijing University of Technology, Beijing 100124, China

Received 13 July 2013; Accepted 16 August 2013

Academic Editors: M. Escobedo, G. Mantica, and W. Sun

Copyright © 2013 Jing-jing Tan and Cao-zong Cheng. 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 consider the boundary value problems with Dirichlet-type boundary conditions of nonlinear fractional differential equation in Banach space. The existence of the solution to the boundary value problems is established. Our analysis relies on the Sadovskii fixed point theorem. As an application, we give an example to demonstrate our results.

1. Introduction

Fractional differential equations have been of great interest recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications in various sciences, such as physics, mechanics, chemistry, and engineering, (e.g., [19]).

Consequently, the fractional calculus and its applications in various fields of science and engineering have received much attention and have developed very rapidly. Jiang and Yuan [10], by using fixed point theorem on the cone, discussed the existence and multiplicity of solutions of the nonlinear fractional differential equation boundary value problem as follows: whereis a real number andis standard Riemann-Liouville fractional derivative. The authors in [11] consider the same boundary value problem. They derived the corresponding Green function and obtained the existence of solutions of this problem.

As far as we know, the nonlinear integer order differential equation for the Dirichlet boundary value problem has been studied extensively (e.g., [1217]). However, only a few papers have dealt with the boundary value problem for fractional differential equation, especially, in Banach spaces. The authors in [18] studied the existence of positive solutions of second-order two-point boundary value problem as follows: in Banach spaces. The authors in [19], by using the Mönch fixed point theorems, obtained the same results.

Motivated by the results mentioned above, we discuss the following boundary value problem (BVP for short): in Banach space, whereis the zero element of,is a real number,is standard Riemann-Liouville fractional derivative,, andis continuous. We establish an existence result of BVP in Banach spaces. The technique relies on the properties of the Kuratowski noncompactness measure and and Sadovskii fixed point theorem. To the best of our knowledge, this is the first paper considering the existence of solutions to Dirichlet-type value problems of fractional order in Banach spaces.

2. Preliminaries

For the convenience of the reader, we present here the necessary definitions and preliminary facts which are used throughout this paper.

Definition 1 (see [1]). The Riemann-Liouville fractional integral of orderof a functionis given by provided that the right side is pointwise defined on, where.

Definition 2 (see [1]). The Riemann-liouville fractional derivative of orderof a continuous functionis given by where,denotes the integer part of the number, provided that the right side is pointwise defined on.

Lemma 3 (see [2]). Let. If we assume that, then the fractional differential equation hasas unique solutions, whereis the smallest integer greater than or equal to.

Lemma 4 (see [2]). Assume thatwith a fractional derivative of orderthat belongs to. Then for some, whereis the smallest integer greater than or equal to.

Definition 5 (Kuratowski noncompactness measure). Let be a real Banach space, let be a bounded set in. We denote is called Kuratowski noncompactness measure of, wheredenote the diameters of. Obviously.

Definition 6. Letbe real Banach spaces,, andbe a continuous and bounded operator.
is called a-set contraction operator if there exists a constant, such thatfor any bounded setin. When,is called a strict set contraction operator.
Letbe a nonrelative compact, bounded subset in.is called a condensation if.

Remark 7. A strict set contraction operator is condensation.
Now, we denote the Banach space of continuous functionsbywith the maximal normThe basic space used in this paper is: equipped with norm. It is easy to see thatis a Banach space.

A mapis called a solution of BVP if it satisfies (3). For a bounded subsetof Banach space, letbe the Kuratowski noncompactness measure of. In this paper, the Kuratowski noncompactness measure in, , and is denoted by,  and  , respectively. The following properties of the Kuratowski noncompactness measure and Sadovskii fixed point theorem are needed for our discussion.

Lemma 8 (see [20]). Ifis bounded and equicontinuous. Thenis continuous onand, wherefor each  .

Lemma 9 (see [21] (Sadovskii)). Letbe a bounded, closed, and convex subset of the Banach space. If the operatoris condensing, thenhas a fixed point in.

3. Main Result

In order to discuss the BVP, the preliminary lemmas are given in this section.

For convenience, let us list some conditions. There exist nonnegative functionssuch that For any,,is uniformly continuous on, whereis the zero element ofand.There exists a positive constantwith,  such thatfor alland all bounded subsetin.

Lemma 10 (see [2]). Givenand, then the problem has a unique solution satisfying

Lemma 11 (see [2]). Suppose that conditionis satisfied, then BVP is equivalent to integral equation: For any, we define the operatorby

Remark 12. Lemma 11 indicates that the solution of BVP coincides with the fixed point of the operator.is well defined andIndeed, from condition, we have By (14) and (15), we get that

Lemma 13. Suppose conditionsandare satisfied, thenis continuous and bounded.

Proof. First, by (14) and (16), we getfor any. Thus,is bounded. Next, we will prove thatis continuous on. TakeandHence,is a bounded subset of, that is, there existssuch thatfor all. Taking the limit, we have.
In addition, by (14) we have It follows from conditionthat for any, there existssuch that Therefore, for any,  and, by (17) and (18), we have Thus, we conclude that; namely,is continuous, and the conclusion of lemma follows.

Lemma 14. Let conditionbe satisfied and letbe a bounded subset of. Thenis equicontinuous on.

Proof. In order to show thatis equicontinuous on, we only need to testify the following conclusion.
For any,, there exists asuch that In fact, from condition, it follows that() nonnegative functionsare bounded in, that is, there exist positive constantsand, such thatand,().
In view of the boundedness of, there existssuch thatfor any. Without loss of generality, for any,with, by (14), we can find that This ensures thatis equicontinuous on. If, we can also get the same result. Thus,is equicontinuous on.

The main result of this paper is as follows.

Theorem 15. Let conditionsbe satisfied. Then the BVP has at least one solution belonging to.

Proof. We only need to prove that the the operatorhas a fixed point in. By condition, we can choose a real numbersuch that and let Frist we prove that. In fact, for any, by (16), we have From Lemma 13, it follows that.
Choose, that is,is the convex closure ofin. Clearly,is nonempty, bounded, convex, and closed subset of. By Lemma 14, it follows thatis equicontinuous on, together with the definition of , it follows thatare equicontinuous on.
Now we show thatis a strict set contraction operator fromto.
Observing thatand, together with Lemma 13 we know thatis bounded and continuous. Finally, we prove that there exists a constantsuch that, for.
In fact, by (14), conditionand Lemma 14, applying [22, Lemma 2.6], we have wherefor each. Thus, by (25), we need only to prove that, forBy conditionand the definition of, we knoware equicontinuous on. Thus, by virtue of Lemma 8 and condition, we get where. For any given, there exists a partition with Now for, chooseand a partitionsuch that Clearly,, where. For any, by (27) and (28), we have which impliesand, thus,Sinceis arbitrary, we get It follows from (25), (26), and (30) that Thus Consequently where. From condition, it follows that. Therefore,is a strict set contraction operator fromto, obviouslyis condensing too. It follows from Lemma 9 thathas at least one fixed point in, that is, the BVP has at least one solution in.

4. Example

Now we consider the system of scalar nonlinear fractional differential equations to illustrate our results.

Let with the normEvidentlyis a Banach space. Consider the boundary value problem: System (35) can be regard as a boundary value problem of the form (3), where

Next, we show that conditions()–()are satisfied. Clearly,and With the aid of simple computation, we have Note thatTakeand. Clearly, conditionis satisfied. It is easy to see that conditionis also satisfied.

Finally, we verify condition. Denote, where Then we can obtain thatfor any bounded set. Indeed, letbe bounded, that is, there exists, such that, where. Then we have, for each, which implied thatis bounded. And by the diagonal method, we can choose a subsequencesuch that From (40), it follows that that is,.

For any, (40) and (42) imply that there existssuch that On the other hand, from (41) we obtain that there existssuch that It follows from (43), (44), and the definition of the norm inthat This means thatas and sois relatively compact for any bounded. Hence Consequently, we arrive at since we conclude that conditionis satisfied for.

Therefore, all the conditions of Theorem 15 are satisfied. An application of Theorem 15 implies that problem (35) has a solution.

5. Conclusion

In this paper, we present some sufficient conditions which ensure the existence of solutions to fractional differential equation for Dirichlet-type boundary value problems. Applying the Sadovskii fixed point theorem, we establish some new existence criteria for boundary value problems (3) in Banach space. Although, for the fractional differential equation for the Dirichlet boundary value problem (3), only a few papers have dealt with the boundary value problem for fractional differential equations, especially in Banach space. In this aspect, our work fills up the deficiency. As applications, examples are presented to illustrate the main results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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