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

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., [1–9]). 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:

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: 0+ ( ) + ( , ( )) = 0, 0 < < 1, (0) = (1) = 0, where 1 < ≤ 2 is a real number and 0+ is 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.
Motivated by the results mentioned above, we discuss the following boundary value problem (BVP for short): in Banach space , where is the zero element of , 1 < ≤ 2 is a real number, 0+ is standard Riemann-Liouville fractional derivative, = [0, 1], and : × → is 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 Dirichlettype value problems of fractional order in Banach spaces.

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

Remark 7.
A strict set contraction operator is condensation. Now, we denote the Banach space of continuous functions : → by [ , ] with the maximal norm ‖ ‖ = max ∈ ‖ ( )‖. The basic space used in this paper is: is called a solution of BVP if it satisfies (3). For a bounded subset of Banach space , let ( ) be 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.

Main Result
In order to discuss the BVP, the preliminary lemmas are given in this section. For convenience, let us list some conditions.
In view of the boundedness of , there exists > 0 such that ‖ ‖ ≤ for any ∈ . Without loss of generality, for any ∈ , 1 , 2 ∈ with 1 < 2 , by (14), we can find that  The main result of this paper is as follows.
Proof. We only need to prove that the the operator has a fixed point in [ , ]. By condition ( 1 ), we can choose a real number such that and let =: Frist we prove that ⊂ . In fact, for any ∈ , by (16), we have From Lemma 13, it follows that ⊂ . Choose = co ( ), that is, is the convex closure of in [ , ]. Clearly, is nonempty, bounded, convex, and closed subset of . By Lemma 14, it follows that ( )( )/(1 + −1 ) is equicontinuous on [0, 1], together with the definition of , it follows that ( )/(1 + −1 ) are equicontinuous on [0, 1]. Now we show that is a strict set contraction operator from to .

Example
Now we consider the system of scalar nonlinear fractional differential equations to illustrate our results. Let with the norm ‖ ‖ = sup | |. Evidently is a Banach space. Consider the boundary value problem: Then we can obtain that ( On the other hand, from (41) we obtain that there exists > 0 such that This means that ‖ (2) ( , ( ) ) − V‖ → 0 as → ∞ and so (2) ( , ) is relatively compact for any bounded ⊂ . Hence ( (2) ( , )) = 0, ∀ ∈ , ⊂ .

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.