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., [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: where is a real number and 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. As far as we know, the nonlinear integer order differential equation for the Dirichlet boundary value problem has been studied extensively (e.g., [12–17]). 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 M？nch 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 , where is the zero element of , is a real number, is standard Riemann-Liouville fractional derivative, , 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 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 order of
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