Existence results for boundary value problems of nonlinear fractional differential equations

In this paper, we consider the existence of solutions for the nonlinear fractional differential equation (C)D(0+)(alpha)u(t) + r(C)D(0+)(alpha-1)u(t) = f (t, u(t)), t is an element of (0,1) with the boundary value conditions u(0) = u(1), u(xi) = eta, xi is an element of (0, 1), where D-C(0+)alpha and D-C(0+)alpha-1 are the standard Caputo derivative with 1 < alpha <= 2, r not equal 0. By using the contraction mapping principle and the Schauder fixed point theorem, some existence results are obtained. In addition, Lemma 2.6 in this paper is a valuable tool in seeking solvability of the fractional differential equations. (C) 2011 Elsevier Ltd. All rights reserved.