Existence of solutions for fractional differential equations with integral boundary conditions

Abstract

In this paper, we study boundary-value problems for the following nonlinear fractional differential equations involving the Caputo fractional derivative: CD0+x(t) = f (t, x(t), CD0+x(t)), t ∈ [0, 1], x(0) + x(0) = y(x),10 x(t) dt = m, x(0) = x(0) = x(n-1)(0) = 0, where CD?0+, CD?0+ are the Caputo fractional derivatives, f : [0, 1]×R×R R is a continuous function, y : C([0, 1],R)R is a continuous function and m ∈ R, n - 1 < α < n (n 2), 0 < β < 1 is a real number. By means of the Banach fixed-point theorem and the Schauder fixed-point theorem, some solutions are obtained, respectively. As applications, some examples are presented to illustrate our main results. © 2014 Yan et al.; licensee Springer.