Existence of positive solutions for the boundary value problem of nonlinear fractional differential equations

In this paper, we investigate the nonlinear differential equation of fractional order. (c)D(0+)(x) u(t) = f (t, u(t), u'(t)), 1 < x <= 2, 0 < t < 1, where (c)D(0+)(x) is the Caputo fractional derivative, subject to the boundary conditions. u(0) + u'(0) = 0, u(1) + u'(1) = 0. By means of Schauder's fixed point theorem and an extension of Krasnoselskii's fixed point theorem in a cone, some results on the existence of positive solutions are obtained. (C) 2011 Elsevier B.V. All rights reserved.