Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation

In this paper, we consider the properties of Green's function for the nonlinear fractional differential equation boundary value problem D-0+(alpha) u(t) = f(t, u(t)), 0 < t < 1, u(0) = u(1) = u'(0) = u'(1) = 0, where 3 < alpha <= 4 is a real number, and D-0+(alpha) is the standard Riemann-Liouville differentiation. As an application of Green's function, we give some multiple positive solutions for singular and nonsingular boundary value problems, and we also give the uniqueness of solution for a singular problem by means of the Leray-Schauder nonlinear alternative, a fixed-point theorem on cones and a mixed monotone method. (C) 2009 Elsevier Ltd. All rights reserved.