Existence and uniqueness of strictly nondecreasing and positive solution for a fractional three-point boundary value problem

In this paper, we consider the following nonlinear fractional three-point boundary value problem D(0+)(alpha)u(t) + f(t, u(t)) = 0, 0 < t < 1, 3 < alpha <= 4, u(0) = u'(0) = u ''(0) = 0, u ''(1) = beta u ''(eta), where D-0+(alpha) is the standard Riemann-Liouville fractional derivative. By using a fixed point theorem in partially ordered sets, we obtain sufficient conditions for the existence and uniqueness of positive and nondecreasing solution to the above boundary value problem. Crown Copyright (C) 2011 Published by Elsevier Ltd. All rights reserved.