Some existence results on nonlinear fractional differential equations

Abstract

In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundaryvalue problem Dαu(t) = f(t, u(t)) with a Riemann-Liouville fractional derivative via the different boundary-value problems u(0) = u(T), and the three-point boundary condition u(0) = β1u(η) and u(T) = β2u(η), where T > 0, t ∈ I = [0, T], 0 < α < 1, 0 < η < T, 0 < β? 1 < β2 < 1. © 2013 The Author(s) Published by the Royal Society. All rights reserved.