Three-point boundary value problems of fractional functional differential equations with delay

In this paper, we study three-point boundary value problems of the following fractional functional differential equations involving the Caputo fractional derivative: (C)D(alpha)u(t) = f(t, u(t), (C)D(beta)u(t)), 0 < t < 1, u'(0) = 0, u'(1) = lambda u'(eta), where D-C(alpha), D-C(beta) denote Caputo fractional derivatives, 2 < alpha < 3, 0 < beta < 1, eta is an element of (0, 1), 1 < lambda < 1/2 eta. We use the Green function to reformulate boundary value problems into an abstract operator equation. By means of the Schauder fixed point theorem and the Banach contraction principle, some existence results of solutions are obtained, respectively. As an application, some examples are presented to illustrate the main results.