Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation

We investigate the fractional differential equation u '' + A(c)D(alpha)u = f (t, u, D-c(mu) u, u') subject to the boundary conditions u'(0) = 0, u(T)+ au'(T) = 0. Here alpha is an element of (1, 2), mu is an element of (0, 1), f is a Caratheodory function and D-c is the Caputo fractional derivative. Existence and uniqueness results for the problem are given. The existence results are proved by the nonlinear Leray-Schauder alternative. We discuss the existence of positive and negative solutions to the problem and properties of their derivatives.