Existence of solutions for a class of fractional boundary value problems via critical point theory

In this paper, by the critical point theory, a new approach is provided to study the existence of solutions to the following fractional boundary value problem: {d/dt(1/2 D-0(t)-beta(u'(t)) + 1/2(t)D(T)(-beta)(mu'(t)) + del F(t, u(t)) = 0, u a.e.t is an element of [0, T], u(0) = u(T) = 0, where D-0(t)-beta and D-t(T)-beta are the left and right Riemann-Liouville fractional integrals of order 0 <= beta < 1 respectively, F : [0, T] x R-N -> R is a given function and del F(t, x) is the gradient of F at x. Our interest in this problem arises from the fractional advection-dispersion equation (see Section 2). The variational structure is established and various criteria on the existence of solutions are obtained. (C) 2011 Elsevier Ltd. All rights reserved.