EXISTENCE OF SOLUTIONS TO FRACTIONAL BOUNDARY-VALUE PROBLEMS WITH A PARAMETER

This article concerns the existence of solutions to the fractional boundary-value problem - d/dt (1/2(0)D(t)(-beta) + 1/2(t)D(T)(-beta))mu' (t) = lambda u(t) + del F (t, u(t)), a.e t is an element of [0, T], u(0) = 0, u,(T) = 0. First for the eigenvalue problem associated with it, we prove that there is a sequence of positive and increasing real eigenvalues; a characterization of the first eigenvalue is also given. Then under different assumptions on the nonlinearity F(t, u), we show the existence of weak solutions of the problem when lambda lies in various intervals. Our main tools are variational methods and critical point theorems.