Existence and uniqueness of nontrivial solutions for eigenvalue boundary value problem of nonlinear fractional differential equation

In this paper, we study the existence and uniqueness of a nontrivial solution to eigenvalue problems for the following nonlinear fractional differential equation of the form (Formula Presented) where (Formula Presented) is a parameter, (Formula Presented) are two standard Riemann–Liouville fractional derivatives, (Formula Presented) is continuous, and (Formula Presented) is Lebesgue integrable. We obtain several sufficient conditions of the existence and uniqueness of nontrivial solution of the above eigenvalue problems when (Formula Presented) is in some interval. Our approach is based on the Leray–Schauder nonlinear alternative. In addition, some examples are included to demonstrate the main result. © 2013, Università degli Studi di Ferrara.