Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions

This paper deals with the study of the existence and non-existence of solutions of a three-parameter family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the <mml:semantics>alpha</mml:semantics>-Riemann-Liouville fractional derivative, with <mml:semantics>alpha is an element of(1,2]</mml:semantics>. To deduce the existence and non-existence results, we first study the linear equation, by deducing the main properties of the related Green functions. We obtain the optimal set of parameters where the Green function has constant sign. After that, by means of the index theory, the nonlinear boundary value problem is studied. Some examples, at the end of the paper, are showed to illustrate the applicability of the obtained results.