On (k, ψ)-Hilfer Fractional Differential Equations and Inclusions with Mixed (k, ψ)-Derivative and Integral Boundary Conditions

In this paper we study single-valued and multi-valued (k, psi)-Hilfer-type boundary value problems of fractional order in (1,2], subject to nonlocal boundary conditions involving (k, psi)-Hilfer-type derivative and integral operators. The results for single-valued case are established by using Banach and Krasnosel'skii fixed point theorems as well as Leray-Schauder nonlinear alternative. In the multi-valued case, we establish an existence result for the convex valued right-hand side of the inclusion via Leray-Schauder nonlinear alternative for multi-valued maps, while the second one when the right-hand side has non-convex values is obtained by applying Covitz-Nadler fixed point theorem for multi-valued contractions. Numerical examples illustrating the obtained theoretical results are also presented.