Nonlocal Integro-Multi-Point (k, ψ)-Hilfer Type Fractional Boundary Value Problems

Abstract

In this paper we investigate the criteria for the existence of solutions for single-valued as well as multi-valued boundary value problems involving (k,ψ)-Hilfer fractional derivative operator of order in (1,2], equipped with nonlocal integral multi-point boundary conditions. For the single-valued case, we rely on fixed point theorems due to Banach and Krasnosel’skiĭ, and Leray–Schauder alternative to establish the desired results. The existence results for the multi-valued problem are obtained by applying the Leray–Schauder nonlinear alternative for multi-valued maps for convex-valued case, while the nonconvex-valued case is studied with the aid of Covit–Nadler’s fixed point theorem for multi-valued contractions. Numerical examples are presented for the illustration of the obtained results.