Nonlocal ψ-Hilfer Generalized Proportional Boundary Value Problems for Fractional Differential Equations and Inclusions

Abstract

In this paper, we establish existence and uniqueness results for a new class of boundary value problems involving the ψ-Hilfer generalized proportional fractional derivative operator, supplemented with mixed nonlocal boundary conditions including multipoint, fractional integral multiorder and derivative multiorder operators. The given problem is first converted into an equivalent fixed point problem, which is then solved by means of the standard fixed point theorems. The Banach contraction mapping principle is used to establish the existence of a unique solution, while the Krasnosel’skiĭ and Schaefer fixed point theorems as well as the Leray–Schauder nonlinear alternative are applied for obtaining the existence results. We also discuss the multivalued analogue of the problem at hand. The existence results for convex- and nonconvex-valued multifunctions are respectively proved by means of the Leray–Schauder nonlinear alternative for multivalued maps and Covitz–Nadler’s fixed point theorem for contractive multivalued maps. Numerical examples illustrating the obtained results are also presented.