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

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