A Single and Multi‐Valued Problems Involving Mixed (k₁,η)‐Hilfer and (k₂,ϕ)‐Hilfer Fractional Derivatives for the Fractional Navier Problem

ABSTRACT This paper investigates fractional differential equations and inclusions involving two distinct fractional derivatives, specifically the ‐Hilfer and ‐Hilfer fractional derivatives, within the framework of the Navier boundary value problem. For the single‐valued case, we establish the existence and uniqueness of solutions using classical iterative methods, including the Leray–Schauder, Banach, and Krasnoselskii fixed‐point theorems. In the multi‐valued case, we apply the Leray–Schauder nonlinear alternative when the right‐hand side (RHS) of the inclusion has convex values, while the Covitz–Nadler fixed‐point theorem is employed for non‐convex RHS mappings. To illustrate the practical applicability of our results, we present numerical examples that validate our theoretical findings.