Existence and Uniqueness Analysis for (k, ψ)-Hilfer and (k, ψ)-Caputo Sequential Fractional Differential Equations and Inclusions with Non-Separated Boundary Conditions

This paper investigates the existence and uniqueness of solutions to a class of sequential fractional differential equations and inclusions involving the (k,ψ)-Hilfer and (k,ψ)-Caputo derivatives under non-separated boundary conditions. By reformulating the problems into equivalent fixed-point systems, several classical fixed-point theorems, including those of Banach, Krasnosel’skii˘’s, Schaefer, and the Leray–Schauder alternative, are employed to derive rigorous results. The study is further extended to the multi-valued setting, where existence results are established for both convex- and nonconvex-valued multi-functions using appropriate fixed-point techniques. Numerical examples are provided to illustrate the applicability and effectiveness of the theoretical findings.