Existence of Solutions for Caputo Sequential Fractional Differential Inclusions with Nonlocal Generalized Riemann–Liouville Boundary Conditions

In this study, we explore the existence and uniqueness of solutions for a boundary value problem defined by coupled sequential fractional differential inclusions. This investigation is augmented by the introduction of a novel set of generalized Riemann–Liouville boundary conditions. Utilizing Carathéodory functions and Lipschitz mappings, we establish existence results for these nonlocal boundary conditions. Utilizing fixed-point theorems designed for multi-valued maps, we obtain significant existence results for the problem, considering both convex and non-convex values. The derived results are clearly demonstrated with an illustrative example. Numerical examples are provided to validate the theoretical conclusions, contributing to a deeper understanding of fractional-order boundary value problems.