Analytical Study of Variable-Order Fractional Differential Equations With Initial and Terminal Antiperiodic Boundary Conditions

This study investigates the existence, uniqueness, and stability of solutions to Riemann–Liouville fractional differential equations with fractional variable-order and antiperiodic boundary conditions. By employing the Banach fixed point theorem, we establish conditions for the uniqueness of solutions, while Schauder’s fixed point theorem is used to prove their existence in a Banach space. We further demonstrate Ulam–Hyers–Rassias stability, ensuring solution robustness against perturbations. The variable-order framework enables modeling of complex systems with evolving memory, offering advantages over constant-order models in applications such as physics and epidemiology. A concrete example illustrates the practical applicability of our results. This work provides a rigorous theoretical foundation, bridging pure mathematics with potential applications in science and engineering, and sets the stage for future numerical and applied studies.