Variable-Order Fractional Delay Differential Equations with Integral Boundary Values: A Study on Existence, Uniqueness, and Stability

Variable-order fractional differential equations (VO-FDEs) significantly generalize classical fractional calculus by permitting the order of differentiation to vary with time or other system parameters. This flexibility offers a powerful framework for modeling complex phenomena characterized by evolving memory and dynamic heterogeneities, features that are beyond thereach of constant-order models. In this work, we conduct a rigorous analysis of a coupled system of VO-FDEs that incorporates multiple delays and nonlocal integral boundary conditions within the Caputo formalism. Our investigation first establishes sufficient criteria for the existence and uniqueness of solutions using Banach’s and Schauder’s fixed-point theorems. We thenperform a comprehensive stability analysis, deriving explicit conditions for Ulam–Hyers and Ulam–Hyers–Rassias stability to guarantee the robustness of solutions against small perturbations. The practical applicability of our theoretical findings is demonstrated through detailed numerical examples, which serve to validate the efficacy of the proposed framework.