A Unified Framework for Cross-Coupled Delay Systems Under Generalized Power-Law Caputo Fractional Operators

In this study, we address a coupled system of nonlinear fractional delay differential equations subject to cross-coupled multi-point boundary conditions. By utilizing the generalized power Caputo fractional derivative, we present a unified theoretical framework that encompasses several operators—including the Atangana–Baleanu, Caputo–Fabrizio, and weighted Hattaf derivatives—as special cases. This generality ensures that our results remain applicable across a broad family of fractional kernels. We transform the complex delay system into an equivalent integral form to derive sufficient criteria for the existence and uniqueness of solutions via fixed-point theory. Furthermore, we rigorously establish the Ulam–Hyers stability of the system, a critical property for ensuring robustness in the presence of perturbations. Finally, the theoretical findings are validated through a detailed numerical study employing a predictor–corrector scheme adapted for fractional delay systems. The simulations highlight the sensitivity of solutions to the memory kernel and fractional orders and include a systematic exploration of delay effects.