Investigation of coupled system with double delay terms via an exponential kernel differential operator

This work presents a detailed analysis of a coupled system of fractional delay differential equations involving an exponential kernel operator of Mittag-Leffler type. These models realistically capture complex epidemics and biological systems in which memory and time delays are essential. Our system in particular describes the simultaneous action of long-term memory and multiple distinct delays, as in incubation and immune-response lags in diseases. Using tools from functional analysis and contraction mapping theory, we prove the existence and uniqueness of solutions. We also study several Ulam-type stability notions, including Ulam-Hyers and Ulam-Hyers-Rassias stability, to evaluate the robustness of model predictions under small measurement errors or perturbations. Finally, a concrete numerical example confirms the theoretical framework and illustrates its practical applicability.