Coupled implicit fractional integro-differential systems with mixed operators in weighted spaces

This paper investigates a class of coupled nonlinear implicit fractional integro-differential systems featuring Riemann–Liouville derivatives of distinct orders and nonlocal boundary conditions. The system integrates both Volterra- and Fredholm-type integral operators, yielding a highly nonlocal and implicitly coupled structure. To handle the singularities inherent in Riemann–Liouville derivatives, we work in weighted Banach spaces. We reformulate the coupled system as an equivalent system of integral equations and apply fixed-point theory. Existence and uniqueness are established via Banach’s contraction principle. Existence under more general growth conditions is derived via Krasnoselskii’s fixed-point theorem. A comprehensive stability analysis yields criteria for Ulam–Hyers, generalized Ulam–Hyers, and Ulam–Hyers–Rassias stability. An illustrative example and numerical simulations substantiate the theoretical findings.