On the dynamics of nonlinear mixed fractional integro-differential systems with Grünwald–Letnikov operators

In this article, we investigate a class of nonlinear mixed fractional integro-differential equations formulated via the Grünwald–Letnikov fractional derivative under fractional initial conditions expressed through Riemann—Liouville integrals. By employing fixed point techniques, including the Banach contraction principle, Krasnoselskii’s theorem, and Schaefer’s fixed point theorem, sufficient conditions for the existence and uniqueness of mild solutions are established. The analysis is further extended to coupled systems and equations involving multiple nonlinear terms. In addition, different notions of Hyers—Ulam type stability are studied for both single and coupled systems. Several illustrative examples are provided to demonstrate the applicability of the theoretical results.