Qualitative analysis of a coupled system with nonlinear mixed fractional integro-differential equation involving Caputo fractional derivative

This paper introduces a novel model involving an integro-differential equation of fractional order characterized by multiple nonlinear terms with Caputo fractional derivatives. We derive conditions ensuring the existence, uniqueness, and stability of the mild solution. Furthermore, we explore a coupled system that includes the proposed equations and extend our analysis to an equation featuring three nonlinearities. The study utilizes Banach's, Krasnoselskii's, and Schaefer's fixed point theorems. Additionally, we investigate various forms of Hyers-Ulam stability for the equation and the coupled system. To demonstrate the practical implications of our findings, illustrative examples are provided.