Existence and controllability analysis of multi-term fractional coupled systems with generalized $$[\psi ,w]$$-Caputo-Fabrizio operators

This work is a generalization of the coupled system of fractional differential equations governed by the multi-term [Formula: see text]-Caputo-Fabrizio derivatives. Different fractional dynamics are supported by the kernel weight and monotone functions, and the system incorporates nonlinear, nonlocal initial conditions. Banach's and Krasnoselskii's fixed point theorems are applied to prove the existence, uniqueness, and Ulam-Hyers stability theorems. Furthermore, Schauder's fixed point theorem and the controllability Gramian are used to investigate controllability conclusions for both linear and nonlinear scenarios. To demonstrate the system's adaptability and broad applicability, several special cases are discussed. An example is provided to demonstrate how theoretical results are validated. The suggested system is used as a practical application to simulate the dynamics of an epidemic involving susceptible, infected, and recovered populations, proving the framework's applicability and flexibility in real-world problems.