Existence and stability of solution for a coupled system of Caputo–Hadamard fractional differential equations

In our present work, we study a coupled system of Caputo–Hadamard fractional differential equations supplemented with a novel set of initial value conditions involving the $\eta =(t\frac{d}{dt})$ derivatives. We provided sufficient criteria for the existence and stability of the solutions for a coupled system of fractional differential equations by applying the Hyers–Ulam stability theory, the fixed point theorems of Banach, Krasnoselskii, and the Leray–Schauder nonlinear alternative. When computing priori bounds in Leray–Schauder nonlinear alternative and stability of the solutions, a novel Gronwall type inequality related to Hadamard integral is employed. This study investigates the properties of a solution, such as existence, uniqueness, and stability, to a given problem without attempting to solve the exact solution, and its theoretical applications are illustrated by providing an example.