<abstract><p>In this paper, we obtain a new generalized coupled Gronwall inequality through the Caputo fractional integral with respect to another function $ \psi $. Based on this result, we prove the existence and uniqueness of solutions for nonlinear delay coupled $ \psi $-Caputo fractional differential system. Moreover, the Ulam-Hyers stability of solutions for $ \psi $-Caputo fractional differential system is discussed. An example is also presented to demonstrate the application of main results.</p></abstract>

In this paper, we study the existence and uniqueness of solutions for a coupled implicit system involving ψ-Riemann–Liouville fractional derivative with nonlocal conditions. We first transformed the coupled implicit problem into an integral system and then analyzed the uniqueness and existence of this integral system by means of Banach fixed-point theorem and Krasnoselskiis fixed-point theorem. Some known results in the literature are extended. Finally, an example is given to illustrate our theoretical result.