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.

This paper deals with the global Mittag-Leffler stability (GMLS) of Caputo fractional-order fuzzy inertial neural networks with time delay (CFOFINND). Based on Lyapunov stability theory and global fractional Halanay inequalities, the existence of unique equilibrium point and GMLS of CFOFINND have been established. A numerical example is given to illustrate the effectiveness of our results.