Of concern is the Hopfield neural network system comprising discrete as well as distributed delays in the form of a convolution. For a desired convergence rate of the solution to the equilibrium state, we establish sufficient conditions on the delay kernels ensuring this matter. Our result improves an existing one in the literature. The adopted approach is completely different. It relies on a judicious choice of a Lyapunov-like function and careful manipulations.

We expand the Halanay inequality to accommodate fractional-order systems incorporating both discrete and distributed neutral delays. By establishing specific conditions, we demonstrate that the solutions of these systems converge to zero at a Mittag-Leffler rate. Our analysis is versatile, accommodating a wide range of delay kernels. This versatility extends the applicability of our findings to fractional Cohen-Grossberg neural networks, offering valuable insights into their stability and dynamical behavior.