Halanay inequality involving ψ-Caputo derivative and application to a neural network system

The well-known Halanay inequality is generalized here to the fractional-order case. We investigate this inequality in presence of a generalized Caputo fractional derivative in addition to discrete and distributed delays of not necessarily convolution type. The obtained result is then applied to a Hopfield Neural Network system to discuss its stability. This needs proving various lemmas on properties of the considered generalized Caputo derivative. We prove that solutions decay as a Mittag-Leffler function composed with a power function.