We extend the well-known Halanay inequality to the fractional order case
in presence of distributed delays and delays of neutral type (in the
fractional derivative). Both the discrete and distributed neutral delays
are investigated. It is proved that solutions decay toward zero in a
Mittag-Leffler manner under some rather general conditions. Some large
classes of kernels and examples satisfying our assumptions are provided.
We apply our findings to prove Mittag-Leffler stability for solutions of
fractional neutral network systems of Cohen-Grossberg type.
Abstract
A Halanay inequality with distributed delay of non-convolution type is considered. We establish a decay of solutions as a Mittag-Leffler function composed with a logarithmic function. A general sufficient condition is found and a large class of admissible retardation kernels is provided. This needs the preparation of several lemmas on properties of the Hadamard derivative and some basic fractional differential problems with this kind of derivative. The obtained result is then applied to a Hopfield neural network system to discuss its stability.
A neutral fractional Halanay inequality and application to a Cohen–Grossberg neural network system
