Halanay inequality involving Caputo-Hadamard fractional derivative and application

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

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.

Polynomial dissipativity of proportional delayed BAM neural networks

This paper pays close attention to the global polynomial dissipativity (GPD) for proportional delayed BAM neural networks (PDBAMNNs). The global exponential dissipativity (GED) and the global dissipativity (GD) are also talked about. Under the help of novel Lyapunov functionals and a generalized Halanay inequality, a set of dissipative criteria for such systems are led out, together with the global polynomial attracting set (GPAS) and the global attracting set (GAS). Further, the relationship among GPD, GED and GD is unveiled. Finally, a proposed theoretical condition is validated through a simulation experiment.