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.

Dissipativity of Fractional Navier–Stokes Equations with Variable Delay

We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modifying the fractional Halanay inequality and the comparison principle, we investigate the dissipativity of the corresponding system, namely, we obtain the existence of global absorbing set. Besides, some available results are improved in this work. The existence of a global attracting set is still an open problem.

Dissipativity Analysis of Memristor-Based Fractional-Order Hybrid BAM Neural Networks with Time Delays

A new class of memristor-based time-delay fractional-order hybrid BAM neural networks has been put forward. The contraction mapping principle has been adopted to verify the existence and uniqueness of the equilibrium point of the addressed neural networks. By virtue of fractional Halanay inequality and fractional comparison principle, not only the dissipativity has been analyzed, but also a globally attractive set of the new model has been formulated clearly. Numerical simulation is presented to illustrate the feasibility and validity of our theoretical results.