Periodicity and multi-periodicity of generalized Cohen–Grossberg neural networks via functional differential inclusions

Abstract

In this paper, we investigate the periodic dynamical behaviors for a class of general Cohen–Grossberg neural networks with discontinuous right-hand sides and mixed time delays involving both time-varying delays and distributed delays. In view of functional differential inclusions theory, we obtain the existence of global solutions. By means of functional differential inclusions theory and fixed-point theorem of multi-valued maps, the existence of one and multiple positive periodic solutions for the neural networks is given. It is worthy to point out that, without assuming the boundedness or under linear growth condition of the discontinuous neuron activation functions, our results on the existence of one and multiple positive periodic solutions will also be valid. We derive some sufficient conditions for the global exponential stability and convergence of the discontinuous neural networks, in terms of non-smooth analysis theory with generalized Lyapunov approach. Finally, we give some numerical examples to show the applicability and effectiveness of our main results.