Almost periodicity for a class of delayed Cohen-Grossberg neural networks with discontinuous activations.

*(English)*Zbl 1258.92002Summary: The objective of this paper is to investigate the dynamics of a class of delayed Cohen-Grossberg neural networks with discontinuous neuron activations. By means of retarded differential inclusions, we obtain a result on the local existence of solutions, which improves the previous related results for delayed neural networks. It is shown that an M-matrix condition satisfied by the neuron interconnections, can guarantee not only the existence and uniqueness of an almost periodic solution, but also its global exponential stability. It is also shown that the \(M\)-matrix condition ensures that all solutions of the system display a common asymptotic behavior. We prove that the existence interval of the almost periodic solution is \((-\infty , +\infty )\), whereas the existence interval is only proved to be \([0, +\infty )\) in most of the literature. As special cases, we derive results of existence, uniqueness and global exponential stability of a periodic solution for delayed neural networks with periodic coefficients, as well as the similar results of an equilibrium for the systems with constant coefficients. To the author’s knowledge, the results in this paper are the only available results on almost periodicity for Cohen-Grossberg neural networks with discontinuous activations and delays.

##### MSC:

92B20 | Neural networks for/in biological studies, artificial life and related topics |

34K14 | Almost and pseudo-almost periodic solutions to functional-differential equations |

39A24 | Almost periodic solutions of difference equations |

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\textit{J. Wang} and \textit{L. Huang}, Chaos Solitons Fractals 45, No. 9--10, 1157--1170 (2012; Zbl 1258.92002)

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