Anti-periodic solutions for high-order cellular neural networks with mixed delays and impulses

Changjin Xu,Yusen Wu

doi:10.1186/s13662-015-0497-4

Abstract

In this paper, delayed high-order cellular neural networks with impulses are investigated. Some sufficient conditions on the existence and exponential stability of anti-periodic solutions are established. An example with its numerical simulations is presented. Our results are new and complement previously known results.

Highlights

During the past decades, high-order cellular neural networks (HCNNs) have been extensively investigated due to their immense potential of application perspective in various fields such as signal and image processing, pattern recognition, optimization, and many other subjects
The existence of anti-periodic solutions plays a key role in characterizing the behavior of nonlinear differential equations [ – ]
There have been some papers which deal with the problem of existence and stability of anti-periodic solutions

Summary

Introduction

High-order cellular neural networks (HCNNs) have been extensively investigated due to their immense potential of application perspective in various fields such as signal and image processing, pattern recognition, optimization, and many other subjects. There have been some papers which deal with the problem of existence and stability of anti-periodic solutions. It is worthwhile to investigate the existence and stability of anti-periodic solutions for HCNNs with impulses. We study the anti-periodic solution of the following high-order cellular neural network with. N, bi, cij, dij, eijl, Ii(t), gj : R → R, kij : R+ → R+, αjl, βjl : R → R+ are continuous functions, and there exists a constant T > such that bi(t + T) = bi(t), Ii(t + T) = –Ii(t), τij(t + T) = τij(t), αjl(t + T) = αjl(t), cij(t + T)gj(u) = –cij(t)gj(–u), dij(t + T)gj(u) = –dij(t)gj(–u), βjl(t + T) = βjl(t), eijl(t + T)gj(u)gl(u) = –eijl(t)gj(–u)gl(–u).

Ljg dij σ

Thus we can claim that