Periodic solutions for complex-valued neural networks of neutral type by combining graph theory with coincidence degree theory

Zhengqiu Zhang,Jinde Cao

doi:10.1186/s13662-018-1716-6

Zhengqiu Zhang, Jinde Cao

Open Access

https://doi.org/10.1186/s13662-018-1716-6

Copy DOI

Abstract

In this paper, by combining graph theory with coincidence degree theory as well as Lyapunov functional method, sufficient conditions to guarantee the existence and global exponential stability of periodic solutions of the complex-valued neural networks of neutral type are established. In our results, the assumption on the boundedness for the activation function in (Gao and Du in Discrete Dyn. Nat. Soc. 2016:Article ID 1267954, 2016) is removed and the other inequality conditions in (Gao and Du in Discrete Dyn. Nat. Soc. 2016:Article ID 1267954, 2016) are replaced with new inequalities.

Highlights

Time delays have been extensively studied in last decades due to their potential existence in many fields [12, 13, 18,19,20]
Very few studies have been reported on the dynamical behaviors of complexvalued neural networks of neutral type with time delays [1, 29]
This motivates us to carry out a study on dynamical behaviors of complex-valued neural networks of neutral type

Summary

Introduction

Because in a lot of practical applications complex signals often occur and the complexvalued neural networks are preferable and practical, up to now, there has been increasing research interest in the stability of equilibrium point and periodic solutions of complex-valued neural networks, for example, see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] and the references therein. The results on the existence and global exponential stability of periodic solutions for delayed complex-valued neural networks of neutral type have not been obtained by combining coincidence degree theory with graph theory as well as Lyapunov functional method. The contribution of our paper lies in the following two aspects: (1) Combination of coincidence degree theory with graph theory as well as Lyapunov functional is introduced to study the existence and exponential stability of periodic solutions for delayed complexvalued neural networks of neutral type; (2) Novel sufficient conditions to guarantee the existence and global exponential stability of periodic solutions for system (1) are derived by removing the limitation on the boundedness for the activation functions in [1] and replacing inequality conditions in [1] with new inequality conditions.

Preliminaries

Conclusion