Periodic Solutions for Shunting Inhibitory Cellular Neural Networks of Neutral Type with Time-Varying Delays in the Leakage Term on Time Scales

Yongkun Li,Lei Wang,Yu Fei

doi:10.1155/2014/496396

Abstract

A class of shunting inhibitory cellular neural networks of neutral type with time-varying delays in the leakage term on time scales is proposed. Based on the exponential dichotomy of linear dynamic equations on time scales, fixed point theorems, and calculus on time scales we obtain some sufficient conditions for the existence and global exponential stability of periodic solutions for that class of neural networks. The results of this paper are completely new and complementary to the previously known results even if the time scale𝕋=ℝorℤ. Moreover, we present illustrative numerical examples to show the feasibility of our results.

Highlights

As we know, shunting inhibitory cellular neural networks (SCINNs) have been applied in a wide range of practical fields such as psychophysics, speech, perception, robotics, adaptive pattern recognition, and image processing
In [1,2,3], authors consider the existence and stability of almost periodic solutions for SCINNs; in [4, 5], authors consider the existence and stability of periodic solutions for SCINNs; in [5], authors by using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions consider the periodic solution for SCINNs; in [6, 7], authors obtained some sufficient conditions for the existence and stability of an equilibrium point
Another type of time delays, namely, neutral type time delays, which always appears in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar, and so forth, has recently drawn much research attention

Summary

Introduction

As we know, shunting inhibitory cellular neural networks (SCINNs) have been applied in a wide range of practical fields such as psychophysics, speech, perception, robotics, adaptive pattern recognition, and image processing. In a recent paper [16], authors studied the existence and exponential stability for the following SICNN with continuously distributed delays of neutral type: xi󸀠j (t) = −aij (t) xij (t) − ∑ Cikjl (t) Ckl ∈Nr (i,j). Motivated by the above discussions, in this paper, we are concerned with the following SCINNs of neutral type with time-varying delays in the leakage term on time scale T: xiΔj (t) = −aij (t) xij (t − μij (t)) − ∑ Cikjl (t) Ckl ∈Nr (i,j).

Preliminaries

Existence of Periodic Solutions

Exponential Stability of the Periodic Solution

Some Examples