New results on anti-periodic solutions for SICNNs with oscillating coefficients in leakage terms

Abstract

This paper concerns with the anti-periodic solutions for a class of shunting inhibitory cellular neural networks model with oscillating coefficients in leakage terms. By applying contraction mapping fixed point theorem and differential inequality techniques, we establish some sufficient conditions for the existence and exponential stability of anti-periodic solutions for the model, which complement with some recent ones. Moreover, an example and its numerical simulation are given to support the theoretical results.

Introduction

It was well known that the following delayed differential system:

$$x_{\mathit{ij}}^{\prime }\left(t\right)=-a_{\mathit{ij}}\left(t\right)x_{\mathit{ij}}\left(t\right)-\sum _{C_{\mathit{kl}}\in N_r(i,j)}{C}_{\mathit{ij}}^{\mathit{kl}}\left(t\right)f\left(x_{\mathit{kl}}\right(t-{\tau }_{\mathit{kl}}\left(t\right)\left)\right)x_{\mathit{ij}}\left(t\right)-\sum _{B_{\mathit{kl}}\in N_q(i,j)}{B}_{\mathit{ij}}^{\mathit{kl}}\left(t\right){\int }_0^{+\infty }{K}_{\mathit{ij}}\left(u\right)g\left(x_{\mathit{kl}}\right(t-u\left)\right)\mathit{du}{x}_{\mathit{ij}}\left(t\right)+L_{\mathit{ij}}\left(t\right),t\ge 0,$$

has been used to describe shunting inhibitory cellular neural networks (SICNNs), which has been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, image processing, and population dynamics [1], [2], [3], [4], [5], [6]. Here $C_{\mathit{ij}}$ is the cell at the $(i,j)$ position of the lattice, and the r neighborhood $N_r(i,j)$ of $C_{\mathit{ij}}$ is defined as follows:

$$N_r(i,j)=\left\{C_{\mathit{kl}}:\max \left(\right|k-i|,|l-j\left|\right)\le r,1\le k\le m,1\le l\le n\right\},$$

$N_q(i,j)$ is similarly specified. $x_{\mathit{ij}}\left(t\right)$ represents the state of the cell $C_{\mathit{ij}}$, $a_{\mathit{ij}}\left(t\right)$ means the passive decay rate of the cell activity, $C_{\mathit{ij}}^{\mathit{kl}}\left(t\right)$ and $B_{\mathit{ij}}^{\mathit{kl}}\left(t\right)$ describe the connection or coupling strength of postsynaptic activity of the cell $C_{\mathit{kl}}$ transmitted to the cell $C_{\mathit{ij}}$, and $i=1,2,\dots ,m,j=1,2,\dots ,n$. Moreover, the first term in each of the right sides of SICNNs (1.1) corresponds to a stabilizing negative feedback of the system which acts instantaneously without time delay; these terms are variously known as forgetting or leakage terms (see for instance [7], [8], [9], [10], [11], [12], [13]).

On the other hand, the existence and stability of anti-periodic solutions play a key role in characterizing the behavior of nonlinear differential equations (see [14], [15], [16], [17], [18]). The signal transmission process of neural networks can often be described as an anti-periodic process. In particular, assume that the leakage term coefficient function $a_{\mathit{ij}}\left(t\right)$ is not oscillating, i.e.,

$$\underset{t\in \mathbb{R}}{\inf }{a}_{\mathit{ij}}\left(t\right)>0,\mathit{ij}\in J≔\{11,\dots ,1n,21,\dots ,2n,\dots ,m1,\dots ,\mathit{mn}\},$$

some sufficient conditions ensuring the existence and exponential stability of anti-periodic solutions for SICNNs (1.1) and its generalized models have been established in [19], [20] and [21], [22], [23], [24], [25], [26], respectively.

In the past few decades, people have paid much attention to the stability on biological systems with oscillating coefficients appear in linearizations of population dynamics models with seasonal fluctuations, where during some seasons the death or harvesting rates may be greater than the birth rate (see [27], [28], [29], [30], [31]). This motivates us to find the new criteria to guarantee the existence and exponential stability of anti-periodic solutions for (1.1) with oscillating coefficients in leakage terms.

The main purpose of this paper is to establish some sufficient conditions on the existence and exponential stability of anti-periodic solutions for (1.1) without the conditions (1.2). To the best of our knowledge, this has not been done before.

The remaining of this paper is organized as follows. In Section 2, we give some basic definitions and lemmas, which play an important role in Section 3 to establish the existence of anti-periodic solutions of (1.1). Here we also study the global exponential stability of anti-periodic solutions. The paper concludes with an example to illustrate the effectiveness of the obtained results by numerical simulation.