Open Access Research

Global exponential stability of impulsive cellular neural networks with time-varying delays via fixed point theory

Yutian Zhang1* and Qi Luo2

Author Affiliations

1 School of Mathematics & Statistics, Nanjing University of Information Science & Technology, Nanjing, 210044, China

2 School of Information & Control, Nanjing University of Information Science & Technology, Nanjing, 210044, China

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Advances in Difference Equations 2013, 2013:23  doi:10.1186/1687-1847-2013-23


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/23


Received:12 July 2012
Accepted:9 January 2013
Published:28 January 2013

© 2013 Zhang and Luo; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This work is devoted to studying the application of fixed point theory to the stability analysis of complex neural networks. We employ the new method of contraction mapping principle to investigate the stability of impulsive cellular neural networks with time-varying delays. Some novel and concise sufficient conditions are presented to ensure the existence and uniqueness of a solution and the global exponential stability of the considered system at the same time. These conditions are easily checked and do not require the differentiability of delays.

Keywords:
neural network; fixed point; global exponential stability; delay; impulse

1 Introduction

Cellular neural networks (CNNs), proposed by Chua and Yang in 1988 [1,2], have become a hot topic for their numerous successful applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition and computer vision.

Due to the finite switching speed of neurons and amplifiers in the implementation of neural networks, it turns out that time delays should not be neglected; and therefore, the model of delayed cellular neural networks (DCNNs) is put forward, which is naturally of better realistic significance. In fact, besides delay effects, stochastic and impulsive as well as diffusion effects are also likely to exist in the neural networks. Accordingly, many experts are showing a growing interest in the dynamic behavior research of complex CNNs such as impulsive delayed reaction-diffusion CNNs and stochastic delayed reaction-diffusion CNNs, followed by a mass of achievements [3-9] obtained.

Synthesizing the reported results about the complex CNNs, we find that the existing research skill for dealing with the stability is mainly based on Lyapunov theory. However, we also notice that there are still lots of difficulties in the applications of corresponding theory to the specific problems [10-16]. It is therefore necessary to seek some new techniques to overcome those difficulties.

It is inspiring that in a few recent years, Burton and other authors have applied fixed point theory to investigate the stability of deterministic systems and obtained some more applicable results; for example, see the monograph [17] and the papers [18-29]. In addition, more recently, there have been a few papers where fixed point theory is employed to deal with the stability of stochastic (delayed) differential equations; see [10-16,30]. Particularly, in [11-13], Luo used fixed point theory to study the exponential stability of mild solutions for stochastic partial differential equations with bounded delays and with infinite delays. In [14,15], fixed point theory is used to investigate the asymptotic stability in the pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with bounded delays and with infinite delays. In [16], the exponential stability of stochastic Volterra-Levin equations is studied based on fixed point theory. As is known to all, although Lyapunov functions play an important role in Lyapunov stability theory, it is not easy to find the appropriate Lyapunov functions. This difficulty can be avoided by applying fixed point theory. By means of fixed point theory, refs. [11-16] require no Lyapunov functions for stability analysis, and the delay terms need no differentiability.

Naturally, for the complex CNNs which have great application values, we wonder if fixed point theory can be used to investigate the stability, not just the existence and uniqueness of a solution. With this motivation, in the present paper, we aim to discuss the stability of impulsive CNNs with time-varying delays via fixed point theory. It is worth noting that our research skill is contraction mapping theory which is different from the usual method of Lyapunov theory. We use the fixed point theorem to prove the existence and uniqueness of a solution and the global exponential stability of the considered system all at once. Some new and concise algebraic criteria are provided; moreover, these conditions are easy to verify and do not require even the differentiability of delays, let alone the monotone decreasing behavior of delays.

2 Preliminaries

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M1">View MathML</a> denote the n-dimensional Euclidean space and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M2">View MathML</a> represent the Euclidean norm. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M3">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M4">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M5">View MathML</a> corresponds to the space of continuous mappings from the topological space X to the topological space Y.

In this paper, we consider the following impulsive cellular neural network with time-varying delays:

(1)

(2)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8">View MathML</a> and n is the number of neurons in the neural network. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M9">View MathML</a> corresponds to the state of the ith neuron at time t. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M10">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M11">View MathML</a> is the activation function of the jth neuron at time t and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M12">View MathML</a> represents the activation function of the jth neuron at time <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M13">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M14">View MathML</a> corresponds to the transmission delay along the axon of the jth neuron and satisfies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M15">View MathML</a> (τ is a constant). The constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M16">View MathML</a> represents the connection weight of the jth neuron on the ith neuron at time t. The constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M17">View MathML</a> denotes the connection strength of the jth neuron on the ith neuron at time <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M13">View MathML</a>. The constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M19">View MathML</a> represents the rate with which the ith neuron will reset its potential to the resting state when disconnected from the network and external inputs. The fixed impulsive moments <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M20">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21">View MathML</a>) satisfy <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M22">View MathML</a> , and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M23">View MathML</a>. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M24">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M25">View MathML</a> stand for the right-hand and left-hand limit of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M9">View MathML</a> at time <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M20">View MathML</a>, respectively. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M28">View MathML</a> shows the abrupt change of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M9">View MathML</a> at the impulsive moment <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M20">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M31">View MathML</a>.

Throughout this paper, we always assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M32">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21">View MathML</a> . Denote by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M35">View MathML</a> the solution to Eqs. (1)-(2) with the initial condition

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M36">View MathML</a>

(3)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M37">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M38">View MathML</a>.

The solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M39">View MathML</a> of Eqs. (1)-(3) is, for the time variable t, a piecewise continuous vector-valued function with the first kind discontinuity at the points <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M20">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21">View MathML</a>), where it is left-continuous, i.e., the following relations are valid:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M42">View MathML</a>

Definition 2.1 Equations (1)-(2) are said to be globally exponentially stable if for any initial condition <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M43">View MathML</a>, there exists a pair of positive constants λ and M such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M44">View MathML</a>

The consideration of this paper is based on the following fixed point theorem.

Theorem 2.1[31]

Let ϒ be a contraction operator on a complete metric space Θ, then there exists a unique point<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M45">View MathML</a>for which<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M46">View MathML</a>.

3 Main results

In this section, we investigate the existence and uniqueness of a solution to Eqs. (1)-(3) and the global exponential stability of Eqs. (1)-(2) by means of the contraction mapping principle. Before proceeding, we introduce some assumptions listed as follows:

(A1) There exist nonnegative constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M47">View MathML</a> such that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M48">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M49">View MathML</a>

(A2) There exist nonnegative constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M50">View MathML</a> such that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M48">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M52">View MathML</a>

(A3) There exist nonnegative constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M53">View MathML</a> such that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M48">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M55">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M56">View MathML</a>, and let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M57">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8">View MathML</a>) be the space consisting of functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M59">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M60">View MathML</a> satisfies:

(1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M60">View MathML</a> is continuous on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M62">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21">View MathML</a>);

(2) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M64">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M65">View MathML</a> exist; furthermore, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M66">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21">View MathML</a> ;

(3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M68">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M69">View MathML</a>;

(4) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M70">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M71">View MathML</a>, where α is a positive constant and satisfies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M72">View MathML</a>,

here <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M20">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21">View MathML</a>) and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M75">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M76">View MathML</a>) are defined as shown in Section 2. Also, ℋ is a complete metric space when it is equipped with a metric defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M77">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M78">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M79">View MathML</a>.

In what follows, we give the main result of this paper.

Theorem 3.1Assume that the conditions (A1)-(A3) hold. Provided that

(i) there exists a constantμsuch that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M80">View MathML</a>,

(ii) there exist constants<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M81">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M82">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21">View MathML</a> ,

(iii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M85">View MathML</a>,

then Eqs. (1)-(2) are globally exponentially stable.

Proof The following proof is based on the contraction mapping principle, which can be divided into three steps.

Step 1. The mapping is needed to be determined. Multiplying both sides of Eq. (1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M86">View MathML</a> gives, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M87">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M62">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M89">View MathML</a>

which yields, after integrating from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M90">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M91">View MathML</a>) to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M92">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21">View MathML</a>),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M94">View MathML</a>

(4)

Letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M95">View MathML</a> in (4), we have, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M96">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21">View MathML</a>),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M98">View MathML</a>

(5)

Setting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M99">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M91">View MathML</a>) in (5), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M101">View MathML</a>

which generates by letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M95">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M103">View MathML</a>

(6)

Noting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M104">View MathML</a>, (6) can be rearranged as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M105">View MathML</a>

(7)

Combining (5) and (7), we derive that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M106">View MathML</a>

is true for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M107">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21">View MathML</a>). Further,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M109">View MathML</a>

holds for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M107">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M21">View MathML</a>). Hence,

which produces, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M87">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M114">View MathML</a>

(8)

Noting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M115">View MathML</a> in (8), we define the following operator π acting on ℋ for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M116">View MathML</a>:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M117">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M118">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8">View MathML</a>) obeys the rule as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M120">View MathML</a>

(9)

on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M121">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M122">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M123">View MathML</a>.

Step 2. We need to prove <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M124">View MathML</a>. Choosing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M125">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8">View MathML</a>), it is necessary to testify <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M127">View MathML</a>.

First, since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M122">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M76">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M130">View MathML</a>, we immediately know <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M131">View MathML</a> is continuous on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M76">View MathML</a>. Then, for a fixed time <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M87">View MathML</a>, it follows from (9) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M134">View MathML</a>

(10)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M135">View MathML</a>

Owing to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M136">View MathML</a>, we see that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M9">View MathML</a> is continuous on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M62">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139">View MathML</a>). Moreover, as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M140">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M141">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M142">View MathML</a> exist, in addition, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M143">View MathML</a>.

Consequently, when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M62">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139">View MathML</a>) in (10), it is easy to find that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M146">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M147">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M148">View MathML</a>, and so <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M149">View MathML</a> is continuous on the fixed time <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M62">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M151">View MathML</a>). On the other hand, as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M140">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139">View MathML</a>) in (10), it is not difficult to find that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M146">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M147">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M156">View MathML</a>. Furthermore, if letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M157">View MathML</a> be small enough, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M158">View MathML</a>

which implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M159">View MathML</a>. While if letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M160">View MathML</a> be small enough, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M161">View MathML</a>

which yields <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M162">View MathML</a>.

According to the above discussion, we see that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M163">View MathML</a> is continuous on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M62">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M165">View MathML</a>), and for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M140">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139">View MathML</a>), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M168">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M169">View MathML</a> exist; furthermore, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M170">View MathML</a>.

Next, we will prove <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M171">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M71">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8">View MathML</a>. First of all, it is obvious that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M174">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M175">View MathML</a>. In addition, owing to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M176">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M177">View MathML</a>, we know <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M178">View MathML</a>. Then, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M91">View MathML</a>, there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M180">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M181">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M182">View MathML</a>. Choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M183">View MathML</a>. It is derived from (A1) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M184">View MathML</a>

which leads to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M185">View MathML</a>

(11)

Similarly, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M91">View MathML</a>, since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M178">View MathML</a>, there also exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M188">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M189">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M190">View MathML</a>. Select <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M191">View MathML</a>. It follows from (A2) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M192">View MathML</a>

which results in

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M193">View MathML</a>

(12)

Furthermore, from (A3), we know that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M194">View MathML</a>. So,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M195">View MathML</a>

As <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M136">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M197">View MathML</a>. Then, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M198">View MathML</a>, there exists a non-impulsive point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M199">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M200">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M201">View MathML</a>. It then follows from the conditions (i) and (ii) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M202">View MathML</a>

which produces

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M203">View MathML</a>

(13)

From (11), (12) and (13), we deduce <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M204">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M71">View MathML</a>. We therefore conclude that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M206">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M8">View MathML</a>), which means <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M124">View MathML</a>.

Step 3. We need to prove π is contractive. For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M209">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M210">View MathML</a>, we estimate <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M211">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M212">View MathML</a>

Note

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M213">View MathML</a>

(14)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M214">View MathML</a>

(15)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M215">View MathML</a>

(16)

It hence follows from (14), (15) and (16) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M216">View MathML</a>

which implies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M217">View MathML</a>

Therefore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M218">View MathML</a>

In view of the condition (iii), we see π is a contraction mapping, and thus there exists a unique fixed point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M219">View MathML</a> of π in ℋ, which means <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M220">View MathML</a> is the solution to Eqs. (1)-(3) and meets <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M221">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M71">View MathML</a>. This completes the proof. □

Theorem 3.2Assume the conditions (A1)-(A3) hold. Provided that

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M223">View MathML</a>,

(ii) there exist constants<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M81">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M225">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M226">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139">View MathML</a> ,

(iii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M228">View MathML</a>,

then Eqs. (1)-(2) are globally exponentially stable.

Proof Theorem 3.2 is a direct conclusion by letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M229">View MathML</a> in Theorem 3.1. □

Remark 3.1 In Theorem 3.1, we see that it is fixed point theory that deals with the existence and uniqueness of a solution and the global exponential stability of impulsive delayed neural networks at the same time, while the Lyapunov method fails to do this.

Remark 3.2 The presented sufficient conditions in Theorems 3.1-3.2 do not require even the differentiability of delays, let alone the monotone decreasing behavior of delays which is necessary in some relevant works.

Remark 3.3 In [4], the abrupt changes are assumed linear with the coefficient <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M230">View MathML</a>, while in our paper, this restriction is removed and the abrupt changes can be linear and nonlinear. On the other hand, the activation functions in [6] are assumed to satisfy <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M231">View MathML</a>, where f is an activation function. However, in this paper, we relax this restriction and instead suppose an activation function f satisfies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M232">View MathML</a>.

4 Example

Consider the following two-dimensional impulsive cellular neural network with time-varying delays:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M233">View MathML</a>

with the initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M234">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M235">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M236">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M237">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M238">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M239">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M240">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M241">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M242">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M243">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M244">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M245">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M246">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M247">View MathML</a>), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M248">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M249">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139">View MathML</a> , <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M251">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M139">View MathML</a>). It is easy to see that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M253">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M254">View MathML</a> as well as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M255">View MathML</a>.

Select <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M256">View MathML</a> and compute <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/23/mathml/M257">View MathML</a>. From Theorem 3.1, we conclude that this two-dimensional impulsive cellular neural network with time-varying delays is globally exponentially stable.

5 Conclusion

This work aims to seek new methods to study the stability of complex CNNs. From what have been discussed above, we find that the application of fixed point theory to the stability analysis of complex CNNs is successful. We utilize the contraction mapping principle to deal with the existence and uniqueness of a solution and the global exponential stability of the considered system at the same time, for which Lyapunov theory feels helpless. Now that there are different kinds of fixed point theorems and complex neural networks, our future work is to continue the study on the application of fixed point theory to the stability analysis of complex neural networks.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YZ carried out the main part of this manuscript. QL participated in the discussion and gave the example. All authors read and approved the final manuscript.

Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grants 60904028, 61174077 and 71171116.

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