boundedness and stability for nonautonomous cellular neural networks with delay

Boundedness and stability for nonautonomous cellular

neural networks with delayq

Mehbuba Rehim*, Haijun Jiang, Zhidong Teng

Department of Mathematics, Xinjiang University, Urumqi 830046, China

Received 18 November 2002; revised 31 March 2004; accepted 31 March 2004

Abstract

In this paper, a class of nonautonomous cellular neural networks is studied. By constructing a suitable Liapunov functional, applying the

boundedness theorem for general functional–differential equations and the Banach fixed point theorem, a series of new criteria are obtained

on the boundedness, global exponential stability, and existence of periodic solutions.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Boundedness; Global exponential stability; Periodic solution; Liapunov functional; Delay; Cellular neural networks

1. Introduction

Neural network models introduced by Grossberg (1967,

1968a–c) in the 1960s led him to prove global theorems

about their boundedness and convergence, including net-

works with delays. Furthermore, Grossberg (1978) con-

sidered the following very general system

dui

dt¼ diðuiÞ 2biðuiÞ þ

Xn

j¼1

aijgiðujÞ

24

35; i ¼ 1; 2;…; n;

ð1Þ

which, for appropriate choices of the functions di; bi and gi

encompasses a large variety of biological models, including

several types of neural networks. Thereafter, as a special

case of model (1), the neural network model for n neurons

was proposed by Hopfield (1984) with an electrical circuit

implementation, as a continuous extension of a discrete,

two-state neural network model

dui

dt¼ 2biui þ

Xn

j¼1

aijgjðujÞ; i ¼ 1; 2;…; n ð2Þ

and the cellular neural network model was proposed by

Chua and Yang (1988a,b), which represents a class of

recurrent neural networks with local interneuron

connections.

In electronic implementations of neural networks, the

delay parameters must be introduced into the systems.

Therefore, as a generalization of the standard neural

network models, the delayed neural network models

(DNNs) were considered early by Grossberg (1967,

1968a–c) and thereafter, by Marcus and Westervelt

(1989) for the simpler model of just (2) with time delays

in the connection terms and by Roska and Chua (1992);

Roska, Wu, Balsi, and Chua (1992) for the cellular neural

network models with time delays.

In recent years, the dynamical behaviors of the

autonomous DNNs models have been widely investigated.

Many important results on the existence and uniqueness of

equilibrium point, global asymptotic stability and global

exponential stability have been established (see Arik, 2003;

Cao, 1999a,b, 2000a–c, 2001a,b, 2003; Cao & Li, 2000;

Cao & Wang, 2002; 2003; Dong, 2002; Gopalsamy & He,

1994; Huang, Cao, & Wang, 2002; Liao & Wang, 2003;

Lu, 2000; Zhang, 2003; Zhang, Heng, & Prahlad, 2002; and

references cited therein).

However, as we well know, nonautonomous phenom-

ena often occur in many realistic systems. Particularly

when we consider a long-term dynamical behaviors of a

system, the parameters of the system usually will change

with time. In addition, in many applications, the property

of periodic oscillatory solutions of cellular neural net-

works also is of great interest. In fact, there has been

0893-6080/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.neunet.2004.03.009

Neural Networks 17 (2004) 1017–1025

www.elsevier.com/locate/neunet

q Supported by The National Natural Science Foundation of P.R China

(10361004) and The Natural Science Foundation of Xinjiang University.* Corresponding author.

E-mail address: [email protected] (M. Rehim).

http://www.elsevier.com/locate/neunet

considerable research on the nonautonomous neural net-

works. For example, in Cao (1999a, 2000a–c, 2003); Cao

and Li (2000); Cao and Wang (2002); Dong (2002);

Huang, Cao, and Wang (2002); Zhang, Heng, and Prahlad

(2002), and the references cited therein, the delayed

cellular neural networks with the constant coefficients and

periodic variable inputs are studied, where the criteria on

the existence, uniqueness and global stability of periodic

solution are obtained. In Jiang, Li, and Teng (2003); Jiang

and Teng (2004), the cellular neural networks with

varying-time coefficients and delays is studied, by use a

Lyapunov functional method, the technique of matrix

analysis and the technique of inequality analysis, the

authors established a series of the criteria on the

boundedness, global exponential stability and the exist-

ence of periodic solutions. In Dong, Matsui, and Huang

(2002); Guo, Huang, Dai, and Zhang (2003); Zhou, Liu,

and Cao (1999a), the neural network in (2), cellular neural

networks, and BAM neural networks with periodic

variable coefficients and delays were studied, by using

the continuation theorem based on coincidence degree and

the Lyapunov functional method and some new sufficient

conditions ensuring the existence, uniqueness and global

exponential stability of the periodic solution were

obtained. In Chen and Cao (2003); Chen, Huang, and

Cao (2003), delayed cellular neural networks and BAM

neural networks with almost periodic variable coefficients

were studied, by using the Banach fixed point theorem

and constructing suitable Lyapunov functionals, some

sufficient conditions ensuring the existence, uniqueness

and global stability of almost periodic solution were

established.

In this paper, we consider the following nonautonomous

cellular neural networks model with variable time delay

dxiðtÞ

dt¼2 ciðtÞxiðtÞ þ

Xml¼1

Xn

j¼1

aijlðtÞfijlðxjðt 2 tijlðtÞÞÞ þ IiðtÞ;

i ¼ 1; 2;…; n ð3Þ

Our main purpose is to derive a set of new criteria on the

boundedness and global exponential stability of solutions

for the general nonautonomous system (3) by constructing

new Liapunov functionals. As some special cases of system

(3), by using the Banach fixed point theorem, we will further

obtain that the existence, uniqueness and global exponential

stability of periodic solutions for periodic system (3) and the

existence, uniqueness and global exponential stability of

equilibrium points for autonomous system (3). In this paper,

we will not require that all nonlinear response functions

fijlðuÞ in system (3) are bounded on R ¼ ð21;1Þ: In

addition, we also will not require that system (3) must have

equilibrium point. We will see that the results given in this

paper will improve and generalize some well-known

corresponding results, for example, given in Cao (1999a,

2000a–c, 2001b), Cao and Wang (2003); Gopalsamy and

He (1994); Huang, Cao, and Wang, (2002); Lu (2000).

The present paper is organized as follows. In Section 2

we will give a description of system (3). In Sections 3–5, we

will state and prove the main results of this paper. In Section

6, we will illustrate our main results by two examples.

2. Model description

In system (3), the integer n corresponds to the number of

units in a neural network and the integer m corresponds to

the number of neural axons, that is, signals that emit from

the ith unit have m pathways to the jth unit; xi corresponds

to the state of the ith unit at time t; fijlðxjðtÞÞ denotes the

output of the jth unit at time t; aijlðtÞ denotes the strength of

the jth unit on the ith unit at time t 2 tijl; IiðtÞ denotes the

external bias on the ith unit at time t; tijlðtÞ corresponds to

the transmission delay of the ith unit along the l axon of the

jth unit at time t and is a non-negative function; and ciðtÞ

represents the rate with which the ith unit will reset its

potential to the resting state in isolation when disconnected

from the network and external input.

For system (3), we introduce the following main

assumptions.

(H1) ciðtÞ; aijlðtÞ and IiðtÞ (i; j ¼ 1; 2;…; n; l ¼ 1; 2;…;m) are

bounded continuous functions defined on t [ Rþ ¼

½0;1Þ:

(H2) There are constants Lijl . 0 such that for every i; j ¼

1; 2;…; n and l ¼ 1; 2;…;m one has

lfijlðuÞ2 fijlðvÞl# Lijllu2 vl for all u;v[R¼ ð21;1Þ:

(H3) Functions tijlðtÞ (i; j¼ 1;2;…;n; l¼ 1;2;…;m) are non-

negative, bounded and continuously differentiable

defined on Rþ and satisfying inft[Rþ{12 _tijlðtÞ}. 0;

where _tijlðtÞ expresses the derivative of tijlðtÞ with

respect to time t:

(H4) There are positive constants a and jiði¼ 1;2;…;nÞ such

that

jiciðtÞ2Xml¼1

Xn

j¼1

jjLjil

lajilðc21jil ðtÞÞl

12 _tjilðc21jil ðtÞÞ

.a

for all t$ 0 and i¼ 1;2;…;n; where c21ijl ðtÞ is the

inverse function of cijlðtÞ ¼ t2 tijlðtÞ:

Remark 1. We see that assumptions (H1) and (H2) are

general and elementary because the same assumptions are

required in many works on the qualitative analysis of

autonomous and nonautonomous neural networks with

delay, for example, Arik (2003); Cao (1999a,b, 2000a–c,

2001a,b, 2003); Cao and Li (2000); Cao and Wang (2002,

2003); Chen and Cao (2003); Chen, Huang, and Cao (2003);

Dong (2002); Dong et al. (2002); Gopalsamy and He

(1994); Guo et al. (2003); Huang, Cao, and Wang (2002);

Jiang, Li, and Teng (2003); Jiang and Teng (2004);

M. Rehim et al. / Neural Networks 17 (2004) 1017–10251018

Liao and Wang (2003); Lu (2000); Zhang (2003); Zhang,

Heng, and Prahlad (2002); Zhou, Liu, and Chen (2004), and

references cited therein. Assumption (H3) will assure in

system (3) t 2 tijlðtÞ!1 as t !1 and the functions

AijlðtÞ ¼aijlðc

21ijl ðtÞÞ

12 _tijlðc21ijl ðtÞÞ

; i; j¼ 1;2;…;n; l¼ 1;2;…;m

are bounded on Rþ: Assumption (H4) is fundamental and

essential to prove the uniform boundedness, uniform

ultimate boundedness and global exponential stability of

solutions for system (3).

Let t¼ sup{tijlðtÞ : t[Rþ; i; j¼ 1;2;…;n; l¼ 1;2;…;

m}: We denote by C½2t;0� the Banach spaces of continuous

functions fðuÞ¼ ðf1ðuÞ;f2ðuÞ;…;fnðuÞÞ : ½2t;0�!Rn with

the norm

kfk ¼ sup2t#u#0

lfðuÞl; where lfðuÞl ¼Xn

i¼1

lfiðuÞl:

In this paper, we always assume that all solutions of

system (3) satisfy the following initial conditions:

xiðt0 þ uÞ ¼ fiðuÞ; for all u[ ½2t;0�; i ¼ 1;2;…;n; ð4Þ

where t0 [ Rþ and f ¼ ðf1;f2;…;fnÞ [ C½2t; 0�: It is

well known that by the fundamental theory of functional

differential equations (see Burton, 1985; Li & Wen, 1987),

system (3) has a unique solution xðt; t0;fÞ ¼ ðx1ðt; t0;fÞ;

x2ðt; t0;fÞ;…; xnðt; t0;fÞÞ satisfying the initial condition (4).

On the boundedness and global exponential stability of

solutions of system (3), we have the following definitions.

Definition 1. System (3) is said to be uniformly bounded, if

for any constant g . 0 there exists a constant b ¼ bðgÞ . 0

such that for any t0 [ Rþ and f [ C½2t; 0� with kfk # g;

one has

lxiðt; t0;fÞl # b for all t $ t0; i ¼ 1; 2;…; n:

Definition 2. System (3) is said to be uniformly ultimately

bounded, if there is a constant b . 0 such that for

any constant g . 0 there exists a T ¼ TðgÞ . 0 such that

for any t0 [ Rþ and f [ C½2t; 0� with kfk # g; one has

lxiðt; t0;fÞl # b for all t $ t0 þ T ; i ¼ 1; 2;…; n:

Definition 3. System (3) is said to be globally exponentially

stable, if there are constants e . 0 and M $ 1 such that for

any two solutions xðtÞ and yðtÞ with the initial functions

f [ C½2t; 0� and c [ C½2t; 0� at time t ¼ 0; respectively,

one has

lxðtÞ2 yðtÞl # Mkf2 ckexpð2e tÞ for all t $ 0:

Remark 2. If system (3) has a equilibrium point x0 ¼

ðx10; x20;…; xn0Þ; i.e.

2ciðtÞxi0 þXn

j¼1

aijðtÞfjðxj0Þ þXn

j¼1

bijðtÞgjðxj0Þ þ IiðtÞ ¼ 0;

i ¼ 1; 2;…; n

for all t [ Rþ; then Definition 3 is equivalent to the global

exponential stability of equilibrium point x0; that is, for any

solutions xðtÞ with the initial functions f [ C½2t; 0� at time

t ¼ 0; we have

lxðtÞ2 x0l # Mkf2 x0kexpð2e tÞ for all t $ 0:

3. Boundedness

First of all, we introduce the following result on the

uniform boundedness and uniform ultimate boundedness of

solutions for general functional-differential equations. We

consider the following equation

dx

dt¼ Fðt; xtÞ; ð5Þ

where the functional Fðt;fÞ : Rþ £ C½2t; 0�! Rn is con-

tinuous with respect to ðt;fÞ and satisfies the local Lipschitz

condition with respect to f:

Let the functions WiðrÞ : Rþ ! Rþði ¼ 1; 2; 3; 4Þ be

continuous and increasing on Rþ with Wið0Þ ¼ 0 and

WiðrÞ!1 as r !1: Let further the functional Vðt;fÞ :

Rþ £ C½2t; 0�! R be continuous with respect to ðt;fÞ and

satisfy the local Lipschitz condition with respect to f: We

have the following lemma which can be found in Burton

(1985).

Lemma 1. [Theorem 4.2.10, Burton (1985)] Suppose that

there is a functional Vðt;fÞ and functions WiðrÞði ¼

1; 2; 3; 4Þ such that along any solution xðtÞ of Eq. (5)

W1ðlxðtÞlÞ # Vðt; xtÞ # W2ðlxðtÞlÞ þ W3

ðt

t2tW4ðlxðtÞlÞds

� �ð6Þ

and

dVðt; xtÞ

dt# 2W4ðlxðtÞlÞ þ M ð7Þ

for some constant M . 0: Then all solutions of Eq. (5) are

uniformly bounded and uniformly ultimately bounded.

Applying Lemma 1, we have the following result on the

boundedness of solutions of system (3).

Theorem 1. Suppose that (H1)–(H4) hold. Then all

solutions of system (3) are uniformly bounded and uniformly

ultimately bounded.

M. Rehim et al. / Neural Networks 17 (2004) 1017–1025 1019

Proof. : Let xðtÞ ¼ ðx1ðtÞ; x2ðtÞ;…; xnðtÞÞ be any solution of

system (3). We construct a Liapunov functional Vðt; xtÞ as

follows

Vðt; xtÞ ¼Xn

i¼1

jilxiðtÞlþ ji

Xml¼1

Xn

j¼1

Lijl

24

�ðt

t2tijlðtÞ

laijlðc21ijl ðsÞÞl

1 2 _tijlðc21ijl ðsÞÞ

lxjðsÞlds

35:

We let jp ¼ min1#i#nji; jpp ¼ max1#i#nji;

sijl ¼ sups[½2t;0�


12 _tijlðc21ijl ðsÞÞ

; i; j¼ 1;2;…;n; l¼ 1;2;…;m

and s¼max{Lijlsijl : i; j¼ 1;2;…;n; l¼ 1;2;…;m}; We

further choose the functions WiðrÞ :Rþ!Rþði¼ 1–4Þ as

follows: W1ðrÞ ¼ jpr;W2ðrÞ ¼ jppr;W3ðrÞ ¼ jppnma21sr

and W4ðrÞ ¼ar: By directly calculating, we obtain

Vðt;xtÞ$ jpXn

i¼1

lxiðtÞl¼ jplxðtÞl¼W1ðlxðtÞl

and

Vðt;xtÞ# jppXn

i¼1

lxiðtÞlþXml¼1

Xn

j¼1

Lijl

ðt

t2tsijllxjðsÞlds

0@

1A

# jppXn

i¼1

lxiðtÞlþ jppsnma21Xn

i¼1

ðt

t2talxiðsÞlds

¼W2ðlxðtÞlÞþW3

ðt

t2tW4ðlxðtÞlds

� �:

Hence,

W1ðlxðtÞlÞ#Vðt;xtÞ#W2ðlxðtÞlÞþW3

ðt

t2tW4ðlxðsÞlÞds

� �:

This shows that Vðt;xtÞ satisfies condition (6) of

Lemma 1.

Calculating further the right upper Dini derivation of Vðt; xtÞ

with respect to time t; we have

dVðt; xtÞÞ

dt

¼Xn

i¼1

ji sgnxiðtÞ_xiðtÞ þXml¼1

Xn

j¼1

Lijl

laijlðc21ijl ðtÞÞl

1 2 _tijlðc21ijl ðtÞÞ

lxjðtÞl

24

2Xml¼1

Xn

j¼1

LijllaijlðtÞkxjðt 2 tijlðtÞÞl

35

¼Xn

i¼1

ji sgnxiðtÞ 2ciðtÞxiðtÞ

0@

24

þXml¼1

Xn

j¼1

aijlðtÞfijlðxjðt 2 tijlðtÞÞ þ IiðtÞ

1A

þXml¼1

Xn

j¼1

Lijl



lxjðtÞl

2Xml¼1

Xn

j¼1


35

#Xn

i¼1

ji 2ciðtÞlxiðtÞlþXml¼1

Xn

j¼1

laijlðtÞkfijlðxjðt 2 tijlðtÞÞÞ

24

2fijlð0ÞlþXml¼1

Xn

j¼1

laijlðtÞkfijlð0Þlþ lIiðtÞl

þXml¼1

Xn

j¼1

Lijl



lxjðtÞl

2Xml¼1

Xn

j¼1


35

#Xn

i¼1

ji 2ciðtÞlxiðtÞlþXml¼1

Xn

j¼1

LijllaijlðtÞkðxjðt 2 tijlðtÞÞÞl

24

þXml¼1

Xn

j¼1

laijlðtÞkfijlð0Þlþ lIiðtÞl

þXml¼1

Xn

j¼1

Lijl



lxjðtÞl

2Xml¼1

Xn

j¼1


35

# 2Xn

i¼1

jiciðtÞ2Xml¼1

Xn

j¼1

jjLjil


1 2 _tjilðc21jil Þ

24

35lxiðtÞl

þXn

i¼1

ji lIiðtÞlþXml¼1

Xn

j¼1

laijlðtÞkfijlð0Þl

24

35

# 2aXn

i¼1

lxiðtÞlþ M ¼ 2W4ðlxðtÞlÞ þ M;

where

M ¼ supt[Rþ

Xn

i¼1

ji lIiðtÞlþXml¼1

Xn

j¼1

laijlðtÞkfijlð0Þl

24

35:

This shows that Vðt; xtÞ satisfies condition (7) of Lemma 1.

Therefore, by Lemma 1, we obtain that all solutions of

system (3) are defined on Rþ and are uniformly bounded


and uniformly ultimately bounded. This completes the proof

of Theorem 1. A

As some special cases of Theorem 1, we have the

following a series of corollaries of Theorem 1. Firstly, we

consider the case of constant delays and have the following

result.

Corollary 1. Suppose that (H1) and (H2) hold, tijlðtÞ ¼ tijl

(i; j ¼ 1; 2;…; n; l ¼ 1; 2;…;m) are constants for all t [ Rþ

and there exist constants ji . 0ði ¼ 1; 2;…; nÞ and a . 0

such that

jiciðtÞ2Xml¼1

Xn

j¼1

jjLjillajilðt þ tjilÞl . a

for all t [ Rþ and i ¼ 1; 2;…; n: Then all solutions of

system (3) are uniformly bounded and uniformly ultimately

bounded.

Next, we consider the following special case of

system (3)

dxiðtÞ

dt¼2 ciðtÞxiðtÞ þ

Xn

j¼1

aijðtÞfijðxjðtÞÞ

þXn

j¼1

bijðtÞgijðxjðt 2 tijlðtÞÞ þ IiðtÞ;

i ¼ 1; 2…; n; ð8Þ

where all functions ciðtÞ; aijðtÞ; bijðtÞ and IiðtÞ are bounded

and continuous defined on Rþ and tijðtÞ are nonnegative,

bounded and continuously differentiable on Rþ; and

satisfying inft[Rþ{1 2 _tijlðtÞ} . 0: Further, we assume

that functions fijðuÞ and gijðuÞ satisfy the Lipschitz

condition on R with the Lipschitz constants lij and mij:

We have following result.

Corollary 2. Suppose that there exist constants ji . 0 and

a . 0ði ¼ 1; 2;…; nÞ such that

jiciðtÞ2Xn

j¼1

jjljilajiðtÞl2Xn

j¼1

jjmji

lbjiðc21ji ðtÞÞl

1 2 _tjiðc21ji ðtÞÞ

. a

for all t $ 0 and i ¼ 1; 2;…; n: Then all solutions of system

(8) are uniformly bounded and uniformly ultimately

bounded.

At last, as another special case of system (3) we consider

the following autonomous neural networks

dxiðtÞ

dt¼2 cixiðtÞ þ

Xml¼1

Xn

j¼1

aijlfijlðxjðt 2 tijlÞÞ þ Ii;

i ¼ 1; 2;…; n; ð9Þ

where ci; aijl; tijl and Ii (i; j ¼ 1; 2;…; n; l ¼ 1; 2;…;m) are

constants. We further have following result.

Corollary 3. Suppose that (H2) holds and there are

constants ji . 0ði ¼ 1; 2;…; nÞ such that

jici 2Xml¼1

Xn

j¼1

jjLjilllajill . 0; i ¼ 1; 2;…; n: ð10Þ

Then system (9) is uniformly bounded and uniformly

ultimately bounded.

4. Global exponential stability

On the global exponential stability of solutions of system

(3), we have the following theorem.

Theorem 2. Suppose that (H1)–(H4) hold. Then system (3)

is globally exponentially stable.

Proof. : Let xðtÞ ¼ ðx1ðtÞ; x2ðtÞ;…; xnðtÞÞ and yðtÞ ¼

ðy1ðtÞ; y2ðtÞ;…; ynðtÞÞ be any two solutions of system (1)

satisfying the initial conditions xðuÞ ¼ fðuÞ and yðuÞ ¼

cðuÞ for all u [ ½2t; 0�; respectively, where fðuÞ ¼

ðf1ðuÞ;f2ðuÞ;…;fnðuÞÞ and cðuÞ ¼ ðc1ðuÞ;c2ðuÞ;…;

cnðuÞÞ: By (H4), we can choose a positive constant 1 such

that

jiðciðtÞ2 1Þ2 eetXml¼1

Xn

j¼1

jjLjil


1 2 _tjilðc21jil ðtÞÞ

. a

for all t $ 0: Let ziðtÞ ¼ lxiðtÞ2 yiðtÞlði ¼ 1; 2;…; nÞ: Then

from system (3) we can obtain

dziðtÞ

dt# 2ciðtÞziðtÞ

þXml¼1

Xn

j¼1

laijlðtÞkfijlðxjðt 2 tijlðtÞÞÞ

2 fijlðyjðt 2 tijlðtÞÞÞl ð11Þ

for all t $ 0 and i ¼ 1; 2;…; n: We consider another

Liapunov functional as follows:

Uðt;ztÞ ¼Xn

i¼1

ji

ziðtÞe

et

þXml¼1

Xn

j¼1

Lijl

ðt

t2tijlðtÞ

laijlðc21ijl ðsÞl


�zjðsÞe1ðsþtijlðc

21ijl ðsÞÞÞds

�: ð12Þ


Calculating the right upper Dini derivation Uðt; ztÞ along the

solution of (3), from (11) we obtain

dUðt; ztÞ

dt

¼Xn

i¼1

ji _ziðtÞeet þ eziðtÞe

et

24

þXml¼1

Xn

j¼1

Lijl

laijlðc21ðtÞÞl

1 2 _tijlðc21ðtÞÞ

eeðtþtijlðc21ijl ðtÞÞÞzjðtÞ

2Xml¼1

Xn

j¼1

e1tLijllaijlðtÞlzjðt 2 tijlðtÞÞ

35

#Xn

i¼1

ji 2 ðciðtÞ2 1ÞziðtÞe1t

"

þXml¼1

Xn

j¼1

e1tlaijlðtÞkfijlðxjðt 2 tijlðtÞÞ2 fijlðyjðt 2 tijlðtÞÞl

þXml¼1

Xn

j¼1

Lijl



e1ðtþtijlðc21ijl ðtÞÞÞzjðtÞ

2Xml¼1

Xn

j¼1

e1tLijllaijlðtÞkxjðt 2 tijlðtÞÞ2 yjðt 2 tijlðtÞÞl

35

#Xn

i¼1

ji 2 ðciðtÞ2 1ÞziðtÞe1t

"

þXml¼1

Xn

j¼1

e1tLijllaijlðtÞlðxjðt 2 tijlðtÞÞ2 ðyjðt 2 tijlðtÞÞl

þXml¼1

Xn

j¼1

Lijl



e1ðtþtijlðc21ijl ðtÞÞÞzjðtÞ

2Xml¼1

Xn

j¼1

e1tLijllaijlðtÞkxjðt 2 tijlðtÞÞ2 yjðt 2 tijlðtÞÞl

35

# 2Xn

i¼1

e1t jiðciðtÞ2 1Þ

24

2Xml¼1

Xmj¼1

jjLjil


1 2 _tjilðc21jil ðtÞÞ

e1t

35ziðtÞ

# 2aXn

i¼1

ziðtÞ , 0

for all t $ 0: Therefore, we obtain Uðt; ztÞ , Uð0; z0Þ for all

t $ 0: From (12) we have

Uðt; ztÞ $Xn

i¼1

ziðtÞee t for all t $ 0

and

Uð0;z0Þ¼Xn

i¼1

zið0ÞþXml¼1

Xn

j¼1

24

�Lijl

ð0

2tijlð0Þ



zjðsÞeeðsþtijlðc

21ijl ðsÞds�

#Xn

i¼1

zið0ÞþXml¼1

Xn

j¼1

LijlMijl

ð0

2tijlð0ÞzjðsÞds

24

35

#Xn

i¼1

1þXml¼1

Xn

j¼1

LjilMjiltjilð0Þ

24

35 sup

s[½2t;0�

lfiðsÞ2ciðsÞl

ð13Þ

for all t $ 0; where

Mijl ¼ sups[½2t;0�


1 2 _tijlðc21ijl ðsÞÞ

e1ðsþtijlðc21ijl ðsÞÞÞ

" #. 0:

Hence, from (13) we obtain

Xn

i¼1

lxiðtÞ2 yiðtÞl # Me21t sups[½2t;0�

Xn

i¼1

lfiðsÞ2 ciðsÞl

¼ Me2etkf2 ck ð14Þ

for all t $ 0; where M $ 1 is a constant and independent of

any solution of system (3). This implies that all solutions of

(3) are globally exponentially stable. The proof of Theorem

2 is completed. A

As some special cases of Theorem 2, we have the

following a series of corollaries of Theorem 2.

Corollary 4. Suppose that (H1)–(H4) hold, tijlðtÞ ¼ tijl

(i; j ¼ 1; 2;…; n; l ¼ 1; 2;…;m) are constants for all t [ Rþ

and there are constants ji . 0 and a . 0 such that

jiciðtÞ2Xml¼1

Xn

j¼1

jjLjillajilðt þ tjilÞl . a


(3) are globally exponentially stable.

Corollary 5. Suppose that there are constants ji . 0ði ¼

1; 2;…; nÞ and a . 0 such that

jiciðtÞ2Xn

j¼1

jjljilajiðtÞl2Xn

j¼1

jjmji

lbjiðc21ji ðtÞÞl

1 2 _tjiðc21ji ðtÞÞ

. a


(8) are globally exponentially stable.

Corollary 6. Suppose that (H2) and condition (10) hold.

Then all solutions of system (9) are globally exponentially

stable.


Remark 3. In Theorem 2, we give a new Liapunov

functional for nonautonomous cellular neural network

systems with variable time delay. This functional was

constructed by improving the Liapunov functional given in

Teng and Yu (2000).

5. Periodic solutions

In this section, we study the existence and global

exponential stability of periodic solutions of system (3).

We assume that ciðtÞ; aijlðtÞ; tijlðtÞ; and IiðtÞ (i; j ¼

1; 2;…; n; l ¼ 1; 2;…;m) are periodic continuous functions

with period v . 0: We have the following main result.

Theorem 3. Suppose that system (3) is v-periodic and

(H1)–(H4) hold. Then system (3) has a unique v-periodic

solution and all other solutions of system (3) converge

exponentially to it as t !1:

Proof. For any f;c [ C½2t; 0�; let xðt;fÞ ¼ ðx1ðt;fÞ;

x2ðt;fÞ;…; xnðt;fÞÞ and xðt;cÞ ¼ ðx1ðt;cÞ; x2ðt;cÞ;…;

xnðt;cÞÞ be the solutions of system (3) satisfying the initial

conditions xðuÞ ¼ fðuÞ and xðuÞ ¼ cðuÞ for all u [ ½2t; 0�;

respectively. We define xtðfÞ ¼ xðt þ u;fÞ and xtðcÞ ¼

xðt þ u;cÞ for all t $ 0: From (14) we easily get

Xn

i¼1

lxiðt;fÞ2 xiðt;cÞl # Mkf2 cke2e t ð15Þ

for any t $ 0; where e . 0 and M $ 1 are constants. One

can easily obtain from above formula (15) that

kxtðfÞ2 xtðcÞk # Me2eðt2tÞkf2 ck: ð16Þ

for all t $ 0: We can choose a positive integer m such that

Me2eðmv2tÞ #1

9ð17Þ

Now, we define a Poincare mapping P : C½2t; 0�!

C½2t; 0� as follows

Pf ¼ xvðfÞ;

for all f [ C½2t; 0�: By the periodicity of system (3) we

can obtain

PkðfÞ ¼ xkvðfÞ ð18Þ

for any integer k . 0: From (16)–(18) we have

kPmf2 Pmck #1

9kf2 ck

This implies that Pm is a contraction mapping. Hence there

exists a unique fixed point fp such that Pmfp ¼ fp: Note

that

PmðPfpÞ ¼ PðPmfpÞ ¼ Pfp:

This shows that Pfp [ C is also a fixed point of Pm:

Therefore, Pfp ¼ fp: So,

xvðfpÞ ¼ fp

:

Let xðt;fpÞ be the solution of system (3) satisfying the initial

condition xðuÞ ¼ fpðuÞ for all u [ ½2t; 0�: Since xðt þ

v;fpÞ is also a solution of system (3) and

xtþvðfpÞ ¼ xtðxvðf

pÞÞ ¼ xtðfpÞ

for all t $ 0; we obtain

xðt þ v;fpÞ ¼ xðt;fpÞ for all t $ 0:

This shows that xðt;fpÞ is exactly one v-periodic solution of

(3). By (14), it is easy to see that all other solutions of (3)

converge exponentially to it as t !þ1: This completes the

proof of Theorem 3. A

Corollary 7. Suppose that (H2) and condition (10) hold.

Then system (9) has a unique equilibrium point which is

globally exponentially stable.

Proof. In fact, for any constant v . 0; system (9) also is v-

periodic. Hence, it follows from Theorem 3 that system

(9) has a unique v-periodic solution xpðtÞ ¼ ðxp1ðtÞ;

xp2ðtÞ;…; xpnðtÞÞ which is globally exponentially stable.

Hence, we obtain that xpðt þ vÞ ¼ xpðtÞ for all v . 0 and

t [ Rþ: This shows that xpðtÞ ; xpð0Þ for all t [ Rþ:

Therefore, that xpðtÞ is an equilibrium point of the system

(9). This completes the proof.

Remark 4. As consequence of Theorem 3 and combining

Corollaries 4–6, we can obtain a series of corollaries of

Theorem 3 which is similar to Corollaries 4–6.

Remark 5. It is obvious that the results obtained in this

section improve and extend many well-known corresponding

results on the existence, uniqueness and global exponential

stability of periodic solutions for delayed nonautonomous

cellular neural network systems, for example those given in

Cao (1999a, 2000a–c); Huang, Cao, and Wang (2002).

Remark 6. In Cao (1999b); Cao and Wang (2002);

Gopalsamy and He (1994); Huang, Cao, and Wang (2002);

Liao and Wang (2003); Lu (2000), and references cited

therein, for autonomous cellular neural networks with

delays, the authors first obtained the existence and

uniqueness of an equilibrium point, then proved the global

stability of this equilibrium point. However, in this paper we

supply a new method. That is, we first discuss the

boundedness and global stability of the system, then by

applying the existence theorem of periodic solutions for

general-functional differential equations, obtain the exist-

ence and uniqueness of equilibrium point.


6. Examples

In order to illustrate some feature of our main results, in

this section, we will apply our main results to some special

two-dimensional systems.

dx1ðtÞ

dt¼ 2c1ðtÞx1ðtÞ þ a11ðtÞf11ðx1ðt 2 t11ðtÞÞ

þ a12ðtÞf12ðx2ðt 2 t12ðtÞÞ þ I1ðtÞ

dx2ðtÞ

dt¼ 2c2ðtÞx2ðtÞ þ a21ðtÞf21ðx1ðt 2 t21ðtÞÞ

þ a22ðtÞf22ðx2ðt 2 t22ðtÞÞ þ I2ðtÞ

ð19Þ

where fijðxÞ ¼ f ðxÞ ¼ 12ðlx þ 1lþ lx 2 1lÞði; j ¼ 1; 2Þ:

Obviously, fijðxÞ is unbounded on R and satisfies Lipschitz

condition.

Example 1. For system (19), we take c1ðtÞ ¼ 2 2 ðt2=3ð1 þ

t2ÞÞ; c2ðtÞ ¼ ð5 2 t2=3ð1 þ t2ÞÞ; a11ðtÞ ¼16ð1 þ cos tÞ; a21ðtÞ

¼ 112ð21 þ sin tÞ; a12ðtÞ ¼

13ð1 þ sin tÞ; a22ðtÞ ¼

12ð1 þ

cos tÞ; tijðtÞ ¼12

cos t þ e2t þ 1ði; j ¼ 1; 2Þ: Let cijðtÞ ¼ t 2

tijðtÞði; j ¼ 1; 2Þ: Obviously, inft[Rþ{1 2 _tijðtÞ} . 0: We

choose ji ¼ 1ði ¼ 1; 2Þ and a ¼ 13: Then

j1c1ðtÞ2 j1

la11ðc2111 ðtÞÞl

1 2 _t11ðc2111 ðtÞÞ

2 j2


1 2 _t21ðc2121 ðtÞÞ

¼ 2 2t2

4ð1 þ t2Þ2

16l1 þ cosðc21

11 ðtÞÞl1 þ 1

2sinðc21

11 ðtÞÞ þ e2c2111

ðtÞ

21

12l2 1 þ sinðc21

21 ðtÞÞl1 þ 1

2sinðc21


ðtÞ. a

and

j2c2ðtÞ2 j1


1 2 _t12ðc2112 ðtÞÞ

2 j2


1 2 _t22ðc2122 ðtÞÞ

¼ 5 2t2

3ð1 þ t2Þ2

13l1 þ sinðc21

12 ðtÞÞl1 þ 1

2sinðc21


ðtÞÞ

212l1 þ sinðc21

22 ðtÞÞl1 þ 1

2sinðc21


ðtÞ. a

for all t [ Rþ: This shows that assumptions (H1)–(H2) hold.

Therefore, by Theorems 1 and 2, the solutions of systems

(19) are uniformly bounded, uniformly ultimately bounded

and globally exponential stable.

Example 2. For system (19), we take c1ðtÞ ¼ 10 þ sin t;

c2ðtÞ ¼ 4 þ sin t; a11ðtÞ ¼ 1 þ cos t; a12ðtÞ ¼18ð1 þ sin tÞ;

a12ðtÞ ¼ a22ðtÞ ¼14ð1 þ sin tÞ; t11ðtÞ ¼ t22ðtÞ ¼ 1 þ 1

2cos t;

t12ðtÞ ¼ t21ðtÞ ¼ 1 þ 14

e2sin t: Let cijðtÞ ¼ t 2 tijðtÞði; j ¼

1; 2Þ: Obviously, inft[Rþ{1 2 _tijðtÞ} . 0: We choose

ji ¼ 1ði ¼ 1; 2Þ and a ¼ 13: Then

j1c1ðtÞ2 j1


1 2 _t11ðc2111 ðtÞÞ

2 j2


1 2 _t21ðc2121 ðtÞÞ

¼ 10 þ sin t 2l1 þ cosðc21

11 ðtÞÞl1 þ 1

2sinðc21

11 ðtÞÞ

218l1 þ sinðc21

21 ðtÞÞl1 þ 1

4e2sinðc21

21ðtÞÞcosðc21

21 ðtÞÞ. a

and

j2c2ðtÞ2 j1


1 2 _t12ðc2112 ðtÞÞ

2 j2


1 2 _t22ðc2122 ðtÞÞ

¼ 4 þ sin t 214l1 þ sinðc21

12 ðtÞÞl1 þ 1

4e2sinðc21

12ðtÞÞcosðc21

12 ðtÞÞ

214l1 þ sinðc21

22 ðtÞÞl1 þ 1

4cosðc21

22 ðtÞ. a

for all t [ Rþ: Therefore, assumptions (H1)–(H2) hold. By

Theorem 3, system (19) has a unique 2p-periodic solution

and all other solutions of system (19) converge exponen-

tially to it as t !1: A 1–5,8–11,13,17,19–26

Acknowledgements

The authors are grateful to the anonymous referees for

their helpful comments and valuable suggestions which

greatly improve the paper.

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