ARTICLE IN PRESS

Neurocomputing 72 (2009) 1688–1693

Contents lists available at ScienceDirect

Neurocomputing

0925-23

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/neucom

Global exponential stability of impulsive BAM neural networks withdistributed delays

Jie Zhou a,b,�, Shuyong Li b

a College of Science, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, PR Chinab College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068, PR China

a r t i c l e i n f o

Article history:

Received 25 February 2008

Received in revised form

6 June 2008

Accepted 11 August 2008

Communicated by T. Heskesand differential inequality technique to obtain our results. It is shown that, in some case, the stability

Available online 1 October 2008

Keywords:

Global exponential stability

BAM

Distributed delays

Impulsive

12/$ - see front matter & 2008 Elsevier B.V. A

016/j.neucom.2008.08.008

esponding author.

ail address: jiezhou198208@126.com (J. Zhou

a b s t r a c t

In this paper, the existence and uniqueness, and the global exponential stability (GES) of equilibrium

point of a impulsive BAM neural networks with distributed delays are considered. Here we point out,

different from previous methods, we do not construct Lyapunov functional or use the properties of

nonsingular M-matrix. We use some basic analytical technique, such as contraction mapping principle

criteria can be easily checked. Several remarks are worked out to demonstrate the advantage of our

results.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

The bidirectional associative memory (BAM) neural networksmodel, known as an extension of the unidirectional autoassociatorof Hopfield [4,14,16], was first introduced by Kosko [6,7]. This classof networks possesses good application prospects in some fieldssuch as pattern recognition, signal and image process, artificialintelligence. Such applications heavily depend on the dynamicalbehaviors of the neural networks. Thus, the analysis of thedynamical behaviors is a necessary step for practical design ofneural networks.

One of the most investigated problems in dynamical behaviorsof BAM neural networks is the existence, uniqueness and globalstability of the equilibrium point. The property of the globalstability, which means that the domain of attraction of theequilibrium point is the whole space, is of importance fromthe theoretical as well as application of view in several field. Thereexist some results of stability for the constant delays anddistributed delays BAM [17,8,2,1,15,10].

Recently, there has been a somewhat new category of neuralnetworks, which is neither purely continuous-time nor purelydiscrete-time ones; these are called impulsive neural networks.This third category of neural networks displays a combination ofcharacteristics of both the continuous-time and discrete-timesystems [12,13,3,9]. Therefore, it is necessary to consider both

ll rights reserved.

).

impulsive effect and delay effect on the neural networks. In thispaper, inspired by Refs. [17,8,2,1,15,10,12,13,3,9,11], we shall studythe existence, uniqueness and global exponential stability (GES) ofthe equilibrium point of impulsive BAM neural networks withdistributed delays. Here we point out, different from previousmethods, we do not construct Lyapunov functional or use theproperties of nonsingular M-matrix. We use some basic analyticaltechnique, such as contraction mapping principle and differentialinequality technique to obtain our results, so our results aredifferent from the previous.

2. Preliminaries

In the following, we will consider the BAM neural networkswith distributed delays and subjected to impulsive state displace-ments at fixed instants, which can be described by a set of integro-differential equations:

dxiðtÞ

dt¼ �aixiðtÞ þ

Xn

j¼1

pijf jðyjðtÞÞ þXn

j¼1

mij

Z 10

kijðsÞf jðyjðt � sÞÞdsþ Ii;

t40; tatk; i ¼ 1;2; . . . ;n;

DxiðtkÞ ¼ IkðxiðtkÞÞ; i ¼ 1;2; . . .n; k ¼ 1;2; . . . ;

dyjðtÞ

dt¼ �bjyjðtÞ þ

Xn

i¼1

qjigiðxiðtÞÞ þXn

i¼1

nji

Z 10

rjiðsÞgiðxiðt � sÞÞdsþ Jj;

t40; tatk; j ¼ 1;2; . . . ;n;

DyjðtkÞ ¼ JkðyjðtkÞÞ; j ¼ 1;2; . . . ;n; k ¼ 1;2; . . . ;

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

(2.1)

ARTICLE IN PRESS

J. Zhou, S. Li / Neurocomputing 72 (2009) 1688–1693 1689

where DxiðtkÞ ¼ xiðtþ

k Þ � xiðt�k Þ, DyjðtkÞ ¼ yjðt

þ

k Þ � yjðt�k Þ are the

impulses at moments tk and 0ot1ot2o � � � is a strictly increasingsequence such that limk!1tk ¼ þ1; xiðtÞ; yjðtÞ are the state ofneurons, ai40; bj40 represent the passive delay rates; pij; qji;mij;

nji are the synaptic connection strengths, I ¼ ðI1; I2; . . . ; InÞT, J ¼

ðJ1; J2; . . . ; JnÞT are the constant input vectors, f j; gi present the

activation functions of the neuron; the delayed feedbackkijðsÞ; rjiðsÞ are real value nonnegative continuous functions definedon ½0;1Þwith

R10 kijðsÞpkij, and

R10 rjiðsÞprji, kij; rij are nonnegative

constants.And the initial conditions associated with the system (2.1) are

of the form

xiðsÞ ¼ ciðsÞ; s 2 ð�1;0Þ; i ¼ 1;2; . . . ;n,

yjðsÞ ¼ fjðsÞ; s 2 ð�1;0Þ; j ¼ 1;2; . . . ;n,

where ciðsÞ;fjðsÞ are bounded continuous functions on ð�1;0�.As usual in the theory of impulsive differential equations, at

the points of discontinuity tk of the solution t/ðx1ðtÞ;

x2ðtÞ; . . . ; xnðtÞ; y1ðtÞ; y2ðtÞ; . . . ; ynðtÞÞT, denoted by ðxðtÞ; yðtÞÞ. We

assume that ðx1ðtkÞ; x2ðtkÞ; . . . ; xnðtkÞ; y1ðtkÞ; y2ðtkÞ; . . . ; ynðtkÞÞT�

ðx1ðtk � 0Þ, x2ðtk � 0Þ; . . . ; xnðtk � 0Þ; y1ðtk � 0Þ; y2ðtk � 0Þ; . . . ; yn

ðtk � 0ÞÞT. It is clear that, in general, the derivatives x0iðtkÞ do notexist. On the other hand, according to the equalities of (2.1) thereexist the limits x0iðtk � 0Þ. For the above convention, we assumex0iðtkÞ � x0iðtk � 0Þ; y0iðtkÞ � y0iðtk � 0Þ.

For convenience, we can rewrite (2.1) in the form

dxðtÞ

dt¼ �AxðtÞ þ Pf ðyðtÞÞ þM

Z 10

KðsÞf ðyðt � sÞÞdsþ I;

t40; tatk;

DxðtkÞ ¼ IkðxðtkÞÞ; k ¼ 1;2; . . . ;

dyðtÞ

dt¼ �ByðtÞ þ QgðxðtÞÞ þ N

Z 10

RðsÞgðxðt � sÞÞdsþ J;

t40; tatk;

DyðtkÞ ¼ JkðyðtkÞÞ; k ¼ 1;2; . . . ;

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

(2.2)

in which xðtÞ ¼ colfxiðtÞg, yðtÞ ¼ colfyjðtÞg; A ¼ diagfaig;B ¼

diagfbjg; P ¼ ðpijÞn�n, Q ¼ ðqjiÞn�n, M ¼ ðmijÞn�n, N ¼ ðnjiÞn�n;f ¼ colff jg, g ¼ colfgig; KðsÞ ¼ ðkijðsÞÞn�n, RðsÞ ¼ ðrjiðsÞÞn�n; I ¼

colfIig, J ¼ colfJjg;DxðtkÞ ¼ colfDxiðtkÞg, DyðtkÞ ¼ colfDyjðtkÞg;cðsÞ ¼ colfciðsÞg, fðsÞ ¼ colffjðsÞg:

Definition 1. A constant vector ðx�; y�Þ ¼ ðx�1; . . . ; x�n; y�1; . . . ; y

�nÞ

T

is said to be the equilibrium point of the system (2.2), if itsatisfies

Ax� ¼ Pf ðy�Þ þM

Z 10

KðsÞf ðy�Þdsþ I;

By� ¼ Qgðx�Þ þ N

Z 10

RðsÞgðx�Þdsþ J;

8>>><>>>:

(2.3)

when the impulsive jumps Ikð�Þ; Jkð�Þ satisfy Ikðx�Þ ¼ 0; Jkðy

�Þ ¼ 0,where x� ¼ ðx�1; . . . ; x

�nÞ

T; y� ¼ ðy�1; . . . ; y�nÞ

T.

Definition 2. The equilibrium point ðx�; y�Þ of the model (2.2) issaid to be globally exponentially stable, if there exists e40 andEX1 such that

kxðtÞ � x�k2 þ kyðtÞ � y�k2pEðkc� x�k þ kf� y�kÞe�et for all t40,

where ðxðtÞ; yðtÞÞ is any solution of the system (2.2) with initialvalue ðcðsÞ;fðsÞÞ.

Here we recall that the Euclidean vector norm is defined on Rn

by

kxk2 ¼Xn

i¼1

jxij2

!1=2

for x ¼ col fxig 2 Rn,

and the spectral norm [5] which is the matrix norm induced bythe Euclidean vector norm is defined by

kQk2 ¼ ðmaxfl; l is an eigenvalue of QTQgÞ1=2

where Q is an� n-matrix.

For c;f, we define kck ¼ sup�1oto0kcðtÞk2; kfk ¼ sup�1oto0

kfðtÞk2:Suppose that the system (2.2) has a unique equilibrium point

ðx�; y�Þ (which we will prove in the following section), and letuðtÞ ¼ xðtÞ � x�, vðtÞ ¼ yðtÞ � y�; FðvðtÞÞ ¼ f ðyðtÞÞ � f ðy�Þ, GðuðtÞÞ ¼

gðxðtÞÞ � gðx�Þ; CðtÞ ¼ cðtÞ � x�;FðtÞ ¼ fðtÞ � y�, then system (2.2)becomes

duðtÞ

dt¼ �AuðtÞ þ PFðvðtÞÞ þM

Z 10

KðsÞFðvðt � sÞÞds;

t40; tatk;

DuðtkÞ ¼ IkðuðtkÞÞ ¼ DxðtkÞ; k ¼ 1;2; . . . ;

dvðtÞ

dt¼ �BvðtÞ þ QGðuðtÞÞ þ N

Z 10

RðsÞGðuðt � sÞÞds;

t40; tatk;

DvðtkÞ ¼ JkðvðtkÞÞ ¼ DyðtkÞ; k ¼ 1;2; . . . :

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

(2.4)

The initial conditions associated with the system (2.4) are

uðsÞ ¼ CðsÞ; s 2 ð�1;0Þ,

vðsÞ ¼ FðsÞ; s 2 ð�1;0Þ,

where CðsÞ;FðsÞ are bounded continuous functions on ð�1;0�:Throughout this paper, we make the following assumptions:

ðH1Þ

There exist positive constant numbers xj;Zi such that

jf jðy0Þ � f jðy

00Þjpxjjy0 � y00j; j ¼ 1;2; . . . ;n,

jgiðx0Þ � giðx

00ÞjpZijx0 � x00j; i ¼ 1;2; . . . ;n.

ðH2Þ

There exist positive constant number e such thatZ 10

eeskijðsÞdso1;Z 1

0eesrjiðsÞdso1,

there must be constants l1 and l2 such thatZ 10

KðsÞds

��������

2

pZ 1

0eesKðsÞds

��������

2

¼ l1,Z 10

RðsÞds

��������

2

pZ 1

0eesRðsÞds

��������

2

¼ l2.

ðH3Þ

W ¼ maxfðkPk2 þ l1kMk2Þx=a�; ðkQk2 þ l2kNk2ÞZ=a�go1,where x ¼ maxjfxjg, Z ¼maxifZig; a0 ¼ minifaig, b0 ¼

minjfbjg; a� ¼ minfa0; b0g.

ðH4Þ The impulsive operators IkðxðtkÞÞ; JkðyðtkÞÞ satisfy

IkðxðtkÞÞ ¼ �akðxðtkÞ � x�Þ; 0oako2;

JkðyðtkÞÞ ¼ �bkðyðtkÞ � y�Þ; 0obko2;

((2.5)

where ak;bk are constants.

3. Existence and uniqueness of the equilibrium point

Theorem 1. If ðH1Þ, ðH3Þ and ðH4Þ hold, then for any C 2 C and

F 2 C, where C9Cðð�1;0�;RnÞ, there exists some constant D40,

such that

kuðtÞk2oD

2; kvðtÞk2o

D

2i:e:; kuðtÞk2 þ kvðtÞk2oD. (3.1)

That is, the solutions of system (2.4) are uniformly bounded.

Proof. For tatk and any C 2 C, F 2 C, there exists a large enoughnumber D40 such that kCkoD=2; kFkoD=2.

ARTICLE IN PRESS

J. Zhou, S. Li / Neurocomputing 72 (2009) 1688–16931690

If (3.1) is false, then there must be some t140 such that one of

the following holds:

(1)

kuðt1Þk2pD

2; kvðt1Þk2o

D

2,

(2)

kuðt1Þk2oD

2; kvðt1Þk2p

D

2, (3.2)

(3)

kuðt1Þk2pD

2; kvðt1Þk2p

D

2i:e: kuðtÞk2 þ kvðtÞk2pD,

for�1otpt1; tatk. (3.3)

We just prove (3.3), the others are the same as (3.3).

By the assumption ðH1Þ, ðH3Þ, we have

kuðt1Þk2 þ kvðt1Þk2

pe�a0t1kCk þZ t1

0e�a0ðt1�yÞkPk2xkvðyÞk2dy

þ

Z t1

0e�a0ðt1�yÞkMk2x

Z 10

vðy� sÞKðsÞds

��������

2

dy

þ e�b0t1kFk þZ t1

0e�b0ðt1�yÞkQk2ZkuðyÞk2 dy

þ

Z t1

0e�b0ðt1�yÞkNk2Z

Z 10

uðy� sÞRðsÞds

��������

2

dy

pe�a0t1D

2þ

Z t1

0e�a0ðt1�yÞkPk2x

D

2dy

þ

Z t1

0e�a0ðt1�yÞ dykMk2x

D

2

Z 10

KðsÞds

��������

2

þ e�b0t1D

2þ

Z t1

0e�b0ðt1�yÞkQk2Z

D

2dy

þ

Z t1

0e�b0ðt1�yÞdykNk2Z

D

2

Z 10

RðsÞds

��������

2

pe�a0t1D

2þ

D

2kPk2xþ

D

2kMk2xl1

� �Z t1

0e�a0ðt1�yÞ dy

FðS; ZÞ ¼ ðF1ðS; ZÞ; F2ðS; ZÞÞ

F1ðS; ZÞ ¼ Pf ðB�1ZðtÞÞ þM

Z 10

KðyÞf ðB�1ZðtÞÞdyþ I

F2ðS; ZÞ ¼ QgðA�1SðtÞÞ þ N

Z 10

RðyÞgðA�1SðtÞÞdyþ J

���������

8>>><>>>:

9>>>=>>>;

.

þ e�b0t1D

2þ

D

2kQk2Zþ

D

2kNk2Zl2

� �Z t1

0e�b0ðt1�yÞ dy

pe�a0t1D

2þkPk2xþ kMk2xl1Þ

a0ð1� e�a0t1 Þ

D

2þ e�b0t1

D

2

þkQk2Zþ kNk2Zl2Þ

b0ð1� e�b0t1 Þ

D

2oe�a0t1

D

2

þ ð1� e�a0t1 ÞD

2þ e�b0t1

D

2þ ð1� e�b0t1 Þ

D

2¼ D,

which contradicts the equality (3.3), so (3.1) holds.

For t ¼ tk, by ðxTðtkÞ; yTðtkÞÞ ¼ ðx

Tðtk � 0Þ; yTðtk � 0ÞÞ and ðH4Þ;

kuðtk þ 0Þk2 þ kvðtk þ 0Þk2

¼ kxðtk þ 0Þ � x�k2 þ kyðtk þ 0Þ � y�k2

¼ kxðtkÞ þ IkðxðtkÞÞ � x�k2 þ kyðtkÞ þ JkðyðtkÞÞ � y�k2

¼ kð1� akÞðxðtkÞ � x�Þk2 þ kð1� bkÞðyðtkÞ � y�Þk2

okxðtkÞ � x�k2 þ kyðtkÞ � y�k2 ¼ kuðtkÞk2 þ kvðtkÞk2

oD.

The proof is completed. &

Theorem 2. Assume that ðH1Þ, ðH3Þ and ðH4Þ hold, then the system

(2.3) has an unique solution ðx�; y�Þ which is an equilibrium point of

the system (2.2).

Proof. From Theorem 1, all solutions of system (2.2) are bounded,i.e., for any initial value, there exists some constant D040, suchthat

kxðtÞk2oD0; kyðtÞk2oD0.

So

Pf ðyðtÞÞ þM

Z 10

KðsÞf ðyðtÞÞdsþ I

��������

2

pkPk2xkyðtÞk2 þ kMk2xZ 1

0KðsÞyðtÞds

��������

2

þ kIk2

okPk2xD0 þ kMk2xD0l1 þ kIk29p1, (3.4)

QgðxðtÞÞ þ N

Z 10

RðsÞgðxðtÞÞdsþ J

��������

2

pkQk2ZkxðtÞk2 þ kNk2ZZ 1

0RðsÞxðtÞds

��������

2

þ kJk2

okQk2ZD0 þ kNk2ZD0l2 þ kJk29p2. (3.5)

Define SðtÞ ¼ AxðtÞ; ZðtÞ ¼ ByðtÞ such that

SðtÞ ¼ Pf ðB�1ZðtÞÞ þM

Z 10

KðyÞf ðB�1ZðtÞÞdyþ I,

ZðtÞ ¼ QgðA�1SðtÞÞ þ N

Z 10

RðyÞgðA�1SðtÞÞdyþ J.

It follows from (3.4) and (3.5) that the set O R2n defined by

O ¼ fðS; ZÞjkSk2pp1; kZk2pp2g�

Consider a mapping F : O! O defined by

If ðS; ZÞ, ðS̄; Z̄Þ are any two points of O, then

kFðS; ZÞ � FðS̄; Z̄Þk2

¼ Pðf ðB�1ZðtÞÞ � f ðB�1Z̄ðtÞÞÞ þM

Z 10

KðyÞf ðB�1ZðtÞÞdy�����

�

Z 10

KðyÞf ðB�1Z̄ðtÞÞdy�����

2

þ Q ðgðA�1SðtÞÞ � gðA�1S̄ðtÞÞÞ þ N

Z 10

RðyÞgðA�1SðtÞÞdy�����

�

Z 10

RðyÞgðA�1S̄ðtÞÞdy�����

2

pkPk2kf ðB�1ZðtÞÞ � f ðB�1Z̄ðtÞÞk2 þ kMk2

Z 10

KðyÞðf ðB�1ZðtÞÞ

����� f ðB�1Z̄ðtÞÞÞdy2 þ kQk2kgðA

�1SðtÞÞ � gðA�1S̄ðtÞÞk2

þ kNk2

Z 10

RðyÞðgðA�1SðtÞÞ � gðA�1S̄ðtÞÞÞdy����

����2

pkPk2xkB�1k2kZðtÞ � Z̄ðtÞk2 þ kMk2xkB

�1k2kZðtÞ � Z̄ðtÞk2l1

þ kQk2ZkA�1k2kSðtÞÞ � S̄ðtÞk2 þ kNk2ZkA

�1k2kSðtÞÞ � S̄ðtÞk2l2

¼ðkPk2 þ l1kMk2Þx

b0kZðtÞ � Z̄ðtÞk2 þ

ðkQk2 þ l2kNk2ÞZa0

kSðtÞ � S̄ðtÞk2

pgðkZðtÞ � Z̄ðtÞk2 þ kSðtÞ � S̄ðtÞk2Þ

¼ gkðS; ZÞ � ðS̄; Z̄Þk2,

ARTICLE IN PRESS

J. Zhou, S. Li / Neurocomputing 72 (2009) 1688–1693 1691

where g ¼maxfðkPk2 þ l1kMk2Þx=b0, ðkQk2 þ l2kNk2ÞZ=a0g, ob-viously, 0ogo1, then it follows by the well known contractionmapping principle that the mapping F has a unique point ðS�; Z�Þ.We can easily see that (2.3) satisfies, and this completes theproof. &

4. Stability of the equilibrium point of BAM neural networkswith impulses

Theorem 3. Suppose that the conditions ðH1Þ, ðH2Þ, ðH3Þ and ðH4Þ

hold, then the equilibrium point of system (2.4) is globally

exponentially stable.

Proof. We first choose the constant e in ðH2Þ with 0oeoa� suchthat

W 0¼ max

ðkPk2 þ l1kMk2Þxa� � e

;ðkQk2 þ l2kNk2ÞZ

a� � e

� �o1. (4.1)

For any C 2 C, F 2 C, we shall prove that

kuðtÞk2 þ kvðtÞk2pEðkCk þ kFkÞe�et for all t40. (4.2)

To prove (4.2), we first show: for tatk, C 2 C, F 2 C, and b41,the following inequality holds,

kuðtÞk2 þ kvðtÞk2obEðkCk þ kFkÞe�et for all t40; tatk. (4.3)

If (4.3) is false, then there must be some t240, such that

kuðt2Þk2 þ kvðt2Þk2 ¼ bEðkCk þ kFkÞe�et2 (4.4)

and

kuðtÞk2 þ kvðtÞk2pbEðkCk þ kFkÞe�et ; 0ptpt2; tatk. (4.5)

By ðH1Þ–ðH4Þ and (4.1), (4.4), (4.5), we have

kuðt2Þk2 þ kvðt2Þk2

pe�a0t2kCk þZ t2

0e�a0ðt2�yÞkPk2xkvðyÞk2dy

þ

Z t2

0e�a0ðt2�yÞ

Z 10kMk2xkvðy� sÞk2kKðsÞk2ds

� �dy

þ e�b0t2kFk þZ t2

0e�b0ðt2�yÞkQk2ZkuðyÞk2dy

þ

Z t2

0e�b0ðt2�yÞ

Z 10kNk2Zkuðy� sÞk2kRðsÞk2ds

� �dy

pe�a0t2kCk þZ t2

0e�a0ðt2�yÞkPk2xbEkFke�eydy

þ

Z t2

0e�a0ðt2�yÞ

Z 10kMk2xbEkFke�eðy�sÞkKðsÞk2ds

� �dy

þ e�b0t2kFk þZ t2

0e�b0ðt2�yÞkQk2ZbEkCke�eydy

þ

Z t2

0e�b0ðt2�yÞ

Z 10kNk2ZbEkCke�eðy�sÞkRðsÞk2ds

� �dy

pe�b0t2kFk þ kPk2xbEkFkZ t2

0e�a0ðt2�yÞe�eydy

þ kMk2xbEkFkZ t2

0e�a0ðt2�yÞe�eydy

Z 10keesKðsÞk2ds

þ e�a0t2kCk þ kQk2ZbEkCkZ t2

0e�b0ðt2�yÞe�eydy

þ kNk2ZbEkCkZ t2

0e�b0ðt2�yÞe�eydy

Z 10keesRðsÞk2ds

pe�b0t2kFk þ kPk2xbEkFkZ t2

0e�a0ðt2�yÞe�eydy

þ kMk2xbEkFkl1

Z t2

0e�a0ðt2�yÞe�eydy

þ e�a0t2kCk þ kQk2ZbEkCkZ t2

0e�b0ðt2�yÞe�eydy

þ kNk2ZbEkCkl2

Z t2

0e�b0ðt2�yÞe�eydy

¼ bEkFke�et2e�b0t2þet2

bEþ kPk2x

Z t2

0eet2 e�a0ðt2�yÞe�eydy

�

þkMk2xl1

Z t2

0eet2 e�a0ðt2�yÞe�eydy

�

þ bEkCke�et2e�a0t2þet2

bEþ kQk2Z

Z t2

0eet2 e�b0ðt2�yÞe�eydy

�

þkNk2Zl2

Z t2

0eet2 e�b0ðt2�yÞe�eydy

�

¼ bEkFke�et2e�ðb0�eÞt2

bEþ ðkPk2xþ l1xkMk2Þ

1� e�ða0�eÞt2

a0 � e

� �

þ bEkCke�et2e�ða0�eÞt2

bEþ ðkQk2Zþ l2ZkNk2Þ

1� e�ðb0�eÞt2

b0 � e

� �

pbEkFke�et2e�ða

��eÞt2

bEþ ðkPk2 þ l1kMk2Þx

1� e�ða��eÞt2

a� � e

� �

þ bEkCke�et2e�ða

��eÞt2

bEþ ðkQk2Zþ l2ZkNk2Þ

1� e�ða��eÞt2

a� � e

� �

¼ bEkFke�et21

bE�ðkPk2 þ l1kMk2Þx

a� � e

� �e�ða

��eÞt2

�

þðkPk2 þ l1kMk2Þx

a� � e

�

þ bEkCke�et21

bE�ðkQk2 þ l2kNk2ÞZ

a� � e

� �e�ða

��eÞt2

�

þðkQk2 þ l2kNk2ÞZ

a� � e

�obEkFke�et2 þ bEkCke�et2 ¼ bEðkFk þ kCkÞe�et2 ,

which contradicts the equality of (4.4), and so (4.3) holds. Letb! 1, then (4.2) holds. For t ¼ tk, by ðuðtkÞ; vðtkÞÞ ¼ ðuðtk �

0Þ; vðtk � 0Þ and ðH4Þ,

kuðtk þ 0Þk2 þ kvðtk þ 0Þk2okuðtkÞk2 þ kvðtkÞk2pEðkFk þ kCkÞe�etk .

This completes the proof. &

Remark 1. In [1], Arik and Tavsanoglu considered the followingmodel:

duiðtÞ

dt¼ �aiuiðtÞ þ

Xn

j¼1

wijgjðzjðt � tÞÞ þ Ii; i ¼ 1;2; . . . ;n;

dzjðtÞ

dt¼ �bjzjðtÞ þ

Xn

i¼1

vjigiðuiðt � tÞÞ þ Jj; j ¼ 1;2; . . . ;m:

8>>>>>><>>>>>>:

Obviously, this is a BAM neural networks with constant timedelays. This is a special case of (2.1), When DxðtkÞ ¼ 0;DyðtkÞ ¼ 0,pij ¼ qij ¼ 0, and f jðyjðt � sÞ ¼ gjðxjðt � sÞ ¼ f jðyjðt � tÞ ¼ gjðxjðt � tÞ.If we take A ¼ B ¼ ð1Þ;W ¼ V ¼ ð1Þ;a ¼ b ¼ 3=2 andkðyÞ ¼ rðyÞ ¼ 10�y; e ¼ 0:1, by simple calculation, O140;O240,which means the condition of Theorem 1 in [1] failed, but ðH3Þ

holds.

Furthermore, when proving the existence of the equilibrium

point of the model, Arik and Tavsanoglu required that the

activation functions be bounded. Here we do not need this

assumption. We obtain GES for the equilibrium point of our

model, while Arik and Tavsanoglu obtain global asymptotic

stability for the equilibrium point of their model.

ARTICLE IN PRESS

−2 0 2 4 6 8 10 12−1

−0.5

0

0.5

1

1.5

time t

stat

es x

(t), y

(t)

x(t)

y(t)

Fig. 1. Stability of impulsive BAM neural networks (5.1) with distributed delays

where I1 ¼ I2 ¼ 0; ð0;0ÞT is the equilibrium point.

J. Zhou, S. Li / Neurocomputing 72 (2009) 1688–16931692

Remark 2. When DxðtkÞ ¼ 0;DyðtkÞ ¼ 0 and P ¼ Q ¼ 0, the model(2.2) becomes the same as Zhao [1], Zhao [8, Theorem 7] provedthe global asymptotic stability of the equilibrium point if ðH1Þwassatisfied and

rðWÞo1; W ¼ K�1jMjx0 þ K�1

jNjZ0, (4.6)

where K ¼ diagfkig, with ki ¼ minfai; big, jMj ¼ ðjmijjÞn�n,jNj ¼ ðjnijjÞn�n, x0 ¼ diagfxg, Z0 ¼ diagfZg, ðH3Þ is weaker than(4.6). In fact, if we take M ¼ N ¼ ð11

�11 Þ, x1 ¼ x2 ¼ Z1 ¼ Z2 ¼

12,

a1 ¼ a2 ¼ b1 ¼ b2 ¼ 2, rij ¼ kij ¼ e�y; e ¼ 0:1, by simple calcula-tion, ðH3Þ holds. But

W ¼

1

20

01

2

0BBB@

1CCCA

1 1

1 1

! 1

20

01

2

0BBB@

1CCCAþ

1

20

01

2

0BBB@

1CCCA

1 1

1 1

! 1

20

01

2

0BBB@

1CCCA

¼

1

2

1

2

1

2

1

2

0BBB@

1CCCA,

where rðWÞ ¼ 1, the condition (4.6) failed.

Remark 3. When discussing the GES of BAM, many authors usedthe properties of nonsingular M-matrix. Here we point out ourresults do not use these properties. So we can deal with the casewhich does not satisfy M-matrix properties.

Li and Yang [9] discussed the system (2.1) with pij ¼ qji ¼

0; i ¼ 1;2; . . . ;n; j ¼ 1;2; . . . ;n. Li and Yang [9, Theorem 4] proved

the GES of the unique equilibrium point if ðH1Þ, ðH2Þ, ðH4Þ hold

and

S ¼A �TjNj

�LjMj B

!is an M-matrix where T ¼ diagðZ1; . . . ;ZnÞ,

L ¼ diagðx1; . . . ; xnÞ. (4.7)

However, let n ¼ 1; a1 ¼ b1 ¼ 1, M ¼ N ¼ ð1Þ, kðyÞ ¼ rðyÞ ¼ 10�y,

x ¼ Z ¼ 32, e ¼ 0:1.

By simple calculation,

Z 10

e0:1y10�ydy ¼1

ln 10� 0:1¼ l1 ¼ l2

ðH3Þ holds. But S ¼1 �

3

2

�3

21

0BB@

1CCA is not an M-matrix.

Remark 4. When DxðtkÞ ¼ 0;DyðtkÞ ¼ 0, the model (2.2) becomesthe same as Song and Cao [10]. In [10], Song and Cao constructed anew Lyapunov functional, employed the homeomorphism theory,M-matrix theory and Hardy inequality to obtain a sufficientcondition to ensure the existence, uniqueness and GES of theequilibrium point for the model. By the same discussion withRemarks 3, Theorem 3 can deal with the case which does notsatisfy M-matrix properties.

5. Illustrative examples

Example 1. Let n ¼ 1, f jðxÞ ¼ giðxÞ ¼ jxj; kijðsÞ ¼ rjiðsÞ ¼ e�s.Clearly, the activation functions f jð�Þ; gið�Þ satisfy hypothesis ðH1Þ

with xi ¼ Zi ¼ 1 and the kernel kijð�Þ; rjið�Þ satisfy hypothesis ðH2Þ.Consider impulsive BAM neural networks with distributed delaysas follows:

dxðtÞ

dt¼ �AxðtÞ þ Pf ðyðtÞÞ þM

Z 10

KðsÞf ðyðt � sÞÞdsþ I;

t40; tatk;

DxðtkÞ ¼ IkðxðtkÞÞ; k ¼ 1;2; . . . ;

dyðtÞ

dt¼ �ByðtÞ þ QgðxðtÞÞ þ N

Z 10

RðsÞgðxðt � sÞÞdsþ J;

t40; tatk;

DyðtkÞ ¼ JkðyðtkÞÞ; k ¼ 1;2; . . . ;

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

(5.1)

where A ¼ B ¼ ð2:5Þ, P ¼ Q ¼ M ¼ N ¼ ð1Þ, I ¼ 5, J ¼ 3,ak ¼ 1þ 1

2 sinð1þ kÞ, bk ¼ 1þ 23 cosð2k3

Þ;0ot1ot2o � � � is a strictlyincreasing sequence such that limt!1tk ¼ þ1; k 2 Zþ.

By simple calculation, we obtainZ 10

e0:1se�sds ¼1

0:9¼ l1 ¼ l2,

which implies ðH2Þ holds, and

W ¼ maxðkPk2 þ l1kMk2Þx

a�;ðkQk2 þ l2kNk2ÞZ

a�

� �o1,

which implies ðH3Þ holds: obviously, ðH4Þ holds with 0oako2,

0obko2, k 2 Zþ.

So, by Theorem 3, the equilibrium point of system (5.1) is

globally exponentially stable (Fig. 1).

6. Conclusion

In this paper, GES of impulsive BAM neural networks withdistributed delays have been studied. Some sufficient conditionsfor the existence and GES of the equilibrium point have beenestablished. These obtained results are new and they complementpreviously known results. Moreover, an example is given toillustrate the effectiveness of our results. Moreover, our methodsin the paper may be extended for more complex networks.

Acknowledgments

The authors would like to thank the Associate Editor and fourreviewers for their careful reading of this paper and helpful

ARTICLE IN PRESS

J. Zhou, S. Li / Neurocomputing 72 (2009) 1688–1693 1693

comments, which have been very useful for improving the qualityof this paper. This work is supported by National ScienceFoundation of PR China under Grant 10671133 and by ScientificResearch Fund of Sichuan Provincial Education Department.

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Jie Zhou received the Master degree from SichuanNormal University, Chengdu, China. He is with collegeof science ,Sichuan University of Science and Engineer-ing, Zigong, Sichuan 643000, PR China. He is currentlya teaching assistant of college of science in SichuanUniversity of Science and Engineering, Zigong, PRChina.

Shuyong Li received the Ph.D. degree from SichuanUniversity, Chengdu, China, and the Post-DoctoralFellow in School of Science, Xi’an Jiaotong University,Xi’an, China. He was with College of Mathematics andSoftware science at Sichuan Normal University,Cheng-du, China. He is currently a Professor of SichuanNormal university. He also is the author or coauthorof more than 40 journal papers. His research interestsinclude functional differential systems, partial func-tional differential systems and neural networks.