Convergence behavior of delayed bidirectional associative memory cellular neural networks with asymptotically periodic coefficients

Applied Mathematics and Computation 215 (2009) 928–935

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Convergence behavior of delayed bidirectional associative memory cellular neural networks with asymptotically periodic coefficients Junyan Xu *, Lijuan Chen, Zhong Li College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China

a r t i c l e

i n f o

a b s t r a c t This paper considers the convergence behavior of delayed BAM cellular neural networks with asymptotically periodic coefficients. By applying mathematical analysis techniques, some new sufficient conditions are obtained to ensure that solutions of the networks converge to a periodic function. Ó 2009 Elsevier Inc. All rights reserved.

Keywords: CNNs Convergence Asymptotically periodic coefficients Delays

1. Introduction Since the cellular neural networks (CNNs) were introduced by Chua and Roska [1] in 1990, they have been successfully applied in signal and image processing, pattern recognition and optimization. The delayed CNNs can be described by the following differential equations:

x_ i ðtÞ ¼ ci ðtÞhi ðxi ðtÞÞ þ

n X

aij ðtÞfj ðxj ðtÞÞ þ

j¼1

n X

bij ðtÞg j ðxj ðt  sij ðtÞÞÞ þ Ii ðtÞ;

i ¼ 1; 2; . . . ; n;

ð1:1Þ

j¼1

where fi ðxÞ; g i ðxÞ are activation functions, aij ðtÞ is the strength of the jth unit on the ith unit at time t, bij ðtÞ is the strength of the jth unit on the ith unit at time t  sij ðtÞ, and sij ðtÞ P 0 denotes the transmission delay of the ith unit along the axon of the jth unit at the time t, Ii ðtÞ denotes the external bias on the ith unit at the time t, hi ðtÞ is an appropriately behaved function, ci ðtÞ represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the networks and external inputs at the time t. Under the assumptions: ci ; Ii ; aij ; bij : R ! R are continuous periodic functions, i; j ¼ 1; 2; . . . ; n; extensive results on the problem of the existence and stability of periodic solutions for system (1.1) are given in many literature entries (see [2–5] and the references cited therein). However, it is hard to find a strict periodic environment in real world, so, it is worthwhile to investigate the convergence behavior for all solutions of CNNs for more general kind of system. Recently, under the assumption that the coefficients of system (1.1) are all asymptotically periodic functions, by using of Lyapunov method and mathematical analysis techniques used in [6–8], convergence behavior of asymptotically periodic system have been researched. To the best of our knowledge, to this day, still no scholar consider the convergence behavior for all solutions of BAM system with asymptotically periodic coefficients. For more details about the deduction of BAM neural networks, one could refer to [9–21]. In this paper, we propose the delayed BAM cellular neural networks as follows:

8 m m P P > > > < x_ i ðtÞ ¼ ai ðtÞhi ðxi ðtÞÞ þ aij ðtÞfj ðyj ðtÞÞ þ bij ðtÞg j ðyj ðt  sij ðtÞÞÞ þ Ii ðtÞ; j¼1

j¼1

n n > > ~ ðy ðtÞÞ þ P c ðtÞ~f ðx ðtÞÞ þ P d ðtÞg~ ðx ðt  r ðtÞÞÞ þ J ðtÞ; > : y_ j ðtÞ ¼ bj ðtÞh j j ji i i ji i i ji j i¼1

i¼1

* Corresponding author. E-mail addresses: [email protected] (J. Xu), [email protected] (L. Chen), [email protected] (Z. Li). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.037

ð1:2Þ

J. Xu et al. / Applied Mathematics and Computation 215 (2009) 928–935

929

where i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; m correspond to the numbers of neurons in the I-layer and J-layer, respectively. xi ðtÞ; yj ðtÞ ~i ðxÞ are activation functions. denote the state variables associated with the ith and jth neurons, respectively. fj ðxÞ; g j ðxÞ; ~f i ðxÞ; g aij ðtÞ; bij ðtÞ; cji ðtÞ; dji ðtÞ are connection weights. sij ðtÞ; rji ðtÞ P 0 denote transmission delays. Ii ðtÞ; J j ðtÞ denote the input to the ith and jth neurons, respectively. ai ðtÞ; bj ðtÞ : R ! Rþ ; aij ðtÞ; bij ðtÞ; Ii ðtÞ; cji ðtÞ; dji ðtÞ; J j ðtÞ; sij ðtÞ; rji ðtÞ : R ! R are all asymptotically periodic functions, that is

ai ðtÞ ¼ ai ðtÞ þ a~ i ðtÞ; aij ðtÞ ¼ aij ðtÞ þ a~ij ðtÞ; bij ðtÞ ¼ bij ðtÞ þ b~ij ðtÞ; ~j ðtÞ; bj ðtÞ ¼ bj ðtÞ þ b Ii ðtÞ ¼ Ii ðtÞ þ eI i ðtÞ;

cji ðtÞ ¼ cji ðtÞ þ ~cji ðtÞ;

~ ðtÞ; dji ðtÞ ¼ dji ðtÞ þ d ji 

sij ðtÞ ¼ sij ðtÞ þ s~ij ðtÞ; Jj ðtÞ ¼ Jj ðtÞ þ ~Jj ðtÞ; rji ðtÞ ¼ rji ðtÞ þ r~ ji ðtÞ;

  ~ i ðtÞ ¼ where ai ðtÞ; aij ðtÞ; bij ðtÞ; Ii ðtÞ; sij ðtÞ; bj ðtÞ; cji ðtÞ; dji ðtÞ; J j ðtÞ; rji ðtÞ are all continuous x-periodic functions and limt!þ1 a ~ ~ji ðtÞ ¼ e ~j ðtÞ ¼ 0; limt!þ1 ~cji ðtÞ ¼ 0; limt!þ1 d ~ ~ij ðtÞ ¼ 0; limt!þ1 b 0; limt!þ1 aij ðtÞ ¼ 0; limt!þ1 bij ðtÞ ¼ 0; limt!þ1 I i ðtÞ ¼ 0; limt!þ1 s ~ ji ðtÞ ¼ 0: Denote X1 ¼ f1; 2; . . . ; ng; X2 ¼ f1; 2; . . . ; mg. Apparently, we have: (H0) For each 0; limt!þ 1~J j ðtÞ ¼ 0; limt!þ1 r i 2 X1 ; j 2 X2 ,

lim ðbj ðtÞ  bj ðtÞÞ ¼ 0;

lim ðai ðtÞ  ai ðtÞÞ ¼ 0;

t!þ1

lim ðIi ðtÞ  Ii ðtÞÞ ¼ 0;

t!þ1

t!þ1

lim ðJ j ðtÞ  Jj ðtÞÞ ¼ 0;

t!þ1



lim ðaij ðtÞ  aij ðtÞÞ ¼ 0;

t!þ1

lim ðcji ðtÞ  cji ðtÞÞ ¼ 0;

t!þ1

lim ðsij ðtÞ  sij ðtÞÞ ¼ 0;

t!þ1

lim ðbij ðtÞ  bij ðtÞÞ ¼ 0;

t!þ1



lim ðdji ðtÞ  dji ðtÞÞ ¼ 0;

t!þ1

lim ðrji ðtÞ  rji ðtÞÞ ¼ 0:

t!þ1

Then, we can choose constants

s and r such that

s ¼ max f max sij ðtÞg; r ¼ max f max rji ðtÞg: i2X1 ;j2X2 t2½0;þ1Þ

i2X1 ;j2X2 t2½0;þ1Þ

The initial conditions associated with system (1.2) are of the form

xi ðsÞ ¼ uxi ðsÞ;

s 2 ð1; 0; i 2 X1 ;

yj ðsÞ ¼ uyj ðsÞ;

s 2 ð1; 0; j 2 X2 ;

where u ¼ ðux1 ðtÞ; . . . ; uxn ðtÞ; uy1 ðtÞ; . . . ; uym ðtÞÞT 2 Cðð1; 0; Rnþm Þ. Consider the following delayed BAM cellular neural networks with periodic coefficients:

8 m m P P  >  > > x_ i ðtÞ ¼ ai ðtÞhi ðxi ðtÞÞ þ aij ðtÞfj ðyj ðtÞÞ þ bij ðtÞg j ðyj ðt  sij ðtÞÞÞ þ Ii ðtÞ; <

i 2 X1 ;

n n > >  ~ ðy ðtÞÞ þ P c ðtÞ~f ðx ðtÞÞ þ P d ðtÞg~ ðx ðt  r ðtÞÞÞ þ J  ðtÞ; > : y_ j ðtÞ ¼ bj ðtÞh j j i i i i ji j ji ji

j 2 X2 :

j¼1

i¼1

j¼1

ð1:3Þ

i¼1

The initial conditions associated with system (1.3) are of the form

xi ðsÞ ¼ wxi ðsÞ; s 2 ð1; 0;

i 2 X1 ;

yj ðsÞ ¼ wyj ðsÞ; s 2 ð1; 0;

j 2 X2 ;

ð1:4Þ

where w ¼ ðwx1 ðtÞ; . . . ; wxn ðtÞ; wy1 ðtÞ; . . . ; wym ðtÞÞT 2 Cðð1; 0; Rnþm Þ. The organization of this paper is as follows. In Section 2, some new sufficient conditions for the existence and uniqueness of the x-periodic solution of system (1.3) are obtained. In Section 3, we derive some sufficient conditions ensuring that all solutions of system (1.2) converge to a periodic function, which are new and complement previously known results. In Section 4, we shall give an example to illustrate the effectiveness of our results. 2. Conditions and lemma Throughout this paper, we assume that: e ;F ;e ~ 2 C½R; R, and there exist nonnegative constants h ; h e (H1) For each i 2 X1 ; j 2 X2 , ai ðtÞ > 0; bj ðtÞ > 0; hi ; h j i j j F i ; Gj ; G i such that

e ðv Þ e hi ðuÞ  hi ðv Þ h j ðuÞ  h j e; P hi ; Ph j uv uv e e F i ju  v j; jfj ðuÞ  fj ðv Þj 6 F j ju  v j; j f i ðuÞ  f i ðv Þj 6 e e i ju  v j g i ðxÞ  e g i ðv Þj 6 G jg ðuÞ  g ðv Þj 6 Gj ju  v j; j e j

j

for all u; v 2 R; u – v :

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J. Xu et al. / Applied Mathematics and Computation 215 (2009) 928–935

(H2) For each i 2 X1 ; j 2 X2 , there exist constants c1 > 0; c2 > 0; k > 0 and ni > 0; gj > 0, such that for all t > 0, there holds

 ðai ðtÞhi  kÞni þ

m X

jaij ðtÞjF j gj þ

j¼1

e  kÞg þ  ðbj ðtÞ h j j

n X

m X



jbij ðtÞjGj eks gj < c1 < 0;

j¼1

jcji ðtÞj e F i ni þ

n X

i¼1

 e i ekr ni < c < 0: jdji ðtÞj G 2

i¼1

As usual, we introduce the phase space C ¼ Cðð1; 0; Rnþm Þ as a Banach space of continuous mappings from ð1; 0 to Rnþm equipped with the supremum norm defined by

kzðtÞk ¼ sup kzðtÞkðn;gÞ ; 1
where 1 kzðtÞkðn;gÞ ¼ max fjn1 i xi ðtÞj; jgj yj ðtÞjg

ð2:1Þ

i2X1 ;j2X2

for all zðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞ; y1 ðtÞ; . . . ; ym ðtÞÞT 2 Cðð1; 0; Rnþm Þ. Lemma 1. Let ðH1 Þ and ðH2 Þ hold, then system (1.3) has exactly one x-periodic solution z ðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞ; y1 ðtÞ; . . . ; ym ðtÞÞT . Proof. The proof is similar to the method used in [4]. Pick a constant N satisfying N > max

I ¼ max maxfai ðtÞjhi ð0Þj þ

m X

t

i

m X

j¼1

~ ð0Þj þ J ¼ max maxfbj ðtÞjh j t

j

aij ðtÞjfj ð0Þj þ

n X

n

I

J

c1 ; c2

o

, where



ð2:2Þ



ð2:3Þ

bij ðtÞjg j ð0Þj þ Ii ðtÞg;

j¼1

cji ðtÞj~f i ð0Þj þ

n X

i¼1

dji ðtÞjg~i ð0Þj þ J j ðtÞg:

i¼1

Denote

X ¼ fzðhÞ 2 C : kzðhÞk 6 N; kz_ ðhÞk 6 Mg; where M ¼ maxfM1 ; M 2 g and

( M1 ¼ max sup ½ai ðtÞhi þ i

þ

m X

m X

t

 jbij ðtÞjjg j ð0Þjn1 i

jaij ðtÞjF j gj n1 i þ

j¼1

þ

Ii ðtÞn1 i

m X



1  jbij ðtÞjGj gj n1 i N þ ai ðtÞjhi ð0Þjni þ

j¼1

)

m X

jaij ðtÞjjfj ð0Þjn1 i

j¼1

;

j¼1

( n n n X X X  ~ ð0Þjg1 þ ~ þ e i ni g1 N þ b ðtÞjh M2 ¼ max sup ½bj ðtÞh jcji ðtÞj e F i ni g1 jdji ðtÞj G jcji ðtÞjjef i ð0Þjg1 j j j þ j j j j j

þ

n X

t

i¼1

)

i¼1

i¼1



 1 : jdji ðtÞjj e g i ð0Þjg1 j þ J j ðtÞgj

i¼1

It is easy to check that X is a convex compact set. Now, define a map T from X to C by

T : wðsÞ ! zðs þ x; wÞ; where zðtÞ ¼ zðt; wÞ is the solution of system (1.3) with the initial condition (1.4). In the following, we will prove that T X  X, i.e. if w 2 X, then z 2 X. To do that, we define the following function

NðtÞ ¼ sup kzðt þ sÞkðn;gÞ : s2ð1;0

It is easy to see that

kzðtÞkðn;gÞ 6 NðtÞ: Therefore, what we need to do is to prove NðtÞ 6 N for all t > 0. Assume that t0 P 0 is the smallest value such that

kzðt0 Þkðn;gÞ ¼ Nðt 0 Þ ¼ N;

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J. Xu et al. / Applied Mathematics and Computation 215 (2009) 928–935

and

kzðtÞkðn;gÞ 6 N

if t < t 0 :

Let i0 ; j0 be indexes such that

kzðtÞkðn;gÞ ¼ jn1 or kzðtÞkðn;gÞ ¼ jg1 j0 yj0 ðtÞj: i0 xi0 ðtÞj 1 If kzðtÞkðn;gÞ ¼ jn1 i0 xi0 ðtÞj, which implies that gj yj ðtÞ 6 kzðtÞkðn;gÞ , we have

m X djxi0 ðtÞj ai0 j ðt 0 Þfj ðyj ðt 0 ÞÞþ jt¼t0 ¼ sgnðxi0 ðt0 ÞÞfai0 ðt 0 Þhi0 ðxi0 ðt 0 ÞÞ þ dt j¼1 m X

6 ai0 ðt 0 Þ½hi0 jxi0 ðt 0 Þjn1 i0 ni0 þ hi0 ð0Þsgnðxi0 ðt 0 ÞÞ þ

 m j¼1 bi0 j ðt 0 Þg j ðyj ðt 0

 si0 j ðt 0 ÞÞÞ þ Ii0 ðt0 Þg

ai0 j ðt 0 Þ½F j jyj ðt 0 Þjg1 j gj þ jfj ð0Þj

j¼1 m X

þ (



 bi0 j ðt 0 Þ½Gj jyj ðt0  si0 j ðt 0 ÞÞjg1 j gj þ jg j ð0Þj þ Ii0 ðt 0 Þ

j¼1

ai0 ðt 0 Þhi0 ni0 þ

6

m X

ai0 j ðt0 ÞF j gj þ

j¼1

m X

) 

bi0 j ðt 0 ÞGj gj Nðt 0 Þ þ I < c1 N þ I < 0:

ð2:4Þ

j¼1

Similarly, when kzðtÞkðn;gÞ ¼ jg1 j0 yj0 ðtÞj, we also have

 djyj0 ðtÞj  < c2 N þ J < 0: dt t¼t0

ð2:5Þ

(2.4) and (2.5) mean that kzðtÞkðn;gÞ can never exceed N. Thus, kzðtÞkðn;gÞ 6 NðtÞ 6 N for all t > t0 . Moreover, it is easy to see that kz_ ðs þ xÞk 6 M. Therefore, T X  X. The proof is complete. h 3. Convergence behavior In this section, we derive sufficient conditions which guarantee that all solutions of system (1.2) converge to a periodic function. Theorem 1. Let (H1) and (H2) holds. Suppose that z ðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞ; y1 ðtÞ; . . . ; ym ðtÞÞT is the xperiodic solution of system (1.3), then for every solution zðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞ; y1 ðtÞ; . . . ; ym ðtÞÞT of system (1.2) with any initial u ¼ ðux1 ðtÞ; . . . ; uxn ðtÞ; uy1 ðtÞ; . . . ; uym ðtÞÞT 2 Cðð1; 0; Rnþm Þ, there holds

lim jxi ðtÞ  xi ðtÞj ¼ 0;

t!1

i ¼ 1; . . . ; n;

lim jyj ðtÞ  yj ðtÞj ¼ 0;

j ¼ 1; . . . ; m:

t!1

Proof. Let uðtÞ ¼ zðtÞ  z ðtÞ ¼ ðu1 ðtÞ; . . . ; un ðtÞ; unþ1 ðtÞ; . . . ; unþm ðtÞÞT , that is ui ðtÞ ¼ xi ðtÞ  xi ðtÞ; unþj ðtÞ ¼ yj ðtÞ  yj ðtÞ; i 2 X1 ; j 2 X2 . Then

u_ i ðtÞ ¼ ai ðtÞhi ðxi ðtÞÞ þ ai ðtÞhi ðxi ðtÞÞ þ

m X j¼1



m X

aij ðtÞfj ðyj ðtÞÞ 

m X

aij ðtÞfj ðyj ðtÞÞ þ

j¼1

m X

bij ðtÞg j ðyj ðt  sij ðtÞÞÞ

j¼1



bij ðtÞg j ðyj ðt  sij ðtÞÞÞ þ Ii ðtÞ  Ii ðtÞ ¼ ai ðtÞ½hi ðxi ðtÞÞ  hi ðxi ðtÞÞ þ

j¼1

þ

m X

m X

aij ðtÞ½fj ðyj ðtÞÞ  fj ðyj ðtÞÞ

j¼1

bij ðtÞ½g j ðyj ðt  sij ðtÞÞÞ  g j ðyj ðt  sij ðtÞÞÞ þ D1i ðtÞ;

i 2 X1 ;

ð3:1Þ

j¼1

where

D1i ðtÞ ¼ ½ai ðtÞ  ai ðtÞhi ðxi ðtÞÞ þ

m m X X  ½aij ðtÞ  aij ðtÞfj ðyj ðtÞÞ þ ½bij ðtÞ  bij ðtÞg j ðyj ðt  sij ðtÞÞÞ þ ½Ii ðtÞ  Ii ðtÞ: j¼1

j¼1

Similarly, we have

~ ðy ðtÞÞ  h ~ ðy ðtÞÞ þ u_ nþj ðtÞ ¼ bj ðtÞ½h j j j j

n X

cji ðtÞ½~f i ðxi ðtÞÞ  ~f i ðxi ðtÞÞ þ

i¼1

þ D2j ðtÞ;

n X

dji ðtÞ½g~i ðxi ðt  rji ðtÞÞÞ  g~i ðxi ðt  rji ðtÞÞÞ

i¼1

j 2 X2 ;

ð3:2Þ

where

~ j ðy ðtÞÞ þ D2j ðtÞ ¼ ½bj ðtÞ  bj ðtÞh j

n n X X  ½cji ðtÞ  cji ðtÞ~f i ðxi ðtÞÞ þ ½dji ðtÞ  dji ðtÞg~i ðxi ðt  rji ðtÞÞÞ þ ½J j ðtÞ  J j ðtÞ: i¼1

i¼1

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J. Xu et al. / Applied Mathematics and Computation 215 (2009) 928–935

Since z ðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞ; y1 ðtÞ; . . . ; ym ðtÞÞT is the xperiodic solution of system (1.3), together with ðH0 Þ and ðH2 Þ, then 8e > 0, we can choose a sufficient large constant T > 0 such that

jD1i ðtÞj <

c1 4

e; jD2j ðtÞj <

 ðai ðtÞhi  kÞni þ

m X

c2 4

e;

ð3:3Þ

jaij ðtÞjF j gj þ

m X

j¼1

e  kÞg þ  ðbj ðtÞ h j j

n X

jbij ðtÞjGj eks gj < 

c1

< 0;

ð3:4Þ

e i ekr ni <  c2 < 0; jdji ðtÞj G 2

ð3:5Þ

j¼1

jcji ðtÞj e F i ni þ

n X

i¼1

i¼1

2

and

jyj ðt  sij ðtÞÞ  yj ðt  sij ðtÞÞj < K e; K is a certain positive constant

ð3:6Þ

for all t P T; i 2 X1 ; j 2 X2 . Let MðtÞ ¼ max1
MðtÞ > ekt kuðtÞkðn;gÞ :

ð3:7Þ

Then, we claim that

MðtÞ  MðTÞ

ð3:8Þ

for all t P T. By way of contradiction, assume that (3.8) does not hold. Consequently, there exists t1 > T such that Mðt 1 Þ > MðTÞ. So there must exist t 2 ðT; t1  such that 

ekt kuðt  Þkðn;gÞ ¼ Mðt 1 Þ: Since MðtÞ is non-decreasing, we have

Mðt1 Þ P Mðt Þ: 

That is ekt kuðt  Þkðn;gÞ P Mðt Þ, which contradicts (3.7). This contradiction implies that (3.8) holds. It follows that there exists sufficient large t2 > T such that

kuðtÞkðn;gÞ < ekt MðtÞ ¼ ekt MðTÞ < e for all t P t2 . Case (ii). Suppose there exists a point t0 P T such that Mðt 0 Þ ¼ ekt0 kuðt 0 Þkðn;gÞ . Let it ; jt be indexes such that   kuðtÞkðn;gÞ ¼ n1 or kuðtÞkðn;gÞ ¼ g1 jt jyjt ðtÞ  yjt ðtÞj: it jxit ðtÞ  xit ðtÞj  P First, suppose kuðtÞkðn;gÞ ¼ n1 it jxit ðtÞ  xit ðtÞj, K ¼ m 4

j¼1

c1 jbit j ðt 0 ÞjGj

 : Calculating the upper right derivative of eks jxis ðsÞ  xi ðsÞj s

0

along (3.1), in view of (3.3), (3.4) and (3.6) and ðH1 Þ, we have

Dþ ðeks jxis ðsÞ  xis ðsÞjÞjs¼t0 ¼ kekt0 jxit0 ðt 0 Þ  xit ðt 0 Þj þ ekt0 sgnðxit0 ðt0 Þ  xit ðt 0 ÞÞfait0 ðt 0 Þ½hit0 ðxit0 ðt0 ÞÞ 0

hit0 ðxit ðt 0 ÞÞ 0



þ

0

m X

ait0 j ðt 0 Þ½fj ðyj ðt 0 ÞÞ 

fj ðyj ðt0 ÞÞ

j¼1

þ

m X

bit0 j ðt 0 Þ½g j ðyj ðt0  sit0 j ðt 0 ÞÞÞ

j¼1

 g j ðyj ðt 0  sit j ðt0 ÞÞÞ þ D1it ðt0 Þg 0 0 m h i X 6  ait0 ðt 0 Þhit0  k ekt0 juit0 ðt 0 Þjn1 jait0 j ðt 0 ÞjF j ekt0 jyj ðt0 Þ  yj ðt 0 Þjg1 j gj it nit0 þ 0

m X

þ

ksit j ðt0 Þ kðt 0 sit j ðt0 ÞÞ

jbit0 j ðt 0 ÞjGj e

0

e

0

j¼1

jyj ðt 0  sit0 j ðt 0 ÞÞ  yj ðt 0  sit0 j ðt0 ÞÞjg1 j gj

j¼1 m X

þ ( 6

jbit0 j ðt 0 ÞjGj ekt0 jyj ðt 0  sit0 j ðt0 ÞÞ  yj ðt0  sit j ðt 0 ÞÞj þ ekt0 D1it ðt 0 Þ 0

j¼1

0

)

m m h i X X jait0 j ðt0 ÞjF j gj þ jbit0 j ðt 0 ÞjGj eks gj Mðt 0 Þ  ait0 ðt 0 Þhit0  k nit0 þ j¼1

þ ekt0

c1 4

e þ ekt0

c1 4

e<

c1 2

Mðt 0 Þ þ ekt0

j¼1

c1 2

e:

ð3:9Þ

J. Xu et al. / Applied Mathematics and Computation 215 (2009) 928–935  P Secondly, suppose kuðtÞkðn;gÞ ¼ g1 jt jyjt ðtÞ  yjt ðtÞj; K ¼ n 4

( þ

ks

D ðe jyjs ðsÞ 

yjs ðsÞjÞ js¼t0

6

c2 jdjt i ðt 0 Þje Gi

~  kg þ  ½bjt ðt 0 Þh jt jt 0

<

i¼1

c2 2

0

kt 0

Mðt 0 Þ þ e

0

c2 2

. From (3.2), (3.3), (3.5) and (3.6) and ðH1 Þ, we can obtain

0

n X

jcjt i ðt 0 Þj e F i ni þ

i¼1

933

0

n X i¼1

) c kr e jdjt i ðt 0 Þj G i e ni Mðt 0 Þ þ ekt0 2 e 0 2

e:

ð3:10Þ

If Mðt 0 Þ P ekt0 e, (3.9) and (3.10) show that

Dþ MðtÞjt¼t0 < 0: It implies that MðtÞ is strictly decreasing in a small neighborhood ðt 0 ; t 0 þ dÞ. This contradicts that MðtÞ is non-decreasing. Hence

Mðt 0 Þ < ekt0 e and kuðt 0 Þkðn;gÞ ¼ ekt0 Mðt 0 Þ < e:

ð3:11Þ

Furthermore, for any t > t 0 , by the same approach used in the proof of (3.11), we have

MðtÞ < ekt e and kuðtÞkðn;gÞ ¼ ekt MðtÞ < e;

if MðtÞ ¼ ekt kuðtÞkðn;gÞ :

On the other hand, if MðtÞ > ekt kuðtÞkðn;gÞ ; t > t0 , there exists t3 such that

Mðt 3 Þ ¼ ekt3 kuðt3 Þkðn;gÞ ;

kuðt 3 Þkðn;gÞ < e and MðsÞ > eks kuðsÞkðn;gÞ ;

for all s 2 ðt 3 ; t:

Using a similar argument as in the proof of Case(i), we can show that

MðsÞ  Mðt3 Þ is a constant for all s 2 ðt 3 ; t; which implies that

kuðtÞkðn;gÞ < ekt MðtÞ ¼ ekt Mðt 3 Þ ¼ kuðt 3 Þkðn;gÞ ekðtt3 Þ < e: In summary, there must exist T  > 0 such that kuðtÞkðn;gÞ < e holds for all t > T  . The proof is complete. h 4. An example In this section, we give an example to demonstrate the results obtained in previous sections. Example 4.1. Consider the following BAM cellular neural networks with time-varying delays:

    8 3 1 1 2t > _ > > x1 ðtÞ ¼ ðx1 ðtÞ þ x1 ðtÞÞ þ 4 sin t þ 1þt2 f1 ðy1 ðtÞÞ þ cos t þ 1þt2 f2 ðy2 ðtÞÞ > > >      > > 2 1 1 3 > þ 14 sin t þ 2þt þ ðcos t þ 1þt > 2 g 1 y1 t  sin t  1þt 4 2 Þg 2 > > > >       > > 1 > > y2 t  2 cos2 t  1þt þ sin t þ 1þtt 2 ; 2 > > > >     > > > 2t > x_ 2 ðtÞ ¼ ðx2 ðtÞ þ x32 ðtÞÞ þ 13 sin 2t þ 1þtt 2 f1 ðy1 ðtÞÞ þ 14 cos 2t þ 1þt > 2 f2 ðy2 ðtÞÞ > > > >        > > 1 2 1 1 > þ 14 cos 2t þ 1þt þ 19 sin 2t þ 1þ2t < 2 g 1 y1 t  cos t  2þt 2 2 g2      > 2 1 1 > þ cos t þ 1þt  y2 t  12 sin t  1þ2t > 2 2 ; > > > >   > > > y_ ðtÞ ¼ 2y ðtÞ þ 1 sin t þ t ef ðx ðtÞÞ > 1 2 2 > 1 2 1þt 2 > > >   > > > 1 2 2 > e þ 12 cos t þ 1þt > 2 g 2 ðx2 ðt  3 cos t  2þt 2 ÞÞ þ sin t; > > > >   > > 2t e > y_ 2 ðtÞ ¼ 2y2 ðtÞ þ 14 sin t þ 1þt > 2 f 2 ðx2 ðtÞÞ > > > >      > > 1 2 1 : e þ cos t; þ 14 cos t þ 3þt 2 g 2 x2 t  cos t  1þt 2 where f1 ðxÞ ¼ f2 ðxÞ ¼ g 1 ðxÞ ¼ g 2 ðxÞ ¼ e f 1 ðxÞ ¼ e f 2 ðxÞ ¼ e g 1 ðxÞ ¼ e g 2 ðxÞ ¼ arctan x.

ð4:1Þ

934

J. Xu et al. / Applied Mathematics and Computation 215 (2009) 928–935

Note the following BAM cellular neural networks system:

8 x_ 1 ðtÞ ¼ ðx1 ðtÞ þ x31 ðtÞÞ þ 14 sin tf1 ðy1 ðtÞÞ þ cos tf2 ðy2 ðtÞÞ > > > > > > > þ 14 sin tg 1 ðy1 ðt  sin tÞÞ þ cos tg 2 ðy2 ðt  2 cos tÞÞ þ sin t; > > > > > > 3 1 1 > < x_ 2 ðtÞ ¼ ðx2 ðtÞ þ x2 ðtÞÞ þ 3 sin 2tf1 ðy1 ðtÞÞ þ 4 cos 2tf2 ðy2 ðtÞÞ

ð4:2Þ

> þ 19 sin 2tg 1 ðy1 ðt  cos tÞÞ þ 14 cos 2tg 2 ðy2 ðt  12 sin tÞÞ þ cos t; > > > > > > > g 2 ðx2 ðt  3 cos tÞÞ þ sin t; y_ 1 ðtÞ ¼ 2y1 ðtÞ þ 12 sin tef 2 ðx2 ðtÞÞ þ 12 cos t e > > > > > > :_ g 2 ðx2 ðt  cos tÞÞ þ cos t; y2 ðtÞ ¼ 2y2 ðtÞ þ 14 sin tef 2 ðx2 ðtÞÞ þ 14 cos t e where

ai ðtÞ ¼ bj ðtÞ ¼ hi ðtÞ ¼ he j ðtÞ ¼ F j ¼ Gj ¼ eF i ¼ Ge i ¼ 1; i; j ¼ 1; 2; s ¼ 3; r ¼ 4; A¼

ðaij ðtÞÞ22

C¼

ðcji ðtÞÞ22

¼

¼

1 4

1 13

0

1 2

0

1 4

1 4



;

B¼

ðbij ðtÞÞ22

! ;

D¼

ðdji ðtÞÞ22

¼

¼

1 4

1

1 9

1 4

0

1 2

0

1 4

! ;

!

:

1.5

1

1

2

Soulution of x

Soulution of x1

0.5

0

−0.5

−1

−1.5

0

−0.5

−1

0

10

20

30

time t

40

50

−1.5

60

0.8

0.8

0.6

0.6

2

Soulution of y

0.2 0 −0.2

10

20

30

40

time t

50

60

0.2 0 −0.2 −0.4

−0.4 −0.6

0

0.4

0.4

Soulution of y1

0.5

−0.6

0

10

20

30

time t

40

50

60

−0.8

0

10

20

30

40

time t

50

60

70

80

Fig. 1. Dynamics behavior of system (4.1) and (4.2) with the initial conditions ðux1 ðsÞ; ux2 ðsÞ; uy1 ðsÞ; uy2 ðsÞÞ ¼ ð0:1; 1:4; 0:3; 0:8Þ; ðwx1 ðsÞ; wx2 ðsÞ; wy1 ðsÞ; wy2 ðsÞÞ ¼ ð0:1; 1:4; 0:3; 0:8Þ.

J. Xu et al. / Applied Mathematics and Computation 215 (2009) 928–935

Then, we get

0

1  a1 ðtÞ

0

B B 0 1  a2 ðtÞ B P¼B      ðtÞj G e e2 e B jc11 ðtÞj F 1 þ jd11 ðtÞj G 1 jc12 ðtÞj e F 2 þ jd 12 @  ðtÞj G e 1 jc ðtÞj e e2 F 1 þ jd21 ðtÞj G F 2 þ jd jc21 ðtÞj e 22 22

935

1  ðtÞjG ja  ðtÞjG 1 0 11 ðtÞjF 1 þ jb 12 ðtÞjF 2 þ jb ja 1 2 11 12 0 0 12 2 C  ðtÞjG1 ja  ðtÞjG2 C B 4 1C 21 ðtÞjF 1 þ jb 22 ðtÞjF 2 þ jb ja 21 22 C B0 0 9 2 C C¼B C:  C @0 1 0 0A 1  b1 ðtÞ 0 A 0 12 0 0 0 1  b2 ðtÞ

Hence, we have qðPÞ ¼ 56 < 1. Therefore, it follows from the theory of the Mmatrix in [22] that there exist constants c~1 > 0; c~2 > 0 and ni > 0; gj > 0, such that for all t > 0, the following inequalities hold

 ai ðtÞni þ  bj ðtÞgj þ

2 X

jaij ðtÞjF j gj þ

2 X

j¼1

j¼1

2 X

2 X

jcji ðtÞj e F i ni þ

i¼1



jbij ðtÞjGj gj 6 ai ðtÞni þ  e i ni jdji ðtÞj G

i¼1

6 bj ðtÞgj þ

2 X

jaij ðtÞjF j gj þ

2 X

j¼1

j¼1

2 X

2 X

jcji ðtÞj e F i ni þ

i¼1

jbij ðtÞjGj gj <  e c 1 < 0; e i ni <  e jdji ðtÞj G c 2 < 0;

i¼1

i; j ¼ 1; 2. Then, we can choose constants c1 > 0; c2 > 0; 0 < k < 1 such that for all i; j ¼ 1; 2; 8t > 0,

ðai ðtÞ  kÞni þ

2 X

jaij ðtÞjF j gj þ

j¼1

ðbj ðtÞ  kÞgj þ

2 X i¼1

2 X



jbij ðtÞjeks Gj gj < c1 < 0;

j¼1

jcji ðtÞj e F i ni þ

2 X

e i ni < c < 0; jdji ðtÞjekr G 2 

i¼1

which implies that system (4.1) and (4.2) satisfies ðH0 Þ; ðH1 Þ and ðH2 Þ. Hence, from Lemma 1 and Theorem 1, system (4.2) has exactly one 2p-periodic solution. Moreover, all solutions of system (4.1) converge to the periodic solution of system (4.2). Fig. 1 is the numeric simulation of the solution of system (4.1) and (4.2) with initial condition ðux1 ðsÞ; ux2 ðsÞ; uy1 ðsÞ; uy2 ðsÞÞ ¼ ð0:1; 1:4; 0:3; 0:8Þ; ðwx1 ðsÞ; wx2 ðsÞ; wy1 ðsÞ; wy2 ðsÞÞ ¼ ð0:1; 1:4; 0:3; 0:8Þ. Acknowledgements This work was supported by the Foundation of Fujian Education Bureau (JB08030), the National Natural Science Foundation of China (10501007) and the Program for New Century Excellent Talents in Fujian Province University (NCETFJ). References [1] L.O. Chua, T. Roska, Cellular neural networks with nonlinear and delay-type template elements, in: Proceedings of the 1990 IEEE International Woorkshop on Cellular Neural Networks and Their Applications, 1990, pp. 12–25. [2] J. Cao, New results concerning exponential stability and periodic solutions of delayed cellular neural networks with delays, Phys. Lett. A 307 (2003) 136–147. [3] B. Liu, L. huang, Existence and exponential stability of periodic solutions for cellular neural networks with time-varying delays, Phys. Lett. A 349 (2006) 474–483. [4] W. Lu, T. Chen, On periodic dynamical system, Chinese Ann. Math. B (25) (2004) 455–462. [5] K. Yuan, J. Cao, J. Deng, Exponential stability and periodic solution of fuzzy cellular neural networks with time-varying delays, Neurocomputing 69 (2006) 1619–1627. [6] J. Zhou, Q. Li, F. Zhang, Convergence behavior of delayed cellular neural networks without periodic coefficients, Appl. Math. Lett. 21 (2008) 1012–1017. [7] F.Y. Wei, K. Wang, Global stability and asymptotically periodic solution for nonautonomous cooperative Lotka–Volterra diffusion system, Appl. Math. Comput. 182 (2006) 161–165. [8] F.Y. Wei, K. Wang, Asymptotically periodic solution of N-species cooperation system with time delay, Nonlinear Anal. 7 (2006) 591–596. [9] A. Chen, J. Cao, Periodic bi-directional Cohen–Grossberg neural networks with distributed delays, Nonlinear Anal. 66 (2007) 2947–2961. [10] R. McEliece, E. Posner, E. Rodemich, S. Venkatesh, The capacity of the Hopfield associative memory, IEEE Trans. Inform. Theory 33 (1987) 461–482. [11] Y. Kamp, M. Hasler, Recursive Neural Networks for Associative Memory, Wiley, New York, 1990. [12] P.K. Simpson, High ordered and intra-connected bidirectional associative memories, IEEE Trans. Syst. Man Cybern. 20 (1990) 637–653. [13] Ju H. Park, C.H. Park, O.M. Kwon, S.M. Lee, A new stability criterion for bidirectional associative memory neural networks of neutral-type, Appl. Math. Comput. 199 (2) (2008) 716–722. [14] Ju H. Park, O.M. Kwon, On improved delay-dependent criterion for global stability of bidirectional associative memory neural networks with timevarying delays, Appl. Math. Comput. 199 (2) (2008) 435–446. [15] T.J. Zhou, Y.R. Liu, Y.H. Liu, Existence and global exponential stability of periodic solution for discrete-time BAM neural networks, Appl. Math. Comput. 182 (2006) 1341–1354. [16] M.H. Jiang, Y. Shen, X.X. Liao, Global stability of periodic solution for bidirectional associative memory neural networks with varying-time delay, Appl. Math. Comput. 182 (2006) 509–520. [17] Ju H. Park, Robust stability of bidirectional associative memory neural networks with time delays, Phys. Lett. A 349 (6) (2006) 494–499. [18] F.J. Yang, C.L. Zhang, D.Q. Wu, Global stability analysis of impulsive BAM type Cohen–Grossberg neural networks with delays, Appl. Math. Comput. 186 (2007) 932–940. [19] Y.K. Li, Existence and stability of periodic solution for BAM neural networks with distributed delays, Appl. Math. Comput. 159 (2004) 847–862. [20] Y.H. Xia, M. Lin, J. Cao, Existence and exponential stability of almost periodic solution for BAM neural networks with impulse, Dyn. Contin. Discrete Impuls. Syst. A (13) (2006) 248–255. [21] J.Y. Xu, Global exponential p-stability in Cohen–Grossberg-type bidirrectional associative memory neural networks with transmission delays and learning behavior, Appl. Math. Comput. in press.,doi:10.1007/s1290-009-0268-z [22] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Science, Academic Press, New York, 1979.