Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales

Commun Nonlinear Sci Numer Simulat 16 (2011) 3326–3336

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Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales q Yongkun Li ⇑, Jiangye Shu Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 17 May 2010 Received in revised form 2 October 2010 Accepted 4 November 2010 Available online 13 November 2010 Keywords: Anti-periodic solution Global exponential stability Shunting inhibitory cellular neural networks Impulse Time scales

a b s t r a c t By using the method of coincidence degree and constructing suitable Lyapunov functional, several sufficient conditions are established for the existence and global exponential stability of anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scale T. Our results are new even if the time scale T ¼ R or Z. An example is given to illustrate our feasible results. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Since Bouzerdout and Pinter in [1] described SICNNs as a new cellular neural networks (CNNs), SICNNs have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Hence, they have been the object of intensive analysis by numerous authors in recent years. Many important results on the dynamical behaviors of SICNNs have been established and successfully applied to signal processing, pattern recognition, associative memories, and so on. We refer the reader to [2–9] and the references cited therein. In contrast, however, very few results are available on the existence and exponential stability of anti-periodic solutions for neural networks, while the existence of anti-periodic solutions plays a key role in characterizing the behavior of nonlinear differential equations (see [10–14]). Since SICNNs can be analog voltage transmission, and voltage transmission process often a anti-periodic process. Thus, it is worth while to continue to investigate the existence and stability of anti-periodic solutions of SICNNs. In reality, many physical systems undergo abrupt changes at certain moments due to instantaneous perturbations, which lead to impulsive effects. Since the existence of impulses is frequently a source of instability, bifurcation and chaos for neural networks, the impulsive neural networks is an appropriate description of the phenomena of abrupt qualitative dynamical changes of essentially continuous-time systems, see [15–19] and references therein. As well known, both continuous and discrete systems are very important in implementing and applications. Recently, several types of neural networks on time scales have been presented and studied, see, for e.g. [20–23], which can unify the continuous and discrete situations.

q

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183.

⇑ Corresponding author.

E-mail address: [email protected] (Y. Li). 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.11.004

Y. Li, J. Shu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3326–3336

3327

Motivated by all above mentioned, in this paper, we will apply the method of coincidence degree to investigate the existence of anti-periodic solutions to the following impulsive shunting inhibitory cellular neural networks on time scales

8 R þ1 P > xDij ðtÞ ¼ aij ðtÞxij ðtÞ  C kl K ij ðuÞxij ðtÞfij ðxkl ðt  uÞÞDu þ Lij ðtÞ; > ij ðtÞ 0 > < C 2N ði;jÞ r

kl

> > > :

t 2 Tþ ;

Dxij ðt h Þ ¼

t – th ;

xij ðt þh Þ



ð1:1Þ

h 2 N;

xij ðt h Þ

¼ Iijk ðxij ðth ÞÞ;

t ¼ th ;

i ¼ 1; . . . ; m;

j ¼ 1; . . . ; n;

where T is an x2 -periodic time scale which has the subspace topology inherited from the standard topology on R and Tþ ¼ ft 2 T : t P 0g; C ij denotes the cell at the (i, j) position of the lattice, the r-neighborhood Nr(i, j) of Cij is given by

Nr ði; jÞ ¼ fC ij : maxðjk  ij; jl  jjÞ 6 r;

1 6 k 6 m;

1 6 l 6 ng;

xij acts as the activity of the cell Cij, Lij(t) is the external input to Cij, aij(t) > 0 represents the passive decay rate of the cell activity, C kl ij ðtÞ > 0 is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell Cij, and the  activity function fij(  ) is a continuous function representing the output or firing rate of the cell C kl ; xij ðt þ h Þ; xij ðt h Þ represent the right and left limit of xij(th) in the sense of time scales, {th} is a sequence of real numbers such that 0 < t1 < t2 <  < th ? 1 as h ? 1, there exists a positive integer p such that thþp ¼ th þ x2 ; IijðhþpÞ ðxij ðthþp ÞÞ ¼ Iijh ðxij ðth ÞÞ; h 2 N. Without loss of generality, we also assume that ½0; x2 ÞT \ fth : h 2 Ng ¼ ft1 ; t2 ; . . . ; tq g. The main purpose of this paper is to study the existence and global exponential stability of the anti-periodic solutions of system (1.1) by using the method of coincidence degree and Lyapunov method. To the best of the author’s knowledge, this is the first paper to discuss the anti-periodic solutions of shunting inhibitory cellular neural networks with impulse on time scales. P Pn Let xðtÞ ¼ ðx11 ðtÞ; x12 ðtÞ; . . . ; x1n ðtÞ; . . . ; xm1 ðtÞ; xm2 ðtÞ; . . . ; xmn ðtÞÞT 2 CðT; Rmn Þ, we define the norm kxk ¼ m i¼1 j¼1 maxt2 ½0; xT jxij ðtÞj. The initial conditions associated with system (1.1) are of the form

xij ðtÞ ¼ uij ðtÞ;

t 2 ½1; 0T ;

ð1:2Þ

where uij(t), i = 1, 2, . . . , m, j = 1, 2, . . . , n are continuous functions on ½1; 0T . For the sake of convenience, we introduce some notations

aij ¼ min jaij ðtÞj; t2½0;xT

ij ¼ max jaij ðtÞj; a t2½0;xT

( Eij ¼

"

ij xÞ  ðaij x þ 1Þ aij xð1  a

Lij ¼ max jLij ðtÞj;

X

t2½0;xT

Z

C kl ij M fij

Dij ¼ ð1 þ aij xÞ

# jIijh ð0Þj þ xLij ;

h¼1

kgk2 ¼

t2½0;xT

þ1

xjK ij ðuÞjDu þ

0

C kl 2Nr ði;jÞ

" 2q X

kl C kl ij ¼ max jC ij ðtÞj;

Z

x

2q X

#)

qijh

;

h¼1

jgðtÞj2 Dt

1=2 ;

0

where i = 1, 2, . . . , m, j = 1, 2, . . . , n, g is an x-periodic function. Throughout this paper, we assume that kl kl x x (H1) aij 2 CðT; ð0; þ1ÞÞ; aij ðt þ x2 Þ ¼ aij ðtÞ; C kl ij ; Lij 2 CðT; RÞ; C ij ðt þ 2 Þ ¼ C ij ðtÞ; Lij ðt þ 2 Þ ¼ Lij ðtÞ; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n; R þ1 (H2) 0 jK ij ðuÞjDu < þ1; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n; (H3) fij 2 CðR; RÞ; fij ðuÞ ¼ fij ðuÞ, and there exist positive constants M fij and Lfij such that jfij ðuÞj 6 M fij ; jfij ðuÞ  fij ðv Þj 6 Lfij ju  v j; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n; (H4) Iijh 2 CðR; RÞ and there exist positive constants qijh such that

jIijh ðuÞ  Iijh ðv Þj 6 qijh ju  v j for all u; v 2 R; h 2 N; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n. The organization of this paper is as follows. In Section 2, we introduce some definitions and lemmas. In Section 3, by using the method of coincidence degree, we obtain the existence of the anti-periodic solutions of system (1.1). In Section 4, we give the criteria of global exponential stability of the anti-periodic solutions of system (1.1). In Section 5, an example is also provided to illustrate the effectiveness of the main results in Sections 3 and 4. The conclusions are drawn in Section 6. 2. Preliminaries In this section, we shall first recall some basic definitions, lemmas which are used in what follows.

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Definition 2.1 [24]. A time scale T is an arbitrary nonempty closed subset of the real set R with the topology and ordering inherited from R. The forward and backward jump operators r; q : T ! T and the graininess l : T ! Rþ are defined, respectively, by

rðtÞ :¼ inffs 2 T : s > tg; qðtÞ :¼ supfs 2 T : s < tg; lðtÞ :¼ rðtÞ  t: The point t 2 T is called left-dense, left-scattered, right-dense or right-scattered if q(t) = t, q(t) < t, r(t) = t or r(t) > t, respectively. Points that are right-dense and left-dense at the same time are called dense. If T has a left-scattered maximum m, defined Tk ¼ T  fmg; otherwise, set Tk ¼ T. Definition 2.2 [24]. A vector function f : T ! Rn is rd-continuous provided it is continuous at each right-dense point in T and has a left-sided limit at each left-dense point in T. The set of rd-continuous functions f : T ! Rn will be denoted by C rd ðTÞ ¼ C rd ðT; Rn Þ. Definition 2.3 [24]. For a function f : T ! R (the range R of f may be actually replaced by Banach space) the (delta) derivative is defined by

fD ¼

f ðrðtÞÞ  f ðtÞ ; rðtÞ  t

if f is continuous at t and t is right-scattered. If t is not right-scattered then the derivative is defined by

f D ¼ lim s!t

f ðrðtÞÞ  f ðsÞ f ðtÞ  f ðsÞ ; ¼ lim s!t ts rðtÞ  s

provided this limit exists. Definition 2.4 [24]. If FD(t) = f(t), then we define the delta integral by

Z

t

f ðsÞDs ¼ FðtÞ  FðaÞ:

a

Definition 2.5 [24]. If a 2 T; sup T ¼ 1, and f is rd-continuous on ½a; 1ÞT , then we define the improper integral by

Z

Z

1

f ðsÞDs :¼ lim

b!1

a

b

f ðsÞDs;

a

provided this limit exists, and we say that the improper integral converges in this case. If this limit does not exist, then we say that the improper integral diverges. Definition 2.6 [24]. A function p : T ! R is said to be regressive provided 1 + l(t)p(t) – 0 for all t 2 Tk , where l(t) = r (t)  t is the graininess function. The set of all regressive rd-continuous functions f : T ! R is denoted by R while the set Rþ is given by ff 2 R : 1 þ lðtÞf ðtÞ > 0g for all t 2 T. Let p 2 R. The exponential function is defined by

ep ðt; sÞ ¼ exp

Z

t

 nlðsÞ ðpðsÞÞDs ;

s

where nh(z) is the so-called cylinder transformation. Definition 2.7 [25]. For each t 2 T, let N be a neighborhood of t. Then we defined the generalized derivative (or Dini derivative), D+uD(t) to mean that, given  > 0, there exists a right neighborhood N()  N of t such that

uðrðtÞÞ  uðsÞ < Dþ uD ðtÞ þ  rðtÞ  s for each s 2 N(), s > t. In case t is right-scattered and u(t) is continuous at t, this reduces to

Dþ uD ðtÞ ¼

uðrðtÞÞ  uðtÞ : rðtÞ  t

Y. Li, J. Shu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3326–3336

3329

Similar to [26], we shall first give the definition of anti-periodic function on a time scale as following: Definition 2.8. We say that a time scale T is periodic if there exists p > 0 such that if t 2 T, then t  p 2 T. For T – R, the smallest positive p is called the period of the time scale. Let T – R be a periodic time scale with period p. We say that the x x x function f : T ! R is x 2 -anti-periodic if there exists a natural number n such that 2 ¼ np; f ðt þ 2 Þ ¼ f ðtÞ for all t 2 T and 2 is x-anti-periodic if x is the smallest positive number the smallest number such that f ðt þ x Þ ¼ f ðtÞ. If T ¼ R, we say that f is 2 2 2 such that f ðt þ x 2 Þ ¼ f ðtÞ for all t 2 T. Lemma 2.1 [24]. Let p; q 2 R. Then (a) (b) (c) (d) (e)

e0(t, s)  1 and ep(t, t)  1; ep(r(t), s) = (1 + l(t)p(t))ep(t, s); 1 ¼ ep ðt; sÞ, where pðtÞ ¼  1þlpðtÞ ; ep ðt;sÞ ðtÞpðtÞ ep(t, s) ep(s, r) = ep(t, r); eDp ð; sÞ ¼ pep ð; sÞ.

Lemma 2.2 [24]. Assume that f, g : T ! R are delta differentiable at t 2 Tk . Then

ðfgÞD ðtÞ ¼ f D ðtÞgðtÞ þ f ðrðtÞÞg D ðtÞ ¼ f ðtÞg D ðtÞ þ f D ðtÞgðrðtÞÞ: The following lemmas can be found in [27,28], respectively. Lemma 2.3. Let t 1 ; t2 2 ½0; xT . If x : T ! R is x-periodic, then

xðtÞ 6 xðt1 Þ þ

Z

x

jxD ðsÞjDs and xðtÞ P xðt 2 Þ 

Z

0

x

jxD ðsÞjDs:

0

Lemma 2.4 (Cauchy Schwarz inequality on time scale). Let a; b 2 T. For rd-continuous functions f ; g : ½a; bT ! R we have

Z a

b

jf ðtÞjjgðtÞjDt 6

Z

b

2

jf ðtÞj Dt

!1=2 Z

a

b

!1=2

2

jgðtÞj Dt

:

a

Definition 2.9. The anti-periodic solution x ðtÞ ¼ ðx 11 ðtÞ; . . . ; x 1n ðtÞ; . . . ; x m1 ðtÞ; . . . ; x mn ðtÞÞT of system (1.1) with initial value u ðtÞ ¼ ðu 11 ðtÞ; . . . ; u 1n ðtÞ; . . . ; u m1 ðtÞ; . . . ; u mn ðtÞÞT is said to be globally exponentially stable if there exist positive constants k and M = M(k) P 1, for any solution x(t) = (x11(t), . . . , x1n(t), . . . , xm1(t), . . . , xmn(t))T of system (1.1) with the initial value u(t) = (u11 (t), . . . , u1n(t), . . . , um1(t), . . . , umn(t)T, such that

jxij ðtÞ  x ij ðtÞj 6 MðkÞek ðt; aÞku  u k1 ;

8 t > 0;

i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n;

where

ku  u k1 ¼ ku  u k1 ¼ sup max juij ðsÞ  u ij ðsÞj; 1
ði;jÞ

a 2 ½1; 0T :

The following fixed point theorem of coincidence degree is crucial in the arguments of our main results. Lemma 2.5 [29]. Let X; Y be two Banach spaces, X  X be open bounded and symmetric with 0 2 X. Suppose that  – ; and N : X  ! Y is L-compact. Further, we also L : DðLÞ  X ! Y is a linear Fredholm operator of index zero with DðLÞ \ X assume that (H) L x  Nx – k(Lx  N(x)) for all x 2 D(L) \ @ X, k 2 (0, 1]. Then equation L x = N x has at least one solution on DðLÞ \ X. 3. Existence of anti-periodic solutions

Theorem 3.1. Assume that (H1)–(H4) hold. Suppose further that Eij > 0, i = 1, 2, . . . , m, j = 1, 2, . . . , n. Then system (1.1) has at least one x 2 -anti-periodic solution. Proof. Let C k ð½0; x; t1 ; t2 ; . . . ; t q ; t qþ1 ; . . . ; t 2q T ; Rmn Þ ¼ fx : ½0; xT ! Rmn jxðkÞ ðtÞ is a piecewise continuous map with first-class discontinuous points in ½0; xT \ ft h : h 2 Ng and at each discontinuous point it is continuous on the left}, k = 0, 1. Take

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Y. Li, J. Shu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3326–3336

n x x o X ¼ x 2 Cð½0; x; t 1 ; t2 ; . . . ; t q ; tqþ1 ; . . . ; t 2q T ; Rmn Þ : xðt þ Þ ¼ xðtÞ for all t 2 ½0; T 2 2 and

Y ¼ X RðmnÞ q be two Banach spaces equipped with the norms m X n X

kxkX ¼

i¼1

and kykY ¼ kxkX þ kzk for all x 2 X;

jxij j0

z 2 RðmnÞ q ;

j¼1

in which jxij j0 ¼ maxt2½0;xT jxij ðtÞj; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n; k  k is any norm of RðmnÞ q . Set

x ! ðxD ; Dxðt1 Þ; Dxðt 2 Þ; . . . ; Dxðt q ÞÞ;

L : Dom L \ X ! Y; where

n x x o Dom L ¼ x 2 C 1 ½0; x; t1 ; t 2 ; . . . ; t2q T : xðt þ Þ ¼ xðtÞ for all t 2 ½0; T 2 2 and N : X ! Y

11 1 0 1 0 1 0 I11q ðx11 ðt q ÞÞ I111 ðx11 ðt1 ÞÞ I112 ðx11 ðt 2 ÞÞ A11 ðtÞ BB . CC C B. C B. C B. BB .. CC C B .. C B .. C B .. BB CC C B C B C B BB CC C B C B C B BB A1n ðtÞ C B I1n1 ðx1n ðt1 ÞÞ C B I1n2 ðx1n ðt 2 ÞÞ C B I1nq ðx1n ðt q ÞÞ CC BB CC C B C B C B BB . CC C B. C B. C B CC; C; B . C; B . C; . . . ; B .. B. Nx ¼ B BB . CC C B. C B. C B. BB CC C B C B C B BB Am1 ðtÞ C B Im11 ðxm1 ðt 1 ÞÞ C B Im12 ðxm1 ðt 2 ÞÞ C B Im1q ðxm1 ðt q ÞÞ CC BB CC C B C B C B BB . CC C B. C B. C B. BB . CC C B. C B. C B. @@ . AA A @. A @. A @. 00

Amn ðtÞ

Imn1 ðxmn ðt 1 ÞÞ

Imn2 ðxmn ðt 2 ÞÞ

Imnq ðxmn ðt q ÞÞ

where

X

Aij ðtÞ ¼ aij ðtÞxij ðtÞ 

C kl ij ðtÞ

Z

C kl 2N r ði;jÞ

þ1

K ij ðuÞxij ðtÞfij ðxkl ðt  uÞÞDu þ Lij ðtÞ

0

for i = 1, 2, . . . , m, j = 1, 2, . . . , n. It is easy to see that

 Z z ¼ ðg; c1 ; . . . ; cq Þ 2 Y :

Ker L ¼ f0g and Im L ¼

x

 gðsÞDs ¼ 0  Y:

0

Thus dim Ker L = 0 = codim Im L, and L is a linear Fredholm operator of index zero. Define the continuous projector P : X ! Ker L and the averaging projector Q : Y ! Y by

Px ¼

Z

x

xðsÞDs ¼ 0

0

and

 Qz ¼ Q ðg; c1 ; . . . ; cq Þ ¼

1

Z

x

 gðsÞDs; 0; . . . ; 0 :

x

0

Hence Im P= Ker L and Ker Q= Im L=Im (I  Q). Denoting by L1 P : Im L ! DomðLÞ \ KerP the inverse of LjD(L)\KerP, we have

L1 P z ¼

Z

t

gðsÞDs þ

0

X

ck 

t k
1 2

Z

x 2

gðsÞDs 

0

q 1X ck ; 2 k¼1

in which cq+i = ci for all 1 6 i 6 q. Similar to [21], it is not difficult to show that QNðXÞ; L1 P ðI  Q ÞNðXÞ are relatively compact for any open bounded set X  X. Therefore, N is L-compact on X for any open bounded set X  X. In order to apply Lemma 2.5, we need to find an appropriate open bounded subset X in X. Corresponding to the operator equation Lx  Nx = k(Lx  N(x)), k 2 (0, 1], we have

(

1 k xDij ðtÞ ¼ 1þk Gij ðt; xÞ  1þk Gij ðt; xÞ;

Dxij ðth Þ ¼

1 I ðx ðt ÞÞ 1þk ijh i h



t 2 Tþ ;

k I ðxij ðt h ÞÞ; 1þk ijh

t – th ;

h 2 N;

i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n;

ð3:1Þ

3331

Y. Li, J. Shu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3326–3336

where

X

Gij ðt; xÞ ¼ aij ðtÞxij ðtÞ 

C kl ij ðtÞ

Z

þ1

K ij ðuÞxij ðtÞfij ðxkl ðt  uÞÞDu þ Lij ðtÞ

0

C kl 2N r ði;jÞ

and

X

Gij ðt; xÞ ¼ aij ðtÞðxij ðtÞÞ 

C kl ij ðtÞ

Z

þ1

K ij ðuÞðxij ðtÞÞfij ðxkl ðt  uÞÞDu þ Lij ðtÞ

0

C kl 2N r ði;jÞ

for i = 1, 2, . . . , m, j = 1, 2, . . . , m. Set t0 ¼ t þ 0 ¼ 0; t 2qþ1 ¼ x, in view of (3.1), we get

Z

x

jxDij ðtÞjDt ¼

0

2qþ1 X

Z

th

jxDij ðtÞjDt þ

tþ

h¼1

jDxij ðth Þj

h¼1

h1

x 

Z 6

  2q  X   1  k  1 Gij ðt; xÞ  k Gij ðt; xÞDt þ   I I ðx ðt ÞÞ  ðx ðt ÞÞ i ij ijh h ijh h 1 þ k  1 þ k  1þk 1þk

0

h¼1

 6

2q X

1 k þ 1þk 1þk

þ

2q X

k 1þk

h¼1

X

þ

Z

x

2q X

max jGij ðt; xÞj; jGij ðt; xÞj Dt þ

0

h¼1

2q   X Iijh ðxij ðt h ÞÞ  Iijh ð0Þ þ ij jIijh ð0Þj 6 a

Z

þ1

jK ij ðuÞjDu

0

pffiffiffiffiffi ij xkxij k2 þ 6a

X

Z

C kl ij M fij

x

jxij ðtÞjDt þ xLij þ

2q X

qijh jxij j0 þ

h¼1

2q X

jIijh ð0Þj

h¼1

h¼1

i ¼ 1; 2; . . . ; m;

jIijh ð0Þj;

jxij ðtÞjDt

2q X pffiffiffiffiffi jK ij ðuÞjDu xkxij k2 þ xLij þ qijh jxij j0

þ1

0

C kl 2N r ði;jÞ 2q X

Z

x

0

0

C kl 2Nr ði;jÞ

þ

Z

h¼1

C kl ij M fij

 1  Iijh ðxij ðth ÞÞ  Iijh ð0Þ 1þk

j ¼ 1; 2; . . . ; m:

ð3:2Þ

h¼1

Integrating (3.1) from 0 to x, we have   Z x Z x X Z þ1    kl  ¼  1  a ðtÞx ðtÞ D t C ðtÞ K ij ðuÞxij ðtÞfij ðxkl Þðt  uÞÞDuDt ij ij ij   1þk  0 0 0 C kl 2Nr ði;jÞ Z x X Z þ1 k þ C kl K ij ðuÞðxij ðtÞÞfij ðxkl ðt  uÞÞDuDt ij ðtÞ 1 þ k 0 C 2N ði;jÞ 0 r

kl

6

1 þ 1þk X

Z

x

k 1þk

Lij ðtÞDt  0

C kl ij M fij

Z

Z

x

Lij ðtÞDt þ 0

Z

þ1

jK ij ðuÞjDu

x

jxij ðtÞjDt þ xLij

0

0

C kl 2Nr ði;jÞ

 2q 2q  1 X k X  Iijh ðxi ðth ÞÞ  Iijh ðxi ðth ÞÞ  1 þ k h¼1 1 þ k h¼1

(

)  2q 2q X X 1 k þ þ jIijh ðxij ðth ÞÞj; jIijh ðxij ðth ÞÞj max 1þk 1þk h¼1 h¼1 ( ) Z þ1 2q 2q X X X pffiffiffiffiffi 6 kx C kl M jK ðuÞj D u x k þ x L þ max jI ðx ðt ÞÞ  I ð0Þj; jI ðx ðt ÞÞ  I ð0Þj ij ij 2 ij ij h fij ijh ij h ijh ijh ijh ij 

0

C kl 2N r ði;jÞ 2q X

þ

h¼1

h¼1

X

Iijh ð0Þ 6

C kl ij M fij

C kl 2Nr ði;jÞ

i ¼ 1; 2; . . . ; m;

Z

x

aij ðtÞxij ðtÞDt 6

0

Z

jK ij ðuÞjDu xkxij k2 þ xLij þ

0

x

aij ðtÞxij ðfij ÞDt þ

Z

0

x

aij ðtÞ

h¼1 2q X h¼1

Z

0

qijh jxij j0 þ

2q X

jIijh ð0Þj;

h¼1

ð3:3Þ

j ¼ 1; 2; . . . ; m:

From Lemma 2.3, for any fij, gij 2 ½0; xT , we have

Z

pffiffiffiffiffi

þ1

x

0

 jxDij ðtÞjDt Dt

ð3:4Þ

and

Z 0

x

aij ðtÞxij ðtÞDt P

Z 0

x

aij ðtÞxij ðgij ÞDt 

Z

x

aij ðtÞ 0

Z 0

x

 jxDij ðtÞjDt Dt;

ð3:5Þ

3332

Y. Li, J. Shu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3326–3336

Rx

where i = 1, 2, . . . , m, j = 1, 2, . . . , n. Dividing by

1 xij ðfij Þ P R x a ij ðtÞDt 0

Z

x

aij ðtÞxij ðtÞDt 

0

aij ðtÞDt on the two sides of (3.4) and (3.5), respectively, we obtain that

0

Z

x

jxDij ðtÞjDt

ð3:6Þ

jxDij ðtÞjDt;

ð3:7Þ

0

and

xij ðgij Þ 6 R x 0

1 aij ðtÞDt

Z

x

aij ðtÞxij ðtÞDt þ

Z

0

x

0

where i = 1, 2, . . . , m, j = 1, 2, . . . , n. Let tij , tij 2 ½0; xT such that xij ðtij Þ ¼ maxt2½0;xT xij ðtÞ; xij ðt ij Þ ¼ mint2½0;xT xij ðtÞ, by the arbitrariness of fij, gij, we obtain from (3.2)–(3.7) that Z x  Z x Z x Z x   1 1  xij ðt ij Þ P R x aij ðtÞxij ðtÞDt  jxDij ðtÞjDt P  R x aij ðtÞxij ðtÞDt   jxDij ðtÞjDt  a ðtÞ D t a ðtÞ D t 0 0 0 0 ij ij 0 0 " # Z þ1 2q 2q X X X pffiffiffiffiffi 1 C kl jK ij ðuÞjDu xkxij k2 þ xLij þ qijh jxij j0 þ jIijh ð0Þj P ij M fij aij x C 2N ði;jÞ 0 h¼1 h¼1 r kl " # Z þ1 2q 2q X X X ffiffiffiffi ffi pffiffiffiffiffi p kl   aij xkxij k2 þ C ij M fij jK ij ðuÞjDu xkxij k2 þ xLij þ qijh jxij j0 þ jIijh ð0Þj ; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n 0

C kl 2Nr ði;jÞ

h¼1

h¼1

ð3:8Þ

and

1 aij ðtÞDt 0 " X 1

xij ðtij Þ 6 R x 6

aij x "

Z x  Z x   1  þ a ðtÞx ðtÞ D t jxDij ðtÞjDt ij ij   aij ðtÞDt 0 0 0 0 0 # Z þ1 2q 2q X X pffiffiffiffiffi kl C ij M fij jK ij ðuÞjDu xkxij k2 þ xLij þ qijh jxij j0 þ jIijh ð0Þj

Z

x

aij ðtÞxij ðtÞDt þ

Z

x

jxDij ðtÞjDt 6 R x

0

C kl 2N r ði;jÞ

h¼1

X

pffiffiffiffiffi þ aij xkxij k2 þ

Z

C kl ij M fij

0

C kl 2Nr ði;jÞ

i ¼ 1; 2; . . . ; m;

þ1

h¼1

pffiffiffiffiffi jK ij ðuÞjDu xkxij k2 þ xLij þ

2q X

qijh jxij j0 þ

h¼1

2q X

# jIijh ð0Þj ;

h¼1

j ¼ 1; 2; . . . ; n:

ð3:9Þ

Thus, we have from (3.8) and (3.9) that

jxij j0 ¼ max jxij ðtÞj 6 t2½0;xT

"

1

"

aij x

X

C kl 2Nr ði;jÞ

X

pffiffiffiffiffi

ij xkxij k2 þ þ a

C kl ij M fij

C kl ij M fij

C kl 2Nr ði;jÞ

i ¼ 1; 2; . . . ; m;

Z

Z

þ1

0

þ1

0

# 2q 2q X X pffiffiffiffiffi jK ij ðuÞjDu xkxij k2 þ xLij þ qijh jxij j0 þ jIijh ð0Þj h¼1

h¼1

# 2q 2q X X pffiffiffiffiffi jK ij ðuÞjDu xkxij k2 þ xLij þ qijh jxij j0 þ jIijh ð0Þj ; h¼1

h¼1

j ¼ 1; 2; . . . ; n:

ð3:10Þ

In addition, we have that

kxij k2 ¼

Z

x

jxij ðsÞj2 Ds

1=2 6

pffiffiffiffiffi

pffiffiffiffiffi

x max jxij ðtÞj ¼ xjxij j0 ; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n: t2½0;xT

0

By (3.10), we have

" aij xjxij j0 6

X

C kl ij M fij

C kl 2Nr ði;jÞ

"

Z

þ1

0

# 2q 2q X X pffiffiffiffiffi jK ij ðuÞjDu xkxij k2 þ xLij þ qijh jxij j0 þ jIijh ð0Þj h¼1

pffiffiffiffiffi

ij xkxij k2 þ þ aij x a " 6

X

C kl ij M fij

C kl 2N r ði;jÞ

X

C kl 2Nr ði;jÞ

C kl ij M fij

Z

Z 0

þ1

2q 2q X X pffiffiffiffiffi jK ij ðuÞjDu xkxij k2 þ xLij þ qijh jxij j0 þ jIijh ð0Þj h¼1

þ1

0

"

ij xjxij j0 þ þ aij x a

jK ij ðuÞjDuxjxij j0 þ xLij þ

2q X h¼1

X C kl 2Nr ði;jÞ

where i = 1, 2, . . . , m, j = 1, 2, . . . , n. That is,

C kl ij M fij

Z 0

h¼1

qijh jxij j0 þ

2q X

h¼1

#

jIijh ð0Þj

h¼1

þ1

jK ij ðuÞjDuxjxij j0 þ xLij þ

2q X h¼1

qijh jxij j0 þ

2q X h¼1

# jIijh ð0Þj ;

#

Y. Li, J. Shu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3326–3336

(

"

ij xÞ  ðaij x þ 1Þ aij xð1  a

6 ð1 þ aij xÞ

" 2q X

X

C kl ij M fij

C kl 2N r ði;jÞ

Z

þ1

xjK ij ðuÞjDu þ

0

#)

qijh

jxij j0 ¼ Eij jxij j0

h¼1

#

jIijh ð0Þj þ xLij ¼ Dij ;

2q X

3333

i ¼ 1; 2; . . . ; m;

n ¼ 1; 2; . . . ; n:

h¼1

For i = 1, 2, . . . , m, j = 1, 2, . . . , n, denote

(

"

X

ij xÞ þ ðaij x  1Þ aij xð1  a

Eij ¼

C kl ij M fij

C kl 2Nr ði;jÞ

Z

þ1

jK ij ðuÞjDux þ

0

2q X

#)

qijh

:

h¼1

From the assumption of Theorem 3.1 we get

Dij :¼ M ij ; Eij

jxij j0 6

i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n:

Let

M¼

m X n X i¼1

M ij þ 1:

j¼1

Clearly, M is independent of k. Then take

X ¼ fx 2 X : kxkX < Mg: It is clear that X satisfies all the requirement in Lemma 2.5 and the condition (H) is satisfied. In view of all the discussions above, we conclude from Lemma 2.5 that system (1.1) has at least one x2 -anti-periodic solution. This completes the proof. h 4. Global exponential stability of the anti-periodic solution In this section, we will construct some suitable Lyapunov functions to study the global exponential stability of the antiperiodic solution of system (1.1). Theorem 4.1. Assume that (H1)–(H6) hold. Suppose further that (H7) The impulsive operators Iik(xi(t)) satisfy

Iik ðxi ðt k ÞÞ ¼ cik xi ðt k Þ; (H8) There exist constants

(

  aij þ

bij ¼ where M ij ¼

i ¼ 1; 2; . . . ; n;

k 2 N:

 > 0 and g > 0 such that,  Z C kl M fij ij

X C kl 2N r ði;jÞ

Dij Eij

0 6 cik 6 2;

0

þ1

jK ij ðuÞjDu þ Lfij

Z

þ1

jK ij ðuÞje ðu; aÞDuM ij

0

) x 1 þ  < g < 0; 2

; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n:

Then the x2 -anti-periodic solution of system (1.1) is globally exponentially stable. Proof. According to Theorem 3.1 and its proof, we know that system (1.1) has an x2 -anti-periodic solution x ðtÞ ¼ ðx 11 ðtÞ; x 12 ðtÞ; . . . ; x 1n ðtÞ; . . . ; x m1 ðtÞ; x m2 ðtÞ; . . . ; x mn ðtÞÞT with the initial value u ðtÞ ¼ ðu 11 ðtÞ; . . . ; u 1n ðtÞ; . . . ; u m1 ðtÞ; . . . ; u mn ðtÞÞT and jxijj0 6 Mij, suppose that x(t) = (x11(t), x12 (t), . . . , x1n(t), . . . , xm1(t), xm2(t), . . . , xmn(t))T is an arbitrary solution of system (1.1) with the initial value u(t) = (u11(t), . . . , u1n(t), . . . , um1(t), . . . , umn(t))T. Set yij(t) = xij(t)  xij(t). Then it follows from system (1.1) and (H7) that

8 R þ1 P > ðyij ðtÞÞD ¼ aij ðtÞyij ðtÞ  C kl K ij ðuÞxij ðtÞfij ðxkl ðt  uÞÞDu > ij ðtÞ 0 > > C kl 2N r ði;jÞ > > > < R þ1 P C kl K ij ðuÞx ij ðtÞfij ðx kl ðt  uÞÞDu; t – t h ; þ ij ðtÞ 0 > > C kl 2N r ði;jÞ > > > > > : Dy ðt Þ ¼ c y ðt Þ; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n: ij

h

ijh ij

ð4:1Þ

h

For any a 2 ½1; 0T , we consider the Lyapunov functional

V ij ðtÞ ¼ jyij ðtÞje ðt; aÞ;

i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n:

ð4:2Þ

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Y. Li, J. Shu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3326–3336

For t 2 Tþ ; t – th ; h 2 N, calculating the upper right derivative of Vij(t) along the solution of system (4.1), we have

 Z þ1  X  Dþ jV ij ðtÞjD 6 aij ðtÞjyij ðtÞje ðrðtÞ; aÞ þ  C kl K ij ðuÞxij ðtÞfij ðxkl ðt  uÞÞDu ij ðtÞ C 2N ði;jÞ 0 r kl  Z þ1   K ij ðuÞfij ðx kl ðt  uÞÞDux ij ðtÞe ðrðtÞ; aÞ þ jyij ðtÞje ðt; aÞ 0  Z þ1  X  6 aij ðtÞjyij ðtÞje ðrðtÞ; aÞ þ  C kl ðtÞ K ij ðuÞyij ðtÞfij ðxkl ðt  uÞÞDu C 2N ði;jÞ ij 0 r kl  Z þ1    þ K ij ðuÞx ij ðtÞ fij ðxkl ðt  uÞÞ  fij ðx kl ðt  uÞÞ Due ðrðtÞ; aÞ þ jyij ðtÞje ðt; aÞ 0

for i = 1, 2, . . . , m, j = 1, 2, . . . , n. For t 2 Tþ ; t ¼ t h ; h 2 N, we have from (H7) that

jxij ðtþh Þ  x ij j ¼ j1  cih jjxij ðth Þ  x ij j 6 jxij ðt h Þ  x ij j; where i = 1, 2, . . . , m, j = 1, 2, . . . , n. Let M > 1 denote an arbitrary real number and set

ku  u k1 ¼ sup max juij ðsÞ  u ij ðsÞj > 0: 1
ði;jÞ

It follows from (4.2) that

V ij ðtÞ ¼ jyij ðtÞje ðt; aÞ < Mku  u k1 ; 8 t 2 ð1; 0;

i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n:

We claim that

V ij ðtÞ ¼ jyij ðtÞje ðt; aÞ < Mku  u k1 ;

8 t > 0;

i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n:

ð4:3Þ

Contrarily, in view of the arbitrariness of M, there exists ij 2 {11, 12, . . . , 1m, . . . , m1, m2, . . . ,mn} and 0 < tij < t1 such that

V ij ðt ij Þ ¼ Mku  u k1 ;

ð4:4Þ

e 2 f11; 12; . . . ; 1m; . . . ; m1; m2; . . . ; mng n fijg we have and for ij

V eðtÞ < Mku  u k1 ; 8 t 2 ð1; t ij Þ:

ð4:5Þ

ij

Together with (4.1), (4.5) and (4.6) we obtain

 Z þ1  X  06D 6 aij ðtij Þjyij ðt ij Þje ðrðt ij Þ; aÞ þ  C kl ðt Þ K ij ðuÞyij ðtij Þfij ðxkl ðt ij  uÞÞDu ij C 2N ði;jÞ ij 0 r kl  Z þ1    þ K ij ðuÞx ij ðtij Þ fij ðxkl ðtij  uÞÞ  fij ðx kl ðt ij  uÞÞ Due ðrðtij Þ; aÞ þ jyij ðt ij Þje ðt ij ; aÞ 0 Z þ1 X jC kl ðt Þj jK ij ðuÞjM fij Dujyij ðtij Þje ðtij ; aÞ 6 aij ðt ij Þð1 þ lðt ij ÞÞjyij ðt ij Þje ðt ij ; aÞ þ ij ij þ

V Dij ðt ij Þ

0

C kl 2Nr ði;jÞ

 jK ij ðuÞje ðu; aÞLfij jykl ðt ij  uÞje ðt ij  u; aÞDujx ij ðt ij Þj ð1 þ lðt ij ÞÞ þ jyij ðt ij Þje ðtij ; aÞ 0 Z þ1 X C kl jK ij ðuÞjM fij DuMku  u k1 6 aij Mku  u k1 þ ij þ

Z

þ1

Z

þ1

þ ( ¼

0

  aij þ (

06

 x 1 þ  þ Mku  u k1 2 )   Z þ1 Z þ1 X x kl C ij Mfij jK ij ðuÞjDu þ Lfij jK ij ðuÞje ðu; aÞDuM ij 1 þ  Mku  u k1 : 2 0 0 2N ði;jÞ

jK ij ðuÞje ðu; aÞLfij Mku  u k1 DuM ij

C kl

Thus

0

C kl 2Nr ði;jÞ

  aij þ

ð4:6Þ

r

X C kl 2N r ði;jÞ

 Z C kl M fij ij

0

þ1

jK ij ðuÞjDu þ Lfij

Z

þ1

jK ij ðuÞje ðu; aÞDuM ij

0

)  x 1þ  ; 2

which contradicts (H8). Hence, (4.4) holds. It follows that

jxij ðtÞ  x ij ðtÞj < Me ðt; aÞku  u k1 ;

8 t 2 ð1; t 1 Þ;

i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n:

When t = t1, the second expression of (4.1) implies

jyij ðt þ1 Þj ¼ jyij ðt 1 Þ  cij1 yij ðt1 Þj 6 j1  cij1 jjyij ðt1 Þj 6 lim jyij ðtÞj < Me ðt; aÞku  u k1 ; t!t 1

ð4:7Þ

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Y. Li, J. Shu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3326–3336

that is

jxij ðtþ1 Þ  x ij ðt þ1 Þj < Me ðt; aÞku  u k1 ;

ð4:8Þ

where i = 1, 2, . . . , m, j = 1, 2, . . . , n. Similar to the step of (4.3)–(4.7), we can also prove that

jxij ðtÞ  x ij ðtÞj < Me ðt; aÞku  u k1 ;

8 t 2 ½t1 ; t 2 Þ;

i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n:

When t = t2, again, from the second expression of (4.1), we have

jxij ðtþ2 Þ  x ij ðt þ2 Þj < Me ðt; aÞku  u k1 ;

i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n:

By repeating the same procedure, we obtain

jxij ðtÞ  x ij ðtÞj < Me ðt; aÞku  u k1 ; In view of Definition 2.9, the proof. h

x-anti-periodic 2

8 t > 0;

i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n:

solution x*(t) of system (1.1) is global exponentially stable. This completes the

5. An example In this section, we give an example to illustrate that our results are feasible. When T ¼ R, consider the following SICNNs with impulses

8 dx ðtÞ < dtij ¼ aij ðtÞxij ðtÞ  :

P C kl 2N 1 ði;jÞ

C kl ij ðtÞ

R þ1 0

K ij ðuÞxij ðtÞfij ðxkl ðt  uÞÞDu þ Lij ðtÞ;

Dxij ðt h Þ ¼ xij ðt þh Þ  xij ðt h Þ ¼ 0:025xij ðt h Þ;

i ¼ 1; 2;

j ¼ 1; 2;

ð5:1Þ

h ¼ 1; 2;

where

 ðaij Þ2 2 ¼

2:0 þ 0:02j sinð8ptÞj

1:9 þ 0:01j sinð8ptÞj 1:95  0:01j cosð8ptÞj 

ðC ij Þ2 2 ¼

0:6 sinð8ptÞ

0:9 sinð8ptÞ



0:8 cosð8ptÞ 0:5 cosð8ptÞ 

ðLij Þ2 2 ¼

0:6 sinð8ptÞ 0:7 cosð8ptÞ

By calculation, we have M fij ¼

 ðEij Þ2 2 ¼ Take

2:0  0:01j cosð8ptÞj

0:5 cosð8ptÞ 0:4 sinð8ptÞ 1 ; Lfij 2

¼

 0:0381 0:1651 ; 0:1687 0:1648

1 ; 2

f ðuÞ ¼

;

 ;

 ;

1 j sinðuÞj; 2 

ðK ij Þ2 2 ¼

0:2 expf20tg 0:3 expf21tg 0:3 expf27tg 0:4 expf25tg

 ;

q11h ¼ q12h ¼ q21h ¼ q22h ¼ 0:025; h ¼ 1; 2, and  ðDij Þ2 2 ¼

0:225

0:1872

0:2581 0:1485

 ;

 ðMij Þ2 2 ¼

5:9055 1:1339 1:5299 0:9011

 :

 = 0.1, and g = 1, then  ðbij Þ2 2 ¼

 1:8200 1:8469 : 1:4028 1:8196

Now we can see that (H1)–(H8) hold. By Theorems 3.1 and 4.1 the system (5.1) has a 18-anti-periodic solution which is global exponential stable.

6. Conclusions Arising from problems in applied sciences, it is well-known that the existence of anti-periodic solutions plays a key role in characterizing the behavior of nonlinear differential equations as a special periodic solution and have been extensively studied by many authors during the past ten years, see [10–14] and references therein. For example, anti-periodic trigonometric polynomials are important in the study of interpolation problems [30], and anti-periodic wavelets are discussed in [31]. Using the time scales calculus theory, the coincidence degree theory and the Lyapunov functional method, this paper unifies the discrete and continuous time shunting inhibitory neural networks under one frame work and obtained some more generalized results to ensure the existence and global exponential stability of anti-periodic solution which extend some previously known results. The conditions possess highly important significance and are easily checked in practice by simple algebraic methods. In addition, the method in this paper maybe applied to some other systems such as the Hopfield neural networks and BAM neural networks and soon.

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Y. Li, J. Shu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3326–3336

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