Anti-periodic solutions for high-order Hopfield neural networks with impulses

Neurocomputing 138 (2014) 339–346

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

Anti-periodic solutions for high-order Hopfield neural networks with impulses$ Qi Wang a,n, Yayun Fang a, Hui Li a, Lijuan Su a, Binxiang Dai b a b

School of Mathematical Sciences, Anhui University, Hefei 230601, PR China School of Mathematical Sciences and Computing Technology, Central South University, Changsha 410075, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 19 February 2013 Received in revised form 17 January 2014 Accepted 29 January 2014 Communicated by Y. Liu Available online 15 February 2014

In this paper, we consider anti-periodic solutions of high-order Hopfield neural networks (HHNNs) with time-varying delays and impulses. Sufficient conditions for the existence and exponential stability of anti-periodic solutions are established by using Krasnoselski's fixed point theorem and Lyapunov functions with inequality techniques. In the end, example and numerical simulations are given to illustrate our main results. & 2014 Elsevier B.V. All rights reserved.

Keywords: High-order Hopfield neural networks Anti-periodic solution Existence and exponential stability Delays Impulses

1. Introduction It is well known that the existence of anti-periodic solutions plays a key role in characterizing the behavior of nonlinear differential equations (see [1–10]). Due to the fact that high-order Hopfield neural networks have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lowerorder neural networks, high-order Hopfield neural networks have been the object of intensive analysis by numerous authors in recent years. In particular, there have been extensive results on the problem of the existence and stability of equilibrium points, periodic solutions, almost periodic solutions and anti-periodic solutions of high-order Hopfield neural networks (HHNNs) n

n

n

x0i ðtÞ ¼  ci ðtÞxi ðtÞ þ ∑ aij ðtÞg j ðxj ðt  τij ðtÞÞÞ þ ∑ ∑ bijl ðtÞg j ðxj ðt  sijl ðtÞÞÞ j¼1

j¼1l¼1

g l ðxl ðt  υijl ðtÞÞÞþ I i ðtÞ; i ¼ 1; 2; …; n

ð1:1Þ

in the literature [11–18,29–33,36–39] and the references therein.

Impulsive differential equations are mathematical apparatus for simulation of process and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, etc. [19–21]. Consequently, many neural networks with impulses have been studied extensively, and a great deal of literature is focused on the existence and stability of an equilibrium point [22–25]. In [26–28,40], the authors discussed the existence and global exponential stability of periodic solution of a class of neural networks with impulse. In [29], the authors discussed the existence and global exponential stability of anti-periodic solution of a class of cellular neural networks with impulse 8 n n > > x0i ðtÞ ¼  ci ðtÞxi ðtÞ þ ∑ aij ðtÞf j ðxj ðtÞÞ þ ∑ bij ðtÞg j ðxj ðt  τij ðtÞÞÞ þ ui ðtÞ; > > > j¼1 j¼1 > < t Z 0; t at k ; > > > Δxi ðt k Þ ¼ I ik ðt k ; xi ðt k ÞÞ; > > > : x ðtÞ ¼ φ ðtÞ; t A ½  τ; 0; k ¼ 1; 2; …; i ¼ 1; 2; …; n: i

☆

Research supported by the Foundation of Anhui Education Bureau (KJ2012A019, KJ2013A028), NNSF of China (11271371, 11301004), the Research Fund for the Doctoral Program of Higher Education (20103401120002, 20113401110001), 211 Project of Anhui University (02303129, 02303303-33030011, 02303902-39020011, KYXL2012004, XJYJXKC04) and Anhui Provincial NSF (1408085MA02, 1208085MA13, 1308085MA01, 1308085QA15). n Corresponding author. Tel.: þ 86 0551 63861470. E-mail addresses: [email protected] (Q. Wang), [email protected] (Y. Fang), [email protected] (H. Li), [email protected] (L. Su), [email protected] (B. Dai). http://dx.doi.org/10.1016/j.neucom.2014.01.028 0925-2312 & 2014 Elsevier B.V. All rights reserved.

i

ð1:2Þ

However, to the best of our knowledge, there are little results for the existence and stability of anti-periodic solutions of HHNNs (1.1) with impulses. Moreover, HHNNs can be analog voltage transmission, and the voltage transmission process is often an anti-periodic process. Thus, it is worthwhile to continue the investigation of the existence and stability of anti-periodic

340

Q. Wang et al. / Neurocomputing 138 (2014) 339–346

solutions of HHNNs with impulses 8 n > > > x0i ðtÞ ¼  ci ðtÞxi ðtÞ þ ∑ aij ðtÞg j ðxj ðt  τij ðtÞÞÞ > > > j¼1 > > > > n n > > < þ ∑ ∑ bijl ðtÞg j ðxj ðt  sijl ðtÞÞÞg l ðxl ðt  υijl ðtÞÞÞ þ I i ðtÞ; j ¼ 1l ¼ 1

> > > t Z 0; t a t k ; k ¼ 1; 2; …; > > > > > Δxi ðt k Þ ¼ Iik ðt k ; xi ðt k ÞÞ; > > > > : xi ðtÞ ¼ φ ðtÞ; t A ½  τ; 0; i ¼ 1; 2; …; n; i

ð1:3Þ

where n is the number of units in a neural network, xi(t) corresponds to the state vector of the i-th unit at time t; ci ðtÞ 40 represents the rate with which the i-th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, aij(t) and bijl(t) are the first- and second-order connection weights of the neural network, τij ðtÞ Z0; sijl ðtÞ Z 0 and υijl ðtÞ Z 0 correspond to the transmission delays, Ii(t) denotes the external inputs at time t, and gj is the activation function of signal transmission. ci ; aij ; bijl ; g j ; τij ; sijl ; υijl are continuous functions on R; I ik : R2 -R are continuous. τ ¼ maxt A ½0;ω fτij ðtÞ; sijl ðtÞ; υijl ðtÞg is a positive constant. Δxi ðt k Þ ¼ xi ðt kþ Þ  xi ðt k Þ; xi ðt kþ Þ ¼ limh-0 þ xi ðt k þ hÞ; xi ðt k Þ ¼ lim h-0  xi ðt k þhÞ; i ¼ 1; 2; …; k ¼ 1; 2; …; n; t k Z 0 are impulsive moments satisfying t k o t k þ 1 and limk- þ 1 t k ¼ þ1; I ik characterizes the impulsive function at time tk for i-th unit. The outline of the paper is as follows. In Section 2, some preliminaries and basic results are established. In Section 3, we give sufficient conditions for the existence and exponential stability of anti-periodic solutions for system (1.3). In Section 4, we shall give an example to illustrate our results.

Let x ¼ ðx1 ; x2 ; …; xn ÞT A Rn , where T denotes the transposition. The initial conditions associated with system (1.3) are given by the function xðtÞ ¼ φðtÞ; t A ½  τ; 0, where φðtÞ ¼ ðφ1 ; φ2 ; …; φn ÞT ; φi ðtÞ : ½  τ; 0-ð0; þ 1Þ; i ¼ 1; 2; …; n, are continuous with the norm J φ J ¼ sup  τ r t r 0 ð∑ni¼ 1 jφi ðtÞjr Þ1=r , where r 4 1 is a constant. Definition 2.1. A function xðtÞ : ½  τ; αÞ-Rn ; α 4 0 is said to be a solution of system (1.3), if (i) xðtÞ ¼ φðtÞ for  τ rt r0; (ii) x(t) satisfies system (1.3) for t Z 0; (iii) x(t) is continuous everywhere except for some tk and left continuous at t ¼ t k , and the right limit xðt kþ Þ exist for k ¼ 1; 2; …. Definition 2.2. A solution x(t) of (1.3) is said to be ω-anti-periodic solution of (1.3), if (

xðt þ ωÞ ¼  xðtÞ;

t a tk ;

xððt k þ ωÞ þ Þ ¼  xðt kþ Þ;

k ¼ 1; 2; …;

where the smallest positive number ω is called the anti-periodic of function x(t).

2. Preliminaries and basic results

Definition 2.3. Let xðtÞ ¼ ðx1 ; x2 ; …; xn ÞT A Rn be an ω-anti-periodic solution of system (1.3) with initial value φðtÞ ¼ ðφ1 ; φ2 ; …; φn ÞT ; φi ðtÞ : ½  τ; 0-ð0; þ1Þ; i ¼ 1; 2; …; n. If there exist constants λ 40 and M 4 1 such that for every solution xðtÞ ¼ ðx 1 ; x 2 ; …; x n ÞT A Rn of system (1.3) with any initial value φ ðtÞ ¼ ðφ 1 ; φ 2 ; …; φ n ÞT ; φ i ðtÞ : ½  τ; 0-ð0; þ 1Þ; i ¼ 1; 2; …; n

For the sake of convenience, we introduce the following notations:

jxi ðtÞ  x i ðtÞj r M J φ  φ J e  λt ;

ci ¼ min jci ðtÞj; t A ½0;ω

ciþ ¼ max max jci ðtÞj; 0 r i r n t A ½0;ω

aijþ ¼ max jaij ðtÞj; t A ½0;ω

where J φ  φ J ¼ sup  τ r s r 0 max1 r i r n jφi ðsÞ  φ i ðsÞj. Then x(t) is said to be globally exponentially stable.

þ

bijl ¼ max jbijl ðtÞj; uiþ ¼ max max jui ðtÞj; t A ½0;ω 0 r i r n t A ½0;ω Z ω  κ i ¼ exp ci ðθÞ dθ : 0

aij ðt þ ωÞg j ðuÞ ¼  aij ðtÞg j ð  uÞ;

bijl ðt þ ωÞg j ðuÞg l ðuÞ ¼  bijl ðtÞg j ð  uÞg l ð  uÞ;

sijl ðt þ ωÞ ¼ τijl ðtÞ;

υijl ðt þ ωÞ ¼ υijl ðtÞ;

Let PCðRn Þ ¼ fx ¼ ðx1 ; x2 ; …; xn ÞT : R-Rn ; xjðtk ;t k þ 1  A Cððt k ; t k þ 1 ; R Þ; xðt kþ Þ and xðt k Þ exist and xðt k Þ ¼ xðt k Þ; k ¼ 1; 2; …g. Set X ¼ fx : x A PCðRn Þ; xðt þ ωÞ ¼  xðtÞ; xððt k þ ωÞ þ Þ ¼  xðt kþ Þ; t A Rg. Then X is a Banach space with the norm J x J ¼ sup0 r t r ω ð∑ni¼ 1 jxi ðtÞjr Þ1=r . n

Throughout this paper, we have the following assumptions: ðH 1 Þ i; j; l ¼ 1; 2; …; n; k A N, there exists ω 4 0 such that for u A R ci ðt þ ωÞ ¼ ci ðtÞ;

8 t 4 0; i ¼ 1; 2; …; n;

τij ðt þ ωÞ ¼ τij ðtÞ; I ik ðt þ ωÞ ¼  I ik ðtÞ; t; u A R: ð2:1Þ

ðH 2 Þ For i ¼ 1; 2; …; n; k A N, there exists a positive integer q such that I iðk þ qÞ ¼ I ik ; t k þ q ¼ t k þ ω. ðH 3 Þ For each jA f1; 2; …; ng, there are nonnegative constants Lj ; j ¼ 1; 2; …; n and ν such that jg j ðuÞj r ν; jg j ðuÞ  g j ðvÞj r Lj ju  vj; u; v A R. ðH 4 Þ For each iA f1; 2; …; ng; k A N, there exist nonnegative constants dik 4 0 such that

The proof of the following lemma is similar to [29], for the completeness, we list it as follows. Lemma 2.1. Let x ¼ ðx1 ; x2 ; …; xn ÞT be an system (1.3). Then " Z t þω

xi ðtÞ ¼

ω-anti-periodic solution of

n

Gi ðt; sÞ ∑ aij ðsÞg j ðxj ðs  τij ðsÞÞÞ j¼1

t n

#

n

þ ∑ ∑ bijl ðtÞg j ðxj ðs  sijl ðsÞÞÞg l ðxl ðs  υijl ðsÞÞÞ þ I i ðsÞ ds j¼1l¼1

þ

Gi ðt; t k ÞI ik ðt k ; xi ðt k ÞÞ;

∑

t r tk o t þ ω

i ¼ 1; 2; …; n; t A ½0; ω; u; v A R:

jI ik ðt; uÞ  I ik ðt; vÞj r dik ju  vj;

ð2:2Þ

ðH 5 Þ Therehis r 41  such that r i1=r H ¼ ∑qk ¼ 1 ∑ni¼ 1 ðκ i =ð1 þ κ i ÞÞdik o 1.

where

ðH 6 Þ There exist constants η 40 and λ 4 0 such that for all t Z 0

Gi ðt; sÞ ¼ exp

n

n

t

n

λ  ci ðtÞ þ ∑ jaij ðtÞjLj eλτ þ ∑ ∑ jbijl ðtÞjðLj þ Ll Þνeλτ o  η o 0: j¼1

Z

j¼1l¼1

i ¼ 1; 2; …; n:

s

Z ω   ci ðθÞ dθ =ð1 þ exp ci ðθÞ dθÞ ; s A ½t; t þ ω; 0

Q. Wang et al. / Neurocomputing 138 (2014) 339–346

Rs Proof. Let yi ðsÞ ¼ expð t ci ðθÞ dθÞxi ðsÞ, then 8 R Δyi ðsk Þ ¼ expð tsk ci ðθÞ dθÞΔxi ðsk Þ; > > > " > > > n > > y0 ðsÞ ¼ expðR sk c ðθÞ dθÞ ∑ a ðsÞg ðx ðs  τ ðsÞÞÞ < i ij ij j j i t

Z þ Z

n

þ ∑ ∑ bijl ðsÞg j ðxj ðs  sijl ðsÞÞÞg l ðxl ðs  υijl ðsÞÞÞ j ¼ 1l ¼ 1

j¼1

t

n

#

n

þ ∑ ∑ bijl ðsÞg j ðxj ðs  sijl ðsÞÞÞg l ðxl ðs  υijl ðsÞÞÞ þ I i ðsÞ ds: j¼1l¼1

ð2:4Þ Integrating the second equality of (2.3) from t1 to s, where s A ðt 1 ; t 2 , it follows " Z s Z s n expð ci ðθÞ dθÞ ∑ aij ðsÞg j ðxj ðs  τij ðsÞÞÞ yi ðsÞ ¼ yi ðt 1þ Þ þ t 1þ

n

j¼1

t

#

n

þ ∑ ∑ bijl ðsÞg j ðxj ðs  sijl ðsÞÞÞg l ðxl ðs  υijl ðsÞÞÞ þ I i ðsÞ ds j¼1l¼1

¼ yi ðtÞ þ n

Z

Z s

s

exp t

t

c i ðθ Þ dθ

"

#

þ ∑ ∑ bijl ðsÞg j ðxj ðs  sijl ðsÞÞÞg l ðxl ðs  υijl ðsÞÞÞ þ I i ðsÞ ds j¼1l¼1

Z

t1

þexp 0



ci ðθÞ dθ I i1 ðt 1 ; xi ðt 1 ÞÞ:

ð2:5Þ

j¼1

t

t

#

n

þ

∑

t r tk o s

Z

tk

exp t

 ci ðθÞ dθ I ik ðt k ; xi ðt k ÞÞ:

s

s

exp t

t n

c i ðθ Þ dθ

ð2:6Þ

þ

n

t r tk o s

j¼1

#

Lemma 2.2 (Pan and Cao [29]). Let p; q; τ; dk ; k ¼ 1; 2; … be constants and q Z 0; τ 4 0; dk 4 0 and assume that x(t) is a piecewise continuous nonnegative function satisfying 8 þ > < D xðtÞ r pxðtÞ þ qxðtÞ; xðt kþ Þ rdk xðt k Þ; k ¼ 1; 2; …; > : xðtÞ ¼ φðtÞ; t A ½t  τ; t : 0

Z

tk

exp t

 ci ðθÞ dθ I ik ðt k ; xi ðt k ÞÞ:

If there exists α such that for k ¼ 1; 2; …, ln dk =t k ¼ t k  1 r α, and p þ dq þ α o 0, then

R tþω c i ðθ Þ dθ Þ 1 þexpð t (Z Z s " n t þω  ∑ aij ðsÞg j ðxj ðs  τij ðsÞÞÞ exp c i ðθ Þ dθ t

t

n

j¼1

#

þ ∑ ∑ bijl ðsÞg j ðxj ðs  sijl ðsÞÞÞg l ðxl ðs  υijl ðsÞÞÞ þ I i ðsÞ ds j¼1l¼1

sup

jφðtÞjexpð  λðt  t 0 ÞÞ;

Lemma 2.3 (Minkowski inequality). Let r 4 1; aji Z 0; i ¼ 1; 2; …; n; j ¼ 1; 2; …; m. Then we have " !r #1=r !1=r ∑

i¼1

m

∑ aji

j¼1

m

r ∑

j¼1

n

∑ ðaji Þr

:

i¼1

Lemma 2.4 (Young inequality). Let a 4 0; b4 0; p 4 1; q 4 1 and p  1 þ q  1 ¼ 1. Then we have

Lemma 2.5. Let P be a closed convex and nonempty subset of a Banach space X. Let Φ; Ψ be the operators such that

Similar to the proof of Lemma 3 [35], we have ð2:7Þ

1

xi ðtÞ ¼ 

0

3. Main results

In view of xi ðt þ ωÞ ¼  xi ðtÞ, we have

n

It is easy to see that Gi ðt þ ω; s þ ωÞ ¼ Gi ðt; sÞ and 1=ð1 þ κ i Þ r jGi ðt; sÞj r κ i =ð1 þ κ i Þ; s A ½t; t þ ω; i ¼ 1; 2; …; n, where κ i ¼ exp Rω ð 0 ci ðθÞ dθÞ.

Then there exists z A P such that z ¼ Φz þ Ψ z.

∑ aij ðsÞg j ðxj ðs  τij ðsÞÞÞ

n

∑

□

(i) Φx þ Ψ yA P whenever x; y A P, (ii) Φ is compact and continuous, (iii) Ψ is a contraction mapping.

þ ∑ ∑ bijl ðsÞg j ðxj ðs  sijl ðsÞÞÞg l ðxl ðs  υijl ðsÞÞÞ þ I i ðsÞ ds j¼1l¼1

ð2:8Þ

where ε 4 0. The equality holds if and only if ðaεÞp ¼ ðbε  1 Þq .

Note that yi ðtÞ ¼ xi ðtÞ, then Z s  ci ðθÞ dθ xi ðsÞ ¼ xi ðtÞ exp t Z " Z þ

i ¼ 1; 2; …; n:

ab r p  1 ðaεÞp þ q  1 ðbε  1 Þq ;

þ ∑ ∑ bijl ðsÞg j ðxj ðs  sijl ðsÞÞÞg l ðxl ðs  υijl ðsÞÞÞ þ I i ðsÞ ds j¼1l¼1

Gi ðt; t k ÞI ik ðt k ; xi ðt k ÞÞ;

∑

t r tk o t þ ω

Then the proof is completed.

n

Repeating the above procession, for s A ðt k ; t k þ 1 ; k ¼ 1; 2…; p, we have Z s " n Z s yi ðsÞ ¼ yi ðtÞ þ exp ci ðθÞ dθ ∑ aij ðsÞg j ðxj ðs  τij ðsÞÞÞ n

#

where xðtÞ ¼ sups A ½t0  τ;t 0 fxðsÞg; d ¼ sup1 r k o þ 1 fexpðαðt k  t k  1 ÞÞ; expð  αðt k  t k  1 ÞÞg; λ is a unique positive solution of λ þ p þ dq expðλτ Þ þ α ¼ 0.

∑ aij ðsÞg j ðxj ðs  τij ðsÞÞÞ

n

n

n

t A ½t 0  τ ;t 0 

j¼1

ci ðθÞ dθ I ik ðt k ; xi ðt k ÞÞ

j¼1l¼1

þ

xðtÞ r d

n

t

Gi ðt; sÞ ∑ aij ðsÞg j ðxj ðs  τij ðsÞÞÞ

n

ð2:3Þ

t

exp

"

)



þ ∑ ∑ bijl ðtÞg j ðxj ðs  sijl ðsÞÞÞg l ðxl ðs  υijl ðsÞÞÞ þ I i ðsÞ ds

þ I i ð sÞ :

Assume that there exist p impulse points t 1 ; t 2 ; …; t p on ½t; t þ ω. Integrating the second equality of (2.3) from t to t1, it follows Z s " n Z t1 ∑ aij ðsÞg j ðxj ðs  τij ðsÞÞÞ yi ðt 1 Þ  yi ðtÞ ¼ exp ci ðθÞ dθ

tk

j¼1

t

#

n

∑

t r tk o t þ ω

t þω

¼

j¼1

> > > > > > > > :

341

Lemma 3.1. Let 8 > <1 sðf ðtÞÞ ¼ 0 > : 1

f be a differential function on R and 0

if f ðtÞ 4 0 or f ðtÞ ¼ 0 and f ðtÞ o 0; 0

if f ðtÞ ¼ 0 and f ðtÞ ¼ 0; 0

if f ðtÞ o 0 or f ðtÞ ¼ 0 and f ðtÞ 4 0: 0

Then jf ðtÞj ¼ f ðtÞsðf ðtÞÞ; D  jf ðtÞj ¼ f ðtÞsðf ðtÞÞ, where D  jf ðtÞj is the upper left derivative of jf ðtÞj. Proof. For the completeness, we list the proof of Lemma 3.1 as follows. Since f is a differential function on R, f ðt þ hÞ  f ðhÞ ¼ 0 f ðtÞh þ αh, where α ¼ oð1Þ; ðh-0Þ.

342

Q. Wang et al. / Neurocomputing 138 (2014) 339–346

If f ðtÞ 4 0, for sufficiently small h, it follows that f ðt þhÞ 4 0. So   jf ðt þ hÞj  jf ðtÞj f ðt þhÞ  f ðtÞ ¼ lim sup D  f ðtÞ ¼ lim sup   h h h-0 h-0 0

Case 1.2: If xi ðt Þ o 0, calculating the upper left derivative of jxi ðtÞj, from Lemma 3.1 and (1.3), we have n

0 r D  jxi ðt Þj ¼  x0i ðt Þ ¼ ci ðtÞxi ðt Þ  ∑ aij ðt Þg j ðxj ðt  τij ðt ÞÞÞ

0

¼ f ðtÞ ¼ f ðtÞsðf ðtÞÞ:

j¼1

n

If f ðtÞ o 0, for sufficiently small h, it follows that f ðt þhÞ o 0. So

j¼1l¼1

n

  jf ðt þ hÞj  jf ðtÞj  f ðt þ hÞ þ f ðtÞ ¼ lim sup D  f ðtÞ ¼ lim sup   h h h-0 h-0 0

n

 ∑ ∑ bijl ðt Þg j ðxj ðt  sijl ðt ÞÞÞg l ðxl ðt  υijl ðt ÞÞÞ I i ðt Þ r  ci ðt Þjxi ðt Þj þ ∑ jaij ðt Þjjg j ðxj ðt  τ ij ðt ÞÞÞj j¼1

0

¼  f ðtÞ ¼ f ðtÞsðf ðtÞÞ:

n

n

þ ∑ ∑ jbijl ðt Þjjg j ðxj ðt  sijl ðt ÞÞÞg l ðxl ðt  υijl ðt ÞÞÞj þ jI i ðt Þj j¼1l¼1

0

n

If f ðtÞ ¼ 0; f ðtÞ 40, for sufficiently small h, then 0   jf ðt þ hÞj  jf ðtÞj jf ðtÞh þ αhj ¼ lim sup D  f ðtÞ ¼ lim sup   h h h-0 h-0 0

j¼1

0

  jf ðt þ hÞj  jf ðtÞj jαhj 0 ¼ lim sup ¼ 0 ¼ f ðtÞsðf ðtÞÞ: D  f ðtÞ ¼ lim sup h h h-0  h-0  0

If f ðtÞ ¼ 0; f ðtÞ o 0, for sufficiently small h, then 0   jf ðt þ hÞj  jf ðtÞj jf ðtÞh þ αhj ¼ lim sup D  f ðtÞ ¼ lim sup   h h h-0 h-0

which is a contraction and thus (3.2) holds. □ Case 2: If γ 4 jxi ðt k Þj Z γ =ð1 þ dik Þ. Thus, we can assume that there is i A f1; 2; …; ng and t A ðt k ; t k þ 1  such that   γ xi ðt Þ ¼ γ and jxi ðtÞj 4 ; t A ½0; t Þ; i ¼ 1; 2; …; n: 1 þ dik 1 þdik ð3:4Þ Calculating the upper left derivative of jxi ðtÞj, by (3.1), (3.4) and ðH 6 Þ, we have

0

¼ f ðtÞ ¼ f ðtÞsðf ðtÞÞ:

n   0 r D  jxi ðt Þj r  ci ðt Þjxi ðt Þj þ ∑ aij ðt Þg j ðxj ðt  τij ðtÞÞÞj

□

j¼1

    þ ∑ ∑ bijl ðt Þg j ðxj ðt  sijl ðt ÞÞÞg l ðxl ðt  υijl ðt ÞÞÞ þ I i ðt Þj n

Theorem 3.1. Assume that ðH 1 Þ–ðH 6 Þ hold. Suppose that xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; …; xn ðtÞÞT be any solution of system (1.3) with any initial value φðtÞ; 0 o jφi ðtÞj o γ ; t A ½  τ; 0, where ! n n 1 þ dik n ∑ jaij ðtÞjν þ ∑ ∑ jbijl ðtÞjν2 þ jI i ðtÞj ; γ4 ci ðtÞ j ¼ 1 j¼1l¼1 t 40; i ¼ 1; 2; …; n:

ð3:1Þ

Then jxi ðtÞj o γ

and

jxi ðt kþ Þj o

γ ; t 4 0; i ¼ 1; 2; …; n; k A N:

ð3:2Þ

r  ci ðt Þ

n

0 r D  jxi ðt Þj ¼ x0i ðt Þ ¼  ci ðtÞxi ðt Þ þ ∑ aij ðt Þg j ðxj ðt  τij ðt ÞÞÞ j¼1

j¼1l¼1

Proof. Define the operator ðΦxÞðtÞ ¼ ððΦ1 xÞðtÞ; …; ðΦn xÞðtÞÞT ; ðΨ xÞðtÞ ¼ ððΨ 1 xÞðtÞ; …; ðΨ n xÞðtÞÞT ; Z

tþω

"

x A X; t A R;

ð3:5Þ

n

Gi ðt; sÞ ∑ aij ðsÞg j ðxj ðs  τij ðsÞÞÞ j¼1

t n

#

n

þ ∑ ∑ bijl ðtÞg j ðxj ðs  sijl ðsÞÞÞg l ðxl ðs  υijl ðsÞÞÞ þI i ðsÞ ds; j¼1l¼1

ðΨ i xÞðtÞ ¼

∑

Gi ðt; t k ÞI ik ðt k ; xi ðt k ÞÞ;

t r tk o t þ ω

x A X; t A R; i ¼ 1; 2; …; n: ð3:6Þ

n

þ ∑ ∑ bijl ðt Þg j ðxj ðt  sijl ðt ÞÞÞg l ðxl ðt  υijl ðt ÞÞÞ þ I i ðt Þ j¼1l¼1

n

r  ci ðt Þjxi ðt Þj þ ∑ jaij ðt Þjjg j ðxj ðt  τij ðt ÞÞÞj j¼1

n

j¼1

Theorem 3.2. Assume that ðH 1 Þ–ðH 5 Þ hold. Then system (1.3) has at least one ω-anti-periodic solution.

ðΦi xÞðtÞ ¼

Case 1.1: If xi ðt Þ 4 0, calculating the upper left derivative of jxi ðtÞj, from Lemma 3.1 and (1.3), we have

n n n    þ ∑ jaij ðt Þjν þ ∑ ∑ bijl ðt Þν2 þ I i ðt Þj o 0;

Remark 3.1. In view of boundedness of solution x(t) of (1.3) from Theorem 3.1, we know that x(t) can be defined on ½0; þ 1Þ.

jxi ðt Þj ¼ γ

ð3:3Þ

γ

1 þdik

which is a contraction and also shows that (3.2) holds. The proof of Theorem 3.1 is completed.

where

and jxi ðtÞj o γ ; 8 t A ½0; t Þ; i ¼ 1; 2; …; n; k A N:

n

j¼1l¼1

Proof. By the way of contradiction, suppose that (3.2) does not hold. Since xi ðt kþ Þ ¼ xi ðt k Þ þ I ik ðt k ; xi ðt k ÞÞ and by ðH 4 Þ, then jxi ðt kþ Þj ¼ jxi ðt k Þ þ I ik ðt k ; xi ðt k ÞÞj r ð1 þ dik Þjxi ðt k Þj. If jxi ðt kþ Þj ¼ γ , then jxi ðt k Þj Z γ = ð1 þ dik Þ. Case 1: If jxi ðt k Þj ¼ γ . Suppose that there is i A f1; 2; …; ng and t A ðt k  1 ; t k  such that

n

j¼1l¼1

0 r D  jxi ðt Þj o 0;

If f ðtÞ ¼ 0; f ðtÞ ¼ 0, for sufficiently small h, then

The proof is completed.

n

In all by (3.1), (3.3) and ðH 6 Þ, we have

0

¼  f ðtÞ ¼ f ðtÞsðf ðtÞÞ:

0

n

r  ci ðt Þγ þ ∑ jaij ðt Þjν þ ∑ ∑ jbijl ðt Þjν2 þ jI i ðt Þj o 0:

n

þ ∑ ∑ jbijl ðt Þjjg j ðxj ðt  sijl ðt ÞÞÞg l ðxl ðt  υijl ðt ÞÞÞj þ jI i ðt Þj j¼1l¼1

We also define the operator T ¼ Φ þ Ψ : X-X. Choose h  r i1=r þ ðna þ νω þn2 b ν2 ω þ u þ ω þ qIÞ ∑ni¼ 1 1 þκi κ i 4 0; ρZ 1H þ

n

n

n

r  ci ðt Þγ þ ∑ jaij ðt Þjν þ ∑ ∑ jbijl ðt Þjν2 þ jI i ðt Þj o 0: j¼1

j¼1l¼1

þ where a þ ¼ max1 r i;j r n faijþ g; b ¼ max1 r i;j;l r n fbijl g; ν ¼ max1 r i r n fjg j ð0Þjg; u þ ¼ max1 r i r n max0 r t r ωfjI i ðtÞjg; I ¼ max1 r i r n;k ¼ 1;2;…;q fsup0 r t r ω jI ik ðt; 0Þjg.

Q. Wang et al. / Neurocomputing 138 (2014) 339–346

"

We will show that TBρ  Bρ , where Bρ ¼ fx A X : J x J r ρg. By Minkowski inequality, we have J Tx J ¼ J Φx þ Ψ x J Z ( " tþω n  n  ¼ sup ∑  Gi ðt; sÞ ∑ aij ðsÞg j ðxj ðs  τij ðsÞÞÞ  0rt rω i ¼ 1 t j¼1 n

n

r

κi ω n þ ∑ a ν 1 þ κ i j ¼ 1 ij

∑

i¼1

" þ ωu "

n

þ

∑

i¼1

n

þ

∑ n

þ

∑

r

κi

n

∑

i¼1

∑

i¼1

κi ω n n þ 2 ∑ ∑ b ν 1 þ κ i j ¼ 1 l ¼ 1 ijl

!r #1=r

!r #1=r     ∑ I ik ðt k ; xi ðt k ÞÞ  I ik ðt k ; 0Þ

κi

!r #1=r   ∑ I ik ðt k ; 0Þ

r #1=r

1 þ κi

:

!r #1=r q   ∑ dij xi ðt k Þ  yi ðt k Þ

i¼1

"

1 þ κi k ¼ 1 n

∑

i¼1



κ i dik 1 þ κi

r #1=r J x  yJ o J x  yJ ;

\right\left\hskip 12ptit is easy to see that Ψ is a contraction mapping. By Lemma 2.5, system (1.3) has at least one ω-antiperiodic solution. The proof of Theorem 3.2 is completed. □

!r #1=r q   þ ∑ dik xi ðt k Þ þðna þ νω þ n2 b ν2 ω

i¼1

i ¼ 1 1 þ κi

\eqno

q

κi

κi

n

þ

n

κi

k¼1

q

n

þ

j¼1l¼1

(

n

∑

r ∑

κi

þu þ ω þqIÞ ∑

" r

q

1 þ κi

1 þ κi k ¼ 1 " 

j¼1

2

 r )1=r     ¼ sup ∑  ∑ Gi ðt; t k Þ½I ik ðt k ; xi ðt k ÞÞ  I ik ðt k ; yi ðt k ÞÞ  0 r t r ω i ¼ 1 t r t k o t þ ω

r #1=r

κi

 ∑ a νþ ∑ ∑ b ν þu

JΨ xΨ yJ

1 þ κi k ¼ 1

i¼1

"

þ

þ

Hence, by means of Arzela–Ascoli theorem, Φ is compact on Bρ . In the end, for any x; y A Bρ , by ðH 5 Þ, we have

#

1 þ κi k ¼ 1

i¼1

"



n

#

n

r jt 1 t 2 jðna ν þ n2 b ν2 þ u þ Þωc þ ∑

j¼1l¼1

n

n

þ

þ

þ ∑ ∑ bijl ðsÞg j ðxj ðs  sijl ðsÞÞÞg l ðxl ðs  υijl ðsÞÞÞ þ I i ðsÞ ds r )1=r   þ ∑ Gi ðt; t k ÞI ik ðt k ; xi ðt k ÞÞ  t r tk o t þ ω " !r #1=r "

n

343

Theorem 3.3. Assume that ðH 1 Þ–ðH 6 Þ hold. If the following assumptions are satisfied ðH 7 Þ for each iA f1; 2; …; ng, there exist positive constants d ik ; α such that ju þ I ik ðt; uÞ  v  I ik ðt; vÞj rd ik ju  vj;

:

t A ½0; ω; u; v A R;

ð3:7Þ

and

\eqno \right\left\hskip 12ptThen using Hölder inequality, we have "   #1=r q n κ i dik r J Tx J r ∑ ∑ JxJ k ¼ 1 i ¼ 1 1 þ κi "   #1=r n κi r þ þ ðna þ νω þ n2 b ν2 ω þ u þ ω þ qIÞ ∑ i ¼ 1 1 þ κi o J x J r ρ; thus TBρ  Bρ . In view of continuity of g j ; j ¼ 1; 2; …; n, we see that the operator Φ is continuous. For x A Bρ , we have "   #1=r n

þ

J Φ J r ðna þ νω þ n2 b ν2 ω þ u þ ωÞ ∑

i¼1

κi

1 þ κi

r

:

That is to say that Φ is uniformly bounded on Bρ . Next, we prove the compactness of the operator Φ. For t 1 ; t 2 A ½0; ω; x A Bρ , we have J ðΦxÞðt 1 Þ  ðΦxÞðt 2 Þ J Z ( " ω n  n  ¼ sup ∑  jGi ðt 1 ; sÞ  Gi ðt 2 ; sÞj ∑ aij ðsÞg j ðxj ðs  τij ðsÞÞÞ  0rt rω i ¼ 1 0 j¼1 r )1=r   þ ∑ ∑ bijl ðsÞg j ðxj ðs  sijl ðsÞÞÞg l ðxl ðs  υijl ðsÞÞÞ þ I i ðsÞ ds  j¼1l¼1 n

n

1 1 þ κi i¼1 "

r ∑

n

#

n

Rs Z ω R s   e t1 ci ðθÞ dθ  e t2 ci ðθÞdθ  ds  0 # n

n

ln dk r α; k ¼ 1; 2; …; tk  tk  1

Θ1 þ dΘ2 þ α o0;

ð3:8Þ

where dk ¼ maxfd 1k ; d 2k ; …; d nk g; (

Θ2 ¼ d¼

n

max

t A ½0;ω;i ¼ 1;2;…;n

sup 1rko þ1

Θ1 ¼

max

fλ  ci ðtÞg;

t A ½0;ω;i ¼ 1;2;…;n

)

n

∑ ∑ jbijl ðtÞjðLj ν þ Ll νÞ ;

j¼1l¼1

fexpðαðt k t k  1 ÞÞ; expð  αðt k  t k  1 ÞÞg;

λ is a unique positive solution of λ þ Θ1 þ dΘ2 expðλτÞ þ α ¼ 0. Then the ω-anti-periodic solution system of (1.3) is globally exponentially stable. Proof. Let xn ðtÞ ¼ ðxn1 ðtÞ; xn2 ðtÞ; …; xnn ðtÞÞT A Rn be an ω-anti-periodic solution of system (1.3) with initial value φn ðtÞ ¼ ðφn1 ; φn2 ; …; φnn ÞT ; φni ðtÞ : ½  τ; 0-ð0; þ1Þ; i ¼ 1; 2; …; n and xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; …; xn ðtÞÞT A Rn be any solution of system (1.3) with any initial value φðtÞ ¼ ðφ1 ; φ2 ; …; φn ÞT ; φi ðtÞ : ½  τ; 0-ð0; þ 1Þ; i ¼ 1; 2; …; n. Set yi ðtÞ ¼ xi ðtÞ  xni ðtÞ, then 8 n > n 0 > > > yi ðtÞ ¼  ci ðtÞyi ðtÞ þ ∑ aij ðtÞ½g j ðxj ðt  τij ðtÞÞÞ  g j ðxj ðt  τij ðtÞÞÞ > > j¼1 > > > n n > > > < þ ∑ ∑ b ðtÞ½g ðx ðt  s ðtÞÞÞg ðx ðt  υ ðtÞÞÞ j ¼ 1l ¼ 1

ijl

j

j

ijl

l

l

ijl

> > >  g j ðxnj ðt  sijl ðtÞÞÞg l ðxnl ðt  υijl ðtÞÞÞ; t Z0; t a t k ; > > > > þ > yi ðt k Þ ¼ yi ðt k Þ þ I ik ðt k ; xi ðt k ÞÞ  I ik ðt k ; xni ðt k ÞÞ; k ¼ 1; 2; …; > > > > : ϕ ðtÞ ¼ φ ðtÞ  φn ðtÞ; t A ½  τ; 0; i ¼ 1; 2; …; n: i i i ð3:9Þ

þ

 ∑ aijþ ν þ ∑ ∑ bijl ν2 þ u þ j¼1

j¼1l¼1

 Rs Z ω  R s  ci ðθÞdθ  1  t1 ci ðθÞdθ r ∑  e t2 e  ds  i ¼ 1 1 þ κi 0  n

Define the Lyapunov function as n

VðtÞ ¼ ∑ jyi ðtÞjeλt : i¼1

ð3:10Þ

344

Q. Wang et al. / Neurocomputing 138 (2014) 339–346

From the definition of the upper left derivative of V(t) with respect to t and Lemma 3.1, we have for t a t k ! D ðVðtÞÞ ¼ λ

n



n

∑ eλt jyi ðtÞj þeλt D 

∑ jyi ðtÞj

i¼1

n

n

i¼1

i¼1

i¼1

¼ λ ∑ eλt jyi ðtÞj þ eλt ∑ D  jyi ðtÞj n

n

i¼1

i¼1

¼ λ ∑ eλt jyi ðtÞj þ eλt ∑ y0i ðtÞsðyi ðtÞÞ (

n

¼ ∑ eλt n

n

 ci ðtÞyi ðtÞ þ ∑ aij ðtÞ½g j ðxj ðt  τij ðtÞÞÞ g j ðxnj ðt  τij ðtÞÞÞ

i¼1

j¼1

n

þ ∑ ∑ bijl ðtÞ½g j ðxj ðt  sijl ðtÞÞÞg l ðxl ðt  υijl ðtÞÞÞ j¼1l¼1

)

i¼1

n

n

i¼1

i¼1

r  ∑ ci ðtÞjyi ðtÞjeλt þ ∑ n

Consider the following HHNNs with delays and impulse: 8 x1 ðtÞ 1 1 > > þ g 1 ðx1 ðt  sin 2 tÞÞ þ g 2 ðx2 ðt  7 sin 2 tÞÞ x01 ðtÞ ¼  > > 12 32 32 > > > > 1 1 > > > þ g 1 ðx1 ðt  5 sin 2 tÞÞg 2 ðx2 ðt  3 sin 2 tÞÞ þ cos t > > 16 36 > > > > 1 1 1 > > þ sin t  arctanð sin tÞ  arctanð cos tÞ; > > > 36 72 24 > > > > x2 ðtÞ 1 1 > 0 > x þ g g ðx2 ðt  5 sin 2 tÞÞ ðtÞ ¼  ðx ðt  cos tÞÞ þ > 1 1 2 < 12 32 32 2 1 1 1 > þ g 1 ðx1 ðt  2 sin 2 tÞÞg 2 ðx2 ðt 4 sin 2 tÞÞ þ cos t þ sin t > > > 16 20 40 > > > > 5 1 k π > >  arctanð sin tÞ  arctanð cos tÞ; t a ; k ¼ 1; 2; …; > > 144 2 > >     48    > > kπ 1 kπ 1 kπ > > > sin x1 Δx1 ¼  x1 þ ; > > 4 36 2 2 2 > > >        > > k π 1 k π 1 k π > > ; > : Δx2 2 ¼  4 x2 2 þ 36 sin x2 2

ð4:1Þ

n

 g j ðxnj ðt  sijl ðtÞÞÞg l ðxnl ðt  υijl ðtÞÞÞ sðyi ðtÞÞ þ λ ∑ eλt jyi ðtÞj "

4. Example

n

∑ jaij ðtÞjjg j ðxj ðt  τ ij ðtÞÞÞ

j¼1

n

 g j ðxnj ðt  τij ðtÞÞÞj þ ∑ ∑ jbijl ðtÞjjg j ðxj ðt  sijl ðtÞÞÞg l ðxl ðt  υijl ðtÞÞÞ j¼1l¼1

i n  g j ðxnj ðt  sijl ðtÞÞÞg l ðxnl ðt  υijl ðtÞÞÞj eλt þ λ ∑ jyi ðtÞjeλt

    where g 1 ðxÞ ¼ g 2 ðxÞ ¼ 12 ðx þ 1  x  1Þ. It is obvious that þ 1 1 1 þ c1 ¼ 12 ¼ c2 ; L1 ¼ L2 ¼ ν ¼ 1; aij ¼ 32 ; bijl ¼ 16 ; u þ ¼ 19 ; d1k ¼ 5 7 7 1 49 18 ¼ d2k ; d 1k ¼ 9 ¼ d 2k ; dk ¼ 9 . Thus Θ2 ¼ 4 ; d ¼ 81 ; τ ¼ 7; α ¼  0:32, when we choose sufficiently small Θ1 ; λ, then the conditions of Theorems 3.2 and 3.3 will be satisfied. Hence, system (4.1) has exactly one π-anti-periodic solution. Moreover, the π-antiperiodic solution is globally exponentially stable. This fact is verified by the numerical simulation in Figs. 1 and 2.

i¼1

n

i¼1

i¼1

n

n

∑ jaij ðtÞjLj jyj ðt  τij ðtÞÞj

0.15

#

0.1

j¼1

n

þ ∑ ∑ jbijl ðtÞjðLj νjyj ðt  sijl ðtÞÞj þLl νjyl ðt  υijl ðtÞÞjÞ eλt j¼1l¼1

r Θ1 VðtÞ þ Θ2 V ðtÞ;

ð3:11Þ

x1 and x2

n

r ∑ ðλ  ci ðtÞÞjyi ðtÞjeλt þ ∑

"

where V ðtÞ ¼ supt  τ r s r t fV ðsÞg. From ðH 7 Þ, we have n

þ

n

Vðt kþ Þ ¼ ∑ jyi ðt kþ Þjeλk r ∑ d ik jyi ðt kþ Þjeλt k r dk Vðt k Þ: i¼1

τrsr0

0 −0.05

ð3:12Þ

−0.1 −20

i¼1

By Lemma 2.2, there exists L 41 such that   VðtÞ rL sup V ðsÞ e  λt :

0.05

0

20

40

60

80

100

120

140

160

180

time

ð3:13Þ

Fig. 1. The time response of state of state variables x1 ðtÞ; x2 ðtÞ for system (4.1) with impulsive effects and ðφ1 ðsÞ; φ2 ðsÞÞ ¼ ð0; 0Þ; ð  0:05; 0:05Þ; ð 0:1; 0:1Þ, where x1 ðtÞ is blue and x2 ðtÞ is green. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Thus, we have ð3:14Þ

where K ¼ sup  τ r s r 0 fVðsÞg. By Definition 2.3, the ω-anti-periodic solution system of (1.3) is globally exponentially stable. The proof of Theorem 3.3 is completed. □ Similar to [34], we have the following remark. Remark 3.2. From the global exponential stability of ω-antiperiodic solution x(t), we can know that system (1.3) has a unique ω-anti-periodic solution x(t). So ω-anti-periodic solution of (1.3) is well-posed, that is, ω-anti-periodic solution of (1.3) is existent, unique and stable. Remark 3.3. The results in the work show that by means of appropriate impulsive perturbations, we can control the antiperiodic dynamics of these equations.

0.15 0.1

x1 and x2

J yðtÞ J ¼ J xðtÞ  xn ðtÞ J r LK J φðtÞ  φn ðtÞ J e  2λt ;

0.05 0 −0.05 −0.1 −20

0

20

40

60

80

100

120

140

160

180

time Fig. 2. The time response of state of state variables x1 ðtÞ; x2 ðtÞ for system (4.1) without impulsive effects and ðφ1 ðsÞ; φ2 ðsÞÞ ¼ ð0; 0Þ; ð 0:05; 0:05Þ; ð  0:1; 0:1Þ, where x1 ðtÞ is blue and x2 ðtÞ is green. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Q. Wang et al. / Neurocomputing 138 (2014) 339–346

Acknowledgments The author would like to express his sincere appreciation to the reviewer for his/her helpful comments in improving the presentation and quality of the paper. References [1] H. Okochi, On the existence of periodic solutions to nonlinear abstract parabolic equations, J. Math. Soc. Jpn. 40 (3) (1988) 541–553. [2] A.R. Aftabizadeh, S. Aizicovici, N.H. Pavel, On a class of second-order antiperiodic boundary value problems, J. Math. Anal. Appl. 171 (1992) 301–320. [3] S. Aizicovici, M. McKibben, S. Reich, Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities, Nonlinear Anal. 43 (2001) 233–251. [4] Y. Chen, J.J. Nieto, D. O'Regan, Anti-periodic solutions for fully nonlinear firstorder differential equations, Math. Comput. Model. 46 (2007) 1183–1190. [5] R. Wu, An anti-periodic LaSalle oscillation theorem, Appl. Math. Lett. 21 (2008) 928–933. [6] F.J. Delvos, L. Knoche, Lacunary interpolation by anti-periodic trigonometric polynomials, BIT 39 (1999) 439–450. [7] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [8] Y.K. Li, L. Yang, Anti-periodic solutions for Cohen–Grossberg neural networks with bounded and unbounded delays, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3134–3140. [9] G.Q. Peng, L.H. Huang, Anti-periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays, Nonlinear Anal.: RWA 10 (2009) 2434–2440. [10] J.Y. Shao, An anti-periodic solution for a class of recurrent neural networks, J. Comput. Appl. Math. 228 (2009) 231–237. [11] Z. Wang, J. Fang, X. Liu, Global stability of stochastic high-order neural networks with discrete and distributed delays, Chaos Solitons Fractals 36 (2) (2008) 388–396. [12] S. Mohamad, Exponential stability in Hopfield-type neural networks with impulses, Chaos Solitons Fractals 32 (2007) 456–467. [13] Y. Liu, Z. You, Multi-stability and almost periodic solutions of a class of recurrent neural networks, Chaos Solitons Fractals 33 (2007) 554–563. [14] Y. Jiang, B. Yang, J. Wang, C. Shao, Delay-dependent stability criterion for delayed Hopfield neural networks, Chaos Solitons Fractals 39 (2009) 2133–2137. [15] B. Xiao, H. Meng, Existence and exponential stability of positive almost periodic solutions for high-order Hopfield neural networks, Appl. Math. Model. 33 (2009) 532–542. [16] B. Liu, L. Huang, Existence and exponential stability of periodic solutions for a class of Cohen–Grossberg neural networks with time-varying delays, Chaos Solitons Fractals 32 (2007) 617–627. [17] F. Zhang, Y. Li, Almost periodic solutions for higher-order Hopfield neural networks without bounded activation functions, Electron. J. Diff. Equ. 2007 (2007) 1–10. [18] C.X. Ou, Anti-periodic solutions for high-order Hopfield neural networks, Comput. Math. Appl. 56 (2008) 1838–1844. [19] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [20] A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. [21] M.U. Akhmet, On the general problem of stability for impulsive differential equations, J. Math. Anal. Appl. 288 (2003) 182–196. [22] K. Gopalsamy, Stability of artificial neural networks with impulses, Appl. Math. Comput. 154 (2004) 783–813. [23] Z. Guan, G. Chen, On delayed impulsive Hopfield neural networks, Neural Netw. 12 (1999) 273–280. [24] Q. Zhou, Global exponential stability of BAM neural networks with distributed delays and impulses, Nonlinear Anal.: RWA 10 (2009) 144–153. [25] D. Xu, Z. Yang, Impulsive delay differential inequality and stability of neural networks, J. Math. Anal. Appl. 305 (2005) 107–120. [26] Y.Q. Yang, J.D. Cao, Stability and periodicity in delayed cellular neural networks with impulsive effects, Nonlinear Anal.: RWA 8 (2007) 362–374. [27] Y. Li, J. Wang, An analysis on the global exponential stability and the existence of periodic solutions for non-autonomous hybrid BAM neural networks with distributed delays and impulses, Comput. Math. Appl. 56 (2008) 2256–2267. [28] J. Zhang, Z. Gui, Existence and stability of periodic solutions of high-order Hopfield neural networks with impulses and delays, J. Comput. Appl. Math. 224 (2009) 602–613. [29] L.J. Pan, J.D. Cao, Anti-periodic solution for delayed cellular neural networks with impulsive effects, Nonlinear Anal.: RWA 12 (2011) 3014–3027. [30] J.D. Cao, J.L. Liang, J. Lam, Exponential stability of high-order bidirectional associative memory neural networks with time delays, Physica D 199 (2004) 425–436.

345

[31] F.L. Ren, J.D. Cao, Periodic solutions for a class of higher-order Cohen– Grossberg type neural networks with delays, Comput. Math. Appl. 54 (6) (2007) 826–839. [32] Z. Chen, D.H. Zhao, J. Ruan, Dynamic analysis of high-order Cohen–Grossberg neural networks with time delay, Chaos Solitons Fractals 32 (4) (2007) 1538–1546. [33] Z. Chen, D.H. Zhao, X.L. Fu, Discrete analogue of high-order periodic Cohen– Grossberg neural networks with delay, Appl. Math. Comput. 214 (2009) 210–217. [34] Z. Chen, X.L. Fu, D.H. Zhao, Anti-periodic mild attractor of delayed Hopfield neural networks systems with reaction–diffusion terms, Neurocomputing 99 (2013) 372–380. [35] A.P. Chen, J.D. Cao, Existence and attractivity of almost periodic solutions for cellular neural networks with distributed delays and variable coefficients, Appl. Math. Comput. 134 (2003) 125–140. [36] Jia R.W., Yang M.Q., Convergence for HRNNs with unbounded activation functions and time-varying delays in the leakage terms, Neural Process Lett. 39 (2014) 69–79. [37] A.P. Zhang, Existence and exponential stability of anti-periodic solutions for HCNNs with time-varying leakage delays, Adv. Differ. Equ. (2013) 162, 〈http:// www.advancesindifferenceequations.com/content/2013/1/162〉. [38] Y.L. Xu, Anti-periodic solutions for HCNNs with time-varying delays in the leakage terms, Neural Computing and Applications, Springer-Verlag London, 2013, /http://dx.doi.org/10.1007/s00521-012-1330-6S. [39] Liu B.W., An anti-periodic LaSalle oscillation theorem for a class of functional differential equations, J. Comput. Appl. Math. 223 (2) (2009) 1081–1086. [40] B.W. Liu, S.H. Gong, Periodic solution for impulsive cellar neural networks with time-varying delays in the leakage terms, in: Abstract and Applied Analysis, vol. 2013, Hindawi Publishing Corporation, 10 p., Article ID 701087, 〈http://dx.doi.org/10.1155/2013/701087〉.

Qi Wang was born in Anhui, China, in 1974. He received the B.Sc. degree from the Department of Mathematics, Huaibei Normal University, in 1997, the M.Sc. degree from the School of Mathematical Sciences, Anhui University, in 2005, the Ph.D. degree from School of Mathematical Sciences and Computing Technology, Central South University, in 2008, China. He is working as a full associate professor at the School of Mathematical Sciences, Anhui University, since 2009. His major research interests include cellular neural networks, functional differential equations and biomathematics. He has authored and coauthored some 30 publications. He also serves as reviewers for Nonlinear Analysis (RWA, TMA), Applied Mathematics and Computations, Journal of Computational and Applied Mathematics, etc.

Yayun Fang received the B.Sc. degree from the Department of Mathematics, Tongling University, China, in 2012. She is currently working as a master student at the School of Mathematical Sciences, Anhui University. Her major research interests include cellular neural networks, functional differential equations and biomathematics.

Hui Li is currently working as a bachelor student at the School of Mathematical Sciences, Anhui University. His major research interests include cellular neural networks, computational mathematics and bio-mathematics.

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Q. Wang et al. / Neurocomputing 138 (2014) 339–346 Lijuan Su was born in Shandong, China, in 1986. She received the B.S. degree in Applied Mathematics in 2005 and the Ph.D. degree in Computational Mathematics in 2010 from Shandong University, Jinan, China, respectively. Now she is an associate professor of Applied Mathematics at Anhui University, Hefei, China. Her research interests are in the areas of dynamics of neural networks, numerical solution of partial differential equation, applications of fractional calculus, scientific calculation and impulse differential equations.

Binxiang Dai was born in Hunan, China, in 1962. He received the B.S. degree in Mathematics in 1983 from Beijing Normal University, Beijing, China, and the M.S. degree in Applied Mathematics and the Ph.D. degree in Applied Mathematics from Hunan University, Changsha, China, in 1994 and 2001, respectively. Now he is a professor and doctoral advisor of Applied Mathematics at Central South University, Changsha, China. He is the author or co-author of more than 90 journal papers, five edited books. His research interests are in the areas of dynamics of neural networks, and qualitative theory of differential equations and impulse differential equations.