Pseudo almost periodic solutions for neutral type CNNs with continuously distributed leakage delays

Neurocomputing 148 (2015) 445–454

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Pseudo almost periodic solutions for neutral type CNNs with continuously distributed leakage delays$ Bingwen Liu n College of Mathematics and Computer Science, Hunan University of Arts and Science, Changde, Hunan 415000, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 7 November 2013 Received in revised form 3 May 2014 Accepted 9 July 2014 Communicated by W. Lu Available online 23 July 2014

This paper is concerned with a new generalized neutral type cellular neural network model with continuously distributed leakage delays. Some criteria are established for the existence and global exponential stability of pseudo almost periodic solutions for this model by using the exponential dichotomy theory, contraction mapping fixed point theorem and inequality analysis technique. Moreover, an example and its numerical simulation are employed to illustrate the main results. & 2014 Elsevier B.V. All rights reserved.

Keywords: Neutral type cellular neural networks Pseudo almost periodic solution Exponential stability Continuously distributed leakage delay

1. Introduction In the classic study of population dynamics, it is well known that a neural network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths, it is desired to model them by introducing continuously distributed delays over a certain duration of time [1–6]. Recently, H. Zhang [7] considered the existence and stability of almost periodic solutions for the following cellular neural networks (CNNs) with continuously distributed leakage delays: Z 1 n x0i ðtÞ ¼  ci ðtÞ hi ðsÞxi ðt  sÞ ds þ ∑ aij ðtÞf j ðxj ðt  τij ðtÞÞÞ 0

n

þ ∑ bij ðtÞ j¼1

Z 0

j¼1

1

K ij ðuÞg j ðxj ðt  uÞÞ du þ I i ðtÞ;

i ¼ 1; 2; …; n; ð1:1Þ

where ci : R-ð0 þ 1Þ; I i ; aij ; bij ; τij : R-R are almost periodic functions, delay kernels hi(s) and Kij(u) are continuous functions. The description of biological significance of coefficient functions and the delays can also be found in [7]. As is well known, in the study of neural network dynamics, because of the complicated ☆ This work was supported by the construct program of the key discipline in Hunan province (Mechanical Design and Theory), the Scientific Research Fund of Hunan Provincial Natural Science Foundation of PR China (Grant no. 11JJ6006), and the Natural Scientific Research Fund of Hunan Provincial Education Department of PR China (Grant nos. 11C0916, 11C0915). n Tel./fax: þ86 7367186113. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.neucom.2014.07.020 0925-2312/& 2014 Elsevier B.V. All rights reserved.

dynamic properties of the neural cells in the real world, the existing neural network models in many cases cannot characterize the properties of a neural reaction process precisely. Thus, it is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamics for such complex neural reactions. Therefore, delayed neural networks of neutral type are considered to be more accordant with reality (see [5,8–11]). Moreover, neutral type CNNs with continuously distributed delays in leakage terms usually can be described as follows: Z 1 n x0i ðtÞ ¼  ci ðtÞ hi ðsÞxi ðt  sÞ dsþ ∑ aij ðtÞf j ðxj ðt  τij ðtÞÞÞ j¼1

0

n

n

j¼1

j¼1

þ ∑ a~ ij ðtÞf~ j ðx0j ðt  τ~ ij ðtÞÞÞþ ∑ bij ðtÞ n

þ ∑ b~ ij ðtÞ j¼1

Z

1 0

Z 0

1

K ij ðuÞg j ðxj ðt  uÞÞ du

K~ ij ðuÞg~ j ðx0j ðt  uÞÞ du þ I i ðtÞ;

i ¼ 1; 2; …; n: ð1:2Þ

On the other hand, as pointed out by Ait Dads and Ezzinbi in [12], it would be of great interest to study the dynamics of pseudo almost periodic systems with time delay. In addition, in view of the fact that many CNN models display sustained fluctuations, it is thus desirable to construct CNN models capable of producing pseudo almost periodic solution. This is modeled using the pseudo almost periodic notion and the related functions which allow complex repetitive phenomena to be represented as an almost periodic process plus an ergodic component. Furthermore, many researchers have focused their attention on the study of existence

446

B. Liu / Neurocomputing 148 (2015) 445–454

and stability of pseudo almost periodic solutions for delayed cellular neural networks (DCNNs) (see [13–18] and the references cited therein). In particular, the assumption of pseudo almost periodicity of the coefficient functions and the delays in CNNs models is a way of incorporating the time-dependent variability of the environment, especially when the various components of the environment are periodic with not necessarily commensurate periods (e.g. seasonal effects of weather, food supplies, mating habits, and harvesting). However, to the best of our knowledge, no such work has been performed on the dynamic analysis of pseudo almost periodic solution of CNNs (1.2). Motivated by the above discussions, the aim of this work is to prove the existence and global exponential stability of pseudo almost periodic solution for CNNs (1.2). Our approach is based on the exponential dichotomy theory and contraction mapping fixed point theorem developed in [19]. The initial conditions associated with system (1.2) are of the form xi ðsÞ ¼ φi ðsÞ;

x0i ðsÞ ¼ φ0i ðsÞ;

s A ð  1; 0; i ¼ 1; 2; …; n;

ð1:3Þ

where φi ðÞ and φ0i ðÞ are the real-valued bounded continuous functions defined on ð  1; 0. In the following part of this paper, given a bounded continuous function g defined on R, let g þ and g  be defined as g þ ¼ supjgðtÞj; tAR

g  ¼ inf jgðtÞj: tAR

For convenience, we denote by Rn (R ¼ R1 ) the set of all n–dimensional real vectors (real numbers). We will use x ¼ ðx1 ; x2 ; …; xn ÞT ¼ fxi g A Rn to denote a column vector. We let jxj denote the absolute-value vector given by jxj ¼ ðjx1 j; jx2 j; …; jxn jÞT , and define J x J ¼ max1 r i r n jxi j. A matrix or vector A Z 0 means that all entries of A are greater than or equal to zero. A 40 can be defined similarly. For matrices or vectors A and B, A Z B (resp. A 4 B) means that A  B Z 0 (resp. A  B 4 0).

0

Lemma 2.1. Let Bn ¼ ff jf ; f A PAPðR; RÞg equipped with the induced 0 norm defined by ‖f ‖Bn ¼ maxf‖f ‖1 ; ‖f ‖1 g ¼ maxfsupt A R jf ðtÞj; 0 n supt A R jf ðtÞjg, then B is a Banach space. Proof. This Lemma can be proven in a similar way to that in Lemma 2.3 of [14]. But for convenience of reading, we give the þ1 detail proof as follows. Assume that ff p gp ¼ 1 is a Cauchy sequence n in B , then for any ε 4 0, there exists NðεÞ 4 0 such that   0 0 ‖f p  f q ‖Bn ¼ max supjf p ðtÞ  f q ðtÞj; supjf p ðtÞ  f q ðtÞj o ε; t AR

ð2:1Þ By the definition of pseudo almost periodic function, let f p ¼ g p þ φp

where g p A APðR; RÞ; φp A PAP 0 ðR; RÞ and p ¼ 1; 2; …:

From Lemma 3.20 [20, p. 140], we obtain 0

f p ¼ g 0p þ φ0p

where g 0p A APðR; RÞ; φ0p A PAP 0 ðR; RÞ and p ¼ 1; 2; …:

On combining (2.1) with Lemma 5.2 [20, p.57], we deduce that fg p gpþ¼11 ; fg 0p gpþ¼11  APðR; RÞ are the Cauchy sequence, so that fφp gpþ¼11 ; fφ0p gpþ¼11  PAP 0 ðR; RÞ are also the Cauchy sequence. Firstly, we show that there exists g A APðR; RÞ such that g p uniformly converges to g, as p- þ 1. Note that fg p g is the Cauchy sequence in APðR; RÞ. 8 ε 4 0, (NðεÞ, such that 8 p; q Z NðεÞ jg p ðtÞ  g q ðtÞj o ε

for all t A R:

ð2:2Þ fg p ðtÞgpþ¼11

So for fixed t A R, it is easy to see that is the Cauchy number sequence. Thus, the limits of gp(t) exist as p- þ 1 and let gðtÞ ¼ limp- þ 1 g p ðtÞ. In (2.2), let q- þ 1, we have jgðtÞ g p ðtÞj rε

for all t A R; p Z NðεÞ:

ð2:3Þ

Thus, g p uniformly converges to g, as p- þ 1. Moreover, from Theorem 1.9 [21, p. 5], we obtain g A APðR; RÞ. Similarly, we also obtain that there exist g n A APðR; RÞ and φ; φn A BCðR; RÞ, such that jg n ðtÞ g 0p ðtÞj r ε;

2. Preliminary results

8 p; q Z NðεÞ:

t AR

jφðtÞ φp ðtÞj r ε;

jφn ðtÞ  φ0p ðtÞj r ε

for all t A R; p Z NðεÞ;

ð2:4Þ In this section, we shall first recall some basic definitions, lemmas which are used in what follows. In this paper, BCðR; Rn Þ denotes the set of bounded continued functions from R to Rn . Note that ðBCðR; Rn Þ; ‖  ‖1 Þ is a Banach space where ‖  ‖1 denotes the sup norm ‖f ‖1 ≔supt A R J f ðtÞ J . Definition 2.1 (see Zhang [20] and Fink [21]). Let uðtÞ A BCðR; Rn Þ. u (t) is said to be almost periodic on R if, for any ε 4 0, the set Tðu; εÞ ¼ fδ : J uðt þ δÞ  uðtÞ J o ε for all t A Rg is relatively dense, i.e., for any ε 40, it is possible to find a real number l ¼ lðεÞ 4 0 with the property that, for any interval with length lðεÞ, there exists a number δ ¼ δðεÞ in this interval such that J uðt þ δÞ  uðtÞ J oε; for all t A R. n

We denote by APðR; R Þ the set of the almost periodic functions from R to Rn . Precisely, define the class of functions PAP 0 ðR; Rn Þ as follows:  ) ( Z  1 r  f A BCðR; Rn Þ lim jf ðtÞj dt ¼ 0 :  r- þ 1 2r  r A function f A BCðR; Rn Þ is called pseudo almost periodic if it can be expressed as f ¼ h þ φ; where hA APðR; Rn Þ and φ A PAP 0 ðR; Rn Þ. The collection of such functions will be denoted by PAPðR; Rn Þ. The functions h and φ in above definition are respectively called the almost periodic component and the ergodic perturbation of the pseudo almost periodic function f.

which imply that g 0p ) g n ;

φ0p ) φn ;

φp ) φ;

where p- þ 1 and “ ) ” means the uniform convergence. Next, we claim that φ; φn A PAP 0 ðRÞ. Together with (2.4) and the facts that Z 1 r 0 jφp ðsÞj ds ¼ 0; lim jφp ðsÞj ds ¼ 0; p ¼ 1; 2; …; rþ 1 2r r r Z r Z 1 1 1 r jφðsÞj ds r jφðsÞ φp ðsÞj ds þ jφ ðsÞj ds; r 40; p ¼ 1; 2; …; 2r  r 2r  r 2r  r p lim

r- þ 1 Z r

1 2r

Z

r

and 1 2r

Z

r r

jφn ðsÞj ds r

we have 1 r- þ 1 2r

Z

r

lim

1 2r

Z

r r

jφn ðsÞ  φ0p ðsÞj ds þ

jφðsÞj ds ¼ 0;

r

1 r- þ 1 2r

1 2r

Z

Z

r r

r

lim

jφ0p ðsÞj ds;

r 4 0; p ¼ 1; 2; …;

jφn ðsÞj ds ¼ 0:

r n

Hence φ; φn A PAP 0 ðRÞ. Let f ¼ g þ φ, f ¼ g n þ φn , then f ¼ g þ φ A n 0 n PAPðRÞ, f ¼ g n þ φn A PAPðRÞ and f p ) f , f p ) f as p- þ 1. 0 n Finally, we reveal f ¼ f . For t; Δt A R, it follows that Z t þ Δt 0 f p ðsÞ ds ð2:5Þ f p ðt þ ΔtÞ  f p ðtÞ ¼ t

0

In view of the uniform convergence of fp and f p , let p- þ 1 for (2.5), we get Z t þ Δt n f ðt þ ΔtÞ  f ðtÞ ¼ f ðsÞ ds; t

B. Liu / Neurocomputing 148 (2015) 445–454

which implies that R t þ Δt n f ðsÞ ds f ðt þ ΔtÞ  f ðtÞ n 0 ¼ f ðtÞ: ¼ lim f ðtÞ ¼ lim t Δt-0 Δt-0 Δt Δt

and ð2:6Þ

Therefore, in view of (2.3), (2.4) and (2.6), we obtain that the þ1 Cauchy sequence ff p gp ¼ 1  Bn satisfies ‖f p f ‖Bn ⟶0ðp- þ 1Þ; and f A Bn . This yields that Bn is a Banach space. The proof is completed. □

~ jf~ j ðuÞ  f~ j ðvÞj r Lfj ju vj;

0

þ

Bnn ¼ fφjφ A Bn ; φ and φ0 are uniformly continuous on Rg: and Z ci

Proof. Suppose that fxp gpþ¼11 D Bn satisfies ð2:7Þ

p- þ 1

It follows from Lemma 2.1 that x A Bn . We next show that x and x0 are uniformly continuous on R. 8 ε 4 0, from (2.7), we can choose p 4 0 such that   ε ð2:8Þ ‖xp x‖Bn ¼ max supjxp ðtÞ  xðtÞj; supjx0p ðtÞ  x0 ðtÞj o : 3 tAR t AR Note that xp and x0p are uniformly continuous on R. Then, there exists δ ¼ δðεÞ such that ε jxp ðt 1 Þ  xp ðt 2 Þj o ; 3

jx0p ðt 1 Þ x0p ðt 2 Þj o

ε 3

where t 1 ; t 2 AR and jt 1  t 2 j o δ;

which, together with (2.8), implies that jxðt 1 Þ xðt 2 Þj r jxðt 1 Þ  xp ðt 1 Þj þjxp ðt 1 Þ  xp ðt 2 Þj þ jxp ðt 2 Þ  xðt 2 Þj ε ε ε o þ þ 3 3 3 ¼ ε; jx0 ðt 1 Þ  x0 ðt 2 Þj r jx0 ðt 1 Þ  x0p ðt 1 Þj þ jx0p ðt 1 Þ  x0p ðt 2 Þj þjx0p ðt 2 Þ  x0 ðt 2 Þj ε ε ε o þ þ 3 3 3 ¼ ε; where t 1 , t 2 A R and jt 1  t 2 j o δ. Hence, x and x0 are uniformly continuous on R, and Bnn is a closed subset of Bn . This completes the proof of Lemma 2.2. □ 0

Remark 2.1. Let Bnnn ¼ ff jf ; f A PAPðR; Rn Þg, and

1 0

1

hi ðvÞ 0

þ

jK ij ðuÞj du Lgj þ b~ ij

dv  αi þ ciþ

Z

Z

1 0

~

1

hi ðvÞ dv 1  0

!

jK~ ij ðuÞj du Lgj ξj r  αi ;

ci

! αi R1 o 1: 0 hi ðvÞ dv

Lemma 2.3. Assume that assumptions ðA1 Þ and ðA2 Þ hold. Then, for φ A PAPðR; RÞ, the function Z 1 K ij ðuÞg j ðφðt  uÞÞ du A PAPðR; RÞ; i; j ¼ 1; 2; …; n: 0

Proof. Let φ A PAPðR; RÞ. Obviously, ðA1 Þ implies that g j is a uniformly continuous function on R. By using Corollary 5.4 in [20, p. 58], we immediately obtain g j ðφðtÞÞ ¼ χ j1 ðtÞ þ χ j2 ðtÞ A PAPðR; RÞ; where χ j1 A APðR; RÞ, χ j2 A PAP 0 ðR; RÞ, j ¼ 1; 2; …; n. Then, for any ε 4 0, it is possible to find a real number l ¼ lðεÞ 40, for any interval with length l, there exists a number τ ¼ τðεÞ in this interval such that ε R1 jχ j1 ðt þ τÞ  χ j1 ðtÞj o ; 8 t A R; i; j ¼ 1; 2; …; n; 1 þ 0 jK ij ðuÞj du and

and

j¼1

0

Z

bij

Then, Bnn is a closed subset of Bn .

~

jg~ j ðuÞ  g~ j ðvÞj r Lgj ju  vj;

for all u, v A R. ðA2 Þ For i; j A f1; 2; …; ng, the delay kernels K ij ; K~ ij : ½0; þ 1Þ-R are continuous, jK ij ðtÞjeκt and jK~ ij ðtÞjeκt are integrable on ½0; þ 1Þ. ðA3 Þ For each i A f1; 2; …; ng, there exist constants αi 4 0 and ξi 4 0, such that Z 1 Z 1 n þ ~ hi ðvÞ dv þciþ vhi ðvÞ dv þξi 1 ∑ aijþ Lfj þ a~ ij Lfj  ci

Lemma 2.2. Let

lim ‖xp  x‖Bn ¼ 0:

447

1 r- þ 1 2r

Z

r

lim

r

jχ j2 ðvÞj dv ¼ 0;

j ¼ 1; 2; …; n:

It follows that Z 1 Z 1   K ij ðuÞχ j1 ðt þ τ  uÞ du  K ij ðuÞχ j1 ðt  uÞ duj  0 0 Z 1 r jK ij ðuÞjjχ j1 ðt þ τ  uÞ  χ j1 ðt  uÞj du Z0 1 ε R1 jK ij ðuÞj du o 1 þ 0 jK ij ðuÞj du 0 o ε; 8 t A R; i; j ¼ 1; 2; …; n;

0

B ¼ ff jf ¼ ff i gA Bnnn ; f i and f i are uniformly continuous on R; i ¼ 1; 2; …; ng:

It follows from that Bnnn is a Banach space, and B is a closed subset of Bnnn . Thus, B is also a Banach space equipped with the induced 0 norm defined by ‖f ‖B ¼ maxf‖f ‖1 ; ‖f ‖1 g ¼ maxfsupt A R J f ðtÞ J ; 0 supt A R J f ðtÞ J g. For i, j ¼ 1; 2; …; n, it will be assumed that hi : R-½0; þ 1Þ is a R1 continuous function with 0 o 0 hi ðvÞ dv o þ 1 and R1 κv vh ðvÞe dv o þ 1 for a certain positive constant κ, ci : i 0 R-ð0; þ 1Þ is an almost periodic function, τij , τ~ ij : R-½0; þ1Þ and I i , aij , bij , a~ ij , b~ ij : R-R are uniform continuous and pseudo almost periodic on R. We also make the following assumptions which will be used later. ðA1 Þ For each j A f1; 2; …; ng, there exist nonnegative constants ~ ~ Lfj , Lfj , Lgj and Lgj such that jf j ðuÞ  f j ðvÞj r Lfj ju  vj;

jg j ðuÞ  g j ðvÞj rLgj ju vj;

and

Z   1    K ij ðuÞχ j2 ðv  uÞ du dv    r 0 Z r Z 1 1 r lim jK ij ðuÞjjχ j2 ðv  uÞj du dv r- þ 1 2r  r 0 Z 1 Z r 1 jK ij ðuÞj jχ j2 ðv  uÞj dv du ¼ lim r- þ 1 2r 0 r Z 1 Z ru 1 jK ij ðuÞj jχ j2 ðzÞj dz du ¼ lim r- þ 1 2r 0 r u Z 1 Z r þu 1 r lim jK ij ðuÞj jχ j2 ðzÞj dz du r- þ 1 2r 0   r u Z r þu Z 1 1 1 r lim jK ij ðuÞj 1 þ u jχ ðzÞj dz du r- þ 1 0 r 2ðr þ uÞ  r  u j2 Z rþu Z 1 1 r lim jK ij ðuÞjeð1=rÞu jχ ðzÞj dz du r- þ 1 0 2ðr þ uÞ  r  u j2

1 r- þ 1 2r lim

Z

r

448

B. Liu / Neurocomputing 148 (2015) 445–454

Z r lim

r- þ 1

¼0

1

jK ij ðuÞjeκu

0

1 2ðr þ uÞ

Z

rþu ru

jχ j2 ðzÞj dz du

1 where r 4 ; i; j ¼ 1; 2; …; n: κ

Thus, Z 1 K ij ðuÞχ j1 ðt  uÞ du A APðR; RÞ; 0

Z

1

K ij ðuÞχ j2 ðt  uÞ duA PAP 0 ðR; RÞ;

0

which yield Z 1 K ij ðuÞg j ðφðt  uÞÞ du A PAPðR; RÞ;

Let φ A B. Obviously, ðA1 Þ implies that f j is uniformly continuous functions on R for j ¼ 1; 2; …; n. Set f ðt; zÞ ¼ φj ðt  zÞ ðj A f1; 2; …; ngÞ. By Theorem 5.3 in [20, p. 58] and Definition 5.7 in [20, p. 59], we can obtain that f A PAPðR  ΩÞ and f is continuous in z A K and uniformly in t A R for all compact subset K of Ω  R. This, together with τij A PAPðR; RÞ and Theorem 5.11 in [20, p. 60], implies that φj ðt  τij ðtÞÞ A PAPðR; RÞ;

Again from Corollary 5.4 in [14, p. 58], we have i; j ¼ 1; 2; …; n:

0

The proof of Lemma 2.3 is completed.

f j ðξj φj ðt  τij ðtÞÞÞA PAPðR; RÞ;

□

Z

Z

1

hi ðsÞ 0

Definition 2.2. Let xn ðtÞ ¼ ðxn1 ðtÞ; xn2 ðtÞ; …; xnn ðtÞÞT be the pseudo almost periodic solution of system (1.2). If there exist constants α 4 0 and M 4 1 such that for every solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; …; xn ðtÞÞT of system (1.2) with any initial value φðtÞ ¼ ðφ1 ðtÞ; φ2 ðtÞ; …; φn ðtÞÞT satisfying (1.3) ‖xðtÞ  xn ðtÞ‖1 ¼ max fmaxfjxi ðtÞ  xni ðtÞj; jx0i ðtÞ  x0in ðtÞjgg rM 1rirn

8 t 4 0;

t ts

φ0i ðuÞ du ds ¼

Z

1

hi ðsÞ ds φi ðtÞ 

0

1

hi ðsÞφi ðt  sÞ ds APAPðR; RÞ;

0

for all i¼ 1,2,…,n. By combining (3.1)–(3.3) with Lemma 2.3 and Remark 2.2, we obtain Z 1 Z t n ci ðtÞ hi ðsÞ φ0i ðuÞ du dsþ ξi 1 ∑ aij ðtÞf j ðξj φj ðt  τij ðtÞÞÞ ts

0

n

∑

j¼1

j¼1

a~ ij ðtÞf~ j ðξj φ0j ðt  τ~ ij ðtÞÞÞ þ ξi 1

ðξj φj ðt  uÞÞ du Z n þ ξi 1 ∑ b~ ij ðtÞ j¼1

3. Main results

Z

ð3:3Þ

þ ξi 1

where ‖φ  xn ‖0 ¼ maxfsupt r 0 max1 r i r n jφi ðtÞ  xni ðtÞj; supt r 0 max1 r i r n jφ0i ðtÞ  x0in ðtÞjg. Then xn ðtÞ is said to be globally exponentially stable.

1 0

Z

n

∑ bij ðtÞ

j¼1

Theorem 3.1. Let ðA1 Þ, ðA2 Þ and ðA3 Þ hold. Then, there exists a unique continuously differentiable pseudo almost periodic solution of system (1.2) in B. Proof. Set

K ij ðuÞg j

0

K~ ij ðuÞg~ j ðξj φ0j ðt  uÞÞ du þ ξi 1 I i ðtÞ A PAP

For any φ A B, we consider the pseudo almost periodic solution xφ ðtÞ of the following auxiliary pseudo almost periodic differential equations: Z 1 Z 1 Z t x 0i ðtÞ ¼  ci ðtÞ hi ðsÞ ds x i ðtÞ þ ci ðtÞ hi ðsÞ φ0i ðuÞ du ds þ ξi 1

0 n

t s

0

∑ aij ðtÞf j ðξj φj ðt  τij ðtÞÞÞ

j¼1 n

n

j¼1

j¼1

þ ξi 1 ∑ a~ ij ðtÞf~ j ðξj φ0j ðt  τ~ ij ðtÞÞÞ þ ξi 1 ∑ bij ðtÞ

x i ðtÞ ¼ ξi 1 xi ðtÞ; then we can transform (1.2) into the following system: Z 1 n x 0i ðtÞ ¼  ci ðtÞ hi ðsÞx i ðt  sÞ ds þ ξi 1 ∑ aij ðtÞf j ðξj x j ðt  τij ðtÞÞÞ j¼1

0

n

n

j¼1

j¼1

þ ξi 1 ∑ a~ ij ðtÞf~ j ðξj x 0j ðt  τ~ ij ðtÞÞÞ þξi 1 ∑ bij ðtÞ ðξj x j ðt  uÞÞ du Z n þ ξi 1 ∑ b~ ij ðtÞ j¼1

1

0

Z

Z

1

hi ðsÞ ds x i ðtÞ þ ci ðtÞ 0 n

Z

1

K ij ðuÞg j

0

K~ ij ðuÞg~ j ðξj x 0j ðt  uÞÞ du þ ξi 1 I i ðtÞ

1

¼  ci ðtÞ

1

ðR; RÞ; i ¼ 1; 2; …; n:

In this section, we establish sufficient conditions on the existence and global exponential stability of pseudo almost periodic solutions of (1.2).

hi ðsÞ 0

ts

x 0i ðuÞ

j¼1 n

n

j¼1

j¼1

∑ a~ ij ðtÞf~ j ðξj x 0j ðt  τ~ ij ðtÞÞÞ þξi 1 ∑ bij ðtÞ

ðξj x j ðt  uÞÞ du Z n þ ξi 1 ∑ b~ ij ðtÞ j¼1

i ¼ 1; 2; …; n:

0

1

ðξj φj ðt  uÞÞ du Z n þ ξi 1 ∑ b~ ij ðtÞ j¼1

i ¼ 1; 2; …; n:

1 0

Z

1 0

K ij ðuÞg j

K~ ij ðuÞg~ j ðξj φ0j ðt  uÞÞ du þ ξi 1 I i ðtÞ; ð3:4Þ

R1

Then, notice that M½ci ðtÞ 0 hi ðsÞ ds 40, i ¼1, 2, …, n, it is easy to show that the linear system Z 1 x 0i ðtÞ ¼  ci ðtÞ hi ðsÞ ds x i ðtÞ; i ¼ 1; 2; …; n; ð3:5Þ 0

t

du ds

admits an exponential dichotomy on R. Thus, by Theorem 2.2 in [13], we obtain that the system (3.4) has exactly one pseudo almost periodic solution:

1

xφ ðtÞ ¼ ðxφ1 ðtÞ; xφ2 ðtÞ; …; xφn ðtÞÞT

þ ξi 1 ∑ aij ðtÞf j ðξj x j ðt  τij ðtÞÞÞ þ ξi 1

ð3:1Þ

ð3:2Þ f~ j ðξj φ0j ðt  τ~ ij ðtÞÞÞ A PAPðR; RÞ; i; j ¼ 1; 2; …; n: R1 R1 In view of 0 hi ðvÞ dv o þ 1 and 0 vhi ðvÞ dv o þ 1, by using a similar argument as in proof of Lemma 2.3, we can show

0

Z

i; j ¼ 1; 2; …; n:

Similarly, from the uniform continuity of f~ j and φ0j , we obtain

Remark 2.2. Assume that assumptions ðA1 Þ and ðA2 Þ hold. Then, for φ0 A PAPðR; RÞ, by a similar argument as that in the proof of Lemma 2.3, we can get Z 1 K~ ij ðuÞg~ j ðφ0 ðt  uÞÞ du A PAPðR; RÞ; i; j ¼ 1; 2; …; n:

‖φ xn ‖0 e  αt ;

i; j ¼ 1; 2; …; n:

Z 0

K ij ðuÞg j

K~ ij ðuÞg~ j ðξj x 0j ðt  uÞÞ du þ ξi 1 I i ðtÞ;

¼ fxφi ðtÞg Z t  Z Rt R1  c ðuÞ hi ðvÞ dv du 0 e s i ci ðsÞ ¼ 1

Z

1

hi ðvÞ 0

n

n

j¼1

j¼1

s

sv

φ0i ðuÞ du dv

þ ξi 1 ∑ aij ðsÞf j ðξj φj ðs τij ðsÞÞÞ þ ξi 1 ∑ a~ ij ðsÞf~ j ðξj φ0j ðs  τ~ ij ðsÞÞÞ

B. Liu / Neurocomputing 148 (2015) 445–454

þ ξi 1 þ ξi 1

Z

n

∑ bij ðsÞ

j¼1 n

∑ b~ ij ðsÞ

Z

j¼1

 ¼ jðTðφðtÞÞ  TðψðtÞÞÞi j  φ ¼ jðx ðtÞÞi  ðxψ ðtÞÞi j Z t  Z Rt R1   c ðuÞ hi ðvÞ dv du 0 ¼  e s i ci ðsÞ

1

K ij ðuÞg j ðξj φj ðs  uÞÞ du

0 1

0

# ) 0  1 ~ K ij ðuÞg~ j ðξj φj ðs  uÞÞ du þξi I i ðsÞ ds :

1

Z

1

hi ðvÞ

s

sv

0

ðφ0i ðuÞ  ψ 0i ðuÞÞ

n

ð3:6Þ

 du dv þ ξi 1 ∑ aij ðsÞðf j ðξj φj ðs  τij ðsÞÞÞ  f j ðξj ψ j ðs  τij ðsÞÞÞÞ j¼1

Furthermore, ðxφi ðtÞÞ0

449

n

"

Z

þ ξi 1 ∑ a~ ij ðsÞðf~ j ðξj φ0j ðs  τ~ ij ðsÞÞÞ  f~ j ðξj ψ 0j ðs  τ~ ij ðsÞÞÞÞ

Z

1

¼ ci ðtÞ

hi ðvÞ

t

n

tv

0

j¼1

φ0i ðuÞ du dv þ ξi 1 ∑ aij ðtÞf j ðξj φj ðt  τij ðtÞÞÞ

n

n

þ ξi 1 ∑ a~ ij ðtÞf~ j ðξj φ0j ðt  τ~ ij ðtÞÞÞþ ξi 1 ∑ bij ðtÞ j¼1

j¼1

 ðξj φj ðt  uÞÞ du Z n þ ξi 1 ∑ b~ ij ðtÞ j¼1

Z

1

Z

 ci ðtÞ

t

hi ðvÞdv

e

Z

s

n

∑

j¼1

þ ξi 1 ∑ bij ðsÞ j¼1 n

þ ξi 1 ∑ b~ ij ðsÞ j¼1

Z

hi ðvÞ 0 n

∑

j¼1

Z

K ij ðuÞg j

þ ξi 1 ∑ b~ ij ðsÞ

#

Z r

ci ðuÞ

0

" hi ðvÞ dv du

Z

e

 ci ðtÞ 0

tv

ci ðsÞ

Z

1

hi ðvÞ

s

jφ0i ðuÞ

sv

~

Z

n

þ ξi 1 ∑ jbij ðsÞj j¼1

j¼1

Z

j¼1

(Z

# K~ ij ðuÞg~ j ðξj φ0j ðs  uÞÞ du þ ξi 1 I i ðsÞ ds

r

t

Z

∑ bij ðtÞ

j¼1

K ij ðuÞg j

Z

#

t

¼

e Z  αi

j¼1

j¼1

0

K~ ij ðuÞg~ j ðξj φ0j ðt  uÞÞ du þξi 1 I i ðtÞ

þ þ bij

Z

ciþ

Z

1 0

n

vhi ðvÞ dv þ ξi 1 ∑

1

jK ij ðuÞj du 0

þ Lgj þ b~ ij

t

e



1



t

e

Rt s

Z

j¼1

1

0 ci ðuÞ

jK~ ij ðuÞj

R1 0

hi ðvÞ dv du



Rt s



ci ðuÞ

Rt s

R1

hi ðvÞ dv du

0

ci ðuÞ

R1 0

 Z d 

t

Z ci ðuÞ

s

hi ðvÞ dv du

1

0

1

 hi ðvÞ dv du )

 ds ‖φðtÞ  ψðtÞ‖B

   Z t Rt þ R1  c h ðvÞ dv du e s i 0 i ds ‖φðtÞ ψ ðtÞ‖B 1  αi 1 ( ) ! αi ¼ 1 þ R 1 ‖φðtÞ ψ ðtÞ‖B ; ci 0 hi ðvÞ dv

xφi ðtÞ

K ij ðuÞg j

" hi ðvÞ dv du

hi ðvÞ dv  αi Þ ds‖φðtÞ  ψðtÞ‖B

1

1

0

1

0

Z

ð3:7Þ

Z

R1

þ ~ þ a~ ij Lfj

aijþ Lfj

 ðci ðsÞ

K~ ij ðuÞg~ j ðξj φ0j ðt  uÞÞ du þ ξi 1 I i ðtÞ

j¼1

ci ðuÞ

)

~

Z

~  du Lgj ξj ds ‖φðtÞ  ψðtÞ‖B g r

1

0

s

0

#

jK~ ij ðuÞjLgj ξj jφ0j ðs uÞ ψ 0j ðs  uÞj du ds

#

j¼1 n

Rt

1

jK ij ðuÞjLgj ξj jφj ðs  uÞ  ψ j ðs  uÞj du

1



φ0i ðuÞ du dv þ ξi 1 ∑ aij ðtÞf j ðξj φj ðt  τij ðtÞÞÞ



e 

n

n

1 0

þ ξi 1 ∑ jb~ ij ðsÞj

∑ a~ ij ðtÞf~ j ðξj φ0j ðt  τ~ ij ðtÞÞÞ þ ξi 1 ∑ bij ðtÞ

0

 Z ci ðsÞ 0

n

∑ a~ ij ðsÞf~ j ðξj φ0j ðs  τ~ ij ðsÞÞÞ

n

j¼1

hi ðvÞ dv du

j¼1

hi ðvÞ

is bounded on R, and is uniformly which yields that continuous on R, where i ¼ 1; 2; …; n. From ðA1 Þ, ðA2 Þ and the uniform continuity of ci , aij , bij , τij , a~ ij , b~ ij , τ~ ij , I i , φj , φ0j ði; j ¼ 1; 2; …; nÞ, we can show that Z 1 Z t n ci ðtÞ hi ðvÞ φ0i ðuÞ du dv þ ξi 1 ∑ aij ðtÞf j ðξj φj ðt  τij ðtÞÞÞ

1

0

 ψ 0i ðuÞj du dv þ ξi 1 ∑ jaij ðsÞjLfj ξj jφj ðs  τij ðsÞÞ  ψ j ðs  τij ðsÞÞj

1

hi ðvÞ dv xφi ðtÞ;

 ðξj φj ðt  uÞÞ du Z n þξi 1 ∑ b~ ij ðtÞ

R1

n

t

t v

ci ðuÞ

n

Z

n

ðxφi ðtÞÞ0

0

s

0

j¼1

0

0



) #   K~ ij ðuÞðg~ j ðξj φ0j ðs uÞÞ  g~ j ðξj ψ 0j ðs  uÞÞÞ du  ds 

1

1

0

a~ ij ðtÞf~ j ðξj φ0j ðt  τ~ ij ðtÞÞÞþ ξi 1

j¼1 Z 1

Rt

K ij ðuÞðg j ðξj φj ðs  uÞÞ  g j ðξj ψ j ðs  uÞÞÞ du

þ ξi 1 ∑ ja~ ij ðsÞjLfj ξj jφ0j ðs  τ~ ij ðsÞÞ  ψ 0j ðs  τ~ ij ðsÞÞj

1

1

t

1

0

Z

j¼1

K ij ðuÞg j ðξj φj ðs  uÞÞ du

0

 ðξj φj ðt  uÞÞ du Z n þ ξi 1 ∑ b~ ij ðtÞ

þξi 1

n

1

Z

1

¼ ci ðtÞ þ ξi 1

s

R1

aij ðsÞf j ðξj φj ðs  τij ðsÞÞÞ þ ξi 1

n

"

0

Z

j¼1

1

φ0i ðuÞ du dv

sv

þ ξi 1



Rt

1

0



Z

K~ ij ðuÞg~ j ðξj φ0j ðt  uÞÞ du þ ξi 1 I i ðtÞ

0

1

n

þ ξi 1 ∑ bij ðsÞ

j¼1

r

ð3:8Þ

and jT 0 ðφðtÞÞ  T 0 ðψðtÞÞj  ¼ jðT 0 ðφðtÞÞ  T 0 ðψðtÞÞÞi j  Z 1 Z t  ¼  ci ðtÞ hi ðvÞ ðφ0i ðuÞ ψ 0i ðuÞÞ dud v tv

0

n

þ ξi 1 ∑ aij ðtÞðf j ðξj φj ðt  τij ðtÞÞÞ f j ðξj ψ j ðt  τij ðtÞÞÞÞ j¼1

is uniformly continuous on R. This, together with (3.7) and the uniform continuity of xφi , implies that ðxφi ðtÞÞ0 is uniformly continuous on R. Thus, by Corollary 5.6 in [14, p. 59], we get that ðxφi ðtÞÞ0 is a pseudo almost periodic function, and xφ A B. Now, we define a mapping T : B-B by setting φ

TðφÞðtÞ ¼ x ðtÞ;

8 φ A B:

We next prove that the mapping T is a contraction mapping of the B. In fact, in view of (3.6), (A1) and (A3), for φ, ψ A B, we have jTðφðtÞÞ  Tðψ ðtÞÞj

n

þ ξi 1 ∑ a~ ij ðtÞðf~ j ðξj φ0j ðt  τ~ ij ðtÞÞÞ  f j ðξj ψ 0j ðt  τ~ ij ðtÞÞÞÞ j¼1 n

þ ξi 1 ∑ bij ðtÞ j¼1 n

þ ξi 1 ∑ b~ ij ðtÞ j¼1

Z

1

 ci ðtÞ 0

Z

1

K ij ðuÞðg j ðξj φj ðt  uÞÞ  g j ðξj ψ j ðt  uÞÞÞ du

0

Z 0

1

# K~ ij ðuÞðg~ j ðξj φ0j ðt uÞÞ  g~ j ðξj ψ 0j ðt  uÞÞÞ du

#)   hi ðvÞ dv ½ðxφ ðtÞÞi ðxψ ðtÞÞi  

450

B. Liu / Neurocomputing 148 (2015) 445–454

(" ciþ

r

Z

1

vhi ðvÞ 0

dv þ ξi 1

n

∑



j¼1

aijþ Lfj

n

þ ξi 1 ∑ b~ ij ðtÞ

~

þ þ a~ ij Lfj

 ~ jK~ 1j ðuÞj du Lgj ξj ‖φðtÞ 0 0 ! ) Z 1 αi þ ‖φðtÞ  ψðtÞ‖B hi ðvÞ dv 1  þ R 1  ψðtÞ‖B þ ci ci 0 hi ðvÞdv 0 ( Z ! Z 1 1 αi  þ r ci hi ðvÞ dv  αi þ ci hi ðvÞ dv 1  þ R 1 ci 0 hi ðvÞdv 0 0  ‖φðtÞ  ψðtÞ‖B : ð3:9Þ Z

þ þ bij

1

jK 1j ðuÞj du

þ Lgj þ b~ ij

Z

1

j¼1

Z

(

K ¼ max ( ci

( max

1rirn

Z

1

0

ciþ

0 n

Z

Z

1

hi ðsÞ 0

t

ts

y0i ðuÞ du ds

þ ξi 1 ∑ aij ðtÞðf j ðxj ðt  τij ðtÞÞÞ f j ðxnj ðt  τij ðtÞÞÞÞ j¼1 n

þ ξi 1 ∑ a~ ij ðtÞðf~ j ðx0j ðt  τ~ ij ðtÞÞÞ  f~ j ðxnj 0 ðt  τ~ ij ðtÞÞÞÞ j¼1 n

þ ξi 1 ∑ bij ðtÞ j¼1 n

j¼1

Z

1

K ij ðuÞðg j ðxj ðt  uÞÞ  g j ðxnj ðt  uÞÞÞ du

0

Z

1

K~ ij ðuÞðg~ j ðx0j ðt uÞÞ  g~ j ðxjn0 ðt  uÞÞÞ du;

0

ð3:11Þ

α R1 i ; max 1rirn 0 hi ðvÞdv

hi ðvÞ dv  αi þ ciþ

Z

1

hi ðvÞ dv 1  0

ciþ

!)) α R1 i o 1; 0 hi ðvÞ dv

where i ¼ 1; 2; …; n. Define continuous functions Γ i ðωÞ and Π i ðωÞ by setting Z 1 Z 1 n hi ðvÞ dv þ ωþ ciþ vhi ðvÞeωv dv þξi 1 ∑ aijþ Γ i ðωÞ ¼ ci 0

n

and Π i ðωÞ ¼

1þ

Theorem 3.2. Suppose that all conditions in Theorem 3.1 are satisfied. Moreover, assume that ! R1 Z 1 n ciþ 0 hi ðvÞ dv vhi ðvÞ dv þ ξi 1 ∑ aijþ Lfj ξj þ ξi 1 1þ  R 1 ciþ ci 0 hi ðvÞ dv 0 j¼1 Z 1 Z 1 n n n þ þ þ f~ 1 ~ ∑ bij jK ij ðuÞj du Lgj ξj þξi 1 ∑ b~ ij  ∑ a ij Lj ξj þ ξi j¼1

i ¼ 1; 2; …; n:

ωτijþ

~

j¼1

Z

1

jK ij ðuÞj 0

jK~ ij ðuÞjeωu du Lgj ξj ; ~

n

þ

j¼1

Z

j¼1

Z

j¼1

1 0

1

jK ij ðuÞjeωu

0

# jK~ ij ðuÞjeωu du Lgj ξj ; ~

where t 4 0, ω A ½0; κ, i¼ 1, 2, …, n. Then, from ðA3 Þ and (3.10), we have Z 1 Z 1 n Γ i ð0Þ ¼  ci hi ðvÞ dv þciþ vhi ðvÞ dv þξi 1 ∑ aijþ Lfj ξj 0

þ ξi 1

0

n

∑

j¼1

þ ~ a~ ij Lfj ξj þ ξi 1

n

þ þ ξi 1 ∑ b~ ij

Z

j¼1

r  αi o0; and Π i ð0Þ ¼

1 0

n

∑

j¼1

j¼1

Z

þ bij

1 0

jK ij ðuÞj du Lgj ξj

jK~ ij ðuÞj du Lgj ξj ~

i ¼ 1; 2; …; n;

!" Z R1 1 ciþ 0 hi ðvÞ dv 1þ  R 1 vhi ðvÞ dv ciþ ci 0 hi ðvÞ dv 0 n

n

~

þ þ ξi 1 ∑ aijþ Lfj ξj þ ξi 1 ∑ a~ ij Lfj ξj j¼1

þ ξi 1

T

yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞ; …; yn ðtÞÞ

¼ ðξ1 1 ðx1 ðtÞ  xn1 ðtÞÞ; ξ2 1 ðx2 ðtÞ  xn2 ðtÞÞ; …; ξn 1 ðxn ðtÞ  xnn ðtÞÞÞT :

n

n

ð3:10Þ

Proof. By Theorem 3.1, (1.2) has a unique continuously differentiable pseudo almost periodic solution xn ðtÞ ¼ ðxn1 ðtÞ; xn2 ðtÞ; …; xnn ðtÞÞT . Suppose that xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; …; xn ðtÞÞT is an arbitrary solution of (1.2) associated with initial value φðtÞ ¼ ðφ1 ðtÞ; φ2 ðtÞ; …; φn ðtÞÞT satisfying (1.3). Let

0

j¼1

ω~τ þ þ þ ξi 1 ∑ a~ ij Lfj ξj e ij þ ξi 1 ∑ bij

þ du Lgj ξj þ ξi 1 ∑ b~ ij

0

Then system (1.2) has at least one pseudo almost periodic solution xn ðtÞ. Moreover, xn ðtÞ is globally exponentially stable.

1

þ ξi 1 ∑

þ bij

!" Z R1 1 n ciþ 0 hi ðvÞ dv R vhi ðvÞeωv dv þ ξi 1 ∑ aijþ ciþ 1  ci 0 hi ðvÞ dv  ω 0 j¼1

Lfj ξj e

By (3.4) and (3.6), xnn satisfies (3.4). So (1.2) has a unique continuously differentiable almost periodic solution xn ¼ ðξ1 xnn 1 ðtÞ; T nn ξ2 xnn 2 ðtÞ; …; ξn xn ðtÞÞ . The proof of Theorem 3.1 is now completed. □

! g~ ~  jK ij ðuÞj du Lj ξj o 1;

Z

j¼1

Txnn ¼ xnn :

0

j¼1

n

ω~τ þ þ ~ a~ ij Lfj ξj e ij

þ eωu du Lgj ξj þ ξi 1 ∑ b~ ij

which implies that the mapping T : B⟶B is a contraction mapping. According to the fact that B is also a Banach space, we obtain that the mapping T possesses a unique fixed point

j¼1

þ ξi 1 ∑

Lfj ξj e

‖TðφðtÞÞ  TðψðtÞÞ‖B r K‖φðtÞ  ψðtÞ‖B ;

T nn nn xnn ¼ ðxnn 1 ðtÞ; x2 ðtÞ; …; xn ðtÞÞ A B;

0

n

ωτijþ

which, together with (3.6) and (3.7), yield

j¼1

K~ ij ðuÞðg~ j ðx0j ðt uÞÞ  g~ j ðxjn0 ðt  uÞÞÞ du

0

hi ðsÞ ds yi ðtÞ þ ci ðtÞ

þ ξi 1 ∑ b~ ij ðtÞ

)

1

1

1

¼  ci ðtÞ

From ðA3 Þ, we have α o 1; 0o1 þ R 1 i ci 0 hi ðvÞ dv and

Z

n

∑

j¼1

þ bij

du

1 0

#

~ Lgj ξj

j¼1

Z

o 1;

n

þ jK ij ðuÞj du Lgj ξj þξi 1 ∑ b~ ij j¼1

Z

1

jK~ ij ðuÞj

0

i ¼ 1; 2; …; n;

Then y0i ðtÞ ¼  ci ðtÞ

Z

1

n

hi ðsÞyi ðt  sÞ ds þξi 1 ∑ aij ðtÞðf j ðxj ðt  τij ðtÞÞÞ f j ðxnj ðt  τij ðtÞÞÞÞ j¼1

0

n

þ ξi 1 ∑ a~ ij ðtÞðf~ j ðx0j ðt  τ~ ij ðtÞÞÞ  f~ j ðxjn0 ðt  τ~ ij ðtÞÞÞÞ j¼1 n

þ ξi 1 ∑ bij ðtÞ j¼1

Z

1 0

K ij ðuÞðg j ðxj ðt  uÞÞ  g j ðxnj ðt  uÞÞÞ du

which, together with the continuity of Γ i ðÞ and Π i ðÞ, imply that we can choose a constant λ A ð0; minfκ; min1 r i r n ci gÞ such that Z 1 Z 1 n λτ þ Γ i ðλÞ ¼  ci hi ðvÞ dv þ λ þ ciþ vhi ðvÞeλv dv þ ξi 1 ∑ aijþ Lfj ξj e ij 0

j¼1

0

n

~

þ

n

λ~τ þ þ þ ξi 1 ∑ a~ ij Lfj ξj e ij þ ξi 1 ∑ bij j¼1

j¼1

Z 0

1

jK ij ðuÞjeλu du Lgj ξj

B. Liu / Neurocomputing 148 (2015) 445–454 n

þ þ ξi 1 ∑ b~ ij

Z

j¼1

¼ ðci

and Π i ðλÞ ¼

Z

1

0

~

hi ðvÞ dv  λÞ

0

!" Z R1 1 n ciþ 0 hi ðvÞ dv R vhi ðvÞeλv dv þ ξi 1 ∑ aijþ ciþ 1  ci 0 hi ðvÞ dv  λ 0 j¼1 Z 1 n n λτijþ λ~τ ijþ þ þ f~ f 1  1 ∑ a~ ij Lj ξj e þ ξi ∑ bij jK ij ðuÞjeλu Lj ξj e þ ξi du Lgj ξj þ ξi 1 ∑

j¼1

¼

βi ¼ ciþ

ð3:12Þ

1þ

n

1þ

þ b~ ij

j¼1

Z

1

0

#

0

1 0

þ ξi 1

0

!

R1 ciþ 0 hi ðvÞ dv R1 βi o1; 0 hi ðvÞ dv  λ

ð3:13Þ

þ ξi 1 ∑ aij ðsÞðf j ðxj ðs  τij ðsÞÞÞ f j ðxnj ðs  τij ðsÞÞÞÞ

þ ξi 1

n

∑ a~ ij ðsÞðf~ j ðx0j ðs  τ~ ij ðsÞÞÞ  f~ j ðxjn0 ðs  τ~ ij ðsÞÞÞÞ

j¼1

Z

n

∑ bij ðsÞ

j¼1

þ ξi 1 ∑ b~ ij ðsÞ

∑

j¼1 n

þ þ ξi 1 ∑ b~ ij j¼1

Z

1 0

n

∑

j¼1

þ bij

Z

jK~ ij ðuÞjeλu du Lgj ξj ; ~

1 0

Z

j¼1

λτijþ

j¼1

þ ξi 1

sv

0

n

þ ξi 1

ci

λ~τ þ þ ~ a~ ij Lfj ξj e ij

K~ ij ðuÞðg~ j ðx0j ðs  uÞÞ  g~ j ðxnj 0 ðs  uÞÞÞ du;

ð3:19Þ Rs R1 ci ðuÞ hi ðvÞ dv du 0 0 , and integratMultiplying both sides of (3.19) by e ing on ½0; t, we get Rt R1  c ðuÞ hi ðvÞ dv du 0 yi ðtÞ ¼ yi ð0Þe 0 i " Z t Z 1 Z s Rt R1  ci ðuÞ hi ðvÞ dv du s 0 þ e hi ðvÞ y0i ðuÞ du dv ci ðsÞ

~

vhi ðvÞeλv dv þξi 1 ∑ aijþ Lfj ξj e n

0

j¼1

jK~ ij ðuÞjeλu du Lgj ξj

n

1

s A ½0; t; t A ½0; θ:

n

Z

Z

j¼1

! βi R1  1 o 0; 0 hi ðvÞ dv  λ

ci

j¼1

where

n

þ ξi 1 ∑ b~ ij ðsÞ

jK~ ij ðuÞjeλu du Lgj ξj

1

451

K ij ðuÞðg j ðxj ðs  uÞÞ  g j ðxnj ðs  uÞÞÞ du

1 0

# K~ ij ðuÞðg~ j ðx0j ðs  uÞÞ  g~ j ðxnj 0 ðs  uÞÞÞ du ds;

t A ½0; θ:

1

λu

jK ij ðuÞje

du

0

Lgj ξj

Thus, with the help of (3.18), we have  Rθ R1   c ðuÞ hi ðvÞ dv du 0 jyi ðθÞj ¼ yi ð0Þe 0 i " Z Z R R

i ¼ 1; 2; …; n:

θ

þ

e



θ

s

ci ðuÞ

1

hi ðvÞ dv du

0

0

Let   ‖φ  x ‖ξ ¼ max sup max ξi 1 jφi ðtÞ  xni ðtÞj; sup max ξi 1 jφ0i ðtÞ  xin0 ðtÞj n

tr0 1rirn

t r0 1rirn

ð3:14Þ and M be a constant such that R1 c hi ðvÞ dv  λ M4 i 0 4 1 for all i ¼ 1; 2; …; n; βi 1 βi R o 0;  M ci 01 hi ðvÞ dv λ

ci

β R1 i o1 0 hi ðvÞ dv  λ

sv

y0i ðuÞ du dv

j¼1

þ ξi 1

þ ξi 1

ð3:16Þ

0

s

n

ð3:15Þ

for all i ¼ 1; 2; …; n;

hi ðvÞ

þ ξi 1 ∑ aij ðsÞðf j ðxj ðs  τij ðsÞÞÞ  f j ðxnj ðs  τij ðsÞÞÞÞ

þ ξi 1

which, together with (3.12), yields

Z

1

ci ðsÞ

n

∑ a~ ij ðsÞðf~ j ðx0j ðs  τ~ ij ðsÞÞÞ  f~ j ðxjn0 ðs  τ~ ij ðsÞÞÞÞ

j¼1

Z

n

∑ bij ðsÞ

j¼1 n

∑ b~ ij ðsÞ

1

0

Z

j¼1

1 0

K ij ðuÞðg j ðxj ðs  uÞÞ  g j ðxnj ðs uÞÞÞ du #    0 n 0 ~ K ij ðuÞðg~ j ðxj ðs  uÞÞ  g~ j ðxj ðs  uÞÞÞ du ds 

R1 c h ðvÞ dv θ r ð‖φ  xn ‖ξ þ εÞe i 0 i  Z Z θ Rθ R1  c ðuÞ hi ðvÞ dv du 0 þ e s i ciþ 0

1 0

vhi ðvÞeλv dv Mð‖φ  xn ‖ξ

n

‖yðtÞ‖1 o ð‖φ  xn ‖ξ þ εÞe  λt o Mð‖φ  xn ‖ξ þ εÞe  λt

j¼1

for all t A ð  1; 0:

In the following, we will show ‖yðtÞ‖1 o Mð‖φ  xn ‖ξ þ εÞe  λt

for all t 4 0:

n

~

þ εÞe  λs þ ξi 1 ∑ jaij ðsÞjLfj ξj jyj ðs  τij ðsÞÞj þξi 1 ∑ ja~ ij ðsÞjLfj ξj

Consequently, for any ε 4 0, it is obvious that

ð3:17Þ

Otherwise, there must exist i A f1; 2; …; ng and θ 4 0 such that ( ‖yðθÞ‖1 ¼ maxfjyi ðθÞj; jy0i ðθÞjg ¼ Mð‖φ  xn ‖ξ þεÞe  λθ ; ð3:18Þ ‖yðtÞ‖1 o Mð‖φ  xn ‖ξ þεÞe  λt for all t A ð  1; θÞ:

j¼1

n

jy0j ðs  τ~ ij ðsÞÞj þ ξi 1 ∑ jbij ðsÞj n

þ ξi 1 ∑ jb~ ij ðsÞj

Z

j¼1

Z

j¼1

1

0

0

K ij ðuÞLgj ξj jyj ðs  uÞj du

# ~ K~ ij ðuÞLgj ξj jy0j ðs uÞj du ds

R1 hi ðvÞ dv θ 0 r ð‖φ  x ‖ξ þ εÞe  Z Z θ Rθ R1  c ðuÞ hi ðvÞ dv du 0 þ e s i ciþ n

1

 ci

0

1 0

vhi ðvÞeλv dv Mð‖φ  xn ‖ξ

n

Note that Z 1 hi ðvÞ dv yi ðsÞ y0i ðsÞ þ ci ðsÞ 0 Z 1 Z s hi ðvÞ y0i ðuÞ du dv ¼ ci ðsÞ sv

0

þ εÞe  λs þ ξi 1 ∑ jaij ðsÞjLfj ξj Mð‖φ  xn ‖ξ þ εÞe  λðs  τij ðsÞÞ j¼1

þ ξi 1

n

þ ξi 1 ∑ jbij ðsÞj

n

j¼1 n

∑ a~ ij ðsÞðf~ j ðx0j ðs  τ~ ij ðsÞÞÞ  f~ j ðxnj 0 ðs  τ~ ij ðsÞÞÞÞ

j¼1 n

þ ξi 1 ∑ bij ðsÞ j¼1

Z

1 0

K ij ðuÞðg j ðxj ðs  uÞÞ  g j ðxnj ðs  uÞÞÞ du

þ ξi 1

~

∑ ja~ ij ðsÞjLfj ξj Mð‖φ  xn ‖ξ þ εÞe  λðs  τ~ ij ðsÞÞ

j¼1

j¼1

þ ξi 1 ∑ aij ðsÞðf j ðxj ðs  τij ðsÞÞÞ f j ðxnj ðs  τij ðsÞÞÞÞ þ ξi 1

n

n

∑ jb~ ij ðsÞj

j¼1

Z

1 0

Z

0

1

jK ij ðuÞjLgj ξj Mð‖φ  xn ‖ξ þ εÞe  λðs  uÞ du # g~ n  λðs  uÞ ~ jK ij ðuÞjLj ξj Mð‖φ  x ‖ξ þ εÞe du

R1 c h ðvÞ dv θ ds rð‖φ  xn ‖ξ þ εÞe i 0 i  Z 1 Z θ Rθ R1  c ðuÞ hi ðvÞ dv du 0 þ e s i ciþ vhi ðvÞeλv dv e  λs 0

0

452

B. Liu / Neurocomputing 148 (2015) 445–454

0.106 0.105 0.104 0.103 0.102 0.101 0.1 0.099 0.098

0

20

40

60

80

0

20

40

60

80

100

120

140

160

180

200

0.102 0.1 0.098 0.096 0.094 0.092 0.09

100

120

140

160

180

200

T

T

Fig. 1. Numerical solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞ of system (4.1) for initial value φðtÞ  ð0:1; 0:09Þ . n

λτijþ

þξi 1 ∑ aijþ Lfj ξj e j¼1 n

þ

þξi 1 ∑ bij j¼1 n

þ þξi 1 ∑ b~ ij j¼1

Z

1

0

Z

1 0

0

θ

ci

0

e 2

R1 0

1 0

R1

n

ds βi Mð‖φ  x ‖ξ þ εÞ

j¼1

þ þ ξi 1 ∑ b~ ij

R1 0

Z

ciþ

Þ

þ

ci

1 0

"

1

R1

λτijþ

0

‖yðθÞ‖1 ¼ maxfjyi ðθÞj; jy0i ðθÞjg ¼ jy0i ðθÞj ¼ Mð‖φ  xn ‖ξ þ εÞe  λθ : Z

n

~

þ

þ

j¼1 n

þ ∑ b~ ij

j¼1

Z

1 0

Z

1 0

jK ij ðuÞjeλu du Lgj ξj ) g~ λu ~ jK ij ðuÞje du Lj ξj Mð‖φ  xn ‖ξ þεÞe  λθ

ð3:21Þ

hi ðvÞ 0

θ θv

jy0i ðuÞj du dv

þξi 1 ∑ jaij ðθÞjjf j ðxj ðθ  τij ðθÞÞÞ  f j ðxnj ðθ  τij ðθÞÞÞj j¼1 n

þξi 1 ∑ ja~ ij ðθÞjjf~ j ðx0j ðθ  τ~ ij ðθÞÞÞ  f~ j ðxnj 0 ðθ  τ~ ij ðθÞÞÞj n

þξi 1 ∑ jbij ðθÞjj j¼1 n

þξi 1 ∑ jb~ ij ðθÞjj j¼1

Z

1 0

Z

1 0

Z

1

hi ðvÞ dv 0

!

R1 1 β ðλ  ci hi ðvÞ dvÞθ 0   R1 i e M ci 0 hi ðvÞ dv  λ !# R1 cþ hi ðvÞ dv þ1 o Mð‖φ  xn ‖ξ þ εÞe  λθ ; þ βi  i R 10 ci 0 hi ðvÞ dv λ

 Z

1

jK ij ðuÞjLgj ξj jyj ðθ  uÞj du

λ~τ þ þ ξi 1 ∑ a~ ij Lfj ξj e ij

þ ξi 1 ∑ bij þ ξi 1

From (3.13) and (3.16), (3.19) and (3.20) yield hi ðvÞ dv jyi ðθÞj þ ci ðθÞ

0

! R1 1 βi ðλ  ci hi ðvÞ dvÞθ 0   R1 e M ci 0 hi ðvÞ dv  λ

r Mð‖φ  xn ‖ξ þ εÞe  λθ ½ciþ

1

1

# Z 1 n βi vhi ðvÞeλv dv þ ξi 1 ∑ aijþ Lfj ξj þ ciþ hi ðvÞ dv λ 0 j¼1

n

ð3:20Þ

and

j¼1

Z

j¼1

jyi ðθÞj o Mð‖φ  xn ‖ξ þ εÞe  λθ ;

0 n

þ

~

hi ðvÞ dv

e

which, together with (3.16) and (3.18), implies that

Z

~

jK~ ij ðuÞjLgj ξj jy0j ðθ  uÞj du

0

# hi ðvÞ dvÞθ

Z

j¼1

r

M

βi ðλ  ci ð1  e hi ðvÞ dv  λ

vhi ðvÞeλv dv Mð‖φ xn ‖ξ þ εÞ

j¼1

n

(

¼ Mð‖φ  xn ‖ξ þ εÞe  λθ " ! R1 1 βi ðλ  ci hi ðvÞ dvÞθ 0   R1  e M ci 0 hi ðvÞ dv  λ # βi R þ  1 ; ci 0 hi ðvÞ dv  λ

jy0i ðθÞj r ci ðθÞ

0

n

jy0j ðθ  τ~ ij ðθÞÞj þ ξi 1 ∑ bij

hi ðvÞ dvÞθ

0

1

n

n

hi ðvÞ dv  λÞs

ðλ  ci  λθ 4e

Z

hi ðvÞ dv jyi ðθÞj þ ciþ

j¼1

hi ðvÞ dv θ

ðci

Z

þ e  λθ þξi 1 ∑ aijþ Lfj ξj jyj ðθ  τij ðθÞÞj þ ξi 1 ∑ a~ ij Lfj ξj

~ K~ ij ðuÞLgj ξj eλu du e  λs ds Mð‖φ  xn ‖ξ þ εÞ

0

R1

r ciþ

#

r ð‖φ  xn ‖ξ þεÞe Z R1 c h ðvÞ dv θ þe i 0 i

þ

þ

K ij ðuÞLgj ξj eλu du e  λs

R1 

r Mð‖φ  x ‖ξ þ εÞe

~

j¼1

 ci

n

n

þ λ~τ e  λs þ ξi 1 ∑ a~ ij Lfj ξj e ij e  λs

K ij ðuÞðg j ðxj ðθ  uÞÞ g j ðxnj ðθ uÞÞÞ duj K~ ij ðuÞðg~ j ðx0j ðθ  uÞÞ  g~ j ðxnj 0 ðθ  uÞÞÞ duj

which contradicts (3.21). Hence, (3.17) holds. Letting ε⟶0 þ , we have from (3.17) that ‖yðtÞ‖1 r M‖φ xn ‖ξ e  λt

for all t 4 0;

which implies ‖xðtÞ  xn ðtÞ‖1 r M‖φ  xn ‖0 e  λt This completes the proof.

□

for all t 4 0:

B. Liu / Neurocomputing 148 (2015) 445–454

4. Example

n

þ

þ ξi 1 ∑ bij

In this section, we give an example to demonstrate the results obtained in the previous sections.

Example 4.1. Consider the following neutral type CNNs with continuously distributed leakage delays: 8 > > > x01 ðtÞ > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <

¼

I i ðtÞ ¼

  pffiffiffi 1 1 ; 4þ sin t þ sin 2t þ 1000 1 þ t2

i ¼ 1; 2:

Obviously, c1 ðtÞ ¼ c2 ðtÞ ¼

1 ; 11:2

h1 ðsÞ ¼ h2 ðsÞ ¼ e  4s ; 2

a11 ðtÞ ¼ b11 ðtÞ ¼ a~ 11 ðtÞ ¼ b~ 11 ðtÞ ¼

sin t ; 8000

a12 ðtÞ ¼ b12 ðtÞ ¼ a~ 12 ðtÞ ¼ b~ 12 ðtÞ ¼

sin t ; 72 000

a21 ðtÞ ¼ b21 ðtÞ ¼ a~ 21 ðtÞ ¼ b~ 21 ðtÞ ¼

cos t ; 8000

a22 ðtÞ ¼ b22 ðtÞ ¼ a~ 22 ðtÞ ¼ b~ 22 ðtÞ ¼

cos t ; 72 000

~ Lgi ¼ Lfi

~

¼ Lgi ¼ 1;

Z

Z

1

1

jK ij ðsÞj ds ¼ 0

jK~ ij ðsÞj dsr 1;

i; j ¼ 1; 2:

0

Let ξ1 ¼ ξ2 ¼ 1. Then, Z

1 0

hi ðvÞ dv þ ciþ

þ ξi 1

2

∑

j¼1

þ bij

Z

1

ci

ciþ

Z

1 0

2

j¼1

0

Z

2

1 0

# jK~ ij ðuÞj du Lgj ξj ~

i ¼ 1; 2:

Hence, from Theorem 3.2, system (4.1) has exactly one pseudo almost periodic solution xn ðtÞ. Moreover, all solutions and their derivatives of solutions for (4.1) with initial conditions (1.2)

ð4:1Þ

1

jK ij ðuÞj du 0

α R1 1  0:3176; 0 hi ðvÞ dv hi ðvÞ dv  αi þ ciþ

Z

Lgj ξj þ ξi 1

2

þ ∑ b~ ij

j¼1

j¼1

Z

1

Remark 4.1. In [9–18], only neutral type CNNs without leakage delays are studied. One can observe that all results there and in the references cited therein cannot be applicable. Moreover, though [7,8,5,22–35] considered neural networks with leakage delays, the problem of pseudo almost periodic solutions is not touched and hence the results there cannot directly be applied to (4.1) either.

Acknowledgments The author would like to express the sincere appreciation to the editor and reviewers for their helpful comments in improving the presentation and quality of the paper.

0

References

~ jK~ ij ðuÞj du Lgj ξj

i ¼ 1; 2;

1

hi ðvÞ dv 1 0

converge exponentially to xn ðtÞ and xn ðtÞ, respectively. The exponential convergent rate is about 0.001. The fact is verified by the numerical simulation in Fig. 1.

~

ciþ

α R1 i 0 hi ðvÞ dv

!  0:0251;

i ¼ 1; 2;

and 1þ

j¼1

Z

þ vhi ðvÞ dvþ ξi 1 ∑ aijþ Lfj ξj þ ξi 1 ∑ a~ ij Lfj ξj

pffiffiffi π 1 1 1 1 1 1 1 r þ   þ þ þ þ 4 11:2 8 4000 36 000 4000 36 000 11:2   0:278 o  0:27; i ¼ 1; 2;

1

0

n

þ jK ij ðuÞj du Lgj ξj þ ξi 1 ∑ b~ ij

0

f i ðxÞ ¼ f~ i ðxÞ ¼ g i ðxÞ ¼ g~ i ðxÞ ¼ jxj;

 ci

1



where

Lfi ¼

o 0:05;

Z

1 R 1  4s2 sin t f ðx1 ðt  2 sin 2 tÞÞ e x1 ðt  sÞ ds þ 11:2 0 8000 1 sin t sin t R 1 f ðx2 ðt  sin 4 tÞÞ þ j sin uje  u g 1 ðx1 ðt  uÞÞ du þ 72 000 2 8000 0 sin t R 1 sin t ~ 0 2 f ðx ðt  2 sin tÞÞ j cos uje  u g 2 ðx2 ðt  uÞÞ du þ þ 72 000 0 8000 1 1 sin t ~ 0 sin t R 1 f ðx ðt  sin 4 tÞÞ þ j sin uje  u g~ 1 ðx01 ðt uÞÞ du þ 72 000 2 2 8000 0 sin t R 1 j cos uje  u g~ 2 ðx02 ðt  uÞÞ du þ I 1 ðtÞ; þ 72 000 0 1 R 1  4s2 cos t f ðx1 ðt  2 sin 2 tÞÞ  e x2 ðt  sÞ ds þ 11:2 0 8000 1 cos t cos t R 1 f ðx2 ðt  sin 4 tÞÞ þ j sin uje  u g 1 ðx1 ðt  uÞÞ du þ 72 000 2 8000 0 cos t R 1 cos t ~ 0 2 f ðx ðt  2 sin tÞÞ j cos uje  u g 2 ðx2 ðt  uÞÞ du þ þ 72 000 0 8000 1 1 cos t ~ 0 cos t R 1 f ðx ðt  sin 4 tÞÞ þ j sin uje  u g~ 1 ðx01 ðt  uÞÞ du þ 72 000 2 2 8000 0 cos t R 1 j cos uje  u g~ 2 ðx02 ðt  uÞÞ du þ I 2 ðtÞ; þ 72 000 0

¼

> > > x0 ðtÞ > > > 2 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > :

j¼1

453

!" Z R1 1 n n ciþ 0 hi ðvÞ dv þ ~ R ciþ vhi ðvÞ dvþ ξi 1 ∑ aijþ Lfj ξj þ ξi 1 ∑ a~ ij Lfj ξj 1  ci 0 hi ðvÞ dv 0 j¼1 j¼1

[1] J.H. Park, Further result on asymptotic stability criterion of cellular neural networks with time-varying discrete and distributed delays, Appl. Math. Comput. 182 (2006) 1661–1666. [2] X. Li, C. Ding, Q. Zhu, Synchronization of stochastic perturbed chaotic neural networks with mixed delays, J. Frank. Inst. 347 (2010) 1266–1280. [3] O. Kwon, J. Park, Delay-dependent stability for uncertain cellular neural networks with discrete and distribute time-varying delays, J. Frank.n Inst. 345 (2008) 766–778. [4] L. Wang, Stability of Cohen–Grossberg neural networks with distributed delays, Appl. Math. Comput. 160 (2005) 93–110. [5] C. Zhao, Z. Wang, Exponential convergence of a SICNN with leakage delays and continuously distributed delays of neutral type, Neural Process. Lett. http://dx. doi.org/10.1007/s11063-014-9341-1, in press. [6] A. Zhang, New results on exponential convergence for cellular neural networks with continuously distributed leakage delays, Neural Process. Lett. http://dx.doi.org/10.1007/s11063-014-9348-7, in press. [7] H. Zhang, Existence and stability of almost periodic solutions for CNNs with continuously distributed leakage delays, Neural Comput. Appl. 2014 (24) (2014) 1135–1146.

454

B. Liu / Neurocomputing 148 (2015) 445–454

[8] Z. Chen, A shunting inhibitory cellular neural network with leakage delays and continuously distributed delays of neutral type, Neural Comput. Appl. 2013 (23) (2013) 2429–2434. [9] B. Xiao, Existence and uniqueness of almost periodic solutions for a class of Hopfield neural networks with neutral delays, Appl. Math. Lett. 22 (2009) 528–533. [10] S. Mandal, N.C. Majee, Existence of periodic solutions for a class of Cohen– Grossberg type neural networks with neutral delays, Neurocomputing 74 (6) (2011) 1000–1007. [11] L. Li, Z. Fang, Y. Yang, A shunting inhibitory cellular neural network with continuously distributed delays of neutral type, Nonlinear Anal. Real World Appl. 13 (2012) 1186–1196. [12] E. Ait Dads, K. Ezzinbi, Pseudo almost periodic solutions of some delay differential equations, J. Math. Anal. Appl. 201 (1996) 840–850. [13] F. Chérif, Existence and global exponential stability of pseudo almost periodic solution for SICNNs with mixed delays, J. Appl. Math. Comput. 39 (2012) 235–251. [14] B.W. Liu, Pseudo almost periodic solutions for CNNs with continuously distributed leakage delays, Neural Process. Lett. http://dx.doi.org/10.1007/ s11063-014-9354-9, in press. [15] Y. Li, C. Wang, X. Li, Existence and global exponential stability of almost periodic solution for high-order BAM neural networks with delays on time scales, Neural Process. Lett. 39 (2014) 247–268. [16] H. Zhang, Y. Xia, Existence and exponential stability of almost periodic solution for Hopfield-type neural networks with impulse, Chaos Solitons Fractals 37 (4) (2008) 1076–1082. [17] S. Abbas, Pseudo almost periodic sequence solutions of discrete time cellular neural networks, Nonlinear Anal. Model. Control 14 (3) (2009) 283–301. [18] S. Abbas, Y. Xia, Existence and attractivity of k-almost automorphic sequence solution of a model of cellular neural networks with delay, Acta Math. Sci. Ser. B Engl. Ed. 33 (1) (2013) 290–302. [19] C. Zhang, Pseudo almost periodic solutions of some differential equations II, J. Math. Anal. Appl. 7–192 (1995) 543–561. [20] C. Zhang, Almost Periodic Type Functions and Ergodicity, Kluwer Academic/ Science Press, Beijing, 2003. [21] A.M. Fink, Almost periodic differential equations, in: Lecture Notes in Mathematics, vol. 377, Springer, Berlin, 1974. [22] O. Kwon, S. Lee, J. Park, Improved results on stability analysis of neural networks with time-varying delays: novel delay-dependent criteria, Mod. Phys. Lett. B 24 (2010) 775–789. [23] O. Kwon, J. Park, Exponential stability analysis for uncertain neural networks with interval time-varying delays, Appl. Math. Comput. 212 (2009) 530–541. [24] O. Kwon, J. Park, Delay-dependent stability for uncertain cellular neural networks with discrete and distribute time-varying delays, J. Frank. Inst. 345 (2008) 766–778.

[25] S. Haykin, Neural Networks, Prentice-Hall, NJ, 1994. [26] B. Kosok, Neural Networks and Fuzzy Systems, Prentice-Hall, New Delhi, 1992. [27] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht, 1992. [28] K. Gopalsamy, Leakage delays in BAM, J. Math. Anal. Appl. 325 (2007) 1117–1132. [29] B. Liu, Global exponential stability for BAM neural networks with timevarying delays in the leakage terms, Nonlinear Anal.: Real World Appl. 14 (2013) 559–566. [30] S. Peng, Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms, Nonlinear Anal.: Real World Appl. 11 (2010) 2141–2151. [31] Z. Chen, J. Meng, Exponential convergence for cellular neural networks with time-varying delays in the leakage terms, Abstr. Appl. Anal. 2012 (941063) (2012) 1–11. [32] Z. Chen, M. Yang, Exponential convergence for HRNNs with continuously distributed delays in the leakage terms, Neural Comput. Appl. 23 (2013) 2221–2229. [33] W. Xiong, J. Meng, Exponential convergence for cellular neural networks with continuously distributed delays in the leakage terms, Electron. J. Qual. Theory Differ. Equ. 2013 (10) (2013) 1–12. [34] H. Zhang, J. Shao, Existence and exponential stability of almost periodic solutions for CNNs with time-varying leakage delays, Neurocomputing 121 (9) (2013) 226–233. [35] H. Zhang, J. Shao, Almost periodic solutions for cellular neural networks with time-varying delays in leakage terms, Appl. Math. Comput. 219 (24) (2013) 11471–11482.

Bingwen Liu was born in Hunan Province, China, in 1971. He was received the B.S. degree in Mathematics, from the Hunan Normal University, Changsha, China, in 1994, and the M.S. and Ph.D. degrees in Applied Mathematics from the Hunan University, Changsha, China, in 2000 and 2005, respectively. He is currently a Professor in the College of Mathematics and Computer Science, Hunan University of Arts and Science, Changde, Hunan 415000, P. R. China. He is also the author or coauthor of more than 60 journal papers. His research interests include nonlinear dynamic systems, neural networks, functional differential equations, almost periodic differential equation.