Almost automorphic solution for neutral type high-order Hopfield neural networks with delays in leakage terms on time scales

Applied Mathematics and Computation 242 (2014) 679–693

Contents lists available at ScienceDirect

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

Almost automorphic solution for neutral type high-order Hopfield neural networks with delays in leakage terms on time scales q Yongkun Li ⇑, Li Yang Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, People’s Republic of China

a r t i c l e

i n f o

Keywords: Almost automorphic solutions High-order Hopfield neural networks Exponential stability Leakage terms Time scales

a b s t r a c t In this paper, using the existence of the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, some sufficient conditions for the existence and global exponential stability of almost automorphic solutions for a class of neutral type high-order Hopfield neural networks (HHNNs) with delays in leakage terms on time scales are obtained. Finally, a numerical example is provided to illustrate the feasibility of our results and our results also show that the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new even when the time scale T ¼ R or Z and show that the delays in the leakage term do harm to the existence and global exponential stability of almost automorphic solutions. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Since the high-order Hopfield neural networks (HHNNs) have stronger approximation property, faster convergence rate, great stronger capacity and higher fault tolerance, they have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. These applications heavily rely on the dynamics of the neural networks. Besides, due to the finite speed of information processing, the existence of time delays frequently causes oscillation, divergence, or instability in neural networks. Therefore, it is of prime importance to consider the delay effects on the dynamics of neural networks. Up to now, neural networks with various types of delays have been widely investigated by many authors, the reader may refer to [1–6] and reference therein. For example, in [7–9], authors studied the robust stability, global asymptotic stability or exponential stability of HHNNs with delays, respectively; in [10,11], by using coincidence degree theory as well as a priori estimates and Lyapunov functional or by employing a fixed point theorem, authors studied the existence and global exponential stability of periodic solutions for delayed HHNNs, respectively; in [12], the author obtained sufficient conditions on the existence and exponential stability of anti-periodic solutions for HHNNs with delays; in [13,14], by using a fixed point theorem, Lyapunov functional method and differential inequality techniques, authors obtained sufficient conditions on the existence and exponential stability of almost periodic solutions for HHNNs with delays.

q

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11361072.

⇑ Corresponding author.

E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.amc.2014.06.052 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

680

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

In fact, 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 [15], many authors investigated the dynamical behaviors of neutral type neural networks. For example, authors in [16–27] studied the stability, periodic solutions and almost periodic solutions for different classes of neutral type neural networks, respectively. Very recently, a leakage delay, which is the time delay in the leakage term of the systems and a considerable factor affecting dynamics for the worse in the systems, is being put to use in the problem of stability for neural networks (see [28,29]). Such time delays in the leakage term are difficult to handle but has great impact on the dynamical behavior of neural networks. Therefore, it is meaningful to consider neural networks with time delays in leakage terms. For example, authors of [30–32] studied the stability of some classes neural networks with leakage delays; authors of [33,34] studied the equilibrium point of two classes fuzzy neural networks with delays in leakage terms; authors of [35] studied the global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage term; authors of [36,37] studied almost periodic solutions of some classes neural networks with delays in leakage terms. Also, as we all know that both continuous time and discrete time neural networks have equal importance in various applications. But it is troublesome to study the dynamical properties for continuous and discrete time systems, respectively. Hence, the theory of time scales, which was initialed by Hilger [38] in his Ph.D. thesis to contain both difference and differential calculus in a consistent way, has recently received a lot of attention from many scholars. By choosing the time scale to be the set of real numbers and the set of integers, results on time scales yield results concerning with differential equations and difference equations, respectively. Besides, results on time scales can also be extended to other types of equations, for example, q-difference equations. Therefore, it is significant to study the dynamical behaviors of neural networks on time scales. For instance, in [39,40], authors studied periodic solutions to some classes of BAM neural networks on time scales; in [41,42], authors studied almost periodic solutions of some classes of neural networks on time scales; in [43,44], authors studied anti-periodic solutions of two classes of neural networks on time scales, respectively. In reality, almost periodicity is universal than periodicity. Moreover, almost automorphic functions, which were introduced by Bochner, are much more general than almost periodic functions. Almost automorphic solutions in the context of differential equations were studied by several authors (see [45–47] and reference therein). In [48,49], authors studied almost automorphic solutions for two classes of neural networks, respectively. For more details about almost automorphic functions, we refer to the recent book [50]. And the concept of the almost automorphic function on time scales, which is the central question in this paper, was first introduced by Wang and Li [51], as a natural generalization of the well-known almost periodicity. They also established some results about the existence of almost automorphic solutions to dynamic equations on time scales. In [52], authors also obtained some results on almost automorphic solutions of dynamic equations on time scales. However, to the best of our knowledge, there is no paper published on the almost automorphic solutions of neutral type high-order Hopfield neural networks with time-varying delays in leakage terms on time scales. Motivated by the above, in this paper, we consider the following neutral type high-order Hopfield neural networks with time-varying delays in leakage terms on time scales

xDi ðtÞ ¼ ci ðtÞxi ðt  si ðtÞÞ þ

n n X X aij ðtÞfj ðxj ðt  dij ðtÞÞÞ þ eij ðtÞhj ðxDj ðt  hij ðtÞÞÞ j¼1

j¼1

n X n X bijl ðtÞg j ðxj ðt  rijl ðtÞÞÞg l ðxl ðt  fijl ðtÞÞÞ þ Ii ðtÞ; þ

t 2 T;

ð1:1Þ

j¼1 l¼1

where T is an almost periodic time scale, i ¼ 1; 2; . . . ; n; n corresponds to the number of units in a neural network; xi ðtÞ corresponds to the state vector of the ith unit at time t; ci ðtÞ represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs; aij ðtÞ and bijl ðtÞ are the first-order and second-order connection weights of the neural network; si ðtÞ > 0 with t  si ðtÞ 2 T for all t 2 T denotes time delay in the leakage term; dij ðtÞ P 0; hij ðtÞ P 0; rijl ðtÞ P 0, fijl ðtÞ P 0 correspond to the transmission delays and satisfy that for t 2 T; t  dij ðtÞ 2 T; t  hij ðtÞ 2 T; t  rijl ðtÞ 2 T, t  fijl ðtÞ 2 T; Ii ðtÞ denote the external inputs at time t; fj ; hj and g j are the activation functions of signal transmission, j ¼ 1; 2; . . . ; n. Our main purpose of this paper is to study the existence and global exponential stability of almost automorphic solutions for (1.1). Our results of this paper are completely new even when the time scale T ¼ R or Z. Our numerical example also shows that the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors. This fact provides a theoretical basis for the numerical simulation of continuous-time neural network systems. For convenience, we denote ½a; bT ¼ ftjt 2 ½a; b \ Tg. For an almost automorphic function f : T ! R, denote f þ ¼ supt2T jf ðtÞj; f  ¼ inf t2T jf ðtÞj. We denote by R the set of real numbers, by Rþ the set of positive real numbers, by X a real Banach space with the norm jj  jj. The initial condition of (1.1) is the following

xi ðsÞ ¼ ui ðsÞ; n

xDi ðsÞ ¼ uDi ðsÞ; s 2 ½h; 0T ;

o 1 þ þ þ þ where h ¼ max max16i6n sþ i ; maxði;jÞ fdij ; hij ; rijl ; fijl g ; ui 2 C ð½h; 0T ; RÞ; i ¼ 1; 2; . . . ; n.

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

681

Remark 1.1. If we take T ¼ R, then (1.1) reduces to the following form

x0i ðtÞ ¼ ci ðtÞxi ðtÞ þ

n n X X aij ðtÞfj ðxj ðt  dij ðtÞÞÞ þ eij ðtÞhj ðx0j ðt  hij ðtÞÞÞ j¼1

j¼1

n X n X bijl ðtÞg j ðxj ðt  rijl ðtÞÞÞg l ðxl ðt  fijl ðtÞÞÞ þ Ii ðtÞ; þ

t 2 R;

ð1:2Þ

j¼1 l¼1

if we take T ¼ Z, then (1.1) reduces to the following form

xi ðt þ 1Þ  xi ðtÞ ¼ ci ðtÞxi ðtÞ þ

n n X X aij ðtÞfj ðxj ðt  dij ðtÞÞÞ þ eij ðtÞhj ðxj ðt þ 1  hij ðt þ 1ÞÞ  xj ðt  hij ðtÞÞÞ j¼1

j¼1

n X n X bijl ðtÞg j ðxj ðt  rijl ðtÞÞÞg l ðxl ðt  fijl ðtÞÞÞ þ Ii ðtÞ; þ

t 2 Z:

ð1:3Þ

j¼1 l¼1

To the best of our knowledge, there is no paper published on the existence and exponential stability of almost automorphic solutions for (1.2) and (1.3). This paper is organized as follows: In Section 2, we introduce some notations and definitions and state some preliminary results which are needed in later sections. In Section 3, we establish some sufficient conditions for the existence of almost automorphic solutions of (1.1) and prove that the almost automorphic solution is globally exponentially stable. In Section 4, we give an example to illustrate the feasibility of our results obtained in previous sections. We draw a conclusion in Section 5. 2. Preliminaries In this section, we introduce some definitions and state some preliminary results. Definition 2.1 [53]. Let T be a nonempty closed subset (time scale) of 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 and lðtÞ ¼ rðtÞ  t: Lemma 2.1 [53]. Assume that p; q : T ! R are two regressive functions, then (i) (ii) (iii) (iv)

e0 ðt; sÞ  1 and ep ðt; tÞ  1; 1 ep ðt; sÞ ¼ ep ðs;tÞ ¼ ep ðs; tÞ; ep ðt; sÞep ðs; rÞ ¼ ep ðt; rÞ; ðep ðt; sÞÞD ¼ pðtÞep ðt; sÞ.

Lemma 2.2 [53]. Let f ; g be D-differentiable functions on T, then D

(i) ðm1 f þ m2 gÞ ¼ m1 f D þ m2 g D , for any constants m1 ; m2 ; D (ii) ðfgÞ ðtÞ ¼ f D ðtÞgðtÞ þ f ðrðtÞÞg D ðtÞ ¼ f ðtÞg D ðtÞ þ f D ðtÞgðrðtÞÞ; Lemma 2.3 [53]. Assume that pðtÞ P 0 for t P s, then ep ðt; sÞ P 1. Definition 2.2 [53]. A function p : T ! R is called regressive provided 1 þ lðtÞpðtÞ – 0 for all t 2 Tk ; p : T ! R is called positively regressive provided 1 þ lðtÞpðtÞ > 0 for all t 2 Tk . The set of all regressive and rd-continuous functions p : T ! R will be denoted by R ¼ RðT; RÞ and the set of all positively regressive functions and rd-continuous functions will be denoted by Rþ ¼ Rþ ðT; RÞ. Lemma 2.4 [53]. Suppose that p 2 Rþ , then (i) ep ðt; sÞ > 0, for all t; s 2 T; (ii) if pðtÞ 6 qðtÞ for all t P s; t; s 2 T, then ep ðt; sÞ 6 eq ðt; sÞ for all t P s.

Lemma 2.5 [53]. If p 2 R and a; b; c 2 T, then

½ep ðc; ÞD ¼ p½ep ðc; Þr

682

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

and

Z

b

pðtÞep ðc; rðtÞÞDt ¼ ep ðc; aÞ  ep ðc; bÞ:

a

Lemma 2.6 [53]. Let a 2 Tk ; b 2 T and assume that f : T  Tk ! R is continuous at ðt; tÞ, where t 2 Tk with t > a. Also assume that f D ðt; Þ is rd-continuous on ½a; rðtÞ. Suppose that for each e > 0, there exists a neighborhood U of s 2 ½a; rðtÞ such that

jf ðrðtÞ; sÞ  f ðs; sÞ  f D ðt; sÞðrðtÞ  sÞj 6 ejrðtÞ  sj;

8s 2 U;

D

where f denotes the derivative of f with respect to the first variable. Then Rt Rt (i) gðtÞ :¼ a f ðt; sÞDs implies g D ðtÞ :¼ a f D ðt; sÞDs þ f ðrðtÞ; tÞ; Rb Rb D (ii) hðtÞ :¼ t f ðt; sÞDs implies h ðtÞ :¼ t f D ðt; sÞDs  f ðrðtÞ; tÞ. In the following, we recall some definitions of almost automorphic functions on time scales. Definition 2.3 [54]. A time scale T is called an almost periodic time scale if

P :¼ fs 2 R : t  s 2 T; 8t 2 Tg – f0g: In this paper, we restrict our discussion on almost periodic time scales. Definition 2.4 [51]. Let T be an almost periodic time scale. (i) A function f ðtÞ : T ! X is said to be almost automorphic, if for any sequence fsn g1 n¼1 P, there is a subsequence 1 fsn g1 n¼1 fsn gn¼1 such that

gðtÞ ¼ lim f ðt þ sn Þ n!1

is well defined for each t 2 T and

lim gðt  sn Þ ¼ f ðtÞ

n!1

for each t 2 T. Denote by AAðT; XÞ the set of all such functions; (ii) A continuous function f : T  X ! X is said to be almost automorphic, if f ðt; xÞ is almost automorphic in t 2 T uniformly in x 2 B, where B is any bounded subset of X. Denote by AAðT  X; XÞ the set of all such functions.

Lemma 2.7 [52]. Let f ; g 2 AAðT; XÞ. Then we have the following (i) f þ g 2 AAðT; XÞ; (ii) af 2 AAðT; XÞ for any constant a 2 R; (iii) if u : X ! Y is a continuous function, then the composite function u f : T ! Y is almost automorphic.

Lemma 2.8. Let f 2 AAðT  X; XÞ and f satisfies the Lipschitz condition in x 2 X uniformly in t 2 T. If u 2 AAðT; XÞ, then f ðt; uðtÞÞ is almost automorphic. 1 1 Proof. For any sequence fsn g1 n¼1 P, there is a subsequence fsn gn¼1 fsn gn¼1 such that

gðt; xÞ ¼ lim f ðt þ sn ; xÞ n!1

is well defined for each t 2 T; x 2 B, where B is any bounded subset of X and

lim gðt  sn ; xÞ ¼ f ðt; xÞ

n!1

1 for each t 2 T and x 2 B. Since u 2 AAðT; XÞ, there exists a subsequence ffn g1 n¼1 fsn gn¼1 such that

wðtÞ ¼ lim uðt þ fn Þ n!1

is well defined for each t 2 T and

lim wðt  fn Þ ¼ uðtÞ

n!1

683

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

for each t 2 T. Since f satisfies the Lipschitz condition in x 2 X uniformly in t 2 T, there exists a positive constant Lf such that

jf ðt þ fn ; uðt þ fn ÞÞ  gðt; wðtÞÞj 6 jf ðt þ fn ; uðt þ fn ÞÞ  f ðt þ fn ; wðtÞÞj þ jf ðt þ fn ; wðtÞÞ  gðt; wðtÞÞj 6 Lf juðt þ fn Þ  wðtÞj þ jf ðt þ fn ; wðtÞÞ  gðt; wðtÞÞj ! 0;

n ! 1:

1 1 Hence, for any sequence fsn g1 n¼1 P, there is a subsequence ffn gn¼1 fsn gn¼1 such that

gðt; wðtÞÞ ¼ lim f ðt þ fn ; uðt þ fn ÞÞ n!1

is well defined for each t 2 T. Using similar arguments as above, we obtain

lim gðt  fn ; wðt  fn ÞÞ ¼ f ðt; uðtÞÞ

n!1

for each t 2 T. That is, f ðt; uðtÞÞ is almost automorphic. The proof of Lemma 2.8 is completed. h Definition 2.5 [54]. Let x 2 Rn and AðtÞ be a n  n matrix-valued function on T, the linear system

xD ðtÞ ¼ AðtÞxðtÞ;

t2T

ð2:1Þ

is said to admit an exponential dichotomy on T if there exist positive constants ki ; ai ; i ¼ 1; 2, projection P and the fundamental solution matrix XðtÞ of (2.1) satisfying

jXðtÞPX 1 ðsÞj 6 k1 ea1 ðt; sÞ;

s; t 2 T; t P s;

jXðtÞðI  PÞX 1 ðsÞj 6 k2 ea2 ðs; tÞ;

s; t 2 T; t 6 s;

where j  j is a matrix norm on T, that is, if A ¼ ðaij Þnn , then we can take jAj ¼

P P n n i¼1

j¼1 jaij j

2

12

.

Lemma 2.9 [52]. Suppose that AðtÞ 2 AAðT; Rnn Þ such that fA1 ðtÞgt2T and fðI þ lðtÞAðtÞÞ1 gt2T are bounded. Moreover, suppose that g 2 AAðT; Rn Þ and (2.1) admits an exponential dichotomy, then the following system

xD ðtÞ ¼ AðtÞxðtÞ þ gðtÞ

ð2:2Þ

n

has a solution xðtÞ 2 AAðT; R Þ and xðtÞ is expressed as follows

xðtÞ ¼

Z

t

XðtÞPX 1 ðrðsÞÞgðsÞDs 

1

Z

þ1

XðtÞðI  PÞX 1 ðrðsÞÞgðsÞDs;

t

where XðtÞ is the fundamental solution matrix of (2.1), I denotes the n  n-identity matrix. Lemma 2.10 [54]. Let ci ðtÞ > 0 and ci ðtÞ 2 Rþ , 8 t 2 T. If

  e > 0; min inf ci ðtÞ ¼ m

16i6n

t2T

then the linear system

xD ðtÞ ¼ diagðc1 ðtÞ; c2 ðtÞ; . . . ; cn ðtÞÞxðtÞ

ð2:3Þ

admits an exponential dichotomy on T. Definition 2.6. Let x ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞT be an almost automorphic solution of (1.1) with initial value

u ðsÞ ¼ ðu1 ðsÞ; u2 ðsÞ; . . . ; un ðsÞÞT . If there exist positive constants k with k 2 Rþ and M > 1 such that for an arbitrary solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞT of (1.1) with initial value uðsÞ ¼ ðu1 ðsÞ; u2 ðsÞ; . . . ; un ðsÞÞT satisfies jjx  x jj 6 Mjju  u jjek ðt; t 0 Þ;

t 0 2 ½h; 1ÞT ; t P t 0 :

Then the solution x ðtÞ is said to be globally exponentially stable. 3. Main results In this section, we state and prove our main results. Let X ¼ ff 2 C 1 ðT; RÞjf ; f D 2 AAðT; Rn Þg with the norm jjf jjX ¼ maxfjf j1 ; jf D j1 g, where jf j1 ¼ max16i6n fiþ ; jf D j1 Rt T þ ¼ max16i6n ðfiD Þ . Then X is a Banach space. Let u0 ðtÞ ¼ ðu01 ðtÞ; u02 ðtÞ; . . . ; u0n ðtÞÞ , where u0i ðtÞ ¼ 1 eai ðt; rðsÞÞI  i ðsÞDs, i ¼ 1; 2; . . . ; n and L be a constant satisfying L P max jju0 jjX ; max16j6n fjfj ð0Þjg; max16j6n fjhj ð0Þjg; max16j6n fjg j ð0Þjg .

684

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

Theorem 3.1. Suppose that ðH1 Þ ci 2 CðT; Rþ Þ with ci 2 Rþ and inf t2T f1  lðtÞci ðtÞg ¼ ~c > 0, aij ; eij ; bijl ; Ii 2 CðT; RÞ; dij ; hij ; rijl ; fijl 2 CðT; Rþ Þ are almost automorphic, where i; j; l ¼ 1; 2; . . . ; n; ðH2 Þ fj ; hj ; g j 2 CðR; RÞ and there exist positive constants N j ; Lj ; Lhj ; Hj such that jg j ðuÞj 6 N j ; jfj ðuÞ  fj ðv Þj 6 Lj ju  v j; jhj ðuÞ  hj ðv Þj 6 Lhj ju  v j; jg j ðuÞ  g j ðv Þj 6 Hj ju  v j, where u; v 2 R; j ¼ 1; 2; . . . ; n; n~   o n  o  þ cþ ~i 6 1 ; max16i6n ci ; 1 þ ci c < 1, where h ~i ¼ cþ sþ þ Pn aþ ðLj Þðþ1Þ þ eþ ðLh þ 1Þ þ Pn bþ N l ðH3 Þ max16i6n chi ; 1 þ ci h i j j¼1 l¼1 ijl i i ij ij c c 2  i i i Pn  þ Pn þ i þ þ h ðHj þ 1ÞÞ, ci ¼ cþ j¼1 aij Lj þ eij Lj þ l¼1 bijl ðN j Hl þ Hj N l Þ , i ¼ 1; 2; . . . ; n, i si þ   then (1.1) has a unique almost automorphic solution in X0 ¼ u 2 X j jju  u0 jjX 6 L . Proof. For any given u 2 X , we consider the following system

xDi ðtÞ ¼ ci ðtÞxi ðtÞ þ F i ðt; uÞ þ Ii ðtÞ;

ð3:1Þ

where

F i ðt; uÞ ¼ ci ðtÞ

Z

t

tsi ðtÞ

uDi ðsÞDs þ

n n X X aij ðtÞfj ðuj ðt  dij ðtÞÞÞ þ eij ðtÞhj ðuDj ðt  hij ðtÞÞÞ j¼1

j¼1

n X n X bijl ðtÞg j ðuj ðt  rijl ðtÞÞÞg l ðul ðt  fijl ðtÞÞÞ; þ

i ¼ 1; 2; . . . ; n:

j¼1 l¼1

It follows from Lemma 2.10 that the linear system

xDi ðtÞ ¼ ci ðtÞxi ðtÞ;

i ¼ 1; 2; . . . ; n

admits an exponential dichotomy on T. Thus, by Lemma 2.9, we obtain that system (3.1) has exactly one almost automorphic solution as follows

xu i ðtÞ ¼

Z

t

1

eci ðt; rðsÞÞðF i ðs; uÞ þ Ii ðsÞÞDs;

i ¼ 1; 2; . . . ; n:

For u 2 X , then jjujjX 6 jju  u0 jjX þ jju0 jjX 6 2L. Define the following operator T

U : X ! X ; ðu1 ; u2 ; . . . ; un ÞT ! ðxu1 ; xu2 ; . . . ; xun Þ : First we show that for any u 2 X , we have Uu 2 X . Note that, for i ¼ 1; 2; . . . ; n, we have

 Z s  n n X X  jF i ðs; uÞj ¼ ci ðsÞ uDi ð#ÞD# þ aij ðsÞfj ðuj ðs  dij ðsÞÞÞ þ eij ðsÞhj ðuDj ðs  hij ðsÞÞÞ  ssi ðsÞ j¼1 j¼1   n n n   XX X  þ bijl ðsÞg j ðuj ðs  rijl ðsÞÞÞg l ðul ðs  fijl ðsÞÞÞ 6 cþi sþi jjujjX þ aþij jfj ðuj ðs  dij ðsÞÞÞ  fj ð0Þj þ jfj ð0Þj  j¼1 l¼1 j¼1 n n X n   X   X þ bijl Nl jg j ðuj ðs  rijl ðsÞÞÞ  g j ð0Þj þ jg j ð0Þj þ eij ðsÞ jhj ðuDj ðs  hij ðsÞÞÞ  hj ð0Þj þ jhj ð0Þj þ j¼1

6

j¼1 l¼1

! n  n  n X n n X n   X X X X þ þ þ þ þ þ h aij Lj þ eij Lj aþij jfj ð0Þj þ eþij jhj ð0Þj þ bijl Nl Hj jjujjX þ bijl N l jg j ð0Þj jjujjX þ ci si þ "

j¼1

6 2L cþi sþi þ

j¼1

n X

n X þ aþij ðLj þ 1Þ þ eþij ðLhj þ 1Þ þ bijl Nl ðHj þ 1Þ

j¼1

!#

j¼1 l¼1

j¼1 l¼1

:

l¼1

Therefore, we have

Z 

   Uðu  u0 Þ ðtÞ ¼   i

t

1

 Z  eci ðt; rðsÞÞF i ðs; uÞDs 6

t

1

eci ðt; rðsÞÞjF i ðs; uÞjDs 6 2L

n n X X þ þ aþij ðLj þ 1Þ þ eþij ðLhj þ 1Þ þ bijl Nl ðHj þ 1Þ j¼1

l¼1

!#

2L~hi Ds 6  ; ci

Z

t 1

" n X cþi sþi eci ðt; rðsÞÞ

i ¼ 1; 2; . . . ; n:

j¼1

685

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

On the other hand, for i ¼ 1; 2; . . . ; n, we have

Z  D   Z t  t 

D      eci ðt; rðsÞÞF i ðs; uÞDs  ¼ F i ðt; uÞ  ci ðtÞ eci ðt; rðsÞÞF i ðs; uÞDs  Uðu  u0 Þ i ðtÞ ¼    1 1 t  Z t cþ 6 jF i ðt; uÞj þ jci ðtÞj eci ðt; rðsÞÞjF i ðs; uÞjDs 6 2L~hi 1 þ i : ci 1 In view of ðH3 Þ, we have

(

~hi  cþ ; 1 þ i ~hi  ci ci

jjUu  u0 jjX 6 2L max 16i6n

) 6 L:

That is, U/ 2 X0 . Next, we show that U is a contraction. For any u ¼ ðu1 ; u2 ; . . . ; un ÞT ; w ¼ ðw1 ; w2 ; . . . ; wn ÞT 2 X0 , for i ¼ 1; 2; . . . ; n, denote

Gi ðs; u; wÞ ¼ ci ðtÞ

Z

s





uD ð#Þ  wD ð#Þ D# þ

ssi ðsÞ

n   X aij ðsÞ fj ðuj ðs  dij ðsÞÞÞ  fj ðwj ðs  dij ðsÞÞÞ j¼1

n   X eij ðsÞ hj ðuDj ðs  hij ðsÞÞÞ  hj ðwDj ðs  hij ðsÞÞÞ þ j¼1

and

K i ðs; u; wÞ ¼

n X n   X bijl ðsÞ g j ðuj ðs  rijl ðsÞÞÞg l ðul ðs  fijl ðsÞÞÞ  g j ðwj ðs  rijl ðsÞÞÞg l ðwl ðs  fijl ðsÞÞÞ : j¼1 l¼1

Note that, for i ¼ 1; 2; . . . ; n,

 Z s  n   X D

 jGi ðs; u; wÞj ¼ ci ðtÞ u ð#Þ  wD ð#Þ D# þ aij ðsÞ fj ðuj ðs  dij ðsÞÞÞ  fj ðwj ðs  dij ðsÞÞÞ  ssi ðsÞ j¼1  ! n n   X X  þ eij ðsÞ hj ðuDj ðs  hij ðsÞÞÞ  hj ðwDj ðs  hij ðsÞÞÞ  6 cþi sþi þ ðaþij Lj þ eþij Lhj Þ jju  wjjX  j¼1 j¼1 and

  X n n     X jK i ðs; u; wÞj ¼  bijl ðsÞ g j ðuj ðs  rijl ðsÞÞÞg l ðul ðs  fijl ðsÞÞÞ  g j ðwj ðs  rijl ðsÞÞÞg l ðwl ðs  fijl ðsÞÞÞ    j¼1 l¼1 6

n X n X þ bijl jg j ðuj ðs  rijl ðsÞÞÞjjg l ðul ðs  fijl ðsÞÞÞ  g l ðwl ðs  fijl ðsÞÞÞj þ jg j ðuj ðs  rijl ðsÞÞÞ j¼1 l¼1

 g j ðwj ðs  rijl ðsÞÞÞjjg l ðwl ðs  fijl ðsÞÞÞj 6

n X n X

þ bijl Nj Hl þ Hj Nl jju  wjjX : j¼1 l¼1

Then, we have

Z  jðUu  UwÞi ðtÞj ¼ 

t

1

"

 Z  eci ðt; rðsÞÞðGi ðs; u; wÞ þ K i ðs; u; wÞÞDs 6

 cþi sþi þ

n X j¼1

aþij Lj þ eþij Lhj þ

n X

þ

t

bijl ðNj Hl þ Hj Nl Þ

l¼1

eci ðt; rðsÞÞ !#

1

Dsjju  wjjX 6

ci ci

jju  wjjX ;

and

  eci ðt; rðsÞÞðGi ðs; u; wÞ þ K i ðs; u; wÞÞDt  1   Z t   ¼ Gi ðt; u; wÞ þ K i ðt; u; wÞ  ci ðtÞ eci ðt; rðsÞÞðGi ðs; u; wÞ þ K i ðs; u; wÞÞDs

  Z   D ðU/  UfÞi ðtÞ ¼ 

t

1

Z t 6 jGi ðt; u; wÞj þ jK i ðt; u; wÞj þ jci ðtÞj eci ðt; rðsÞÞjGi ðs; u; wÞ þ K i ðs; u; wÞjDs 1  cþ 6 1 þ i ci jju  wjjX ; i ¼ 1; 2; . . . ; n: ci

i ¼ 1; 2; . . . ; n

686

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

By ðH3 Þ, we have that jjUu  Uwjj < jju  wjj. Hence, U is a contraction. Therefore, U has a fixed point in X0 , that is, (1.1) has a unique almost automorphic solution in X0 . The proof of Theorem 3.1 is completed. h Theorem 3.2. Let ðH1 Þ  ðH3 Þ hold. Then the almost automorphic solution of system (1.1) is globally exponentially stable. Proof. According to Theorem 3.1, we know that (1.1) has an almost automorphic solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞT with initial condition uðsÞ ¼ ðu1 ðsÞ; u2 ðsÞ; . . . ; un ðsÞÞT . Suppose that yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞ; . . . ; yn ðtÞÞT is an arbitrary solution of (1.1) with initial condition wðsÞ ¼ ðw1 ðsÞ; w2 ðsÞ; . . . ; wn ðsÞÞT . Denote uðtÞ ¼ ðu1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞÞT , where ui ðtÞ ¼ yi ðtÞ  xi ðtÞ; i ¼ 1; 2; . . . ; n. Then it follows from (1.1) that

uDi ðtÞ ¼ ci ðtÞui ðt  si ðtÞÞ þ

n X

aij ðtÞ fj ðyj ðt  dij ðtÞÞÞ  fj ðxj ðt  dij ðtÞÞÞ j¼1

n n X n   X X eij ðtÞ hj ðyDj ðt  hij ðtÞÞÞ  hj ðxDj ðt  hij ðtÞÞÞ þ bijl ðtÞ g j ðyj ðt  rijl ðtÞÞÞg l ðyl ðt  fijl ðtÞÞÞ þ j¼1

j¼1 l¼1

g j ðxj ðt  rijl ðtÞÞÞg l ðxl ðt  fijl ðtÞÞÞ ;

i ¼ 1; 2; . . . ; n:

ð3:2Þ

The initial condition of (3.2) is

wi ðsÞ ¼ ui ðsÞ  wi ðsÞ;

wDi ðsÞ ¼ uDi ðsÞ  wDi ðsÞ;

s 2 ½h; 0T ; i ¼ 1; 2; . . . ; n:

Rewrite (3.2) in the form

uDi ðtÞ ¼ ci ðtÞui ðtÞ þ ci ðtÞ

Z

t

tsi ðtÞ

uDi ðsÞDs þ

n X

aij ðtÞ fj ðyj ðt  dij ðtÞÞÞ  fj ðxj ðt  dij ðtÞÞÞ j¼1

n n X n   X X aij ðtÞ hj ðyDj ðt  hij ðtÞÞÞ  hj ðxDj ðt  hij ðtÞÞÞ þ bijl ðtÞðg j ðyj ðt  rijl ðtÞÞÞg l ðyl ðt  fijl ðtÞÞÞ þ j¼1

j¼1 l¼1

 g j ðxj ðt  rijl ðtÞÞÞg l ðxl ðt  fijl ðtÞÞÞÞ;

i ¼ 1; 2; . . . ; n:

ð3:3Þ

Then, it follows from (3.3) that for i ¼ 1; 2; . . . ; n and t P t0 ; t0 2 ½h; 0T ,

ui ðtÞ ¼ ui ðt 0 Þeci ðt; t 0 Þ þ

Z

(

t

t0

eci ðt; rðsÞÞ ci ðsÞ

Z

s

ssi ðsÞÞ

uDi ð#ÞD# þ

n X

aij ðsÞ fj ðyj ðs  dij ðsÞÞÞ  fj ðxj ðs  dij ðsÞÞÞ j¼1

n n X n   X X þ eij ðsÞ hj ðyDj ðs  hij ðsÞÞÞ  hj ðxDj ðs  hij ðsÞÞÞ þ bijl ðsÞ g j ðyj ðs  rijl ðsÞÞÞg l ðyl ðs  fijl ðsÞÞÞ j¼1

j¼1 l¼1

)

g j ðxj ðs  rijl ðsÞÞÞg l ðxl ðs  fijl ðsÞÞÞ Ds:

ð3:4Þ

For x 2 R, let Ci ðxÞ and Hi ðxÞ be defined by

Ci ðxÞ ¼ ci  x  expfxsup lðsÞg cþi sþi expfxsþi g þ s2T

n n X X aþij Lj expfxdþij g þ eþij Lhj expfxhþij g j¼1

n n X X þ þ bijl Nj Hl expfxfþijl g þ bijl Nl Hj expfxrþijl g þ j¼1

j¼1

!

j¼1

and





Hi ðxÞ ¼ ci  x  cþi expfxsup lðsÞg þ ci  x s2T



cþi þi

s expfxs

þ i g

! n n n n X X X X þ þ þ þ þ þ þ h þ þ aij Lj expfxdij g þ eij Lj expfxhij g þ bijl Nl Hj expfxrijl g þ bijl Nj Hl expfxfijl g ; j¼1

j¼1

j¼1

where i ¼ 1; 2; . . . ; n. In view of ðH3 Þ, for i ¼ 1; 2; . . . ; n, we have that

Ci ð0Þ ¼ ci  cþi sþi þ

n n n X X X þ aþij Lj þ eþij Lhj þ bijl ðNl Hj þ Nj Hl Þ j¼1

j¼1

! >0

j¼1

and

Hi ð0Þ ¼ ci  ðcþi þ ci Þ cþi sþi þ

n X j¼1

aþij Lj þ

n n X X þ eþij Lhj þ bijl ðNl Hj þ Nj Hl Þ j¼1

j¼1

! > 0:

j¼1

687

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

Since Ci ðxÞ; Hi ðxÞ are continuous on ½0; þ1Þ and Ci ðxÞ; Hi ðxÞ ! 1, as x ! þ1, so there exist xi ; xi > 0 such that Ci ðxi Þ ¼Hi ðxi Þ ¼ 0 and Ci ðxÞ > 0 for x 2 ð0; xi Þ, Hi ðxÞ > 0 for x 2 ð0; xi Þ; i ¼ 1; 2; . . . ; n. By choosing a positive constant a ¼ min x1 ; x2 ; . . . ; xn ; x1 ; x2 ; . . . ; xn , we have Ci ðaÞ P 0; Hi ðaÞ P 0; i ¼ 1; 2; . . . ; n. Hence, we can choose a positive con  stant 0 < a < min a; min16i6n fc i g such that

Ci ðaÞ > 0;

Hi ðaÞ > 0; i ¼ 1; 2; . . . ; n;

which implies that expfasuplðsÞg s2T ci 

cþi þi expf

s

a

as

þ i gþ

! n n n   X X X þ þ þ þ þ þ h þ aij Lj expfadij g þ eij Lj expfahij g þ bijl N l Hj expfarijl g þ Nj Hl expfafijl g <1 j¼1

j¼1

j¼1

and 0 @1 þ

cþi expfasuplðsÞg s2T

ci  a

1 A cþ sþ expfasþ g þ i i i

! n n n n X X X X þ þ aþij Lj expfadþij g þ eþij Lhj expfahþij g þ bijl N l Hj expfarþijl g þ bijl N j Hl expfafþijl g < 1; j¼1

j¼1

j¼1

j¼1

where i ¼ 1; 2; . . . ; n. Let

8 9 < = ci   : M ¼ max P P þ 16i6n : þ þ ci si þ nj¼1 aþij Lj þ eþij Lhj þ nl¼1 bijl ðNj Hl þ Hj Nl Þ ; It follows from ðH3 Þ that M > 1. Thus, we can obtain that

1 < M

expfasuplðsÞg s2T ci 

a

cþi þi expf

s

as

þ i gþ

! n n n   X X X þ þ þ þ þ þ h þ aij Lj expfadij g þ eij Lj expfahij g þ bijl Nl Hj expfarijl g þ Nj Hl expfafijl g : j¼1

j¼1

j¼1

Moreover, we have that ea ðt; t0 Þ > 1, where t 2 ½h; t 0 T . Hence, it is obvious that

jjujjX 6 Mea ðt; t0 Þku  wkX ;

8 t 2 ½h; t 0 T :

We claim that

jjujjX 6 Mea ðt; t0 Þku  wkX ;

8 t 2 ðt0 ; þ1ÞT :

ð3:5Þ

To prove this claim, we show that for any p > 1, the following inequality holds

jjujjX < pMea ðt; t 0 Þku  wkX ;

8 t 2 ðt 0 ; þ1ÞT ;

ð3:6Þ

which implies that, for i ¼ 1; 2; . . . ; n, we have

jui ðtÞj < pMea ðt; t 0 Þku  wkX ;

8 t 2 ðt 0 ; þ1ÞT

ð3:7Þ

juDi ðtÞj < pMea ðt; t 0 Þku  wkX ;

8 t 2 ðt 0 ; þ1ÞT :

ð3:8Þ

and

By way of contradiction, assume that (3.6) does not hold. Firstly, we consider the following two cases. Case One: (3.7) is not true and (3.8) is true. Then there exists t1 2 ðt 0 ; þ1ÞT and i0 2 f1; 2; . . . ; ng such that

jui0 ðt 1 Þj P pMea ðt 1 ; t0 Þku  wkX ; juk ðtÞj < pMea ðt; t 0 Þku  wkX ;

jui0 ðtÞj < pMea ðt; t0 Þku  wkX ;

t 2 ðt 0 ; t 1 ÞT ;

for k – i0 ; t 2 ðt 0 ; t1 T ; k ¼ 1; 2; . . . ; n:

Therefore, there must be a constant a1 P 1 such that

jui0 ðt 1 Þj ¼ a1 pMea ðt 1 ; t0 Þku  wkX ; juk ðtÞj < a1 pMea ðt1 ; t 0 Þku  wkX ;

jui0 ðtÞj < a1 pMea ðt; t 0 Þku  wkX ; for k – i0 ; t 2 ðt 0 ; t1 T ; k ¼ 1; 2; . . . ; n:

t 2 ðt 0 ; t1 ÞT ;

688

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

Note that, in view  of (3.4), we have that

( Z Z t1  s  jui0 ðt1 Þj ¼ ui0 ðt 0 Þeci ðt1 ; t 0 Þ þ eci ðt 1 ; rðsÞÞ cþi0 0 0  ssi t0

0

ðsÞÞ

uDi0 ð#ÞD# þ

n X

ai0 j ðsÞ fj ðyj ðs  di0 j ðsÞÞÞ  fj ðxj ðs  di0 j ðsÞÞÞ j¼1

n n X n   X X þ ei0 j ðsÞ hj ðyDj ðs  hi0 j ðsÞÞÞ  hj ðxDj ðs  hi0 j ðsÞÞÞ þ bi0 jl ðsÞ g j ðyj ðs  ri0 jl ðsÞÞÞg l ðyl ðs  fi0 jl ðsÞÞÞ j¼1

j¼1 l¼1

  g j ðxj ðs  ri0 jl ðsÞÞÞg l ðxl ðs  fi0 jl ðsÞÞÞ Ds 6 eci ðt 1 ; t0 Þku  wkX þ a1 pMea ðt 1 ; t 0 Þku  wkX 0 Z ( Z n  t1 s X  þ g  e ðt ; rðsÞÞea ðt 1 ; rðsÞÞ ci0 ea ðrðsÞ; #ÞD# þ aþi0 j Lj ea ðrðsÞ; s  di0 j ðsÞÞ  t0 ci0 1 ssi ðsÞÞ j¼1 0  )  n n X n X X

 þ h þ eþi0 j Lj ea ðrðsÞ; s  hi0 j ðsÞÞ þ bi0 jl N j Hl ea ðrðsÞ; s  fi0 j ðsÞÞ þ Nl Hj ea ðrðsÞ; s  ri0 j ðsÞÞj Dsg   j¼1 j¼1 l¼1 Z t1 n  6 eci ðt1 ; t 0 Þku  wkX þ a1 pMea ðt 1 ; t 0 Þku  wkX  g  eci a ðt1 ; rðsÞÞ cþi0 sþi0 ea ðrðsÞ; s  si0 ðsÞÞ 0

0

t0

n X

n X

n X n X

j¼1

j¼1

j¼1 l¼1

aþi0 j Lj ea ðrðsÞ; s  di0 j ðsÞÞ þ

þ

eþi0 j Lhj ea ðrðsÞ; s  hi0 j ðsÞÞ þ

þ bi0 jl Nj Hl ea ðrðsÞ; s  fi0 j ðsÞÞ

Z t1   eci a ðt 1 ; rðsÞÞ þNl Hj ea ðrðsÞ; s  ri0 j ðsÞÞ Dsgj 6 eci ðt 1 ; t0 Þku  wkX þ a1 pMea ðt 1 ; t 0 Þku  wkX j 0 0 t0 (   X     n n X  cþi0 sþi0 exp aðsþi0 þ suplðsÞÞ þ aþi0 j Lj exp aðdþi0 j þ suplðsÞÞ þ eþi0 j Lhj exp aðhþi0 j þ suplðsÞÞ s2T

s2T

j¼1

s2T

j¼1

     ) þ þ þ bi0 jl  Nj Hl exp aðfi0 j þ suplðsÞÞ þ N l Hj exp aðri0 j þ suplðsÞÞ Dsgj

n X n X

þ

s2T

j¼1 l¼1

(

¼ a1 pMea ðt 1 ; t0 Þku  wkX

1

a1 pM

eci

s2T

" n n o X n o cþi0 sþi0 exp asþi0 þ aþi0 j Lj exp adþi0 j a ðt 1 ; t 0 Þ þ exp asuplðsÞ 

0

s2T

j¼1

n n X n n o X n o n o X þ þ eþi0 j Lhj exp ahþi0 j þ bi0 jl ðNj Hl exp afþi0 j þ Nl Hj exp arþi0 j Þ j¼1

j¼1 l¼1

    1 < a1 pMea ðt 1 ; t 0 Þku  wkX eci a ðt 1 ; rðsÞÞDs eðci aÞ ðt1 ; t 0 Þ þ exp asuplðsÞ  0 0 M s2T t "0 n n n X n n o X n o X n o X  n o þ  cþi0 sþi0 exp asþi0 þ aþi0 j Lj exp adþi0 j þ eþi0 j Lhj exp ahþi0 j þ bi0 jl N j Hl exp afþi0 j Z

t1

j¼1

j¼1

j¼1 l¼1

 Z t1

1 þNl Hj exp arþi0 j ¼ a1 pMea ðt 1 ; t 0 Þku  wkX ðci0  aÞ eðci aÞ ðt1 ; rðsÞÞDs 0 ðci0  aÞ t0   82 > > n n <6 1 exp asuplðsÞ n o X n o X n o s2T  6  cþi0 sþi0 exp asþi0 þ aþi0 j Lj exp adþi0 j þ eþi0 j Lhj exp ahþi0 j  4 > M c i0 a > j¼1 j¼1 :   !# exp a sup l ðsÞ n n  n o n o   n o XX þ s2T bi0 jl Nj Hl exp afþi0 j þ Nl Hj exp arþi0 j cþi0 sþi0 exp asþi0 eðci aÞ ðt 1 ; t0 Þ þ þ 0 c i0  a j¼1 l¼1 !) n n n X n n o X n o X  n o n o X þ þ þ þ þ þ h þ þ ai0 j Lj exp adi0 j þ ei0 j Lj exp ahi0 j þ bi0 jl Nj Hl exp afi0 j þ Nl Hj exp ari0 j n

o

j¼1

j¼1

  a1 pMea ðt 1 ; t 0 Þku  wkX ;

j¼1 l¼1

which is a contradiction. Case Two: (3.8) is not true and (3.7) is true. Then there exists t 2 2 ðt0 ; þ1ÞT and i1 2 f1; 2; . . . ; ng such that

juDi1 ðt2 Þj P pMea ðt 2 ; t 0 Þjju  wjjX ; juDk ðtÞj < pMea ðt; t0 Þjju  wjjX ;

juDi1 ðtÞj < pMea ðt; t 0 Þjju  wjjX ;

t 2 ðt 0 ; t2 ÞT ;

for k – i1 ; t 2 ðt 0 ; t2 T ; k ¼ 1; 2; . . . ; n:

Hence, there must be a constant a2 P 1 such that

juDi1 ðt2 Þj ¼ a2 pMea ðt 2 ; t 0 Þjju  wjjX ;

juDi1 ðtÞj < a2 pMea ðt; t0 Þjju  wjjX ;

t 2 ðt 0 ; t 2 ÞT ;

689

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

juDk ðtÞj < a2 pMea ðt; t0 Þjju  wjjX ;

for k – i1 ; t 2 ðt 0 ; t 2 T ; k ¼ 1; 2; . . . ; n:

Note that, in view of (3.4), we have  Z t2   juDi1 ðt2 Þj ¼ ci1 ðt2 Þui1 ðt0 Þeci ðt2 ;t 0 Þ þ ci1 ðt2 Þ 1  t 2  si

1

ðt2 Þ

uDi1 ðsÞDs þ

n X

ai1 j ðt2 Þ fj ðyj ðt2  di1 j ðt2 ÞÞÞ  fj ðxj ðt2  di1 j ðt2 ÞÞÞ j¼1

n n X n   X X bi1 jl ðt2 Þ g j ðyj ðt2  ri1 jl ðt2 ÞÞÞg l ðyl ðt2  fi1 jl ðt2 ÞÞÞ þ ei1 j ðt2 Þ hj ðyDj ðt2  hi1 j ðt2 ÞÞÞ  hj ðxDj ðt 2  hi1 j ðt 2 ÞÞÞ þ j¼1

j¼1 l¼1

g j ðxj ðt2  ri1 jl ðt2 ÞÞÞg l ðxl ðt 2  fi1 jl ðt2 ÞÞÞ  ci1 ðt2 Þ þ

Z

t2

(

Z

eci ðt2 ; rðsÞÞ ci1 ðsÞ 1

t0

s

ssi ðsÞÞ

uDi1 ð#ÞD#

1

n n   X

X ai1 j ðsÞ fj ðyj ðs  di1 j ðsÞÞÞ  fj ðxj ðs  di1 j ðsÞÞÞ þ ei1 j ðsÞ hj ðyDj ðs  hi1 j ðsÞÞÞ  hj ðxDj ðs  hi1 j ðsÞÞÞ j¼1

j¼1

)  n X n  X

 þ bi1 jl ðsÞ g j ðyj ðs  ri1 jl ðsÞÞÞg l ðyl ðs  fi1 jl ðsÞÞÞ  g j ðxj ðs  ri1 jl ðsÞÞÞg l ðxl ðs  fi1 jl ðsÞÞÞ Ds  j¼1 l¼1 6 cþi1 eci ðt 2 ; t0 Þku  wkX þ a2 pMea ðt2 ;t0 Þku  wkX cþi1

Z

1

s

ea ðrðsÞ;#ÞD# þ

ssi ðsÞ

j¼1

1

þ

n X

eþi1 j Lhj ea ð

rðt2 Þ; t2  hi1 j ðt2 ÞÞ þ

j¼1

þcþi1

(Z

a2 pMea ðt2 ; t0 Þku  wkX

n X n X

þ bi1 jl Nj Hl ea ð

rðt2 Þ; t2  fi1 j ðt2 ÞÞ þ Nl Hj ea ðrðt2 Þ; t2  ri1 j ðt2 ÞÞ

j¼1 l¼1 t2

"

eci ðt2 ; rðsÞÞea ðt2 ; rðsÞÞ  cþi1 1

t0

n X aþi1 j Lj ea ðrðt 2 Þ;t2  di1 j ðt2 ÞÞ

Z

s

ea ðrðsÞ;hÞDh þ

ssi ðsÞ



!

n X aþi1 j Lj ea ðrðsÞ; s  di1 j ðsÞÞ j¼1

1

# ) n n X n X X

þ þ eþi1 j Lhj ea ðrðsÞ;s  hi1 j ðsÞÞ þ bi1 jl Nj Hl ea ðrðsÞ; s  fi1 j ðsÞÞ þ Nl Hj ea ðrðsÞ; s  ri1 j ðsÞÞj Ds j¼1

j¼1 l¼1 n X

6 cþi1 eci ðt 2 ; t0 Þku  wkX þ a2 pMea ðt2 ;t0 Þku  wkX cþi1 sþi1 ea ðt2 ;t 2  si1 ðt 2 ÞÞ þ 1

aþi1 j Lj ea ðrðt2 Þ; t2  di1 j ðt2 ÞÞ

j¼1

n n X n X X

þ þ eþi1 j Lhj ea ðrðt2 Þ; t2  hi1 j ðt2 ÞÞ þ bi1 jl Nj Hl ea ðrðt2 Þ; t2  fi1 j ðt2 ÞÞ þ Nl Hj ea ðrðt2 Þ; t2  ri1 j ðt2 ÞÞ j¼1

(Z

þcþi1 a2 pMea ðt2 ; t0 Þku  wkX

j¼1 l¼1 t2

t0

eci

1

" þ þ a ðt 2 ; rðsÞÞ  c i1 si1 ea ðrðsÞ;s  si1 ðsÞÞ þ

!

n X aþi1 j Lj ea ðrðsÞ;s  di1 j ðsÞÞ j¼1

# ) n n X n X X

þ þ eþi1 j Lhj ea ðrðsÞ;s  hi1 j ðsÞÞ þ bi1 jl Nj Hl ea ðrðsÞ; s  fi1 j ðsÞÞ þ Nl Hj ea ðrðsÞ; s  ri1 j ðsÞÞj Ds j¼1

j¼1 l¼1

6 cþi1 eci ðt 2 ; t0 Þku  wkX þ a2 pMea ðt2 ;t0 Þku  wkX cþi1 sþi1 expfasþi1 g þ 1

n X aþi1 j Lj expfadþi1 j g j¼1

!  n n X n   X X þ þ þ þ h þ þ ei1 j Lj expfahi1 j g þ bi1 jl Nj Hl expfafi1 j g þ Nl Hj expfari1 j g  1 þ cþi1 expfasuplðsÞÞg j¼1



Z

eci

1

t0

þ

s2T

j¼1 l¼1



t2 a ðt 2 ;

n X

eþi1 j Lhj expf

rðsÞÞDs < a2 pMea ðt2 ;t0 Þku  wkX

a

j¼1

hþi1 j g þ

n X n X

þ bi1 jl



Nj Hl expfa

( þ c i1 M

eci

fþi1 j g þ Nl Hj expf

1

a ðt 2 ; t 0 Þ þ

cþi1 sþi1 expfasþi1 g þ

n X aþi1 j Lj expfadþi1 j g j¼1

!  arþi1 j g  ð1 þ cþi1 expfasuplðsÞÞg

j¼1 l¼1

s2T

  82 > >  Z t2 <6 1 exp asuplðsÞ  n o s2T eci a ðt2 ; rðsÞÞDsÞ 6 a2 pMea ðt2 ; t0 Þku  wkX 6 cþi0 sþi0 exp asþi0   4M  1 > c i0 a t0 > : !# n n n X n n o X n o X  n o n o X þ ciþ eðci þ aþi0 j Lj exp adþi0 j þ eþi0 j Lhj exp ahþi0 j þ bi0 jl Nj Hl exp afþi0 j þ Nl Hj exp arþi0 j j¼1

0

B þB @1 þ

ciþ exp 1



j¼1

1

j¼1 l¼1

asuplðsÞ C n n n o X n o X n o s2T C cþ sþ exp asþ þ aþi0 j Lj exp adþi0 j þ eþi0 j Lhj exp ahþi0 j i0 i0 i0 A c i0  a j¼1 j¼1

n X n   X þ þ bi0 jl Nj Hl expfafþi0 j g þ Nl Hj expfarþi0 j g j¼1 l¼1

!) < a2 pMea ðt2 ;t 0 Þku  wkX ;

1

0

aÞ ðt 2 ;t 0 Þ

690

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

1

x 10 −3 x1 x2

0.5

0

−0.5

−1

−1.5

0

10

20

time t

30

40

50

Fig. 1. Continuous situation (T ¼ R): x1 ; x2 with time t.

2

x 10

−4

1.5

x 2

1 0.5 0 −0.5 −1 −1

−0.5

0

0.5

1

1.5

x1

2

x 10 −4

Fig. 2. Continuous situation (T ¼ R): x1 ; x2 .

which is also a contradiction. By the above two cases, for other cases of negative proposition of (3.6), we can obtain a contradiction. Therefore, (3.6) holds. Let p ! 1, then (3.5) holds. We can take k ¼ a, then k > 0 and k 2 Rþ . Hence, we have that

jjujjX 6 Mku  wkX ek ðt; t0 Þ;

t 2 ½h; 1ÞT ; t P t0 ;

which means that, the almost automorphic solution xðtÞ of (1.1) is globally exponentially stable. The proof of Theorem 3.2 is completed. h Remark 3.1. According to Theorems 3.1 and 3.2, we see that the existence and global exponential stability of almost automorphic solutions for system (1.1) only depends on time delays si (the delays in the leakage term) and does not depend on time delays dij ; hij ; rijl and fijl , which means that the delays in the leakage term do harm to the existence and global exponential stability of almost automorphic solutions. Remark 3.2. If we take

si ðtÞ  0; eij ðtÞ  0, dij ðtÞ ¼ rijl ðtÞ ¼ fijl ðtÞ  0, then (1.1) reduces to the following form

xDi ðtÞ ¼ ci ðtÞxi ðtÞ þ

n n X n X X aij ðtÞfj ðxj ðtÞÞ þ bijl ðtÞg j ðxj ðtÞÞg l ðxl ðtÞÞ þ Ii ðtÞ; j¼1

j¼1 l¼1

t 2 T:

ð3:9Þ

691

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

1

x 10

−3 x1 x2

0.5

0

−0.5

−1

−1.5

0

10

20

time t

30

40

50

Fig. 3. Discrete situation (T ¼ Z): x1 ; x2 with time t.

2

x 10

−4

1.5

x 2

1 0.5 0 −0.5 −1 −1

−0.5

0

0.5 x1

1

1.5 x 10

2 −4

Fig. 4. Discrete situation (T ¼ Z): x1 ; x2 .

In [52], the authors studied the existence of almost automorphic solutions to (3.9) under the assumption that fj is bounded and they did not study the stability of the almost automorphic solution to (3.9). So, our results of this paper improve and extend this corresponding result in [52] since we removed the assumption of the boundedness of fj ðxÞ and we also study the global exponential stability of the almost automorphic solution.

4. Numerical example In this section, we present an example to illustrate the feasibility of our results obtained in Section 3. Example 4.1. Let n ¼ 2. Consider the following neural networks on almost periodic time scale T:

xDi ðtÞ ¼ ci ðtÞxi ðt  si ðtÞÞ þ

2 2 X X aij ðtÞfj ðxj ðt  dij ðtÞÞÞ þ eij ðtÞhj ðxDj ðt  dij ðtÞÞÞ j¼1

j¼1

2 X 2 X bijl ðtÞg j ðxj ðt  rijl ðtÞÞÞg l ðxl ðt  fijl ðtÞÞÞ þ Ii ðtÞ; þ j¼1 l¼1

t 2 T; i ¼ 1; 2;

ð4:1Þ

692

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

where

f1 ðu1 Þ ¼ g 1 ðu1 Þ ¼ sin 0:2u1 ; a11 ðtÞ ¼ 0:1 þ 0:05 cos t;

f 2 ðu2 Þ ¼ g 2 ðu2 Þ ¼ sin 0:1u2 ;

pffiffiffi a12 ðtÞ ¼ 0:25 þ 0:01 cos 2t;

pffiffiffi a22 ðtÞ ¼ 0:05 þ 0:02 cos 2t; e11 ðtÞ ¼ 0:2 þ 0:05 cos t;

pffiffiffi c1 ðtÞ ¼ 0:6 þ 0:3 sin 3t;

pffiffiffi e12 ðtÞ ¼ 0:35 þ 0:01 cos 2t;

pffiffiffi e22 ðtÞ ¼ 0:04 þ 0:01 cos 2t; pffiffiffi I1 ðtÞ ¼ 0:03 þ 0:02 sin 3t;

pffiffiffi a21 ðtÞ ¼ 0:03 þ 0:02 cos 3t; 1 c2 ðtÞ ¼ 0:5 þ 0:1 sin t; 3 pffiffiffi e21 ðtÞ ¼ 0:03 þ 0:02 cos 3t;

pffiffiffi

1 3

s1 ðtÞ ¼ 0:2 þ 0:1 sin 3t; s2 ðtÞ ¼ 0:3 þ 0:1 sin t; pffiffiffi I2 ðtÞ ¼ 0:04 þ 0:01 cos 2t;

pffiffiffi b111 ðtÞ ¼ b222 ðtÞ ¼ 0:12 þ 0:01 sin 2t;

4 b112 ðtÞ ¼ b212 ðtÞ ¼ 0:02 þ 0:01 cos t; 3

pffiffiffi b121 ðtÞ ¼ b221 ðtÞ ¼ 0:12 þ 0:02 cos 3t;

pffiffiffi b122 ðtÞ ¼ b211 ðtÞ ¼ 0:03 þ 0:02 sin 5t:

By calculating, we have

L1 ¼ H1 ¼ N1 ¼ 0:2;

L2 ¼ H2 ¼ N2 ¼ 0:1;

aþ21 ¼ 0:05;

aþ22 ¼ 0:07;

cþ1 ¼ 0:9;

eþ11 ¼ 0:25;

eþ12 ¼ 0:36;

eþ21 ¼ 0:05;

Iþ1 ¼ 0:05; þ

þ

Iþ2 ¼ 0:05;

b121 ¼ b221 ¼ 0:14;

þ

þ

aþ11 ¼ 0:15;

c1 ¼ 0:3;

þ

cþ2 ¼ 0:6; c2 ¼ 0:4;

eþ22 ¼ 0:05;

b111 ¼ b222 ¼ 0:13;

þ

aþ12 ¼ 0:26;

sþ1 ¼ 0:3; sþ2 ¼ 0:4; þ

b112 ¼ b212 ¼ 0:03;

þ

b122 ¼ b211 ¼ 0:05:

We can verify that all assumptions in Theorems 3.1 and 3.2 are satisfied. Therefore, we have that (1.1) has an almost automorphic solution, which is globally exponentially stable (see Figs. 1–4). 5. Conclusion Using the existence of the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, we obtain some sufficient conditions for the existence and exponential stability of almost automorphic solutions for a class of neutral type high-order Hopfield neural networks with time-varying delays in leakage terms on time scales. To the best of our knowledge, this is the first paper to study the almost automorphic solution for neutral type high-order Hopfield neural networks with time-varying delays in leakage terms on time scales. The method of this paper can be applied to study other neural networks, such as BAM neural networks, Shunting Inhibitory cellular neural networks and so on. References [1] J.H. Park, C.H. Park, O.M. Kwon, S.M. Lee, A new stability criterion for bidirectional associative memory neural networks of neutral-type, Appl. Math. Comput. 199 (2008) 716–722. [2] R. Rakkiyappan, P. Balasubramaniam, New global exponential stability results for neutral type neural networks with distributed time delays, Neurocomputing 71 (2008) 1039–1045. [3] R. Rakkiyappan, P. Balasubramaniam, LMI conditions for global asymptotic stability results for neutral-type neural networks with distributed time delays, Appl. Math. Comput. 204 (2008) 317–324. [4] L. Hua, H.J. Gao, W.X. Zheng, Novel stability of cellular neural networks with interval time-varying delay, Neural Networks 21 (10) (2008) 1458–1463. [5] J.D. Cao, G. Feng, Y.Y. Wang, Multistability and multiperiodicity of delayed Cohen–Grossberg neural networks with a general class of activation functions, Physica D: Nonlinear Phenom. 237 (13) (2008) 1734–1749. [6] Y.K. Li, Global exponential stability of BAM neural networks with delays and impulses, Chaos Solitons Fract. 24 (2005) 279–285. [7] X.F. Liao, J.B. Yu, Robust stability for interval Hopfield neural networks with time delays, IEEE Trans. Neural Networks 9 (1998) 1042–1046. [8] B.J. Xu, X.Z. Liu, X.X. Liao, Global asymptotic stability of high-order Hopfield type neural networks with time delays, Comput. Math. Appl. 45 (2003) 1729–1737. [9] B.J. Xu, X.Z. Liu, X.X. Liao, Global exponential stability of high order Hopfield type neural network, Appl. Math. Comput. 174 (1) (2006). [10] H. Xiang, K.M. Yan, B.Y. Wang, Existence and global exponential stability of periodic solution for delayed high-order Hopfield-type neural networks, Phys. Lett. A 352 (4–5) (2006). [11] Y.K. Li, L.L. Zhao, P. Liu, Existence and exponential stability of periodic solution of high-Order Hopfield neural network with delays on time scales, Discrete Dyn. Nat. Soc. 2009 (2009) (Article ID 573534, 18 p). [12] C.X. Ou, Anti-periodic solutions for high-order Hopfield neural networks, Comput. Math. Appl. 56 (2008) 1838–1844.

Y. Li, L. Yang / Applied Mathematics and Computation 242 (2014) 679–693

693

[13] Y.H. Yua, M.S. Cai, Existence and exponential stability of almost-periodic solutions for high-order Hopfield neural networks, Math. Comput. Modell. 47 (2008) 943–951. [14] B. Xiao, H. Meng, Existence and exponential stability of positive almost periodic solutions for high-order Hopfield neural networks, Appl. Math. Modell. 33 (1) (2009) 532–542. [15] J.H. Park, C.H. Park, O.M. Kwon, S.M. Lee, A new stability criterion for bidirectional associative memory neural networks of neutral-type, Appl. Math. Comput. 199 (2008) 716–722. [16] R. Rakkiyappan, P. Balasubramaniam, New global exponential stability results for neutral type neural networks with distributed time delays, Neurocomputing 71 (2008) 1039–1045. [17] J. Liu, G. Zong, New delay-dependent asymptotic stability conditions concerning BAM neural networks of neutral type, Neurocomputing 72 (2009) 2549–2555. [18] R. Samidurai, S.M. Anthoni, K. Balachandran, Global exponential stability of neutral-type impulsive neural networks with discrete and distributed delays, Nonlinear Anal.: Hybrid Syst. 4 (2010) 103–112. [19] R. Rakkiyappan, P. Balasubramaniam, J. Cao, Global exponential stability results for neutral-type impulsive neural networks, Nonlinear Anal. Real World Appl. 11 (2010) 122–130. [20] Y.K. Li, L. Zhao, X. Chen, Existence of periodic solutions for neutral type cellular neural networks with delays, Appl. Math. Modell. 36 (2012) 1173–1183. [21] C. Bai, Global stability of almost periodic solutions of Hopfield neural networks with neutral time-varying delays, Appl. Math. Comput. 203 (2008) 72– 79. [22] 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. [23] K. Wang, Y. Zhu, Stability of almost periodic solution for a generalized neutral-type neural networks with delays, Neurocomputing 73 (2010) 3300– 3307. [24] P. Balasubramaniam, V.M. Revathi, J.H. Park, L2  L1 Filtering for neutral Markovian switching system with mode-dependent time-varying delays and partially unknown transition probabilities, Appl. Math. Comput. 219 (2013) 9524–9542. [25] S. Lakshmanan, J.H. Park, H.Y. Jung, O.M. Kwon, R. Rakkiyappan, A delay partitioning approach to delay-dependent stability analysis for neutral type neural networks with discrete and distributed delays, Neurocomputing 111 (2013) 81–89. [26] J.H. Park, Synchronization of cellular neural networks of neutral type via dynamic feedback controller, Chaos Solitons Fract. 42 (2009) 1299–1304. [27] S.M. Lee, O.M. Kwon, J.H. Park, A novel delay-dependent criterion for delayed neural networks of neutral type, Phys. Lett. A 374 (2010) 1843–1848. [28] X. Li, J. Cao, Delay-dependent stability of neural networks of neutral type with time delay in the leakage term, Nonlinearity 23 (2010) 1709–1726. [29] P. Balasubramanian, G. Nagamani, R. Rakkiyappan, Passivity analysis for neural networks of neutral type with Markovian jumping parameters and time delay in the leakage term, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 4422–4437. [30] P. Balasubramaniam, M. Kalpana, R. Rakkiyappan, Global asymptotic stability of BAM fuzzy cellular neural networks with time delay in the leakage term, discrete and unbounded distributed delays, Math. Comput. Modell. 53 (5–6) (2011) 839–853. [31] L. Duan, L.H. Huang, Global exponential stability of fuzzy BAM neural networks with distributed delays and time-varying delays in the leakage terms, Neural Comput. Appl. 23 (1) (2013) 171–178. [32] B.W. Liu, Global exponential stability for BAM neural networks with time-varying delays in the leakage terms, Nonlinear Anal. Real World Appl. 14 (1) (2013) 559–566. [33] X. Li, R. Rakkiyappan, P. Balasubramanian, Exstence and global stability analysis of equilibrium of fuzzy cellular neural networks with time delay in the leakage term under impulsive perturbations, J. Frankl. Inst. 348 (2011) 135–155. [34] Y.K. Li, L. Yang, L.J. Sun, Existence and exponential stability of an equilibrium point for fuzzy BAM neural networks with time-varying delays in leakage terms on time scales, Adv. Difference Equ. 2013 (2013) 218. [35] S.G. Peng, Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms, Nonlinear Anal. Real World Appl. 11 (3) (2010) 2141–2151. [36] H. Zhang, J.Y. Shao, Almost periodic solutions for cellular neural networks with time-varying delays in leakage terms, Appl. Math. Comput. 219 (24) (2013) 11471–11482. [37] Y.K. Li, Y.Q. Li, Existence and exponential stability of almost periodic solution for neutral delay BAM neural networks with time-varying delays in leakage terms, J. Frankl. Inst. 350 (9) (2013) 2808–2825. [38] S. Hilger, Analysis on measure chains – a unified approach to continuous and discrete calculus, Result. Math. 18 (1990) 18–56. [39] Y.K. Li, X. Chen, L. Zhao, Stability and existence of periodic solutions to delayed Cohen–Grossberg BAM neural networks with impulses on time scales, Neurocomputing 72 (2009) 1621–1630. [40] Z.Q. Zhang, K.Y. Liu, Existence and global exponential stability of a periodic solution to interval general bidirectional associative memory (BAM) neural networks with multiple delays on time scales, Neural Networks 24 (2011) 427–439. [41] Y.K. Li, C. Wang, Almost periodic solutions of shunting inhibitory cellular neural networks on time scales, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 3258–3266. [42] Y.K. Li, L. Yang, Almost periodic solutions for neutral-type BAM neural networks with delays on time scales, J. Appl. Math. (2013) (Article ID 942309, 13 p). [43] Y.K. Li, L. Yang, W.Q. Wu, Anti-periodic solutions for a class of Cohen–Grossberg neural networks with time-varying delays on time scales, Int. J. Syst. Sci. 42 (7) (2011) 1127–1132. [44] Y.K. Li, J.Y. Shu, Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 3326–3336. [45] G.M. N’Guérékata, Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, Semigroup Forum 69 (2004) 80–86. [46] J.A. Goldstein, G.M. N’Guérékata, Almost automorphic solutions of semilinear evolution equations, Proc. Am. Math. Soc. 133 (2005) 2401–2408. [47] K. Ezzinbi, S. Fatajou, G.M. N’Guérékata, Almost automorphic solutions for some partitial functional differential equations, J. Math. Anal. Appl. 328 (2007) 344–358. [48] F. Chérif, Z.B. Nahia, Global attractivity and existence of weighted pseudo almost automorphic solution for GHNNs with delays and variable coefficients, Gulf J. Math. 1 (2013) 5–24. [49] F. Chérif, Sufficient conditions for global stability and existence of almost automorphic solution of a class of RNNs, Differ. Equ. Dyn. Syst. 22 (2) (2014) 191–207. [50] G.M. N’Guérékata, Topics in Almost Automorphy, Springer, New York, 2005. [51] C. Wang, Y.K. Li, Weighted pseudo almost automorphic functions with applications to abstract dynamic equations on time scales, Annales Polonici Mathematici 108 (2013) 225–240. [52] C. Lizama, J.G. Mesquita, Almost automorphic solutions of dynamic equations on time scales, J. Funct. Anal. 265 (2013) 2267–2311. [53] M. Bohner, A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser, Boston, 2001. [54] Y.K. Li, C. Wang, Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales, Abstract Appl. Anal. 22 (2011) (Article ID 341520).