Almost periodic solutions for neutral type BAM neural networks with distributed leakage delays on time scales

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Almost periodic solutions for neutral type BAM neural networks with distributed leakage delays on time scales Hui Zhou, Zongfu Zhou, Wei Jiang

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S0925-2312(15)00034-X http://dx.doi.org/10.1016/j.neucom.2015.01.013 NEUCOM15051

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Neurocomputing

Received date: 14 August 2014 Revised date: 23 November 2014 Accepted date: 7 January 2015 Cite this article as: Hui Zhou, Zongfu Zhou, Wei Jiang, Almost periodic solutions for neutral type BAM neural networks with distributed leakage delays on time scales, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.01.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Almost periodic solutions for neutral type BAM neural networks with distributed leakage delays on time scales Hui Zhou1† Zongfu Zhou2

∗ Wei Jiang2

1. School of Mathematics and Statistics, Hefei Normal University, Hefei, 230601, China 2. School of Mathematical Science, Anhui University, Hefei, 230601, China

Abstract: This paper is concerned with a class of neutral type BAM neural networks with distributed leakage delays on time scales. By using the exponential dichotomy theory, Lyapunov functional method and contraction mapping principle, some sufficient conditions for the existence and global exponential stability of the almost periodic solutions for this model are obtained. An example is given to illustrate effectiveness of the obtained theoretical results. Keywords: Distributed Leakage Delay; Almost Periodic Solution; Neutral BAM Neural Network; Time Scale.

1 . Introduction Bidirectional associative memory (BAM) neural networks were first introduced by Ksoko [1]. The dynamical properties for BAM neural networks have been extensively investigated due to their applications in many fields such as pattern recognition, optimization problems, image processing, associative memories and many other fields. Since time delays frequently cause oscillation or instability, some dynamic properties of various time delayed neural networks have been widely studied by many mathematics and engineer communities [2-29]. Neutral type neural networks were always considered in many works because the systems contain some information about complex neural reactions. So the neutral type neural networks are studied to be more accordant with the reality [4, 7, 13, 15, 17-20 ]. In [7], the authors considered the existence and exponential stability of almost periodic solution for neutral delay BAM neural networks with time-varying delays in leakage terms. As the authors in [2, 3, 5] pointed out, the time-delays in the leakage terms have a great impact on the dynamic of neural networks. There were some works about the neural networks with timedelays in leakage terms, the readers can refer to [2, 3, 5, 7,12, 13, 16, 17]. It is well known that the theory of time scales unifies and extends that of discrete and continuous analysis [30-32]. The theory of dynamic equations on time scales is undergoing a rapid development as it provides a powerful tool to generalize the discussion of these systems on time scales. The dynamic systems on time scales have tremendous applications ∗ This

work is supported by the National Natural Science Fundation of China (11371027, 11201109) and the Natural Science

Fundation of Anhui Province (1208085MA13, KJ2012Z335). † Corresponding author E-mail address: [email protected].

1

in some mathematical models, such as neural networks, population dynamics, physics technology, and so on. Since Li and Wang [33] introduced the concept of almost periodic functions on time scales, some studies of almost periodic type systems on time scales were published [34-39]. Wang [37] discussed almost periodic impulsive BAM neural networks with variable delays on time scales, and established some conditions for the existence and stability of almost periodic solution of the systems on time scales. Until now, to the best of our knowledge, there are few published papers considering the neutral type almost periodic BAM neural networks with distributed leakage delays on time scales. Motivated by the above discussions, in this paper, we consider a class of neutral type BAM neural networks with distributed leakage delays on time scales: ⎧ m ∞  ⎪ ∇ ⎪ x (t) = −a (t) k (s)x (t − s)∇s + αij (t)fj (yj (t − τij (t))) ⎪ i i i i 0 ⎪ ⎪ j=1 ⎪ ⎪ m ⎪  ⎪ ⎪ + pij (t)gj (yj∇ (t − σij (t))) + Ii (t), i = 1, 2, · · · , n, ⎨ j=1

n ∞ ⎪ ∇ ⎪ ˜j (s)yj (t − s)∇s +  βji (t)f˜i (xi (t − δji (t))) y (t) = −b (t) k ⎪ j j ⎪ 0 ⎪ i=1 ⎪ ⎪ n ⎪  ⎪ ∇ ⎪ + qji (t)˜ gi (xi (t − ηji (t))) + Jj (t), j = 1, 2, · · · , m, ⎩

(1.1)

i=1

where n and m are the number of neurons in layers; xi (t) and yj (t) represent the the state of vector of the ith neuron and the jth neuron at time t; ai (t) > 0 and bj (t) > 0 stand the rate at which the ith neuron and the jth neuron reset their potential to the resting state in isolation when they are disconnected from the network and external inputs at time t; fj , gj , f˜i and g˜i are the input-output activation functions; αij (t) and pij (t) mean the elements of feedback templates at time t, βji (t) and qji (t) mean the elements of feed-forward templates at time t; continuous leakage delays kernel functions ki > 0 and k˜j > 0 satisfy that ki (s)eιs and k˜j (s)e˜ιs are intigrable on R+ for certain positive constants ι and ˜ι ; τij , σij , δji and ηji denote the transmission delays and satisfy t − τij (t) ∈ T, t − σij (t) ∈ T, t − δji (t) ∈ T and t − ηji (t) ∈ T; Ii and Jj are biases of the ith neuron and the jth neuron, i = 1, 2, · · · , n, j = 1, 2, · · · , m. The main aim of this paper is to study the existence and global exponential stability of almost periodic solutions for Eq. (1.1). The approach to prove the main results will be based on the theory of exponential dichotomy, Lyapunov functional method and fixed point theorems. Let T denote an almost periodic time scale. For convenience, denote by R = (−∞, +∞) and R+ = (0, +∞), for an almost periodic function f : T → R, set f + = sup |f (t)| and f − = inf |f (t)|. t∈T 1

T

t∈T

Set X = {φ = (ϕ1 , ϕ2 , · · · , ϕn , ψ1 , ψ2 , · · · , ψn ) |ϕi , ψj ∈ C (T, R), ϕi , ψj are almost periodic functions on T, i = 1, 2, · · · , n, j = 1, 2, · · · , m} with the norm φ = max{|ϕ|1 , |ψ|1 }, where |ϕ|1 = max{|ϕ|0 , |ϕ∇ |0 }, |ψ|1 = + ∇ ∇ + ∇ ∇ + max{|ψ|0 , |ψ ∇ |0 }, |ϕ|0 = max1≤i≤n ϕ+ i , |ϕ |0 = max1≤i≤n (ϕi ) , |ψ|0 = max1≤i≤n ψi , |ψ |0 = max1≤i≤n (ψi ) ,

C 1 (T, R) is the set of continuous functions with nabla derivatives on T. Then X is a Banach space. The initial conditions associated with Eq. (1.1) are xi (s) = ϕi (s), yj (s) = ψj (s), s ∈ (−∞, 0]T = {t|t ∈ (−∞, 0] ∩ T}, where ϕi , ψj ∈ C 1 ((−∞, 0]T , R), i = 1, 2, · · · , n, j = 1, 2, · · · , m. The rest of this paper is organized as follows: In Section 2, we introduce some definitions and notations, and state some preliminary results, which can be used to prove our results. In Section 3, we establish some sufficient conditions which ensure the existence and uniqueness of the almost periodic solution of Eq. (1.1). The global exponential stability of the system is derived in Section 4. An example is given to illustrate the feasibility of the obtained results in Section 5. 2

2. Preliminaries In this section, let us recall some definitions and lemmas of time scales , which are of importance in proving the main results of this paper. Let T be a time scale which is a closed subset of R. For t ∈ T, the forward and backward jump operators σ, ρ : T → T, respectively, defined by, σ(t) = inf{s ∈ T : s > t}, ρ(t) = sup {s ∈ T : s < t}. The point t ∈ T is called left-dense if t > inf T and ρ(t) = t, left-scattered if ρ(t) < t, right-dense if t < sup T and σ(t) = t, and right-scattered if σ(t) > t. If T has a right-scattered minimum m, set Tk := T\{m}, otherwise Tk := T. The backwards graininess ν : Tk → [0, +∞) is defined by ν(t) = t − ρ(t). A function f : T → R is called left-dense continuous (ld-continuous) provided it is continuous at left-dense point in T and its right-side limit exist at right-dense points in T. Definition2.1([30]). Let f : T → R be a function and t ∈ Tk . Then define f ∇ (t) to be the number (provided it exists) with the property that given any ε > 0, there exits a neighborhood U of t such that |f (ρ(t)) − f (s) − f ∇ (t)(ρ(t) − s)| ≤ ε|ρ(t) − s|, for all s ∈ U, f ∇ is called to be the nabla derivative of f at t. If f is ld-continuous, then there exists a function F such that F ∇ (t) = f (t), and we define  b f (t)∇t = F (b) − F (a). a

The function p is ν−regressive if 1 − ν(t)p(t) = 0 for all t ∈ Tk . Let us denote Rν = {p : T → R|p is ld-continous and ν-regressive}, Rν+ = Rν+ (T → R) = {p ∈ Rν : 1 − ν(t)p(t) > 0, for all t ∈ T}. The nabla exponential function is defined by  eˆp (t, s) = exp{

t s

ξˆν (τ )(p(τ ))∇τ },

for s, t ∈ T, where p ∈ Rν , the ν−-cylinder transformation is expressed by ⎧ ⎨ − log(1−hz) , if h = 0, h ˆ ξh (z) = ⎩ z, if h = 0. Definition2.2([30]). If p, q ∈ Rν , then a circle plus addition is defined by (p ⊕ν q)(t) := p(t) + q(t) − p . p(t)q(t)ν(t), for all t ∈ Tk . For p ∈ Rν , we defined a circle minus p by ν p := − 1−νp

Lemma2.1 ([30, 34]). If p, q ∈ Rν , and s, t, r ∈ T. Then (i) eˆ0 (t, s) ≡ 1 and eˆp (t, t) ≡ 1; ep (t, s); (ii) eˆp (ρ(t), s) = (1 − ν(t)p(t))ˆ ep (s, t); (iii) eˆp (t, s) = 1/ˆ ep (r, s) = eˆp (t, s); (iv) eˆp (t, r)ˆ ep (t, s). (v) (ˆ ep (t, s))∇ = p(t)ˆ Lemma2.2 ([30]). Let f, g be nabla differential functions on T. Then (i) (v1 f + v2 g)∇ = v1 f ∇ + v2 g ∇ , for any constants v1 , v2 ; (ii) (f g)∇ = f ∇ (t)g(t) + f (ρ(t))g ∇ (t) = f (t)g ∇ (t) + f ∇ (t)g(ρ(t)); t t (iii) If f and f ∇ are continuous, then ( a f (t, s)∇s)∇ = f (ρ(t), t) + a f (t, s)∇s. 3

Definition2.3([30]). A time scale T is called an almost periodic time scale if

:= {τ ∈ R : t ± τ ∈ T, ∀t ∈ T} =

{0}.

Definition2.4([33]). Let T be an almost periodic time scale . A function f ∈ C(T, R) is called an almost periodic function if the ε−translation set of f E{ε, f } = {τ ∈



: |f (t + τ ) − f (t)| < ε, ∀t ∈ T}

is a relatively dense set in T for all ε > 0; that is, for any given ε > 0, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains a τ (ε) ∈ E{ε, f } such that |f (t + τ ) − f (t)| < ε, ∀t ∈ T, where τ is called the ε−translation number of and l(ε) is called the inclusion length of E{ε, f }. Definition2.5([35]). Let A(t) be an n × n matrix-valued function on T. Then the linear system x∇ (t) = A(t)x(t), t ∈ T,

(2.1)

is said to admit an exponential dichotomy on T if there exist constants ki , αi , i = 1, 2, projection P and the fundamental solution matrix X(t) of Eq.(2.1) satisfying X(t)P X −1 (s) ≤ k1 eˆν α1 (t, s), s, t ∈ T, t ≥ s, X(t)(I − P )X −1 (s) ≤ k2 eˆν α2 (t, s), s, t ∈ T, t ≤ s. Lemma2.3 ([34, 37]). If the linear system (2.1) admits an exponential dichotomy, then the almost periodic system x∇ (t) = A(t)x(t) + g(t), t ∈ T,

(2.2)

has a unique almost periodic solution x(t), and 

t

x(t) =

X(t)P X

−1

−∞

 (ρ(s))g(s)∇s −

+∞ t

X(t)(I − P )X −1 ρ(s)g(s)∇s,

where X(t) is the fundamental solution matrix of Eq. (2.1). Lemma2.4 ([34, 37]). Assume ci (t) ∈ Rν+ (i = 1, 2, · · · , n) are almost periodic on T, and min {inf ci (t)} = 1≤i≤n t∈T

m ¯ > 0. Then the linear system x∇ (t) = diag(−c1 (t), −c2 (t), · · · , −cn (t))x(t) admits an exponential dichotomy on T. ∗ T Definition2.6. Let z ∗ = (x∗1 , x∗2 , ..., x∗n , y1∗ , y2∗ , ..., ym ) be the almost periodic solution of system (1.1). If

there exists a constant λ > 0 such that for every solution z(t) = (x1 (t), x2 (t), ..., xn (t), y1 (t), y2 (t), ..., ym (t))T of Eq. (1.1) with any initial value φ(s) = (ϕ1 (s), ϕ2 (s), ..., ϕn (s), ψ1 (s), ψ2 (s), ..., ψm (s))T satisfying xi (t) − x∗i (t) = O(ˆ eν λ (t, 0)), yj (t) − yj∗ (t) = O(ˆ eν λ (t, 0)), i = 1, 2, · · · , n, j = 1, 2, · · · , m. Then the solution z ∗ (t) is said to be global exponential stability. Throughout this paper, we assume that the following conditions hold:

4

(H1 ) ai (t), bj (t), τij (t), σij (t), δji (t), ηji (t) ∈ C(T, R+ ) are all almost periodic functions, i = 1, 2, · · · , n, j = 1, 2, · · · , m.

˜ (H2 ) fj , gj , f˜i , g˜i ∈ C(R, R) and there exist positive constants Lfj , Lgj , Lfj and Lgj˜ such that

|fj (u) − fj (v)| ≤ Lfj |u − v|, |gj (u) − gj (v)| ≤ Lgj |u − v|, ˜

|f˜j (u) − f˜j (v)| ≤ Lfj |u − v|, |˜ gj (u) − g˜j (v)| ≤ Lgj˜ |u − v|, for all u, v ∈ R, i = 1, 2, · · · , n, j = 1, 2, · · · , m.

3. Existence of almost periodic solutions In this section, we will state and prove our main results concerning the existence of almost periodic solutions of Eq. (1.1).

t 0 (t))T , where ϕ0i (t) = −∞ eˆ−ai (t, ρ(s))Ii (s)∇s, Let φ0 (t) = (ϕ01 (t), ϕ02 (t), ..., ϕ0n (t), ψ10 (t), ψ20 (t), ..., ψm t ψj0 (t) = −∞ eˆ−bj (t, ρ(s))Jj (s)∇s, i = 1, 2, · · · , n, j = 1, 2, · · · , m. Let L be a constant satisfying max{φ0 , gi (0)|} ≤ L. max |fj (0)|, max |gj (0)|, max |f˜i (0)|, max |˜

1≤j≤m

1≤j≤m

1≤j≤m

1≤j≤m

(H3 ) For each i ∈ {1, 2, · · · , n}, j ∈ {1, 2, · · · , m}, there exist positive constants ξi and ξ¯j such that ∞ b+ a+ Ω j i )Θi }, max { b−  ∞ kj (u)∇u , (1+ b− )Ωj }} < 12 , where Θi = a+ max{ max { a−  ∞Θki (u)∇u , (1+ a− i 0 ki (u)u∇u+ 1≤i≤n

ξi−1

i

0

i

1≤i≤n

i

j

0

j

j

m n m ¯−1 n   ξi−1  g + f + + ∞ ¯−1  ξi (β + Lf˜+p+ Lg˜ )+ ξj  (β + + (αij +p+ Lj +p+ ξ¯j (αij ij ), Ωj = bj 0 kj (u)u∇u+ ξj ji i ji i ji ij Lj )+ 2 2

j=1

j=1

+ ). qji

i=1

i=1

Theorem 3.1 Assume that the conditions (H1 )-(H3 ) hold. Then Eq.(1.1) has a unique almost periodic solution in X0 = {φ ∈ X| φ − φ0  ≤ L}, φ(t) = (ϕ1 (t), ϕ2 (t), · · · , ϕn (t), ψ1 (t), ψ2 (t), · · · , ψm (t))T . Proof. Set −1 x ¯i (t) = ξi−1 xi (t), y¯j (t) = ξ¯j yj (t),

then Eq. (1.1) can be transformed into the following system ⎧ ∞ ∞ t ⎪ x ¯∇ ¯i (t) + ai (t) 0 ki (s) t−s x ¯∇ ⎪ i (t) = −ai (t) 0 ki (s)∇s x i (u)∇u∇s ⎪ ⎪ m ⎪  ⎪ −1 ⎪ +ξi αij (t)fj (ξ¯j y¯j (t − τij (t))) ⎪ ⎪ ⎪ j=1 ⎪ ⎪ m ⎪  ⎪ ⎪ +ξi−1 pij (t)gj (ξ¯j y¯j∇ (t − σij (t))) + ξi−1 Ii (t), i = 1, 2, · · · , n, ⎨ j=1 ∞ ∞ t ⎪ ∇ ⎪ (t) = −b (t) k˜j (s)∇s y¯j (t) + bj (t) 0 k˜j (s) t−s y¯j∇ (u)∇u∇s y ¯ i ⎪ j 0 ⎪ ⎪ n ⎪ ⎪ ¯−1  βji (t)f˜i (ξi x ⎪ + ξ ¯i (t − δji (t))) ⎪ j ⎪ ⎪ i=1 ⎪ ⎪ m  ⎪ ⎪ ¯−1 ⎩ +ξ¯j−1 qji (t)˜ gi (ξi x ¯∇ i (t − ηji (t))) + ξj Jj (t), j = 1, 2, · · · , m.

(3.1)

i=1

For any given φ ∈ X, we consider the following auxiliary almost periodic equation: ⎧ ∞ ⎨ x ¯∇ ¯i (t) + Fi (t, ϕ, ψ) + ξi−1 Ii (t), i = 1, 2, · · · , n, i (t) = −ai (t) 0 ki (s)∇s x  ⎩ y¯∇ (t) = −bi (t) ∞ k˜j (s)∇s y¯j (t) + Gj (t, ϕ, ψ) + ξ¯−1 Jj (t), j = 1, 2, · · · , m, j

j

0

5

(3.2)

where

⎧ m ∞ t ⎪ −1  ⎪ Fi (t, ϕ, ψ) = ai (t) 0 ki (s) t−s ϕ∇ αij (t)fj (ξ¯j ψj (t − τij (t))) ⎪ i (u)∇u∇s + ξi ⎪ ⎪ j=1 ⎪ ⎪ m ⎪  ⎪ ⎪ +ξi−1 pij (t)gj (ξ¯j ψj∇ (t − σij (t))), ⎨ j=1

n ∞ t  ⎪ ⎪ Gj (t, ϕ, ψ) = bj (t) 0 k˜j (s) t−s ψj∇ (u)∇u∇s + ξ¯j−1 βji (t)f˜i (ξi ϕi (t − δji (t))) ⎪ ⎪ ⎪ i=1 ⎪ ⎪ m ⎪  ⎪ ⎪ +ξ¯j−1 qji (t)˜ gi (ξi ϕ∇ ⎩ i (t − ηji (t))). i=1

For i = 1, 2, · · · , n, j = 1, 2, · · · , m, since a− i

∞ 0

ki (s)∇s > 0, b− j

∞ 0

(3.3)

k˜j (s)∇s > 0, and from Lemma 2.4, it

follows that the linear system ⎧ ∞ ⎨ x ¯∇ ¯i (t), i = 1, 2, · · · , n, i (t) = −ai (t) 0 ki (s)∇s x  ∞ ⎩ y¯∇ (t) = −bi (t) k˜j (s)∇s y¯j (t), j = 1, 2, · · · , m, j 0 admits an exponential dichotomy on T. Therefore, by Lemma 2.3, we have that system (3.2) has a unique almost periodic solution, which can be expressed as follows: ⎧ ⎨ xϕ (t) =  t eˆ  ∞ (t, ρ(s))[Fi (s, ϕ, ψ) + ξi−1 Ii (s)]∇s, i = 1, 2, · · · , n, i −∞ −ai 0 ki (u)∇u  ⎩ y ψ (t) = t eˆ  ∞ ˜ (t, ρ(s))[Gj (s, ϕ, ψ) + ξ¯−1 Jj (s)]∇s, j = 1, 2, · · · , m. j

−∞ −bj

0

(3.4)

j

kj (u)∇u

For φ ∈ X0 , we have φ ≤ φ − φ0  + φ0  ≤ 2L. Define the following nonlinear operator Φ by ϕ ϕ ψ ψ ψ T Φ : X0 → X0 , (ϕ1 , ϕ2 , · · · , ϕn , ψ1 , ψ2 , · · · , ψm )T → (xϕ 1 , x2 , · · · , xn , y1 , y2 , · · · , ym ) ,

(3.5)

ψ where xϕ i , yj , (i =, 1, 2, · · · , n, j = 1, 2, · · · , m) are given by (3.4). Firstly, we show that Φφ ∈ X0 , for any

φ ∈ X0 . gi (0)|} ≤ L, we have By (H2 ) and max{ max |fj (0)|, max |gj (0)|, max |f˜i (0)|, max |˜ 1≤j≤m

1≤j≤m

1≤j≤m

1≤j≤m

|fj (ξ¯j ψj (t − τij (t)))| ≤ |fj (ξ¯j ψj (t − τij (t))) − fj (0)| + |fj (0)| ≤ Lfj |ξ¯j ψj (t − τij (t))| + |fj (0)| ≤ Lf ξ¯j |ψ|0 + |fj (0)| ≤ Lf ξ¯j |φ| + |fj (0)| ≤ 2L(Lf ξ¯j + 1 ), j

j

j

2

and |gj (ξ¯j ψj∇ (t − σij (t)))|

≤ |gj (ξ¯j ψj∇ (t − σij (t))) − gj (0)| + |gj (0)| ≤ Lgj |ξ¯j ψj∇ (t − σij (t))| + |gj (0)| ≤ Lg ξ¯j |ψ ∇ |0 + |gj (0)| ≤ Lg ξ¯j |φ| + |gj (0)| ≤ 2L(Lg ξ¯j + 1 ), j

j

j

2

similarly, we have 1 1 ˜ gj (ξ¯j ψj∇ (t − σij (t)))| ≤ 2L(Lgj˜ ξ¯j + ). |f˜i (ξi ϕi (t − δji (t)))| ≤ 2L(Lfi ξi + ), |˜ 2 2

6

From (3.3), for i = 1, 2, · · · , n, we get m ∞ t −1  αij (t)fj (ξ¯j ψj (t − τij (t))) |Fi (t, ϕ, ψ)| = |ai (t) 0 ki (s) t−s ϕ∇ i (u)∇u∇s + ξi +ξi−1

j=1

≤ ai (t) +ξi−1

j=1

m 

pij (t)gj (ξ¯j ψj∇ (t

∞ 0

m 

ki (s)

t−s

−1 |ϕ∇ i (u)|∇u∇s + ξi

m  j=1

αij (t)|fj (ξ¯j ψj (t − τij (t)))|

pij (t)|gj (ξ¯j ψj∇ (t − σij (t)))|

≤

j=1  + ∞ ai 0

≤

∞ 2L[a+ i 0

= 2L[a+ i

t

− σij (t)))|

ki (s)s∇s |ϕ∇ |0 + 2ξi−1

∞ 0

ki (s)s∇s +

ξi−1

ki (s)s∇s + ξi−1

m  j=1 m  j=1

m  j=1

m  g¯ + 1 αij L(Lfj ξ¯j + 21 ) + 2ξi−1 p+ ij L(Lj ξj + 2 )

+ αij (Lfj ξ¯j

+

1 2)

+

ξi−1

g + f L j + p+ ξ¯j (αij ij Lj ) +

m 

j=1

g¯ p+ ij (Lj ξj

j=1 m ξi−1  + (αij 2 j=1

+ 12 )]

+ p+ ij )]

:= 2Θi L, i = 1, 2, · · · , n,

(3.6)

and similarly, we obtain n ∞ ¯−1  ξi (β + Lf˜ + q + Lg˜ ) + ˜ |Gj (t, ϕ, ψ)| ≤ 2L[b+ j 0 ki (s)s∇s + ξj ji i ji i i=1

ξ¯j−1 2

n  i=1

+ + (βji + qji )]

:= 2Ωj L, j = 1, 2, · · · , m.

(3.7)

Then, by (3.5)-(3.7), we obtain t |(Φφ − φ0 )i (t)| = | −∞ eˆ−ai 0∞ ki (u)∇u (t, ρ(s))Fi (s, ϕ, ψ)∇s| t ≤ −∞ eˆ−ai 0∞ ki (u)∇u (t, ρ(s))|Fi (s, ϕ, ψ)|∇s t ≤ −∞ eˆ−ai 0∞ ki (u)∇u (t, ρ(s))2Θi L∇s ≤ and |(Φφ − φ0 )n+j (t)|

a− i

2Θi L ∞ , ki (u)∇u 0

i = 1, 2, · · · , n,

(3.8)

t = | −∞ eˆ−bj  ∞ k˜j (u)∇u (t, ρ(s))Gj (s, ϕ, ψ)∇s| 0 t ≤ −∞ eˆ−bj  ∞ k˜j (u)∇u (t, ρ(s))|Gj (s, ϕ, ψ)|∇s 0 t ≤ −∞ eˆ−bj  ∞ k˜i (u)∇u (t, ρ(s))2Ωj L∇s 0

≤

b− j

2Ω L ∞ j ˜i (u)∇u , k 0

j = 1, 2, · · · , m.

(3.9)

On the other hand, by (3.5)-(3.7), we get t |(Φφ − φ0 )∇ ˆ−ai 0∞ ki (u)∇u (t, ρ(s))Fi (s, ϕ, ψ)∇s)∇ | i (t)| = |( −∞ e ∞ t = |Fi − 0 ki (u)∇u ai (t) −∞ eˆ−ai 0∞ ki (u)∇u (t, ρ(s))Fi (s, ϕ, ψ)∇s| ∞ t ≤ |Fi | + 0 ki (u)∇u ai (t) −∞ eˆ−ai 0∞ ki (u)∇u (t, ρ(s))|Fi (s, ϕ, ψ)|∇s ≤ |Fi |(1 + = 2(1 + and |(Φφ − φ0 )∇ n+j (t)|

≤ ≤ =

a+ i )Θi L, a− i

i = 1, 2, · · · , n,

(3.10)

t

eˆ  ∞ ˜i (u)∇u (t, ρ(s))Gj (s, ϕ, ψ)∇s)∇ | −∞ −bj 0 k ∞ t |Gj − 0 k˜j (u)∇u bj (t) −∞ eˆ−bj  ∞ k˜j (u)∇u (t, ρ(s))Gj (s, ϕ, ψ)∇s| 0 ∞ t |Gj | + 0 k˜j (u)∇u bj (t) −∞ eˆ−bj  ∞ k˜j (u)∇u (t, ρ(s))|Gj (s, ϕ, ψ)|∇s 0 b+ i |Gj |(1 + b− ) i b+ j )Ωj L, j = 1, 2, · · · , m. 2(1 + b−

= |( =

a+ i ) a− i

j

7

(3.11)

It follows from (H3 ), (3.8)—(3.11) that Φφ − φ0 

2Θi L = max{ max { a−  ∞ , 2(1 + k (u)∇u 1≤i≤n

i

0

i

a+ 2Ω L i )Θi L}, max { b−  ∞ k˜j (u)∇u , 2(1 a− j 1≤j≤m j 0 i

+

b+ j

b− j

)Ωj L}}

≤ L, which yields that Φφ ∈ X0 . Next, we show that Φ is a contraction mapping. For φ = (ϕ1 , ϕ2 , · · · , ϕn , ψ1 , ψ2 , · · · , ψm )T , φ¯ = (ϕ¯1 , ϕ¯2 , · · · , ϕ¯n , ψ¯1 , ψ¯2 , · · · , ψ¯m )T ∈ X0 , and for i = 1, 2, · · · , n, denote by ⎧ ∞ s ⎪ Fi1 (s, ϕ, ϕ) ¯ = ai (s) 0 ki (u) s−u (ϕ∇ ¯∇ ⎪ i (v) − ϕ i (v))∇v∇u, ⎪ ⎪ m ⎪  ⎨ 2 −1 ¯ ¯ Fi (s, ψ, ψ) = ξi αij (s)[fj (ξj ψj (s − τij (s))) − fj (ξ¯j ψ¯j (s − τij (s)))], j=1 ⎪ ⎪ m ⎪ ⎪ 3 ¯ = ξ −1  pij (s)[gj (ξ¯j ψ ∇ (s − σij (s))) − gj (ξ¯j ψ¯∇ (s − σij (s)))]. ⎪ ⎩ Fi (s, ψ, ψ) j j i

(3.12)

j=1

Then for i = 1, 2, · · · , n, we obtain ¯ i (t)| |(Φφ − Φφ)

t ¯ + F 3 (s, ψ, ψ))∇s| ¯ = | −∞ eˆ−ai 0∞ ki (u)∇u (t, ρ(s))(Fi1 (s, ϕ, ϕ) ¯ + Fi2 (s, ψ, ψ) i t ¯ + |F 3 (s, ψ, ψ)|∇s ¯ ¯ + |Fi2 (s, ψ, ψ)| ≤ −∞ eˆ−ai 0∞ ki (u)∇u (t, ρ(s))(|Fi1 (s, ϕ, ϕ)| i m t ∞  + f¯ ¯1 ≤ −∞ eˆ−ai 0∞ ki (u)∇u (t, ρ(s))(a+ ¯ 1 + ξi−1 αij Lj ξj |ψ − ψ| i 0 ki (u)u∇u|ϕ − ϕ| m 

+ξi−1 ≤

a− i

j=1

j=1

g¯ p+ ij Lj ξj |ψ

¯ 1 )∇s − ψ|

∞ ∞1 [a+ i 0 k (u)∇u i 0

ki (u)u∇u + ξi−1

m  g + f ¯ L j + p+ ξ¯j (αij ij Lj )]φ − φ,

j=1

(3.13)

and ¯ ∇ (t)| = |( |(Φφ − Φφ) i

t

eˆ −∞ −ai

∞ 0

1 ¯ ki (u)∇u (t, ρ(s))(Fi (s, ϕ, ϕ)

∇ ¯ + F 3 (s, ψ, ψ))∇s) ¯ + Fi2 (s, ψ, ψ) | i

¯ + F 3 (t, ψ, ψ) ¯ ¯ + Fi2 (t, ψ, ψ) = |Fi1 (t, ϕ, ϕ) i ∞ t ¯ + F 3 (s, ψ, ψ)]∇s| ¯ ¯ + Fi2 (s, ψ, ψ) −ai (t) 0 ki (u)∇u −∞ eˆ−ai 0∞ ki (u)∇u (t, ρ(s))[Fi1 (s, ϕ, ϕ) i ¯ + |F 3 (t, ψ, ψ)| ¯ ¯ + |Fi2 (t, ψ, ψ)| ≤ |Fi1 (t, ϕ, ϕ)| i ∞ t ¯ + |F 3 (s, ψ, ψ)|]∇s ¯ ¯ + |Fi2 (s, ψ, ψ)| +ai (t) 0 ki (u)∇u −∞ eˆ−ai 0∞ ki (u)∇u (t, ρ(s))[|Fi1 (s, ϕ, ϕ)| i m m    ∞ g¯ + f¯ ¯ ≤ a+ ¯ 1 + ξi−1 ( αij Lj ξj + p+ ij Lj ξj )| ψ − ψ|1 i 0 ki (u)u∇u|ϕ − ϕ| j=1 j=1 m m  g¯ −1  + f ¯ ¯ ki (u)u∇u + ξi ( αij Lj ξj + p+ ij Lj ξj )]φ − φ j=1 j=1 m m ∞   g¯ + f¯ ¯ ≤ a+ ¯ 1 + ξi−1 ( αij Lj ξj + p+ i 0 ki (u)u∇u|ϕ − ϕ| ij Lj ξj )| ψ − ψ|1 j=1 j=1 m m ∞  a+ g¯ −1  + f ¯ i ¯ + a− [a+ αij Lj ξj + p+ i 0 ki (u)u∇u + ξi ( ij Lj ξj )]φ − φ i j=1 j=1 m m ∞  a+ g −1  ¯ + f i ¯ ≤ [a+ Lj + p+ ξj (αij i 0 ki (u)u∇u + ξi ij Lj )](1 + a− )φ − φ. i j=1 j=1

∞ a+ i [a+ + a− i 0 i

(3.14)

By using a similar way, for j = 1, 2, · · · , m, we have ¯ n+j (t)| ≤ |(Φφ − Φφ)

b− j

 + ∞ ∞ 1 [b j 0 kj (u)∇u 0

kj (u)u∇u + ξ¯j−1

n  i=1

˜

g ˜ + f ¯ ξi (βji L i + p+ ji Li )]φ − φ,

(3.15)

and ¯ ∇ (t)| ≤ [b+ |(Φφ − Φφ) n+j j

∞ 0

ki (u)u∇u + ξ¯j−1

n  i=1

˜

+ f ξi βji Li +

8

n  i=1

g ˜ p+ ji Lj )](1 +

b+ j

b− j

¯ )φ − φ.

(3.16)

Hence, by (H3 ), we obtain max{ max { a−  ∞ k1 1≤i≤n

∞ [a+ i 0

i

0

i (u)∇u

ki (u)u∇u +

[a+ i ξi−1

∞ 0

ki (u)u∇u + ξi−1

m  g + f L j + p+ ξ¯j (αij ij Lj )],

j=1

m m   g + f Lj + p+ ξ¯j (αij ij Lj )](1 +

a+ i )}, a− i

j=1 j=1 n   g ˜ + ∞ + f˜ 1  max { b− ∞ k (u)∇u [bj 0 kj (u)u∇u + ξ¯j−1 ξi (βji L i + p+ ji Li )], j 1≤i≤n j 0 i=1 n n ∞  b+ ˜ −1  + f˜ + g j ¯ [b+ k (u)u∇u + ξ ξ β L + p L i i ji i j 0 j ji j )](1 + b− )}} < 1. j i=1 i=1

(3.17)

¯ < φ − φ. ˜ Thus, this implies the mapping Φ is a contraction It follows from (3.13) – (3.17) that Φφ − Φφ mapping. Therefore, by Brouwer’s fixed point theorem, Φ has a fixed point φ∗ ∈ X0 such that Φφ∗ = φ∗ . That is to say that (3.1) has a unique almost periodic solution φ∗ ∈ X0 . So, there exists a unique almost periodic solution φ∗ of Eq. (1.1) in X0 . The proof is complete.

4. Global exponential stability of almost periodic solution In this section, we will show that the solution of Eq. (1.1) is global exponential stability. In order to obtain the result, we need to suppose further that : (H4 ) For t ∈ (0, ∞)T , there exist positive constants λ ∈ Rν+ , γi and χj such that ⎧ m   g ⎪ − ∞ + ∞ −1  ¯ + f ⎪ L j + p+ ξj χj [αij ⎨ −[ai 0 ki (s)∇s − ai 0 ki (s)s∇s]γi + ξi ij Lj ] < 0, j=1

n   ⎪ g ˜ − ∞ + ∞ −1  + f˜ ⎪ ξi γi [βji L j + p+ ⎩ −[bj 0 k˜i (s)∇s − bj 0 k˜i (s)s∇s]χj + ξ¯j ji Lj ] < 0, i=1

and

⎧ ∞  t + ∞ ⎪ [(1 − νλ)a+ ˆλ (t, ω)∇ω∇s)] ⎪ i 0 ki (s)∇s + (λ + (1 − νλ)ai 0 ki (s) t−s e ⎪ ⎪ m ⎪  f g ⎪ + + ⎪ eλ (ρ(t), 0)(ξi γi )−1 ξ¯j χj [αij Lj eˆλ (0, t − τij (t)) + pij Lj eˆλ (0, t − σij (t))] < 1, ⎨ +ˆ j=1 ∞ ∞ t ⎪ ⎪ k˜j (s)∇s + (λ + (1 − νλ)b+ k˜j (s) t−s eˆλ (t, ω)∇ω∇s)] [(1 − νλ)b+ ⎪ j ij 0 0 ⎪ ⎪ n  ⎪ + f˜ + g ⎪ ⎩ +ˆ eλ (ρ(t), 0)(ξ¯j χj )−1 ξi γi [βji Li eˆλ (0, t − δji (t)) + qji Lj˜ eˆλ (0, t − ηji (t))] < 1. i=1

Theorem 4.1 Let (H1 )–(H4 ) hold. Then the almost solution of Eq.(1.4) is global exponential stability. Proof. By Theorem 3.1, (1.1) has an almost periodic solution z ∗ (t) = (x∗1 (t), x∗2 (t), · · · , x∗n (t), ∗ ∗ (t))T with initial condition φ∗ (s) = (ϕ∗1 (s), ϕ∗2 (s), · · · , ϕ∗n (t), ψ1∗ (s), ψ2∗ (s), · · · , ψm (s))T . y1∗ (t), y2∗ (t), · · · , ym

Suppose that z(t) = (x1 (t), x2 (t), · · · , xn (t), y1 (t), y2 (t), · · · , ym (t))T is an arbitrary solution of Eq. (1.1) with initial condition φ(s) = (ϕ1 (s), ϕ2 (s), · · · , ϕn (t), ψ1 (s), ψ2 (s), · · · , ψm (s))T . For i = 1, 2, · · · , n, j = 1, 2, · · · , m, let

⎧ ⎨ u (t) = ξ −1 (x (t) − x∗ (t)) i i i i ⎩ vj (t) = ξ¯−1 (yj (t) − y ∗ (t))

and

(4.1)

j

j

⎧ ⎪ F¯j (t) = fj (vj (t − τij (t))) − fj (vj∗ (t − τij (t))), ⎪ ⎪ ⎪ ⎪ ⎨ G ¯ j (t) = gj (v ∇ (t − σij (t))) − gj (v ∗ ∇ (t − σij (t))), j j ⎪ H ¯ i (t) = f˜i (ui (t − δji (t))) − f˜i (u∗ (t − δji (t))), ⎪ i ⎪ ⎪ ⎪ ⎩ Q ¯ (t) = g˜ (u∇ (t − η (t))) − g˜ (u∗ ∇ (t − η (t))). i

i

i

ji

i

9

i

ji

(4.2)

Then

⎧ ⎪ u∇ ⎪ i (t) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ vj∇ (t) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∞ ∞ t −ai (t) 0 ki (s)∇s ui (t) + ai (t) 0 ki (s) t−s u∇ i (ω)∇ω∇s m m  −1  −1 ¯ j (t), i = 1, 2, · · · , n, +ξi αij (t)F¯j (t) + ξi pij (t)G j=1 j=1 ∞ ∞ t −bi (t) 0 k˜j (s)∇s v¯j (t) + bj (t) 0 k˜j (s) t−s v¯j∇ (ω)∇ω∇s m n  ¯ i (t) + ξ¯−1  qji (t)Q ¯ i (t), j = 1, 2, · · · , m. +ξ¯j−1 βji (t)H j i=1

(4.3)

i=1

Define Xi (t) = eˆλ (t, 0)ui (t),

Yj (t) = eˆλ (t, 0)vj (t),

i = 1, 2, · · · , n, j = 1, 2, · · · , m.

(4.4)

It follows from (4.3) and (4.4)that Xi∇ (t)

= λˆ eλ (t, 0)ui (t) + eˆλ (ρ(t), 0)u∇ i (t) ∞ ∞ t = λXi (t) + eˆλ (ρ(t), 0)[−ai (t) 0 ki (s)∇s ui (t) + ai (t) 0 ki (s) t−s u∇ i (ω)∇ω∇s m m   ¯ j (t)] +ξi−1 αij (t)F¯j (t) + ξi−1 pij (t)G j=1 j=1 ∞ ∞ t = λXi (t) − (1 − νλ)ai (t) 0 ki (s)∇sXi (t) + (1 − νλ)ai (t) 0 ki (s) t−s eˆλ (t, ω)Xi∇ (ω)∇ω∇s m  ¯ j (t)), i = 1, 2, · · · , n. +ˆ eλ (ρ(t), 0)ξi−1 (αij (t)F¯j (t) + pij (t)G (4.5) j=1

Similarly, Yj∇ (t)

∞ ∞ t = λYj (t) − (1 − νλ)bi (t) 0 kj (s)∇sYj (t) + (1 − νλ)bi (t) 0 kj (s) t−s eˆλ (t, ω)Yj∇ (ω)∇ω∇s n  ¯ i (t) + qji (t)Q ¯ i (t)), j = 1, 2, · · · , m. +ˆ eλ (ρ(t), 0)ξ¯j−1 (βji (t)H (4.6) i=1

Define two continuous functions Γi (κ) and Γj (κ), i = 1, 2, · · · , n, j = 1, 2, · · · , m, set ⎧ ∞ ∞ t ⎪ Γi (κ) = −[(1 − νκ)a− ki (s)∇s − (κ + (1 − νκ)a+ ki (s) t−s eˆκ (θ, ω)∇ω∇s)]γi ⎪ i i 0 0 ⎪ ⎪ m ⎪  g ⎪ + f ⎪ eκ (ρ(t, 0)ξi−1 Lj eˆκ (0, t − τij (θ)) + p+ ˆκ (0, t − σij (t))], ξ¯j χj [αij ⎨ +ˆ ij Lj e j=1 ∞  t + ∞˜ ⎪ ˜ ⎪ ˆκ (t, ω)∇ω∇s)]χj Γj (κ) = −[(1 − νκ)b− ⎪ j 0 ki (s)∇s − (κ + (1 − νκ)bj 0 ki (s) t−s e ⎪ ⎪ n  ⎪ ˜ f g ˜ −1 + + ⎪ ⎩ +ˆ eκ (ρ(t), 0)ξ¯j ξi γi [βji Lj eˆκ (0, t − δji (t)) + qji Lj eˆκ (0, t − ηji (t))].

(4.7)

i=1

Then, for i = 1, 2, · · · , n, j = 1, 2, · · · , m, ⎧ m   g ⎪ − ∞ + ∞ −1  ¯ + f ⎪ L j + p+ ξj χj [αij ⎨ Γi (0) = −[ai 0 ki (s)∇s − ai 0 ki (s)s∇s)]γi + ξi ij Lj ] < 0, j=1

n   ⎪ − ∞ + ∞ −1  + f˜ + g ⎪ ξi γi [βji Lj + qji Lj˜ ] < 0, ⎩ Γj (0) = −[bj 0 k˜i (s)∇s − bj 0 k˜i (s)s∇s)]χj + ξ¯j i=1

which implies that there exists a constant λ > 0 such that ⎧ ∞  θ + ∞ ⎪ −[(1 − νλ)a− ˆλ (θ, ω)∇ω∇s)]γi ⎪ i 0 ki (s)∇s − (λ + (1 − νλ)ai 0 ki (s) θ−s e ⎪ ⎪ m ⎪  f g ⎪ + ⎪ eλ (ρ(θ), 0)ξi−1 Lj eˆλ (0, θ − τij (θ)) + p+ ˆλ (0, θ − σij (θ))] < 0, i = 1, 2, · · · , n, ξ¯j χj [αij ⎨ +ˆ ij Lj e j=1    ∞˜ θ + ∞˜ ⎪ ⎪ ˆλ (θ, ω)∇ω∇s)]χj −[(1 − νλ)b− ⎪ j 0 ki (s)∇s − (λ + (1 − νλ)bj 0 ki (s) θ−s e ⎪ ⎪ n  ⎪ ˜ + f + g ⎪ ⎩ +ˆ eλ (ρ(θ), 0)ξ¯j−1 ξi γi [βji Lj eˆλ (0, θ − δji (θ)) + qji Lj˜ eˆλ (0, θ − ηji (θ))] < 0, j = 1, 2, · · · , m. i=1

Denote M = max{ max {|Xi (s)|, |Xi∇ (s)|}, max {|Yj (s)|, |Yj∇ (s)|}, s ∈ (−∞, 0]T }. 1≤i≤n

1≤j≤m

10

(4.8)

Then, for any t ∈ (−∞, 0]T , there exists K > 0 such that ⎧ ⎨ |X (t)| ≤ M < Kγ , |X ∇ (t)| ≤ M < Kγ , i = 1, 2, · · · , n, i i i i ⎩ |Yj (t)| ≤ M < Kχj , |Y ∇ (t)| ≤ M < Kχj , j = 1, 2, · · · , m. j We claim that

⎧ ⎨ |X (t)| < Kγ , |X ∇ (t)| < Kγ , t ∈ T, i = 1, 2, · · · , n, i i i i ⎩ |Yj (t)| < Kχj , |Y ∇ (t)| < Kχj , t ∈ T, j = 1, 2, · · · , m. j

For otherwise, there exist indices i ∈ {1, 2, · · · , n}, j ∈ {1, 2, · · · , m} and a first time θ ∈ (0, ∞)T such that one of the following cases: (i) |Xi (θ)| = Kγi and |Xi (t)| < Kγi for all t ∈ (−∞, θ)T , and |Xi∇ (t)| < Kγi , |Yj (t)| < Kχj , |Yj∇ (t)| < Kχj for all t ∈ (−∞, θ]T ; (ii) |Xi∇ (θ)| = Kγi and |Xi∇ (t)| < Kγi for all t ∈ (−∞, θ)T , and |Xi (t)| < Kγi , |Yj (t)| < Kχj , |Yj∇ (t)| < Kχj for all t ∈ (−∞, θ]T ; (iii) |Yj (θ)| = Kχj and |Yj (t)| < Kχj for all t ∈ (−∞, θ)T , and |Xi (t)| < Kγi , |Xi∇ (t)| < Kγi , |Yj∇ (t)| < Kχj for all t ∈ (−∞, θ]T ; (iv) |Yj∇ (θ)| = Kχj and |Yj∇ (t)| < Kχj for all t ∈ (−∞, θ)T , and |Xi (t)| < Kγi , |Xi∇ (t)| < Kγi , |Yj (t)| < Kχj for all t ∈ (−∞, θ]T . Assume (i) holds, then either 

(i) Xi (θ) = Kγi , Xi∇ (θ) ≥ 0 and |Xi (t)| < Kγi for all t ∈ (−∞, θ)T , and |Xi∇ (t)| < Kγi , |Yj (t)| < Kχj , |Yj∇ (t)| < Kχj for all t ∈ (−∞, θ]T ; or 

(i) Xi (θ) = −Kγi , Xi∇ (θ) ≤ 0 and |Xi (t)| < Kγi for all t ∈ (−∞, θ)T , and |Xi∇ (t)| < Kγi , |Yj (t)| < Kχj , |Yj∇ (t)| < Kχj for all t ∈ (−∞, θ]T . 

For (i) , from (H4 ) and Eq. (4.5), we get 0

∞ ≤ Xi∇ (θ) = λXi (θ) − (1 − νλ)ai (θ) 0 ki (s)∇sXi (θ) ∞ θ +(1 − νλ)ai (θ) 0 ki (s) θ−s eˆλ (θ, ω)Xi∇ (ω)∇ω∇s m  ¯ j (θ)] +ˆ eλ (ρ(θ), 0)ξi−1 [αij (θ)F¯j (θ) + pij (t)G j=1 ∞ ∞ θ + ˆλ (θ, ω)|Xi∇ (ω)|∇ω∇s ≤ −[(1 − νλ)a− i 0 ki (s)∇s − λ]Xi (θ) + ai (1 − νλ) 0 ki (s) θ−s e m  + f¯ +ˆ eλ (ρ(θ), 0)ξi−1 [αij Lj ξj eˆλ (0, θ − τij (θ))|Yj (θ − τij (θ))| j=1

g¯ ˆλ (0, θ − σij (θ))|Yj∇ (θ − τij (θ))|] +p+ ij Lj ξj e ∞  θ + ∞ ˆλ (θ, ω)Kγi ∇ω∇s < −[(1 − νλ)a− i 0 ki (s)∇s − λ]Kγi + (1 − νλ)ai 0 ki (s) θ−s e m  g¯ + f¯ +ˆ eλ (ρ(θ), 0)ξi−1 [αij Lj ξj eˆλ (0, θ − τij (θ))Kχj + p+ ˆλ (0, θ − σij (θ))Kχj ] ij Lj ξj e j=1 ∞  θ + ∞ ˆλ (θ, ω)∇ω∇s)]γi = {−[(1 − νλ)a− i 0 ki (s)∇s − (λ + (1 − νλ)ai 0 ki (s) θ−s e m −1  ¯ + f + g +ˆ eλ (ρ(θ), 0)ξi ξj χj [αij Lj eˆλ (0, θ − τij (θ)) + pij Lj eˆλ (0, θ − σij (θ))]}K < 0, j=1

which is a contradiction.

11



For (i) , from (H4 ) and (4.5), we get 0

∞ ≥ Xi∇ (θ) ≥ −[(1 − νλ)ai (θ) 0 ki (s)∇s − λ]Xi (θ) ∞ θ −ai (θ)(1 − νλ) 0 ki (s) θ−s eˆλ (θ, ω)|Xi∇ (ω)|∇ω∇s m  −ˆ eλ (ρ(θ), 0)ξi−1 [αij (θ)Lfj ξ¯j eˆλ (0, θ − τij (θ))|Yj (θ − τij (θ))|

j=1 g¯ +pij (θ)Lj ξj eˆλ (0, θ − σij (θ))|Yj∇ (θ − τij (θ))|] ∞  θ + ∞ ˆλ (θ, ω)Kγi ∇ω∇s > [(1 − νλ)a− i 0 ki (s)∇s − λ]Kγi − (1 − νλ)ai 0 ki (s) θ−s e m  g¯ + f¯ −ˆ eλ (ρ(θ), 0)ξi−1 [αij Lj ξj eˆλ (0, θ − τij (θ))Kχj + p+ ˆλ (0, θ − σij (θ))Kχj ] ij Lj ξj e j=1    ∞ θ + ∞ ˆλ (θ, ω)∇ω∇s)]γi = {[(1 − νλ)a− i 0 ki (s)∇s − (λ + (1 − νλ)ai 0 ki (s) θ−s e m  g + f −ˆ eλ (ρ(θ), 0)ξi−1 Lj eˆλ (0, θ − τij (θ)) + p+ ˆλ (0, θ − σij (θ))]}K > 0, ξ¯j χj [αij ij Lj e j=1

which is a contradiction. So (i) is not valid. Assume (ii) holds, from (H4 )and (4.5), we get Kγi =

|Xi∇ (θ)| ≤ (1 − νλ)ai (θ)

∞

ki (s)∇s|Xi (θ)| ∞ θ +λ|Xi (θ)| + ai (θ)(1 − νλ) 0 ki (s) θ−s eˆλ (θ, ω)|Xi∇ (ω)|∇ω∇s m  +ˆ eλ (ρ(θ), 0)ξi−1 [αij (θ)Lfj ξ¯j eˆλ (0, θ − τij (θ))|Yj (θ − τij (θ))| 0

j=1 g¯ +pij (θ)Lj ξj eˆλ (0, θ − σij (θ))|Yj∇ (θ − τij (θ))|] ∞  θ + ∞ ˆλ (θ, ω)Kγi ∇ω∇s < [(1 − νλ)a+ i 0 ki (s)∇s + λ]Kγi + (1 − νλ)ai 0 ki (s) θ−s e m −1  + f¯ + g¯ +ˆ eλ (ρ(θ), 0)ξi [αij Lj ξj eˆλ (0, θ − τij (θ))Kχj + pij Lj ξj eˆλ (0, θ − σij (θ))Kχj ] j=1 ∞  θ + ∞ ˆλ (θ, ω)∇ω∇s)] = {[(1 − νλ)a+ i 0 ki (s)∇s + (λ + (1 − νλ)ai 0 ki (s) θ−s e m  f g + +ˆ eλ (ρ(θ), 0)(ξi γi )−1 Lj eˆλ (0, θ − τij (θ)) + p+ ˆλ (0, θ − σij (θ))]}Kγi ξ¯j χj [αij ij Lj e j=1

< Kγi ,

which is a contradiction. So (ii) is not valid. Similarly, we can also show that (iii) and (iv) are also not valid. Then, it follows that |xi (t) − |x∗i (t)| ≤ eν λ (t, 0)Kγi , |yi (t) − |yi∗ (t)| ≤ eν λ (t, 0)Kχj , t ∈ T, i = 1, 2, · · · , n, j = 1, 2, · · · , m. So the almost periodic solution of Eq. (1.1) is global exponential stability. The proof is completed.

5. An example In this section, we give an example to demonstrate the effectiveness of the obtained results. Example 5.1. In system (1.1), we take n = m = 2, and the corresponding coefficients are listed as follows: a1 (t) = b1 (t) =

5+| sin t| , 16

a2 (t) = b2 (t) =

5+| cos t| , 16

2 k1 (s) = k2 (s) = k˜1 (s) = k˜2 (s) = e−6s ,

sin t cos t 6000 , β11 (t) = β12 (t) = β21 (t) = β22 (t) = 6000 , sin t cos t , q11 (t) = q12 (t) = q21 (t) = q22 (t) = 4000 , p11 (t) = p12 (t) = p21 (t) = p22 (t) = 4000 sin2 (t) τ11 (t) = τ12 (t) = τ21 (t) = τ22 (t) = δ11 (t) = δ12 (t) = δ21 (t) = δ22 (t) = 1000 , 2 (t) σ11 (t) = σ12 (t) = σ21 (t) = σ22 (t) = η11 (t) = η12 (t) = η21 (t) = η22 (t) = cos 1000 ,

α11 (t) = α12 (t) = α21 (t) = α22 (t) =

f1 (x) = f2 (x) = f˜1 (x) = f˜2 (x) = g1 (x) = g2 (x) = g˜1 (x) = g˜2 (x) = |x|. We consider the case T = R, and ρ(t) = t, ν(t) = 0. Let ξ1 = ξ2 = ξ¯1 = ξ¯2 = γ1 = γ2 = χ1 = χ2 = 1, λ = 10−5 . It is not difficult to check that the assumptions (H2 )-(H4 ) are all satisfied, we omit here. Hence, from 12

Theorem 3.1 and Theorem 4.1, the system (1.1) has exactly one almost periodic solution, which is global exponential stability.

6. Conclusions. In this paper, we have investigated a class of neutral BAM neural networks with distributed leakage delays on time scales, and then derived some criteria ensuring the existence and global exponential stability of almost periodic solutions for this model. One can observe that all results in [37, 38] and in the references cited therein cannot directly applied to the system (1.1). Generally, it is not easy to solve the influence of the neutral type delays in neuronal networks on time scales, here we handle this problem with the method of exponential dichotomy theory and Lyapunov functional, which may be further applied to study other almost periodic type neural networks on time scales, thus, the results of this paper are novel. As an interesting problem, other almost automorphic type neural networks on time scales are deserved to study by applying some suitable methods, we will leave it as our future work.

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