Neural Networks 50 (2014) 98–109

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Existence and global exponential stability of periodic solution for high-order discrete-time BAM neural networks Ancai Zhang a , Jianlong Qiu b,∗ , Jinhua She c a

School of Automobile Engineering, Linyi University, Linyi, Shandong 276005, China

b

School of Science, Linyi University, Linyi, Shandong 276005, China

c

School of Computer Science, Tokyo University of Technology, Hachioji, Tokyo 192-0982, Japan

article

info

Article history: Received 14 June 2013 Received in revised form 28 September 2013 Accepted 10 November 2013 Keywords: Bidirectional associative memory (BAM) High-order neural networks Discrete-time Young’s inequality Exponential stability

abstract This paper concerns the existence and exponential stability of periodic solution for the high-order discrete-time bidirectional associative memory (BAM) neural networks with time-varying delays. First, we present the criteria for the existence of periodic solution based on the continuation theorem of coincidence degree theory and the Young’s inequality, and then we give the criteria for the global exponential stability of periodic solution by using a non-Lyapunov method. After that, we give a numerical example that demonstrates the effectiveness of the theoretical results. The criteria presented in this paper are easy to verify. In addition, the proposed analysis method is easy to extend to other high-order neural networks. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction It is well known that the dynamic behaviors of neural networks have been deeply investigated in the past few years (Cao & Wang, 2002; Huang, Chen, Huang, & Cao, 2007; Huang, Huang, & Li, 2008). In 1987, Kosko presented a new type of neural networks called bidirectional associative memory (BAM) networks. It is composed of neurons arranged in two layers: the X -layer and the Y -layer. Since the BAM neural networks have many applications in different fields, for example, in pattern recognition, signal and image processing, automatic control, etc., they have been extensively studied over the last a couple of decades (Senan, Arik, & Liu, 2012; Zhang & Liu, 2011; Zhou, Chen, & Zhou, 2005). However, most researchers concentrated on low-order BAM neural networks and did not consider the high-order connected terms (Balasubramaniam, Kalpana, & Rakkiyappan, 2011; Cao & Jiang, 2004; Yang, Zhang, & Wu, 2007). Since a low-order neural network has intrinsic limitations in convergence rate, storage capacity, and fault tolerance, it is necessary to add high-order interactions to neural networks in order to solve these problems.

∗

Corresponding author. Tel.: +86 0539 8766103; fax: +86 0539 8766103. E-mail addresses: [email protected] (A. Zhang), [email protected], [email protected], [email protected] (J. Qiu), [email protected] (J. She). 0893-6080/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.neunet.2013.11.005

A high-order neural network has bigger approximation ability, faster convergence rate, larger storage capacity and higher fault tolerance than a low-order one (Wang, Lu, Ji, & Wang, 2011), and the characteristics of a high-order neural network have been intensively investigated recently (Chen, Zhao, & Ruan, 2007; Lou & Cui, 2007; Qiu, 2010). The model of a high-order BAM neural network with periodic coefficients and continuously distributed delays is m2  dx (t )     1i  = − a ( t ) x ( t ) + b ( t ) f x t − r ( t )  1i 1i 2ij 2j 2j 2ij   dt  j =1   m2  m2         + c ( t ) f x t − τ ( t )  2ijl 2j 2j 2ij    j =1 l =1       × f x (t − τ (t )) + I (t ), 2l

2l

2il

1i

m1      dx2j (t )   = −a2j (t )x2j (t ) + b1ji (t )f1i x1i t − r1ji (t )    dt  i=1   m1  m1         + c1jil (t )f1i x1i t − τ1ji (t )     i= 1 l =1     × f1l x1l t − τ2jl (t ) + I2j (t ),

(1)

where t ≥ 0, i = 1, 2, . . . , m1 , j = 1, 2, . . . , m2 ; m1 and m2 are the numbers of neurons in the X -layer and Y -layer, respectively.

A. Zhang et al. / Neural Networks 50 (2014) 98–109

To our knowledge, there have no studies been reported on the properties of high-order discrete-time BAM neural networks until now. This motivated us to carry out a study in this paper. In this study, we discuss the existence and exponential stability of periodic solution for the following high-order discrete-time BAM neural networks

The above equations can be written in a reduced form dxki (t ) dt

= −aki (t )xki (t ) +

p(3−k)



b(3−k)ij (t )f(3−k)j

j =1

(3−k) p (3−k)    p c(3−k)ijl (t )f(3−k)j × x(3−k)j t − r(3−k)ij (t ) + j =1

l=1

1xki (n) = −Aki (n)xki (n) +

 t − τ(3−k)ij (t )



p(3−k)





× x(3−k)j    × f(3−k)l x(3−k)l t − τ(3−k)il (t ) + Iki (t ),

m1 , m2 ,



(2)

k = 1, k = 2;

aki (t ) > 0; xki (t ) is the activation of the i-th neuron in the k layer at the time t; fki is the signal transmission function of the i-th neuron in the k layer; bkji (t ) and ckjil (t ) are the first- and second-order synoptic weights of the neural network, rkji (t ) (≥0) and τkji (t ) (≥0) stand for transmission delays; Iki (t ) corresponds to the external input of the i-th neuron in the k layer; and aki (t ), bkji (t ), ckjil (t ), rkji (t ), τkji (t ), and Ikij (t ) are T -periodic functions in R. Numerous attempts have been made to show the properties of the system (2). Cao et al. examined the existence and global exponential stability of the equilibrium with constant coefficients (Cao, Liang, & Lam, 2004). Ho et al. studied the dynamic behaviors with impulse terms (Ho, Liang, & Lam, 2006), and Li et al. considered the problem of impulsive effects on the stability (Li, Lin, Liao, & Huang, 2011). Note that a high-order Hopfield-type neural network is a special case of (2). In other words, (2) becomes a highorder Hopfield-type neural network when k = 1 or k = 2. Yi et al. studied the convergence behavior of a high-order Hopfield-type neural network in Yi, Shao, Yu, and Xiao (2008). They presented some sufficient conditions that ensure all solutions of the network converging to zero under a bounded condition on high-order signal transmission functions. Since discretization is needed in the implementation of a continuous-time neural network, it is of both theoretical and practical importance to study the dynamics of a discrete-time neural network. Recently, discussions on the properties of loworder discrete-time BAM neural networks have thrown new light on the subject. For example, Liang et al. studied the following discrete-time BAM neural network (Liang, Cao, & Ho, 2005)

 m    cij fj (yj (n − k(n))) + Ii ,  xi (n + 1) = ai xi (n) + j =1

m     dij gi (xi (n − l(n))) + Jj , yj (n + 1) = bj yj (n) +

(3)

i=1

p(3−k)



w(3−k)ji f(3−k)j

j=1

 ×

+∞  l =1

j=1



l =1



× x(3−k)j (n − τ(3−k)ij (n))   × f(3−k)l x(3−k)l (n − τ(3−k)il (n)) + Iki (n),

(5)

where 1xki (n) = xki (n + 1) − xki (n), r(3−k)ij (n) and τ(3−k)ij (n) are periodic non-negative integers, and 0 < Aki (n) < 1. (5) is obtained by applying the discretizing method in Mohamad and Gopalsamy (2003) to the continuous-time system (2). Initial conditions of (5) are xki (n) = φki (n),

n ∈ [−τ , 0]Z , i = 1, 2, . . . , p(k), k = 1, 2, (6)

where φki (·) is a continuous function,

  τ = max max{rkji (n)}, max{τkji (n)} , k,i,j

n∈IN

n∈IN

IN = {0, 1, 2, . . . , N − 1} , N is a positive integer,

  [a, b]Z = a, a + 1, . . . , b − 1, b for a, b ∈ Z , and a ≤ b. First, we introduce some notations for convenience. Aki = min{Aki (n)}, n∈IN

bkji = max{|bkji (n)|}, n∈IN

c kjil = max{|ckjil (n)|}, n∈IN

I ki = max{|Iki (n)|}, n∈IN

Z0+ = {0, 1, 2, . . .} ,

M = max sup{|fki (x)|}, k,i

 u(n) =

x∈R N −1 1 

N n =0

u(n),

where u(n) is an N-periodic sequence. We also make the following assumptions.

Assumption 2. There exist non-negative constants Lki and eki such that fki2 (x) ≤ Lki |x| + eki ,

∀x ∈ R, i = 1, 2, . . . , p(k), k = 1, 2.

Assumption 3. There exist a real number p (>2) and non-negative constants Lki and eki such that

|fki (x)|p ≤ Lki |x| + eki ,

∀x ∈ R, i = 1, 2, . . . , p(k), k = 1, 2.

Assumption 4. fki (x) (i = 1, 2, . . . , p(k), k = 1, 2) are continuous and bounded in R. Assumption 5. There exist a non-negative constant Lki such that

 g(3−k)j (l)x(3−k)j (n − l)

(3−k) p (3−k)   p × x(3−k)j (n − r(3−k)ij (n)) + c(3−k)ijl (n)f(3−k)j

Assumption 1. Aki (n), b(3−k)ij (n), c(3−k)ijl (n), r(3−k)ij (n), τ(3−k)ij (n), and Iki (n) are N-periodic sequences on Z0+ .

where ai ∈ (0, 1), bj ∈ (0, 1), and k(n) and l(n) are positive integers. They used a linear matrix inequality and obtained some sufficient conditions for the existence, uniqueness, and global exponential stability of the equilibrium point. Zhou et al. investigated the existence and global exponential stability of periodic solution of the following discrete-time BAM neural networks (Zhou & Liu, 2006) xki (n + 1) = αki (n)xki (n) +

b(3−k)ij (n)f(3−k)j

j =1

where i = 1, 2, . . . , p(k); k = 1, 2 (k = 1 for X -layer and k = 2 for Y -layer); p(k) =

99

+ Iki (n).

(4)

|fki (x) − fki (y)| ≤ Lki |x − y|, ∀x, y ∈ R, i = 1, 2, . . . , p(k), k = 1, 2.

100

A. Zhang et al. / Neural Networks 50 (2014) 98–109

The rest of the paper is organized as follows. Section 2 presents criteria for the existence of a periodic solution without considering bounded conditions on the signal transmission functions. Section 3 shows sufficient conditions on the global stability of a periodic solution, then, a numerical example is given in Section 4 to demonstrate the effectiveness of our theoretical results. Finally, some concluding remarks are given in Section 5.

Proof. Let α = β1 . It is easy to find an α for which α > 1 holds. So,

  α−α 1

we can take a large enough constant C > 0 that ensures αC ≥ e. From Lemma 5, we have

 C  α−α 1

Lx + e ≤ Lx +

α

  α−α 1

2. Existence of a periodic solution

Definition 1. The periodic solution of (5), x∗ (n) = [x∗11 (n), . . . , x∗1m1 (n), x∗21 (n), . . . , x∗2m2 (n)]T , is global exponential stable if there exist constants µ > 1 and L > 0 such that

≤

|xki (n) − x∗ki (n)| ≤ L∥φ − φ ∗ ∥µ−n ,

=

n ∈ Z0+ ,

where x(n) is a solution of (5) with the initial value (6), φ ∗ (n) =  ∗ T ∗ ∗ ∗ φ11 (n), . . . , φ1m (n), φ21 (n), . . . , φ2m (n) is the initial value of 1 2 ∗ x∗ (n), and ∥φ − φ ∗ ∥ = maxk,i maxn∈[−τ ,0]Z |φki (n) − φki (n)|. Definition 2 (Su, Jiang, & Zhang, 2006). Let B = sEn − A, s ∈ R, A ∈ Rn×n , and En be the identity matrix of size n. Suppose s > 0 and A ≥ 0. If the spectral radius of A, ρ(A), satisfies ρ(A) ≤ s, then B is called an M-matrix; and if ρ(A) < s, then B is called a non-singular M-matrix. Here, A ≥ 0 means that the items of A are non-negative. From Definition 2, it is easy to get Lemma 1. Lemma 1. Let A ≥ 0 be a real n × n matrix and ρ(A) < 1. Then, En − A is a non-singular M-matrix. Lemma 2 (Su et al., 2006). Let A = (aij )n×n with aij ≤ 0, i, j = 1, 2, . . . , n, and i ̸= j. The following statements are equivalent:

C

α

L



α L

α



C



   α−1 1 x + 1

C αC

  α−1 1 α x+

C

α

,

x ≥ 0.

It follows that



Lx + e

 α1

  α−1 1

L

≤

C

x+

C

α

,

x ≥ 0.

That is,



Lx + e

β

≤

L C

x + R(C , β),

where R(C , β) = proof.

  α−1 1 C

α

x ≥ 0,

=

β   1−β βC . This completes the



Theorem 1. Assume that Assumptions 1 and 2 hold and that Em1 +m2 − H is a non-singular M-matrix, where



0 H1

H2 0



,

(1) A is a non-singular M-matrix;

H =

 T  T (2) There exists a vector ξ = ξ1 , ξ2 , . . . , ξn > 0, 0, . . . , 0 such that Aξ > 0; and (3) A−1 ≥ 0.

H1 = (h1ij )m2 ×m1 , H2 = (h2ij )m1 ×m2 ,

Lemma 3 (Continuation Theorem, Gaines & Mawhin, 1977). Let Ω be an open bounded set in Banach space X, L : DomL ⊂ X −→ X is a Fredholm operator with index zero, P : DomL ⊂ X −→ KerL, Q : X −→ X/ImL are two projectors, and N : Ω −→ X is ¯ . Moreover, we assume that the following conditions L-compact on Ω are satisfied: (1) Lx ̸= λN x, ∀x ∈ ∂ Ω ∩ DomL, λ ∈ (0, 1); (2) QNx ̸= 0, ∀x ∈ ∂ Ω ∩ Ker  L; and (3) deg QN , Ω ∩ KerL, 0 ̸= 0.

¯. Then the equation Lx = N x has at least one solution in Ω Lemma 4 (Monotonicity of Spectral Radius, Su et al., 2006). If A, B ∈ Rn×n and 0 ≤ A ≤ B, then ρ(A) ≤ ρ(B). Lemma 5. Suppose that α ≥ 1 is a constant. Then (1 + x)α ≥ 1 + α x for x ≥ 0. Lemma 6. Suppose that L ≥ 0, 0 < β < 1 and e are constants. Then, there exists a constant C > 0 such that

(Lx + e)β ≤

L C

x + R(C , β),

x ≥ 0,

where R(C , β) is a constant that is dependent on C and β .

   α x + 1 C α−1

C α

  α−α 1

 L

αL   C  α−1 1 x + 1



α

α





C

= We first present some definitions and lemmas in this section before giving our main results.



=

 C  α−α 1

(m1 +m2 )×(m1 +m2 )

p(3−k)

h(3−k)ij =

 

c (3−k)ijl + c (3−k)ilj

l=1

 L(3−k)j 2Aki

,

k = 1, 2.

Then the system (5) has at least one N-periodic solution. Proof. Let



T

X = x(n) = x11 (n), . . . , x1m1 (n), x21 (n), . . . , x2m2 (n)



x(n + N ) = x(n), ∀n ∈ Z

,



and

∥x ∥ =

p(k) 2   k=1 i=1

max |xki (n)|, n∈IN

∀x ∈ X.

So, X is a Banach space when it endows with the norm ∥ · ∥. For x(n) ∈ X, we define the following mappings

Lx(n) = 1x(n) = 1x11 (n), . . . , 1x1m1 (n),



T 1x21 (n), . . . , 1x2m2 (n) ,  N x(n) =  x(n) =  x11 (n), . . . , x1m (n), T 1  x21 (n), . . . , x2m2 (n) ,  P x(n) = Qx(n) =  x(n) =  x11 (n), . . . , x1m1 (n), T  x21 (n), . . . , x2m2 (n) ,

A. Zhang et al. / Neural Networks 50 (2014) 98–109

101

a small ε > 0 such that (Em1 +m2 − H )η > ε Bη, that is

where

∆xki (n) = xki (n + 1) − xki (n),

 xki (n) = −Aki (n)xki (n) +

 xki (n) =

p(3−k)



N −1 1 

N n =0



× x(3−k)j

b(3−k)ij (n)f(3−k)j

0      B=  b1ij

 n − r(3−k)ij (n)

 

 j =1







p(3−k) p(3−k)

+



A2i

c(3−k)ijl (n)f(3−k)j

l =1

|fki (x)| ≤

It is not difficult to show that KerL = Rm1 +m2 ,

A1i

 m1 ×m2

0

   ∈ R(m1 +m2 )×(m1 +m2 ) .  

m2 ×m1

Lki C

C

|x| +

2

,

∀x ∈ R,

(9) 2

where C is a constant that satisfies Cki ≤ ε and C4 ≥ eki for i = 1, 2, . . . , p(k) and k = 1, 2. Combining Assumption 2, (7), and (9) yields

i = 1, 2, . . . , p(k), k = 1, 2

|xki (n + 1)| ≤ (1 − λAki )|xki (n)|  p (3−k) b(3−k)ij L(3−k)j 1 +λ |x(3−k)j (n − r(3−k)ij (n))| +

L

 N −1    x(n) x(n) ∈ X, xki (n) = 0 ,  n =0 

dim KerL = codimImL = m1 + m2 ,

p(3−k) p(3−k)

P2 = P,

×

  j =1

and

Q2 = Q,

ImP = KerL,

ImL = KerQ = Im(I − Q),

n−1 

x(s) −

s=0

N −1 n−1 1 

N n=0 s=0

x(s),

c (3−k)ijl |f(3−k)j (x(3−k)j (n − τ(3−k)ij (n)))|2



l=1



× (n − τ(3−k)ij (n))| +

 C b(3−k)ij

× e(3−k)j +

2

j =1

+ I ki

ε b(3−k)ij |x(3−k)j (n − r(3−k)ij (n))|

 + I ki .

Denote |xki (nki )| = maxn∈IN |xki (n)|. The above inequality yields

|xki (nki )| ≤ (1 − λAki )|xki (nki )| + λAki   p (3−k) b(3−k)ij × ε + h(3−k)ij Aki

j =1

× |x(3−k)j (n(3−k)j )| + λAki δki ,

xki (n + 1) = (1 − λAki (n))xki (n)

(10)

where b(3−k)ij (n)f(3−k)j (x(3−k)j (n − r(3−k)ij (n)))

p(3−k) p(3−k)

 

p(3−k)

 δki = Aki

j =1

j =1

2



p(3−k) p(3−k)  1   c (3−k)ijl + c (3−k)ilj 2 j=1 l=1

p(3−k)

gives



+

 C b(3−k)ij

p(3−k) p(3−k)  1   c (3−k)ijl + c (3−k)ilj L(3−k)j |x(3−k)j 2 j=1 l=1

 

T assume that x(n) = x11 (n), . . . , x1m1 (n), x21 (n), . . . , x2m2 (n) is a solution of the equation Lx(n) = λN x(n) for λ ∈ (0, 1). This



j =1

+



p(3−k)



≤ (1 − λAki )|xki (n)| + λ

∀x(n) ∈ ImL.

Kp (I − Q)N Ω is compact for any open bounded set Ω in X . The definition of L-compact Gaines and Mawhin (1977) gives that N is L-compact on Ω . Now, we apply Lemma 3 to derive the result. First, it is sufficient to construct a proper open bounded set Ω such that Lx(n) ̸= λN x(n) for each λ ∈ (0, 1) and x(n) ∈ ∂ Ω ∩ DomL. In fact, this is exactly the first assumption in Lemma 3. We

p(3−k) 2

j =1

Since X is a finite-dimensional Banach space, and QN and  Kp (I − Q)N are both continuous, QN Ω is bounded and

p(3−k)

2

+ |f(3−k)l (x(3−k)l (n − τ(3−k)il (n)))|

where I is the identity mapping. It follows that L is a Fredholm operator with index zero, and P and Q are two projectors. Furthermore, the generalized inverse of L, Kp : ImL −→ DomL ∩ KerL, defines as Kp x(n) =

C

j =1

is closed in X,

+



 ImL =

+λ

b2ij

It follows from Lemma 2 and (8) that Em1 +m2 − (H + ε B) is a nonsingular M-matrix. From Assumption 2, we easily obtain

  × x(3−k)j n − τ(3−k)ij (n)    × f(3−k)l x(3−k)l n − τ(3−k)il (n) + Iki (n).



(8)

where

j =1



 η > 0,

Em1 +m2 − H + ε B

xki (n),

−1

 C b(3−k)ij j =1

c(3−k)ijl (n)f(3−k)j (x(3−k)j (n − τ(3−k)ij (n)))

p(3−k)

×

l =1

p(3−k) 1 

2

j =1





c (3−k)ijl + c (3−k)ilj e(3−k)j + I ki .



l=1

 × f(3−k)l (x(3−k)l (n − τ(3−k)il (n))) + λIki (n).

2

+

(7)

From Lemma 2 and the fact that Em1 +m2 − H is a non-singular M-matrix, we know that there exists a vector η = [η11 , . . . , η1m1 , η21 , . . . , η2m2 ]T > 0 such that (Em1 +m2 − H )η > 0. So, there exists

Set F = δ11 , . . . , δ1m1 , δ21 , . . . , δ2m2





Em1 +m2 − (H + ε B)



T

. It follows from (10) that

|x11 (n11 )|, . . . ,

T |x1m1 (n1m1 )|, |x21 (n21 )|, . . . , |x2m2 (n2m2 )| ≤ F .

(11)

102

A. Zhang et al. / Neural Networks 50 (2014) 98–109

Since Em1 +m2 − (H + ε B) is a non-singular M-matrix, we obtain

T

ξ = ξ11 , . . . , ξ1m1 , ξ21 , . . . , ξ2m2  −1  = Em1 +m2 − (H + ε B) F + [1, . . . , 1, 1, . . . , 1]T > 0.  (11) implies that |x11 (n11 )|, . . . , |x1m1 (n1m1 )|, |x21 (n21 )|, . . . , T |x2m2 (n2m2 )| < ξ , that is, 

|xki (nki )| < ξki i = 1, 2, . . . , p(k), k = 1, 2. 2 p(k) Set d = i=1 ξki . So, d is independent of λ and k=1 ∥x ∥ =

which contradicts (13). That gives

QN x(n) ̸= 0,

∀x(n) ∈ ∂ Ω ∩ KerL.

So, the second condition of Lemma 3 is satisfied. Next, we define a continuous function S : Ω × [0, 1] −→ X S (x, µ) = S (x11 , µ), . . . , S (x1m1 , µ),



T

S (x21 , µ), . . . , S (x2m2 , µ) , where S (xki , µ) = −µ Aki (n)xki (n) + (1 − µ) QN x(n)





k = 1, 2; i = 1, 2, . . . , p(k).

p(k) 2  

|xki (nki )| < d.

(12)

k=1 i=1

,

If S (x, µ) = 0, ∀x(n) ∈ ∂ Ω ∩ KerL, then x(n) = α = [α11 , . . . , α1m1 , α21 , . . . , α2m2 ]T is a constant vector in Rm1 +m2 with ∥α∥ = d, and

Consequently, we take an open bounded subset to be



ki

S (xki , µ) = S (αki , µ) = −µ Aki (n)αki + (1 − µ)



Ω = x(n)|x(n) ∈ X, ∥x∥ < d .

 × − Aki (n)αki +

From (12), we easily get

p(3−k)



 b(3−k)ij (n)f(3−k)j (α(3−k)j )

j =1

Lx ̸= λN x,

∀x ∈ ∂ Ω ∩ DomL, λ ∈ (0, 1).

p(3−k) p(3−k)

This verifies the first condition of Lemma 3. Second, if x(n) ∈ ∂ Ω ∩ KerL, then x(n) = α = [α11 , . . . , α1m1 , α21 , . . . , α2m2 ]T is a constant vector in Rm1 +m2 , p(k) 2  

∥x∥ = ∥α∥ =

|αki | = d,

(13)

k=1 i=1

and

QN x(n)



 ki

= − Aki (n)αki +

p(3−k)



j =1

 

+

j=1

for i = 1, 2, . . . , p(k), and k = 1, 2. It follows that

αki =  b(3−k)ij (n)f(3−k)j (α(3−k)j )

 QN x(n) ki = QN αki = 0,

j =1

 (14)

≤

i = 1, 2, . . . , p(k), k = 1, 2.

p(3−k)





Aki

b(3−k)ij |f(3−k)j (α(3−k)j )|

j=1

+

 

 b(3−k)ij (n)f(3−k)j (α(3−k)j )

p(3−k) p(3−k)

  j =1

j =1

 c(3−k)ijl (n)f(3−k)j (α(3−k)j )

p(3−k)

≤

Following the same deduction procedure shown above yields



b(3−k)ij

j =1

Aki

 ε + h(3−k)ij |α(3−k)j | + δki ,

i = 1, 2, . . . , p(k), k = 1, 2. It implies that (Em1 +m2

p(k) 2   k=1 i=1





b(3−k)ij

j =1

Aki

 ε + h(3−k)ij |α(3−k)j | + δki .

In the same manner, we obtain ∥α∥ < d that contradicts ∥α∥ = d. So, we can conclude that S (x, µ) ̸= 0 for every x ∈ ∂ Ω ∩ KerL and µ ∈ [0, 1]. Applying the homotopy invariance theorem gives deg(QN , Ω ∩ KerL, 0) = deg(S (·, 0), Ω ∩ KerL, 0)

 − (H + εB)) |α11 |, . . . , |α1m1 |, |α21 |, . . . ,

T |α2m2 | ≤ F . Thus, we have ∥α∥ =

l=1

   c (3−k)ijl f(3−k)j (α(3−k)j ) 

    × f(3−k)l (α(3−k)l ) + I ki 

l =1

× f(3−k)l (α(3−k)l ) + Iki (n).

|αki | < ξki ,

p(3−k)



1

p(3−k) p(3−k)

j=1

|αki |
0.  So, (22) gives |x11 (n11 )|, . . . , |x1m1 (n1m1 )|, |x21 (n21 )|, . . . , |x2m2 T (n2m2 )| < η, that is, |xki (nki )| < ηki , i = 1, 2, . . . , p(k), k = 1, 2. 2 p(k) Set d = i=1 ηki . It is clear that d is independent of λ in (7) k =1

∥x ∥ =

p(k) 2  

|xki (nki )| < d.

(23)

We take an open bounded subset

′

′

|fki (x)|q ≤ ε|x| + R,

(19)

  Ω = x(n)|x(n) ∈ X , ∥x∥ < d .

Combining (7), (9), and Young’s inequality yields

From (23), we easily get

|xki (n + 1)| ≤ (1 − λAki )|xki (n)|  p (3−k) +λ ε b(3−k)ij |x(3−k)j (n − r(3−k)ij (n))|

Lx ̸= λN x,

 c (3−k)ijl

′

|f(3−k)j (x(3−k)j (n − τ(3−k)ij (n)))|p p′

l =1

|f(3−k)j (x(3−k)j (n − τ(3−k)ij (n)))|

q′



q′ p(3−k)

∀x ∈ ∂ Ω ∩ DomL, λ ∈ (0, 1).

Following the same analytic procedure as that in Theorem 1, it is not difficult to obtain the result of Theorem 2. This completes the proof. 

j =1 p(3−k) p(3−k)



R + I ki .

c (3−k)ijl

l =1

Let Θ = σ11 , . . . , σ1m1 , σ21 , . . . , σ2m2 that





k=1 i=1

|fki (x)|p ≤ ε|x| + R, |fki (x)| ≤ ε|x| + R.

j=1



and

Thus, it is easy to get

+



Since Em1 +m2 − ε Υ is a non-singular M-matrix,

   p′    Lki |x| + eki p ≤ ε|x| + R,  1    Lki |x| + eki p ≤ ε|x| + R,

 

(21)

j =1

j =1

where C > 0 is a large enough constant, and R1 , R2 , and R3 are constants.   Set R = max R1 , R2 , R3 . It follows from (16), (17), (18), and Assumption 3 that

+

(20)

Denote |xki (nki )| = maxn∈IN |xki (n)|. From (20), we have

σki = Aki

It follows from Lemma 2 and (15) that Em1 +m2 − ε Υ is a nonsingular M-matrix. From Lemma 6, we have

+

.

j =1



 pp′

c (3−k)ijl R

l =1





k = 1, 2.



 

where p(3−k)

l =1



p(3−k)

′

< 1, 0 < pp < 1. Choose a vector T  ζ = ζ11 , . . . , ζ1m1 , ζ21 , . . . , ζ2m2 > 0 and a small real number ε > 0 such that Em1 +m2 ζ = ζ > εΥ ζ , that is,

p(3−k) p(3−k)

j =1

Proof. Since p > 2, there exists p′ such that 2 < p′ < p. So, it is easy to obtain 1 < q′ < 2 < p′ < p for which p1′ + q1′ = 1 holds. It follows that 0
0 with   |φki (n) − φki∗ (n)| such that Em1 +m2 − D ξ > 0,

exists a vector ξ =

ξki > maxn∈[−τ ,0]Z that is, p(3−k)



d(3−k)ij ξ(3−k)j < ξki ,

i = 1, 2, . . . , p(k), k = 1, 2.

(27)

j=1

We define a function Fki (µ) to be Fki (µ) = −ξki +

  (3−k) µN +τ 1 − (1 − Aki )N p d(3−k)ij ξ(3−k)j  N 1 − (1 − Aki )µ j =1

for µ ≥ 1, i = 1, 2, . . . ,p(k), and k = 1, 2. Clearly, Fki (µ) is a continuous function on 1, (1 − Aki )−1 and Fki (1) < 0. So, there exist a constant ϱ > 0 and µ ∈ 1, (1 − Aki )−1 such that Fki (µ) < −ϱ for i = 1, 2, . . . , p(k) and k = 1, 2, that is,





 (3−k)  µN +τ 1 − (1 − Aki )N p d(3−k)ij ξ(3−k)j < ξki − ϱ. (28)  N 1 − (1 − Aki )µ j =1  T Set z (n) = z11 (n), . . . , z1m1 (n), z21 (n), . . . , z2m2 (n) and zki (n) n = yki (n)µ , i = 1, 2, . . . , p(k) for k = 1, 2. Since yki (m − rkij (m)) = µrkij (m)−m zki (m − rkij (m)) ≤ µτ −m zki (m − rkij (m)), yki (m − τkij (m)) = µτkij (m)−m zki (m − τkij (m)) ≤ µτ −m zki (m − τkij (m)), and µn = µn−m−1 · µm+1 , it follows from (26) that

Then the system (5) has exactly one N-periodic solution. Moreover, it is globally exponentially stable.

 zki (n) ≤

n 

 µ [1 − Aki (n − u)] zki (0) + µτ +1

u =1

Proof. From Corollary 2, it is easy to obtain that the system (5) has at least one N-periodic solution

T

x∗ (n) = x∗11 (n), . . . , x∗1m1 (n), x∗21 (n), . . . , x∗2m2 (n) .



Let

T

x(n) = x11 (n), . . . , x1m1 (n), x21 (n), . . . , x2m2 (n)



 

for n ∈ Z0+ , i = 1, 2, . . . , p(k), and k = 1, 2. Since Em1 +m2 − D is a non-singular M-matrix, it follows from Lemma 2 that there

3. Uniqueness and exponential stability

∆kij

 p (3−k)  1 − Aki (n − u) yki (0) + L(3−k)j

n  

 ×

n −1 

p(3−k)



L(3−k)j

j =1 z (m) (3−k)ij

△

m=0



n−m−1



 µ [1 − Aki (n − u)]

.

(29)

u=1

Let Z (n) = maxk,i ξki−1 zki (n) . We will show limn−→∞ Z (n) < +∞ below. To do this, we assume that it is not true. So, there exists a positive integer sequence {ns } that satisfies the following





A. Zhang et al. / Neural Networks 50 (2014) 98–109

conditions

According to the definition of D, we get

(a) N < n1 < n2 < · · · < ns < · · · , ns → +∞, s → +∞; (b) Z (n) ≤ Z (ns ), n ∈ [−τ , ns ]Z ; and (c) lims→+∞ Z (ns ) = +∞.

 αkij ≤

Since ns > N and  s = 1, 2, . . ., there exists a positive integer µs such that ns ∈ µs N , (µs + 1)N Z . Denote

 Λ1ki =

ns 

1 − Aki

µ [1 − Aki (ns − u)] zki (0),







×

ns −µs N −1



L(3−k)j

= Z (ns )ξ(3−k)j





ns −m−1



m=ns −µs N



×

  µ 1 − Aki (ns − u) ,

max

 ×

ˆ 3kij , L(3−k)j Λ

= Z (ns )ξ(3−k)j

zkj (n),



Γ(3−k)ij (m) = |b(3−k)ij (m)| + M l =1   × |c(3−k)ijl (m)| + |c(3−k)ilj (m)| .

×

 (µ[1 − Aki (ns − u)])  (µ[1 − Aki (ns − u)])

 µs N −1   v=1

 (31)



N −m−1



×

L(3−k)j σ(3−k)j

(µ[1 − Aki (ns − u)])

(v− 1)N −1



 ×

j =1

(µ[1 − Aki (ns − u)])

u =1

ns −m−1  Γ(3−k)ij (m) µ(1 − Aki )



N −1 

L(3−k)j σ(3−k)j

j =1

≤ µ Z (ns )ξ(3−k)j αkij N

Γ(3−k)ij (m).

(32)

≤ µN Z (ns )ξ(3−k)j αkij

m=0

ns −1

n s −1

=

ns −N −1



ω(m) =



ω(m) +

m=ns −N

ω(m)

m=ns −2N



ω(m)

m=ns −µs N

µs ns −(v− 1)N −1   v=1

ns −(v−1)N −1



m=ns −v N

ω(m) =

N −1 

ω(m),

(33)

αkij =

ω(ns − v N + m),

(34)

m=0

m=0



  u =0

(µ[1 − Aki (ns − u)])

u =1

  (v−1)N −1 µ 1 − Aki

[1 − Aki (ns − u)]

µ

N −1

Z (ns )ξ(3−k)j αkij 1 −



 µs N

1 − Aki µ

1 − µ 1 − Aki

1 − Aki

 

N



(36)

Λ3kij

 (3−k)  µN +τ 1 − (1 − Aki )N p ≤ d(3−k)ij ξ(3−k)j Z (ns )  N 1 − (1 − Aki )µ j =1   < ξki − ϱ Z (ns ).

(37)

From (31), (32), and (37), we have



N −m−1

Γ(3−k)ij (ns + m)



Combining (30), (36), and (28) yields

m=ns −v N

where ω(·) is an any continuous function. Let N −1 

v=1 µs 

(v− 1)N −1



  N  µN −1 1 − 1 − Aki ≤  d(3−k)ij ξ(3−k)j Z (ns ).   N  1 − 1 − Aki µ L(3−k)j

ns −(µs −1)N −1

+ ··· +

µs 

v=1

In addition, it is easy to verify that

=

Γ(3−k)ij (ns + m)

m=0

u =0

ns −µs N −1

m=ns −µs N

Γ(3−k)ij (ns + m)

u =1

= Z (ns )ξ(3−k)j

ns    µ 1 − Aki zki (0) u=1 n = µ 1 − Aki s zki (0) < zki (0),

≤ µτ + 1

v N −m−1

(v− 1)N −1

×



Λ1ki ≤



µs  N −1 



From (29), we have zki (ns ) ≤ Λ1ki + Λ2ki + Λ3ki . Since µ 1 − Aki < 1 and −1 ≤ ns − µs N − 1 < N − 1, we easily get

m=0 p(3−k)

(µ[1 − Aki (ns − u)])

u=(v−1)N



×



v=1 m=0





(µ[1 − Aki (ns − u)])

(v− 1)N −1

(30)

p(3−k)



u=1

p(3−k)

Λ2ki ≤ µτ +1

Γ(3−k)ij (m)

m=ns −v N

u=(v−1)N

j =1 n∈[−τ ,N −1]Z

 

ns −m−1



u =1

p(3−k)

µs ns −(v− 1)N −1   v=1



△z(3(m−)k)ij



µ 1 − Aki (ns − u) 

u=1

u =1 ns −1



Γ(3−k)ij (m)

m=ns −µs N ns −m−1

p(3−k)

(35)

n s −1

 

.

Thus, the condition (b), (33)–(35) give that

△z(3(m−)k)ij j =1 m=0   ns − m−1    × µ 1 − Aki (ns − u) ,

σkj =



ˆ 3kij ≤ Z (ns )ξ(3−k)j Λ



Λ2ki = µτ +1

Λ3ki = µτ +1

1 − (1 − Aki )N d(3−k)ij



L(3−k)j

u=1

ˆ 3kij = Λ

105

.

zki (ns ) < Ξki + ξki − ϱ Z (ns ),





i = 1, 2, . . . , p(k), k = 1, 2,

106

A. Zhang et al. / Neural Networks 50 (2014) 98–109

where

and

Ξki = zki (0) + µ

τ +1

p(3−k)



L(3−k)j σ(3−k)j

j =1

N −1 

Γ(3−k)ij (m).

m=0

Thus, we have

ξki−1 zki (ns ) < ξki−1 Ξki + 1 − ξki−1 ϱ Z (ns ), 



i = 1, 2, . . . , p(k), k = 1, 2.

(38)

If we denote h = maxk,i ξki and Ξ = maxk,i ξki−1 Ξki , then we obtain Z (ns ) ≤ Ξ + (1 − h−1 ϱ)Z (ns ) from (38). This gives Z (ns ) ≤ hϱ−1 Ξ , which contradicts that lims→+∞ Z (ns ) = +∞. Therefore, limn→+∞ Z (n) < +∞, which implies that there exists a constant L1 > 0 such that Z (n) ≤ L1 for n ∈ Z0+ . So, we have ξki−1 zki (n) ≤ Z (n) ≤ L1 , that is,





|xki (n) − x∗ki (n)| ≤ ξki L1 µ−n ,

n ∈ Z0+ .

(39)

∗ (n)|, Since ξki > maxn∈[−τ ,0]Z |φki (n) − φki

h > max k,i

max

n∈[−τ ,0]Z

|φki (n) − φki∗ (n)| = ∥φ − φ ∗ ∥.

Thus, there exists a constant L2 > 1 such that h = L2 ∥φ − φ ∥ and L1 L2 > 1. From (39), we have ∗

|xki (n) − x∗ki (n)| ≤ L∥φ − φ ∗ ∥µ−n , i = 1, 2, . . . , p(k), k = 1, 2,

n ∈ Z0+ ,

where L = L1 L2 . This completes the proof.

N −1   N −m v  1−w = w −1 v, (1 − w) 1 − (1 − w)N m=0



N −1



n∈IN

N −1

|v(n + m)|



|v(m)|

m=0 m=0   =   (1 − w) 1 − (1 − w)N (1 − w) 1 − (1 − w)N  N |v|  , = (1 − w) 1 − (1 − w)N

G(v, w) ≤

= where v = maxn∈IN |v(n)| and |v| we set 

0 U = U1

 V =

0 V1

U2 0 V2 0

 (m1 +m2 )×(m1 +m2 )

 (m1 +m2 )×(m1 +m2 )

1 N

N −1

m=0

|v(m)|. Therefore, if

, (40)

,

where U1 = (u1ij )m2 ×m1 , U2 = (u2ij )m1 ×m2 , V1 = (v1ij )m2 ×m1 , V2 = (v2ij )m1 ×m2 ,

   m2       c 2ijl + c 2ilj A1i −1 L2j ,  u1ij = b2ij + M l =1   m1        c 1ijl + c 1ilj A2i −1 L1j , u2ij = b1ij + M l =1

then 0 ≤ D1 ≤ U1 ,

0 ≤ D2 ≤ U2 ,

0 ≤ D 1 ≤ V1 ,

0 ≤ D2 ≤ V2 .

It means that 0 ≤ D ≤ U and 0 ≤ D ≤ V . So, Lemma 4 gives ρ(D) ≤ ρ(U ) and ρ(D) ≤ ρ(V ). Thus, combining Lemma 4 and Corollary 3 further gives two corollaries below. Corollary 4. Assume that Assumptions 1, 4 and 5 hold and ρ(U ) < 1. Then the system (5) has exactly one N-periodic solution. Moreover, the solution is globally exponentially sable. Corollary 5. Assume that Assumptions 1, 4 and 5 hold and ρ(V ) < 1. Then the system (5) has exactly one N-periodic solution. Moreover, the solution is globally exponentially sable.

This section presents two examples that demonstrate the validity of our theoretical results.



Note that

max

(42)

4. Illustrative examples

Corollary 3. Assume that Assumptions 1, 4 and 5 hold and ρ(D) < 1. Then the system (5) has at exactly one N-periodic solution. Moreover, the solution is globally exponentially sable.

G(v, w) ≤

   m2      NL2j | c | + | c | | b | + M  2ij 2ijl 2ilj   l=1     , v =  N  1ij  (1 − A1im) 1 − (1 − A1i )   1   | c1ijl | + | c1ilj | NL1j | b1ij | + M    l=1   ,     N  v2ij = 1 − A2i 1 − 1 − A2i

4.1. Example 1 Let us consider a model of (5) with p(1) = p(2) = 1 as    △x11 (n) = −A11 (n)x11 (n) + b211 (n)f21 x21 (n − 1)       2 + c2111 (n)f21 x21 (n − 2) + I11 (n),    △ x ( n ) = − A ( n )x21 (n) + b111 (n)f11 x11 (n − 1) 21   21    2 + c1111 (n)f11 x11 (n − 2) + I21 (n),

where △x11 (n) = x11 (n + 1) − x11 (n), △x21 (n) = x21 (n + 1) − x21 (n), and

 1 2 − cos nπ  , A21 (n) = , A11 (n) =   3 3    1 cos nπ  b211 (n) = c1111 (n) = , b111 (n) = c2111 (n) = − , 15 20  cos n π    I11 (n) = I21 (n) = , f11 (x) = sin(x),   5  f21 (x) = cos(x). Thus, the functions Aki (n), bkji (n), ckjil (n), and Iki (n) satisfy the Assumption 1 with N = 2 for i = j = 1 and k = 1, 2. In addition, it is easy to get that f11 (x) and f21 (x) satisfy the Assumptions 4 and 5 with M = 1 and L11 = L21 = 1. We also get A11 = A21 =

1 3

1 | b111 | = | c2111 | = ,

,

20

| b211 | = | c1111 | =

1 15

.

Submitting (44) into (42) and (40) yields

 (41)

(43)

0 V = 9 10

297



300  . 0

(44)

A. Zhang et al. / Neural Networks 50 (2014) 98–109

Fig. 1. Simulation results for the solution of (43) with initial condition (45).

A simple calculation gives that the eigenvalues of V are ±0.9439. So, the spectral radius of V is ρ(V ) = 0.9439 (