# New Results for Periodic Solution of High-Order BAM Neural Networks with Continuously Distributed Delays and Impulses

Authors: Chang-Bo Yang; Ting-Zhu Huang; Jin-Liang Shao
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

ByM-matrix theory, inequality techniques, and Lyapunov functional method, certain sufficient conditions are obtained to ensure the existence, uniqueness, and global exponential stability of periodic solution for a new type of high-order BAM neural networks with continuously distributed delays and impulses. These novel conditions extend and improve some previously known results in the literature. Finally, an illustrative example and its numerical simulation are given to show the feasibility and correctness of the derived criteria.


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## Body

## 1. Introduction

As is well known, during the hardware implementation of neural networks, time delays are inevitable due to finite switching speeds of the amplifiers and communication time, which may bring about complex influence on the system such as oscillation and instability [1, 2]. On the other hand, impulsive effects wildly exist in many realistic networks [3, 4], which may be caused by witching phenomenon, sudden changes, or other unexpected noise. Therefore, it is more appropriate to consider delay and impulsive effects when modeling neural networks, and many researches on various kinds of neural networks with delays, impulses, or both of them have been available [5–12]. (See Figures 1(a), 1(b), and 1(c)).It is confirmed that the system (51) has a unique π-periodic solution. Here (x1,y1) and (x2,y2) denote two pairs of solution of system (51) with different initial conditions ϕ1(s)=(-e0.01s+0.4,e0.01s-0.4)T and ϕ2(s)=(0.3,-0.3)T for s∈[-1,0] and γ=0.5,tk-tk-1=π,k=1,2,….
(a)
State response of the impulse-free system (51)
(b)
State response of the impulsive system (51)
(c)
Phase plots of system (51) without and with impulses, respectivelyBidirectional associative memory (BAM) neural networks, as an extension of the unidirectional autoassociator of Hofield neural network [13], was firstly introduced by Kosko [14]. Due to its wide application in pattern recognition, associative memory, image, and signal processing, BAM neural networks with delays and impulses have been extensively studied in the past few decades [15–22]. In addition, it is worth noting that high-order neural networks structures have advantages of stronger storage capacity, faster convergence rate, and higher fault tolerance, and these merits have been successfully used in pattern recognition [23]. Thus, it is important to investigate BAM neural networks with high-order terms, which is called high-order BAM neural networks.In this paper, we will consider a new type of high-order BAM neural networks with continuously distributed delays and impulses, which can be described by the following integrodifferential equations:(1)dxi(t)dt=-ai(t)xi(t)+∑j=1m‍cij(t)∫0∞‍Kij(s)fj(yj(t-s))ds+∑j=1m‍∑q=1m‍sijq(t)∫0∞‍Rijq(s)fj(yj(t-s))fq(yq(t-s))ds+Ii(t),t≠tk,xi(tk+)=xi(tk-)+Δxi(tk)=Hik(xi(tk-)),i=1,2,…,n,k∈ℤ≜{1,2,…},dyj(t)dt=-bj(t)yj(t)+∑i=1n‍dji(t)∫0∞‍K~ji(s)gi(xi(t-s))ds+∑i=1n‍∑p=1n‍ejip(t)∫0∞‍R~jip(s)gi(xi(t-s))gp+∑i=1n‍∑p=1n‍ejip(t)×(xp(t-s))ds+Jj(t),t≠tk,yj(tk+)=yj(tk-)+Δyj(tk)=Ejk(yj(tk-)),j=1,2,…,m,k∈ℤ≜{1,2,…},
where Δxi(tk) and Δyj(tk) are the impulses at moments tk and t1<t2<⋯ is a strictly increasing sequence such that limk→∞tk=∞. And xi(t) and yj(t) are the activations of the ith neuron and the jth neuron, respectively; ai(t)>0 and bj(t)>0 denote the passive decay rates; cij(t),dji(t),sijq(t),ejip(t) are the first- and second-order connection weights of the neural networks, respectively; Ii(t) and Jj(t) are the external inputs.Clearly, system (1) is a more general form of BAM neural networks, which has been widely applied in areas of science and engineering [24], such as neurobiology, image classification, and image recognition. In recent years, studies of such kind of neural networks with delays and impulses have received considerable interest, and some results have been reported in [25–30]. In particular, authors in [25–28] have discussed the stability of equilibrium point for a kind of impulsive high-order BAM neural networks with discrete delays by different methods, such as linear matrix inequality (LMI), Razumikhin technique. Subsequently, Huo et al. [29] and Yang [30] studied the existence of periodic solution and its exponential stability for an impulsive high-order BAM neural network with discrete delays by using the theory of coincidence degree and Lyapunov functional method. However, to the best of our knowledge, there are few results on the existence, uniqueness, and global exponential stability of periodic solution for system (1) with continuously distributed delays.The main propose of this paper is to study the periodicity of system (1) with distributed delays and general impulsive effects. It should be noticed that some new criteria on the existence and uniqueness of periodic solution for system (1) are established by combining the general ∥·∥r (see Notations) and analytical techniques, which is different from the conventional continuation theorem of coincidence degree theory used in [29, 30]. In addition, it is worth mentioning that the impulsive part in this paper is not necessarily bounded and linear, which makes its applications more extensive.The rest of this paper is organized as follows. In Section2, some assumptions, definitions, and important lemmas are given. In Section 3, the main results and some remarks are presented. In Section 4, an example and its numerical simulation are provided. Finally, some conclusions are summarized in Section 5.


## 2. Preliminaries

Notations.
Throughout this paper,ℜ and ℜn denote the set of real numbers and n-dimensional vector space, respectively. The symbol (·)T denotes the transpose of a vector or a matrix. Take ∥x∥r=(∑i=1n‍|xi|r)1/r with integer r∈[1,∞). Clearly, ∥·∥1,∥·∥2 are special cases of ∥·∥r with r=1,2, respectively, which are used to investigate the dynamics of various kinds of neural networks in [6–8, 10, 12, 15–17, 19, 21, 22, 25–31]. Denote C*≜{ϕ:(-∞,0]→ℜn+m∣ϕ(s) is continuous for all but at most countable points s∈(-∞,0] and at these points s∈(-∞,0], ϕ(s+) and ϕ(s-) exist, ϕ(s)=ϕ(s-)·} and define the norm ∥ϕ∥Δ by
(2)∥ϕ∥Δ=(∑i=1n‍sups∈(-∞,0]|ϕxi(s)|r+∑j=1m‍sups∈(-∞,0]|ϕyj(s)|r)1/r,
where ϕ=(ϕxT,ϕyT)T=(ϕx1,ϕx2,…,ϕxn,ϕy1,ϕy2,…,ϕym)T, and then C* is a Banach space with topology of the uniform convergence. In addition, system (1) is supplemented with initial values
(3)xi(s)=ϕxi(s),s∈(-∞,0],i=1,2,…,n,yj(s)=ϕyj(s),s∈(-∞,0],j=1,2,…,m.
As usual, we have the following assumptions for system (1).(S1) Functions ai(t),aj(t),cij(t),dji(t),sijq(t),ejip(t),Ii(t),andJj(t) are ω-periodic and bounded on [0,+∞) such that ai(t)≥ai->0,bj(t)≥bj->0,|cij(t)|≤cij+,|dji(t)|≤dji+,|sijq(t)|≤sijq+,|ejip(t)|≤ejip+,|Ii(t)|≤Ii,|Jj(t)|≤Jj for i,p=1,2,…,n,j,q=1,2,…,m.(S2) The activation functions gi(·),gp(·),fj(·), and fq(·) are bounded and Lipschitz continuous on ℜ; that is, there exist positive numbers Mjf,Mqf,Mig,Mpg and Lig,Ljf such that
(4)|fj(x)|≤Mjf,|fq(x)|≤Mqf,|fj(x)-fj(y)|≤Ljf|x-y|,|gi(x)|≤Mig,|gp(x)|≤Mpg,|gi(x)-gi(y)|≤Lig|x-y|,
for i,p=1,2,…,n,j,q=1,2,…,m and x,y∈ℜ.(S3)  The delay kernel functions Kij(·),K~ji(·),Rijq(·), and R~ijp(·) are piecewise continuous functions from [0,∞) to [0,∞) and satisfy Kij(s)≤𝒦(s),K~ji(s)≤𝒦(s),Rijq(·)≤ℛ(s), and R~ijp(s)≤ℛ(s) for s∈[0,∞), i,p=1,2,…,n,j,q=1,2,…,m, where 𝒦(s),ℛ(s) satisfy
(5)∫0∞‍𝒦(s)ds=1,∫0∞‍eɛs𝒦(s)ds<+∞,∫0∞‍ℛ(s)ds=1,∫0∞‍eɛsℛ(s)ds<+∞,in whichɛ denotes some positive constant number. For more information on these delay kernels, one can refer to [5, 9, 15, 18, 22].(S4)  Ω is an M-matrix, where r≥1,
(6)Ω=(rA-(r-1)G-LfΠ-LgΓrB-(r-1)F),A=diag(a1-,a2-,…,an-),B=diag(b1-,b2-,…,bm-),G=diag(G1,G2,…,Gn),F=diag(F1,F2,…,Fm),Lg=diag(L1g,L2g,…,Lng),Lf=diag(L1f,L2f,…,Lmf),Gi=∑j=1m[cij++∑q=1m‍(sijq++siqj+)Mqf]Ljf,Fj=∑i=1n[dji++∑p=1n‍(ejip++ejpi+)Mpg]Lig,Π=(Πij)n×m=cij++∑q=1m(sijq++siqj+)Mqf,Γ=(Γji)m×n=dji++∑p=1n(ejip++ejpi+)Mpg.Definition 1.
A functionz(t,ϕ)=(x1(t,ϕ),x2(t,ϕ),…, xn(t,ϕ),y1(t,ϕ),y2(t,ϕ),…,ym(t,ϕ))T∈ℜn+m is said to be the solution of system (1) with initial condition ϕ∈C* if the following two conditions are satisfied.(1)
z(t,ϕ) is piecewise continuous with first kind discontinuity at the points tk,k∈ℤ. Moreover, z(t,ϕ) is left continuous at each of the discontinuity points.(2)
z(t,ϕ) satisfies system (1) for t≥0 and z(s)=ϕ(s) for s∈(-∞,0].Definition 2.
The periodic solutionz*(t,ϕ*) of system (1) is said to be globally exponentially stable, if there exist constants ℳ≥1 and α>0 such that any other solution z(t,ϕ) of system (1) satisfies
(7)∥z(t,ϕ)-z*(t,ϕ*)∥r≤ℳe-αt∥ϕ-ϕ*∥Δ,t≥0.Lemma 3 (see [32]).
LetA∈ℤn×n, where ℤn×n is a set of n×n matrices with nonpositive off-diagonal elements. A is an M-matrix if and only if there exists a positive vector λ=(λ1,λ2,…,λn)T such that ∑j=1n‍ajiλj>0 or ∑j=1n‍aijλj>0.Lemma 4.
Assume that assumptions (S3) and (S4) hold; then there exist positive constants ɛ and λ1,λ2,…,λn+m such that
(8)Pi(ɛ)=[-ɛ+rai--(r-1)Gi]λi-∑j=1m‍λn+jdji+Lig∫0∞‍𝒦(s)eɛsds-∑j=1m‍∑p=1n‍λn+j(ejip++ejpi+)MpgLig×∫0∞‍ℛ(s)eɛsds≥0,Qj(ɛ)=(-ɛ+rbj--(r-1)Fj)λn+j-∑i=1n‍λicij+Ljf∫0∞‍𝒦(s)eɛsds-∑i=1n‍∑q=1m‍λi(sijq++siqj+)MqfLjf×∫0∞‍ℛ(s)eɛsds≥0,
for i,p=1,2,…,n,j,q=1,2,…,m.Proof.
Construct the aided functions as follows:(9)Pi(ϑi)=[-ϑi+rai--(r-1)Gi]λi-∑j=1m‍λn+jdji+Lig∫0∞‍𝒦(s)eϑisds-∑j=1m‍∑p=1n‍λn+j(ejip++ejpi+)MpgLig∫0∞‍ℛ(s)eϑisds,Qj(μj)=(-μj+rbj--(r-1)Fj)λn+j-∑i=1n‍λicij+Ljf∫0∞‍𝒦(s)eμjsds-∑i=1n‍∑q=1m‍λi(sijq++siqj+)MqfLjf∫0∞‍ℛ(s)eμjsds
for i,p=1,2,…,n,j,q=1,2,…,m. Using Lemma 3 and assumptions((S3)-(S4)), it is easy to deduce that (8) hold by similar proof in [7, 15–18]. For concise, it is omitted here.Lemma 5.
Let the integerr≥1; then the inequality holds as follows:
(10)(n+m)(1-r)/r∑i=1n+m‍|zi|≤(∑i=1n+m|zi|r‍)1/r
for all z=(z1,z2…,zn+m)T∈ℜn+m.Proof.
Obviously, the inequality (10) with r=1 is trivial. When r>1, consider the aided function g(x)=xr,x≥0. It is claimed that g(x) is convex since g′′(x)=r(r-1)xr-1>0, r>1 for x>0. Let x=(1/(n+m))∑i=1n+m‍|zi|; by Jensen's inequality, we have
(11)1(n+m)r(∑i=1n+m‍|zi|)r≤1n+m∑i=1n+m‍|zi|r,
which implies that the inequality (10) holds. This completes the proof.


## 3. Main Results

Firstly, letz(t,ϕ)=(x1(t,ϕ),x2(t,ϕ),…,xn(t,ϕ),y1(t,ϕ),y2(t,ϕ),…,ym(t,ϕ))T and z(t,φ)=(x1(t,φ),x2(t,φ),…,xn(t,φ),y1(t,φ),y2(t,φ),…,ym(t,φ))T be any two solutions of system (1) through ϕ,ψ∈C*, respectively; then we have the following useful lemma.Lemma 6.
Under assumptions ((S1)–(S4)), if the following two conditions hold: (S5)
Hik(·),Ejk(·) are Lipschitz continuous on ℜ; that is, there exist positive constants Hik,Ejk such that |Hik(x)-Hik(y)|≤Hik|x-y|,|Ejk(x)-Ejk(y)|≤Ejk|x-y| for i=1,2,…,n,j=1,2,…,m,k∈ℤ and x,y∈ℜ;(S6)
there existsθ such that lnηkr/(tk-tk-1)≤θ<ɛ, where t0=0, ηk=max{1,max1≤i≤nHik,max1≤j≤mEjk},k∈ℤ, and the scalar ɛ is estimated by (8).Then, the following inequality holds:
(12)∥z(t,ϕ)-z(t,φ)∥r≤ℳe-αt∥ϕ-φ∥Δ,t≥0,
where the constants ℳ≥1 and α>0 are to be determined later.Proof.
To be convenient, let(13)Xi(t)=|xi(t,ϕ)-xi(t,φ)|,i=1,2,…,n,Yj(t)=|yj(t,ϕ)-yj(t,φ)|,j=1,2,…,m.
It follows from ((S1)–(S3)) that
(14)D+Xi(t)≤-ai-Xi(t)+∑j=1m‍cij+Ljf∫0∞‍𝒦(s)Yj(t-s)ds+∑j=1m‍∑q=1m‍sijq+×∫0∞‍ℛ(s)(fj(xj(t-s,ϕ))fq(xq(t-s,ϕ))∫0∞‍ℛ(s)-fj(xj(t-s,φ))fq(xq(t-s,ϕ))∫0∞‍ℛ(s)+fj(xj(t-s,φ))fq(xq(t-s,ϕ))∫0∞‍ℛ(s)-fj(xj(t-s,φ))fq(xq(t-s,φ)))ds≤-ai-Xi(t)+∑j=1m‍cij+Ljf∫0∞‍𝒦(s)Yj(t-s)ds+∑j=1m‍∑q=1m‍(sijq++siqj+)MqfLjf∫0∞‍ℛ(s)Yj(t-s)ds,
for t≥0,t≠tk,k∈ℤ, i=1,2,…,n. Similarly, we have
(15)D+Yj(t)≤-bj-Yj(t)+∑j=1m‍dji+Lig∫0∞‍𝒦(s)Xi(t-s)ds+∑i=1n‍∑p=1n‍(ejip++ejpi+)MpgLig∫0∞‍ℛ(s)Xi(t-s)ds,
for t≥0,t≠tk,k∈ℤ, j=1,2,…,m. Also,
(16)Xi(tk+0)=|Hik(xi(tk,ϕ))-Hik(xi(tk,φ))|≤HikXi(tk),k∈ℤ,i=1,2,…,n,Yj(tk+0)=|Ejk(yj(tk,ϕ))-Ejk(yj(tk,φ))|≤EjkYj(tk),k∈ℤ,j=1,2,…,m.
Now define(17)Ui(t)=eɛt(Xi(t))r,i=1,2,…,n,Vj(t)=eɛt(Yj(t))r,j=1,2,…,m.
By using Young inequality apbq≤pa+qb, where a,b,p,q>0 and 1/p+1/q=1, it follows from (14)-(15) that
(18)D+Ui(t)≤(ɛ-rai-)Ui(t)+reɛt∑j=1m‍cij+Ljf×∫0∞‍𝒦(s)((Xi(t))r)(r-1)/r((Yj(t-s))r)1/rds+reɛt∑j=1m‍∑q=1m‍(sijq++siqj+)MqfLjf×∫0∞‍ℛ(s)((Xi(t))r)(r-1)/r((Yj(t-s))r)1/rds≤(ɛ-rai-+(r-1)Gi)Ui(t)+∑j=1m‍cij+Ljf∫0∞‍𝒦(s)eɛsVj(t-s)ds+∑j=1m‍∑q=1m‍(sijq++siqj+)MqfLjf∫0∞‍ℛ(s)eɛsVj(t-s)ds,
for t≥0,t≠tk,k∈ℤ, i=1,2,…,n. Similarly, we have
(19)D+Vj(t)≤(ɛ-rbj-+(r-1)Fj)Vj(t)+∑i=1n‍dji+Lig∫0∞‍𝒦(s)eɛsUi(t-s)ds+∑i=1n‍∑p=1n‍(ejip++ejpi+)MpgLig∫0∞‍ℛ(s)eɛsUi(t-s)ds,
for t≥0,t≠tk,k∈ℤ, j=1,2,…,m. Also
(20)Ui(tk+0)≤HikrUi(tk),k∈ℤ,i=1,2,…,n,Vj(tk+0)≤EjkrVj(tk),k∈ℤ,j=1,2,…,m.
Consider the candidate Lyapunov-Krasovskii functional as follows:
(21)𝕍(t)=𝕍1(t)+𝕍2(t),
where
(22)𝕍1(t)=∑i=1n‍λi[Ui(t)+∑j=1m‍cij+Ljf=∑i=1n‍λi×∫0∞‍𝒦(s)eɛs(∫t-st‍Vj(z)dz)ds=∑i=1n‍λi+∑j=1m‍∑q=1m‍(sijq++siqj+)MqfLjf=∑i=1n‍λi×∫0∞‍ℛ(s)eɛs(∫t-st‍Vj(z)dz)ds{Ui(t)+∑j=1m‍cij+Ljf}],𝕍2(t)=∑j=1n‍λn+j[Vj(t)+∑i=1n‍dji+Lig=∑j=1n‍λn+j×∫0∞‍𝒦(s)eɛs(∫t-st‍Ui(z)dz)ds=∑j=1n‍λn+j+∑i=1n‍∑p=1n‍(ejip++ejpi+)MpgLig=∑j=1n‍λn+j×∫0∞‍ℛ(s)eɛs(∫t-st‍Ui(z)dz)ds{Vj(t)+∑i=1n‍dji+Lig}].
When t≠tk,k∈ℤ, calculating the upper right Dini derivative of 𝕍1(t) along the solutions of system (1), we get
(23)D+𝕍1(t)=∑i=1n‍λi[{×∫0∞‍ℛ(s)eɛs(Vj(t)-Vj(t-s))ds}D+Ui(t)=∑i=1n‍λi+∑j=1m‍cij+Ljf∫0∞‍𝒦(s)eɛs(Vj(t)-Vj(t-s))ds=∑i=1n‍λi+∑j=1m‍∑q=1m‍(sijq++siqj+)MqfLjf=∑i=1n‍λi×∫0∞‍ℛ(s)eɛs(Vj(t)-Vj(t-s))ds]≤∑i=1n‍λi[{×∫0∞‍ℛ(s)eɛsdsVj(t)}(ɛ-rai-+(r-1)Gi)Ui(t)+∑j=1m‍cij+Ljf∫0∞‍𝒦(s)eɛsdsVj(t)+∑j=1m‍∑q=1m‍(sijq++siqj+)MqfLjf×∫0∞‍ℛ(s)eɛsdsVj(t)].
Similarly, we have
(24)D+𝕍2(t)≤∑j=1n‍λn+j[{×∫0∞‍ℛ(s)eɛsdsUi(t)}(ɛ-rbj-+(r-1)Fj)Vj(t)+∑i=1n‍dji+Lig∫0∞‍𝒦(s)eɛsdsUi(t)+∑i=1n‍∑p=1n‍(ejip++ejpi+)MpgLig×∫0∞‍ℛ(s)eɛsdsUi(t)].
Therefore, by Lemma 4, we obtain that
(25)D+𝕍(t)≤D+𝕍1(t)+D+𝕍2(t)≤-∑i=1n‍Pi(ɛ)Ui(t)-∑j=1m‍Qj(ɛ)Vj(t)≤0.
When t=tk,k∈ℤ, we have
(26)𝕍(tk+0)=∑i=1n‍λi[{×∫0∞‍ℛ(s)eɛs(∫tk+0-stk+0‍Vj(z)dz)ds}Ui(tk+0)+∑j=1m‍cij+Ljf∫0∞‍𝒦(s)eɛs(∫tk+0-stk+0‍Vj(z)dz)ds+∑j=1m‍∑q=1m‍(sijq++siqj+)MqfLjf×∫0∞‍ℛ(s)eɛs(∫tk+0-stk+0‍Vj(z)dz)ds],∑j=1n‍λn+j[{×∫0∞‍ℛ(s)eɛs(∫tk+0-stk+0‍Ui(z)dz)ds}Vj(tk+0)+∑i=1n‍dji+Lig∫0∞‍𝒦(s)eɛs(∫tk+0-stk+0‍Ui(z)dz)ds+∑i=1n‍∑p=1n‍(ejip++ejpi+)MpgLig×∫0∞‍ℛ(s)eɛs(∫tk+0-stk+0‍Ui(z)dz)ds],≤∑i=1n‍λi[ηkrUi(t)+∑j=1m‍cij+Ljf∫0∞‍𝒦eɛs(∫t-st‍Vj(z)dz)ds+∑j=1m‍∑q=1m‍(sijq++siqj+)MqfLjf×∫0∞‍ℛ(s)eɛs(∫t-st‍Vj(z)dz)ds{ηkrUi(t)+∑j=1m‍cij+Ljf∫0∞‍𝒦eɛs(∫t-st‍Vj(z)dz)ds}],∑j=1n‍λn+j[{×∫0∞‍ℛ(s)eɛs(∫t-st‍Ui(z)dz)ds}ηkrVj(t)×∑j=1n‍λn+j+∑i=1n‍dji+Lig∫0∞‍𝒦(s)eɛs(∫t-st‍Ui(z)dz)ds×∑j=1n‍λn+j+∑i=1n‍∑p=1n‍(ejip++ejpi+)MpgLig×∑j=1n‍λn+j×∫0∞‍ℛ(s)eɛs(∫t-st‍Ui(z)dz)ds]≤ηkr𝕍(tk).
Now, we claim that
(27)𝕍(tk+0)≤ηkrηk-1r⋯η1r𝕍(t0),t∈(tk-1,tk],k∈ℤ.
In fact, for t∈(t0,t1], noticing that η1≥1 and (25), we have
(28)η1r𝕍(t1)≤η1r𝕍(t0).
On the other hand, from (26), we have
(29)𝕍(t1+0)≤η1r𝕍(t1).
Combining (28) and (29), we obtain
(30)𝕍(t1+0)≤η1r𝕍(t0),
which implies that (27) holds for k=1. Assume that (27) holds for k=m, that is,
(31)𝕍(tm+0)≤ηmrηm-1r⋯η1r𝕍(t0).
Then, for t∈(tm,tm+1], from (25), we have
(32)𝕍(tm+1)≤𝕍(tm+0).
On the other hand, from (26), we have
(33)𝕍(tm+1+0)≤ηm+1r𝕍(tm+1).
From (31)–(33), we obtain
(34)𝕍(tm+1+0)≤ηm+1rηmr⋯η1r𝕍(t0).
This shows that (27) holds for k=m+1. Hence, by mathematical induction, (27) holds for all k∈ℤ. Combining (25) and (27), we obtain
(35)𝕍(t)≤𝕍(tk+0)≤ηkrηk-1r,⋯,η1r𝕍(t0),
for all t∈(tk,tk+1],k∈ℤ. Noticing that ηkr≤eθ(tk-tk-1),k∈ℤ in (S6), we have
(36)𝕍(t)≤𝕍(t0)eθ(tk-tk-1)eθ(tk-1-tk-2)⋯eθ(t1-t0)≤𝕍(0)eθt,
for all t∈(tk,tk+1],k∈ℤ. On the other hand, it follows from (21) that
(37)𝕍(t)≥λ-[∑i=1n‍Ui(t)+∑j=1m‍Vj(t)]=λ-[∑i=1n|xi(t,ϕ)-xi(t,φ)|r{+∑j=1m‍|yj(t,ϕ)-yj(t,φ)|r}+∑j=1m|yj(t,ϕ)-yj(t,φ)|r]eɛt,𝕍(0)=∑i=1n‍λi[{+∑j=1m‍cij+Ljf∫0∞‍𝒦(s)eɛs(∫-s0‍Vj(z)dz)ds}Ui(0)+∑j=1m‍cij+Ljf∫0∞‍𝒦(s)eɛs(∫-s0‍Vj(z)dz)ds+∑j=1m‍∑q=1m‍(sijq++siqj+)MqfLjf×∫0∞‍ℛ(s)eɛs(∫-s0‍Vj(z)dz)ds{+∑j=1m‍cij+Ljf∫0∞‍𝒦(s)eɛs(∫-s0‍Vj(z)dz)ds}]+∑j=1n‍λn+j[{+∑i=1n‍dji+Lig∫0∞‍𝒦(s)eɛs(∫-s0‍Ui(z)dz)ds}Vj(0)+∑i=1n‍dji+Lig∫0∞‍𝒦(s)eɛs(∫-s0‍Ui(z)dz)ds+∑i=1n‍∑p=1n‍(ejip++ejpi+)MpgLig×∫0∞‍ℛ(s)eɛs(∫-s0‍Ui(z)dz)ds{+∑i=1n‍dji+Lig∫0∞‍𝒦(s)eɛs(∫-s0‍Ui(z)dz)ds}]≤λ+Λ{∑j=1msups∈(-∞,0]|ϕxi(s)-ψxi(s)|r+∑i=1nsups∈(-∞,0]|ϕyj(s)-ψyj(s)|r{∑j=1m‍sups∈(-∞,0]|ϕxi(s)-ψxi(s)|r}},
where
(38)λ+=max{λ1,λ2,…,λn+m},λ-=min{λ1,λ2,…,λn+m},Λ=max{{∫0∞‍ℛ(s)eɛs(∫-s0‍eɛzdz)}[1+∑i=1n‍max1≤j≤mdji+Lig∫0∞‍𝒦(s)eɛs(∫-s0‍eɛzdz)ds+∑i=1n‍∑p=1n‍max1≤j≤m(ejip++ejpi+)MpgLig×∫0∞‍ℛ(s)eɛs(∫-s0‍eɛzdz)ds{1+∑i=1n‍max1≤j≤mdji+Lig∫0∞‍𝒦(s)eɛs(∫-s0‍eɛzdz)ds}],[1+∑j=1m‍max1≤i≤ncij+Ljf∫0∞‍𝒦(s)eɛs(∫-s0‍eɛzdz)ds+∑j=1m‍∑q=1m‍max1≤i≤n(sijq++siqj+)MqfLjf×∫0∞‍ℛ(s)eɛs(∫-s0‍eɛzdz)ds{1+∑j=1m‍max1≤i≤ncij+Ljf∫0∞‍𝒦(s)eɛs(∫-s0‍eɛzdz)ds}]}.
Together with (36)-(37), we have
(39)∑i=1n‍|xi(t,ϕ)-xi(t,φ)|r+∑j=1m‍|yj(t,ϕ)-yj(t,φ)|r≤λ+λ-Λe-(ɛ-θ)t∥ϕ-φ∥Δr,
for all t≥0. Let ℳ=((λ+/λ-)Λ)1/r, α=(ɛ-θ)/r and then we have
(40)∥z(t,ϕ)-z(t,φ)∥r≤ℳe-αt∥ϕ-φ∥Δ.
This completes the proof.In the following, we will study the existence, uniqueness, and global exponential stability of periodic solution of system (1) by exploiting Lemmas 5 and 6.Theorem 7.
Assume that assumptions (S1)–(S6) hold, then system (1) has a unique ω-periodic solution, which is globally exponentially stable.Proof.
Firstly, we prove the existence of periodic solution of system (1). To this end, let z(t,φ)=(x1(t,φ),x2(t,φ),…,xn(t,φ),y1(t,φ), y2(t,φ),…,ym(t,φ))T be an arbitrary solution of system (1) through (0,φ), where φ∈C*. Define z(t,ϕ)=z(t+ω,φ), where ϕ=z(s+ω,φ),s≤0. We can know that ϕ∈C* and z(t,ϕ)=(x1(t,ϕ),x2(t,ϕ),…,xn(t,ϕ),y1(t,ϕ),y2(t,ϕ),…,ym(t,ϕ))T is also a solution of system (1) through (0,ϕ). By virtue of Lemma 6, we have
(41){∑i=1n‍|xi(t,ϕ)-xi(t,φ)|r+∑j=1m‍|yi(t,ϕ)-yi(t,φ)|r}1/r≤ℳe-αt∥ϕ-φ∥Δ,
for t≥0. So, we have
(42){{+∑j=1m‍|yj(t+ω,φ)-yj(t,φ)|r}∑i=1n‍|xi(t+ω,φ)-xi(t,φ)|r+∑j=1m‍|yj(t+ω,φ)-yj(t,φ)|r}1/r≤ℳe-αt∥ϕ-φ∥Δ,
for t≥0. It follows from Lemma 5 that
(43)∑i=1n‍|xi(t+ω,φ)-xi(t,φ)|+∑j=1m‍|yj(t+ω,φ)-yj(t,φ)|≤(n+m)1-1/rℳe-αt∥ϕ-φ∥Δ.
Noticing that for i=1,2,…,n,
(44)xi(t+kω,φ)=xi(t,φ)+∑s=1k‍[xi(t+sω,φ)-xi(t+(s-1)ω,φ)].
It follows from (43)-(44) that
(45)limk→∞xi(t+kω,φ)=xi(t,φ)+limk→∞∑s=1k‍[xi(t+sω,φ)-xi(t+(s-1)ω,φ)]≤xi(t,φ)+(n+m)(1-1/r)ℳ∥ϕ-φ∥Δlimk→∞∑s=1k‍e-α(t+(s-1)ω)≤xi(t,φ)+(n+m)(1-1/r)ℳe-αt∥ϕ-φ∥Δ∑s=1∞‍e-α(s-1)ω<∞,
which implies that limk→∞xi(t+kω,φ) exists. Similar to (44) and (45), we obtain that limk→∞yj(t+kω,φ) exists. Let
(46)z*(t,φ*)=(x1*(t,φ*),x2*(t,φ*),…,xn*(t,φ*),y1*(t,φ*),y2*(t,φ*),…,ym*(t,φ*))T,
where limk→∞xi(t+kω,φ)=xi*(t,φ*),limk→∞yj(t+kω,φ)=yj*(t,φ*) for i=1,2,…,n,j=1,2,…,m. Then z*(t,φ*) is an ω-periodic solution for system (1).
Secondly, we prove the uniqueness of periodic solution of system (1). Assume that z(t,ζ)=(x1(t,ζ),x2(t,ζ),…,xn(t,ζ),y1(t,ζ),y2(t,ζ),…,ym(t,ζ))T is another ω-periodic solution of system (1) through (0,ζ), where ζ∈C*. By a minor modification of the proof of (43), we have
(47)∑i=1n‍|xi*(t,φ*)-xi(t,ζ)|+∑j=1m‍|yj*(t,φ*)-yj(t,ζ)|=∑i=1n‍|xi*(t+kω,φ*)-xi(t,ζ)|+∑j=1m‍|yj*(t+kω,φ*)-yj(t,ζ)|≤(n+m)1-1/rℳe-α(t+kω)∥φ*-ζ∥Δ,t≥0.
Taking k→∞, we have
(48)xi*(t,φ*)=xi(t,ζ),t≥0,i=1,2,…,n,yj*(t,φ*)=yj(t,ζ),t≥0,j=1,2,…,m,
which implies that system (1) has a unique ω-periodic solution.
Finally, sincez*(t,φ*) is a unique ω-periodic solution of system (1), let z(t,ψ) be any other solution of system (1) through (0,ψ). From Lemma 6, we obtained that
(49)∥z(t,ψ)-z*(t,φ*)∥r≤ℳe-αt∥ψ-φ*∥Δ,t≥0,
where ψ,φ*∈C* and α,ℳ are the same as defined in Lemma 6. It follows from Definition 2 that the ω-periodic solution z*(t,φ*) is globally exponentially stable. Up to now, we conclude that system (1) has a unique ω-periodic solution z*(t,φ*), which is globally exponentially stable. This completes the proof.Remark 8.
In assumption (S5), we only assume that the impulsive operators Hik(·) and Ejk(·) are Lipschitz continuous, which remove the usual assumptions that the boundedness and linearity of the impulsive operators are required in [18, 19, 21, 29–31].  Thus, our results have wider adaptive range. Particularly, if we take the linear operators Δxi(tk)=-γikxi(tk-)  and Δyj(tk)=-αjkyj(tk-) as considered in [18, 19, 21, 29–31], that is,(S7)
{Hik(xi(tk-))=(1-γik)xi(tk-),γik∈(0,2),i=1,2,…,n,k=1,2,…,Ejk(yj(tk-))=(1-αjk)yj(tk-),αjk∈(0,2),j=1,2,…m,k=1,2,…,then we have Hik=|1-γik|<1,Ejk=|1-αjk|<1. So we can choose ηk=1 and θ=0 to satisfy assumption (S6). In this case, we have the following interesting corollary.Corollary 9.
Assume that assumptions ((S1)–(S4)) and (S7) hold; then system (1) has a unique ω-periodic solution, which is globally exponentially stable.Remark 10.
Note that whentk-tk-1=∞,k=1,2,… in assumption (S6), which implies that there are no impulsive effects on system (1). Correspondingly, we call system (1) an impulse-free. In this case, we have the following corollary.Corollary 11.
Assume that assumptions ((S1)–(S4)) hold; then the impulse-free system (1) has a unique ω-periodic solution, which is globally exponentially stable.Remark 12.
Clearly, based on the general∥·∥r and Lemma 5, a general criterion ensuring the existence of periodic solution and its global exponential stability of system (1) with and without impulses has been established. Compared with results in [6, 7, 15–17], it is easy to see that our results are extended and improved because their results can be viewed as the special case of r=1 in assumption (S4). In addition, since the nonnetwork parameter r is introduced in the condition (S4), it can allow much broader applications for designing the circuit of a convergent impulsive network.Remark 13.
In assumption (S3), if the kernel is a delta function of the form:
(50)Kij(s)=Rijq(s)=δ(s-τ),K~ji(s)=R~jip(s)=δ(s-σ),
where τ≥0 and σ≥0, then system (1) with continuously distributed delays reduces to the model with discrete delays in [29]. According to Lemma 3, we know that the condition (H5) of Theorem 3.1 in [29] implies that Ω with r=1 is an M-matrix but not vice versa. Thus, our results are new and complementary to their results.


## 4. An Example

In this section, an example and its numerical simulation are given to illustrate the correctness of the obtained theoretical results.An Example. Consider the following high-order BAM neural networks with infinite distributed delays and impulses:
(51)dx1(t)dt=-(8+sin2t)x1(t)+(2+cos2t)∫0∞‍e-sf1(y1(t-s))ds+2sin2t∫0∞‍se-sf1(y1(t-s))f1(y1(t-s))ds+cos2t,t≠tk,x1(tk+)=x1(tk-)+Δx1(tk)=H1k(x1(tk-)),k=1,2,…,dy1(t)dt=-(10-3cos2t)y1(t)+(1+sin2t)∫0∞‍e-sg1(x1(t-s))ds+2cos2t∫0∞‍se-sg1(x1(t-s))g1(x1(t-s))ds+sin2t,t≠tk,y1(tk+)=y1(tk-)+Δy1(tk)=E1k(y1(tk-)),k=1,2,…,
where f1(u)=g1(u)=tanh(u). By simple calculation, we obtain that L1g=L1f=M1f=M1g=1 and
(52)Σ=(7-7-66+r).
If the integer r≥1, then Σ is an M-matrix. Thus, assumptions ((S1)–(S4)) are satisfied for system (51). For the impulsive part, the following two cases are considered.Case 1.
Whentk-tk-1=∞,k=1,2,…, by Corollary 11, we conclude that the impulse-free system (51) has a unique π-periodic solution, which is globally exponentially stable.Case 2.
When the impulsive parts are taken as the nonlinear operators such thatΔx1(tk)=γtan(x1(tk-)),γ∈(0,2) and Δy1(tk)=γtan(y1(tk-)),γ∈(0,2), that is,
(53)H1k(xi(tk-))=x1(tk-)-γtan(x1(tk-)),γ∈(0,2),k=1,2,…,E1k(yj(tk-))=y1(tk-)-γtan(y1(tk-)),γ∈(0,2),k=1,2,…,
which satisfy the assumption (S5) with Lipschitz constants H1k=E1k≤|1-γ|<1,k=1,2,…. So we can choose ηk=1 and θ=0 to satisfy the assumption (S6). According to Theorem 7, we conclude that the impulsive system (51) has a unique π-periodic solution, which is globally exponentially stable. However, in this case, results in [18, 19, 21, 29–31] are ineffective because the function tan(·) is nonlinear. Moreover, the condition (H5) of Theorem 3.1 in [29] cannot be applied to system (51), since b1--c11+L1f-2s111+M1fL1f=0≯0.


## 5. Conclusions

In this paper, we have studied the existence, uniqueness, and global exponential stability of periodic solution for a kind of high-order BAM neural networks with continuously distributed delays and general impulses. It should be noted that some extended and improved criteria have been derived by exploiting the general∥·∥r, Lemma 5, and the Lyapunov functional method. In addition, these criteria are in terms of M-matrix, which can be easily checked by many equivalent conditions listed in [32]. Finally, an example and its numerical simulation are given to show the feasibility and correctness of the obtained results.



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