# Exponential Stability of Antiperiodic Solution for BAM Neural Networks with Time-Varying Delays

Authors: Xiaofei Li; Chuan Qin; Quanxin Zhu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this paper, a kind of BAM neural networks with leakage delays in the negative feedback terms and time-varying delays in activation functions was considered. By constructing a suitable Lyapunov function and using inequality techniques, some sufficient conditions to ensure the existence and exponential stability of antiperiodic solutions of these neural networks were derived. These conditions extend some results recently appearing in recent papers. Lastly, an example is given to show the feasibility of these conditions.


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## Body

## 1. Introduction

It is known that neural networks have been applied in numerous fields, such as pattern recognition, classification, associative memory, optimization, signal and image processing, parallel computation, and nonlinear optimization problems. Up to now, there are many works focusing on the dynamical nature of various kinds of neural networks, such as stability, periodic solution, almost periodic solution, bifurcation, and chaos (see [1–11]). However, since significant time delays are ubiquitous, it is necessary to introduce delays into communication channels which leads to delayed neural networks models. Kosko [12] proposed a new kind of neural networks named bidirectional associative memory (BAM) neural networks with time delays which was given by the following:(1)dxitdt=-aixit+∑j=1majifjyjt-σjit+Ii,dyitdt=-biyit+∑j=1mbjigjxjt-τjit+Ji,where i=1,…,m,m is the number of neurons in layers. In model (1), xi and yi stand for the activations of ith neurons; positive constants ai and bi represent the rate with which the ith neuron will reset its potential to the resting state in isolation when they are disconnected from the network and the external inputs at time t; time delays τji(t) and σji(t) are nonnegative functions; and Ii and Ji denote the components of external input source introduced from outside the network.The authors in this paper applied this model to image processing.There are many papers about BAM neural networks and stochastic BAM neural networks; for details, see [13–18].It is known that the existence-uniqueness and stability is an important theoretical problem in the field of dynamics systems and one can find this topic in these papers [19–22]. Always, leakage delays appear in the negative feedback term of neural networks. Based on this, Gopalsmay [23] proposed the following BAM neural networks and studied the stability of the equilibrium and periodic solutions(2)dxitdt=-aixit-δ1i+∑j=1majifjyjt-σji+Ii,dyitdt=-biyit-δ2i+∑j=1mbjigjxjt-τji+Ji.In this system, time delays δ1i and δ2i are positive constants, where i=1,…,m. By constructing Lyapunov-Kravsovskii functionals, using inequalities and M-matrices technology, the author gave two sets of delay dependent sufficient conditions on the existence of a unique equilibrium as well as its asymptotic and exponential stability to system (2). Because of time-varying delays in the real world, Liu [24] proposed the following BAM neural networks and discussed the global exponential stability of the network:(3)dxitdt=-aixit-δ1i+∑j=1maijfjyjt-σjit+Ii,dyitdt=-biyit-δ2i+∑j=1mbijgjxjt-τjit+Ji.By constructing a Lyapunov functional, some sufficient conditions on the global exponential stability of the equilibrium for system (3) were established. There are few papers considering the variable external input. It is significative to consider time-varying delays and external input in neural networks.Recently, Li et al. [25] considered the following BAM neural networks with time-varying external input:(4)dxitdt=-aixit+∑j=1msjitfjxjt,yjt-τji+Iitdyitdt=-biyit+∑j=1mtjitgjxjt-δji,yjt+Jit.To the best of our knowledge, there are few papers focusing on the existence and stability of antiperiodic solution to BAM neural networks with time-varying delays in the leakage terms in the negative feedback term. Motivated by the above discussions, in this paper, we propose a kind of BAM neural networks with time-varying delays and external input as follows:(5)dxitdt=-aixit-δit+∑j=1majitfjxjt+∑j=1mbjitgjyjt-τjit+Iit,dyitdt=-biyit-σit+∑j=1msjithjyjt+∑j=1mtjitφjxjt-ξjit+Jit.We can easily find that model (5) extends the above models. In model (5), we consider the time-varying leakage delays in negative feedback terms and time-varying delays in activation functions. And the activation functions in model (5) are more general. If the parameters δi(t) are all equal to 0, then model (5) will deduce to special model in (4). And if δi(t) are all constants, then model (5) will extend the models in (2) and (3). Therefore, it is important to investigate the stability of model (5). In this paper, by constructing a suitable Lyapunov functional, we give some sufficient conditions to ensure the existence and global exponential stability of antiperiodic solutions of system (5).The rest of this paper is organized as follows. In Section2, we introduce some notations and give some lemmas. In Section 3, we obtain some sufficient conditions on the existence and global exponential stability of antiperiodic solution of system (5). In Section 4, we give some examples to illustrate the efficiency. In Section 5, some conclusions are given.


## 2. Notations and Preliminary Results

First, we give some notations. We denote(6)δi+=supt∈Rδit,aji+=supt∈Rajit,bji+=supt∈Rbjit,τji+=supt∈Rτjit,Ii+=supt∈RIit,σi+=supt∈Rσit,sji+=supt∈Rsjit,tji+=supt∈Rtjit,ξji+=supt∈Rξjit,Ji+=supt∈RJit.For vector U=(u1,…,um)T and matrix M=(mij)m×m, we define the following norms: (7)U=∑i=1mui21/2,M=∑i,j=1mmij21/2,respectively. For (8)ϕs=ϕ1s,…,ϕmsT,ϕis∈C-δ,0,R,i=1,…,m,ψs=ψ1s,…,ψmsT,ψis∈C-τ,0,R,i=1,…,m,where δ=max1≤i,j≤m{δi+,ξji+},τ=max1≤i,j≤m{σi+,τji+}, we denote (9)ϕ=sup-δ≤s≤0∑i=1mϕis21/2,ψ=sup-τ≤s≤0∑i=1mψis21/2.The initial conditions of the system (5) are given by(10)xi0s=ϕis,-δ≤s≤0,yi0s=ψis,-τ≤s≤0.Let(x(t),y(t))T be the solution of system (5) with initial conditions (10), where x(t)=(x1(t),…,xm(t))T, y(t)=(y1(t),…,ym(t))T. We say the solution (x(t),y(t))T is T-antiperiodic if xi(t+T)=-xi(t), yi(t+T)=-yi(t)(i=1,…,m) for all t∈R, where T is a positive constant.Throughout this paper, we assume that the following conditions hold.(H1) Forfj,gj,hj,φj∈C(R,R),j=1,…,m, there exist positive constants Ljf,Ljg,Ljh,Ljφ and Mjf,Mjg,Mjh,Mjφ such that (11)fju-fjv≤Ljfu-v,fju≤Mjf,gju-gjv≤Ljgu-v,gju≤Mjg,hju-hjv≤Ljhu-v,hju≤Mjh,φju-φjv≤Ljφu-v,φju≤Mjφ.for all u,v∈R.(H2) For allt,u∈R, (12)ajit+Tfju=-ajitfj-u,bjit+Tgju=-bjitgi-u,sjit+Thiu=-sjithi-u,tjit+Tφiu=-tjitφi-u,Iit+T=-Iit,Jit+T=-Jit,δit+T=δit,σit+T=σit,where i,j=1,…,m and T is a positive constant.Definition 1.
The solution(x∗(t),y∗(t))T of system (5) is said to be globally exponentially stable if there exist constants β>0 and M>1 such that (13)∑i=1mxit-xi∗t2+∑i=1myit-yi∗t2≤Me-βtϕ-ϕ∗2+ψ-ψ∗2for each solution (x(t),y(t))T of system (5).Lemma 2 (see [25]).
Let(14)A=-ai00-bj,α=min1≤i,j≤mai,bj,and then(15)exp⁡At≤2e-αt,∀t≥0.Lemma 3.
Suppose that (H3)(16)-2ai+2ai2δi++tji+Ljφ21-ςj+aiδi++1∑j=1mbji+Ljg2ξj+∑j=1maji+Ljf2ξj+Ljf21-ξj+∑j=1mbiσi+tji+Ljφ21-ςj<0,-2bi+2bi2σi++bji+Ljg21-ξj+biσi++1∑j=1mtji+Ljφ2ςj+∑j=1msji+Ljh2ςj+Ljh21-ςj+∑j=1maiδi+bji+Ljg21-ξj<0,where 0≤ξj,ςj<1(j=1,…,m) are any constants. Then there exists β>0 such that (17)β-2ai+ai2δi+eβδi++1+tji+Ljφ21-ςjeβξji++aiδi++1∑j=1mbji+Ljg2ξj+∑j=1maji+Ljf2ξj+Ljf21-ξj+∑j=1mbiσi+tji+Ljφ21-ςjeβξji+≤0,β-2bi+bi2σi+eβσi++1+bji+Ljg21-ξjeβτji++biσi++1∑j=1mtji+Ljφ2ςj+∑j=1msji+Ljh2ςj+Ljh21-ςj+∑j=1maiδi+bji+Ljg21-ξjeβτji+≤0.Proof.
Let(18)ϱ1iβ=β-2ai+ai2δi+eβδi++1+tji+Ljφ21-ςjeβξji++aiδi++1∑j=1mbji+Ljg2ξj+∑j=1maji+Ljf2ξj+Ljf21-ξj+∑j=1mbiσi+tji+Ljφ21-ςjeβξji+,ϱ2iβ=β-2bi+bi2σi+eβσi++1+bji+Ljg21-ξjeβτji++biσi++1∑j=1mtji+Ljφ2ςj+∑j=1msji+Ljh2ςj+Ljh21-ςj+∑j=1maiσi+bji+Ljg21-ξjeβτji+.Clearly, ϱ1i(β),ϱ2i(β)(i=1,…,m) are continuously differential functions and satisfy that (19)dϱ1iβdβ>0,limβ→+∞ϱ1iβ=+∞,ϱ1i0<0,dϱ2iβdβ>0,limβ→+∞ϱ2iβ=+∞,ϱ2j0<0.According to the intermediate value theorem, it is clear that there exist constants βi>0,βi∗>0 such that (20)ϱ1iβi=0,ϱ2iβi∗=0,i=1,…,m.Let β0=min{β1,…,βm,β1∗,…,βm∗}; then it follows that β0>0 and (21)ϱ1iβ0≤0,ϱ2iβ0≤0,i=1,…,m.The proof of Lemma 3 is complete.Lemma 4.
Suppose that (H1) holds true. Then, for any solution(x(t),y(t))T of system (5), there exists a constant (22)γ=1-2αai2δi++bi2σi+-12ϕ2+ψ2+2αaiδi+∑j=1maji+Mjf+aiδi+∑j=1mbji+Mjg+aiδi+Ii++∑j=1maji+Mjf+∑j=1mbji+Mjg+Ii++biσi+∑j=1msji+Mjh+biσi+∑j=1mtji+Mjϕ+biσi+Ji++∑j=1msji+Mjh+∑j=1mtji+Mjϕ+Ji+,such that(23)xit≤γ,yit≤γ,i=1,…,m,∀t>0.Proof.
From system (5), we conclude that(24)dxitdt=-aixit+ai∫t-δitt-aixis-δis+∑j=1majisfjxjs+∑j=1mbjisgjyjs-τjis+Iisds+∑j=1majitfjxjt+∑j=1mbjitgjyjt-τjit+Iit,dyitdt=-biyit+bi∫t-σitt-biyis-σis+∑j=1msjishjyjs+∑j=1mtjisφjxjs-σjis+Jisds+∑j=1msjithjyjt+∑j=1mtjitφjxjt-ξjit+Jit. Let(25)ziit=xityit,A=-ai00-bi,Biit=IitJit,Fxit,yit=f1xit,yitf2xit,yit, where(26)f1xit,yit=ai∫t-δitt-aixis-δis+∑j=1majisfjxjs+∑j=1mbjisgjyjs-τjis+Iisds+∑j=1majitfjxjt+∑j=1mbjitgjyjt-τjit,f2xit,yit=bi∫t-σitt-biyis-σis+∑j=1msjishjyjs+∑j=1mtjisφjxjs-σjis+Jisds+∑j=1msjithjyjt+∑j=1mtjitφjxjt-ξjit, and then system (5) has the following form:(27)zii′t≤Aziit+Fxit,yit+Biit. By (27), we get (28)ziit≤eAtzii0+∫0teAt-sFxis,yjs+Biisds. In view of Lemma 2, we have(29)ziit≤2e-αtzii0+2∫0te-αt-sFxis,yis+Biisds≤2ϕ2+ψ2+2α1-e-αtaiδi+aixit-δit+∑j=1maji+Mjf+∑j=1mbji+Mjg+Ii++∑j=1maji+Mjf+∑j=1mbji+Mjg+Ii++biσi+biyit-σit+∑j=1msji+Mjh+∑j=1mtji+Mjφ+Ji++∑j=1msji+Mjh+∑j=1mtji+Mjφ+Ji+≤2ϕ2+ψ2+2αai2δi+ziit+aiδi+∑j=1maji+Mjf+aiδi+∑j=1mbji+Mjg+aiδi+Ii++∑j=1maji+Mjf+∑j=1mbji+Mjg+Ii++bi2σi+ziit+biσi+∑j=1msji+Mjh+biσi+∑j=1mtji+Mjφ+biσi+Ji++∑j=1msji+Mjh+∑j=1mtji+Mjφ+Ji+.Let (30)γ=1-2αai2δi++bi2σi+-12ϕ2+ψ2+2αaiδi+∑j=1maji+Mjf+aiδi+∑j=1mbji+Mjg+aiδi+Ii++∑j=1maji+Mjf+∑j=1mbji+Mjg+Ii++biσi+∑j=1msji+Mjh+biσi+∑j=1mtji+Mjφ+biσi+Ji++∑j=1msji+Mjh+∑j=1mtji+Mjφ+Ji+,and it follows from (29) that zii(t)≤γ. That is, xit≤γ and yit≤γ. This completes the proof of Lemma 4.


## 3. Main Results

In this section, we give our main results for system (5).Theorem 5.
Suppose that (H1)-(H3) are satisfied. Then any solution(x∗(t),y∗(t))T of system (5) is globally exponentially stable.Proof.
First, we denoteui(t)=xi(t)-xi∗(t),vi(t)=yi(t)-yi∗(t),i=1,…,m. By system (5), we have (31)duitdt=-aiuit+ai∫t-δitt-aiuis-δis+∑j=1majisfjxjs-fjxj∗s+∑j=1mbjisgjyjs-τjis-gjyj∗s-τjisds+∑j=1majitfjxjt-fjxj∗t+∑j=1mbjitgjyjs-τjis-gjyj∗s-τjis,dvitdt=-bivit+bi∫t-σitt-bivis-σis+∑j=1msjishjyjs-hjyj∗s+∑j=1mtjisφjxjs-ξjis-φjxj∗s-ξjisds+∑j=1msjithjyjt-hjyj∗t+∑j=1mtjitφjxjt-ξjit-φjxj∗t-ξjit, which leads to (32)12dui2tdt=-aiui2t+aiuit∫t-δitt-aiuis-δis+∑j=1majisfjxjs-fjxj∗s+∑j=1mbjisgjyjs-τjis-gjyj∗s-τjisds+uit∑j=1majitfjxjt-fjxj∗t+uit∑j=1mbjitgjyjs-τjis-gjyj∗s-τjis,12dvi2tdt=-bivi2t+bivit∫t-σitt-bivis-σis+∑j=1msjishjyjs-hjyj∗s+∑j=1mtjisφjxjs-ξjis-φjxj∗s-ξjisds+vit∑j=1msjithjyjt-hjyj∗t+vit∑j=1mtjitφjxjt-ξjit-φjxj∗t-ξjit. Then(33)dui2tdt≤-2aiui2t+2aiuitδi+aiuit-δit+∑j=1maji+Ljfujt+∑j=1mbji+Ljgvjt-τjit+2uit∑j=1maji+Ljfujt+2uit∑j=1mbji+Ljgvjt-τjit≤-2aiui2t+aiδi+aiui2+ui2t-δit+∑j=1maji+Ljf2ξjui2t+Ljf21-ξjuj2t+∑j=1mbji+Ljg2ξjui2t+Ljg21-ξjvj2t-τjit+∑j=1maji+Ljf2ξjui2t+Ljf21-ξjuj2t+∑j=1mbji+Ljg2ξjui2t+Ljg21-ξjvj2t-τjit. With the same method, we have(34)dvi2tdt≤-2bivi2t+biσi+bivi2+vi2t-σit+∑j=1msji+Ljh2ςjvi2t+Ljh21-ςjvj2t+∑j=1mtji+Ljφ2ςjvi2t+Ljφ21-ςjuj2t-ξjit+∑j=1msji+Ljh2ςjvi2t+Ljh21-ςjvj2t+∑j=1mtji+Ljφ2ςjvi2t+Ljφ21-ςjuj2t-ξjit,where 0≤ξj,ςj<1,j=1,…,m. Now we define a Lyapunov function as follows: (35)Vt=eβt∑i=1mui2t+∑i=1mvi2t+∑i=1m∑j=1maiδi+bij+Ljg21-ξj∫t-τjitteβs+τjitvj2sds+∑i=1mai2δi+∫t-δitteβs+δitui2sds+∑i=1mbij+Ljg21-ξj∫t-τjitteβs+τjitvj2sds+∑i=1mbi2σi+∫t-σitteβs+σitvi2sds+∑i=1mtij+Ljφ21-ςj∫t-ξjitteβs+ξjituj2sds+∑i=1m∑j=1mbiσi+tij+Ljφ21-ςj∫t-ξjitteβs+ξjituj2sds,where β is defined by Lemma 3. Differentiating V(t) along solutions to system (5), together with (33) and (34), we have (36)dVtdt≤βeβt∑i=1mui2t+∑i=1mvi2t+eβt∑i=1m-2aiui2t+aiδi+aiui2t+aiui2t-δit+∑j=1maji+Ljf2ξjui2t+Ljf21-ξjuj2t+∑j=1mbji+Ljg2ξjui2t+Ljg21-ξjvj2t-τjit+∑j=1maji+Ljf2ξjui2t+Ljf21-ξjuj2t+∑j=1mbji+Ljg2ξjui2t+Ljg21-ξjvj2t-τjit+eβt∑i=1m-2bivit+biσi+bivi2t-σit+∑j=1msji+Ljh2ςjvi2t+Ljh21-ςjvj2t+∑j=1mtji+Ljφ2ςjvi2t+Ljh21-ςjuj2t-ξjit+∑i=1mai2δi+eβt+βδituit-eβtuit-δit+∑i=1mbij+Ljg21-ξjeβt+βτjitvjt-eβtvj2t-δit+∑i=1mbi2σi+eβt+βσitvi2t-eβtvi2t-σit+∑i=1m∑j=1mbiσi+tij+Ljφ21-ςjeβt+βξjituj2t-eβtuj2t-ξjit+∑i=1mtij+Ljφ21-ςjeβt+βξjituj2t-eβtuj2t-ξjit+∑i=1m∑j=1maiδi+bij+Ljg21-ξjeβt+βτjitvjt-eβtvjt-τjit=eβt∑i=1mβ-2ai+ai2δi+eβδi++1+tji+Ljφ21-ςjeβξji++aiδi++1∑j=1mbji+Ljg2ξj+aiδi++1∑j=1maji+Ljf2ξj+Ljf21-ξj+∑j=1mbiσi+tji+Ljφ21-ςjeβξji+ui2t+eβt∑i=1mβ-2bi+bi2σi+eβσi++1+bji+Ljg21-ξjeβτji++biσi++1∑j=1mtji+Ljφ2ςj+biσi++1∑j=1msji+Ljh2ςj+Ljh21-ςj+∑j=1maiδi+bji+Ljg21-ξjeβτji+vi2t.In view of Lemma 3, we have dV(t)/dt≤0. That is to say, V(t)≤V(0) for all t>0. Thus(37)eβt∑i=1mui2t+∑j=1mvj2t≤Vt≤V0≤ϕ-ϕ∗2+ψ-ψ∗2+1β∑i=1mai2δi+eβδi+ϕ-ϕ∗2+1β∑i=1m∑j=1maiδi+bji+Ljg21-ξjeβτji+ψ-ψ∗2+1β∑i=1mbji+Ljg21-ξjeβτji+ψ-ψ∗2+1β∑i=1mbi2σi+eβσi+ψ-ψ∗2+1β∑i=1mtji+Ljφ21-ςjeβτji+ϕ-ϕ∗2+1β∑i=1m∑j=1mbiσi+tji+Ljφ21-ςjeβξji+ϕ-ϕ∗2=1+1β∑i=1mai2δi+eβδi++1β∑i=1m∑j=1mbiσi+tji+Ljφ21-ςjeβξji++1β∑i=1m∑j=1mbiσi+tji+Ljφ21-ςjeβξji+ϕ-ϕ∗2+1+1β∑i=1mbi2σi+eβσi++1β∑i=1mbji+Ljg21-ξjeβτji++1β∑i=1m∑j=1maiδi+bji+Ljg21-ξjeβτji+ψ-ψ∗2.Let (38)M=max⁡1+1β∑i=1mai2δi+eβδi++1β∑i=1m∑j=1mbiσi+tji+Ljφ21-ςjeβξji++1β∑i=1m∑j=1mbiσi+tji+Ljφ21-ςjeβξji+,1+1β∑i=1mbi2σi+eβσi++1β∑i=1mbji+Ljg21-ξjeβτji++1β∑i=1m∑j=1maiδi+bji+Ljg21-ξjeβτji+>1.By (37), one has (39)∑i=1mui2t+∑i=1mvi2t≤Me-βtφ-φ∗2+ψ-ψ∗2for all t>0. Then (40)∑i=1mxit-xi∗t2+∑i=1myit-yi∗t2≤Me-βtϕ-ϕ∗2+ψ-ψ∗2for all t>0. Thus the solution (x∗(t),y∗(t))T of system (5) is globally exponentially stable.Theorem 6.
Suppose that (H1)–(H3) hold. Then system (5) has exactly one T-antiperiodic solution which is globally stable.Proof.
By system (5) and (H2), for each k∈N, we get (41)ddt-1k+1xit+k+1T=-1k+1-aixit+k+1T-δit+k+1T+∑j=1majit+k+1Tfjxit+k+1T+∑j=1mbjit+k+1Tgjyit+k+1T-τjit+k+1T+Iit+k+1T=-ai-1k+1xit+k+1T-δit+∑j=1majitfj-1k+1xjt+k+1T+∑j=1mbjitgj-1k+1yjt+k+1T-τjit+Iit. In a similar way, we have (42)ddt-1k+1yit+k+1T=-bi-1k+1yit+k+1T-σit+∑j=1msjithj-1k+1yjt+k+1T+∑j=1mtjitφj-1k+1xjt+k+1T-ξjit+Jit. Let (43)x¯t=-1k+1x1t+k+1T,…,-1k+1xmt+k+1TT,y¯t=-1k+1y1t+k+1T,…,-1k+1ymt+k+1TT.It follows that, for any k∈N,(x¯T(t),y¯T(t))T is also the solution of system (5). If the initial functions ϕi(s),ψi(s),i=1,…,m, are bounded, we conclude from Theorem 5 that there exists a constant γ>1 such that (44)-1k+1xit+k+1T--1kxit+kT≤Me-βt+kTsup-τ≤s≤0∑i=1mxit+T+xis2≤γe-βt+kT,where t+kT>0,i=1,…,m, and β is a positive constant. For any k∈N we have (45)-1k+1xit+k+1T=xit+∑j=0k-1j+1xit+j+1T--1jxit+jT.Then (46)-1k+1xit+k+1T≤xit+∑j=0k-1j+1xjt+j+1T--1jxjt+jT.In view of Lemma 4, we know that the solutions of system (5) are bounded. In view of (4) and (5), we can easily know that {(-1)k+1xi(t+(k+1)T)} uniformly converges to a continuous function x∗(t)=(x1∗(t),…,xm∗(t))T on any compact set of R. In a similar way, we can easily prove that {(-1)k+1yi(t+(k+1)T)} uniformly converges to a continuous function y∗(t)=(y1∗(t),…,ym∗(t))T on any compact set of R. Now we will show that (x∗(t),y∗(t))T is T-antiperiodic solution of system (5). Since (47)x∗t+T=limk→∞-1kxt+T+kT=-limk+1→∞-1k+1xt+k+1T=-x∗t.thus x∗(t) is T-antiperiodic solution. Similarly, y∗(t) is also T-antiperiodic solution. Thus we know that (x∗(t),y∗(t))T is a solution of system (5). In fact, together with the continuity of the right side of system (5), letting k→∞, we can easily get (48)dxi∗tdt=-aixi∗t-δit+∑j=1majitfjxj∗t+∑j=1mbjitgjyj∗t-τjit+Iit,dyi∗tdt=-biyi∗t-σit+∑j=1msjithjyj∗t+∑j=1mtjitφjxj∗t-ξjit+Jit.Therefore, (x∗(t),y∗(t))T is the T-periodic solution of system (5). Finally, by applying Theorem 5, it is easy to check that (x∗(t),y∗(t))T is globally exponentially stable. The proof of Theorem 6 is completed.Remark 7.
There are a large number of papers about neural networks with delays. The main topic of these papers is exponential stability. For example, the authors in [26–28] considered the exponential stability of neural networks with constants delays. The authors in [29–31] considered the exponential stability of neural networks with time-varying delays and [32–35] with distributed delays. But these are few results on stability of BAM neural networks with leakage terms. This paper constructs a kind of BAM neural networks with time-varying delay and leakage terms and the results in our paper extend the results in the above papers.


## 4. An Example

In this section, to illustrate the efficiency of our results obtained in Section3, we give an example. Consider the following BAM neural networks with time-varying delays and external input:(49)dx1tdt=-a1x1t-δ1t+∑j=12aj1tfjxjt+∑j=12bj1tgjyjt-τj1t+I1t,dx2tdt=-a2x2t-δ2t+∑j=12aj2tfjxjt+∑j=12bj2tgjyjt-τj2t+I2t,dy1tdt=-b1y1t-σ1t+∑j=12sj1thjyjt+∑j=12tj1tφjxjt-ξj1t+J1t,dy2tdt=-b2y2t-σ2t+∑j=1msj2thjyjt+∑j=12tj2tφjxjt-ξj2t+J2t, where (50)a1a2b1b2=2222,δ1tδ2tσ1tσ2t=0.04sin⁡t0.04sin⁡t0.03cos⁡t0.03cos⁡t,a11ta12ta21ta22t=0.2cos⁡t0.3sin⁡t0.2cos⁡t0.3sin⁡t,b11tb12tb21tb22t=0.1cos⁡t0.2sin⁡t0.1cos⁡t0.2sin⁡t,s11ts12ts21ts22t=0.2sin⁡t0.3cos⁡t0.2sin⁡t0.3cos⁡t,I1I2J1J2=0.3sin⁡t0.3sin⁡t0.4cos⁡t0.4cos⁡t.We also set (51)fju=gju=hju=φju=12u+1-u-1,j=1,2.Then we find that Ljf=Ljg=Ljh=Ljφ=1, and Mjf=Mjg=Mjh=Mjφ=1,j=1,2. Also we have (52)δ1+δ2+σ1+σ2+=0.040.040.030.03,aj1+aj2+bj1+bj2+=0.20.30.10.2,sj1+sj2+tj1+tj2+=0.20.30.10.2.for j=1,2.It is easy to verify that(53)-2a1+2a12δ1++tj1+Ljφ21-ςj+a1δ1++1∑j=12bj1+Ljg2ξj+∑j=12aj1+Ljf2ξj+Ljf21-ξj+∑j=12b1σ1+tj1+Ljφ21-ςj=-2.488<0,-2a2+2a22δ2++tj2+Ljφ21-ςj+a2δ2++1∑j=12bj2+Ljg2ξj+∑j=12aj2+Ljf2ξj+Ljf21-ξj+∑j=12b2σ2+tj2+Ljφ21-ςj=-1.728<0,-2b1+2b12σ1++bj1+Ljg21-ξj+b1σ1++1∑j=12tj1+Ljφ2ςj+∑j=12sj1+Ljh2ςj+Ljh21-ςj+∑j=12a1δ1+bj1+Ljg21-ξj=-2.584<0,-2b2+2b22σ2++bj2+Ljg21-ξj+b2σ2++1∑j=12tj2+Ljφ2ςj+∑j=12sj2+Ljh2ςj+Ljh21-ςj+∑j=12a2δ2+bj2+Ljg21-ξj=-1.832<0.Then all the conditions (H1)–(H3) hold. Thus system (49) has exactly one π-antiperiodic solution which is globally exponentially stable.The numerical result is shown in Figure1. And system (49) is exponentially stable when all inputs are equal to 0, which can also be seen from Figure 2.Figure 1
The state response of BAM neural network in (49).Figure 2
The state response of BAM neural network with inputs equal to 0 in (49).


## 5. Conclusions

In this paper, a kind of BAM neural networks with time-varying delays and external input has been dealt with. By constructing a suitable Lyapunov-Kravsovskii functional and using matrix theory and inequality technique, a series of sufficient criteria to guarantee the existence and global exponential stability of antiperiodic solutions for this system have been established. And the criteria are easy to check in practice.We also give an example to illustrate the feasibility and effectiveness. In real life, many BAM neural networks are affected by external perturbation. There are many papers concerned with stochastic BAM neural networks, e.g., [36–41]. In the future, we will give our research on these networks.



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