# Existence and Exponential Stability of Periodic Solution of High-Order Hopfield Neural Network with Delays on Time Scales

Authors: Yongkun Li; Lili Zhao; Ping Liu
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

On time scales, by using the continuation theorem of coincidence degree theory and constructing some suitable Lyapunov functions, the periodicity and the exponential stability are investigated for a class of delayed high-order Hopfield neural networks (HHNNs), which are new and complement of previously known results. Finally, an example is given to
show the effectiveness of the proposed method and results.


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## Body

## 1. Introduction

Consider the following HHNNs with time-varying delays:(1.1)dxi(t)dt=-ci(t)xi(t)+∑j=1naij(t)fj(xj(t-γij(t)))+∑j=1n∑l=1nbijl(t)gj(xj(t-σijl(t)))gl(xl(t-vijl(t)))+Ii(t),i=1,2,…,n,
where n corresponds to the number of units in a neural network, xi(t) corresponds to the state vector of the ith unit at the time t, ci(t) represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, aij(t) and bijl(t) are the first- and second-order connection weights of the neural network, γij(t)≥0,σijl(t)≥0, and vijl(t)≥0 correspond to the transmission delays, Ii(t) denote the external inputs at time t, and fj and gj are the activation functions of signal transmission.Due to the fact that high-order neural networks have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks, high-order neural networks have been the object of intensive analysis by numerous authors in the recent years. In particular, there have been extensive results on the problem of the existence and stability of equilibrium points and periodic solutions of HHNNs (1.1) in the literature. We refer the reader to [1–9] and the references cited therein. In fact, both continuous and discrete systems are very important in implementing and applications. The theory of calculus on time scales (see [10, 11] and references cited therein) was initiated by Stefan Hilger in his Ph.D. thesis in 1988 [12] in order to unify continuous and discrete analysis, and it has a tremendous potential for application and has recently received much attention since his foundational work. Therefore, it is meaningful to study that on time scales which can unify the continuous and discrete situations.Our purpose of this paper is to consider the model(1.2)xiΔ(t)=-ci(t)xi(t)+∑j=1naij(t)fj(xj(t-γij(t)))+∑j=1n∑l=1nbijl(t)gj(xj(t-σijl(t)))gl(xl(t-vijl(t)))+Ii(t),i=1,2,…,n,t∈𝕋,
where 𝕋 is an ω-periodic time scale which has the subspace topology inherited from the standard topology on ℝ,I(t)=(I1(t),I2(t),…,In(t))T is an input periodic vector function with period ω; that is, there exists ω>0 such that Ii(t+ω)=Ii(t)(i=1,2,…,n) for all t∈(0,+∞)⋂𝕋, and x=(x1,x2,…,xn)T∈ℝn,f(x)=(f1(x1),f2(x2),…,fn(xn))T, and g(x)=(g1(x1),g2(x2),…,gn(xn))T is the activation function of the neurons.System (1.2) is supplemented with initial values given by
(1.3)xi(s)=φi(s),s∈[-θ,0]⋂𝕋,i=1,2,…,n,
where φi(·) denotes continuous ω-periodic function defined on [-θ,0]⋂𝕋,θ=max{γ,σ,v},γ=max1≤i,j≤n{γij+},σ=max1≤i,j,l≤n{σijl+},v=max1≤i,j,l≤n{vijl+},γij+=maxt∈[0,ω]⋂𝕋γij(t),σijl+=maxt∈[0,ω]⋂𝕋σijl(t),vijl+=maxt∈[0,ω]⋂𝕋vijl(t),i,j,l=1,2,…,n. To the best of our knowledge, this is the first paper to study the stability and existence of periodic solutions of (1.2).Throughout this paper, we assume the following.(H1)
Fori,j,l=1,2,…,n,ci(t),aij(t),bijl(t),Ii(t),γij(t),σijl(t),vijl(t) are positive continuous periodic functions with period ω>0, and ci(t) is regressive.(H2)
There exist positive constantsMj,Nj,j=1,2,…,n such that |fj(x)|≤Mj,|gj(x)|≤Nj for j=1,2,…,n,x∈ℝ.(H3)
Functionsfj(u),gj(u)(j=1,2,…,n) satisfy the Lipschitz condition; that is, there exist constants Lj,Hj>0 such that |fj(u1)-fj(u2)|≤Lj|u1-u2|,|gj(u1)-gj(u2)|≤Hj|u1-u2|,j=1,2,…,n.


## 2. Preliminaries

In this section, we will first recall some basic definitions and lemmas which are used in what follows.Let𝕋 be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators σ,ρ:𝕋→𝕋 and the graininess μ:𝕋→ℝ+ are defined, respectively, by
(2.1)σ(t)=inf{s∈𝕋:s>t},ρ(t)=sup{s∈𝕋:s<t},μ(t)=σ(t)-t.A pointt∈𝕋 is called left-dense if t>inf𝕋 and ρ(t)=t, left-scattered if ρ(t)<t, right-dense if t<sup𝕋 and σ(t)=t, and right-scattered if σ(t)>t. If 𝕋 has a left-scattered maximum m, then 𝕋k=𝕋∖{m}; otherwise 𝕋k=𝕋. If 𝕋 has a right-scattered minimum m, then 𝕋k=𝕋∖{m}; otherwise 𝕋k=𝕋.A functionf:𝕋→ℝ is right-dense continuous provided that it is continuous at right-dense point in 𝕋 and its left-side limits exist at left-dense points in 𝕋. If f is continuous at each right-dense point and each left-dense point, then f is said to be continuous function on 𝕋.Fory:𝕋→ℝ and t∈𝕋k, we define the delta derivative of y(t), yΔ(t) to be the number (if it exists) with the property that for a given ε>0, there exists a neighborhood U of t such that
(2.2)|[y(σ(t))-y(s)]-yΔ(t)[σ(t)-s]|<ε|σ(t)-s|
for all s∈U.Ify is continuous, then y is right-dense continuous, and if y is delta differentiable at t, then y is continuous at t.A functionr:𝕋→ℝ is called regressive if
(2.3)1+μ(t)r(t)≠0
for all t∈𝕋k.Ifr is regressive function, then the generalized exponential function er is defined by
(2.4)er(t,s)=exp{∫stξμ(τ)(r(τ))Δτ}fors,t∈𝕋,
with the cylinder transformation
(2.5)ξh(z)={Log(1+hz)h,ifh≠0,z,ifh=0.
Let p,q:𝕋→ℝ be two regressive functions, we define
(2.6)p⊕q:=p+q+μpq,⊝p:=-p1+μp,p⊝q:=p⊕(⊝q).
Then the generalized exponential function has the following properties.Lemma 2.1 (see [10]).
Assume thatp,q:𝕋→ℝ are two regressive functions, then  (i)
e0(t,s)≡1 and ep(t,t)≡1;(ii)
ep(σ(t),s)=(1+μ(t)p(t))ep(t,s);(iii)
ep(t,σ(s))=ep(t,s)/(1+μ(s)p(s));(iv)
1/ep(t,s)=e⊝p(t,s);(v)
ep(t,s)=1/ep(s,t)=e⊝p(s,t);(vi)
ep(t,s)ep(s,r)=ep(t,r);(vii)
ep(t,s)eq(t,s)=ep⊕q(t,s);(viii)
ep(t,s)/eq(t,s)=ep⊝q(t,s).Lemma 2.2 (see [10]).
Assume thatf,g:𝕋→ℝ are delta differentiable at t∈𝕋k, then
(2.7)(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t)=f(t)gΔ(t)+fΔ(t)g(σ(t)).Lety be right-dense continuous. If YΔ(t)=y(t), then one defines the delta integral by
(2.8)∫aty(s)Δs=Y(t)-Y(a).Lemma 2.3.
Ifa,b∈𝕋,α,β∈ℝ and f,g∈C(𝕋,ℝ), then  (1)
∫ab[αf(t)+βg(t)]Δt=α∫abf(t)Δt+β∫abg(t)Δt;(2)
Iff(t)≥0 for all a≤t<b, then ∫abf(t)Δt≥0;(3)
If|f(t)|≤g(t) on [a,b):={t∈𝕋:a≤t<b}, then |∫abf(t)Δt|≤∫abg(t)Δt.In this paper, one assumes thatk¯=min{[0,+∞)⋂𝕋}. Clearly, from Lemma 2.3, one can obtain Lemma 2.4.Lemma 2.4.
Iff,g∈C(𝕋,ℝ), and f(t)≤g(t) on t∈[k¯,k¯+ω), then ∫k¯k¯+ωf(t)Δt≤∫k¯k¯+ωg(t)Δt.In the proof of our main result, one will use the following three lemmas which can be found in [13, 14].Lemma 2.5 (see [13]).
Lett1,t2∈[k¯,k¯+ω]⋂𝕋, and t∈𝕋. If f:𝕋→ℝ is ω-periodic, then f(t)≤f(t1)+∫k¯k¯+ω|fΔ(s)|Δs, and f(t)≥f(t2)-∫k¯k¯+ω|fΔ(s)|Δs.Lemma 2.6 ([14], Cauchy-Schwarz inequality on time scales).
Leta,b∈𝕋. For rd-continuous functions f,g:[a,b]→ℝ, one has (∫ab|f(t)g(t)|Δt)2≤∫ab|f(t)|2Δt∫ab|g(t)|2Δt.Lemma 2.7.
Letr:𝕋→ℝ be right-dense continuous and regressive. a∈𝕋, and ya∈ℝ. The unique solution of the initial value problem
(2.9)yΔ(t)=r(t)y(t)+h(t),y(a)=ya
is given by
(2.10)y(t)=er(t,a)ya+∫ater(t,σ(s))h(s)Δs.For the sake of convenience, one introduces the following notations:(2.11)ci¯=1ω∫k¯k¯+ωci(t)Δt,aij¯=1ω∫k¯k¯+ωaij(t)Δt,Ii¯=1ω∫k¯k¯+ωIi(t)Δt,bijl¯=1ω∫k¯k¯+ωbijl(t)Δt,ci+=maxt∈[k¯,k¯+ω]⋂𝕋|ci(t)|,ci-=mint∈[k¯,k¯+ω]⋂𝕋|ci(t)|,aij+=maxt∈[k¯,k¯+ω]⋂𝕋|aij(t)|,bijl+=maxt∈[k¯,k¯+ω]⋂𝕋|bijl(t)|,Ii+=maxt∈[k¯,k¯+ω]⋂𝕋|Ii(t)|,Ai(t)=-ci(t)xi(t)+∑j=1naij(t)fj(xj(t-γij(t)))+∑j=1n∑l=1nbijl(t)gj(xj(t-σijl(t)))gl(xl(t-vijl(t)))+Ii(t),i,j,l=1,2,…n.
Obliviously, for system (1.2), finding the periodic solutions is equivalent to finding those of the following boundary-value problem:(2.12)xiΔ(t)=-ci(t)xi(t)+∑j=1naij(t)fj(xj(t-γij(t)))+∑j=1n∑l=1nbijl(t)gj(xj(t-σijl(t)))gl(xl(t-vijl(t)))+Ii(t),t∈[k¯,k¯+ω]∩𝕋,i=1,2,…,n,xi(k¯)=xi(k¯+ω),i=1,2,…,n.
Now, one states Mawhin's continuous theorem [15].Theorem 2.8.
LetX and Z be two Banach spaces and let L be a Fredholm mapping of index zero. let  Ω⊂𝕏 be an open bounded set and let N:Ω¯→ℤ be a continuous operator which is L-compact on Ω¯. Assume that (1)
for eachλ∈(0,1),x∈∂Ω∩DomL,Lx≠λNx;(2)
for eachx∈∂Ω∩KerL,QNx≠0;(3)
deg(JNQx,Ω∩KerL,0)≠0, where JQN:KerL→KerL.Then Lx=Nx has at least one solution in Ω¯∩DomL.In order to apply Theorem2.8 to system (2.12), we first define(2.13)X=Z={x=(x1,x2,…,xn)T∈C(𝕋,ℝn):x(t+ω)=x(t),t∈𝕋},∥x∥=∥(x1,x2,…,xn)T∥=∑i=1nmaxt∈[k¯,k¯+ω]⋂𝕋|xi(t)|
for any x∈X(orZ). Then X and Z are Banach spaces with the norm ∥·∥. Let
(2.14)Nx=(A1(t),A2(t),…,An(t))T,x∈X(2.15)Lx=(x1Δ,x2Δ,…,xnΔ)T(2.16)Px=(1ω∫k¯k¯+ωx1(t)Δt,1ω∫k¯k¯+ωx2(t)Δt,…,1ω∫k¯k¯+ωxn(t)Δt)T,Qz=(1ω∫k¯k¯+ωz1(t)Δt,1ω∫k¯k¯+ωz2(t)Δt,…,1ω∫k¯k¯+ωzn(t)Δt)T,z∈Z.
Then it follows that(2.17)KerL={(x1,x2,…,xn)T∈𝕏:(x1,x2,…,xn)T=(w1(t),w2(t),…,wn(t))T∈ℝn,t∈𝕋},ImL={z∈ℤ:(1ω∫k¯k¯+ωz1(t)Δt,1ω∫k¯k¯+ωz2(t)Δt,…,1ω∫k¯k¯+ωzn(t)Δt)T=0}
is closed in Z, dim KerL=n=codimImL. It is not difficult to show that P and Q are continuous and satisfy ImP=KerL,ImL=KerQ=Im(I-Q). It is easy to see that ImL is closed in Z, which leads to the following lemma.Lemma 2.9.
LetL and N be defined by (2.14) and (2.15), respectively, then L is a Fredholm operator of index zero.Lemma 2.10.
LetL and N be defined by (2.14) and (2.15), respectively, suppose that Ω is an open bounded subset of DomL, then N is L-compact on Ω¯.Proof.
Through an easy computation, we find that the inverseKP:ImL→KerP⋂DomL of LP has the form
(2.18)(KPz)(t)=∫k¯tz(s)Δs-1ω∫k¯k¯+ω∫k¯tz(s)ΔsΔt.
Thus, the expression of QNx is
(2.19)QNx=(1ω∫k¯k¯+ωA1(t)Δt,1ω∫k¯k¯+ωA2(t)Δt,…,1ω∫k¯k¯+ωAn(t)Δt)T,
and then
(2.20)KP(I-Q)Nx=(∫k¯tA1(s)Δs⋮∫k¯tAn(s)Δs)-(1ω∫k¯k¯+ω∫k¯tA1(s)ΔsΔt⋮1ω∫k¯k¯+ω∫k¯tAn(s)ΔsΔt)-((t-k¯ω-1(ω)2∫k¯k¯+ω(t-k¯)Δt)∫k¯k¯+ωA1(s)Δs⋮(t-k¯ω-1(ω)2∫k¯k¯+ω(t-k¯)Δt)∫k¯k¯+ωAn(s)Δs).
Thus, QN and KP(I-Q)N are continuous. Since X is a Banach space, it is not difficult to show that KP(I-Q)N(Ω¯)¯ is compact. Moreover, QN(Ω¯) is bounded. Thus, N is L-compact on Ω¯ for any open bounded set Ω⊂X. The proof of Lemma 2.10 is completed.


## 3. Existence of Periodic Solution

In this section, we study the existence of periodic solution of (1.2) based on Mawhin's continuation theorem.Theorem 3.1.
Assume that(H1)-(H2) hold, then system (1.2) has at least one ω-periodic solution.Proof.
Based on the Lemma2.9 and Lemma 2.10, now, what we need to do is just to search for an appropriate open, bounded subset Ω for the application of the continuation theorem. Corresponding to the operator equation Lx=λNx,λ∈(0,1), we have
(3.1)xiΔ(t)=λ[-ci(t)xi(t)+∑j=1naij(t)fj(xj(t-γij(t)))+∑j=1n∑l=1nbijl(t)gj(xj(t-σijl(t)))gl(xl(t-vijl(t)))+Ii(t)∑j=1naij(t),t∈𝕋⋂[k¯,k¯+ω]],i=1,2,…,n,xi(k¯)=xi(k¯+ω),i=1,2,…,n.
For the sake of convenience, defined ∥x∥2 by
(3.2)∥x∥2=(∫k¯k¯+ω|x(t)|2Δt)1/2
for x∈C(𝕋,ℝ). Suppose that (x1(t),x2(t),…,xn(t))T∈X is a solution of system (3.1) for a certain λ∈(0,1). Integrating (3.1) over [k¯,k¯+ω], we obtain
(3.3)∫k¯k¯+ωAi(t)Δt=0.
Hence
(3.4)∫k¯k¯+ωci(s)xi(s)Δs=[∫k¯k¯+ω∑j=1naij(t)fj(xj(t-γij(t)))+∑j=1n∑l=1nbijl(t)gj(xj(t-σijl(t)))gl(xl(t-vijl(t)))+Ii(t)]Δt,i=1,2,…,n.
Let ζi,ηi∈𝕋⋂[k¯,k¯+ω], such that xi(ζi)=inft∈𝕋⋂[k¯,k¯+ω]xi(t),xi(ηi)=supt∈𝕋⋂[k¯,k¯+ω]xi(t). Then by (3.4) andLemma 2.4, we have
(3.5)ωci¯xi(ζi)≤∫k¯k¯+ω[|∑j=1naij(s)fj(xj(s-γij(s)))+∑j=1n∑l=1nbijl(s)gj(xj(s-σijl(s)))gl(xl(s-vijl(s)))+Ii(s)|]Δs≤∫k¯k¯+ω∑j=1n|aij(s)fj(xj(s-γij(s)))||fj(xj(s-γij(s)))|Δs+∫k¯k¯+ω∑j=1n∑l=1nfj(xj(s-))|bijl(s)∥gj(xj(s-σijl(s)))∥gl(xl(s-vijl(s)))|Δs+∫k¯k¯+ωIi(s)Δs≤ω[∑j=1naij+Mj+∑j=1n∑l=1nbijl+NjNl+Ii+],i=1,2,…,n.
Hence xi(ζi)≤(1/ci¯){[∑j=1naij+Mj+∑j=1n∑l=1nbijl+NjNl+Ii+]}:=Bi,i=1,2,…,n. By (3.4) and Lemma 2.4, we can also have
(3.6)ωci¯xi(ηi)≥-∫k¯k¯+ω[|∑j=1naij(s)fj(xj(s-γij(s)))+∑j=1n∑l=1nbijl(s)gj(xj(s-σijl(s)))gl(xl(s-vijl(s)))+Ii(s)|]Δs≥-∫k¯k¯+ω∑j=1n|aij(s)fj(xj(s-γij(s)))||fj(xj(s-γij(s)))|Δs-∫k¯k¯+ω∑j=1n∑l=1n|bijl(s)fj(xj(s-γij(s)))||gj(xj(s-σijl(s)))||gl(xl(s-vijl(s)))|Δs-∫k¯k¯+ω|Ii(s)|Δs≥-ω[∑j=1naij+Mj+∑j=1n∑l=1nbijl+NjNl+Ii+],i=1,2,…,n.
Hence xi(ηi)≥(-1/ci¯){[∑j=1naij+Mj+∑j=1n∑l=1nbijl+NjNl+Ii+]}=-Bi,i=1,2,…,n. From (3.1), (3.4), and Lemma 2.6, we have
(3.7)∫k¯k¯+ω|xiΔ(t)|Δt≤∫k¯k¯+ω|ci(t)||xi(t)|Δt+∫k¯k¯+ω∑j=1n|aij(t)||fj(xj(t-γij(t)))|Δt+∫k¯k¯+ω∑j=1n∑l=1n|bijl(t)fj(xj(s-γij(s)))||gj(xj(t-σijl(t)))||gl(xl(t-vijl(t)))|Δt+∫k¯k¯+ω|Ii(t)|Δt≤(∫k¯k¯+ω|ci(t)|2Δt)1/2(∫k¯k¯+ω|xi(t)|2Δt)1/2+∑j=1n(∫k¯k¯+ω|aij(t)|2Δt)1/2(∫k¯k¯+ω|fj(xj(t-γij(t)))|2Δt)1/2+∑j=1n∑l=1nbijl+(∫k¯k¯+ω|gj(xj(t-σijl(t)))|2Δt)1/2(∫k¯k¯+ω|gl(xl(t-σijl(t)))|2Δt)1/2+Ii+ω≤(ω)1/2ci+∥xi∥2+∑j=1nωaij+Mj+∑j=1n∑l=1n(ω)bijl+NjNl+Ii+ω.
From Lemma 2.7 and (3.1), for i=1,2,…,n, we can obtain
(3.8)xi(t)=e-λci(t)(t,k¯)xi(k¯)+∫k¯tλe-λci(t)(t,σ(s))[∑j=1naij(s)fj(xj(s-γij(s)))+∑j=1n∑l=1nbijl(s)gj(xj(s-σijl(s)))gl(xl(s-vijl(s)))+Ii(s)]Δs.
Hence
(3.9)|xi(t)|≤|xi(k¯)|+∑j=1n∫k¯k¯+ω|aij(s)fj(xj(s-γij(s)))||fj(xj(s-γij(s)))|Δs+∑j=1n∑l=1n∫k¯k¯+ω|bijl(s)fj(xj(s-γij(s)))||gj(xj(s-σijl(s)))||gl(xl(s-vijl(s)))|Δs+∫k¯k¯+ω|Ii(s)|Δs≤|xi(k¯)|+∑j=1naij¯Njω+∑j=1n∑l=1nbijl¯NjNlω+Ii¯ω=:ui,i=1,2,…,n,
that is,
(3.10)∥xi∥2=(∫k¯k¯+ω|xi(t)|2Δt)1/2≤ui(ω)1/2,i=1,2,…,n.
Substituting (3.10) into (3.7), we have
(3.11)∫k¯k¯+ω|xiΔ(t)|Δt≤ωci+ui+∑j=1nωaij+Mj+∑j=1n∑l=1n(ω)bijl+NjNl+Ii+ω,i=1,2,…,n.
From Lemma 2.5, we have
(3.12)xi(t)≤xi(ζi)+∫k¯k¯+ω|xiΔ(t)|Δt,i=1,2,…,n,xi(t)≥xi(ηi)-∫k¯k¯+ω|xiΔ(t)|Δt,i=1,2,…,n.
From (3.5), (3.6) and (3.11), there exist positive constants ξi(i=1,2,…,n) such that for t∈[k¯,k¯+ω]⋂𝕋,|xi(t)|≤ξi,i=1,2,…,n. Clearly, ξi(i=1,2,…,n) is independent of λ. Denote H*=∑i=1nξi+C, where C>0 is taken sufficiently large so that
(3.13)min1≤i≤nci¯H*>nmax1≤i≤n(|Ii¯|+∑j=1n|aij¯|Mj+∑j=1n∑l=1n|bijl¯|NjNl).
Now we takeΩ={(x1(t),x2(t),…,xn(t))T:∥(x1(t),x2(t),…,xn(t))T∥<H*}. Thus (1) of Theorem 2.8 is satisfied. When (x1(t),x2(t),…,xn(t))T∈∂Ω⋂ℝn,(x1(t),x2(t),…,xn(t))T is a constant vector in ℝn with |x1|+|x2|+⋯+|xn|=H*, then
(3.14)QN(x1(t),x2(t),…,xn(t))T=(-ci¯xi(t)+∑j=1naij¯fj(xj(t-γij(t)))+∑j=1n∑l=1nbijl¯gj(xj(t-σijl(t)))gl(xl(t-vijl(t)))+Ii¯)n×1.
Therefore
(3.15)∥QN(x1(t),x2(t),…,xn(t))T∥=∑i=1n|ci¯xi(t)-∑j=1naij¯fj(xj(t-γij(t)))-∑j=1n∑l=1nbijl¯gj(xj(t-σijl(t)))gl(xl(t-vijl(t)))-Ii¯|≥∑i=1nci¯|xi(t)|-∑i=1n∑j=1n|aij¯|Mj-∑i=1n∑j=1n∑l=1nbijl¯NjNl-∑i=1n|Ii¯|≥∑i=1n(ci¯|xi(t)|)-∑i=1n(|Ii¯|+∑j=1n|aij¯|Mj+∑j=1n∑l=1nbijl¯NjNl)≥min1≤i≤n(ci¯)∑i=1n|xi(t)|-nmax1≤i≤n(|Ii¯|+∑j=1n|aij¯|Mj+∑j=1n∑l=1n|bijl¯|NjNl)>0.
Consequently,
(3.16)QN(x1(t),x2(t),…,xn(t))T≠(0,0,…,0)T,
for (x1(t),x2(t),…,xn(t))T∈∂Ω⋂KerL. This satisfies condition (2) of Theorem 2.8. Define Ψ:KerL×[0,1]→X by
(3.17)Ψ(x1,x2,…,xn,μ)=-μ(x1,x2,…,xn)T+(1-μ)QN(x1(t),x2(t),…,xn(t))T
When (x1(t),x2(t),…,xn(t))T∈∂Ω⋂KerL,(x1,x2,…,xn)T is a constant vector in ℝn with ∑i=1n|xi|=H*, we easily have Ψ(x1,x2,…,xn,μ)≠(0,0,…,0)T. Therefore
(3.18)deg(QN(x1(t),x2(t),…,xn(t))T,Ω⋂KerL,(0,0,…,0)T)=deg((-x1(t),-x2(t),…,-xn(t)T,Ω⋂KerL,(0,0,…,0)T)≠0.
Condition (3) of Theorem 2.8 is also satisfied. Thus, by Theorem 2.8 we can obtain that Lx=Nx has at least one solution in X. That is, system (1.2) has at least one ω-periodic solution. The proof is complete.


## 4. Global Exponential Stability of Periodic Solution

In this section, we will construct suitable Lyapunov functions to study the global exponential stability of the periodic solution of (1.2) on time scales. So first we will introduce some definitions.Definition 4.1 (see [12]).
A functionf from 𝕋 to ℝ is positively regressive if 1+μ(t)f(t)>0 for every t∈𝕋.Denoteℜ+ is the set of positively regressive functions from 𝕋 to ℝ, and denote 𝕋+=[0,+∞)⋂𝕋.Definition 4.2.
The periodic solutionx*(t) of system (1.2) is said to be exponentially stable if there exists a positive constant α with -α∈ℜ+ such that for every δ∈𝕋, there exists N=N(δ)≥1 such that the solution x(t) of (1.2) through (δ,φ(δ)) satisfies ∥x(t)-x*(t)∥≤N∥φ(δ)-x*(δ)∥e-α(t,δ),t∈𝕋+ where ∥φ(δ)-x*(δ)∥=∑i=1nmaxδ∈[-θ,0]⋂𝕋|φi(δ)-xi*(δ)|.Theorem 4.3.
Assume that(H1)–(H3) hold. Suppose further that there exists n positive constants ϵi>0,i=1,2,…,n such that    (H4)
(4.1)-ϵici-+∑j=1naij+Mjϵj+∑j=1n∑l=1nbijl+(HjNlϵj+HlNjϵl)<0,i=1,2,…,n,then the ω-periodic solution of system (1.2) is globally exponentially stable.Proof.
According to Theorem3.1, we know that (1.2) has an ω-periodic solution x*(t)=(x1*(t),x2*(t),…,xn*(t))T. Suppose that x(t)=(x1(t),x2(t),…,xn(t))T is an arbitrary solution of (1.2). Then it follows from system (1.2) that
(4.2)(xi(t)-xi*(t))Δ=-ci(t)(xi(t)-xi*(t))+∑j=1naij(t)(fj(xj(t-γij(t)))-fj*(xj(t-γij(t))))+∑j=1n∑l=1nbijl(t)(gj(xj(t-σijl(t)))gl(xl(t-vijl(t))))-gj(xj*(t-σijl(t)))gl(xl*(t-vijl(t))),i=1,2,…,n,
with initial values given by
(4.3)xi(s)=φi(s),s∈[-θ,0]⋂𝕋,i=1,2,…,n,
where θ is defined as in(1.3). If (H4) holds, it can always find a smallenough constant α>0 satisfying ∀t∈𝕋,1-μ(t)α>0, namely, -α∈ℜ+ such that
(4.4)(-ci-+α)ϵi+∑j=1naij+Mjeα(t,t-γij(t))ϵj+∑j=1n∑l=1nbijl+[HjNlϵjeα(t,t-σijl(t))+HlNjϵleα(t,t-vijl(t))]<0,i=1,2,…,n.
We define a Lyapunov function H=(h1,h2,…,hn)T by hi=eα(t,δ)|xi(t)-xi*(t)|,δ∈[-θ,0]⋂𝕋,i=1,2,…,n. In view of (4.2), weobtain
(4.5)[hiΔ(t)]+=αeα(t,δ)|xi(t)-xi*(t)|+eα(σ(t),δ)sign(xi(t)-xi*(t))×{∑j=1n-gj(xj*(t-σijl(t)))gl(xl*(t-vijl(t)))-ci(t)(xi(t)-xi*(t))+∑j=1naij(t)(fj(xj(t-γij(t)))-fj(xj*(t-γij(t))))+∑j=1n∑l=1nbijl(t)(gj(xj(t-σijl(t)))gl(xl(t-vijl(t))))-gj(xj*(t-σijl(t)))gl(xl*(t-vijl(t)))∑j=1n}≤αeα(t,δ)|xi(t)-xi*(t)|+eα(σ(t),δ)×{∑j=1n-ci(t)|xi(t)-xi*(t)|+∑j=1n|aij(t)||xj(t-γij(t))-xj*(t-γij(t))|Mj+∑j=1n∑l=1n|bijl(t)|[HjNl|xj(t-σijl(t))-xj*(t-σijl(t))|+HlNj|xl(t-vijl(t))-xl*(t-vijl(t))||xj(t-σijl(t))-xj*(t-σijl(t))|]∑j=1n}≤eα(σ(t),δ)×{xj(t-γij(t))(-ci(t)+α)|xi(t)-xi*(t)|+∑j=1n|aij(t)||xj(t-γij(t))-xj*(t-γij(t))|Mj+∑j=1n∑l=1n|bijl(t)|[HjNl|xj(t-σijl(t))-xj*(t-σijl(t))|+HlNj|xl(t-vijl(t))-xl*(t-vijl(t))||xj(t-σijl(t))-xj*(t-σijl(t))|]∑j=1n}≤[1+μ(t)α]{(-ci-+α)hi(t)+∑j=1naij+Mjeα(t,t-γij(t))hj(t-γij(t))+∑j=1n∑l=1nbijl+×[HjNleα(t,t-σijl(t))hj(t-σijl(t))+HlNjeα(t,t-vijl(t))hl(t-vijl(t))HjNleα(t,t-σijl(t))hj(t-σijl(t))]∑j=1n},i=1,2…,n.
Defining the curve ρ={ω̲(l):ω̲i=ϵil,l>0,i=1,2,…,n} and the set Ω(ω̲)={u̲:0≤u̲≤ω̲}, Si(ω̲)={u̲∈Ω(ω̲):u̲i=ω̲},i=1,2…,n. It is obvious that if l<l̲, then Ω(ω̲(l))⊂Ω(ω̲(l̲)). We will prove that the zero solution of (4.2) is exponential stable, namely, there exists a constant β>0 such that
(4.6)∥x(t)-x*(t)∥≤N(δ)e-β(t,δ)∥φ(t)-x*(t)∥,t≥0.
Let ϵM=max1≤i≤n{ϵi},ϵm=min1≤i≤n{εi},l0=(1-δ)|φi-xi*|0/ϵm, where -δ≥0 is a constant, |φi-xi*|0=max1≤i≤n{maxδ∈[-θ,0]⋂𝕋|φi(δ)-xi*(δ)|}. Then, {|H|:|H|=eα(t,δ)|φ(δ)-x*(δ)|,-θ≤t≤δ≤0}⊂Ω(ω̲(l0)), namely,
(4.7)|hi(δ)|=eα(t,δ)|φi(δ)-xi*(δ)|<ϵil0,-θ≤δ≤0,i=1,2,…,n.
We can claim that |hi(t)|<ϵil0 for t>0,i=1,2,…,n. If it is not true, then there exist some i and t1(ti>0) such that |hi(t1)|=ϵil0,[hiΔ(t1)]+≥0 and |hi(t)|≤ϵil0 for -θ≤t≤t1,i=1,2,…,n. However, from (4.4) and (4.5), we get
(4.8)[hiΔ(t1)]+≤[1+μ(t)α][(-ci-+α)ϵi+∑j=1naij+Mjeα(t,t-γij(t))ϵj+∑j=1n∑l=1nbijl+[HjNleα(t,t-σijl(t))ϵj+HlNjeα(t,t-vijl(t))ϵl]]l0<0
for t>0,i=1,2,…,n, this is a contradiction. So |hi(t)|<ϵil0, for t≥0. Also
(4.9)|xi(t)-xi*(t)|<eΘα(t,δ)ϵil0≤1ϵmeΘα(t,δ)ϵi(1-δ)|φi-xi*|0,i=1,2,…,n,t≥0,
which means that
(4.10)∥x(t)-x*(t)∥≤1ϵmeΘα(t,δ)ϵM(1-δ)∥φ(δ)-x*(δ)∥.
Denote -β=Θα=-α/(1+μα)∈ℜ+,N=N(δ)=(ϵM/ϵm)(1-δ)>1, in view of (4.10), we have
(4.11)∥x(t)-x*(t)∥≤Ne-β(t,δ)∥φ(s)-x*(s)∥.
From Definition 4.2, the periodic solution of system (1.2) is globally exponentially stable. The proof is complete.


## 5. An Example

Let𝕋=ℝ, in this case, xΔ(t)=dx(t)/dt. Consider the following equation
(5.1)dxi(t)dt=-ci(t)xi(t)+∑j=12aij(t)fj(xj(t-γij(t)))+∑j=12∑l=12bijl(t)gj(xj(t-σijl(t)))gl(xl(t-vijl(t)))+Ii(t),i=1,2,
where
(5.2)f1(x1)=sin(121/2x1),f2(x2)=sin(123/2x2),g1(x1)=arctan(121/2x1),g2(x2)=arctan(123/2x2).
Obviously, fi(xi),gi(xi)(i=1,2) satisfy (H1) and (H2), and
(5.3)H1=H2=M1=M2=1,N1=N2=π2.
Take
(5.4)a11(t)=1+cos(2πt),a12(t)=2+cos(2πt),a21(t)=2+cos(2πt),a22(t)=3+cos(2πt),c1(t)=20+5sin(2πt),c2(t)=33+16sin(2πt),I1(t)=1+sin(2πt),I2(t)=1+cos(2πt),b111(t)=b222(t)=14+14sin(2πt),b112(t)=b212(t)=13+13cos(2πt),b121(t)=b221(t)=15+15cos(2πt),b122(t)=b211(t)=16+16sin(2πt).
One can verifies that (H1) is satisfied, and ω=1,c1-=15,c2-=17,a11+=2,a12+=3,a21+=3,a22+=4,b111+=b222+=1/2,b112+=b212+=2/3,b121+=b221+=2/5,b122+=b211+=1/3, so, if we take ϵ1=ϵ2=1, we can obtain
(5.5)-ϵ1c1-+∑j=12a1j+Mjϵj+∑j=12∑l=12b1jl+(HjNlϵj+HlNjϵl)=-15+5+1910π<0,-ϵ2c2-+∑j=12a2j+Mjϵj+∑j=12∑l=12b2jl+(HjNlϵj+HlNjϵl)=-17+7+1910π<0.
Condition (4.2) is satisfied. From Theorem 3.1 and 4.3, we know that (5.1) has at least one 1-periodic solution and this solution is exponential stability.


## 6. Conclusion

Sufficient conditions are derived to guarantee the stability and existence of periodic solutions for a class of delayed high-order Hopfield neural networks on time scales. To the best of our knowledge, the results presented here have been not appeared in the related literature. In fact, both continuous and discrete systems, are very important in implementing and applications. But it is troublesome to study the existence and stability of periodic solutions for continuous and discrete systems respectively. Therefore, it is meaningful to study that on time scales which can unify the continuous and discrete situations. Also, our methods used in this paper may be applied to some other systems.



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*Source: 573534-2009-09-23.xml*