# Stability and Existence of Periodic Solutions for Cellular Neural Networks with State Dependent Delays on Time Scales

Authors: Yaping Ren; Yongkun Li
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

We study delayed cellular neural networks on time scales. Without assuming the boundedness of the activation functions, we establish the exponential stability and existence of periodic solutions. The results in this paper are completely new even in case of the time scale𝕋=ℝ or ℤ and improve some of the previously known results.


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## Body

## 1. Introduction

Consider the following cellular neural networks with state-dependent delays on time scales:(1.1)xiΔ(t)=-ci(t)xiσ(t)+∑j=1n‍aij(t)fj(x(t))+∑j=1n‍bij(t)fj(xj(ξij(t,x(t))))+Ii(t),t∈𝕋,
where i=1,2,…,n, 𝕋 is an ω-periodic time scale which has the subspace topology inherited from the standard topology on ℝ, xiσ(t)=xi(σ(t)), σ(t) will be defined in the next section, x(t)=(x1(t),x2(t),…,xn(t)), n corresponds to the number of units in the neural network, xi(t) corresponds to the state of the ith unit at time t, fj(xj(t)) denotes the output of the jth unit on ith unit at time t, bij denotes the strength of the jth unit on the ith unit at time ξij(t,x(t)), Ii denotes the external bias on the ith unit at time t, t-ξij(t,x(t)) corresponds to the transmission delay along the axon of the jth unit, ci 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.It is well known that the cellular neural networks have been successfully applied to signal processing, pattern recognition, optimization, and associative memories, especially in image processing and solving nonlinear algebraic equations. They have been widely studied both in theory and applications [1–3]. Many results for the existence of their periodic solutions and the exponential convergence properties for cellular neural networks have been reported in the literatures. See, for instance, [4–17] and references cited therein.In fact, continuous and discrete systems are very important in implementation and applications. It is well known that the theory of time scales has received a lot of attention which was introduced by Stefan Hilger in order to unify continuous and discrete analysis. Therefore, it is meaningful to study dynamic systems on time scales which can unify differential and difference systems see [18–28].When𝕋=ℝ, ξij(t,·)≡t-τij(t), (1.1) reduces to
(1.2)xi′(t)=-ci(t)xi(t)+∑j=1n‍aij(t)fj(xj(t))+∑j=1n‍bij(t)fj(xj(t-τij(t)))+Ii(t),
where i=1,2,…,n. By using Mawhin’s continuation theorem and Liapunov functions, the authors [6, 14] obtained the existence and stability of periodic solutions of (1.2), respectively.Furthermore, (1.1) also covers discrete system (for when 𝕋=ℤ,ξij(n,·)≡n-τij(n); see [15])
(1.3)xi(n+1)=xi(n)e-ci(n)h+θi(h)∑j=1m‍aij(n)fj(xj(n))+θi(h)∑j=1m‍bij(n)fj(xj(n-τij(n)))+θi(h)Ii(n),
where θi(h)=(1-e-ci(n)h)/ci(n), i=1,2,…,m. In [15], the author firstly obtained the discrete-time analogue of (1.3) by the semidiscretization technique [29, 30], and then some sufficient conditions for the existence and global asymptotical stability of periodic solutions of (1.3) were established by using Mawhin’s continuation theorem and Liapunov functions.However, in [5, 13–15], the activation functions fj, j=1,2,…,n are assumed to be bounded. Our main purpose of this paper is to establish the stability and existence of periodic solutions of (1.1) without assuming the boundedness of the activation functions.For the sake of convenience, we denote(1.4)c¯i=maxt∈[0,ω]𝕋|ci(t)|,c_i=mint∈[0,ω]𝕋|ci(t)|,a¯ij=maxt∈[0,ω]𝕋|aij(t)|,b¯ij=maxt∈[0,ω]𝕋|bij(t)|,I¯i=maxt∈[0,ω]𝕋|Ii(t)|,i,j=1,2,…,n.Throughout this paper, we assume that(H1)
ξij(·,·)∈C(𝕋×ℝn,[0,∞)𝕋) is ω-periodic with respect to its first argument and satisfies ξij(t,x)≤τ for all (t,x)∈𝕋×ℝn, ci∈C(𝕋,(0,+∞)), aij, bij, Ii∈C(𝕋,ℝ), i, j=1,2,…,n are ω-periodic functions;(H2)
fj∈C(ℝ,ℝ), and there exists a positive number Mj such that |fj(x)-fj(y)|⩽Mj|x-y| for all x, y∈ℝ, j=1,2,…,n.The initial conditions of system (1.1) are of the following form:
(1.5)xi(s)=ϕi(s),i=1,2,…,n,s∈[-τ,0]𝕋,τ=max1⩽i⩽n,1⩽j⩽nmaxt∈[0,ω]𝕋|τij(t)|,
where ϕi∈C([-τ,0]𝕋,ℝ), i=1,2,…,n.The organization of this paper is as follows. In Section2, we introduce some lemmas and definitions and state some preliminary results needed in later sections, which will be used in latter sections. In Section 3, we will study the existence of periodic solutions of system (1.1) by using the method of coincidence degree. In Section 4, we will derive sufficient conditions to ensure that the periodic solutions of (1.1) are globally exponentially stable. In Section 5, an example is also provided to illustrate the effectiveness of the main results in Sections 3 and 4. The conclusions are drawn in Section 6.


## 2. Preliminaries

In this section, we will introduce some notations and definitions and state some preliminary results.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. The set of continuous functions f:𝕋→ℝ will be denoted by C(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|∀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.Lety be right dense continuous. If YΔ(t)=y(t), then we define the delta integral by ∫aty(s)Δs=Y(t)-Y(a).Definition 2.1 (see [31]).
We say that a time scale𝕋 is periodic if there exists p>0 such that if t∈𝕋, then t±p∈𝕋. For 𝕋≠ℝ, the smallest positive p is called the period of the time scale.Definition 2.2 (see [31]).
Let𝕋≠ℝ be a periodic time scale with period p. We say that the function f:𝕋→ℝ is periodic with period ω if there exists a natural number n such that ω=np,f(t+ω)=f(t) for all t∈𝕋, and ω is the smallest number such that f(t+ω)=f(t). If 𝕋=ℝ, we say that f is periodic with period ω>0 if ω is the smallest positive number such that f(t+ω)=f(t) for all t∈𝕋.If𝕋 is ω periodic, then σ(t+ω)=σ(t)+ω and μ(t) is an ω-periodic function.Definition 2.3 (see [32]).
A functionp:𝕋→ℝ is said to be regressive provided that 1+μ(t)p(t)≠0 for all t∈𝕋k, where μ(t)=σ(t)-t is the graininess function. The set of all regressive rd-continuous functions f:𝕋→ℝ is denoted by ℛ while the set ℛ+ is given by {f∈ℛ:1+μ(t)f(t)>0} for all t∈𝕋. Let p∈ℛ. The exponential function is defined by
(2.3)ep(t,s)=exp(∫st‍ξμ(τ)(p(τ))Δτ),
where ξh(z) is the so-called cylinder transformation.Letp,q:𝕋→ℝ be two regressive functions, and we define
(2.4)p⊕q:=p+q+μpq,⊖p:=-p1+μp,p⊖q:=p⊕(⊖q).
Then the generalized exponential function has the following properties.Lemma 2.4 (see [32]).
Assume thatp,q:𝕋→ℝ are two regressive functions, and then(1)
ep(σ(t),s)=(1+μ(t)p(t))ep(t,s);(2)
ep(t,s)=1/ep(s,t)=e⊖p(s,t);(3)
ep(t,s)ep(s,r)=ep(t,r).Lemma 2.5 (see [32]).
Assume thatf, g:𝕋→ℝ are delta differentiable at t∈𝕋k. Then
(2.5)(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t)=f(t)gΔ(t)+fΔ(t)g(σ(t)).Lemma 2.6 (see [32]).
Ifa,  b∈𝕋, α, β∈ℝ, and f, g∈C(𝕋,ℝ), then (1)
∫ab‍[αf(t)+βg(t)]Δt=α∫ab‍f(t)Δt+α∫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.Lemma 2.7 (see [33]).
Lett1, t2∈[0,ω]𝕋. If x:𝕋→ℝ is ω periodic, then
(2.6)x(t)⩽x(t1)+∫0ω‍|xΔ(s)|Δs,x(t)⩾x(t2)-∫0ω‍|xΔ(s)|Δs.Definition 2.8.
The periodic solutionx*(t)=(x1*(t),x2*(t),…,xn*(t))T of system (1.1) is said to be exponentially stable if there exists a positive constant α with ⊖α∈ℛ+ such that for every δ∈[-τ,0]𝕋, and there exists N=N(δ)⩾1 such that the solution x(t)=(x1(t),x2(t),…,xn(t))T of system (1.1) through (δ,x(δ)) satisfies
(2.7)∑i=1n‍|xi(t)-xi*(t)|⩽N∥ϕ(δ)-x*(δ)∥e⊖α(t,δ),t∈𝕋+:=[0,∞)𝕋,
where ∥ϕ(δ)-x*(δ)∥=∑i=1nmaxδ∈[-τ,0]𝕋|ϕi(δ)-xi*(δ)|.In order to show that there exists at least oneω-periodic solution of system (1.1), we need the following concepts and result which are cited from [34].Let𝕏,𝕐 be Banach spaces, L:DomL⊂𝕏→dim𝕐 be a linear mapping, and N:𝕏→𝕐 be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL=codimImL<+∞ and ImL is closed in 𝕐. If L is a Fredholm mapping of index zero and there exists continuous projector P:𝕏→𝕏 and Q:𝕐→𝕐 such that ImP=KerL,KerQ=Im(I-Q), it follows that mapping L|DomL∩KerP:(I-P)𝕏→ImL is invertible. We denote the inverse of that mapping by KP. If Ω is an open bounded subset of 𝕏, the mapping N will be called L compact on Ω¯ if QN(Ω¯) is bounded and KP(I-Q)N:Ω¯→𝕏 is compact.Lemma 2.9 (see [34]).
Let𝕏,𝕐 be two Banach spaces and Ω⊂𝕏 be open bounded and symmetric with 0∈Ω. Suppose that L:D(L)⊂𝕏→𝕐 is a linear Fredholm operator of index zero with D(L)∩Ω-≠∅, and N:Ω-→𝕐 is L compact. Further, one also assumes that(H)
Lx-Nx≠λ(-Lx-N(-x)) for all x∈D(L)∩∂Ω, λ∈(0,1].
Then equationLx=Nx has at least one solution on D(L)∩Ω-.Definition 2.10 (see [35]).
A real matrixP=(pij)n×n is said to be a nonsingular M matrix if pij≤0, i, j=1,2,…,n, i≠j, and all successive principal minors of P are positive.ForA=(aij)m×n, B=(bij)m×n∈ℝmn, A≥B(A>B) means that each pair of corresponding elements of A and B such that aij≥bij(aij>bij).Lemma 2.11 (see [35]).
Assume thatP is a nonsingular M matrix and Py≤h, then y≤P-1h.


## 3. Existence of Periodic Solutions

In this section, by Lemma2.9, we will study the existence of at least one periodic solution of system (1.1).Theorem 3.1.
Suppose that (H1)-(H2) holds,E=(eij)n×n is a nonsingular M matrix, where
(3.1)eij=z{ri+diii=j,diji≠j,
and ri=c_i(1-ωc¯i)/(1+c_iω), dij=-(a¯ij+b¯ij)Mj, i,j=1,2,…,n, then system (1.1)  has at least one ω-periodic solution.Proof.
Let(3.2)𝕏=𝕐={x=(x1,x2,…,xn)T∈C(𝕋,ℝ):x(t+ω)=x(t),t∈𝕋}
with the norm defined by ∥x∥=max1≤i≤n|xi|0, where |xi|0=maxt∈[0,ω]𝕋|xi(t)|, then 𝕏 and 𝕐 are Banach spaces.
Set(3.3)Lx=xΔ(t),Px=Qx=1ω∫0ω‍x(t)Δt,x∈𝕏,
and ℕ:𝕏→𝕐(3.4)Nxi(t)=-ci(t)xiσ(t)+∑j=1n‍aij(t)fj(xj(t))+∑j=1n‍bij(t)fj(xj(ξij(t,x(t))))+Ii(t),
where i=1,2,…,n. Obviously, KerL=ℝn, ImL={x∈𝕏:∫0ω‍x(s)Δs=0} is closed in 𝕐 and
(3.5)dimKerL=n=codimImL.
Hence, L is a Fredholm mapping of index zero. Furthermore, similar to the proof of Theorem  3.4 in [21], one can easily show that N is L compact on Ω- with any open bounded set Ω⊂X. Corresponding to the operator equation Lx-Nx=λ(-Lx-N(-x)), λ∈(0,1], we have
(3.6)xiΔ(t)=11+λ[-ci(t)xiσ(t)+∑j=1n‍aij(t)fj(xj(t))+∑j=1n‍bij(t)fj(xj(ξij(t,x(t))))+Ii(t)]-λ1+λ[ci(t)xiσ(t)+∑j=1n‍aij(t)fj(-xj(t))+∑j=1n‍bij(t)fj(-xj(ξij(t,x(t))))+Ii(t)],
where i=1,2,…,n.
Suppose thatx=(x1,x2,…,xn)T is a solution of system (3.6) for some λ∈(0,1]. In view of (3.6) and (H2), we have
(3.7)∫0ω‍|xiΔ(t)|Δt≤∫0ω‍|ci(t)xiσ(t)|Δt+∫0ω‍|Ii(t)|Δt+11+λ∫0ω‍|[∑j=1n‍a¯ijfj(xj(t))+∑j=1n‍b¯ijfj(xj(ξij(t,x(t))))]|Δt+λ1+λ∫0ω‍|[∑j=1n‍a¯ijfj(-xj(t))+∑j=1n‍b¯ijfj(-xj(ξij(t,x(t))))]|Δt⩽∫0ω‍c¯i|xiσ(t)|Δt+∫0ω‍[∑j=1n‍a¯ij(Mj|xj(t)|+|fj(0)|)+∑j=1n‍b¯ij(Mj|xj(ξij(t,x(t)))|+|fj(0)|)+I¯i]Δt⩽c¯iω|xi|0+ω∑j=1n‍(a¯ij+b¯ij)Mj|xj|0+ω[∑j=1n‍(a¯ij+b¯ij)|fj(0)|+I¯i]⩽c¯iω|xi|0+ω∑j=1n‍(a¯ij+b¯ij)Mj|xj|0+ωαi,i=1,2,…,n,
where αi=∑j=1n‍(a¯ij+b¯ij)|fj(0)|+I¯i.
Integrating both sides of (3.6) from 0 to ω, we obtain that
(3.8)∫0ω‍ci(t)xiσ(t)Δt=11+λ∫0ω‍[∑j=1n‍aij(t)fj(xj(t))+∑j=1n‍bij(t)fj(xj(ξij(t,x(t))))+Ii(t)]Δt-λ1+λ∫0ω‍[∑j=1n‍aij(t)fj(-xj(t))+∑j=1n‍bij(t)fj(-xj(ξij(t,x(t))))+Ii(t)]Δt,
where i=1,2,…,n. Then we have from (3.7) that
(3.9)|∫0ω‍ci(t)xiσ(t)Δt|⩽∫0ω‍[∑j=1n‍|aij(t)fj(xj(t))|+∑j=1n‍|bij(t)fj(xj(ξij(t,x(t))))|+|Ii(t)|]Δt⩽ω∑j=1n‍(a¯ij+b¯ij)Mj|xj|0+ωαi,i=1,2,…,n.
From Lemma 2.7, for any t1i, t2i∈[0,ω]𝕋, i=1,2,…,n, we have
(3.10)∫0ω‍ci(t)xiσ(t)Δt⩽∫0ω‍ci(t)xi(t1i)Δt+∫0ω‍ci(t)(∫0ω‍|xiΔ(t)|Δt)Δt,∫0ω‍ci(t)xiσ(t)Δt⩾∫0ω‍ci(t)xi(t2i)Δt-∫0ω‍ci(t)(∫0ω‍|xiΔ(t)|Δt)Δt,i=1,2,…,n.
Dividing by ∫0ω‍ci(t)Δt on the two sides of the inequalities above, we obtain that
(3.11)-xi(t1i)⩽-1∫0ω‍ci(t)Δt∫0ω‍ci(t)xiσ(t)Δt+∫0ω‍|xiΔ(t)|Δt,(3.12)xi(t2i)⩽1∫0ω‍ci(t)Δt∫0ω‍ci(t)xiσ(t)Δt+∫0ω‍|xiΔ(t)|Δt,i=1,2,…,n.
Let|xi(t0i)|=maxt∈[0,ω]𝕋|xi(t)|, i=1,2,…,n. If for some i=1,2,…,n, xi(t0i)⩾0, we choose t2i=t0i. Hence maxt∈[0,ω]𝕋|xi(t)|=xi(t2i). From (3.12), we have
(3.13)c_i|xi|0⩽c_i(1∫0ω‍ci(t)Δt∫0ω‍ci(t)xiσ(t)Δt+∫0ω‍|xiΔ(t)|Δt)⩽1ω|∫0ω‍ci(t)xiσ(t)Δt|+c_i∫0ω‍|xiΔ(t)|Δt.
If for some i=1,2,…,n, xi(t0i)⩽0, we choose t1i=t0i. Hence maxt∈[0,ω]𝕋|xi(t)|=-xi(t1i). From (3.11), we also have (3.13).
By using (3.7) and (3.9) and (3.13), for i=1,2,…,n, we obtain that
(3.14)c_i|xi|0⩽∑j=1n‍(a¯ij+b¯ij)Mj|xj|0+αi+c¯iω|xi|0+c_iω∑j=1n‍(a¯ij+b¯ij)Mj|xj|0+c_iωαi,
so,
(3.15)c_i(1-ωc¯i)(1+c_iω)|xi|0⩽∑j=1n‍(a¯ij+b¯ij)Mj|xj|0+αi,
namely,
(3.16)ri|xi|0-∑j=1n‍dij|xj|0⩽αi,i=1,2,…,n.
Denote the following:
(3.17)|x|0=(|x1|0,|x2|0,…,|xn|0)T,C=(α1,α2,…,αn)T.
Thus (3.16) is rewritten in the matrix form
(3.18)E|x|0⩽C.
From the conditions of Theorem 3.1, E is a nonsingular M matrix, therefore
(3.19)|x|0⩽E-1C:=(B1,B2,…,Bn)T,
that is, |xi|0⩽Bi, i=1,2,…,n.
TakeA=max1≤i≤nBi+1 and
(3.20)Ω={x∈𝕏:∥x∥<A}.
It is clear that Ω satisfies all the requirements in Lemma 2.9, and condition (H) is satisfied. In view of all the discussions above, we conclude from Lemma 2.9 that system (1.1) has at least one ω-periodic solution. This completes the proof.From Theorem3.1, when 𝕋=ℝ,ξij(t,·)≡t-τij(t), we have the following.Corollary 3.2.
Under assumptions of Theorem3.1, system (1.2) has at least one ω-periodic solution.When𝕋=ℤ, ξij(n,·)≡n-τij(n), from Theorem 3.1, we have the following.Corollary 3.3.
Under assumptions of Theorem3.1, system (1.3) has at least one ω-periodic solution.


## 4. Global Exponential Stability of the Periodic Solution

In this section, we will establish some results for the global exponential stability of the periodic solution of (1.1). The following lemma is essential.Lemma 4.1 (see [36]).
Letp:𝕋→ℝ be rd-continuous and regressive. Suppose that f:𝕋→ℝ is rd-continuous, δ∈𝕋, and y0∈ℝ. Then y is the unique solution of the initial value problem
(4.1)yΔ(t)+p(t)yσ(t)=h(t),y(δ)=y0,
if and only if
(4.2)y(t)=e⊖p(t,δ)y0+∫δt‍e⊖p(t,s)h(s)Δs.Theorem 4.2.
Assume(H1), (H2), and (H4) hold. Furthermore, suppose that(H5)
fort∈[0,ω]𝕋, i=1,2,…,n, ci(t)-M^>0, where M^=max1≤i≤n{∑j=1n‍a-ijMj+∑j=1n‍b-ijMj}
holds. Then theω-periodic solution of system (1.1) is globally exponentially stable.Proof.
From Theorem3.1, we see that system (1.1) has at least one ω-periodic solution x*(t)=(x1*(t),x2*(t),…,xn*(t))T. Let x(t)=(x1(t),x2(t),…,xn(t))T be an arbitrary solution of system (1.1). Let y(t)=x(t)-x*(t), and then we obtain that
(4.3)yiΔ(t)=-ci(t)yiσ(t)+∑j=1n‍aij(t)[f(xj(t))-f(xj*(t))]+∑j=1n‍bij(t)[fj(xj(ξij(t,x(t))))-fj(xj*(ξij(t,x(t))))],i=1,2,…,n
with initial values given by
(4.4)yi(s)=φi(s)-xi*(s),s∈[-τ,0]𝕋,i=1,2,…,n,
where each φi∈C[(-τ,0]𝕋,ℝ] is bounded.
Leth(t)=(h1(t),h2(t),…,hn(t))T and
(4.5)hi(t)=∑j=1n‍aij(t)(f(xj(t))-f(xj*(t)))+∑j=1n‍bij(t)[fj(xj(ξij(t,x(t))))-fj(xj*(ξij(t,x(t))))],i=1,2,…,n.
From Lemma 4.1, then
(4.6)yi(t)=e⊖ci(t,δ)y0+∫δt‍e⊖ci(t,s)hi(s)Δs,i=1,2,…,n,δ∈(-∞,0]𝕋,t≥δ
is the unique solution of (4.3) with initial value y(δ)=y0. From (4.6), we have
(4.7)|yi(t)|e⊖ci(t,δ)≤|y0|+∫δt‍1e⊖ci(s,δ)|hi(s)|Δs,i=1,2,…,n,t≥δ
and, according to (4.5), we can obtain that
(4.8)∥h∥≤(∑j=1n‍a-ijMj+∑j=1n‍b-ijMj)∥y∥≤M^∥y∥,i=1,2,…,n,t≥δ.
Thus, by (4.7), we have
(4.9)∥y∥e⊖ci(t,δ)≤∥y0∥+∫δt‍M^∥y∥e⊖ci(s,δ)Δs,i=1,2,…,n,t≥δ.
By virtue of Gronwall’s inequality ([32], Corollary  6.7), we obtain
(4.10)∥y∥e⊖ci(t,δ)≤∥y0∥eM^(t,δ),i=1,2,…,n,t≥δ.
Hence we have
(4.11)∥y∥≤∥y0∥eM^(t,δ)e⊖ci(t,δ)≤∥y0∥e⊖ci⊕M^(t,δ)=∥y0∥e-(ci-M^)/(1+μci)(t,δ)≤∥y0∥e-γ(t,δ),t≥δ,
where γ=min1≤i≤n{(ci-M^)/(1+μci)}, i=1,2,…,n. We can rewrite (4.10) as follows
(4.12)∑i=1n‍∥xi(t)-xi*(t)∥≤∥ϕ(δ)-x*(δ)∥e-γ(t,δ),t≥δ.
Therefore, the ω-periodic solution of system (1.1) is globally exponentially stable. This completes the proof.From Theorem4.2, when 𝕋=ℝ,ξij(t,·)≡t-τij(t), we have the following.Corollary 4.3.
Under assumptions of Theorem4.2, all the ω-periodic solutions of system (1.2) are globally exponentially stable.When𝕋=ℤ, ξij(n,·)≡n-τij(n), we have the following.Corollary 4.4.
Under assumptions of Theorem4.2, all the ω-periodic solutions of system (1.3) are globally exponentially stable.Remark 4.5.
Corollaries3.2 and 4.3 improve the corresponding results obtained in [4–6, 13, 14] and are different from those in [8, 11, 12, 16, 17] and the references cited therein. Therefore, the results of this paper are new, and they complement previously known results. Corollaries 3.3 and 4.4 improve the results obtained in [15].


## 5. An Example

Consider the following cellular neural networks with delays:(5.1)xiΔ(t)=-ci(t)xi(t)+∑j=12‍aij(t)fj(xj(t))+∑j=12‍bij(t)fj(xj(t-τij(t)))+Ii(t),
where t∈𝕋, i=1, 2, 𝕋 is a 1-periodic time scale, c1(t)=1/4+(1/8)sin(2πt), c2(t)=1/3+(1/6)cos(2πt), a11(t)=sin2(πt), a12(t)=a21(t)=(1/4)sin4(πt), a22(t)=|cos(πt)|, b11(t)=2cos2(πt), b12(t)=b21(t)=(1/8)cos6(πt), b22(t)=(1/2)|sin(πt)|, f1(x1)=(1/112)x + (1/56)sin((1/2)x1)+3, f2(x2)=(1/126)x+(1/42)cos((2/3)x2)-11, I1(t)=sin6(πt), I2(t)=3cos(2πt), τ11(t)=ecos(2πt), τ12(t)=3sin2(πt), τ21(t)=ln(sin4(πt)), τ22(t)=3/7+5cos6(πt).By calculating, we havec¯1=3/8, c_1=1/8, c¯2=1/2, c_2=1/6, M1=1/56, M2=1/42, a¯11=1, a¯12=a¯21=1/4, a¯22=1, b¯11=2, b¯12=b¯21=1/8, b¯22=1/2. It is not difficult to verify that (H1)-(H2) are satisfied.Also by calculating, we have that(5.2)E=(eij)2×2=(128-1112-15228128)
is a nonsingular M matrix. If we take ξ1=ξ2=1, we can obtain that
(5.3)-ξ1c_1+∑j=12‍ξj(a¯1j+b¯1j)Mj=-7112<0,-ξ2c_2+∑j=12‍ξj(a¯2j+b¯2j)Mj=-1671344<0.
The condition (H3) is satisfied. By Theorems 3.1 and 4.2, we know that system (5.1) has at least one 1-periodic solution, and this solution is globally exponentially stable.


## 6. Conclusion

In this paper, without assuming the boundedness of the activation functions, we establish the stability and existence of periodic solutions of cellular neural networks with delays on time scales. Our results obtained in this paper are completely new even in case of the time scale𝕋 = ℝ or ℤ. Besides, our method used in this paper may be used to study other type neural networks such as BAM neural networks, high-order Hopfield neural networks, and Cohen-Grossberg neural networks.



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*Source: 386706-2012-09-26.xml*