# Periodic Solutions for Shunting Inhibitory Cellular Neural Networks of Neutral Type with Time-Varying Delays in the Leakage Term on Time Scales

Authors: Yongkun Li; Lei Wang; Yu Fei
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

A class of shunting inhibitory cellular neural networks of neutral type with time-varying delays in the leakage term on time scales is proposed. Based on the exponential dichotomy of linear dynamic equations on time scales, fixed point theorems, and calculus on time scales we obtain some sufficient conditions for the existence and global exponential stability of periodic solutions for that class of neural networks. The results of this paper are completely new and complementary to the previously known results even if the time scale𝕋=ℝ or ℤ. Moreover, we present illustrative numerical examples to show the feasibility of our results.


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## Body

## 1. Introduction

As we know, shunting inhibitory cellular neural networks (SCINNs) have been applied in a wide range of practical fields such as psychophysics, speech, perception, robotics, adaptive pattern recognition, and image processing. Hence, they have been the object of intensive analysis by numerous authors in recent years. In particular, there have been extensive results on the problem of the existence and stability of periodic solutions and almost periodic solutions for SCINNs in the literature. For example, in [1–3], authors consider the existence and stability of almost periodic solutions for SCINNs; in [4, 5], authors consider the existence and stability of periodic solutions for SCINNs; in [5], authors by using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions consider the periodic solution for SCINNs; in [6, 7], authors obtained some sufficient conditions for the existence and stability of an equilibrium point.Recently, another type of time delays, namely, neutral type time delays, which always appears in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar, and so forth, has recently drawn much research attention. There are some results on the stability and the existence of periodic solutions to delayed neural networks of neutral type, for example in [8–13], by using the Lyapunov functions and the linear matrix inequality approach, authors studied the asymptotic stability or exponential stability of the equilibrium point for delayed neural networks of neutral type and in [14, 15], by using the theory of abstract continuation theorem of k-set contractive operator, authors studied the existence of periodic solutions for delayed cellular neural networks of neutral type and Hopfield neural networks with neutral delays, respectively. In a recent paper [16], authors studied the existence and exponential stability for the following SICNN with continuously distributed delays of neutral type:
(1)xij′(t)=-aij(t)xij(t)-∑Ckl∈Nr(i,j)‍Cijkl(t)×∫0+∞‍Kij(u)f[xkl(t-u)]duxij(t)-∑Ckl∈Ns(i,j)‍Dijkl(t)×∫0+∞‍Jij(u)g[xkl′(t-u)]duxij(t)+Iij(t),
where Cij is the cell at the (i,j) position of the lattice, the r neighborhood Nr(i,j) of Cij is defined as follows:
(2)Nr(i,j)={Ckl:max⁡{|k-i|,|l-j|}≤r,1≤k≤m,1≤l≤nCkl},
where xij represents the state of the cell Cij, the coefficient aij(t)>0 means the passive decay rate of the passive decay rate of the cell activity, and Cijkl(t) and Dijkl(t) describe the connection or coupling strength of postsynaptic activity of the cell Ckl transmitted to the cell Cij.Very recently, a leakage delay, which is the time delay in the leakage term of the systems and a considerable factor affecting dynamics for the worse in the systems, is being put to use in the problem of stability for neural networks [17, 18]. However, so far, very little attention has been paid to neural networks with time delay in the leakage (or “forgetting”) term [19–25]. Such time delays in the leakage term are difficult to handle but have great impact on the dynamical behavior of neural networks.Also, it is well known that both continuous time and discrete time neural networks have equal importance in various applications. Moreover, the theory of calculus on time scales was initiated by Stefan Hilger [26] in his Ph.D. thesis in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has recently received much attention since his foundational work. For instance, in [27], the authors studied antiperiodic solutions to impulsive SICNNs with distributed delays on time scales. In [28], the authors studied almost periodic solutions of SCINNs on time scales. However, to the best of our knowledge, there is no paper published on the existence and stability of periodic solutions for SCINNs of neutral type with the time delay in the leakage term.Motivated by the above discussions, in this paper, we are concerned with the following SCINNs of neutral type with time-varying delays in the leakage term on time scale𝕋:
(3)xijΔ(t)=-aij(t)xij(t-μij(t))-∑Ckl∈Nr(i,j)‍Cijkl(t)×∫0+∞‍Kij(u)f[xkl(t-u)]Δuxij(t)-∑Ckl∈Ns(i,j)‍Dijkl(t)×∫0+∞‍Jij(u)g[xklΔ(t-u)]Δuxij(t)+Iij(t),×∫0+∞‍Jij(u)g[xklΔ(t-u)]Δuxij(t)+6t∈𝕋+,
where 𝕋 is a periodic time scale and 𝕋+=𝕋∩[0,+∞), Cij is the cell at the (i,j) position of the lattice, the r neighborhood Nr(i,j) of Cij is defined as follows:
(4)Nr(i,j)={Ckl:max⁡{|k-i|,|l-j|}≤r,1≤k≤m,1≤l≤n},xij represents the state of the cell Cij, the coefficient aij(t)>0 means the passive decay rate of the passive decay rate of the cell activity, and Cijkl(t) and Dijkl(t) describe the connection or coupling strength of postsynaptic activity of the cell Ckl transmitted to the cell Cij.The initial condition associated with (3) is defined as follows:
(5)xij(s)=ψij(s),s∈(-∞,0]𝕋,
where i=1,2,…,n, j=1,2,…,m, and ψij(s) are Δ-differentiable.Clearly, if𝕋=ℝ and μij≡0, then (3) reduces to (1).Our main purpose of this paper is to study the existence and global exponential stability of periodic solutions to (3) by using the exponential dichotomy of linear dynamic equations on time scales and some inequality technics. Our results of this paper are completely new and complementary to the previously known results even if the time scale 𝕋=ℝ or ℤ. Our methods used in this paper are different from those used in [14, 15, 19] and can be used to study other types' delayed neural networks of neutral type with delays in the leakage term.For convenience, we denote[a,b]𝕋:={t∣t∈[a,b]∩𝕋}. And we introduce the following notations:
(6)aijM=sup⁡t∈ℝ|aij(t)|,aijm=inf⁡t∈ℝ|aij(t)|,μijM=sup⁡t∈ℝ|μij(t)|,Cijkl¯=sup⁡t∈ℝ|Cijkl(t)|,Dijkl¯=sup⁡t∈ℝ|Dijkl(t)|,Iij¯=sup⁡t∈ℝ|Iij(t)|,lllllliilli=1,2,…,n,j=1,2,…,m.Throughout this paper, we assume that the following conditions hold:(H1)
Cijkl(t),Dijkl(t),Iij(t)∈C(𝕋,ℝ),aij(t)∈C(𝕋,ℝ+), and μij(t)∈C(𝕋,𝕋+) are all ω-periodic functions, for u∈(0,+∞)𝕋 and t∈𝕋,t-u∈𝕋 and for t∈𝕋, t-μij(t)∈𝕋, i=1,2,…,n, j=1,2,…,m.(H2)
fj,gj∈C(ℝ,ℝ) and there exist positive constants L,l such that
(7)|fj(u)-fj(v)|≤L|u-v|,|gj(u)-gj(v)|≤l|u-v|,for allu,v∈ℝ, j=1,2,…,m.(H3)
Fori=1,2,…,n, j=1,2,…,m, the delay kernels Kij,Jij:[0,∞)∩𝕋→ℝ are continuous and integrable with
(8)0≤∫0∞‍|Kij(s)|Δs≤Kij0,0≤∫0∞‍|Jij(s)|Δs≤Jij0.This paper is organized as follows: in Section2, we introduce some notations and definitions and state some preliminary results which are needed in later sections. In Section 3, we establish some sufficient conditions for the existence of periodic solutions of (3). In Section 4, we prove that the periodic solution obtained in Section 3 is globally exponentially stable. In Section 5, we give examples to illustrate the feasibility of our results obtained in previous sections.


## 2. Preliminaries

In this section, we introduce some definitions and state some preliminary results.Definition 1 (see [29]).
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
(9)σ(t)=inf⁡{s∈𝕋:s>t},ρ(t)=sup⁡{s∈𝕋:s<t},μ(t)=σ(t)-t.Definition 2 (see [29]).
A pointt∈𝕋 is called left-dense if t>inf⁡𝕋 and ρ(t)=t, left-scattered if ρ(t)<t, right-dense if t<sup⁡𝕋 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.Definition 3 (see [30]).
One says 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.
Throughout this paper, we restrict our discussions in periodic time scales.Definition 4 (see [29]).
A functionr:𝕋→ℝ is called regressive if
(10)1+μ(t)r(t)≠0,
for all t∈𝕋k. The set of all regressive and rd-continuous functions r:𝕋→ℝ will be denoted by ℛ=ℛ(𝕋)=ℛ(𝕋,ℝ); one defines the set ℛ+=ℛ+(𝕋,ℝ)={r∈ℛ,1+μ(t)r(t)>0,∀t∈𝕋}. If r is regressive function, then the generalized exponential function er is defined by
(11)er(t,s)=exp⁡{∫st‍ξμ(τ)(r(τ))Δτ},fors,t∈𝕋,
with the cylinder transformation
(12)ξh(z)={Log(1+hz)hifh≠0,zifh=0.
Let p,q:𝕋→ℝ be two regressive functions, we define
(13)p⊕q:=p+q+μpq,⊖p:=-p1+μp,p⊖q:=p⊕(⊖q).
Then the generalized exponential function has the following properties.Lemma 5 (see [29]).
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,s)=(1/ep(s,t))=e⊖p(s,t);(iv)
ep(t,s)ep(s,r)=ep(t,r);(v)
ep(t,s)eq(t,s)=ep⊕q(t,s).Lemma 6 (see [29]).
Assume thatf,g:𝕋→ℝ are delta differentiable at t∈𝕋k, then (i)
(ν1f+ν2g)Δ=ν1fΔ+ν2gΔ for any constants ν1,ν2;(ii)
(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t)=f(t)gΔ(t)+fΔ(t)g(σ(t));(iii)
iffΔ≥0, then f is nondecreasing.Lemma 7 (see [29]).
Assume thatp(t)≥0 for t≥s, then ep(t,s)≥1.Lemma 8 (see [29]).
Suppose thatp∈ℛ+, then (i)
ep(t,s)>0, for all t,s∈𝕋;(ii)
ifp(t)≤q(t) for all t≥s, t,s∈𝕋, then ep(t,s)≤eq(t,s) for all t≥s.Definition 9 (see [29]).
Ifa∈𝕋, sup⁡𝕋=∞ and f is rd-continuous on [a,∞), then one defines the improper integral by
(14)∫a∞‍f(t)Δt=lim⁡b→∞∫ab‍f(t)Δt,
provided this limit exists, and one says that the improper integral converges in this case. If this limit does not exist, then one says that the improper integral diverges.Definition 10 (see [29]).
Ifa∈𝕋, inf⁡𝕋=-∞, and f is rd-continuous on (-∞,a), then one defines the improper integral by
(15)∫-∞a‍f(t)Δt=lim⁡b→-∞∫ba‍f(t)Δt,
provided this limit exists, and we say that the improper integral converges in this case. If this limit does not exist, then one says that the improper integral diverges.Lemma 11 (see [29]).
Leta∈𝕋k and b∈𝕋 and assume that f:𝕋×𝕋k→ℝ is continuous at (t,t), where t∈𝕋k with t>a. Also assume that fΔ(t,·) is rd-continuous on [a,σ(t)]. Suppose that for each ɛ>0, there exists a neighborhood U of τ∈[a,σ(t)] such that
(16)|f(σ(t),τ)-f(s,τ)-fΔ(t,τ)(σ(t)-s)|≤ɛ|σ(t)-s|,∀s∈U,
where fΔ denotes the derivative of f with respect to the first variable. Then (i)
g(t):=∫at‍f(t,τ)Δτ implies gΔ(t):=∫at‍fΔ(t,τ)Δτ+f(σ(t),t);(ii)
h(t):=∫tb‍f(t,τ)Δτ implies hΔ(t):=∫at‍fΔ(t,τ)Δτ-f(σ(t),t).Definition 12 (see [31]).
LetX∈ℝn and A(t) be a n×n matrix-valued function on 𝕋, the linear system
(17)XΔ(t)=A(t)X(t),t∈𝕋,
is said to admit an exponential dichotomy on 𝕋 if there exist positive constants ki and αi, i=1,2, projection P, and the fundamental solution matrix X(t) of (17) satisfying
(18)|X(t)PX-1(σ(s))|≤k1e⊖α1(t,s),s,t∈𝕋,t≥s,|X(t)(I-P)X-1(σ(s))|≤k2e⊖α2(s,t),s,t∈𝕋,t≤s,
where |·| is a matrix norm on 𝕋; that is, if A=(aij)n×m, then we can take |A|=(∑i=1n∑j=1m|aij|2)1/2.Lemma 13 (see [31]).
If (17) admits an exponential dichotomy, then the following ω-periodic system:
(19)xΔ(t)=A(t)x(t)+g(t),t∈𝕋,
has an ω-periodic solution as follows:
(20)x(t)=∫-∞t‍X(t)PX-1(σ(s))g(s)Δs-∫t+∞‍X(t)(I-P)X-1(σ(s))g(s)Δs,
where X(t) is the fundamental solution matrix of (17).Lemma 14 (see [31]).
IfA(t) is a uniformly bounded rd-continuous n×n matrix-valued function on 𝕋 and there is a δ>0 such that
(21)|aii(t)|-∑j≠i‍|aij(t)|-12μ(t)(∑j=1n‍|aij(t)|)2-δ2μ(t)≥2δ,t∈𝕋,i=1,2,…,n,
then (17) admits an exponential dichotomy on 𝕋.Lemma 15 (see [29]).
Ifp∈ℛ and a,b,c∈𝕋, then
(22)[ep(c,·)]Δ=-p[ep(c,·)]σ,∫ab‍p(t)ep(c,σ(t))Δt=ep(c,a)-ep(c,b).


## 3. Existence of Periodic Solutions

In this section, we will state and prove the sufficient conditions for the existence of periodic solutions of (3).Set𝕏={φ={φij}∣φij∈C1(𝕋,ℝ), φ is an ω-periodic function on 𝕋,i=1,2,…,n,j=1,2,…,m} with the norm ∥φ∥=max⁡{|φ|0,|φΔ|0}, where |φ|0=max⁡i,jmax⁡t∈[0,ω]𝕋|φij(t)|, |φΔ|0=max⁡i,jmax⁡t∈[0,ω]𝕋|φijΔ(t)|, C1(𝕋,ℝ) is the set of continuous functions with continuous Δ-derivatives on 𝕋; then 𝕏 is a Banach space. Let φ0(t)={φij0(t)}, where φij0(t)=∫-∞t‍e-aij(t,σ(s))Iij(s)Δs, i=1,2,…,n, j=1,2,…,m and let A be a constant satisfying A≥max⁡{∥φ0∥,max⁡1≤j≤n|fj(0)|,max⁡1≤j≤n|gj(0)|}.Theorem 16.
Let(H1)–(H3) hold. Suppose that (H4)
there exists a positive constantδ such that
(23)aij(t)-12μ(t)aij2(t)-δ2μ(t)≥2δ,t∈𝕋,i=1,2,…,n,j=1,2,…,m;(H5)
more over(24)max⁡1≤i≤n1≤j≤m⁡{θij0aijm,(1+aijMaijm)θij0}≤12,max⁡1≤i≤n1≤j≤m{θijaijm,(1+aijMaijm)θij}<1,where i=1,2,…,n, j=1,2,…,m,
(25)θij0=aijMμijM+∑Ckl∈Nr(i,j)‍Cijkl¯Kij0(2AL+|f(0)|)+∑Ckl∈Ns(i,j)‍Dijkl¯Jij0(2Al+|g(0)|),θij=aijMμijM+∑Ckl∈Nr(i,j)‍Cijkl¯Kij0(4AL+|f(0)|)+∑Ckl∈Ns(i,j)‍Dijkl¯Jij0(4Al+|g(0)|).
Then (3) has a unique periodic solution in 𝕏0={φ∈𝕏∣∥φ-φ0∥≤A}.Proof.
Rewrite (3) in the form
(26)xijΔ(t)=-aij(t)xij(t)+aij(t)∫t-μij(t)t‍xijΔ(s)Δs-∑Ckl∈Nr(i,j)‍Cijkl(t)×∫0+∞‍Kij(u)f[xkl(t-u)]Δuxij(t)-∑Ckl∈Ns(i,j)‍Dijkl(t)×∫0+∞‍Jij(u)g[xklΔ(t-u)]Δuxij(t)+Iij(t),
where i=1,2,…,n, j=1,2,…,m. The initial condition associated with (3) is defined as follows:
(27)xij(s)=ψij(s),s∈(-∞,0]𝕋,
where i=1,2,…,n, j=1,2,…,m, and ψij(s) are Δ-differentiable. For any given φ∈𝕏, we consider the following periodic system:
(28)xijΔ(t)=-aij(t)xij(t)+aij(t)∫t-μij(t)t‍φijΔ(s)Δs-∑Ckl∈Nr(i,j)‍Cijkl(t)×∫0+∞‍Kij(u)f[φkl(t-u)]Δuφij(t)-∑Ckl∈Ns(i,j)‍Dijkl(t)×∫0+∞‍Jij(u)g[φklΔ(t-u)]Δuφij(t)+Iij(t)=-aij(t)xij(t)+Fijφ(t)+Iij(t),i=1,2,…,n,j=1,2,…,m,
where
(29)Fijφ(t)=aij(t)∫t-μij(t)t‍φijΔ(s)Δs-∑Ckl∈Nr(i,j)‍Cijkl(t)×∫0+∞‍Kij(u)f[φkl(t-u)]Δuφij(t)-∑Ckl∈Ns(i,j)‍Dijkl(t)×∫0+∞‍Jij(u)g[φklΔ(t-u)]Δuφij(t),i=1,2,…,n,j=1,2,…,m.
Since(H4) holds, it follows from Lemma 14 that the linear system
(30)xijΔ(t)=-aij(t)xij(t),i=1,2,…,n,j=1,2,…,m,
admits an exponential dichotomy on 𝕋. Thus, by Lemma 13, we obtain that (28) has an ω-periodic solution, which is expressed as follows:
(31)xijφ(t)=∫-∞t‍e-aij(t,σ(s))(Fijφ(s)+Iij(s))Δs,i=1,2,…,n,j=1,2,…,m.
For φ∈𝕏0, then ∥φ∥≤∥φ-φ0∥+∥φ0∥≤2A. Define the following nonlinear operator:
(32)Φ:𝕏0⟶𝕏0,φ={φij}⟶xφ={xijφ}.
First we show that, for anyφ∈𝕏0, Φ(φ)∈𝕏0. Note that
(33)|Fijφ(s)|≤aijMμijM∥φ∥+∑Ckl∈Nr(i,j)‍Cijkl¯×∫0+∞‍|Kij(u)||f[φkl(s-u)]|Δu|φij(s)|+∑Ckl∈Ns(i,j)‍Dijkl¯×∫0+∞‍|Jij(u)||g[φklΔ(s-u)]|Δu|φij(s)|≤aijMμijM∥φ∥+∑Ckl∈Nr(i,j)‍Cijkl¯×∫0+∞‍|Kij(u)|Δu(L∥φ∥+|f(0)|)∥φ∥+∑Ckl∈Ns(i,j)‍Dijkl¯∫0+∞‍|Jij(u)|Δu(l∥φ∥+|g(0)|)∥φ∥≤(aijMμijM+∑Ckl∈Nr(i,j)‍Cijkl¯Kij0(L∥φ∥+|f(0)|)+∑Ckl∈Ns(i,j)‍Dijkl¯Jij0(l∥φ∥+|g(0)|))∥φ∥≤(aijMμijM+∑Ckl∈Nr(i,j)‍Cijkl¯Kij0(2AL+|f(0)|)+∑Ckl∈Ns(i,j)‍Dijkl¯Jij0(2Al+|g(0)|))2A≤2Aθij0,i=1,2,…,n,j=1,2,…,m.
So
(34)|(Φ(φ-φ0))ij(t)|=|∫-∞t‍e-aij(t,σ(s))Fijφ(s)Δs|≤∫-∞t‍e-aij(t,σ(s))|Fijφ(s)|Δs≤∫-∞t‍e-aijm(t,σ(s))2Aθij0Δs≤2Aθij0aijm,i=1,2,…,n,j=1,2,…,m.
On the other hand, we have
(35)|(Φ(φ-φ0)Δ)ij(t)|=|(∫-∞te-aij(t,σ(s))Fijφ(s)Δs‍)tΔ|=|Fijφ(t)-aij(t)∫-∞t‍e-aij(t,σ(s))Fijφ(s)Δs|≤|Fijφ(t)|+|aij(t)|∫-∞t‍e-aij(t,σ(s))|Fijφ(s)|Δs≤2Aθij0aijm+aijM2Aθij0aijm=(1+aijMaijm)2Aθij0,i=1,2,…,n,j=1,2,…,m.
In view of (H5), we have
(36)∥Φ(φ-φ0)∥≤max⁡1≤i≤n1≤j≤m⁡{2Aθij0aijm,(1+aijMaijm)2Aθij0}≤A;
that is, Φ(φ)∈𝕏0. Next, we show that Φ is a contraction. For φ,ψ∈𝕏0, i=1,2,…,n, j=1,2,…,m, denote
(37)Gij1(s,φ,ψ)=-∑Ckl∈Nr(i,j)‍Cijkl(s)×[∫0+∞‍Kij(u)f[φkl(s-u)]Δuφij(s)-∫0+∞‍Kij(u)f[ψkl(s-u)]Δuψij(s)],i=1,2,…,n,j=1,2,…,m,Gij2(s,φ,ψ)=aij(s)∫s-μij(s)s‍[φijΔ(u)-ψijΔ(u)]Δu-∑Ckl∈Ns(i,j)‍Dijkl(s)×[∫0+∞‍Jij(u)g[φklΔ(s-u)]Δuφij(s)-∫0+∞‍Jij(u)g[ψklΔ(s-u)]Δuψij(s)],i=1,2,…,n,j=1,2,…,m.
So
(38)|Gij1(s,φ,ψ)|=|-∑Ckl∈Nr(i,j)‍Cijkl(s)×[∫0+∞‍Kij(u)f[φkl(s-u)]Δuφij(s)cci-∫0+∞‍Kij(u)f[ψkl(s-u)]Δuψij(s)]∑Ckl∈Nr(i,j)‍|≤∑Ckl∈Nr(i,j)‍|Cijkl(s)|×|∫0+∞‍Kij(u)[f(φkl(s-u))-f(ψkl(s-u))]×Δuφij(s)cci-∫0+∞‍Kij(u)f(ψkl(s-u))Δu(ψij(s)-φij(s))∫|≤∑Ckl∈Nr(i,j)‍|Cijkl(s)|×{∫0+∞‍|Kij(u)|f(φkl(s-u))ccii-f(ψkl(s-u))|Δu||φij(s)|ccii+∫0+∞‍|Kij(u)|(|f(ψkl(s-u))-f(0)|+|f(0)|)cccccccccci×Δu|ψij(s)-φij(s)|∫0+∞‍}≤∑Ckl∈Nr(i,j)‍Cijkl¯×(L∫0+∞‍|Kij(u)|Δu∥φ-ψ∥∥φ∥ccii+∫0+∞‍|Kij(u)|Δu(L∥ψ∥+|f(0)|)∥φ-ψ∥)≤∑Ckl∈Nr(i,j)‍Cijkl¯(Kij0(4AL+|f(0)|))∥ψ-φ∥,cccccccccccccccccccccii=1,2,…,n,j=1,2,…,m.
Similarly,
(39)|Gij2(s,φ,ψ)|≤(aijMμijM+∑Ckl∈Ns(i,j)‍Dijkl¯(Jij0(4Al+|g(0)|)))∥ψ-φ∥,
so
(40)|Gij1(s,φ,ψ)+Gij2(s,φ,ψ)|≤|Gij1(s,φ,ψ)|+|Gij2(s,φ,ψ)|≤(aijMμijM+∑Ckl∈Nr(i,j)‍Cijkl¯Kij0(4AL+|f(0)|)ccicci+∑Ckl∈Ns(i,j)‍Dijkl¯Jij0(4Al+|g(0)|))∥ψ-φ∥=θij∥ψ-φ∥,i=1,2,…,n,j=1,2,…,m,|(Φφ-Φψ)ij(t)|=|∫-∞t‍e-aij(t,σ(s))(Gij1(s,φ,ψ)+Gij2(s,φ,ψ))Δs|≤∫-∞t‍e-aij(t,σ(s))|Gij1(s,φ,ψ)+Gij2(s,φ,ψ)|Δs≤θijaijm∥ψ-φ∥,i=1,2,…,n,j=1,2,…,m,(41)|(Φφ-Φψ)ijΔ(t)|=|(∫-∞t‍e-aij(t,σ(s))(Gij(1)(t,φ,ψ)+Gij(2)(t,φ,ψ)))tΔ|=|∫Gij(1)(t,φ,ψ)+Gij(2)(t,φ,ψ)-aij(t)c×∫-∞t‍e-aij(t,σ(s))(Gij(1)(t,φ,ψ)+Gij(2)(t,φ,ψ))Δs|≤|Gij(1)(t,φ,ψ)+Gij(2)(t,φ,ψ)|∫+|aij(t)|i×∫-∞t‍e-aij(t,σ(s))|Gij(1)(t,φ,ψ)+Gij(2)(t,φ,ψ)|Δs≤(1+aijMaijm)θij∥ψ-φ∥,i=1,2,…,n,j=1,2,…,m.
By (H5), we have
(42)max⁡1≤i≤n1≤j≤m⁡{1aijmθij,(1+aijMaijm)θij}∥ψ-φ∥<∥ψ-φ∥,
which implies that
(43)∥Φφ-Φψ∥<∥φ-ψ∥.
It follows that Φ is a contraction. Therefore, Φ has a fixed point in 𝕏0; that is, (3) has a unique periodic solution in 𝕏0. This completes the proof of Theorem 16.


## 4. Exponential Stability of the Periodic Solution

In this section, we will discuss the exponential stability of the periodic solution of (3).Definition 17.
The periodic solutionx(t)={xij(t)} of system (3) with initial value ψ(t)={ψij(t)} is said to be globally exponentially stable, if there exists a positive constant λ with ⊖λ∈ℛ+ and M>1 such that every solution y(t)={yij(t)} of (3) with any initial value φ(t)={φij(t)} satisfies
(44)∥x(t)-y(t)∥1≤Me⊖λ(t,t0)∥ϕ∥,∀t∈(0,+∞)𝕋,
where
(45)∥x(t)-y(t)∥1=max⁡{max⁡ij{|xij(t)-yij(t)|},ccccccccmax⁡ij{|xijΔ(t)-yijΔ(t)|}},∥ϕ∥=max⁡{sup⁡s∈(-∞,0]𝕋max⁡ij{|ψij(s)-φij(s)|},ccccccccsup⁡s∈(-∞,0]𝕋max⁡ij{|ψijΔ(s)-φijΔ(s)|}},ccccicccci=1,2,…,n,j=1,2,…,m,t0=max⁡{(-∞,0]𝕋}.Theorem 18.
Suppose that all of the conditions in Theorem16 are satisfied; then the unique periodic solution of (3) is globally exponentially stable.Proof.
By Theorem16, (3) has an ω-periodic solution x(t)={xij(t)} with initial condition ψ(t)={ψij(t)}, i=1,2,…,n, j=1,2,…,m. Suppose that y(t)={yij(t)} is an arbitrary solution of (3), associated with the initial value φ(t)={φij(t)}, i=1,2,…,n, j=1,2,…,m. Let z(t)=y(t)-x(t) and ϕ(t)=φ(t)-ψ(t). Then it follows from system (3) that
(46)zijΔ(t)+aij(t)zij(t)=aij(t)∫t-μij(t)t‍zijΔ(s)Δs-∑Ckl∈Nr(i,j)‍Cijkl(t)×[∫0+∞‍Kij(u)f[xkl(t-u)]Δuxij(t)cci-∫0+∞‍Kij(u)f[ykl(t-u)]Δuyij(t)]-∑Ckl∈Ns(i,j)‍Dijkl(t)×[∫0+∞‍Jij(u)g[xklΔ(t-u)]Δuxij(t)xixxxxx-∫0+∞‍Jij(u)g[yklΔ(t-u)]Δuyij(t)],
where zij(t)=yij(t)-xij(t), i=1,2,…,n, j=1,2,…,m and the initial condition of (46) is
(47)ϕij(s)=φij(s)-ψij(s),s∈(-∞,0]𝕋,i=1,2,…,n,j=1,2,…,m.
LetSij and Tij be defined as follows:
(48)Sij(β)=aijm-β-exp⁡(βsup⁡s∈𝕋(μ(s)))Aij(β),Tij(β)=aijm-β-(aijMexp⁡(βsup⁡s∈𝕋(μ(s)))+aijm-β)Aij(β),
where
(49)Aij(β)=aijMμijM+∑Ckl∈Nr(i,j)‍Cijkl¯[(AL+|f(0)|)∫0+∞‍|Kij(u)|Δuxxxxxxccxxxxxcixxixxxxx+AL∫0+∞‍|Kij(u)|exp⁡(βu)Δu]+∑Ckl∈Ns(i,j)‍Dijkl¯[(Al+|g(0)|)∫0+∞‍|Jij(u)|Δuxxxxxccxxcxxxxxxi+Al∫0+∞‍|Jij(u)|exp⁡⁡(βu)Δu],xxxxxxxxxxxxxxxxxi=1,2,…,n,j=1,2,…,m.
By (H5), for i=1,2,…,n, j=1,2,…,m, we have
(50)Sij(0)=aijm-Aij(0)≥aijm-θij0>0,Tij(0)=aijm-(aijM+aijm)Aij(0)≥aijm-(aijM+aijm)θij0>0.
Since Sij and Tij are continuous on [0,+∞) and Sij(β),Tij(β)→+∞ as β→+∞, there exist λij>0 and ςij>0 such that Sij(λij)=0 and Sij(β)>0 for all β∈(0,λij), Tij(ςij)=0, and Tij(β)>0 for all β∈(0,ςij). Take λ*=min⁡1≤i≤n,1≤j≤m{λij,ςij}; then
(51)Sij(λ*)≥0,Tij(λ*)≥0,i=1,2,…,n,j=1,2,…,m.
So, we can choose a positive constant 0<λ<min⁡{λ*,aijm,i=1,2,…,n,j=1,2,…,m} such that
(52)Sij(λ)>0,Tij(λ)>0,i=1,2,…,n,j=1,2,…,m,
which implies that
(53)exp⁡(λsup⁡s∈𝕋(μ(s)))Aij(λ)aijm-λ<1,i=1,2,…,n,j=1,2,…,m,(54)(1+aijMexp⁡⁡(λsup⁡s∈𝕋(μ(s)))aijm-λ)Aij(λ)<1,i=1,2,…,n,j=1,2,…,m.
Let M=max⁡1≤i≤n,1≤j≤m{aijm/Aij(0)}; by (H5), we have M>1.
It is obvious that(55)∥Z(t)∥1=∥ϕ(t)∥1≤M∥ϕ∥e⊖λ(t,t0),∀t∈(-∞,0]𝕋,
where λ∈ℛ+. In the following, we will show that
(56)∥Z(t)∥1=∥ϕ(t)∥1≤M∥ϕ∥e⊖λ(t,t0),∀t∈(0,+∞)𝕋.
holds for all t∈(0,+∞)𝕋.
For this aim, we show that for anyP>1,
(57)∥Z(t)∥1<PM∥ϕ∥e⊖λ(t,t0),∀t∈(0,+∞)𝕋.
If (57) is not true, then there must be t1∈(0,+∞)𝕋 such that
(58)∥Z(t1)∥=max⁡{∥Z(t1)∥,∥ZΔ(t1)∥}=max⁡1≤i≤n1≤j≤m⁡{|Zij(t1)|,|ZijΔ(t1)|}≥PM∥ϕ∥e⊖λ(t1,t0),∥Z(t)∥≤PM∥ϕ∥e⊖λ(t,t0),t∈(-∞,t1]𝕋,t0∈(-∞,0)𝕋.
Therefore, there must exist a constant c≥1 such that
(59)∥Z(t1)∥=max⁡{∥Z(t1)∥,∥ZΔ(t1)∥}=max⁡{|Zij(t1)|,|ZijΔ(t1)|}=cPM∥ϕ∥e⊖λ(t1,t0),(60)∥Z(t)∥≤cPM∥ϕ∥e⊖λ(t,t0),t∈(-∞,t1]𝕋,t0∈(-∞,0)𝕋.
Multiplying both sides of (46) by e-aij(t0,σ(s)) and integrating over [t0,t], by Lemma 15, for i=1,2,…,n, j=1,2,…,m, we get
(61)zij(t)=ϕij(t0)e-aij(t,t0)+∫t0t‍e-aij(t,σ(s))×{aij(s)∫s-μij(s)s‍xijΔ(u)Δu-∑Ckl∈Nr(i,j)‍Cijkl(s)cccc×[∫0+∞‍Kij(u)f[xkl(s-u)]Δuxij(s)cccccc-∫0+∞‍Kij(u)f[ykl(s-u)]Δuyij(s)]ccc-∑Ckl∈Ns(i,j)‍Dijkl(s)ccc×[∫0+∞‍Jij(u)g[xklΔ(s-u)]Δuxij(s)cccccc-∫0+∞‍Jij(u)g[yklΔ(s-u)]Δuyij(s)]∑Ckl∈Nr(i,j)‍}Δs.
It follows from (60), (61), and the assumptions of the theorem that
(62)|zij(t1)|≤∥ϕ∥e-aij(t1,t0)+∫t0t1‍e-aij(t1,σ(s))×{aijM∫s-μij(s)s‍|zijΔ(u)|Δu+∑Ckl∈Nr(i,j)‍Cijkl¯ccii×|∫0+∞‍Kij(u)f[xkl(s-u)]Δuxij(s)cciicc-∫0+∞‍Kij(u)f[ykl(s-u)]Δuyij(s)|ccicci+∑Ckl∈Ns(i,j)‍Dijkl¯ccci×|∫0+∞‍Jij(u)g[xklΔ(s-u)]Δuxij(s)cccccccccc-∫0+∞‍Jij(u)g[yklΔ(s-u)]Δuyij(s)|∑2%1%}Δs≤∥ϕ∥e-aij(t1,t0)+∫t0t1‍e-aij(t1,σ(s))×{aijM∫s-μij(s)s‍|zijΔ(u)|Δucicci+∑Ckl∈Nr(i,j)‍Cijkl¯cccii×[AL∫0+∞‍|Kij(u)||xkl(s-u)-ykl(s-u)|Δuccccciic+(AL+|f(0)|)ccccciic×∫0+∞‍|Kij(u)|Δu|xkl(s)-ykl(s)|]cicci+∑Ckl∈Ns(i,j)‍Dijkl¯cccii×[Al∫0+∞‍|Jij(u)||xklΔ(s-u)-yklΔ(s-u)|Δucccccicc+(Al+|g(0)|)cccccicc×∫0+∞‍|Jij(u)|Δu|xkl(s)-ykl(s)|]aijM∫s-μij(s)s‍|zijΔ(u)|Δu}Δs≤∥ϕ∥e-aij(t1,t0)+cPM∥ϕ∥e⊖λ(t1,t0)aijMμijM+∫t0t1‍cPM∥ϕ∥e-aij(t1,σ(s))e⊖λ(t1,t0)eλ(t1,σ(s))×{∑Ckl∈Nr(i,j)‍Cijkl¯cccc×[AL∫0+∞‍|Kij(u)|eλ(σ(s),s-u)ccccccc+(AL+|f(0)|)∫0+∞‍|Kij(u)|eλ(σ(s),s)Δu]cccc+∑Ckl∈Ns(i,j)‍Dijkl¯cccc×[Al∫0+∞‍|Jij(u)|eλ(σ(s),s-u)ccccccc+(Al+|g(0)|)ccccccc×∫0+∞‍|Jij(u)|eλ(σ(s),s)Δu]∑Ckl∈Nr(i,j)‍Cijkl¯}Δs≤cPM∥ϕ∥e⊖λ(t1,t0)×{∑Ckl∈Nr(i,j)‍Cijkl¯1cPMe-aij(t1,t0)e⊖λ(t0,t1)+aijMμijMccici+∫t0t1‍e-aij(t1,σ(s))eλ(t1,t0)cic×{∑Ckl∈Nr(i,j)‍Cijkl¯cccccci×[AL∫0+∞‍|Kij(u)|cccccccccccccccci×exp⁡(λ(u+sup⁡s∈𝕋μ(s)))Δucccccccciii+(AL+|f(0)|)cccccccciii×∫0+∞‍|Kij(u)|exp⁡(λμ(s))Δu]cccccci+∑Ckl∈Ns(i,j)‍Dijkl¯cccccci×[Al∫0+∞‍|Jij(u)|exp⁡(λ(u+sup⁡s∈𝕋μ(s)))Δucccciccccii+(Al+|g(0)|)cccciccccii×∫0+∞‍|Jij(u)|exp⁡(λμ(s))Δu(λ(u+sup⁡s∈𝕋μ(s)))]∑Ckl∈Nr(i,j)‍Cijkl¯}}≤cPM∥ϕ∥e⊖λ(t1,t0)×{∑Ckl∈Nr(i,j)‍Cijkl¯1Me-aij⊕λ(t1,t0)+aijMμijM+exp⁡(λsup⁡s∈𝕋μ(s))ccici×{∑Ckl∈Nr(i,j)‍Cijkl¯ccccicic×[AL∫0+∞‍|Kij(u)|exp⁡(λu)Δucccccccccccccccccc+(AL+|f(0)|)ccccicicccii×∫0+∞‍|Kij(u)|Δu]ccoocicc+∑Ckl∈Ns(i,j)‍Dijkl¯cooccicc×[Al∫0+∞‍|Jij(u)|exp⁡(λu)Δu+(Al+|g(0)|)ccccccoiccii×∫0+∞‍|Jij(u)|Δu]∑Ckl∈Nr(i,j)‍Cijkl¯}}∫t0t1‍e-aij(t1,σ(s))Δs≤cPM∥ϕ∥e⊖λ(t1,t0)×{(1M-exp⁡⁡(λsup⁡s∈𝕋(μ(s)))Aij(λ)aijm-λ)×e-aij⊕λ(t1,t0)ccci+exp⁡(λsup⁡s∈𝕋(μ(s)))Aij(λ)aijm-λ(1M-exp⁡⁡(λsup⁡s∈𝕋(μ(s)))Aij(λ)aijm-λ)}<cPM∥ϕ∥e⊖λ(t1,t0).
A direct differentiation of (61) gives
(63)zijΔ(t)=-aij(t)zij(t)+aij(t)∫t-μij(t)t‍zijΔ(s)Δs-∑Ckl∈Nr(i,j)‍Cijkl(t)×[∫0+∞‍Kij(u)f[xkl(t-u)]Δuxij(t)ciii-∫0+∞‍Kij(u)f[ykl(t-u)]Δuyij(t)]-∑Ckl∈Ns(i,j)‍Dijkl(t)×[∫0+∞‍Jij(u)g[xklΔ(t-u)]Δuxij(t)ciii-∫0+∞‍Jij(u)g[yklΔ(t-u)]Δuyij(t)],i=1,2,…,n,j=1,2,…,m.
Thus, we have by (53), (54), (60), and (63)
(64)|zijΔ(t1)|≤aijM|zij(t1)|+aijM∫t1-μij(t1)t1‍|zijΔ(s)|Δs+∑Ckl∈Nr(i,j)‍Cijkl¯×[AL∫0+∞‍|Kij(u)||xkl(t1-u)-ykl(t1-u)|Δuccc+(AL+|f(0)|)ccc×∫0+∞‍|Kij(u)|Δu|xkl(t1)-ykl(t1)|]+∑Ckl∈Ns(i,j)‍Dijkl¯×[Al∫0+∞‍|Jij(u)||xklΔ(t1-u)-yklΔ(t1-u)|Δuccc+(Al+|g(0)|)∫0+∞‍|Jij(u)|Δu|xkl(t1)-ykl(t1)|]≤aijM|zij(t1)|+PM∥ϕ∥e⊖λ(t1,t0)×{aijMμijM+∑Ckl∈Nr(i,j)‍Cijkl¯ccci×[AL×∫0+∞‍|Kij(u)|eλ(t1,t1-u)Δuccccccii+(AL+|f(0)|)∫0+∞‍|Kij(u)|eλ(t1,t1)Δu]ccci+∑Ckl∈Ns(i,j)‍Dijkl¯ccci×[Al∫0+∞‍|Jij(u)|eλ(t1,t1-u)Δucccccii+(Al+|g(0)|)∫0+∞‍|Jij(u)|eλ(t1,t1)Δu]∑Ckl∈Ns(i,j)‍Dijkl¯}≤aijM|zij(t1)|+cPM∥ϕ∥e⊖λ(t1,t0)×{aijMμijM+∑Ckl∈Nr(i,j)‍Cijkl¯cccci×[AL∫0+∞‍|Kij(u)|exp⁡⁡(λu)Δucccccccc+(AL+|f(0)|)∫0+∞‍|Kij(u)|Δu]cccci+∑Ckl∈Ns(i,j)‍Dijkl¯cccci×[Al∫0+∞‍|Jij(u)|exp⁡⁡(λu)Δucccccccc+(Al+|g(0)|)∫0+∞‍|Jij(u)|Δu]∑Ckl∈Ns(i,j)‍Dijkl¯}≤aijM|zij(t1)|+cPM∥ϕ∥e⊖λ(t1,t0)Aij(λ)≤aijM|∫t0t1‍cPM∥ϕ∥e⊖λ(t1,t0)×{∫1Me-aij⊕λ(t1,t0)+exp⁡(λsup⁡s∈𝕋μ(s))Aij(λ)ccc×∫t0t1‍e-aij⊕λ(t1,σ(s))Δs}|+cPM∥ϕ∥e⊖λ(t1,t0)Aij(λ)≤cPM∥ϕ∥e⊖λ(t1,t0)×{aijMMe-aij⊕λ(t1,t0)+aijMexp⁡(λsup⁡s∈𝕋μ(s))Aij(λ)cccc×∫t0t1‍e-aij⊕λ(t1,σ(s))Δs+Aij(λ)}≤cPM∥ϕ∥e⊖λ(t1,t0)×{aijMMe-aij⊕λ(t1,t0)+aijMexp⁡(λsup⁡s∈𝕋μ(s))Aij(λ)cccc×(e-aij⊕λ(t1,t0)-1λ-aijm)+Aij(λ)}≤cPM∥ϕ∥e⊖λ(t1,t0)×{aijMMe-aij⊕λ(t1,t0)-aijMexp⁡(λsup⁡s∈𝕋μ(s))Aij(λ)cccc×(e-aij⊕λ(t1,t0)-1aijm-λ)+Aij(λ)}≤cPM∥ϕ∥e⊖λ(t1,t0)×{aijMMe-aij⊕λ(t1,t0)+aijMexp⁡(λsup⁡s∈𝕋μ(s))Aij(λ)ccc×(e-aij⊕λ(t1,t0)-1aijm-λ)+Aij(λ)}≤cPM∥ϕ∥e⊖λ(t1,t0)×{(1M-exp⁡⁡(λsup⁡s∈𝕋(μ(s)))Aij(λ)aijm-λ)cccc×aijMe-aij⊕λ(t1,t0)cccc+(aijMexp⁡⁡(λsup⁡s∈𝕋(μ(s)))aijm-λ+1)Aij(λ)}<cPM∥ϕ∥e⊖λ(t1,t0).
In view of (62) and (64), we obtain
(65)∥Z(t)∥1<cPM∥ϕ∥e⊖λ(t1,t0),
which contradicts (59), and so (57) holds. Letting P→1, then (56) holds. Hence, the almost periodic solution of system (3) is globally exponentially stable. The global exponential stability implies that the periodic solution is unique. The proof is complete.


## 5. Some Examples

Consider the following SICNNs model:(66)xijΔ(t)=-aij(t)xij(t-μij(t))-∑Ckl∈Nr(i,j)‍Cijkl(t)×∫0+∞‍Kij(u)f[xkl(t-u)]Δuxij(t)-∑Ckl∈Ns(i,j)‍Dijkl(t)×∫0+∞‍Jij(u)g[xklΔ(t-u)]Δuxij(t)+Iij(t),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiit∈𝕋,
where f(x)=2cos⁡x, g(x)=2|x|,Kij(u)=e-4u, and Jij(u)=e-2u, i,j=1,2,3.Example 1.
In system (66), let 𝕋=ℝ; then μ(t)≡0, and take
(67)[a11(t)a12(t)a13(t)a21(t)a22(t)a23(t)a31(t)a32(t)a33(t)]=[4+|sint|5+|sin2t|8+|sint|6+|cos⁡t|6+|sint|7+|cos⁡t|8+|cos⁡t|8+|sint|5+|sin2t|],[c11(t)c12(t)c13(t)c21(t)c22(t)c23(t)c31(t)c32(t)c33(t)]=[0.01|sin2t|0.02|sin3t|0.01|sin4t|0.01|sin2t|0.01|sin3t|0.01|sin4t|0.01|sin2t|0.01|sin3t|0.01|sin4t|],[d11(t)d12(t)d13(t)d21(t)d22(t)d23(t)d31(t)d32(t)d33(t)]=[0.01|cos⁡2t|0.01|cos⁡3t|0.02|cos⁡4t|0.01|cos⁡2t|0.01|cos⁡3t|0.01|cos⁡4t|0.01|cos⁡2t|0.01|cos⁡3t|0.01|cos⁡4t|],[I11(t)I12(t)I13(t)I21(t)I22(t)I23(t)I31(t)I32(t)I33(t)]=[sintsintcos⁡tsintcos⁡tcos⁡tcos⁡tcos⁡tcos⁡t],[μ11(t)μ12(t)μ13(t)μ21(t)μ22(t)μ23(t)μ31(t)μ32(t)μ33(t)]=[0.02|cos⁡2t|0.01|cos⁡3t|0.01|cos⁡4t|0.011|cos⁡2t|0.001|cos⁡3t|0.01|cos⁡4t|0.01|cos⁡2t|0.01|cos⁡3t|0.02|cos⁡4t|].
Let r=s=1, A=1/2, and δ=1. Obviously, L=l=2, f(0)=2 and g(0)=0, Kij0=1/4, Jij0=1/2,
(68)[a11Ma12Ma13Ma21Ma22Ma23Ma31Ma32Ma33M]=[569778996],[a11ma12ma13ma21ma22ma23ma31ma32ma33m]=[458667885],[c11¯c12¯c13¯c21¯c22¯c23¯c31¯c32¯c33¯]=[0.020.020.010.010.010.010.010.010.01],[d11¯d12¯d13¯d21¯d22¯d23¯d31¯d32¯d33¯]=[0.010.010.020.010.010.010.010.010.01],[I11¯I12¯I13¯I21¯I22¯I23¯I31¯I32¯I33¯]=[111111111],[μ11Mμ12Mμ13Mμ21Mμ22Mμ23Mμ31Mμ32Mμ33M]=[0.020.010.010.0110.0010.010.010.010.02].
So
(69)[θ110θ120θ130θ210θ220θ230θ310θ320θ330]=[0.20.210.190.2170.2170.220.170.210.2],[θ11θ12θ13θ21θ22θ23θ31θ32θ33]=[0.270.320.2650.3170.3720.3250.230.30.26].It is easy to check that(70)A≥∥φ0∥,max⁡1≤i≤n1≤j≤m⁡{θij0aijm,(1+aijMaijm)θij0}≤12,max⁡1≤i≤n1≤j≤m{θijaijm,(1+aijMaijm)θij}<1.
Therefore, all the conditions in Theorems 16 and 18 are satisfied. Hence (66) has a 2π-periodic solution, which is globally exponentially stable (we give numerical simulations in Figures 1, 2, and 3 by taking x11,x12, and x13 to illustrate our results).Figure 1
Phase response of state variablesx11,x12 in Example 1.Figure 2
Phase response of state variablesx11 and x13 in Example 1.Figure 3
Phase response of state variablesx12 and x13 in Example 1.Example 2.
In system (66), let 𝕋=ℤ, then μ(t)≡1, and take(71)[a11(t)a12(t)a13(t)a21(t)a22(t)a23(t)a31(t)a32(t)a33(t)]=[0.8+0.1|sin3t|0.7+0.1|sin2t|0.7+0.1|sint|0.7+0.2|cos⁡t|0.8+0.1|cos⁡t|0.8+0.1|sint|0.7+0.2|cos⁡t|0.7+0.1|sint|0.8+0.1|cos⁡2t|],[c11(t)c12(t)c13(t)c21(t)c22(t)c23(t)c31(t)c32(t)c33(t)]=[0.01|sin2t|0.02|sin3t|0.01|sin4t|0.02|sin2t|0.01|sin3t|0.01|cos⁡4t|0.01|sin2t|0.01|sin3t|0.02|cos⁡4t|],[d11(t)d12(t)d13(t)d21(t)d22(t)d23(t)d31(t)d32(t)d33(t)]=[0.01|cos⁡2t|0.02|sin3t|0.01|sin4t|0.01|cos⁡2t|0.01|cos⁡3t|0.01|cos⁡4t|0.01|cos⁡2t|0.01|sin2t|0.02|cos⁡4t|],[I11(t)I12(t)I13(t)I21(t)I22(t)I23(t)I31(t)I32(t)I33(t)]=[0.08sint+0.03cos⁡t0.02sint+0.01cos⁡2t0.02cos⁡t0.01sint0.02cos⁡t0.06cos⁡t0.01cos⁡t0.01sint+0.02sint0.01cos⁡t],[μ11(t)μ12(t)μ13(t)μ21(t)μ22(t)μ23(t)μ31(t)μ32(t)μ33(t)]=[0.02|sint|0.01|sin2t|0.02|cos⁡t|0.01|cos⁡2t|0.01|sint|0.01|cos⁡t|0.02|sin2t|0.01|cos⁡t|0.02|sin2t|].Let r=s=1, A=1/2, and δ=0.1. Obviously, L=l=2, f(0)=2 and g(0)=0, Kij0=1/4, Jij0=1/2,
(72)[a11Ma12Ma13Ma21Ma22Ma23Ma31Ma32Ma33M]=[0.90.80.90.90.90.90.90.80.9],[a11ma12ma13ma21ma22ma23ma31ma32ma33m]=[0.80.70.70.70.80.80.70.70.8],[c11¯c12¯c13¯c21¯c22¯c23¯c31¯c32¯c33¯]=[0.010.020.010.020.010.010.010.010.02],[d11¯d12¯d13¯d21¯d22¯d23¯d31¯d32¯d33¯]=[0.010.020.010.010.010.010.010.010.02],[I11¯I12¯I13¯I21¯I22¯I23¯I31¯I32¯I33¯]=[0.110.030.020.010.020.060.010.030.01],[μ11Mμ12Mμ13Mμ21Mμ22Mμ23Mμ31Mμ32Mμ33M]=[0.020.010.020.010.010.010.020.010.02].
So
(73)[θ110θ120θ130θ210θ220θ230θ310θ320θ330]=[0.1280.1580.1180.1590.2290.1690.1080.1580.118],[θ11θ12θ13θ21θ22θ23θ31θ32θ33]=[0.2080.2880.1930.2690.3890.2890.1330.2680.193].
It is easy to check that(74)A≥∥φ0∥,max⁡1≤i≤n1≤j≤m⁡{θij0aijm,(1+aijMaijm)θij0}≤12,(75)max⁡1≤i≤n1≤j≤m{θijaijm,(1+aijMaijm)θij}<1.
Therefore, all the conditions in Theorems 16 and 18 are verified. Hence (66) has a 2π-periodic solution, which is globally exponentially stable(we give numerical simulations in Figures 4, 5, and 6 by taking x11, x12, and x13 to illustrate our results).Figure 4
Phase response of state variablesx11, x12 in Example 2.Figure 5
Phase response of state variablesx11, x13 in Example 2.Figure 6
Phase response of state variablesx12, x13 in Example 2.Remark. In [27], authors studied the existence and stability of antiperiodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales by using a continuation theorem of coincidence degree theory and constructing a suitable Lyapunov functions, and in [28], authors investigated the existence and stability of almost periodic solutions to shunting inhibitory cellular neural networks on time scales by the exponential dichotomy of linear dynamic equations on time scales and constructing a suitable Lyapunov functions. However, our methods used in this paper are different from those used in [27, 28]; also, the results obtained in [27, 28] cannot be applied to our examples here.



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