# Existence of Periodic Solutions for Shunting Inhibitory Cellular Neural Networks with Neutral Delays

Authors: Ninghua Chen
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

This paper considers the existence of periodic solutions for shunting inhibitory cellular neural networks (SICNNs) with neutral delays. By applying the theory of abstract continuation theorem ofk-set contractive operator and some analysis technique, a new result on the existence of periodic solutions is obtained.


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## Body

## 1. Introduction

It is well known that SICNNs have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. The dynamical behaviors of SICNNs with delays have been widely investigated in recent years. A large number of important results on the dynamical behaviors of SICNNs have been established and successfully applied to signal processing, pattern recognition, associative memories, and so on. In particular, there exist many results on the existence and stability of periodic and almost periodic solutions for SICNNs with delays. We refer the reader to [1–4] and references cited therein.On the other hand, the stability analysis of various neutral delay-differential systems has drawn much research attention [5–7]. The theory of neutral delay-differential systems is of both theoretical and practical interest. For a large class of electrical networks containing lossless transmission lines, the describing equations can be reduced to neutral delay-differential equations; such networks arise in high speed computers where nearly lossless transmission lines are used to interconnect switching circuits. Also, the neutral systems often appear in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar.Motivated by the above, we consider shunting inhibitory cellular neural networks with neutral delays described by(1)x˙ij(t)=-aij(t)xij(t)-∑Ckl∈Nr(i,j)‍Cijkl(t)fij(xkl(t-τkl(t)))xij(t)+bij(t)x˙ij(t-σij(t))+Iij(t),
where i=1,2,…,m,j=1,2,…,n, Cij is the cell at the (i,j) position of the lattice and the r-neighborhood Nr(i,j) of Cij is given as
(2)Nr(i,j)={Ckl:max(|k-i|,|l-j|)000≤r,1≤k≤m,1≤l≤n}.xij(t) is the activity of the cell, Cij, aij(t)≥0 is the passive decay rate of the cell activity, Iij(t) is the external input to Cij, Cijkl≥0 is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell Cij, and the activity function fij(xkl(t)) is a continuous function representing the output or firing rate of cell Ckl.By using the Lyapunov functions and linear matrix inequality approach (LMI), most authors studied the asymptotic stability or exponential stability of the equilibrium point (see [7–10]). However, few papers have been published on the existence of periodic solutions or almost periodic solutions for neutral type neural networks ([11–14]). Thus, it is worthwhile to study the existence of periodic solutions for neutral type neural networks with delays. And to the author’s best knowledge, there is no published paper considering the existence of periodic solutions for SICNNs with neutral delays.The initial conditions of (1) are of the form
(3)xij(t)=φij(s),s∈(-∞,0],0000i=1,…,m,j=1,…,n,
where φij∈C((-∞,0],R).For the sake of convenience, we denote(4)Ω1={1,2,…,m},Ω2={1,2,…,n},h¯=maxt∈[0,ω]|h(t)|,h_=mint∈[0,ω]|h(t)|,h~=1ω∫0ω‍h(s)ds,∥h∥2=(∫0ω‍|h(t)|2dt)1/2,∥h˙∥2=(∫0ω‍|h˙(t)|2dt)1/2,
where h is a continuous ω-periodic function.The main purpose of this paper is to obtain some sufficient conditions for the existence of the periodic solutions for system (1). By applying the theory of abstract continuation theorem of k-set contractive operator and some analysis technique, we derive some new sufficient conditions ensuring the existence of periodic solutions for system (1). Moreover, numerical examples are provided to illustrate the feasibility of the new results.


## 2. Conditions and Lemma

Throughout this paper, fori∈Ω1,j∈Ω2, we assume that(H1)
aij(t),Cijkl(t),bij(t),Iij(t),τij(t) are continuous ω-periodic functions;(H2)
functionσij(t) is nonnegative, bounded, ω-periodic, and continuously differential defined on R+ and inft∈R+{1-σ˙ij(t)}>0, where σ˙ij(t) expresses the derivative of σij(t);(H3)
there exists positive constantαij such that |fij(u)|≤αij, for all u,v∈R.In order to study (1), we should make some preparations.LetE be a Banach space. For a bounded subset A⊂E, let
(5)αE(A)=inf{⋃i‍δ>0:thereisafinitenumberofsubsets00000000Ai⊂AsuchthatA=⋃i‍Aianddiam(Ai)≤δ}
denote the (Kuratowski) measure of noncompactness, where diam (Ai) denotes the diameter of the set Ai. Let X, Y be two Banach spaces and Ω be a bounded open subset of X. A continuous and bounded map N:Ω¯→Y is called k-set contractive if for any bounded set S⊂Ω we have αY(N(S))≤kαX(S), where k is a constant. In addition, for a Fredholm operator L:X→Y with index zero, according to [15], we define
(6)l(L)=sup{a≥0:aαX(S)≤αY(L(S)),000000∀boundedsubsetS⊂Xa≥0:aαX(S)≤αY(L(S))}.Lemma 1 (see [16]).
LetL:X→Y be a Fredholm operator with index zero, and let r∈Y be a fixed point. Suppose that N:Ω→Y is a k-set contractive with k<l(L), where Ω⊂X is bounded, open, and symmetric about 0∈Ω. Furthermore, we suppose that(1)
Lx≠λNx+λr for x∈∂Ω,λ∈(0,1),(2)
[QN(x)+Qr,x]·[QN(-x)+Qr,x]<0 for x∈∂Ω∩KerL, where [·,·] is some bilinear form on Y×X and Q is the project of Y onto Coker  (L).
Then there is ax∈Ω¯ such that Lx-Nx=r.In order to use Lemma1 to establish the existence of periodic solutions of (1), we set
(7)Y=Cω={xin,…,xm1…,xmn)Tx∈C(R,R):x(t+ω)=x(t),00i0x=(x11,…,x1n,…,xi1,…,00000000xin,…,xm1,…,xmn)T}
with the norm defined by ∥x∥=∑i=1m∑j=1n|xij|0, where |xij|0=maxt∈[0,ω]|xij(t)| and
(8)X=Cω1={xin,…,xm1…,xmn)Tx∈C1(R,R):x(t+ω)=x(t),0i00x=(x11,…,x1n,…,xi1,…,00000000xin,…,xm1,…,xmn)T}
with the norm defined by ∥x∥1=max{∥x∥,∥x˙∥}, where ∥x˙∥=∑i=1m∑j=1n|x˙ij|0, |x˙ij|0=maxt∈[0,ω]|x˙ij(t)|. Then X, Y are Banach spaces.LetL:X→Y  defined by
(9)Lx=dxdt=(x˙11,…,x˙1n,…,x˙i1,…,00000000x˙in,…,x˙m1,…,x˙mn)T
and N:X→Y defined by the following:
(10)Nxij=-aij(t)xij(t)-∑Ckl∈Nr(i,j)‍Cijkl(t)fij(xkl(t-τkl(t)))xij(t)+bij(t)x˙ij(t-σij(t)),
for i∈Ω1,j∈Ω2.It is easy to see thatL is a Fredholm operator with index zero. Clearly, (1) has a ω-periodic solution if and only if Lx=Nx+r for some x∈X, where
(11)r≔I(t)=(I11(t),…,I1n(t),…,Ii1(t),…,000000000Iin(t),…,Im1(t),…,Imn(t))T.Lemma 2 (see [17]).
The differential operatorL is a Fredholm operator with index zero and satisfies l(L)≥1.Lemma 3.
Ifk=maxi∈Ω1,j∈Ω2{b-ij}<1, then N:Ω¯→Y is a k-set contractive map, where Ω⊂X.Proof.
The proof of Lemma3 is similar to Lemma  3 in [14] and will be omitted here.Lemma 4 (see [13]).
Lett1,t2∈[0,ω]. If x:[0,ω]→R is ω-periodic, then
(12)x(t)≤x(t1)+∫0ω‍|x˙(s)|ds,x(t)≥x(t2)-∫0ω‍|x˙(s)|ds.Lemma 5 (see [18]).
Suppose thatσ∈X and σ˙(t)<1 for all t∈[0,ω], then the function t-σ(t) has an inverse ν∈C(R,R) with ν(a+ω)=ν(a)+ω, for all a∈R.Throughout this paper, we assume thatσij∈X and σ˙ij(t)<1,i∈Ω1,j∈Ω2. So that σij(t) has a unique inverse, and we set νij(t) to represent the inverse of function t-σij(t).


## 3. Main Results

Set(13)pij2=maxt∈R{11-σ˙ij(νij(t))},A=diag(a_11,…,a_1n,…,a_m1,…,a_mn),B=diag(b¯11p11,…,b¯1np1n,…,b¯m1pm1,…,b¯mnpmn),D=diag(1-b¯11p11,…,1-b¯1np1n,…,00000001-b¯m1pm1,…,1-b¯mnpmn),C=diag(∑Ckl∈Nr(i,j)‍C¯11klα11,…,∑Ckl∈Nr(i,j)‍C¯1nklα1n,…,00000000∑Ckl∈Nr(i,j)‍C¯m1klαm1,…,∑Ckl∈Nr(i,j)‍C¯mnklαmn),I=(∥I11∥2,…,∥I1n∥2,…,∥Im1∥2,…,∥Imn∥2)T,Z=(∥x11∥2,…,∥x1n∥2,…,∥xm1∥2,…,∥xmn∥2)T,W=(∥x˙11∥2,…,∥x˙1n∥2,…,∥x˙m1∥2,…,∥x˙mn∥2)T.Theorem 6.
Suppose that(H1)–(H3) hold, furthermore, assume that(H4)
mini∈Ω1,j∈Ω2{1-b¯ijpij,1-b¯ij}>0 and maxi∈Ω1,j∈Ω2{a~ij-∑Ckl∈Nr(i,j)C~ijklαij}>0,(H5)
A-C-BD-1(A+C) is nonsingular M-matrix, then system (1) has at least one ω-periodic solution.Proof.
We consider the operator equationLx=λNx+λr, λ∈(0,1), where L,N defined by (9) and (10), respectively. Corresponding to system (1), for i∈Ω1,j∈Ω2, we have
(14)x˙ij(t)=λ[∑Ckl∈Nr(i,j)‍-aij(t)xij(t)0000i0-∑Ckl∈Nr(i,j)‍Cijkl(t)fij(xkl(t-τkl(t)))xij(t)0000i0+bij(t)x˙ij(t-σij(t))+Iij(t)∑Ckl∈Nr(i,j)‍].
Suppose that (x11(t),…,x1n(t),…,xm1(t),…,xmn(t))T is a solution of system (14) for a parameter λ∈(0,1). Multiplying both sides of system (14) by xij(t) and integrating over [0,ω], from (H3) and Cauchy-Schwarz inequality, we obtain
(15)a_ij∥xij∥22≤∫0ωaij(t)xij2(t)dt=∫0ω[-∑Ckl∈Nr(i,j)‍Cijkl(t)fij(xkl(t-τkl(t)))xij(t)00000000000+bij(t)x˙ij(t-σij(t))+Iij(t)∑Ckl∈Nr(i,j)‍]xij(t)dt≤∑Ckl∈Nr(i,j)‍C¯ijklαij∫0ωxij2(t)dt+b¯ij∫0ω|xij(t)||x˙ij(t-σij(t))|dt+∫0ω|xij(t)||Iij(t)|dt≤∑Ckl∈Nr(i,j)‍C¯ijklαij∥xij∥22+b¯ij∥xij∥2(∫0ω|x˙ij(t-σij(t))|2dt)1/2+(∫0ω|Iij(t)|2dt)1/2(∫0ω|xij(t)|2dt)1/2,
and according to Lemma 5, we have
(16)(∫0ω|x˙ij(t-σij(t))|2dt)1/2=(∫0ω11-σ˙ij(νij(s))|x˙ij(s)|2ds)1/2≤pij∥x˙ij∥2,i∈Ω1,j∈Ω2.
That is,
(17)a_ij∥xij∥2≤∑Ckl∈Nr(i,j)‍C¯ijklαij∥xij∥2+b¯ijpij∥x˙ij∥2+∥Iij∥2,i∈Ω1,j∈Ω2.
From the above inequality, we obtain
(18)(a_ij-∑Ckl∈Nr(i,j)‍C¯ijklαij)∥xij∥2≤b¯ijpij∥x˙ij∥2+∥Iij∥2,00000i0i∈Ω1,j∈Ω2,
which gives
(19)(A-C)Z≤BW+I.
Multiplying both sides of (14) by x˙ij(t) and integrating over [0,ω], for i∈Ω1,j∈Ω2, we have
(20)∥x˙ij∥22=λ∫0ωx˙ij(t)[∑Ckl∈Nr(i,j)‍-aij(t)xij(t)00000000000000000-∑Ckl∈Nr(i,j)‍Cijkl(t)fij(xkl(t-τkl(t)))00000000000000000000000000×xij(t)+bij(t)x˙ij0000000000000000×(t-σij(t))+Iij(t)∑Ckl∈Nr(i,j)‍]dt≤(a¯ij+∑Ckl∈Nr(i,j)‍C¯ijklαij)(∫0ω|x˙ij(t)|2dt)1/2×(∫0ω|xij(t)|2dt)1/2+b¯ij(∫0ω|x˙ij(t)|2dt)1/2×(∫0ω|x˙ij(t-σij(t))|2dt)1/2+(∫0ω|Iij(t)|2dt)1/2(∫0ω|x˙ij(t)|2dt)1/2≤(a¯ij+∑Ckl∈Nr(i,j)‍C¯ijklαij)∥xij∥2∥x˙ij∥2+b¯ijpij∥x˙ij∥22+∥x˙ij∥2∥Iij∥2.
Thus
(21)∥x˙ij∥2≤(a¯ij+∑Ckl∈Nr(i,j)‍C¯ijklαij)∥xij∥2+b¯ijpij∥x˙ij∥2+∥Iij∥2,
that is,
(22)(1-b¯ijpij)∥x˙ij∥2≤(a¯ij+∑Ckl∈Nr(i,j)‍C¯ijklαij)∥xij∥2+∥Iij∥2,
where 1-b¯ijpij>0,i∈Ω1,j∈Ω2. Then we may rewrite (22) as
(23)DW≤(A+C)Z+I⟹W≤D-1(A+C)Z+D-1I.
Substituting (23) into (19), we get
(24)(A-C)Z≤B(D-1(A+C)Z+D-1I)+I⟹(A-C-BD-1(A+C))Z≤BD-1I+I.
By (H5), we have Z≤H+≜(h11+,…,h1n+,…,hm1+,…,hmn+)T. That is,
(25)∥xij∥2≤hij+,i∈Ω1,j∈Ω2,
where H+=(A-C-BD-1(A+C))-1(BD-1I+I). Substituting (25) into (23), we have W≤G+≜(g11+,…,g1n+,…,gm1+,…,gmn+)T. That is,
(26)∥x˙ij∥2≤gij+,i∈Ω1,j∈Ω2,
where G+=D-1(A+C)H++D-1I.
Integrating both sides of (14), from 0 to ω, we obtain
(27)0=-∫0ωaij(t)xij(t)dt+∫0ω[-∑Ckl∈Nr(i,j)‍Cijkl(t)fij(xkl(t-τkl(t)))xij(t)000000000+bij(t)x˙ij(t-σij(t))+Iij(t)∑Ckl∈Nr(i,j)‍]dt.
Using Cauchy-Schwarz inequality, we have
(28)|∫0ωaij(t)xij(t)dt|≤∑Ckl∈Nr(i,j)‍∫0ωC¯ijklαij|xij(t)|dt+∫0ωb¯ij|x˙ij(t-σij(t))|dt+∫0ω|Iij(t)|dt≤∑Ckl∈Nr(i,j)‍C¯ijklαij∥xij∥2ω+b¯ijpij∥x˙ij∥2ω+∥Iij∥2ω.
In view of (14), we have
(29)∫0ω|x˙ij(t)|dt≤∫0ωa¯ij|xij(t)|dt+∑Ckl∈Nr(i,j)‍∫0ωC¯ijklαij|xij(t)|dt+∫0ωb¯ij|x˙ij(t-σij(t))|dt+∫0ω|Iij(t)|dt≤a¯ij∥xij∥2ω+∑Ckl∈Nr(i,j)‍C¯ijklαij∥xij∥2ω+b¯ijpij∥x˙ij∥2ω+∥Iij∥2ω.
From Lemma 4, for any t1ij,t2ij∈[0,ω], i∈Ω1,j∈Ω2, we have
(30)∫0ωaij(t)xij(t)dt≤∫0ωaij(t)xij(t1ij)dt+∫0ωaij(t)(∫0ω‍|x˙ij(t)|dt)dt,∫0ωaij(t)xij(t)dt≥∫0ωaij(t)xij(t2ij)dt-∫0ωaij(t)(∫0ω‍|x˙ij(t)|dt)dt.
Dividing by ∫0ωaij(t)dt on the two sides of the above inequalities, we obtain that
(31)xij(t1ij)≥1∫0ωaij(t)dt∫0ωaij(t)xij(t)dt-∫0ω‍|x˙ij(t)|dt,xij(t2ij)≤1∫0ωaij(t)dt∫0ωaij(t)xij(t)dt+∫0ω‍|x˙ij(t)|dt.
Let t¯ij,t_ij∈[0,ω],i∈Ω1,j∈Ω2, such that xij(t¯ij)=maxt∈[0,ω]xij(t),xij(t_ij)=mint∈[0,ω]xij(t), then by (25)–(31), we have
(32)xij(t¯ij)≤1a_ijω∫0ω|aij(t)xij(t)|dt+∫0ω‍|x˙ij(t)|dt≤1a_ijω(∑Ckl∈Nr(i,j)‍C¯ijklαij∥xij∥2000000000+b¯ijpij∥x˙ij∥2+∥Iij∥2∑Ckl∈Nr(i,j)‍)+ω[(a¯ij+∑Ckl∈Nr(i,j)‍C¯ijklαij)∥xij∥200000000+b¯ijpij∥x˙ij∥2+∥Iij∥2(a¯ij+∑Ckl∈Nr(i,j)‍C¯ijklαij)]≤1a_ijω(∑Ckl∈Nr(i,j)‍C¯ijklαijhij+0000000000+b¯ijpijgij++∥Iij∥2∑Ckl∈Nr(i,j)‍)+ω[(a¯ij+∑Ckl∈Nr(i,j)‍C¯ijklαij)hij+000000000+b¯ijpijgij++∥Iij∥2(a¯ij+∑Ckl∈Nr(i,j)‍C¯ijklαij)]≜Mij(1).
Similarly,(33)xij(t_ij)≥-[1a_ijω∫0ω|aij(t)xij(t)|dt+∫0ω‍|x˙ij(t)|dt]≥-{1a_ijω(∑Ckl∈Nr(i,j)‍C¯ijklαijhij+0000000000000+b¯ijpijgij++∥Iij∥2∑Ckl∈Nr(i,j)‍)00000+ω[(a¯ij+∑Ckl∈Nr(i,j)‍C¯ijklαij)hij+00000000000+b¯ijpijgij++∥Iij∥2∑Ckl∈Nr(i,j)‍](∑Ckl∈Nr(i,j)‍C¯ijklαijhij+}=-Mij(1).
Thus, for alli∈Ω1,j∈Ω2, |xij(t)|≤Mij(1), which yields that |xij|0≤Mij(1). Again from (14), we get
(34)|x˙ij|0≤a¯ij|xij|0+∑Ckl∈Nr(i,j)‍C¯ijklαij|xij|0+b¯ij|x˙ij|0+I¯ij≤(a¯ij+∑Ckl∈Nr(i,j)‍C¯ijklαij)Mij(1)+b¯ij|x˙ij|0+I¯ij,
which gives |x˙ij|0≤((a¯ij+∑Ckl∈Nr(i,j)C¯ijklαij)Mij(1)+I¯ij)/(1-b¯ij)≜Mij(2).
DenoteM(1)=∑i=1m∑j=1nMij(1), M(2)=∑i=1m∑j=1nMij(2), M(3)=maxi∈Ω1,j∈Ω2{|I~ij|/(a~ij-∑Ckl∈Nr(i,j)C~ijklαij)}. Let M>max{M(1),M(2),M(3)}, Ω={x∈Cω1:∥x∥1<M}, and define a bounded bilinear form [·,·] on Cω×Cω1 by [y,x]=∫0ωy(t)x(t)dt. Also we define Q:y→Coker(L) by y→∫0ωy(t)dt. Obviously, {x∣x∈KerL∩∂Ω}={x∣x≡Morx≡-M}. Without loss of generality, we may assume that x≡M. Let Pij=-a~ijM-∑Ckl∈Nr(i,j)C~ijklfij(M)M+I~ij,Qij=a~ijM+∑Ckl∈Nr(i,j)C~ijklfij(-M)M+I~ij, i∈Ω1,j∈Ω2. Thus
(35)[QN(x)+Q(r),x][QN(-x)+Q(r),x]=M2ω2·(P11Q11⋮PmnQmn).
Considering assumption (H4), we know M>maxi∈Ω1,j∈Ω2{|I~ij|/(a~ij-∑Ckl∈Nr(i,j)C~ijklαij)}>0, and it follows that a~ijM>∑Ckl∈Nr(i,j)C~ijklαijM+I~ij>∑Ckl∈Nr(i,j)C~ijkl|fij(M)|M+I~ij. So we have [QN(xij)+Q(rij),xij][QN(-xij)+Q(rij),xij]<0, for all i∈Ω1,j∈Ω2.
Therefore, by using Lemma1, we obtain that (1) has at least one ω-periodic solution. The proof is complete.


## 4.  Examples

In this section, we give two examples to demonstrate the results obtained in previous sections.Consider the following shunting inhibitory cellular neural networks with neutral delays:(36)x˙ij(t)=-aij(t)xij(t)-∑Ckl∈Nr(i,j)‍Cijkl(t)fij(xkl(t-τkl(t)))xij(t)+bij(t)x˙ij(t-σij(t))+Iij(t),
where i=1,…,m,j=1,…,n.Example 1.
Letm=n=2, fij(u)=tanhu=(eu-e-u)/(eu+e-u). We take
(37)(aij)3×3=(4+0.2sin3t5+0.1cost6+sin2t5-cost),(Cij)3×3=(0.3-0.1sint1+0.2cost0.6|sint|0.3-0.2sint),(bij)3×3=(0.03sint0.10.150.1sin2t),(Iij)3×3=(1.6cost+sint3tant20.5),(τij)3×3=(sintcost0.25cost7sintsin0.1t),(σij)3×3=(1+sin0.5t0.8cost0.3sintsin0.2t).
Obviously, (H1)–(H3) hold and
(38)∑Ckl∈N1(1,1)‍C11kl¯=∑Ckl∈N1(1,2)‍C12kl¯=∑Ckl∈N1(2,1)‍C21kl¯=∑Ckl∈N1(2,2)‍C22kl¯=2.7,p11=1.414, p12=2.236, p21=1.195, and p22=1.118.
Letαij=1, we have A=diag(3.8,4.9,5,4),
(39)B=diag(0.042,0.224,0.179,0.112),C=diag(2.7,2.7,2.7,2.7),D=diag(0.958,0.776,0.821,0.888),A-C-BD-1(A+C)=diag(0.815,0.006,0.621,0.455).
It is easy to see that (H4) and (H5) hold. Therefore, all conditions of Theorem 6 are satisfied, then system (36) has at least one 2π-periodic solution.Example 2.
Letm=n=3, fij(u)=(1/2)sinu. We take
(40)(aij)3×3=(4+sin3t5+cost6-sint5+sin3t6+sin2t4+sint6+cos5t4+sint5+cost),(Cij)3×3=(0.1|cos3t|0.2|cost|0.1|sint|0.2|sin2t|0.3|sint|0.4|cost|0.1|cos5t|0.1|sint|0.2|sint|),(bij)3×3=(0.01sint0.02cos5t0.01sin2t0.02sin3t0.03sin2t0.01sin5t0.03cos3t0.01sint0.02cos3t),(Iij)3×3=(2+sintcostsintsintcost1+costcostsin2tsint+cost),(τij)3×3=(0.1sint0.5cost0.6sint0.8sint0.1sint0.4sint0.5cost0.15sint0.55cost),(σij)3×3=(0.6cost0.2cost0.15sint0.2sint0.75sint0.4cos2t0.8cost0.9sint0.75sint).
Obviously, (H1)–(H3) hold and
(41)∑Ckl∈N1(1,1)‍C11kl¯=0.8,∑Ckl∈N1(1,2)‍C12kl¯=1.3,∑Ckl∈N1(1,3)‍C13kl¯=1,∑Ckl∈N1(2,1)‍C21kl¯=1,∑Ckl∈N1(2,2)‍C22kl¯=1.7,∑Ckl∈N1(2,3)‍C23kl¯=1.3,∑Ckl∈N1(3,1)‍C31kl¯=0.7,∑Ckl∈N1(3,2)‍C32kl¯=1.3,∑Ckl∈N1(3,3)‍C33kl¯=1,p11=1.581, p12=1.118, p13=1.085, p21=1.118, p22=2, p23=2.236, p31=2.236, p32=3.162, and p33=2.
Letαij=1/2, we have A=diag(3,4,5,4,5,3,5,3,4),
(42)B=diag(0.016,0.022,0.011,0.022,0.06,00000000.022,0.067,0.032,0.04),C=diag(0.4,0.65,0.5,0.5,0.85,0.65,0.35,0.65,0.5),D=diag(0.984,0.978,0.989,0.978,0.94,00000000.978,0.933,0.968,0.96),A-C-BD-1(A+C)=diag(2.545,3.245,4.439,3.399,3.777,000000000002.268,4.266,2.229,3.313).
It is easy to see that (H4) and (H5) hold. Therefore, all conditions of Theorem 6 are satisfied, then system (36) has at least one 2π-periodic solution.


## 5. Conclusion

In this paper, a class of neutral type shunting inhibitory cellular neural networks is investigated by means of an abstract continuous theorem ofk-set contractive operator and some analysis technique. Sufficient conditions are derived to ensure the existence of periodic solutions. To the best of our knowledge, the results presented here have not appeared in the related literature. Also, illustrative examples are given to demonstrate the effectiveness of the results obtained.



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*Source: 143706-2013-10-27.xml*