# Asymptotic Stability of Impulsive Cellular Neural Networks with Infinite Delays via Fixed Point Theory

Authors: Yutian Zhang; Yuanhong Guan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We employ the new method of fixed point theory to study the stability of a class of impulsive cellular neural networks with infinite delays. Some novel and concise sufficient conditions are presented ensuring the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium at the same time. These conditions are easily checked and do not require the boundedness and differentiability of delays.


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## Body

## 1. Introduction

Cellular neural networks (CNNs), proposed by Chua and Yang in 1988 [1, 2], have become a hot topic for their numerous successful applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision.Due to the finite switching speed of neurons and amplifiers in the implementation of neural networks, it turns out that the time delays should not be neglected, and therefore, the model of delayed cellular neural networks (DCNNs) is put forward, which is naturally of better realistic significances. In fact, besides delay effects, stochastic and impulsive as well as diffusing effects are also likely to exist in neural networks. Accordingly many experts are showing a growing interest in the research on the dynamic behaviors of complex CNNs such as impulsive delayed reaction-diffusion CNNs and stochastic delayed reaction-diffusion CNNs, with a result of many achievements [3–9] obtained.Synthesizing the reported results about complex CNNs, we find that the existing research methods for dealing with stability are mainly based on Lyapunov theory. However, we also notice that there are still lots of difficulties in the applications of corresponding results to specific problems; correspondingly it is necessary to seek some new techniques to overcome those difficulties.Encouragingly, in recent few years, Burton and other authors have applied the fixed point theory to investigate the stability of deterministic systems and obtained some more applicable results; for example, see the monograph [10] and papers [11–22]. In addition, more recently, there have been a few publications where the fixed point theory is employed to deal with the stability of stochastic (delayed) differential equations; see [23–29]. Particularly, in [24–26], Luo used the fixed point theory to study the exponential stability of mild solutions to stochastic partial differential equations with bounded delays and with infinite delays. In [27, 28], Sakthivel used the fixed point theory to investigate the asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with bounded delays and with infinite delays. In [29], Luo used the fixed point theory to study the exponential stability of stochastic Volterra-Levin equations.Naturally, for complex CNNs which have high application values, we wonder if we can utilize the fixed point theory to investigate their stability, not just the existence and uniqueness of solution. With this motivation, in the present paper, we aim to discuss the stability of impulsive CNNs with infinite delays via the fixed point theory. It is worth noting that our research skill is the contraction mapping theory which is different from the usual method of Lyapunov theory. We employ the fixed point theorem to prove the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium all at once. Some new and concise algebraic criteria are provided, and these conditions are easy to verify and, moreover, do not require the boundedness and differentiability of delays.


## 2. Preliminaries

LetRn denote the n-dimensional Euclidean space and let∥·∥ represent the Euclidean norm. 𝒩≜{1,2,…,n}. R+=[0,∞). C[X,Y] corresponds to the space of continuous mappings from the topological space X to the topological space Y.In this paper, we consider the following impulsive cellular neural network with infinite delays:(1)dxi(t)dt=-aixi(t)+∑j=1nbijfj(xj(t))+∑j=1ncijgj(xj(t-τj(t))),t≥0,t≠tk,(2)Δxi(tk)=xi(tk+0)-xi(tk)=Iik(xi(tk)),k=1,2,…,
where i∈𝒩 and n is the number of neurons in the neural network. xi(t) corresponds to the state of the ith neuron at time t. fj(·),gj(·)∈C[R,R] denote the activation functions, respectively.  τj(t)∈C[R+,R+] corresponds to the known transmission delay satisfying τj(t)→∞ and t-τj(t)→∞ as t→∞. Denote ϑ=inf{t-τj(t),t≥0,j∈𝒩}. The constant bij represents the connection weight of the jth neuron on the ith neuron at time t. The constant cij denotes the connection strength of the jth neuron on the ith neuron at time t-τj(t). The constant ai>0 represents the rate with which the ith neuron will reset its potential to the resting state when disconnected from the network and external inputs. The fixed impulsive moments tk (k=1,2,…) satisfy 0=t0<t1<t2<⋯ and limk→∞tk=∞. xi(tk+0) and xi(tk-0) stand for the right-hand and left-hand limits of xi(t) at time tk, respectively. Iik(xi(tk)) shows the abrupt change of xi(t) at the impulsive moment tk and Iik(·)∈C[R,R].Throughout this paper, we always assume thatfi(0)=gi(0)=Iik(0)=0 for i∈𝒩 and k=1,2,…. Thereby, problem (1) and (2) admits a trivial equilibrium x=0.Denote byx(t)≜x(t;s,φ)=(x1(t;s,φ1),…,xn(t;s,φn))T∈Rn the solution to (1) and (2) with the initial condition
(3)xi(s)=φi(s),ϑ≤s≤0,i∈𝒩,
where φ(s)=(φ1(s),…,φn(s))T∈Rn and φi(s)∈C[[ϑ,0],R]. Denote |φ|=sups∈[ϑ,0]∥φ(s)∥.The solutionx(t)≜x(t;s,φ)∈Rn of (1)–(3) is, for the time variable t, a piecewise continuous vector-valued function with the first kind discontinuity at the points tk (k=1,2,…), where it is left continuous; that is, the following relations are valid:
(4)xi(tk-0)=xi(tk),xi(tk+0)=xi(tk)+Iik(xi(tk)),i∈𝒩,k=1,2,….Definition 1.
The trivial equilibriumx=0 is said to be stable, if, for any ε>0, there exists δ>0 such that for any initial condition φ(s)∈C[[ϑ,0],Rn] satisfying|φ|<δ:
(5)∥x(t;s,φ)∥<ε,t≥0.Definition 2.
The trivial equilibriumx=0 is said to be asymptotically stable if the trivial equilibrium x=0 is stable, and for any initial condition φ(s)∈C[[ϑ,0],Rn], limt→∞∥x(t;s,φ)∥=0 holds.The consideration of this paper is based on the following fixed point theorem.Theorem 3 (see [30]).
LetΥ be a contraction operator on a complete metric space Θ, then there exists a unique point ζ∈Θ for which Υ(ζ)=ζ.


## 3. Main Results

In this section, we will consider the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium by means of the contraction mapping principle. Before proceeding, we introduce some assumptions listed as follows.(A1)
There exist nonnegative constantslj such that, for any η,υ∈R,
(6)|fj(η)-fj(υ)|≤lj|η-υ|,j∈𝒩.(A2)
There exist nonnegative constantskj such that, for any η,υ∈R,
(7)|gj(η)-gj(υ)|≤kj|η-υ|,j∈𝒩.(A3)
There exist nonnegative constantspjk such that, for any η,υ∈R,
(8)|Ijk(η)-Ijk(υ)|≤pjk|η-υ|,j∈𝒩,k=1,2,….Letℋ=ℋ1×⋯×ℋn, and let ℋi (i∈𝒩) be the space consisting of functions ϕi(t):[ϑ,∞)→R, where ϕi(t) satisfies the following: (1)
ϕi(t) is continuous on t≠tk (k=1,2,…);(2)
limt→tk-ϕi(t) and limt→tk+ϕi(t) exist; furthermore, limt→tk-ϕi(t)=ϕi(tk) for k=1,2,…;(3)
ϕi(s)=φi(s) on s∈[ϑ,0];(4)
ϕi(t)→0 as t→∞;here tk (k=1,2,…) and φi(s) (s∈[ϑ,0]) are defined as shown in Section 2. Also ℋ is a complete metric space when it is equipped with the following metric:
(9)d(q¯(t),h¯(t))=∑i=1nsupt≥ϑ|qi(t)-hi(t)|,
where q¯(t)=(q1(t),…,qn(t))∈ℋ and h¯(t)=(h1(t),…,hn(t))∈ℋ.In what follows, we will give the main result of this paper.Theorem 4.
Assume that conditions (A1)–(A3) hold. Provided that(i)
there exists a constantμ such that infk=1,2,…{tk-tk-1}≥μ,(ii)
there exist constantspi such that pik≤piμ for i∈𝒩 and k=1,2,…,(iii)
λ*≜∑i=1n{(1/ai)maxj∈𝒩|bijlj|+(1/ai)maxj∈𝒩|cijkj|}+maxi∈𝒩{pi(μ+(1/ai))}<1,(iv)
maxi∈𝒩{λi}<1/n, where λi=(1/ai)∑j=1n|bijlj|+(1/ai)∑j=1n|cijkj|+pi(μ+(1/ai)),then the trivial equilibrium x=0 is asymptotically stable.Proof.
Multiplying both sides of (1) with eait gives, for t>0 and t≠tk,
(10)deaitxi(t)=eaitdxi(t)+aixi(t)eaitdt=eait{∑j=1nbijfj(xj(t))+∑j=1ncijgj(xj(t-τj(t)))}dt,
which yields after integrating from tk-1+ε (ε>0) to t∈(tk-1,tk) (k=1,2,…)
(11)xi(t)eait=xi(tk-1+ε)eai(tk-1+ε)+∫tk-1+εteais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τj(s)))}ds.
Lettingε→0 in (11), we have
(12)xi(t)eait=xi(tk-1+0)eaitk-1+∫tk-1teais×{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τj(s)))}ds,
for t∈(tk-1,tk) (k=1,2,…). Setting t=tk-ε (ε>0) in (12), we get
(13)xi(tk-ε)eai(tk-ε)=xi(tk-1+0)eaitk-1+∫tk-1tk-εeais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τj(s)))}ds,
which generates by letting ε→0(14)xi(tk-0)eaitk=xi(tk-1+0)eaitk-1+∫tk-1tkeais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τj(s)))}ds.
Notingxi(tk-0)=xi(tk), (14) can be rearranged as
(15)xi(tk)eaitk=xi(tk-1+0)eaitk-1+∫tk-1tkeais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τj(s)))}ds.
Combining (12) and (15), we reach that
(16)xi(t)eait=xi(tk-1+0)eaitk-1+∫tk-1teais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τj(s)))}ds
is true for t∈(tk-1,tk] (k=1,2,…). Further,
(17)xi(t)eait=xi(tk-1)eaitk-1+∫tk-1teais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τj(s)))}ds+Ii(k-1)(xi(tk-1))eaitk-1
holds for t∈(tk-1,tk] (k=1,2,⋯). Hence,
(18)xi(tk-1)eaitk-1=xi(tk-2)eaitk-2+∫tk-2tk-1eais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τj(s)))}ds+Ii(k-2)(xi(tk-2))eaitk-2,⋮xi(t2)eait2=xi(t1)eait1+∫t1t2eais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τj(s)))}ds+Ii1(xi(t1))eait1,xi(t1)eait1=φi(0)+∫0t1eais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τj(s)))}ds,
which produces, for t>0,
(19)xi(t)=φi(0)e-ait+e-ait∫0teais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τj(s)))}ds+e-ait∑0<tk<t{Iik(xi(tk))eaitk}.
Notexi(0)=φi(0) in (19). We then define the following operator π acting on ℋ, for y¯(t)=(y1(t),…,yn(t))∈ℋ:
(20)π(y¯)(t)=(π(y1)(t),…,π(yn)(t)),
where π(yi)(t):[ϑ,∞)→R(i∈𝒩) obeys the rules as follows:
(21)π(yi)(t)=φi(0)e-ait+e-ait∫0teais{∑j=1nbijfj(yj(s))+∑j=1ncijgj(yj(s-τj(s)))}ds+e-ait∑0<tk<t{Iik(yi(tk))eaitk},
on t≥0 and π(yi)(s)=φi(s) on s∈[ϑ,0].
The subsequent part is the application of the contraction mapping principle, which can be divided into two steps.
Step
1. We need to prove π(ℋ)⊂ℋ. Choosing yi(t)∈ℋi (i∈𝒩), it is necessary to testify π(yi)(t)⊂ℋi.
First, sinceπ(yi)(s)=φi(s) on s∈[ϑ,0] and φi(s)∈C[[ϑ,0],R], we know π(yi)(s) is continuous on s∈[ϑ,0]. For a fixed time t>0, it follows from (21) that
(22)π(yi)(t+r)-π(yi)(t)=Q1+Q2+Q3+Q4,
where
(23)Q1=φi(0)e-ai(t+r)-φi(0)e-ait,(24)Q2=e-ai(t+r)∫0t+reais∑j=1nbijfj(yj(s))ds-e-ait∫0teais∑j=1nbijfj(yj(s))ds,Q3=e-ai(t+r)∫0t+reais∑j=1ncijgj(yj(s-τj(s)))ds-e-ait∫0teais∑j=1ncijgj(yj(s-τj(s)))ds,Q4=e-ai(t+r)∑0<tk<(t+r){Iik(yi(tk))eaitk}-e-ait∑0<tk<t{Iik(yi(tk))eaitk}.
Owing toyi(t)∈ℋi, we see that yi(t) is continuous on t≠tk (k=1,2,…); moreover, limt→tk-yi(t) and limt→tk+yi(t) exist, and limt→tk-yi(t)=yi(tk).
Consequently, whent≠tk (k=1,2,…) in (22), it is easy to find that Qi→0 as r→0 for i=1,…,4, and so π(yi)(t) is continuous on the fixed time t≠tk (k=1,2,…).
On the other hand, ast=tk (k=1,2,…) in (22), it is not difficult to find that Qi→0 as r→0 for i=1,2,3. Furthermore, if letting r<0 be small enough, we derive
(25)Q4=e-ai(tk+r)∑0<tm<(tk+r)Iim(yi(tm))eaitm-e-aitk∑0<tm<tkIim(yi(tm))eaitm={e-ai(tk+r)-e-aitk}×∑0<tm<tk{Iim(yi(tm))eaitm},
which implies limr→0-Q4=0 as t=tk. While lettingr>0tend to zero gives
(26)Q4=e-ai(tk+r)∑0<tm<(tk+r)Iim(yi(tm))eaitm-e-aitk∑0<tm<tkIim(yi(tm))eaitm=e-ai(tk+r){∑0<tm<tk{Iim(yi(tm))eaitm}+Iik(yi(tk))eaitk∑0<tm<tk}-e-aitk∑0<tm<tk{Iim(yi(tm))eaitm}={e-ai(tk+r)-e-aitk}×∑0<tm<tk{Iim(yi(tm))eaitm}+e-ai(tk+r)Iik(yi(tk))eaitk,
which yields limr→0+Q4=e-aitkIik(yi(tk))eaitk as t=tk.
According to the above discussion, we find thatπ(yi)(t):[ϑ,∞)→R is continuous on t≠tk (k=1,2,…); moreover, limt→tk-π(yi)(t) and limt→tk+π(yi)(t) exist; in addition, limt→tk-π(yi)(t)=π(yi)(tk)≠limt→tk+π(yi)(t).
Next, we will proveπ(yi)(t)→0 as t→∞. For convenience, denote
(27)π(yi)(t)=J1+J2+J3+J4,t>0,
where J1=φi(0)e-ait, J2=e-ait∫0teais∑j=1nbijfj(yj(s))ds, J4=e-ait∑0<tk<t{Iik(yi(tk))eaitk}, andJ3=e-ait∫0teais∑j=1ncijgj(yj(s-τj(s)))ds.
Due toyj(t)∈ℋj (j∈𝒩), we know limt→∞yj(t)=0. Then for any ε>0, there exists a Tj>0 such that t≥Tj implies |yj(t)|<ε. Choose T*=maxj∈𝒩{Tj}. It is derived from (A1) that, for t≥T*,
(28)J2≤e-ait∫0teais∑j=1n{|bijlj||yj(s)|}ds=e-ait∫0T*eais∑j=1n{|bijlj||yj(s)|}ds+e-ait∫T*teais∑j=1n{|bijlj||yj(s)|}ds≤e-ait∑j=1n{|bijlj|sups∈[0,T*]|yj(s)|}{∫0T*eaisds}+ε∑j=1n{|bijlj|}e-ait∫T*teaisds≤e-ait∑j=1n{|bijlj|sups∈[0,T*]|yj(s)|}×{∫0T*eaisds}+εai∑j=1n{|bijlj|}.
Moreover, as limt→∞e-ait=0, we can find a T¯¯>0 for the given ε such that t≥T¯¯ implies e-ait<ε, which leads to
(29)J2≤ε{∑j=1n{|bijlj|sups∈[0,T*]|yj(s)|}×{∫0T*eaisds}+1ai∑j=1n{|bijlj|}},t≥max{T*,T¯¯};
namely,
(30)J2→0ast→∞.
On the other hand, sincet-τj(t)→∞ as t→∞, we get limt→∞yj(t-τj(t))=0. Then for any ε>0, there also exists a Tj′>0 such that s≥Tj′ implies |yj(s-τj(s))|<ε. Select T¯=maxj∈𝒩{Tj′}. It follows from (A2) that
(31)J3≤e-ait∫0teais∑j=1n{|cijkj||yj(s-τj(s))|}ds=e-ait∫0T¯eais∑j=1n{|cijkj||yj(s-τj(s))|}ds+e-ait∫T¯teais∑j=1n{|cijkj||yj(s-τj(s))|}ds≤∑j=1n{|cijkj|sups∈[ϑ,T¯]|yj(s)|}e-ait∫0T¯eaisds+ε∑j=1n{|cijkj|}e-ait∫T¯teaisds≤e-ait∑j=1n{|cijkj|sups∈[ϑ,T¯]|yj(s)|}∫0T¯eaisds+εai∑j=1n{|cijkj|},
which results in
(32)J3→0ast→∞.
Furthermore, from (A3), we know that|Iik(yi(tk))|≤pik|yi(tk)|. So
(33)J4≤e-ait∑0<tk<t{pik|yi(tk)|eaitk}.
Asyi(t)∈ℋi, we have limt→∞yi(t)=0. Then for any ε>0, there exists a nonimpulsive point Ti>0 such that s≥Ti implies |yi(s)|<ε. It then follows from conditions (i) and (ii) that
(34)J4≤e-ait{∑0<tk<Ti{pik|yi(tk)|eaitk}+∑Ti<tk<t{pik|yi(tk)|eaitk}}≤e-ait∑0<tk<Ti{pik|yi(tk)|eaitk}+e-aitpiε∑Ti<tk<t{μeaitk}≤e-ait∑0<tk<Ti{pik|yi(tk)|eaitk}+e-aitpiε{∑Ti<tr<tk{eaitr(tr+1-tr)}+μeaitk∑Ti<tr<tk}≤e-ait∑0<tk<Ti{pik|yi(tk)|eaitk}+e-aitpiε(∫Titeaisds+μeait)≤e-ait∑0<tk<Ti{pik|yi(tk)|eaitk}+εpiai+piεμ,
which produces
(35)J4→0ast→∞.
From (30), (32), and (35), we deduce π(yi)(t)→0 as t→∞ for i∈𝒩. We therefore conclude that π(yi)(t)⊂ℋi (i∈𝒩) which means π(ℋ)⊂ℋ.
Step
2. We need to prove π is contractive. For y¯=(y1(t),…,yn(t))∈ℋ and z¯=(z1(t),…,zn(t))∈ℋ, we estimate
(36)|π(yi)(t)-π(zi)(t)|≤I1+I2+I3,
where I1=e-ait∫0teais∑j=1n[|bij||fj(yj(s))-fj(zj(s))|]ds,I3=e-ait∑0<tk<t{eaitk|Iik(yi(tk))-Iik(zi(tk))|},andI2=e-ait∫0teais∑j=1n[|cij||gj(yj(s-τj(s)))-gj(zj(s-τj(s)))|]ds.
Note(37)I1≤e-ait∫0teais∑j=1n[|bijlj||yj(s)-zj(s)|]ds≤maxj∈𝒩|bijlj|∑j=1n{sups∈[0,t]|yj(s)-zj(s)|}e-ait∫0teaisds≤1aimaxj∈𝒩|bijlj|∑j=1n{sups∈[0,t]|yj(s)-zj(s)|},I2≤e-ait∫0teais×∑j=1n[|cijkj||yj(s-τj(s))-zj(s-τj(s))|]ds≤maxj∈𝒩|cijkj|∑j=1n{sups∈[ϑ,t]|yj(s)-zj(s)|}e-ait∫0teaisds≤1aimaxj∈𝒩|cijkj|∑j=1n{sups∈[ϑ,t]|yj(s)-zj(s)|},I3≤e-ait∑0<tk<t{eaitkpik|yi(tk)-zi(tk)|}≤pie-aitsups∈[0,t]|yi(s)-zi(s)|∑0<tk<t{eaitkμ}≤pie-aitsups∈[0,t]|yi(s)-zi(s)|×{∑0<tr<tk{eaitr(tr+1-tr)}+eaitkμ}≤pisups∈[0,t]|yi(s)-zi(s)|e-ait×{∫0teaisds+eaitμ}≤pi(μ+1ai)sups∈[0,t]|yi(s)-zi(s)|.
It hence follows from (37) that
(38)|π(yi)(t)-π(zi)(t)|≤1aimaxj∈𝒩|bijlj|×∑j=1n{sups∈[0,t]|yj(s)-zj(s)|}+1aimaxj∈𝒩|cijkj|∑j=1n{sups∈[ϑ,t]|yj(s)-zj(s)|}+pi(μ+1ai)sups∈[0,t]|yi(s)-zi(s)|,
which implies
(39)supt∈[ϑ,T]|π(yi)(t)-π(zi)(t)|≤1aimaxj∈𝒩|bijlj|∑j=1n{sups∈[ϑ,T]|yj(s)-zj(s)|}+1aimaxj∈𝒩|cijkj|∑j=1n{sups∈[ϑ,T]|yj(s)-zj(s)|}+pi(μ+1ai)sups∈[ϑ,T]|yi(s)-zi(s)|.
Therefore,(40)∑i=1nsupt∈[-τ,T]|π(yi)(t)-π(zi)(t)|≤λ*∑j=1n{sups∈[ϑ,T]|yj(s)-zj(s)|}.
In view of condition (iii), we seeπ is a contraction mapping, and, thus there exists a unique fixed point y¯*(·) of π in ℋ which means the transposition of y¯*(·) is the vector-valued solution to (1)–(3) and its norm tends to zero as t→∞.
To obtain the asymptotic stability, we still need to prove that the trivial equilibriumx=0 is stable. For any ε>0, from condition (iv), we can find δ satisfying 0<δ<ε such that δ+maxi∈𝒩{λi}ε≤ε/n. Let |φ|<δ. According to what has been discussed above, we know that there exists a unique solution x(t;s,φ)=(x1(t;s,φ1),…,xn(t;s,φn))T to (1)–(3); moreover,
(41)xi(t)=π(xi)(t)=J1+J2+J3+J4,t≥0;
here J1=φi(0)e-ait, J2=e-ait∫0teais∑j=1nbijfj(xj(s))ds, J3=e-ait∫0teais∑j=1ncijgj(xj(s-τj(s)))ds, and  J4=e-ait∑0<tk<t{Iik(xi(tk))eaitk}.
Suppose there existst*>0 such that ∥x(t*;s,φ)∥=ε and ∥x(t;s,φ)∥<ε as 0≤t<t*. It follows from (41) that
(42)|xi(t*)|≤|J1(t*)|+|J2(t*)|+|J3(t*)|+|J4(t*)|.
As
(43)|J1(t*)|=|φi(0)e-ait*|≤δ,|J2(t*)|≤e-ait*∫0t*eais∑j=1n|bijljxj(s)|ds<εai∑j=1n|bijlj|,|J3(t*)|≤e-ait*∫0t*eais×∑j=1n|cijkjxj(s-τj(s))|ds<εai∑j=1n|cijkj|,|J4(t*)|≤pie-ait*∑0<tk<t*{μ|xi(tk)|eaitk}<εpie-ait*{∫0t*eaisds+μeait*}≤εpi(μ+1ai),
we obtain |xi(t*)|<δ+λiε.
So∥x(t*;s,φ)∥2=∑i=1n{|xi(t*)|2}<∑i=1n{|δ+λiε|2}≤n|δ+maxi∈𝒩{λi}ε|2≤ε2. This contradicts the assumption of ∥x(t*;s,φ)∥=ε. Therefore, ∥x(t;s,φ)∥<ε holds for all t≥0. This completes the proof.Corollary 5.
Assume that conditions (A1)–(A3) hold. Provided that(i)
infk=1,2,…{tk-tk-1}≥1,(ii)
there exist constantspi such that pik≤pi for i∈𝒩 and k=1,2,…,(iii)
∑i=1n{(1/ai)maxj∈𝒩|bijlj|+(1/ai)maxj∈𝒩|cijkj|}+maxi∈𝒩{pi(1+(1/ai))}<1,(iv)
maxi∈𝒩{λi′}<1/n, where λi′=(1/ai)∑j=1n|bijlj|+(1/ai)∑j=1n|cijkj|+pi(1+(1/ai)),then the trivial equilibrium x=0 is asymptotically stable.Proof.
Corollary5 is a direct conclusion by letting μ=1 in Theorem 4.Remark 6.
In Theorem4, we can see it is the fixed point theory that deals with the existence and uniqueness of solution and the asymptotic analysis of trivial equilibrium at the same time, while Lyapunov method fails to do this.Remark 7.
The presented sufficient conditions in Theorems4 and Corollary 5 do not require even the boundedness and differentiability of delays, let alone the monotone decreasing behavior of delays which is necessary in some relevant works.Provided thatIik(·)≡0, (1) and (2) will become the following cellular neural network with infinite delays and without impulsive effects:
(44)dxi(t)dt=-aixi(t)+∑j=1nbijfj(xj(t))+∑j=1ncijgj(xj(t-τj(t))),i∈𝒩,t≥0,
where ai, bij, cij, fj(·), gj(·), τj(t), andxi(t) are the same as defined in Section 2. Obviously, (44) also admits a trivial equilibrium x=0. From Theorem 4, we reach the following.Theorem 8.
Assume that conditions (A1)-(A2) hold. Provided that(i)
∑i=1n{(1/ai)maxj∈𝒩|bijlj|+(1/ai)maxj∈𝒩|cijkj|}<1,(ii)
maxi∈𝒩{λi′′}<1/n, where λi′′=(1/ai)∑j=1n|bijlj|+(1/ai)∑j=1n|cijkj|,then the trivial equilibrium x=0 is asymptotically stable.


## 4. Example

Consider the following two-dimensional impulsive cellular neural network with infinite delays:(45)dxi(t)dt=-aixi(t)+∑j=12bijfj(xj(t))+∑j=12cijgj(xj(t-τj(t))),t≥0,t≠tk,Δxi(tk)=xi(tk+0)-xi(tk)=arctan(0.4xi(tk)),k=1,2,…,
with the initial conditions x1(s)=cos(s), x2(s)=sin(s) on -1≤s≤0, where τj(t)=0.4t+1, a1=a2=7, bij=0, c11=3/7, c12=2/7, c21=0, c22=1/7, fj(s)=gj(s)=(|s+1|-|s-1|)/2, and tk=tk-1+0.5k.It is easy to see thatμ=0.5, lj=kj=1, and pik=0.4. Let pi=0.8 and compute
(46)∑i=12{1aimaxj=1,2|cijkj|}+maxi=1,2{pi(μ+1ai)}<1,maxi∈𝒩{λi}<12,
where λi=(1/ai)∑j=1n|cijkj|+pi(μ+(1/ai)). From Theorem 4, we conclude that the trivial equilibrium x=0 of this two-dimensional impulsive cellular neural network with infinite delays is asymptotically stable.


## 5. Conclusions

This work is devoted to seeking new methods to investigate the stability of complex neural networks. From what has been discussed above, we find that the fixed point theory is feasible. With regard to a class of impulsive cellular neural networks with infinite delays, we utilize the contraction mapping principle to deal with the existence and uniqueness of solution and the asymptotic analysis of trivial equilibrium at the same time, for which Lyapunov method feels helpless. Now that there are different kinds of fixed point theorems and complex neural networks, our future work is to continue the study on the application of fixed point theory to the stability analysis of complex neural networks.



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*Source: 427827-2013-03-25.xml*