# Stability of Impulsive Neural Networks with Time-Varying and Distributed Delays

Authors: Qi Luo; Xinjie Miao; Qian Wei; Zhengxin Zhou
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This work is devoted to investigating the stability of impulsive cellular neural networks with time-varying and distributed delays. We use the new method of fixed point theory to obtain some new and concise sufficient conditions to ensure the existence and uniqueness of solution and the global exponential stability of trivial equilibrium. The presented algebraic criteria are easily checked and do not require the differentiability of delays.


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## Body

## 1. Introduction

Since cellular neural networks (CNNs) were proposed by Chua and Yang in 1988 [1, 2], many researchers have put great effort into this subject due to their numerous successful applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision.Owing to the finite switching speed of amplifiers, there is no doubt that time delays exist in the communication and response of neurons. Moreover, as neural networks usually have a spatial extent due to the presences of a multitude of parallel pathways with a variety of axon sizes and lengths, there is a distribution of conduction velocities along these pathways and a distribution of propagation designed with discrete delays. Therefore, a more appropriate and ideal way is to incorporate continuously distributed delays with a result that a more effective model of cellular neural networks with time-varying and distributed delays proposed.In fact, beside delay effects, stochastic and impulsive as well as diffusing effects are also likely to exist in neural networks. So far, there have been many results [3–11] on the study of dynamic behaviors of complex CNNs such as impulsive delayed reaction-diffusion CNNs and stochastic delayed reaction-diffusion CNNs. Summing up the existing researches on the stability of complex CNNs, we see that the primary method is Lyapunov theory. However, there are also lots of difficulties in the applications of corresponding theories to specific problems. It is therefore necessary to seek some new methods to deal with the stability in order to overcome those difficulties.Recently, it is inspiring that Burton and other authors have applied the fixed point theory to investigate the stability of deterministic systems and obtained some more applicable conclusions, for example, see the monograph [12] and the work in [13–24]. In addition, more recently, there have been a few papers where the fixed point theory is employed to investigate the stability of stochastic (delayed) differential equations, for instance, see [25–31]. Precisely, in [26–28], Luo used the fixed point theory to study the exponential stability of mild solutions for stochastic partial differential equations with bounded delays and with infinite delays. In [29, 30], Sakthivel used the fixed point theory to discuss the asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with bounded delays and with infinite delays. In [31], Luo used the fixed point theory to study the exponential stability of stochastic Volterra-Levin equations. We wonder if we can obtain some new and more applicable stability criteria of complex CNNs by applying the fixed point theory.With this motivation, in this paper, we aim to discuss the global exponential stability of impulsive CNNs with time-varying and distributed delays. It is worth noting that our research technique is based on the contraction mapping principle rather than the usual method of Lyapunov theory. We deal with, by employing the fixed point theorem, the existence and uniqueness of solution and the global exponential stability of trivial equilibrium at the same time, for which Lyapunov method feels helpless. The obtained stability criteria are easily checked and do not require the differentiability of delays.


## 2. Preliminaries

LetRn denote the n-dimensional Euclidean space and ∥·∥ represent the Euclidean norm 𝒩≜{1,2,…,n} and 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 networks with time-varying and distributed delays:(1)dxi(t)dt=-aixi(t)+∑j=1nbijfj(xj(t))+∑j=1ncijgj(xj(t-τij(t)))+∑j=1ndij∫0ρ(t)σj(xj(t-θ))dθt≥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, and σj denote the activation functions, respectively. 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 constants bij, cij, and dij represent the connection weights of the jth neuron to the ith neuron, respectively. τij(t) and ρ(t) correspond to the transmission delays meeting 0≤τij(t)≤τ (τ=constant) and 0≤ρ(t)≤ρ (ρ=constant). 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 impulsive perturbation of the ith neuron at the impulsive moment tk.Throughout this paper, we always assume thatfi(0)=gi(0)=σi(0)=Iik(0)=0 for i∈𝒩 and k=1,2,…. Thereby, problems (1) and (2) admit 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),-m*≤s≤0,i∈𝒩,
where m*=max{τ,ρ}, φi(s)∈C[[-m*,0],R] and φ(s)=(φ1(s),…,φn(s))T∈Rn.The solutionx(t)≜x(t;s,φ)∈Rn to (1)–(3) is, for the time variable t, a piecewise continuous vector-valued function with the first-kind discontinuous points tk (k=1,2,…), where it is left-continuous; that is, the following relations are true:
(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 globally exponentially stable if for any initial condition φ(s)∈C[[-m*,0],Rn], there exists a pair of positive constants λ and M such that
(5)∥x(t;s,φ)∥≤Me-λt,∀t≥0.
The consideration of this paper is based on the following fixed point theorem.Theorem 2 (see [32]).
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, for (1)–(3), use the contraction mapping principle to prove the existence and uniqueness of the solution and the global exponential stability of trivial equilibrium all at once. Before proceeding, we firstly introduce some assumptions 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,….(A4)
There exist nonnegative constantsωj such that for any η,υ∈R,
(9)|σj(η)-σj(υ)|≤ωj|η-υ|,j∈𝒩.Letℋ=ℋ1×⋯×ℋn, and let ℋi (i∈𝒩) be the space embracing functions ϕi(t):[-m*,+∞)→R, wherein 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; moreover, limt→tk-ϕi(t)=ϕi(tk) for k=1,2,…,(3)
ϕi(s)=φi(s) on s∈[-m*,0],(4)
eαtϕi(t)→0 as t→∞, where α=const and 0<α<mini∈𝒩{ai},wheretk and φi(s) are defined as shown in Section 2. Also ℋ is a complete metric space when it is equipped with a metric defined by
(10)d(q¯(t),h¯(t))=∑i=1nsupt≥-m*|qi(t)-hi(t)|,
where q¯(t)=(q1(t),…,qn(t))∈ℋ and h¯(t)=(h1(t),…,hn(t))∈ℋ.Theorem 3.
Assume that conditions (A1)–(A4) 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)
∑i=1n{(1/ai)maxj∈𝒩|bijlj|+(1/ai)maxj∈𝒩|cijkj|+(ρ/ai)maxj∈𝒩|ωjdij|}+maxi∈𝒩{pi(μ+(1/ai))}≜χ<1,
and then the trivial equilibriumx=0 is globally exponentially stable.Proof.
Multiplying both sides of (1) with eait gives, for t>0 and t≠tk,
(11)deaitxi(t)=eaitdxi(t)+aixi(t)eaitdt=eait{∑j=1nbijfj(xj(t))+∑j=1ncijgj(xj(t-τij(t)))+∑j=1ndij∫0ρ(t)σj(xj(t-θ))dθ}dt,
which yields after integrating from tk-1+ε (ε>0) to t∈(tk-1,tk) (k=1,2,…) that
(12)xi(t)eait=xi(tk-1+ε)eai(tk-1+ε)+∫tk-1+εteais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(xj(s-θ))dθ}ds.
Lettingε→0 in (12), we have, for t∈(tk-1,tk) (k=1,2,…),
(13)xi(t)eait=xi(tk-1+0)eaitk-1+∫tk-1teais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(xj(s-θ))dθ}ds.
Settingt=tk-ε (ε>0) in (13), we get
(14)xi(tk-ε)eai(tk-ε)=xi(tk-1+0)eaitk-1+∫tk-1tk-εeais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(xj(s-θ))dθ}ds,
which generates by letting ε→0(15)xi(tk-0)eaitk=xi(tk-1+0)eaitk-1+∫tk-1tkeais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(xj(s-θ))dθ}ds.
Notingxi(tk-0)=xi(tk), (15) can be rearranged as
(16)xi(tk)eaitk=xi(tk-1+0)eaitk-1+∫tk-1tkeais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(xj(s-θ))dθ}ds.
Combining (13) and (16), we derive that
(17)xi(t)eait=xi(tk-1+0)eaitk-1+∫tk-1teais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(xj(s-θ))dθ}ds
is true for t∈(tk-1,tk] (k=1,2,…). Hence, we get, for t∈(tk-1,tk] (k=1,2,…),
(18)xi(t)eait={xi(tk-1)+Ii(k-1)(xi(tk-1))}eaitk-1+∫tk-1teais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(xj(s-θ))dθ}ds=xi(tk-1)eaitk-1+∫tk-1teais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(xj(s-θ))dθ}ds+Ii(k-1)(xi(tk-1))eaitk-1,
which results in
(19)xi(tk-1)eaitk-1=xi(tk-2)eaitk-2+∫tk-2tk-1eais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(xj(s-θ))dθ}ds+Ii(k-2)(xi(tk-2))eaitk-2⋮xi(t2)eait2=xi(t1)eait1+∫t1t2eais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(xj(s-θ))dθ}ds+Ii1(xi(t1))eait1,xi(t1)eait1=φi(0)+∫0t1eais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(xj(s-θ))dθ}ds.
We therefore conclude, fort>0,
(20)xi(t)=φi(0)e-ait+e-ait∫0teais{∑j=1nbijfj(xj(s))+∑j=1ncijgj(xj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(xj(s-θ))dθ}ds+e-ait∑0<tk<t{Iik(xi(tk))eaitk}.
Note thatxi(0)=φi(0) in (20). We then define the following operator π acting on ℋ, for y¯(t)=(y1(t),…,yn(t))∈ℋ:
(21)π(y¯)(t)=(π(y1)(t),…,π(yn)(t)),
where π(yi)(t):[-m*,+∞)→R (i∈𝒩) obeys the rule as follows:
(22)π(yi)(t)=φi(0)e-ait+e-ait∫0teais{∑j=1nbijfj(yj(s))+∑j=1ncijgj(yj(s-τij(s)))+∑j=1ndij∫0ρ(s)σj(yj(s-θ))dθ}ds+e-ait∑0<tk<t{Iik(yi(tk))eaitk}
on t≥0 and π(yi)(s)=φi(s) on s∈[-m*,0].
In what follows, we will apply the contraction mapping principle to prove the existence and uniqueness of solution and the global exponential stability of trivial equilibrium at the same time. The subsequent proof can be divided into two steps.
Step 1. We need to prove that π(ℋ)⊂ℋ. For yi(t)∈ℋi (i∈𝒩), it is necessary to show that π(yi)(t)⊂ℋi. As defined above, we see that π(yi)(s)=φi(s) on s∈[-m*,0]. Owing to the continuity of φi(s) on s∈[-m*,0], we immediately know that π(yi)(t) is continuous on t∈[-m*,0].
Choose a fixed timet>0, and it is then derived from (22) that
(23)π(yi)(t+r)-π(yi)(t)=Q1+Q2+Q3+Q4+Q5,t>0,
where,
(24)Q1=φi(0)e-ai(t+r)-φi(0)e-ait,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-τij(s)))ds-e-ait∫0teais∑j=1ncijgj(yj(s-τij(s)))ds,Q4=e-ai(t+r)∫0t+reais∑j=1ndij∫0ρ(s)σj(yj(s-θ))dθds-e-ait∫0teais∑j=1ndij∫0ρ(s)σj(yj(s-θ))dθds,Q5=e-ai(t+r)∑0<tk<(t+r){Iik(yi(tk))eaitk}-e-ait∑0<tk<t{Iik(yi(tk))eaitk}.
Sinceyi(t)∈ℋi, we know that yi(t) is continuous on t≠tk (k=1,2,…); moreover, limt→tk-yi(t) and limt→tk+yi(t) exist, in addition, limt→tk-yi(t)=yi(tk).
Lettingt≠tk (k=1,2,…) in (23), it is easy to see that Qi→0 as r→0 for i=1,…,5. Thus, π(yi)(t+r)-π(yi)(t)→0 as r→0 holds on t>0 and t≠tk (k=1,2,…).
Lettingt=tk (k=1,2,…) in (23), it is not difficult to find that Qi→0 as r→0 for i=1,…,4. Letting r<0 be small enough, we compute
(25)Q5=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-Q5=0. Letting r>0 be small enough, we have
(26)Q5=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}-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 implies limr→0+Q5=e-aitkIik(yi(tk))eaitk.
According to the above discussion, we see thatπ(yi)(t):[-m*,+∞)→R is continuous on ≠tk(k=1,2,…), while for t=tk (k=1,2,…), limt→tk-π(yi)(t) and limt→tk+π(yi)(t) exist; moreover, limt→tk-π(yi)(t)=π(yi)(tk)≠limt→tk+π(yi)(t).
Next, we will prove thateαtπ(yi)(t)→0 as t→∞ for i∈𝒩. To begin with, we give the expression of eαtπ(yi)(t) as follows:
(27)eαtπ(yi)(t)=W1+W2+W3+W4+W5,t>0,
WhereW1=φi(0)e-(ai-α)t,W2=eαte-ait∫0teais∑j=1nbijfj(yj(s))ds,W5=eαte-ait∑0<tk<t{Iik(yi(tk))eaitk},W3=eαte-ait∫0teais∑j=1ncijgj(yj(s-τij(s)))ds, andW4=eαte-ait∫0teais∑j=1ndij∫0ρ(s)σj(yj(s-θ))dθds.
First, it is obvious thatlimt→∞W1=0 as ai-α>0. Furthermore, for yj(t)∈ℋj (j∈𝒩), we see limt→∞eαtyj(t)=0. Then, for any ε>0, there exists a Tj>0 such that s≥Tj implies |eαsyj(s)|<ε. Choose T*=maxj∈𝒩{Tj}. It is derived form (A1) that
(28)W2≤eαte-ait∫0teais∑j=1n{|bijlj||yj(s)|}ds=eαte-ait∫0teaise-αs∑j=1n{|bijlj|eαs|yj(s)|}ds=e-(ai-α)t∫0T*e(ai-α)s∑j=1n{|bijlj|eαs|yj(s)|}ds+e-(ai-α)t∫T*te(ai-α)s∑j=1n{|bijlj|eαs|yj(s)|}ds≤e-(ai-α)t∑j=1n{|bijlj|sups∈[0,T*]|eαsyj(s)|}{∫0T*e(ai-α)sds}+ε∑j=1n{|bijlj|}e-(ai-α)t∫T*te(ai-α)sds≤e-(ai-α)t∑j=1n{|bijlj|sups∈[0,T*]|eαsyj(s)|}{∫0T*e(ai-α)sds}+εai-α∑j=1n{|bijlj|},
which leads to W2→0 as t→∞.
Similarly, for the givenε>0 above, there also exists a Tj′>0 such that s≥Tj′-τ implies |eαsyj(s)|<ε. Select T^=maxj∈𝒩{Tj′}. It follows from (A2) that
(29)W3≤eαte-ait∫0teais∑j=1n{|cijkj||yj(s-τij(s))|}ds≤e-(ai-α)t×∫0teaise-α{s-τ}×∑j=1n{|cijkj|eα[s-τij(s)]|yj(s-τij(s))|}ds=eατe-(ai-α)t×∫0T^e(ai-α)s∑j=1n{|cijkj|eα[s-τij(s)]|yj(s-τij(s))|}ds+eατe-(ai-α)t×∫T^te(ai-α)s∑j=1n{|cijkj|eα[s-τij(s)]|yj(s-τij(s))|}ds≤eατ∑j=1n{|cijkj|sups∈[-τ,T^]|eαsyj(s)|}×e-(ai-α)t∫0T^e(ai-α)sds+eατε∑j=1n{|cijkj|}e-(ai-α)t∫T^te(ai-α)sds≤eατ∑j=1n{|cijkj|sups∈[-τ,T^]|eαsyj(s)|}×e-(ai-α)t∫0T^e(ai-α)sds+eατεai-α∑j=1n{|cijkj|},
which results in W3→0 as t→∞. In addition, it is derived from (A4) that
(30)W4≤eαte-ait∫0teais∑j=1n{dij∫0ρωj|yj(s-θ)|dθ}ds=eαte-ait×∫0teaise-αs∑j=1n{dij∫0ρeαθωjeα(s-θ)|yj(s-θ)|dθ}ds≤eαte-ait×∫0teaise-αs×∑j=1n{dijsupζ∈[s-ρ,s]{eαζ|yj(ζ)|}∫0ρeαθωjdθ}ds.
Sinceeαζ|yj(ζ)|→0 as ζ→∞, we know that, for any ε>0, there exists a Tj′′>0 such that ζ>Tj′′-ρ implies eαζ|yj(ζ)|<ε. Selecting T¯=maxj∈𝒩{Tj′′}, it follows from (30) that
(31)W4≤e(α-ai)t×∫0T¯e(ai-α)s∑j=1n{dijsupζ∈[s-ρ,s]{eαζ|yj(ζ)|}×∫0ρeαθωjdθsupζ∈[s-ρ,s]}ds+e(α-ai)t×∫T¯te(ai-α)s∑j=1n{dijsupζ∈[s-ρ,s]{eαζ|yj(ζ)|}×∫0ρeαθωjdθsupζ∈[s-ρ,s]}ds≤eαρα{∑j=1ndijωjsupζ∈[-ρ,T¯]{eαζ|yj(ζ)|}}×e(α-ai)t∫0T¯e(ai-α)sds+eαρα∑j=1n{dijωjsupζ∈[T¯-ρ,t]{eαζ|yj(ζ)|}}×e(α-ai)t∫T¯te(ai-α)sds≤e(α-ai)teαρα{∑j=1ndijωjsupζ∈[-ρ,T¯]{eαζ|yj(ζ)|}}×∫0T¯e(ai-α)sds+ε∑j=1n{dijωj}eαρα(ai-α),
which yields W4→0 as t→∞.
Furthermore, from (A3), we see that|Iik(xi(tk))|≤pik|yi(tk)|. So,
(32)W5≤eαte-ait∑0<tk<t{pik|yi(tk)|eaitk}.
Asyi(t)∈ℋi, we have limt→∞eαtyi(t)=0. Then, for any ε>0, there exists a nonimpulsive point Ti>0 such that s≥Ti implies |eαsyi(s)|<ε. It then follows from conditions (i) and (ii) that
(33)W5≤eαte-ait{∑0<tk<Ti{pik|yi(tk)|eaitk}+∑Ti<tk<t{pik|yi(tk)|eαtke(ai-α)tk}}≤eαte-ait∑0<tk<Ti{pik|yi(tk)|eaitk}+eαte-aitpiε∑Ti<tk<t{μe(ai-α)tk}≤e-(ai-α)t∑0<tk<Ti{pik|yi(tk)|eaitk}+e-(ai-α)tpiε{∑Ti<tr<tk{e(ai-α)tr(tr+1-tr)}+μe(ai-α)tk∑Ti<tr<tk}≤e-(ai-α)t∑0<tk<Ti{pik|yi(tk)|eaitk}+e-(ai-α)tpiε∫Tite(ai-α)sds+e-(ai-α)tpiεμe(ai-α)t≤e-(ai-α)t∑0<tk<Ti{pik|yi(tk)|eaitk}+piεai-α+piεμ,
which means that W5→0 as t→∞.
Now, we can derive from (27) that eαtπ(yi)(t)→0 as t→∞ for i∈𝒩. It is therefore concluded that π(yi)(t)⊂ℋi which results in π(ℋ)⊂ℋ.
Step 2. We need to prove that π is contractive. For z¯=(z1(t),…,zn(t))∈ℋ and y¯=(y1(t),…,yn(t))∈ℋ, we estimate
(34)|π(yi)(t)-π(zi)(t)|≤J1+J2+J3+J4,
where
(35)J1=e-ait∫0teais∑j=1n[|bij||fj(yj(s))-fj(zj(s))|]ds,J2=e-ait∫0teais∑j=1n[|cij||gj(yj(s-τij(s)))-gj(zj(s-τij(s)))|]ds,J3=e-ait∫0teais∑j=1n|dij|∫0ρ(s)|σj(yj(s-θ))-σj(zj(s-θ))|dθds,J4=e-ait∑0<tk<t{eaitk|Iik(yi(tk))-Iik(zi(tk))|}.
Note that(36)J1≤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)|},J2≤e-ait∫0teais∑j=1n[|cijkj||yj(s-τij(s))-zj(s-τij(s))|]ds≤maxj∈𝒩|cijkj|∑j=1n{supξ∈[-τ,t]|yj(ξ)-zj(ξ)|}×e-ait∫0teaisds≤1aimaxj∈𝒩|cijkj|∑j=1n{supξ∈[-τ,t]|yj(ξ)-zj(ξ)|},J3≤e-ait∫0teais∑j=1n{|dij|∫0ρ(s)ωj|yj(s-θ)-zj(s-θ)|dθ∫0ρ(s)ωj|yj(s-θ)}ds≤e-ait∫0teais∑j=1n{|dij|supξ∈[s-ρ,s]|yj(ξ)-zj(ξ)|×∫0ρ(s)ωjdθsupξ∈[s-ρ,s]}ds≤e-ait∫0teais∑j=1n{ωj|dij|supξ∈[-ρ,t]|yj(ξ)-zj(ξ)|ρ(s)supξ∈[-ρ,t]}ds≤maxj∈𝒩{ωj|dij|}∑j=1n{supξ∈[-ρ,t]|yj(ξ)-zj(ξ)|}×e-ait∫0teaisρ(s)ds≤ρaimaxj∈𝒩{ωj|dij|}∑j=1n{supξ∈[-ρ,t]|yj(ξ)-zj(ξ)|},J4≤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 is then derived from (36) that
(37)supt∈[-m*,T]|π(yi)(t)-π(zi)(t)|≤1aimaxj∈𝒩|bijlj|∑j=1n{sups∈[-m*,T]|yj(s)-zj(s)|}+1aimaxj∈𝒩|cijkj|∑j=1n{supξ∈[-m*,T]|yj(ξ)-zj(ξ)|}+ρaimaxj∈𝒩{ωj|dij|}∑j=1n{supξ∈[-m*,T]|yj(ξ)-zj(ξ)|}+pi(μ+1ai)sups∈[-m*,T]|yi(s)-zi(s)|,
which means that
(38)∑i=1nsupt∈[-m*,T]|π(yi)(t)-π(zi)(t)|≤χ∑j=1n{sups∈[-m*,T]|yj(s)-zj(s)|},
where
(39)χ≜∑i=1n{1aimaxj∈𝒩|bijlj|+1aimaxj∈𝒩|cijkj|+ρaimaxj∈𝒩|ωjdij|}+maxi∈𝒩{pi(μ+1ai)}.
In view of condition (iii), we know thatπ is a contraction mapping, and hence, there exists a unique fixed point y¯(·) of π in ℋ which means that y¯T(·) is the solution to (1)–(3) and eαt∥y¯T(·)∥→0 as t→∞. This completes the proof.Lemma 4.
Assume conditions (A1)–(A4) 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|+(ρ/ai)maxj∈𝒩|ωjdij|}+maxi∈𝒩{pi(1+(1/ai))}≜χ<1,
then the trivial equilibriumx=0 is globally exponentially stable.Proof.
Lemma4 is a direct conclusion by letting μ=1 in Theorem 3.Remark 5.
In Theorem3, we use the fixed point theorem to prove the existence and uniqueness of solution and the global exponential stability of trivial equilibrium all at once, while Lyapunov method fails to do this.Remark 6.
The presented sufficient conditions in Theorem3 and Lemma 4 do not require even the differentiability of delays, let alone the monotone decreasing behavior of delays which is necessary in some relevant works.


## 4. Example

Consider the following two-dimensional impulsive cellular neural network with time-varying and distributed delays.(40)dxi(t)dt=-aixi(t)+∑j=12bijfj(xj(t))+∑j=12cijgj(xj(t-τij(t)))+∑j=12dij∫0ρ(t)σj(xj(t-θ))dθ,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 -m*≤s≤0, where τij(t)=0.8+0.4cos(t), ρ(t)=0.5+0.3sin(t), m* is defined as shown in (3), a1=a2=7, bij=0, c11=0, c12=1/7, c21=-1/7, c22=-1/7, d11=3/7, d12=2/7, d21=0, d22=1/7, fj(s)=gj(s)=σj(s)=(|s+1|-|s-1|)/2, and tk=tk-1+0.5k.It is easily to find thatμ=0.5, lj=kj=ωj=1, and pik=0.4. Let pi=0.8 and compute
(41)∑i=12{1aimaxj∈𝒩|bijlj|+1aimaxj∈𝒩|cijkj|+ρaimaxj∈𝒩|ωjdij|}+maxi∈𝒩{pi(μ+1ai)}<1.From Theorem3, we conclude that the trivial equilibrium x=0 of this two-dimensional impulsive cellular neural network with time-varying and distributed delays is globally exponentially stable.


## 5. Conclusions

This article is a new attempt of applying the fixed point theory to the stability analysis of impulsive neural networks with time-varying and distributed delays, which is different from the existing relevant publications where Lyapunov theory is the main technique. From what have been discussed above, we see that the contraction mapping principle is effective for not only the investigation of the existence and uniqueness of solution but also for the stability analysis of trivial equilibrium. In the future, we will continue to explore the application of other kinds of fixed point theorems to the stability research of complex neural networks.



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*Source: 325310-2013-05-12.xml*