# Dynamics of Cohen-Grossberg Neural Networks with Mixed Delays and Impulses

Authors: Xinsong Yang; Chuangxia Huang; Defei Zhang; Yao Long
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

Impulsive Cohen-Grossberg neural networks with bounded and unbounded delays (i.e., mixed delays) are investigated. By using the Leray-Schauder fixed point theorem, differential inequality techniques, and constructing suitable Lyapunov functional, several new sufficient conditions on the existence and global exponential stability of periodic solution for the system are obtained, which improves some of the known results. An example and its numerical simulations are employed to illustrate our feasible results.


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## Body

## 1. Introduction

In the recent years, dynamics of the Cohen-Grossberg neural networks (CGNNs) [1] has been extensively studied because of their immense potentials of application perspective
in different areas such as pattern recognition, parallel computing, associative memory, combinational
optimization, and signal and image processing [2–6]. The authors of
[7–11] have studied the stability of
equilibrium point or periodic solution of CGNNs with time-varying delays due to the transmission delays
during the communication between neurons which will affect the dynamical behavior of
neural networks. Considering the distant past also has influence on the recent behavior of the state,
the authors of [12] investigated the stability of equilibrium point for
CGNNs with continuously distributed delays. Impulsive effects are also likely to exist in the neural
networks, that is, the state of the networks is subject to instantaneous perturbations and experiences
abrupt change at certain moments. Authors of [13–16] have studied the
stability of equilibrium point for impulsive delay CGNNs. However, the activation functions of CGNNs in
[12–16] are bounded, and the
restrictions on impulses in [13, 17] are
very strong, which limit CGNNs' applications [7].In theory and applications, global stability of periodic solution of CGNNs is of great
importance since the global stability of equilibrium points can be considered
as a special case of periodic solution with random period. Moreover, CGNNs
model is one of the most popular and typical neural network models. Some other
models, such as Hopfield-type neural networks,
cellular neural networks, and bidirectional associative memory neural networks,
are special cases of the model [14]. To our best knowledge, few authors have
considered the existence and global exponential stability of periodic solutions
for CGNNs with mixed delays and impulses. Therefore, it is necessary to
consider the existence and global exponential stability of periodic solution
for impulsive CGNNs with mixed delays.The main methods used in this paper are
Leray-Schauder's fixed point theorem, differential inequality techniques, and
Lyapunov functional. Several new sufficient conditions are obtained for the
existence and global exponential stability of periodic solution for impulsive
CGNNs with mixed delays. Moreover, we discharge some restrictive conditions on the
activation functions of the neurons, such as boundedness, monotonicity, and
differentiable, we offer the precise convergence
index, and give the weak conditions on impulses.The rest of this paper is organized as follows. In Section2, we introduce some notations
and definitions and state some preliminary results needed in later sections. We
then study, in Section 3, the existence of periodic solutions of impulsive
CGNNs with mixed delays by using Leray-Schauder's fixed point theorem. In
Section 4, with the help of Lyapunov functional, we will derive sufficient
conditions for the global exponential stability of the periodic solution. At
last, an example and its numerical simulations are
employed to illustrate the feasible results of this paper.


## 2. Preliminaries

Consider the following CGNNs with mixed delays and
impulses:(2.1)ui′(t)=−ai(ui(t)){bi(ui(t))−∑j=1n[bij(t)fj(uj(t−τij(t)))+cij(t)∫−∞tkij(t−s)gj(uj(s))ds]+Ii(t)},t≠tk,Δui(tk)=ui(tk+)−ui(tk−)=γikui(tk),i=1,2,…,n,
where Δui(tk)(i=1,2,…,n) are the impulses at moments tk, ui(t) is left continuous at time tk, and the right limit exists at tk, that is, ui(tk)=ui(tk−) and ui(tk+) exist and 0<t1<t2<⋯ is a strictly increasing sequence such that limk→+∞tk=+∞; ui(t) is the state of neuron; ai(ui(t)) and bi(ui(t)) represent an amplification function at time t and an appropriately behaved function at time t,
respectively; B=(bij(t))n×n and C=(cij(t))n×n are connection matrices; I(t)=(I1(t),I2(t),…,In(t))T is the input function; and g(u)=(g1(u1),g2(u2),…,gn(un))T are the activation functions of the neurons.Throughout this paper, we assume that(H1)
bij(t),cij(t),τij(t)≥0, Ii(t) are continuous ω-periodic functions, ω>0 is a constant, i,j=1,2,…,n;(H2)
ai(u) is continuous and there exist positive
constants a¯i and a¯i such that 0<a¯i≤ai(u)≤a¯i, u∈R, i=1,2,…,n;(H3)
there is a positive constantli such that bi′(u)≥li, bi′(u) denotes the derivative of bi(u),u∈R,
and bi(0)=0, i=1,2,…,n;(H4)
{γik} and {tk} are ω-periodic sequence, that is, there exists a
positive constant p such that t(k+p)=tk+ω, γi(k+p)=γik, i=1,2,…,n;(H5)
fj,gj∈C(R,R),
there are positive constants Ljf>0, Ljg>0,
such that |gj(x)−gj(y)|≤Ljg|x−y|, |fj(x)−fj(y)|≤Ljf|x−y|∀x,y∈R,j=1,2,…,n;(H6)
the delay kernelskij:[0,+∞)→R are continuous, integrable and there exist
positive constants βij such that(2.2)∫0+∞|kij(s)|ds≤βij,i,j=1,2,…,n;(H7)
(2.3)max1≤i≤nsup0≤t≤ω{∑j=1na¯j[|bij(t)|Ljf+|cij(t)|Ljgβij]qi}=θ<1,max1≤i≤n{θ+a¯ia¯i(1−e−qiω)∑k=1p|bik|}<1, where qi=a¯ili;(H8)
there exists a constantα0>0 such that(2.4)∫0+∞|kij(s)|eα0sds<+∞,i,j=1,2,…,n.From(H2),
the antiderivative of  1/ai(ui) exists. We choose an antiderivative hi(ui) of 1/ai(ui) that satisfies hi(0)=0.
Obviously, (d/dui)hi(ui)=1/ai(ui).
By ai(ui)>0,
we obtain that hi(ui) is strictly monotone increasing about ui.
In view of derivative theorem for inverse function, the inverse function hi−1(ui) of hi(ui) is differential and (d/dui)hi−1(ui)=ai(ui).
By (H3),
composition function bi(hi−1(z)) is differential. Denote xi(t)=hi(ui(t)), it is easy to see that xi′(t)=ui′(t)/ai(ui(t)) and ui(t)=hi−1(xi(t)).
Substituting these equalities into system (2.1), we get(2.5)xi′(t)=−bi(hi−1(xi(t)))+∑j=1n[bij(t)fj(hj−1(xj(t−τij(t))))+cij(t)∫−∞tkij(t−s)gj(hj−1(xj(s)))ds]−Ii(t),t≠tk,Δxi(tk)=hi((1+γik)hi−1(xi(tk)))−xi(tk)≐ri(xi(tk)),i=1,2,…,n,
which can be rewritten as
(2.6)xi′(t)=−di(xi(t))xi(t)+∑j=1n[bij(t)fj(hj−1(xj(t−τij(t))))+cij(t)∫−∞tkij(t−s)gj(hj−1(xj(s)))ds]−Ii(t),t≠tk,Δxi(tk)=ri(xi(tk)),i=1,2,…,n,
where di(xi(t))≜bi(hi−1(z))′|z=ξi, bi(hi−1(z))′|z=ξi denote the
derivative of bi(hi−1(z)) at point z=ξi, z∈R, ξi is between 0 and xi(t).The existence and global exponential stability of
periodic solution for system (2.1)
are equivalent
to the existence and global exponential stability of periodic solution for
system (2.5) or (2.6). So, we investigate the the existence and global
exponential stability of periodic solution for system (2.6).From the definition ofhi−1(u),
using Lagrange mean value theorem, one gets(2.7)|hi−1(x)−hi−1(y)|=|(hi−1)′(ξ)(x−y)|≤a¯i|x−y|,∀x,y∈R,
where ξ is between x and y. Moreover, we have
(2.8)|ri(xi(tk))|≤1a¯i|(1+γik)hi−1(xi(tk))−ui(tk)|=|γik|a¯i|hi−1(xi(tk))|≤a¯ia¯i|γik||xi(tk)|.For convenience, we denoteτ=max{τij(t):t∈[0,ω],i,j=1,2,…,n}, x=(x1,x2,…,xn)T to be a column vector, in which the symbol (T) denotes the transpose of a vector.Definition 2.1.
A functionx∈(R,Rn) is said to be a solution of (2.6) if the
following conditions are satisfied.
(i)
x(t) is absolutely continuous on each (tk,tk+1), k∈N.(ii)
For eachk∈N, x(tk+) and x(tk−) exist and x(tk)=x(tk−).(iii)
x(t) satisfies (2.6) for almost everywhere and at
impulsive points, tk may have discontinuity of the first kind.
The initial conditionϕ=(ϕ1,…,ϕn)T of (2.6) is of the form
(2.9)xi(s)=ϕi(s),s∈(−∞,0],i=1,…,n,
where ϕi(s),i=1,2,…,n are continuous functions.Definition 2.2.
Letx*(t) be an ω-periodic solution of (2.6) with initial value ϕ*=(ϕ1*,…,ϕn*)T∈C((−∞,0];Rn).
If there exist constants α>0, M≥1 such that for every solution x(t) of (2.6) with initial value ϕ∈C((−∞,0];Rn),
(2.10)|xi(t)−xi*(t)|≤M∥ϕ−ϕ*∥e−αt,∀t>0,i=1,2,…,n,
where ∥ϕ−ϕ*∥=sups≤0max1≤i≤n|ϕi(s)−ϕi*(s)|.
Then, x*(t) is said to be globally exponentially stable.Lemma 2.3 (Leray-Schauder).
LetE be a Banach space, and let the operator A:E→E be completely continuous. If the set {∥x∥∣x∈E,x=λAx,0<λ<1} is bounded, then A has a fixed point in T, where
(2.11)T={x∣x∈E,∥x∥≤R},R=sup{∥x∥∣x=λAx,0<λ<1}.


## 3. Existence of Periodic Solutions

Lemma 3.1.
Suppose that(H1)–(H5) hold and let x(t) be an ω-periodic solution of system (2.6).
Then,
(3.1)xi(t)=∫0ωwix(t,s)[∑j=1n[bij(s)fj(hj−1(xj(s−τij(s))))+cij(s)∫−∞skij(s−v)gj(hj−1(xj(v)))dv]−Ii(s)]ds+∑k=1pwix(t,tk)ri(xi(tk)),t∈[0,ω],i=1,2,…,n,
where
(3.2)wix(t,s)=11−e−∫0ωdi(xi(v))dv{e−∫stdi(xi(v))dv,0≤s≤t≤ω,e−(∫0ωdi(xi(v))dv−∫tsdi(xi(v))dv),0≤t<s≤ω.Proof.
Lettq≤t<tq+1, q≤p.
From the first expression of (2.6), we have(3.3)(xi(t)e∫0tdi(xi(s))ds)′={∑j=1n[bij(t)fj(hj−1(xj(t−τij(t))))+cij(t)∫−∞tkij(t−s)gj(hj−1(xj(s)))ds]−Ii(t)}e∫0tdi(xi(s))ds.Integrating (3.3) on intervals (0,t1−),(t1+,t2−),…,(tq+,t) and adding all of them, by the second formula
of (2.6), we have
(3.4)xi(t)=e−∫0tdi(xi(s))dsxi(0)+∑k=1qe−∫tktdi(xi(s))dsri(xi(tk))+∫0t{∑j=1n[bij(s)fj(hj−1(xj(s−τij(s))))+cij(s)∫−∞skij(s−v)gj(hj−1(xj(v)))dv]−Ii(s)}e−∫stdi(xi(v))dvds.
Since xi(ω)=xi(0),
from (3.4), we obtain
(3.5)xi(0)=11−e−∫0ωdi(xi(v))dv{∑k=1pe−∫tkωdi(xi(s))dsri(xi(tk))+∫0ω{∑j=1n[bij(s)fj(hj−1(xj(s−τij(s))))+cij(s)∫−∞skij(s−v)gj(hj−1(xj(v)))dv]−Ii(s)}×e−∫sωdi(xi(v))dvds}.
Substituting (3.5) into (3.4),
we obtain (3.1). This completes the proof.In order to use Lemma2.3, we take PC(J,Rn)={x(t)∣x:J=[0,ω]→Rn as piecewise continuous periodic
solution at t≠tk,
and x(tk+),x(tk−)=x(tk) exist at t=tk,k=1,2,…,p}.
Then, PC(J,Rn) is a Banach space with the norm
(3.6)∥x∥=max1≤i≤n{|xi|0},|xi|0=sup0≤t≤ω|xi(t)|,i=1,2,…,n.Set a mappingΦ:PC(J,Rn)→PC(J,Rn) by setting(3.7)(Φx)(t)=x(t),t∈J=[0,ω],where(3.8)(Φx)i(t)=∫0ωwix(t,s)[∑j=1n[bij(s)fj(hj−1(xj(s−τij(s))))+cij(s)∫−∞skij(s−v)gj(hj−1(xj(v)))dv]−Ii(s)]ds+∑k=1pwix(t,tk)ri(xi(tk)).It is easy to know the fact that the existence ofω-periodic solution of (2.6) is equivalent to
the existence of fixed point of the mapping Φ in PC(J,Rn).Lemma 3.2.
Suppose that(H1)–(H7) hold. Then, Φ:PC(J,Rn)→PC(J,Rn) is completely continuous.Proof.
First, we show thatΦ:PC(J,Rn)→PC(J,Rn) is continuous. For any ε>0,
we take 0<δ<ε/max1≤i≤n{θ+∑k=1pηik/(1−e−qiω)},
where ηik=(a¯i/a¯i)|1+γik|+1.
Then, for all x,y∈PC(J,Rn) and ∥x−y∥<δ, we have
(3.9)∥Φx−Φy∥=max1≤i≤nsup0≤t≤ω{|∫0ωwix(t,s)×∑j=1n[bij(s)[fj(hj−1(xj(s−τij(s))))−fj(hj−1(yj(s−τij(s))))]+cij(s)∫−∞skij(s−v)[gj(hj−1(xj(v)))−gj(hj−1(yj(v)))]dv]ds+∑k=1pwix(t,tk)[ri(xi(tk))−ri(yi(tk))]|}≤max1≤i≤nsup0≤t≤ω{∫0ωwix(t,s)qiθds+∑k=1pηik1−e−qiω}∥x−y∥<max1≤i≤n{θ+∑k=1pηik1−e−qiω}δ<ε.
Hence, Φ is continuous.
Next, we showΦ maps bounded set into bounded set. For any x∈PC(J,Rn) with ∥x∥<D,
where D is some positive constant, we have
(3.10)∥Φx∥=max1≤i≤nsup0≤t≤ω{|∫0ωwix(t,s)×[∑j=1n[bij(s)fj(hj−1(xj(s−τij(s))))+cij(s)∫−∞skij(s−v)gj(hj−1(xj(v)))dv]−Ii(s)]ds+∑k=1pwix(t,tk)ri(xi(tk))|}<max1≤i≤n{θ+a¯ia¯i(1−e−qiω)∑k=1p|γik|}D+max1≤i≤n{1qi[∑j=1n[b¯ij|fj(0)|+c¯ijβij|gj(0)|]+I¯i]},where b¯ij=maxt∈[0,ω]|bij(t)|, c¯ij=maxt∈[0,ω]|cij(t)|, I¯i=maxt∈[0,ω]|Ii(t)|.
Equation (3.10) implies that ∥Φx∥ is uniformly bounded for any ∥x∥<D.
Hence, {Φx∣x∈PC(J,Rn)} is a family of uniformly bounded and
equicontinuous functions on R.
By using the Arzela-Ascoli theorem, Φ:PC(J,Rn)→PC(J,Rn) is compact. Therefore, Φ:PC(J,Rn)→PC(J,Rn) is completely continuous. This completes the
proof.Theorem 3.3.
Suppose that(H1)–(H7) hold. Then, system (2.6) has an ω-periodic solution.Proof.
Letx∈PC(J,Rn), t∈J.
We consider the operator equation
(3.11)x=λΦx,λ∈(0,1).
If x is a solution of (3.11), for t∈J, we obtain
(3.12)∥x∥≤∥Φx∥≤max1≤i≤n{θ+a¯ia¯i(1−e−qiω)∑k=1p|γik|}∥x∥+max1≤i≤n{1qi[∑j=1n[b¯ij|fj(0)|+c¯ijβij|gj(0)|]+I¯i]},
where this and max1≤i≤n{θ+(a¯i/a¯i(1−e−qiω))∑k=1p|γik|}<1 imply that
(3.13)∥x∥≤max1≤i≤n{(1/qi)[∑j=1n[b¯ij|fj(0)|+c¯ijβij|gj(0)|]+I¯i]}1−max1≤i≤n{θ+(a¯i/a¯i(1−e−qiω))∑k=1p|γik|}≜R.This shows that ∥x∥ of (3.11) is bounded, which
is independent of λ∈(0,1).
In view of Lemma 2.3, we obtain that Φ has a fixed point. Hence, system (2.6) has one ω-periodic solution with ∥x∥≤R.
This completes the proof.


## 4. Global Exponential Stability of Periodic Solution

In this section, we will construct some suitable
Lyapunov functionals to derive sufficient conditions which ensure that (2.6)
has a unique ω-periodic solution, and all solutions of (2.6)
exponentially converge to its unique ω-periodic solution.Theorem 4.1.
Assume that(H1)–(H8)
hold and
(H9)
lnMk/(tk−tk−1)≤η<α, where Mk=max1≤i≤n{(a¯i/a¯i)|1+γik|,1}, η is a positive constant, α  is given in the
proof of this theorem, t0=tp−ω.
Then, system
(2.6) has exactly one ω-periodic solution, which is globally
exponentially stable with the convergence index α−η.Proof.
By Theorem3.3, there exists an ω-periodic solution x*(t)=(x1*(t),x2*(t),…,xn*(t))T of (2.6) with initial value ϕ*=(ϕ1*,…,ϕn*)T. Suppose that x(t) is an arbitrary solution of system (2.6) with
initial value ϕ=(ϕ1,…,ϕn)T.
Set z(t)=(z1(t),z2(t),…,zn(t))T=x(t)−x*(t), then, from system (2.5), we
have
(4.1)zi′(t)=−[bi(hi−1(xi(t)))−bi(hi−1(xi*(t)))]+∑j=1n[bij(t)[fj(hj−1(xj(t−τij(t))))−fj(hj−1(xj*(t−τij(t))))]+cij(t)∫−∞tkij(t−s)[gj(hj−1(xj(s)))−gj(hj−1(xj*(s)))]ds],t≠tk,zi(tk+)=hi((1+bik)hi−1(xi(tk)))−hi((1+bik)hi−1(xi*(tk))),i=1,2,…,n.By θ<1,
we have(4.2)−qi+∑j=1na¯j[|bij(t)|Ljf+|cij(t)|Ljgβij]<0,i=1,2,…,n.Set(4.3)hi(λ)=λ−qi+∑j=1na¯j[|bij(t)|Ljfeλτ+|cij(t)|Ljg∫0+∞|kij(s)|eλsds],i=1,2,…,n.
Clearly, hi(λ),i=1,2,…,n are continuous functions on R.
Since(4.4)hi(0)≤−qi+∑j=1na¯j[|bij(t)|Ljf+|cij(t)|Ljgβij]<0,dhi(λ)dλ=1+∑j=1na¯j[τ|bij(t)|Ljfeλτ+|cij(t)|Ljg∫0+∞|kij(s)|seλsds]>0,
and hi(+∞)=+∞,
hence hi(λ),i=1,2,…,n are strictly monotone increasing functions.
Therefore, for any i∈{1,2,…,n} and t≥0,
there is a unique λ(t) such that
(4.5)λ(t)−qi+∑j=1na¯j[|bij(t)|Ljfeλ(t)τ+|cij(t)|Ljg∫0+∞|kij(s)|eλ(t)sds]=0.
Let λi*=inft≥0{λ(t)∣λ(t)−qi+∑j=1na¯j[|bij(t)|Ljfeλ(t)τ+|cij(t)|Ljg∫0+∞|kij(s)|eλ(t)sds]=0}.
Obviously, λi*≥0,i=1,2,…,n. Now, we will prove that λi*>0,i=1,2,…,n.
Suppose this is not true, from (4.2), there exists a positive constant η such that
(4.6)inft≥0,1≤i≤n{qi−∑j=1na¯j[|bij(t)|Ljf+|cij(t)|Ljgβij]1+∑j=1na¯j[1.5τ|bij(t)|Ljf+|cij(t)|Ljgβij]}≥η.
Pick small ε>0,
then there exists ti≥0 such that(4.7)0<λi*(ti)<ε<η,∫0+∞|kij(s)|eεsds<βij+η.
Let us recall the inequality ex<1+1.5x for sufficiently small x>0, then, we obtain(4.8)0=λi*(ti)−qi+∑j=1na¯j[|bij(ti)|Ljfeλi*(ti)τ+|cij(ti)|Ljg∫0+∞|kij(s)|eλi*(ti)sds]<η−qi+∑j=1na¯j[|bij(ti)|Ljf(1+1.5ητ)+|cij(ti)|Ljg(βij+η)]≤−qi+∑j=1na¯j[|bij(ti)|Ljf+|cij(ti)|Ljgβij]+qi−∑j=1na¯j[|bij(ti)|Ljf+|cij(ti)|Ljgβij]1+∑j=1na¯j[1.5τ|bij(ti)|Ljf+|cij(ti)|Ljgβij]×{1+∑j=1na¯j[1.5τ|bij(ti)|Ljf+|cij(ti)|Ljgβij]}=0,
which is a contradiction, and
hence, λi*>0,i=1,2,…,n.
Letα=min{λ1*,λ2*,…,λn*}.
Obviously, for all t≥0, we have
(4.9)hi(α)=α−qi+∑j=1na¯j[|bij(t)|Ljfeατ+|cij(t)|Ljg∫0+∞|kij(s)|eαsds]≤0,i=1,2,…,n.
It is obvious that(4.10)|zi(t)|≤∥ϕ−ϕ*∥≤∥ϕ−ϕ*∥e−αt,fort∈(−∞,0],i=1,2,…,n,
where ∥ϕ−ϕ*∥ is defined as that in Definition 2.2.
Define the Lyapunov functionalV=(V1,V2,…,Vn)T by Vi=eαt|zi(t)|, i=1,2,…,n. In view of (4.1), for t≠tk, k=1,2,…, we obtain
(4.11)d+Vi(t)dt≤eαt{(α−qi)|zi(t)|+∑j=1n[|bij(t)|Ljfa¯j|zj(t−τij(t))|+|cij(t)|∫−∞t|kij(t−s)|Ljga¯j|zj(s)|ds]}≤(α−qi)Vi(t)+∑j=1na¯j[|bij(t)|eατLjfVj(t−τij(t))+|cij(t)|Ljg∫−∞t|kij(t−s)|eα(t−s)Vj(s)ds].
We claim that
(4.12)Vi(t)=|zi(t)|eαt≤∥ϕ−ϕ*∥,i=1,2,…,n,∀t∈[0,t1).
Contrarily, there must
exist i0∈{1,2,…,n} and 0<t˜<t1 such that
(4.13)Vi0(t˜)=∥ϕ−ϕ*∥,d+Vi0(t˜)dt>0,Vi(t)≤∥ϕ−ϕ*∥,∀t∈(−∞,t˜],i=1,2,…,n.
Together with (4.11) and (4.13),
we obtain
(4.14)0<d+Vi0(t˜)dt≤∥ϕ−ϕ*∥{α−qi0+∑j=1na¯j[|bi0j(t˜)|eατLjf+|ci0j(t˜)|Ljg∫0+∞|ki0j(s)|eαsds]}.
Hence,
(4.15)0<α−qi0+∑j=1na¯j[|bi0j(t˜)|eατLjf+|ci0j(t˜)|Ljg∫0+∞|ki0j(s)|eαsds],
which contradicts (4.9). Hence,
(4.12) holds. It follows that
(4.16)|zi(t)|≤∥ϕ−ϕ*∥e−αt,∀t∈[0,t1),i=1,2,…,n.
Whent=t1,
from the second expression of (4.1), we have(4.17)|zi(t1+)|≤a¯ia¯i|1+γi1||zi(t1)|≤M1limt→t1−|zi(t1)|≤M1∥ϕ−ϕ*∥e−αt1,i=1,2,…,n.
Similar to the steps of
(4.9)–(4.16), we can also prove that
(4.18)|zi(t)|≤M1∥ϕ−ϕ*∥e−αt,t∈[t1,t2),i=1,2,…,n.
When t=t2, again, from the second expression of (4.1), we have
(4.19)|zi(t2+)|≤a¯ia¯i|1+bi2||zi(t2)|≤M1M2∥ϕ−ϕ*∥e−αt2,i=1,2,…,n.
By repeating the same procedure,
we can deduce the following general result:
(4.20)|zi(t)|≤M1M2⋯Mk∥ϕ−ϕ*∥e−αt,t∈[tk,tk+1),i=1,2,…,n.
By (H9),
we have Mk≤eη(tk−tk−1), which implies that
(4.21)M1M2⋯Mk≤eη(t1−t0)eη(t2−t1),…,eη(tk−tk−1)≤eη(ω−tp)eηt,t∈[tk,tk+1).
So, in light of (4.20) and
(4.21), we have
(4.22)|xi(t)−xi*(t)|=|zi(t)|≤eη(ω−tp)∥ϕ−ϕ*∥e−(α−η)t,∀t>0,i=1,2,…,n.
In view of Definition 2.2, the ω-periodic solution x*(t) of system (2.6) is globally exponentially
stable with the convergence index α−η.
This completes the proof.Remark 4.2.
Note that we have dropped the restriction: the activation
functions are bounded and|1+γik|≤1, which is indispensable in [13, 17].Remark 4.3.
To the best of our knowledge, most of the existing research
papers only give the existence of the convergence index, while we offer the
precise convergence index in Theorem4.1.


## 5. Application

In this section, we give an example to illustrate that
our results are feasible. Consider the following Cohen-Grossberg neural
networks with mixed delays and impulses:(5.1)ui′(t)=−ai(ui(t)){bi(ui(t))−∑j=12[bij(t)fj(uj(t−τij(t)))+cij(t)∫−∞tkij(t−s)gj(uj(s))ds]+Ii(t)},t≠tk,Δui(tk)=ui(tk+)−ui(tk−)=0.02ui(tk),i=1,2,tk=k−0.5,k=1,2,…,
where f1(u)=f2(u)=g1(u)=g2(u)=|u+1|, a1(u)=3+sinu, a2(u)=3−cosu, b1(u)=5u+sinu, b2(u)=5u−cosu, Ii(t)=sinπt, τij(t)=(cosπt)2,kij=sin(t−s)e−2(t−s),i,j=1,2: (5.2)(b11b12b21b22)=(16sinπt216cosπt2110sinπt218cosπt2),(c11c12c21c22)=(16cosπt216sinπt2110sinπt2112cosπt2).We haveω=4,L1f=L2f=L1g=L2g=1, a¯1=a¯2=4, a¯1=a¯2=2, l1=l2=4, βij=1/2: (5.3)θ≈0.1863<1,max1≤i≤2{θ+a¯ia¯i(1−e−qiω)∑k=1p|bik|}≈0.3463<1,lnMk/(tk−tk−1)=ln2.04=0.7129<1.
Submitting λ=1 into (4.9), we have h1(λ)<−3.9<0, h2(λ)<−4.9<0, (see Figures 1(a) and 1(b)). This implies that ln2.04<1<α. Hence, all the conditions needed in Theorem 4.1 are satisfied. Therefore,
system (5.1) has a unique 4-periodic solution, which is globally exponentially
stable (see Figures 1(c)–1(g)).(a) The  trajectory ofh1(λ). (b) The trajectory of h2(λ). (c) Numeric
simulation of (u1,u2). (d)  Numeric simulation of (u1,u2) without impulses. (e) Numeric
simulation of (u1,u2) with impulses. (f) Numeric simulation of (u1,u2) without impulses. (g) Numeric simulation of (u1,u2) with impulses.
(a)(b)(c)(d)(e)(f)(g)Remark 5.1.
System (5.1) is a simple CGNNs with time-varying coefficients,
mixed delays, and impulses. In system (5.1), the activation functions are all
unbounded and |1+bik|=1.02>1, i=1,2. Thus, none of the results in [13–17] can be
applied to (5.1). This implies that the results of this paper are new and
complement previously known results.



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