Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 491268, 17 pages
doi:10.1155/2009/491268
Research Article
Global Exponential Stability of
Delayed Cohen-Grossberg BAM Neural Networks
with Impulses on Time Scales
Yongkun Li,1 Yuchun Hua,1 and Yu Fei2
1
2
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
School of Statistics and Mathematics, Yunnan University of Finance and Economics,
Kunming, Yunnan 650221, China
Correspondence should be addressed to Yongkun Li, yklie@ynu.edu.cn
Received 18 April 2009; Accepted 14 July 2009
Recommended by Patricia J. Y. Wong
Based on the theory of calculus on time scales, the homeomorphism theory, Lyapunov functional
method, and some analysis techniques, sufficient conditions are obtained for the existence,
uniqueness, and global exponential stability of the equilibrium point of Cohen-Grossberg
bidirectional associative memory BAM neural networks with distributed delays and impulses
on time scales. This is the first time applying the time-scale calculus theory to unify the discretetime and continuous-time Cohen-Grossberg BAM neural network with impulses under the same
framework.
Copyright q 2009 Yongkun Li et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
In the recent years, bidirectional associative memory BAM neural networks and CohenGrossberg neural networks CGNNs with their various generalizations have attracted the
attention of many mathematicians, physicists, and computer scientists see 1–17 due to
their wide range of applications in, for example, pattern recognition, associative memory,
and combinatorial optimization. Particularly, as discussed in 18–20, in the hardware
implementation of the neural networks, when communication and response of neurons
happens time delays may occur. Actually, time delays are known to be a possible source
of instability in many real-world systems in engineering, biology, and so forth. see, e.g.,
21 and references therein. However, besides delay effect, impulsive effect likewise exists
in a wide variety of evolutionary processes in which states are changed abruptly at certain
moments of time, involving fields such as medicine and biology, economics, mechanics,
electronics, and telecommunications. As artificial electronic systems, neural networks such
as Hopfield neural networks, bidirectional neural networks, and recurrent neural networks
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Journal of Inequalities and Applications
often are subject to impulsive perturbations which can affect dynamical behaviors of the
systems just as time delays. Therefore, it is necessary to consider both impulsive effect and
delay effect on the stability of neural networks.
As is well known, both continuous and discrete systems are very important in
implementation and applications. However, it is troublesome to study the stability for
continuous and discrete systems, respectively. Therefore, it is worth studying a new method,
such as the time-scale theory, which can unify the continuous and discrete situations.
Motivated by the above discussions, the objective of this paper is to study the global
exponential stability of the following Cohen-Grossberg bidirectional associative memory
networks with impulses and time delays on time scales:
⎡
xiΔ t −ai xi t⎣bi xi t −
m
pji0 fj yj t − τji
j1
−
m
pji1
j1
Δxi tk Ik xi tk ,
∞
⎤
hij sfj yj t − s Δs ri ⎦,
t ≥ 0, t /
tk , t ∈ T,
0
i 1, 2, . . . , n, k 1, 2, . . . ,
1.1
n
yjΔ t −cj yj t dj yj t − qij0 gi xi t − σij
i1
−
Δyj tk Jk yj tk ,
n
i1
qij1
∞
kij sgi xi t − sΔs sj ,
t ≥ 0, t /
tk , t ∈ T,
0
j 1, 2, . . . , m, k 1, 2, . . . ,
where T is a time scale; Ik , Jk : R → R are continuous, xi t, yj t are the states of the
ith neuron from the neural field FX and the jth neuron from the neural field FY at time
t, respectively; fj , gi denote the activation functions of the jth neuron from FY and the ith
neuron from FX , respectively; ri and sj are constants, which denote the external inputs on
the ith neuron from FX and the jth neuron from FY , respectively; τji and σij correspond to
the transmission delays; ai xi t and cj yj t represent amplification functions; bi xi t
and dj yj t are appropriately behaved functions such that the solutions of system 1.1
remain bounded; pji0 , pji1 , qij0 , and qij1 denote the connection strengths which correspond to the
neuronal gains associated with the neuronal activations; Ii and Jj denote the external inputs.
For each interval I of R, we denote that by IT I T, Δxi tk xi t
k −xi t−k , Δyj tk yj t
k −
yj t−k are the impulses at moments tk , and xi t
k , xi t−k , yj t
k , yj t−k i 1, 2, . . . , n, j 1, 2, . . . , m represent the right and left limits of xi t
k and yj t
k in the sense of time scales;
0 < t1 < t2 < · · · < tk → ∞ is a strictly increasing sequence.
The system 1.1 is supplement with initial values given by
xi s ϕi s,
s ∈ −∞, 0T , i 1, 2, . . . , n,
yj s ψj s,
s ∈ −∞, 0T , j 1, 2, . . . , m,
where ϕi , ψj are continuous real-valued functions defined on their respective domains.
1.2
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3
As usual in the theory of impulsive differential equations, at the points of discontinuity
tk of the solution t → x1 t, x2 t, . . . , xn t, y1 t, y2 t, . . . , ym tT we assume that
xi tk xi t−k ,
yj tk yj t−k ,
xiΔ tk xiΔ t−k ,
yjΔ tk yjΔ t−k ,
1.3
for i 1, 2, . . . , n, j 1, 2, . . . , m.
The organization of the rest of this paper is as follows. In Section 2, we introduce
some notations and definitions, and state some preliminary results which are needed in
later sections. In Section 3, by means of homeomorphism theory, we study the existence
and uniqueness of the equilibrium point of system 1.1. In Section 4, by constructing a
suitable Lyapunov function, we establish the exponential stability of the equilibrium of 1.1.
In Section 5, we present an example to illustrate the feasibility and effectiveness of our results
obtained in previous sections.
2. Preliminaries
In this section, we will cite some definitions and lemmas which will be used in the proofs of
our main results.
Let T 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
σt inf{s ∈ T : s > t},
ρt sup{s ∈ T : s < t},
μt σt − t.
2.1
A point t ∈ T is called left dense if t > inf T and ρt t, left scattered if ρt < t,
right dense if t < sup T and σt t, and right scattered if σt > 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.
A function f : T → R is right dense continuous provided that it is continuous at right
dense point in T and its left-side limits exist at left-dense points in T. If f is continuous at
each right dense point and each left-dense point, then f is said to be a continuous function
on T. The set of continuous functions f : T → R will be denoted by CT.
For y : T → R and t ∈ Tk , we define the delta derivative of yt, yΔ t to be the
number if it exists with the property that for a given ε > 0, there exists a neighborhood U
of t such that
yσt − ys − yΔ tσt − s < ε|σt − s|
2.2
for all s ∈ U.
If y is continuous, then y is right dense continuous, and y is delta differentiable at t,
then y is continuous at t.
Let y be right dense continuous. If yΔ t yt, then we define the delta integral by
t
a
ysΔs Y t − Y a.
2.3
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Journal of Inequalities and Applications
Definition 2.1 see 22. For each t ∈ T, let N be a neighborhood of t, then, for V ∈ Crd T ×
Rn , R
, define D
V Δ t, xt to mean that, given ε > 0, there exists a right neighborhood
Nε ⊂ N of t such that
V σt, xσt − V s, xσt − μt, sft, xt
< D
V Δ t, xt ε
μt, s
2.4
for each s ∈ Nε , s > t, where μt, s ≡ σt − s. If t is rd and V t, xt is continuous at t, this
reduce to
D
V Δ t, xt V σt, xσt − V t, xσt
.
σt − t
2.5
Definition 2.2 see 23. If a ∈ T, sup T ∞, and f is right dense continuous on a, ∞, then
we define the improper integral by
∞
ftΔt lim
b
b→∞ a
a
2.6
ftΔt
provided that this limit exists, and we say that the improper integral converges in this case.
If this limit does not exist, then we say that the improper integral diverges.
A function r : T → R is called regressive if
1 μtrt /
0
2.7
for all t ∈ Tk .
If r is regressive function, then the generalized exponential function er is defined by
er t, s exp
t
for s, t ∈ T,
ξμτ rτΔτ
2.8
s
with the cylinder transformation
⎧
⎪
⎨ Log1 hz ,
h
ξh z ⎪
⎩z,
if h / 0,
2.9
if h 0.
Let p, q : T → R be two regressive functions, then we define
p ⊕ q : p q μpq,
p q : p ⊕ q ,
p :
p
.
1 μp
Then the generalized exponential function has the following properties.
2.10
Journal of Inequalities and Applications
5
Lemma 2.3 see 24. Assume that p, q : T → R are two regressive functions, then
i e0 t, s ≡ 1 and ep t, t ≡ 1
ii ep σt, s 1 μtptep t, s
iii ep t, σs ep t, s/1 μsps
iv 1/ep t, s ep t, s
v ep t, s 1/ep s, t ep s, t
vi ep t, sep s, r ep t, r
vii ep t, seq t, s ep⊕q t, s
viii ep t, s/eq t, s epq t, s.
∗ T
of system
Definition 2.4. The equilibrium point u∗ x1∗ , x2∗ , . . . , xn∗ , y1∗ , y2∗ , . . . , ym
1.1 is said to be exponentially stable if there exists a positive constant α such
that for every δ ∈ T, there exists N Nδ ≥ 1 such that the solution ut x1 t, x2 t, . . . , xn t, y1 t, y2 t, . . . , ym tT of 1.1 with initial value
ϕ1 s, ϕ2 s, . . . , ϕn s, ψ1 s, ψ2 s, . . . , ψm sT satisfies
⎤
⎡
n
m
max ϕi δ − xi∗ max ψj δ − yj∗ ⎦.
u − u∗ ≤ Ne−α t, δ⎣
i1
δ∈−∞,0T
j1
δ∈−∞,0T
2.11
Lemma 2.5 see 25. If Hx ∈ CRn
m , Rn
m satisfies the following conditions:
i Hx is injective on Rn
m ,
ii H → ∞ as x → ∞,
then Hx is a homeomorphism of Rn
m onto itself.
For z x1 , x2 , . . . , xn , y1 , y2 , . . . , ym T ∈ Rn
m , we define the norm as
z n
|xi | i1
m yj .
2.12
j1
Throughout this paper, we assume that
H1 ai , cj ∈ CT, R
, and satisfy 0 < ai ≤ ai x ≤ ai , 0 < cj ≤ cj x ≤ cj , ∀x ∈ R, i 1, 2, . . . , n, j 1, 2, . . . , m;
H2 the activation functions fj , gi ∈ CR, R and there exist positive constants Mj , Ni
such that
fj x − fj y ≤ Mj x − y,
for all x, y ∈ R, i 1, . . . , n, j 1, . . . , m;
gi x − gi y ≤ Ni x − y,
2.13
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Journal of Inequalities and Applications
H3 bi , dj ∈ CR, R, bi 0 dj 0 0, i 1, 2, . . . , n, j 1, 2, . . . , m, and there exist
positive constants ηi , ωj such that
bi x − bi y
≥ ηi ,
x−y
dj x − dj y
≥ ωj ,
x−y
∀x / y;
2.14
H4 the kernels hji and
0, ∞T are nonnegative
func ∞ continuousintegral
∞ kij defined on ∞
∞
tions such that 0 hji sΔs 1, 0 shji sΔs < ∞, 0 kij sΔs 1, 0 skij sΔs <
∞.
3. Existence and Uniqueness of the Equilibrium
In this section, using homeomorphism theory, we will study the existence and uniqueness of
the equilibrium point of system 1.1.
∗ T
∈ Rn
m
An equilibrium point of 1.1 is a constant vector x1∗ , x2∗ , . . . , xn∗ , y1∗ , y2∗ , . . . , ym
which satisfies the system
⎡
⎤
m ∗ ∗ pji0 pji1 fj yj∗ ri ⎦ 0,
ai xi ⎣bi xi −
i 1, 2, . . . , n,
j1
3.1
n cj yj∗ dj yj∗ −
qij0 qij1 gi xi∗ sj 0,
j 1, 2, . . . , m,
i1
where the impulsive jumps Ik ·, Jk · satisfy
Ik xi∗ 0,
i 1, 2, . . . , n,
Jk yj∗ 0,
j 1, 2, . . . , m.
3.2
From the assumptions H1 and H4 , it follows that
m pji0 pji1 fj yj∗ ri ,
bi xi∗ i 1, 2, . . . , n,
j1
dj yj∗ n qij0 qij1 gi xi∗ sj ,
3.3
j 1, 2, . . . , m.
i1
Noting that if bi−1 ·, dj−1 · exist and activation functions fj · and gj · are bounded, then
the existence of an equilibrium point of system 1.1 is easily obtained from Brouwer’s fixed
point theorem. We can refer to 2–8.
Journal of Inequalities and Applications
7
Theorem 3.1. Assume that H2 and H3 hold. Suppose further that for each i 1, 2, . . . , n, j 1, 2, . . . , m, the following inequalities are satisfied:
ηi >
m 0
qij qij1 Ni ,
ωj >
j1
n 0
pji pji1 Mj .
3.4
i1
Then there exists a unique equilibrium point of system 1.1.
Proof. Consider a mapping Φ : Rn
m → Rn
m defined by
Φi z bi xi −
m pji0 pji1 fj yj ri ,
i 1, 2, . . . , n,
j1
n Φi z dj yj −
qij0 qij1 gi xi sj ,
3.5
j 1, 2, . . . , m,
i1
where z x1 , x2 , . . . , xn , y1 , y2 , . . . , ym T ∈ Rn
m , Φz Φ1 z, . . . , Φn z, . . . , Φn
m zT ∈
Rn
m . First, we want to show that Φ is an injective mapping on Rn
m . By contradiction,
suppose that there exists a distinct z, z ∈ Rn
m such that Φz Φz, where z x1 , x2 , . . . , xn , y1 , y2 , . . . , ym T ∈ Rn
m and z x1 , x2 . . . , xn , y 1 , y2 , . . . , y m T ∈ Rn
m . Then
it follows from 3.5 that
bi xi − bi xi m pji0 pji1
fj yj − fj yj ,
i 1, 2, . . . , n,
j1
n dj yj − dj yj qij0 qij1 gi xi − gj xi ,
3.6
j 1, 2, . . . , m.
i1
In view of H2 -H3 and 3.6, we have
m n
n 0
ηi |xi − xi | ≤
pji pji1 Mj yj − y j ,
i1
i1 j1
n m
m 0
ωj yj − y j ≤
qij qij1 Ni |xi − xi |.
j1
3.7
j1 i1
Thus, we can obtain
⎡
⎤
n
m m
n 0
1
0
1
⎣ηi − qij qij Ni ⎦|xi − xi | ωj − pji pji Mj yj − y j ≤ 0.
i1
j1
j1
3.8
i1
It follows from 3.4 and 3.8 that |xi − xi | 0 and |yj − y j | 0, i 1, 2, . . . , n, j 1, 2, . . . , m.
That is z z, which leads to a contradiction. Therefore, Φ is an injective on Rn
m .
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Journal of Inequalities and Applications
Then we will prove Φ is a homeomorphism on Rn
m . For convenience, we let Φz
Φz − Φ0, where
i z bi xi −
Φ
m pji0 pji1
fj yj − fj 0 ,
i 1, 2, . . . , n,
j1
n n
j z dj yj −
qij0 qij1 gi xi − gi 0 ,
Φ
3.9
j 1, 2, . . . , m.
i1
→ ∞ as z → ∞. Otherwise there is a sequence {zv } such that zv →
We assert that Φ
v
v T
is bounded as v → ∞, where zv x1v , x2v , . . . , xnv , y1v , y2v , . . . , ym
∈ Rn
m .
∞ and Φz
Noting that
n
n
m i zv pji0 pji1 fj yjv − fj 0
sgn xiv bi xiv − Φ
sgn xiv
i1
i1
≤
j1
m n 0
pji pji1 Mj yjv ,
i1 j1
m
sgn
yjv
m
n n
j zv dj yjv − Φ
qij0 qij1 gi xiv − gi 0
sgn yjv
j1
j1
≤
3.10
i1
n m 0
qij qij1 Ni xiv ,
j1 i1
we have
n
m
i zv sgn yv dj yv − Φ
n
j zv sgn xiv bi xiv − Φ
j
j
i1
j1
m n n m 0
0
≤
pji pji1 Mj yjv qij qij1 Ni xiv .
i1 j1
3.11
j1 i1
On the other hand, we have
n
m
n
j zv i zv sgn yv dj yv − Φ
sgn xiv bi xiv − Φ
j
j
i1
j1
n
n m m
v v ≥
ηi xiv − Φ
ωj yj − Φn
j zv .
i z i1
i1
j1
j1
3.12
Journal of Inequalities and Applications
9
It follows from 3.11 and 3.12 that
⎞
n m n m
v v
v
Θ⎝ xi yj ⎠ ≤
Φi z Φn
j zv ,
⎛
i1
j1
i1
3.13
j1
where
⎧
⎨
⎧
⎨
⎫
⎫
m n ⎬
⎬
Θ min min ηi − qij0 qij1 Ni , min ωj − pji0 pji1 Mj
> 0.
⎩1≤i≤n⎩
⎭ 1≤j≤m
⎭
j1
i1
3.14
That is
zv ≤
%
1%
% v %
%Φz %,
Θ
3.15
which contradicts our choice of {zv }. Hence, Φ satisfies Φ → ∞ as z → ∞.
By Lemma 2.5, Φ is a homeomorphism on Rn
m and there exists a unique point z∗ ∗ T
such that Φz∗ 0. From the definition of Φ, we know that
x1∗ , x2∗ , . . . , xn∗ , y1∗ , y2∗ , . . . , ym
∗
∗
∗
∗
∗
∗
∗ T
z x1 , x2 , . . . , xn , y1 , y2 , . . . , ym
is the unique equilibrium point of 1.1.
4. Global Exponential Stability of the Equilibrium
In this section, we will construct some suitable Lyapunov functions to derive the sufficient
conditions which ensure that the equilibrium of 1.1 is globally exponentially stable.
Theorem 4.1. Assume that (H1 )–( H4 ) hold, suppose further that
H5 for each i 1, 2, . . . , n, j 1, 2, . . . , m, the following inequalities are satisfied:
ai ηi >
m
cj qij0 qij1 Ni ,
j1
cj ωj >
n
ai pji0 pji1 Mj
4.1
i1
H6 the impulsive operators Iik xi t and Jjk yj t satisfy
Iik xi tk −γik xi tk − xi∗ , 0 < γik < 2, i 1, . . . , n, k ∈ Z
,
Jjk yj tk −
γjk yj tk − yj∗ , 0 < γjk < 2, j 1, . . . , m, k ∈ Z
.
4.2
Then the unique equilibrium point of system 1.1 is globally exponentially stable.
Proof. According to Theorem 3.1, we know that 1.1 has a unique equilibrium point z∗ ∗ T
. In view of H6 , it is easy to see that Ii xi∗ 0 and Jj yj∗ 0.
x1∗ , x2∗ , . . . , xn∗ , y1∗ , . . . , ym
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Journal of Inequalities and Applications
Suppose that zt x1 t, x2 t, . . . , xn t, y1 t, y2 t, . . . , ym tT is an arbitrary solution of
1.1. Let ui t xi t − xi∗ , vj t yj t − yj∗ , t ≥ 0, then system 1.1 can be rewritten as
⎡
uΔ
ai ui t⎣bi ui t −
i t −
m
pji0 fj vj t − τji
j1
−
m
pji1
j1
∞
⎤
hji sfj vj t − s Δs − ri ⎦,
0
i 1, 2, . . . , n, t > 0, t /
tk , t ∈ T,
vjΔ t −
cj vj t dj vj t −
4.3
n
qij0 gi ui t − σij
i1
−
n
i1
qij1
∞
kij s
gi ui t − sΔs − sj ,
0
j 1, 2, . . . , m, t > 0, t /
tk , t ∈ T,
where, for i 1, 2, . . . , n, j 1, 2, . . . , m,
ai ui t ai ui t xi∗ ,
cj vj t cj vj t yj∗ ,
dj vj t dj vj t yj∗ − bi yj∗ ,
bi ui t bi ui t xi∗ − bi xi∗ ,
fi vi t fj vj t yj∗ − fj yj∗ ,
4.4
gi ui t gi ui t xi∗ − bi xi∗ .
Also, for all t tk , k ∈ Z
, i 1, 2, . . . , n, j 1, 2, . . . , m,
ui t xi t − x∗ xi tk Ik xi tk − x∗ k
k
i
i
1 − γik xi tk − x∗ ≤ xi tk − x∗ ≤ |ui tk |,
i
i
vj t yj t − y∗ yj tk Ik yj tk − y∗ k
k
j
j
1 − γjk yj tk − yj∗ ≤ yj tk − yj∗ ≤ vj tk .
4.5
Journal of Inequalities and Applications
11
Hence by H1 and H2 , we have
D
|ui t|Δ ≤ −ai ηi |ui t| ai
m 0 pji Mj vj t − τji j1
m ∞
ai pji1 Mj hji svj t − sΔs,
4.6
i 1, 2, . . . , n,
0
j1
n Δ
D
vj t ≤ −cj vj t cj qij0 Ni ui t − σij i1
n cj qij1 Ni
i1
∞
4.7
kij s|ui t − s|Δs,
j 1, 2, . . . , m.
0
Also, for i 1, 2, . . . , n,
xi tk 0 − xi∗ tk 0 xi tk Iik xi tk − xi∗ tk − Iik xi∗ tk 1 − γik xi tk − xi∗ tk , k ∈ Z
,
4.8
xi t − x∗ t
1 − γik xi tk − x∗ tk i k
i
k
≤ xi tk − xi∗ tk , i 1, . . . , n, k ∈ Z
.
4.9
thus
Similarly, we have
∗ yj tk − yj∗ tk yj t
k − yj∗ t
k 1 − γjk
≤ yj tk − yj∗ tk , j 1, 2, . . . , m, k ∈ Z
.
4.10
Let Gi and G∗j be defined by
Gi εi ai ηi − εi −
m
cj qij0 Ni eεi σt, t − σij
j1
−
m
∞
cj qij1 Ni kij seεi σt, t − sΔs,
i 1, 2, . . . , n,
0
j1
4.11
n
G∗j ξj cj ωj − ξj − ai pji0 Mj eξj σt, t − τji
i1
−
n
i1
∞
ai pji1 Mj hji seξj σt, t − sΔs,
0
j 1, 2, . . . , m,
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Journal of Inequalities and Applications
respectively, where εi , ξj ∈ 0, ∞. By H5 , we have
Gi 0 ai ηi −
m
cj qij0 qij1 Ni > 0,
i 1, 2, . . . , n,
j1
n
G∗j 0 cj ωj − ai pji0 pji1 Mj > 0,
4.12
j 1, 2, . . . , m.
i1
Since Gi , G∗j are continuous on 0, ∞ and Gi εi → −∞, G∗j ξj → −∞, as εi → ∞, ξj →
∞, there exist εi∗ , ξj∗ > 0 such that Gi εi∗ 0, G∗j ξj∗ 0 and Gi εi > 0, for εi ∈ 0, εi∗ , G∗j ξj >
0, for ξj ∈ 0, ξj∗ . By choosing α min1≤i≤n,1≤j≤m {εi∗ , ξj∗ }, we obtain
Gi α ai ηi − α −
m
cj qij0 Ni eα σt, t − σij
j1
m
∞
− cj qij1 Ni kij seα σt, t − sΔs
0
j1
≥ 0,
G∗j α
i 1, 2, . . . , n,
n
cj ωj − α − ai pji0 Mj eα σt, t − τji
4.13
i1
n
∞
− ai pji1 Mj hji seα σt, t − sΔs
0
i1
≥ 0,
j 1, 2, . . . , m,
Denote
μi t eα t, δ|ui t|, t ∈ R, i 1, 2, . . . , n,
νj t eα t, δvj t, t ∈ R, j 1, 2, . . . , m,
4.14
4.15
where δ ∈ −∞, 0T . For t > 0, t / tk , k ∈ Z
, i 1, 2, . . . , n, j 1, 2, . . . , m, it follows from
4.6–4.15, we can obtain
Δ
D μΔ
i t αeα t, δ|ui t| eα σt, δD |ui t|
≤ αeα t, δ|ui t| eα σt, δ
⎡
m × ⎣−ai ηi |ui t| ai pji0 Mj vj t − τji j1
⎤
m ∞
ai pji1 Mj hji svj t − sΔs⎦
j1
0
Journal of Inequalities and Applications
13
m ≤ − ai ηi − α μi t ai pji0 Mj eα σt, t − τji νj t − τji
j1
ai
m 1
pji Mj
∞
hji seα σt, t − sνj t − sΔs,
0
j1
n D
νjΔ t ≤ − cj ωj − α νj t cj qij0 Ni eα σt, t − σij μi t − σij
i1
ci
n 1
qij Ni
∞
i1
kij seα σt, t − sμi t − sΔs.
0
4.16
Also,
μi t
k ≤ μi tk , νj t
k ≤ νj tk ,
i 1, 2, . . . , n, j 1, 2, . . . , m, k ∈ Z
.
4.17
Define a Lyapunov function
V t n
⎡
⎣μi t ai
i1
m t
0
pji Mj eα σt, t − τji
νj sΔs
t−τji
j1
⎤
t
m ∞
ai pji1 Mj hji seα σt, t − s
νj zΔzΔs⎦
0
j1
m
νj t cj
j1
t−s
t
0
qij Ni eα σt, t − σij
n 4.18
μi sΔs
t−σij
i1
t
n ∞
1
cj qij Ni kij seα σt, t − s
μi zΔzΔs .
0
i1
t−s
And we note that V t > 0 for t > 0 and V 0 > 0. Calculating the Δ-derivatives of V , we get
D
V Δ t ≤
n
⎡
m ⎣− a ηi − η μi t ai pji0 Mj eα σt, t − τji νj t
i
i1
j1
⎤
m ∞
ai pji1 Mj hji seα σt, t − sνj tΔs⎦
j1
m
0
n − cj ωj − η νj t cj qij0 Ni eα σt, t − σij μi t
j1
i1
n ∞
cj qij1 Ni kij seα σt, t − sμi tΔs
i1
0
14
Journal of Inequalities and Applications
⎡
n
m
− ⎣ai ηi − η − cj qij0 Ni eη σt, t − σij
i1
j1
⎤
m
∞
− cj qij1 Ni kij seα σt, t − sΔs⎦μi t
0
j1
−
m
cj ωj − η −
j1
n
ai pji0 Mj eη σt, t − τji
i1
n
∞
1
− ai pji Mj hji seα σt, t − sΔs νj t
0
i1
−
n
m
Gi η μi t − G∗j η νj t
i1
j1
≤ 0, t > 0, t /
tk , t ∈ T, k ∈ Z
.
4.19
Also,
⎡
n
m tk
⎣μi t
ai pji0 Mj eα σ t
, t
− τji
V tk k
k
k
i1
t
k −τji
j1
νj sΔs
⎤
t
m ∞
k
ai pji1 Mj hji seα σ t
k , t
k − s
νj zΔzΔs⎦
m
t
k −s
0
j1
n tk
0
νj tk qij Ni eα σ tk , tk − σij
j1
t
k −σij
i1
μi sΔs
n ∞
tk
1
cj qij Ni kij seα σ tk , tk − s
μi zΔzΔs
≤
n
⎡
⎣μi tk ai
i1
t
k −s
0
i1
4.20
m tk
0
pji Mj eα σtk , tk − τji
tk −τji
j1
νj sΔs
⎤
tk
m ∞
νj zΔzΔs⎦
ai pji1 Mj hji seα σtk , tk − s
m
j1
νj tk cj
tk
n 0
qij Ni eα σtk , tk − σij
i1
tk −σij
μi sΔs
tk
n ∞
1
μi zΔzΔs
cj qij Ni kij seα σtk , tk − s
i1
V tk ,
tk −s
0
j1
k ∈ Z
.
0
tk −s
Journal of Inequalities and Applications
15
It follows that V t ≤ V 0 for t > 0 and hence, for t > 0, we can obtain
n
m
μi t i1
⎡
n
m 0
0
⎣
μi 0 pji Mj eα σ0, −τji
νj t ≤
j1
i1
νj sΔs
0−τji
j1
⎤
0
m ∞
νj zΔzΔs⎦
ai pji1 Mj hji seα σ0, 0 − s
0
j1
0−s
m
n 0
νj 0 qij0 Ni eα σ0, 0 − σij
j1
4.21
μi sΔs
0−σij
i1
0
n ∞
1
μi zΔzΔs .
cj qij Ni kij seα σ0, 0 − s
0
i1
0−s
In view of 4.14-4.15 and the previous inequality, we have
n m xi t − x∗ t yj t − y∗ t
i
i1
j
j1
⎡ ⎛
n
m
≤ eα t, δ⎣ ⎝1 ci qij0 Ni eα σ0, −σij σij
i1
j1
m
∞
ci qij1 Ni kij seα σ0, −ssΔs
m
j1
0
i1
'
1
&
max ϕi δ − xi∗ δ
δ∈−∞,0T
n
ai pji0 Mj eα σ0, −τji τji
i1
&
n
∞
ai pji1 Mj hji seα σ0, −ssΔs
i1
0
⎤
⎦
∗
max ψj δ − yj δ
δ∈−∞,0T
⎤
⎡
n
m
≤ Neα t, δ⎣
max ϕi δ − xi∗ δ max ψj δ − yj∗ δ⎦,
i1
δ∈−∞,0T
j1
δ∈−∞,0T
4.22
where
⎧
⎨
N max 1 ⎩
m
ci qij0 Ni eα σ0, −σij σij
j1
m
n
∞
ci qij1 Ni kij seα σ0, −ssΔs, 1 ai pji0 Mj eα σ0, −τji τji
j1
n
0
i1
0
∞
ai pji1 Mj hji seα σ0, −ssΔs
The proof is complete.
i1
≥ 1.
4.23
16
Journal of Inequalities and Applications
5. An Example
In this section, we give an example to illustrate our results.
Consider the following Cohen-Grossberg BAM neural networks system with distributed delays and impulses:
(
)
1
x1Δ t − 1 cos x1 t
3
∞
+
*
1 −s
1
e sin 2y1 t − s Δs r1 ,
× 5x1 t − sin 2y1 t − 1 −
4
0 4
t > 0, t / tk , t ∈ T,
Δx1 tk I1 x1 tk ,
k 1, 2, . . . ,
5.1
(
)
1
yΔ t − 1 sin y1 t
3
∞
+
*
1 −s
1
e cos2x1 t − sΔs s1 ,
× 3y1 t − cos2x1 t − 1 −
4
0 4
Δy1 tk J1 y1 tk ,
t > 0, t /
tk , t ∈ T,
k 1, 2, . . . ,
where T R, γ1k 1 1/2 sin1 k, γ1k 1 2/3 cos 2k, k ∈ Z
. A simple computation
0
0
0
0
p11
q11
q11
shows that a c 2/3, a c 4/3, M1 N1 1, η1 5, ω1 3, p11
1/4, 0 < γ1k , γ1k < 2. It is easy to check that all conditions of Theorems 3.1 and 4.1 are satisfied.
Hence, 5.1 has a unique equilibrium point, which is globally exponentially stable.
6. Conclusion
Using the time-scale calculus theory, the homeomorphism theory and the Lyapunov
functional method, some sufficient conditions are obtained to ensure the existence and
the global exponential stability of the unique equilibrium point of Cohen-Grossberg BAM
neural networks with distributed delays and impulses on time scales. This is the first time
applying the time-scale calculus theory to unify and improve impulsive Cohen-Grossberg
BAM neural networks with distributed delays on time scales under the same framework.
The sufficient conditions we obtained can easily be checked in practice by simple algebraic
methods.
Acknowledgments
This work was supported by the National Natural Sciences Foundation of People’s Republic
of China and the Natural Sciences Foundation of Yunnan Province under Grant 04Y239A.
Journal of Inequalities and Applications
17
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