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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 Fei
2
1
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
2
School of Statistics and Mathematics, Yunnan University of Finance and Economics,
Kunming, Yunnan 650221, China
Correspondence should be addressed to Yongkun Li,
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 t heory, 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 discrete-
time 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 Cohen-
Grossberg 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
2 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:
x
Δ
i
t
−a
i
x
i
t
⎡
⎣
b
i
x
i
t
−
m
j1
p
0
ji
f
j
y
j
t − τ
ji
−
m
j1
p
1
ji
∞
0
h
ij
s
f
j
y
j
t − s
Δs r
i
⎤
⎦
,t≥ 0,t
/
t
k
,t∈ T,
Δx
i
t
k
I
k
x
i
t
k
,i 1, 2, ,n, k 1, 2, ,
y
Δ
j
t
−c
j
y
j
t
d
j
y
j
t
−
n
i1
q
0
ij
g
i
x
i
t − σ
ij
−
n
i1
q
1
ij
∞
0
k
ij
s
g
i
x
i
t − s
Δs s
j
,t≥ 0,t
/
t
k
,t∈ T,
Δy
j
t
k
J
k
y
j
t
k
,j 1, 2, ,m, k 1, 2, ,
1.1
where T is a time scale; I
k
,J
k
: R → R are continuous, x
i
t, y
j
t are the states of the
ith neuron from the neural field F
X
and the jth neuron from the neural field F
Y
at time
t, respectively; f
j
,g
i
denote the activation functions of the jth neuron from F
Y
and the ith
neuron from F
X
, respectively; r
i
and s
j
are constants, which denote the external inputs on
the ith neuron from F
X
and the jth neuron from F
Y
, respectively; τ
ji
and σ
ij
correspond to
the transmission delays; a
i
x
i
t and c
j
y
j
t represent amplification functions; b
i
x
i
t
and d
j
y
j
t are appropriately behaved functions such that the solutions of system 1.1
remain bounded; p
0
ji
,p
1
ji
,q
0
ij
, and q
1
ij
denote the connection strengths which correspond to the
neuronal gains associated with the neuronal activations; I
i
and J
j
denote the external inputs.
For each interval I of R, we denote that by I
T
I
T, Δx
i
t
k
x
i
t
k
−x
i
t
−
k
, Δy
j
t
k
y
j
t
k
−
y
j
t
−
k
are the impulses at moments t
k
,andx
i
t
k
,x
i
t
−
k
,y
j
t
k
,y
j
t
−
k
i 1, 2, ,n,j
1, 2, ,m represent the right and left limits of x
i
t
k
and y
j
t
k
in the sense of time scales;
0 <t
1
<t
2
< ···<t
k
→∞is a strictly increasing sequence.
The system 1.1 is supplement with initial values given by
x
i
s
ϕ
i
s
,s∈
−∞, 0
T
,i 1, 2, ,n,
y
j
s
ψ
j
s
,s∈
−∞, 0
T
,j 1, 2, ,m,
1.2
where ϕ
i
, ψ
j
are continuous real-valued functions defined on their respective domains.
Journal of Inequalities and Applications 3
As usual in the theory of impulsive differential equations, at the points of discontinuity
t
k
of the solution t → x
1
t,x
2
t, ,x
n
t,y
1
t,y
2
t, ,y
m
t
T
we assume that
x
i
t
k
x
i
t
−
k
,y
j
t
k
y
j
t
−
k
,x
Δ
i
t
k
x
Δ
i
t
−
k
,y
Δ
j
t
k
y
Δ
j
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.InSection 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
σ
t
inf
{
s ∈ T : s>t
}
,ρ
t
sup
{
s ∈ T : s<t
}
,μ
t
σ
t
− t. 2.1
Apointt ∈ T is called left dense if t>inf T and ρtt, left scattered if ρt <t,
right dense if t<sup T and σtt, and right scattered if σt >t.IfT has a left-scattered
maximum m, then T
k
T \{m}; otherwise T
k
T.IfT has a right-scattered minimum m,
then T
k
T \{m}; otherwise T
k
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.Iff 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 ∈ T
k
, we define the delta derivative of yt,y
Δ
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
y
σ
t
− y
s
− y
Δ
t
σ
t
− s
<ε
|
σ
t
− 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 y
Δ
tyt, then we define the delta integral by
t
a
y
s
Δs Y
t
− Y
a
.
2.3
4 Journal of Inequalities and Applications
Definition 2.1 see 22. For each t ∈ T,letN be a neighborhood of t, then, for V ∈ C
rd
T ×
R
n
, R
, define D
V
Δ
t, xt to mean that, given ε>0, there exists a right neighborhood
N
ε
⊂ N of t such that
V
σ
t
,x
σ
t
− V
s, x
σ
t
− μ
t, s
f
t, x
t
μ
t, s
<D
V
Δ
t, x
t
ε
2.4
for each s ∈ N
ε
, s>t, where μt, s ≡ σt − s.Ift is rd and V t, xt is continuous at t,this
reduce to
D
V
Δ
t, x
t
V
σ
t
,x
σ
t
− V
t, x
σ
t
σ
t
− t
.
2.5
Definition 2.2 see 23.Ifa ∈ T, sup T ∞,andf is right dense continuous on a, ∞, then
we define the improper integral by
∞
a
f
t
Δt lim
b →∞
b
a
f
t
Δt
2.6
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 μ
t
r
t
/
0 2.7
for all t ∈ T
k
.
If r is regressive function, then the generalized exponential function e
r
is defined by
e
r
t, s
exp
t
s
ξ
μτ
r
τ
Δτ
for s, t ∈ T, 2.8
with the cylinder transformation
ξ
h
z
⎧
⎪
⎨
⎪
⎩
Log
1 hz
h
, if h
/
0,
z, if h 0.
2.9
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
.
2.10
Then the generalized exponential function has the following properties.
Journal of Inequalities and Applications 5
Lemma 2.3 see 24. Assume that p, q : T → R are two regressive functions, then
i e
0
t, s ≡ 1 and e
p
t, t ≡ 1
ii e
p
σt,s1 μtpte
p
t, s
iii e
p
t, σs e
p
t, s/1 μsps
iv 1/e
p
t, se
p
t, s
v e
p
t, s1/e
p
s, t e
p
s, t
vi e
p
t, se
p
s, re
p
t, r
vii e
p
t, se
q
t, se
p⊕q
t, s
viii e
p
t, s/e
q
t, se
pq
t, s.
Definition 2.4. The equilibrium point u
∗
x
∗
1
,x
∗
2
, ,x
∗
n
,y
∗
1
,y
∗
2
, ,y
∗
m
T
of system
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 solu-
tion utx
1
t,x
2
t, ,x
n
t,y
1
t,y
2
t, ,y
m
t
T
of 1.1 with initial value
ϕ
1
s,ϕ
2
s, ,ϕ
n
s,ψ
1
s,ψ
2
s, ,ψ
m
s
T
satisfies
u − u
∗
≤ Ne
−α
t, δ
⎡
⎣
n
i1
max
δ∈−∞,0
T
ϕ
i
δ
− x
∗
i
m
j1
max
δ∈−∞,0
T
ψ
j
δ
− y
∗
j
⎤
⎦
. 2.11
Lemma 2.5 see 25. If Hx ∈ CR
nm
, R
nm
satisfies the following conditions:
i Hx is injective on R
nm
,
ii H→∞ as x→∞,
then Hx is a homeomorphism of R
nm
onto itself.
For z x
1
,x
2
, ,x
n
,y
1
,y
2
, ,y
m
T
∈ R
nm
, we define the norm as
z
n
i1
|
x
i
|
m
j1
y
j
.
2.12
Throughout this paper, we assume that
H
1
a
i
,c
j
∈ CT, R
, and satisfy 0 <a
i
≤ a
i
x ≤ a
i
, 0 <c
j
≤ c
j
x ≤ c
j
, ∀x ∈ R,i
1, 2, ,n, j 1, 2, ,m;
H
2
the activation functions f
j
,g
i
∈ CR, R and there exist positive constants M
j
,N
i
such that
f
j
x
− f
j
y
≤ M
j
x − y
,
g
i
x
− g
i
y
≤ N
i
x − y
, 2.13
for all x, y ∈ R,i 1, ,n, j 1, ,m;
6 Journal of Inequalities and Applications
H
3
b
i
,d
j
∈ CR, R,b
i
0d
j
00,i 1, 2, ,n, j 1, 2, ,m, and there exist
positive constants η
i
,ω
j
such that
b
i
x
− b
i
y
x − y
≥ η
i
,
d
j
x
− d
j
y
x − y
≥ ω
j
, ∀x
/
y;
2.14
H
4
the kernels h
ji
and k
ij
defined on 0, ∞
T
are nonnegative continuous integral func-
tions such that
∞
0
h
ji
sΔs 1,
∞
0
sh
ji
sΔs<∞,
∞
0
k
ij
sΔs 1,
∞
0
sk
ij
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.
An equilibrium point of 1.1 is a constant vector x
∗
1
,x
∗
2
, ,x
∗
n
,y
∗
1
,y
∗
2
, ,y
∗
m
T
∈ R
nm
which satisfies the system
a
i
x
∗
i
⎡
⎣
b
i
x
∗
i
−
m
j1
p
0
ji
p
1
ji
f
j
y
∗
j
r
i
⎤
⎦
0,i 1, 2, ,n,
c
j
y
∗
j
d
j
y
∗
j
−
n
i1
q
0
ij
q
1
ij
g
i
x
∗
i
s
j
0,j 1, 2, ,m,
3.1
where the impulsive jumps I
k
·,J
k
· satisfy
I
k
x
∗
i
0,i 1, 2, ,n, J
k
y
∗
j
0,j 1, 2, ,m. 3.2
From the assumptions H
1
and H
4
, it follows that
b
i
x
∗
i
m
j1
p
0
ji
p
1
ji
f
j
y
∗
j
r
i
,i 1, 2, ,n,
d
j
y
∗
j
n
i1
q
0
ij
q
1
ij
g
i
x
∗
i
s
j
,j 1, 2, ,m.
3.3
Noting that if b
−1
i
·,d
−1
j
· exist and activation functions f
j
· and g
j
· 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 H
2
and H
3
hold. Suppose further that for each i 1, 2, ,n,j
1, 2, ,m, the following inequalities are satisfied:
η
i
>
m
j1
q
0
ij
q
1
ij
N
i
,ω
j
>
n
i1
p
0
ji
p
1
ji
M
j
.
3.4
Then there exists a unique equilibrium point of system 1.1.
Proof. Consider a mapping Φ : R
nm
→ R
nm
defined by
Φ
i
z
b
i
x
i
−
m
j1
p
0
ji
p
1
ji
f
j
y
j
r
i
,i 1, 2, ,n,
Φ
i
z
d
j
y
j
−
n
i1
q
0
ij
q
1
ij
g
i
x
i
s
j
,j 1, 2, ,m,
3.5
where z x
1
,x
2
, ,x
n
,y
1
,y
2
, ,y
m
T
∈ R
nm
, ΦzΦ
1
z, ,Φ
n
z, ,Φ
nm
z
T
∈
R
nm
.First,wewanttoshowthatΦ is an injective mapping on R
nm
. By contradiction,
suppose that there exists a distinct z,
z ∈ R
nm
such that ΦzΦz, where z
x
1
,x
2
, ,x
n
,y
1
,y
2
, ,y
m
T
∈ R
nm
and z x
1
, x
2
,x
n
, y
1
, y
2
, ,y
m
T
∈ R
nm
. T hen
it follows from 3.5 that
b
i
x
i
− b
i
x
i
m
j1
p
0
ji
p
1
ji
f
j
y
j
− f
j
y
j
,i 1, 2, ,n,
d
j
y
j
− d
j
y
j
n
i1
q
0
ij
q
1
ij
g
i
x
i
− g
j
x
i
,j 1, 2, ,m.
3.6
In view of H
2
-H
3
and 3.6, we have
n
i1
η
i
|
x
i
− x
i
|
≤
n
i1
m
j1
p
0
ji
p
1
ji
M
j
y
j
− y
j
,
m
j1
ω
j
y
j
− y
j
≤
m
j1
n
i1
q
0
ij
q
1
ij
N
i
|
x
i
− x
i
|
.
3.7
Thus, we can obtain
n
i1
⎡
⎣
η
i
−
m
j1
q
0
ij
q
1
ij
N
i
⎤
⎦
|
x
i
− x
i
|
m
j1
ω
j
−
n
i1
p
0
ji
p
1
ji
M
j
y
j
− y
j
≤ 0. 3.8
It follows from 3.4 and 3.8 that |x
i
− x
i
| 0and|y
j
− 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 R
nm
.
8 Journal of Inequalities and Applications
Then we will prove Φ is a homeomorphism on R
nm
. For convenience, we let
Φz
Φz − Φ0, where
Φ
i
z
b
i
x
i
−
m
j1
p
0
ji
p
1
ji
f
j
y
j
− f
j
0
,i 1, 2, ,n,
Φ
nj
z
d
j
y
j
−
n
i1
q
0
ij
q
1
ij
g
i
x
i
− g
i
0
,j 1, 2, ,m.
3.9
We assert that
Φ→∞as z→∞. Otherwise there is a sequence {z
v
} such that z
v
→
∞ and
Φz
v
is bounded as v →∞, where z
v
x
v
1
,x
v
2
, ,x
v
n
,y
v
1
,y
v
2
, ,y
v
m
T
∈ R
nm
.
Noting that
n
i1
sgn
x
v
i
b
i
x
v
i
−
Φ
i
z
v
n
i1
sgn
x
v
i
m
j1
p
0
ji
p
1
ji
f
j
y
v
j
− f
j
0
≤
n
i1
m
j1
p
0
ji
p
1
ji
M
j
y
v
j
,
m
j1
sgn
y
v
j
d
j
y
v
j
−
Φ
nj
z
v
m
j1
sgn
y
v
j
n
i1
q
0
ij
q
1
ij
g
i
x
v
i
− g
i
0
≤
m
j1
n
i1
q
0
ij
q
1
ij
N
i
x
v
i
,
3.10
we have
n
i1
sgn
x
v
i
b
i
x
v
i
−
Φ
i
z
v
m
j1
sgn
y
v
j
d
j
y
v
j
−
Φ
nj
z
v
≤
n
i1
m
j1
p
0
ji
p
1
ji
M
j
y
v
j
m
j1
n
i1
q
0
ij
q
1
ij
N
i
x
v
i
.
3.11
On the other hand, we have
n
i1
sgn
x
v
i
b
i
x
v
i
−
Φ
i
z
v
m
j1
sgn
y
v
j
d
j
y
v
j
−
Φ
nj
z
v
≥
n
i1
η
i
x
v
i
−
n
i1
Φ
i
z
v
m
j1
ω
j
y
v
j
−
m
j1
Φ
nj
z
v
.
3.12
Journal of Inequalities and Applications 9
It follows from 3.11 and 3.12 that
Θ
⎛
⎝
n
i1
x
v
i
m
j1
y
v
j
⎞
⎠
≤
n
i1
Φ
i
z
v
m
j1
Φ
nj
z
v
, 3.13
where
Θmin
⎧
⎨
⎩
min
1≤i≤n
⎧
⎨
⎩
η
i
−
m
j1
q
0
ij
q
1
ij
N
i
⎫
⎬
⎭
, min
1≤j≤m
ω
j
−
n
i1
p
0
ji
p
1
ji
M
j
⎫
⎬
⎭
> 0. 3.14
That is
z
v
≤
1
Θ
Φ
z
v
,
3.15
which contradicts our choice of {z
v
}. Hence, Φ satisfies Φ→∞as z→∞.
By Lemma 2.5, Φ is a homeomorphism on R
nm
and there exists a unique point z
∗
x
∗
1
,x
∗
2
, ,x
∗
n
,y
∗
1
,y
∗
2
, ,y
∗
m
T
such that Φz
∗
0. From the definition of Φ, we know that
z
∗
x
∗
1
,x
∗
2
, ,x
∗
n
,y
∗
1
,y
∗
2
, ,y
∗
m
T
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 (H
1
)–( H
4
) hold, suppose further that
H
5
for each i 1, 2, ,n, j 1, 2, ,m, the following inequalities are satisfied:
a
i
η
i
>
m
j1
c
j
q
0
ij
q
1
ij
N
i
,c
j
ω
j
>
n
i1
a
i
p
0
ji
p
1
ji
M
j
4.1
H
6
the impulsive operators I
ik
x
i
t and J
jk
y
j
t satisfy
I
ik
x
i
t
k
−γ
ik
x
i
t
k
− x
∗
i
, 0 <γ
ik
< 2,i 1, ,n, k ∈ Z
,
J
jk
y
j
t
k
−γ
jk
y
j
t
k
− y
∗
j
, 0 < γ
jk
< 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,weknowthat1.1 has a unique equilibrium point z
∗
x
∗
1
,x
∗
2
, ,x
∗
n
,y
∗
1
, ,y
∗
m
T
.InviewofH
6
, it is easy to see that I
i
x
∗
i
0andJ
j
y
∗
j
0.
10 Journal of Inequalities and Applications
Suppose that ztx
1
t,x
2
t, ,x
n
t,y
1
t,y
2
t, ,y
m
t
T
is an arbitrary solution of
1.1.Letu
i
tx
i
t − x
∗
i
,v
j
ty
j
t − y
∗
j
,t≥ 0, then system 1.1 can be rewritten as
u
Δ
i
t
−a
i
u
i
t
⎡
⎣
b
i
u
i
t
−
m
j1
p
0
ji
f
j
v
j
t − τ
ji
−
m
j1
p
1
ji
∞
0
h
ji
s
f
j
v
j
t − s
Δs − r
i
⎤
⎦
,
i 1, 2, ,n, t >0,t
/
t
k
,t∈ T,
v
Δ
j
t
−c
j
v
j
t
d
j
v
j
t
−
n
i1
q
0
ij
g
i
u
i
t − σ
ij
−
n
i1
q
1
ij
∞
0
k
ij
s
g
i
u
i
t − s
Δs − s
j
,
j 1, 2, ,m, t>0,t
/
t
k
,t∈ T,
4.3
where, for i 1, 2, ,n, j 1, 2, ,m,
a
i
u
i
t
a
i
u
i
t
x
∗
i
,
b
i
u
i
t
b
i
u
i
t
x
∗
i
− b
i
x
∗
i
,
c
j
v
j
t
c
j
v
j
t
y
∗
j
,
f
i
v
i
t
f
j
v
j
t
y
∗
j
− f
j
y
∗
j
,
d
j
v
j
t
d
j
v
j
t
y
∗
j
− b
i
y
∗
j
, g
i
u
i
t
g
i
u
i
t
x
∗
i
− b
i
x
∗
i
.
4.4
Also, for all t t
k
,k∈ Z
,i 1, 2, ,n, j 1, 2, ,m,
u
i
t
k
x
i
t
k
− x
∗
i
x
i
t
k
I
k
x
i
t
k
− x
∗
i
1 − γ
ik
x
i
t
k
− x
∗
i
≤
x
i
t
k
− x
∗
i
≤
|
u
i
t
k
|
,
v
j
t
k
y
j
t
k
− y
∗
j
y
j
t
k
I
k
y
j
t
k
− y
∗
j
1 − γ
jk
y
j
t
k
− y
∗
j
≤
y
j
t
k
− y
∗
j
≤
v
j
t
k
.
4.5
Journal of Inequalities and Applications 11
Hence by H
1
and H
2
, we have
D
|
u
i
t
|
Δ
≤−a
i
η
i
|
u
i
t
|
a
i
m
j1
p
0
ji
M
j
v
j
t − τ
ji
a
i
m
j1
p
1
ji
M
j
∞
0
h
ji
s
v
j
t − s
Δs, i 1, 2, ,n,
4.6
D
v
j
t
Δ
≤−c
j
v
j
t
c
j
n
i1
q
0
ij
N
i
u
i
t − σ
ij
c
j
n
i1
q
1
ij
N
i
∞
0
k
ij
s
|
u
i
t − s
|
Δs, j 1, 2, ,m.
4.7
Also, for i 1, 2, ,n,
x
i
t
k
0
− x
∗
i
t
k
0
x
i
t
k
I
ik
x
i
t
k
− x
∗
i
t
k
− I
ik
x
∗
i
t
k
1 − γ
ik
x
i
t
k
− x
∗
i
t
k
,k∈ Z
,
4.8
thus
x
i
t
k
− x
∗
i
t
k
1 − γ
ik
x
i
t
k
− x
∗
i
t
k
≤
x
i
t
k
− x
∗
i
t
k
,i 1, ,n, k ∈ Z
.
4.9
Similarly, we have
y
j
t
k
− y
∗
j
t
k
1 − γ
∗
jk
y
j
t
k
− y
∗
j
t
k
≤
y
j
t
k
− y
∗
j
t
k
,j 1, 2, ,m, k∈ Z
.
4.10
Let G
i
and G
∗
j
be defined by
G
i
ε
i
a
i
η
i
− ε
i
−
m
j1
c
j
q
0
ij
N
i
e
ε
i
σ
t
,t− σ
ij
−
m
j1
c
j
q
1
ij
N
i
∞
0
k
ij
s
e
ε
i
σ
t
,t− s
Δs, i 1, 2, ,n,
G
∗
j
ξ
j
c
j
ω
j
− ξ
j
−
n
i1
a
i
p
0
ji
M
j
e
ξ
j
σ
t
,t− τ
ji
−
n
i1
a
i
p
1
ji
M
j
∞
0
h
ji
s
e
ξ
j
σ
t
,t− s
Δs, j 1, 2, ,m,
4.11
12 Journal of Inequalities and Applications
respectively, where ε
i
,ξ
j
∈ 0, ∞.ByH
5
, we have
G
i
0
a
i
η
i
−
m
j1
c
j
q
0
ij
q
1
ij
N
i
> 0,i 1, 2, ,n,
G
∗
j
0
c
j
ω
j
−
n
i1
a
i
p
0
ji
p
1
ji
M
j
> 0,j 1, 2, ,m.
4.12
Since G
i
,G
∗
j
are continuous on 0, ∞ and G
i
ε
i
→−∞,G
∗
j
ξ
j
→−∞,asε
i
→ ∞,ξ
j
→
∞, there exist ε
∗
i
,ξ
∗
j
> 0 such that G
i
ε
∗
i
0,G
∗
j
ξ
∗
j
0andG
i
ε
i
> 0, for ε
i
∈ 0,ε
∗
i
,G
∗
j
ξ
j
>
0, for ξ
j
∈ 0,ξ
∗
j
. By choosing α min
1≤i≤n,1≤j≤m
{ε
∗
i
,ξ
∗
j
},weobtain
G
i
α
a
i
η
i
− α −
m
j1
c
j
q
0
ij
N
i
e
α
σ
t
,t− σ
ij
−
m
j1
c
j
q
1
ij
N
i
∞
0
k
ij
s
e
α
σ
t
,t− s
Δs
≥ 0,i 1, 2, ,n,
G
∗
j
α
c
j
ω
j
− α −
n
i1
a
i
p
0
ji
M
j
e
α
σ
t
,t− τ
ji
−
n
i1
a
i
p
1
ji
M
j
∞
0
h
ji
s
e
α
σ
t
,t− s
Δs
≥ 0,j 1, 2, ,m,
4.13
Denote
μ
i
t
e
α
t, δ
|
u
i
t
|
,t∈ R,i 1, 2, ,n, 4.14
ν
j
t
e
α
t, δ
v
j
t
,t∈ R,j 1, 2, ,m, 4.15
where δ ∈ −∞, 0
T
. For t>0,t
/
t
k
,k∈ Z
,i 1, 2, ,n, j 1, 2, ,m, it follows from
4.6–4.15, we can obtain
D
μ
Δ
i
t
αe
α
t, δ
|
u
i
t
|
e
α
σ
t
,δ
D
|
u
i
t
|
Δ
≤ αe
α
t, δ
|
u
i
t
|
e
α
σ
t
,δ
×
⎡
⎣
−a
i
η
i
|
u
i
t
|
a
i
m
j1
p
0
ji
M
j
v
j
t − τ
ji
a
i
m
j1
p
1
ji
M
j
∞
0
h
ji
s
v
j
t − s
Δs
⎤
⎦
Journal of Inequalities and Applications 13
≤−
a
i
η
i
− α
μ
i
t
a
i
m
j1
p
0
ji
M
j
e
α
σ
t
,t− τ
ji
ν
j
t − τ
ji
a
i
m
j1
p
1
ji
M
j
∞
0
h
ji
s
e
α
σ
t
,t− s
ν
j
t − s
Δs,
D
ν
Δ
j
t
≤−
c
j
ω
j
− α
ν
j
t
c
j
n
i1
q
0
ij
N
i
e
α
σ
t
,t− σ
ij
μ
i
t − σ
ij
c
i
n
i1
q
1
ij
N
i
∞
0
k
ij
s
e
α
σ
t
,t− s
μ
i
t − s
Δs.
4.16
Also,
μ
i
t
k
≤ μ
i
t
k
,ν
j
t
k
≤ ν
j
t
k
,i 1, 2, ,n, j 1, 2, ,m, k ∈ Z
. 4.17
Define a Lyapunov function
V
t
n
i1
⎡
⎣
μ
i
t
a
i
m
j1
p
0
ji
M
j
e
α
σ
t
,t− τ
ji
t
t−τ
ji
ν
j
s
Δs
a
i
m
j1
p
1
ji
M
j
∞
0
h
ji
s
e
α
σ
t
,t− s
t
t−s
ν
j
z
ΔzΔs
⎤
⎦
m
j1
ν
j
t
c
j
n
i1
q
0
ij
N
i
e
α
σ
t
,t− σ
ij
t
t−σ
ij
μ
i
s
Δs
c
j
n
i1
q
1
ij
N
i
∞
0
k
ij
s
e
α
σ
t
,t− s
t
t−s
μ
i
z
ΔzΔs
.
4.18
And we note that V t > 0fort>0andV 0 > 0. Calculating the Δ-derivatives of V ,weget
D
V
Δ
t
≤
n
i1
⎡
⎣
−
a
i
η
i
− η
μ
i
t
a
i
m
j1
p
0
ji
M
j
e
α
σ
t
,t− τ
ji
ν
j
t
a
i
m
j1
p
1
ji
M
j
∞
0
h
ji
s
e
α
σ
t
,t− s
ν
j
t
Δs
⎤
⎦
m
j1
−
c
j
ω
j
− η
ν
j
t
c
j
n
i1
q
0
ij
N
i
e
α
σ
t
,t− σ
ij
μ
i
t
c
j
n
i1
q
1
ij
N
i
∞
0
k
ij
s
e
α
σ
t
,t− s
μ
i
t
Δs
14 Journal of Inequalities and Applications
−
n
i1
⎡
⎣
a
i
η
i
− η −
m
j1
c
j
q
0
ij
N
i
e
η
σ
t
,t− σ
ij
−
m
j1
c
j
q
1
ij
N
i
∞
0
k
ij
s
e
α
σ
t
,t− s
Δs
⎤
⎦
μ
i
t
−
m
j1
c
j
ω
j
− η −
n
i1
a
i
p
0
ji
M
j
e
η
σ
t
,t− τ
ji
−
n
i1
a
i
p
1
ji
M
j
∞
0
h
ji
s
e
α
σ
t
,t− s
Δs
ν
j
t
−
n
i1
G
i
η
μ
i
t
−
m
j1
G
∗
j
η
ν
j
t
≤ 0,t>0,t
/
t
k
,t∈ T,k∈ Z
.
4.19
Also,
V
t
k
n
i1
⎡
⎣
μ
i
t
k
a
i
m
j1
p
0
ji
M
j
e
α
σ
t
k
,t
k
− τ
ji
t
k
t
k
−τ
ji
ν
j
s
Δs
a
i
m
j1
p
1
ji
M
j
∞
0
h
ji
s
e
α
σ
t
k
,t
k
− s
t
k
t
k
−s
ν
j
z
ΔzΔs
⎤
⎦
m
j1
ν
j
t
k
n
i1
q
0
ij
N
i
e
α
σ
t
k
,t
k
− σ
ij
t
k
t
k
−σ
ij
μ
i
s
Δs
c
j
n
i1
q
1
ij
N
i
∞
0
k
ij
s
e
α
σ
t
k
,t
k
− s
t
k
t
k
−s
μ
i
z
ΔzΔs
≤
n
i1
⎡
⎣
μ
i
t
k
a
i
m
j1
p
0
ji
M
j
e
α
σ
t
k
,t
k
− τ
ji
t
k
t
k
−τ
ji
ν
j
s
Δs
a
i
m
j1
p
1
ji
M
j
∞
0
h
ji
s
e
α
σ
t
k
,t
k
− s
t
k
t
k
−s
ν
j
z
ΔzΔs
⎤
⎦
m
j1
ν
j
t
k
c
j
n
i1
q
0
ij
N
i
e
α
σ
t
k
,t
k
− σ
ij
t
k
t
k
−σ
ij
μ
i
s
Δs
c
j
n
i1
q
1
ij
N
i
∞
0
k
ij
s
e
α
σ
t
k
,t
k
− s
t
k
t
k
−s
μ
i
z
ΔzΔs
V
t
k
,k∈ Z
.
4.20
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
i1
μ
i
t
m
j1
ν
j
t
≤
n
i1
⎡
⎣
μ
i
0
m
j1
p
0
ji
M
j
e
α
σ
0
, −τ
ji
0
0−τ
ji
ν
j
s
Δs
a
i
m
j1
p
1
ji
M
j
∞
0
h
ji
s
e
α
σ
0
, 0 − s
0
0−s
ν
j
z
ΔzΔs
⎤
⎦
m
j1
ν
j
0
n
i1
q
0
ij
N
i
e
α
σ
0
, 0 − σ
ij
0
0−σ
ij
μ
i
s
Δs
c
j
n
i1
q
1
ij
N
i
∞
0
k
ij
s
e
α
σ
0
, 0 − s
0
0−s
μ
i
z
ΔzΔs
.
4.21
In view of 4.14-4.15 and the previous inequality, we have
n
i1
x
i
t
− x
∗
i
t
m
j1
y
j
t
− y
∗
j
t
≤ e
α
t, δ
⎡
⎣
n
i1
⎛
⎝
1
m
j1
c
i
q
0
ij
N
i
e
α
σ
0
, −σ
ij
σ
ij
m
i1
c
i
q
1
ij
N
i
∞
0
k
ij
s
e
α
σ
0
, −s
sΔs
max
δ∈−∞,0
T
ϕ
i
δ
− x
∗
i
δ
m
j1
1
n
i1
a
i
p
0
ji
M
j
e
α
σ
0
, −τ
ji
τ
ji
n
i1
a
i
p
1
ji
M
j
∞
0
h
ji
s
e
α
σ
0
, −s
sΔs
max
δ∈−∞,0
T
ψ
j
δ
− y
∗
j
δ
⎤
⎦
≤ Ne
α
t, δ
⎡
⎣
n
i1
max
δ∈−∞,0
T
ϕ
i
δ
− x
∗
i
δ
m
j1
max
δ∈−∞,0
T
ψ
j
δ
− y
∗
j
δ
⎤
⎦
,
4.22
where
N max
⎧
⎨
⎩
1
m
j1
c
i
q
0
ij
N
i
e
α
σ
0
, −σ
ij
σ
ij
m
j1
c
i
q
1
ij
N
i
∞
0
k
ij
s
e
α
σ
0
, −s
sΔs, 1
n
i1
a
i
p
0
ji
M
j
e
α
σ
0
, −τ
ji
τ
ji
n
i1
a
i
p
1
ji
M
j
∞
0
h
ji
s
e
α
σ
0
, −s
sΔs
≥ 1.
4.23
The proof is complete.
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 dis-
tributed delays and impulses:
x
Δ
1
t
−
1
1
3
cos x
1
t
×
5x
1
t
−
1
4
sin
2y
1
t − 1
−
∞
0
1
4
e
−s
sin
2y
1
t − s
Δs r
1
,
t>0,t
/
t
k
,t∈ T,
Δx
1
t
k
I
1
x
1
t
k
,k 1, 2, ,
y
Δ
t
−
1
1
3
sin y
1
t
×
3y
1
t
−
1
4
cos
2x
1
t − 1
−
∞
0
1
4
e
−s
cos
2x
1
t − s
Δs s
1
,
t>0,t
/
t
k
,t∈ T,
Δy
1
t
k
J
1
y
1
t
k
,k 1, 2, ,
5.1
where T R,γ
1k
1 1/2 sin1 k, γ
1k
1 2/3 cos 2k, k ∈ Z
. A simple computation
shows that a
c 2/3, a c 4/3,M
1
N
1
1,η
1
5,ω
1
3,p
0
11
p
0
11
q
0
11
q
0
11
1/4, 0 <γ
1k
, γ
1k
< 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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