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RESEARCH Open Access
Integro-differential inequality and stability of BAM
FCNNs with time delays in the leakage terms and
distributed delays
Xinhua Zhang and Kelin Li
*
* Correspondence:
School of Science, Sichuan
University of Science &
Engineering, Sichuan 643000, PR
China
Abstract
In this paper, a class of impulsive bidirectional associative memory (BAM) fuzzy cellular
neural networks (FCNNs) with time delays in the leakage terms and distributed delays is
formulated and investigated. By establishing an integro-differential inequality with
impulsive initial conditions and employing M-matrix theory, some sufficient conditions
ensuring the existence, uniqueness and global exponential stability of equilibrium point
for impulsive BAM FCNNs with time delays in the leakage terms and distributed delays
are obtained. In particular, the estimate of the exponential convergence rate is also
provided, which depends on the delay kernel functions and system parameters. It is
believed that these results are significant and useful for the design and applications of
BAM FCNNs. An example is given to show the effectiveness of the results obtained here.
Keywords: bidirectional associative memory, fuzzy cellular neural networks, impulses,
distributed delays, global exponential stability
1 Introduction
The bidirectional associative memory (BAM) neural network models were first introduced
by Kosko [1]. It is a special class of recurrent neural networks that can store bipolar vector
pairs. The BAM neural network is composed of neurons arranged in two layers, the
X-layer and Y-layer. The neurons in one layer are fully interconnected to the neurons in
the other layer. Thr ough iterations of forward and backward info rmation flows between
the two layer, it performs a two-way associative search for stored bipolar vector pairs and

generalize the single-layer autoassociative Hebbian correlation to a two-layer pattern-
matched heteroassociative circuits. Therefore, this class of networks possesses good appli-
cation prospects in some fields such as pattern recognition, signal and image process, and
artificial intelligence [2]. In such applications, the stability of networks plays an important
role; it is of significance and necessary to investigate the stability. It is well known, in both
biological and artificial neural networks, the delays arise because of the processing of
information. Time delays may lead to oscillation, divergence or instability which may be
harmful to a system. Therefore, study of neural dynamics with consideration of the
delayed problem becomes extremely important to manufacture high-quality neural net-
works. In recent years, there have been many analytical results for BAM neural networks
with various axonal signal transmission delays, for example, see [3-11] and references
therein. I n addition, except various axonal s ignal transmissi on delays, time delay in the
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>© 2011 Zhang a nd Li; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License ( which permits unre stricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
leakage term also has great impact on the dynamics of neural networks. As pointed out by
Gopalsamy [12,13], time delay in the stabilizing negative feedback term has a tendency to
destabilize a system. Recently, some authors have paid attention to stability analysis of
neural networks with time delays in the leakage (or “forgetting”) terms [12-18].
Since FCNNs were introduced by Yang et. al [19,20], many researchers have done
extensive works on this subject due to their extensive applications in classification of
image processing and pattern recognition. Specially, in the past few years, the stability
analysis on FCNNs with various delays and fuzzy BAM neural networks with transmis-
sion delays has been the highl ight in the neural network field, for example, see [21-27]
and references therein. On the other hand, in respect of the comple xity, 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 such fields as medicine
and biology, economics, me chanics, electronics and telecommunications. Many inte rest-
ing results on impulsive effect have been gained, e.g., Refs. [28-37]. As artificial electronic

systems, neural networks such as CNNs, bidirectional neural networks and recurrent
neural networks 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 netwo rks. To the best of our
knowledge, few authors have considered impulsive BAM FCNNs with time delays in th e
leakage terms and distributed delays.
Motivated by the above discussions, the objective of this paper is to formulate and
study impulsive BAM FCNNs with time delays in the leakage terms and distributed
delays. Under quite general conditions, some sufficient conditions ensuring the exis-
tence, uniqueness and gl obal exponential stability of equilibrium point are obtained by
the topological degree theory, properties of M-matrix, the integro-differential inequality
with impulsive initial conditions and analysis technique.
The paper is organized as follows. In Section 2, the new neural network model is for-
mulated, and the necessary knowledge is provided. The existence and uniqueness of
equilibrium point are presented in Section 3. In Section 4, we give some sufficient con-
ditions of exponential stability of the impulsive BAM FCNNs with time delays in the
leakage terms and distributed delays. An example is given to show the effectiveness of
the results obtained here in Section 5. Finally, in Section 6, we give the conclusion.
2 Model description and preliminaries
In this section, we will consider the model of impulsive BAM FCNNs with time delays in
the leakage terms and distributed delays, it is described by the following functional dif-
ferential equation:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙

x
i
(t)=−a
i
x
i
(t − δ
i
)+
m

j=1
a
ij
g
j
(y
j
(t)) +
m

j=1
˜
a
ij
v
j
+ I
i
+

m
∧
j=1
α
ij
+∞

0
K
ij
(s)g
j
(y
j
(t − s))ds +
m
∨
j=1
˜α
ij
+∞

0
K
ij
(s)g
j
(y
j
(t − s))d

s
+
m
∧
j=1
T
ij
v
j
+
m
∨
j=1
H
ij
v
j
, t = t
k
x
i
(t
+
)=x
i
(t
−
)+P
ik
(x

i
(t
−
)), t = t
k
, k ∈ N = {1, 2, },
˙
y
j
(t)=−b
j
y
j
(t − θ
j
)+
n

i=1
b
ji
f
i
(x
i
(t)) +
n

i=1
˜

b
ji
u
i
+ J
j
+
n
∧
i=1
β
ji
+∞

0
¯
K
ji
(s)f
i
(x
i
(t − s))ds +
n
∨
i=1
˜
β
ji
+∞


0
¯
K
ij
(s)f
i
(x
i
(t − s))ds
+
n
∧
i=1
¯
T
ji
u
i
+
n
∨
i=1
¯
H
ji
u
i
, t = t
k

y
j
(t
+
)=y
j
(t
−
)+Q
j
k
(y
j
(t
−
)), t = t
k
, k ∈ N = {1, 2, },
(1)
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 2 of 18
for i = 1, 2, , n, j = 1, 2, , m, t >0,wherex
i
(t)andy
j
(t) are the states of the ith
neuron and the jth neuron at time t, respectively; δ
i
≥ 0andθ
j

≥ 0 denote the leakage
delays, respectively; f
i
and g
j
denote the signal functions of the ith neuron and the jth
neuron at time t, respectivel y; u
i
, v
j
and I
i
, J
j
denote inputs and bias of the ith neuron
and the jth neur on, respect ively; a
i
>0,b
j
>0,
a
i
j
,
˜
a
i
j
, α
i

j
, ˜α
i
j
, b
j
i
,
˜
b
j
i
, β
j
i
,
˜
β
ji
are constants,
a
i
and b
j
represent the rate with which the ith neuron and the jth neuron will reset
their potential to the resting state in isolation when disconnected from the networks
and external inputs, respectively; a
ij
, b
ji

and
˜
a
i
j
,
˜
b
ji
denote connection weights o f feed-
back template and feedforward template, respectively; a
ij
, b
ji
and
˜
α
i
j
,
˜
β
j
i
denote connec-
tion weights of the distrib uted fuzzy feedback MIN template and the distributed fuzzy
feedback MAX template, respectively;
T
i
j

,
¯
T
ji
and
H
i
j
,
˜
H
ji
are elements of fuzzy feedfor-
ward MIN template and fuzzy feedforward MAX template, respectively; ⋀ and ⋁
denote the fuzzy AND and fuzzy OR operations, respectively; K
ij
(s)and
¯
K
j
i
(s
)
corre-
spond to the delay kernel functions, respectively. t
k
is called impulsive moment and
satisfies 0 <t
1
<t

2
< ,
lim
k
→+∞
t
k
=+
∞
;
x
i
(t
−
k
)
and
x
i
(t
+
k
)
denote the left-hand and right-
hand limits at t
k
, respectively; P
ik
and Q
jk

show impulsive perturbations of the ith neu-
ron and jth neuron at time t
k
, respectively.
We always assume
x
i
(t
+
k
)=x
i
(t
k
)
and
y
j
(t
+
k
)=y
j
(t
k
)
, k Î N . The initial conditions are
given by

x

i
(t )=φ
i
(t ), −∞ ≤ t ≤ 0
,
y
j
(t )=ϕ
j
(t ), −∞ ≤ t ≤ 0
,
where j
i
(t), 
j
(t)(i =1,2, ,n; j =1,2, ,m) are bounded and continuous on (-∞,0],
respectively.
If the impulsive operators P
ik
(x
i
)=0,Q
jk
(y
j
)=0,i =1,2, ,n, j = 1, 2, , m, k Î N,
then system (1) may reduce to the following model:
⎧
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙
x
i
(t )=−a
i
x

i
(t − δ
i
)+
m

j=1
a
ij
g
j
(y
j
(t )) +
m

j=1
˜
a
ij
v
j
+ I
i
+
m
∧
j=1
α
ij

+∞

0
K
ij
(s)g
j
(y
j
(t − s))ds +
m
∨
j=1
˜α
ij
+∞

0
K
ij
(s)g
j
(y
j
(t − s))d
s
+
m
∧
j=1

T
ij
v
j
+
m
∨
j=1
H
ij
v
j
,
˙
y
j
(t )=−b
j
y
j
(t − θ
j
)+
n

i=1
b
ji
f
i

(x
i
(t )) +
n

i=1
˜
b
ji
u
i
+ J
j
+
n
∧
i=1
β
ji
+∞

0
¯
K
ji
(s)f
i
(x
i
(t − s))ds +

n
∨
i=1
˜
β
ji
+∞

0
¯
K
ij
(s)f
i
(x
i
(t − s))ds
+
n
∧
i=1
¯
T
ji
u
i
+
n
∨
i=1

¯
H
ji
u
i
.
(2)
System (2) is called the continuous system of model (1).
Throughout this paper, we make the following assumptions:
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 3 of 18
(H1) For neuron activation functions f
i
and g
j
(i = 1, 2, , n; j =1,2, ,m), there
exist two positive diagonal matrices F =diag(F
1
, F
2
, ,F
n
)andG =diag(G
1
, G
2
, ,
G
m
) such that

F
i
=sup
x=
y




f
i
(x) − f
i
(y)
x − y




, G
j
=sup
x=
y




g
j

(x) − g
j
(y)
x − y




for all x, y Î R (x ≠ y).
(H2) The delay kernels K
ij
: [0, +∞) ® R and
¯
K
j
i
:[0,+∞) →
R
are real-valued piece-
wise continuous, and there exists δ > 0 such that
k
ij
(λ)=

+∞
0
e
λs
|K
ij

(s)|ds,
¯
k
ji
(λ)=

+∞
0
e
λs
|
¯
K
ji
(s)|d
s
Are continuous for l Î [0,δ), i = 1,2, , n, j = 1,2, , m.
(H3) Let
¯
P
k
(
x
)
= x + P
k
(
x
)
and

¯
Q
k
(
y
)
= y + Q
k
(
y
)
be Lipschitz continuous in R
n
and
R
m
, respectively, that is, there exist nonnegative diagnose matrices Γ
k
= diag(g
1k
, g
2k
, ,
g
nk
) and
¯
Γ
k
=diag

(
¯γ
1k
, ¯γ
2k
, , ¯γ
mk
)
such that
|
¯
P
k
(x) −
¯
P
k
(y)|≤Γ
k
|x − y|, for all x, y ∈ R
n
, k ∈ N ,
|
¯
Q
k
(
u
)
−

¯
Q
k
(
v
)
|≤
¯
Γ
k
|u − v|, for all u, v ∈ R
m
, k ∈ N
,
where
¯
P
k
(x)=(
¯
P
1k
(x
1
),
¯
P
2k
(x
2

), ,
¯
P
nk
(x
n
))
T
,
¯
Q
k
(x)=(
¯
Q
1k
(y
1
),
¯
Q
2k
(y
2
), ,
¯
Q
mk
(y
m

))
T
,
P
k
(x)=(P
1k
(x
1
), P
2k
(x
2
), , P
nk
(x
n
))
T
,
Q
k
(
y
)
=
(
Q
1k
(

y
1
)
, Q
2k
(
y
2
)
, , Q
mk
(
y
m
))
T
.
To begin with, we introduce some notation and recall some basic definitions.
PC[J, R
l
]={z(t): J ® R
l
|z(t) is continuous at t ≠ t
k
,
z
(t
+
k
)=z(t

k
)
, and
z
(t
−
k
)
exists for t,
t
k
Î J, k Î N}, where J ⊂ R is an interval, l Î N.
PC ={ψ:(-∞,0]® R
l
| ψ(s)isbounded,andψ(s
+
)=ψ(s)fors Î (-∞,0),ψ(s
-
)exists
for s Î (-∞, 0], j(s
-
)=j(s) for all but at most a finite number of points s Î (-∞, 0]}.
For an m × n matrix A,|A| denotes the absolute value matrix given by |A|=(|a
ij
|)
m
×n
. For A =(a
ij
)

m × n
, B =(b
ij
)
m × n
Î R
m × n
, A ≥ B (A>B) means that each pair of
corresponding elements of A and B such that the inequality a
ij
≥ b
ij
(a
ij
>b
ij
).
Definition 1 A function (x, y)
T
:(-∞,+∞) ® R
n+m
is said to be the special solution of
system (1) with initial conditions
x
(
s
)
= φ
(
s

)
, y
(
s
)
= ϕ
(
s
)
s ∈
(
−∞,0]
,
if the following two conditions are satisfied
(i) (x, y)
T
is piecewise continuous with first kind disc ontinuity at the points t
k
, k Î K.
Moreover,(x, y)
T
is right continuous at each discontinuity point.
(ii) (x, y)
T
satisfies model (1) for t ≥ 0, and x(s)=j(s), y(s)=(s) for s Î (-∞, 0].
Especially, a point (x*, y*)
T
Î R
n+m
is called an equilibrium point of model (1),if(x

(t), y(t))
T
=(x*, y*)
T
is a solution of (1).
Throughout this paper, we always assume th at the impulsive jumps P
k
and Q
k
satisfy
(referring to [28-37])
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 4 of 18
P
k
(
x
∗
)
=0 and Q
k
(
y
∗
)
=0, k ∈ N
,
i.e.,
¯
P

k
(
x
∗
)
= x
∗
and
¯
Q
(
y
∗
)
= y
∗
, k ∈ N
,
(3)
where (x*, y*)
T
is the equilibrium point of continuous systems (2). That is, if (x*, y*)
T
is an equilibrium point of continuous system (2), then (x*, y*)
T
is also the equilibrium
of impulsive system (1).
Definition 2 The equilibrium point (x*, y*)
T
of model (1) is said to be globally expo-

nentially stable, if there exist constants l >0and M ≥ 1 such that
||x
(
t
)
− x
∗
|| + ||y
(
t
)
− y
∗
|| ≤ M
(
||φ − x
∗
|| + ||ϕ − y
∗
||
)
e
−λ
t
for all t ≥ 0, where (x(t), y(t))
T
is any s olution of system (1) with initial value (j(s),
(s))
T
and

|
|x(t) − x
∗
|| =
n

i=1
|x
i
(t ) − x
∗
i
|, ||y(t) − y
∗
|| =
m

j=1
|y
j
(t ) − y
∗
j
|,
|
|φ − x
∗
|| =sup
−∞<s≤0
n


i=1
|φ
i
(s) − x
∗
i
|, ||ϕ − y
∗
|| =sup
−∞<s≤0
m

j
=1
|ϕ
j
(s) − y
∗
j
|
.
Definition 3 A real matrix D =(d
ij
)
n × n
is said to be a nonsingular M-matrix if d
ij
≤
0, i, j = 1, 2, , n, i ≠ j, and all successive principal minors of D are positive.

Lemma 1 [38]Let D =(d
ij
)
n × n
with d
ij
≤ 0(i ≠ j), then the following state ments are
true:
(i) D is a nonsingular M-matrix if and only if D is inverse-positive, that is, D
-1
exists
and D
-1
is a nonnegative matrix.
(ii) D is a nonsingular M-matrix if and only if there exists a positive vector ξ =(ξ
1
, ξ
2
,
, ξ
n
)
T
such that Dξ >0.
Lemma 2 [20]For any positive integer n, let h
j
: R ® Rbeafunction(j = 1, 2, , n),
then we have
|
n

∧
j=1
α
j
h
j
(u
j
) −
n
∧
j=1
α
j
h
j
(v
j
)|≤
n

j=1
|α
j
|·|h
j
(u
j
) − h
j

(v
j
)|
,
|
n
∨
j=1
α
j
h
j
(u
j
) −
n
∨
j=1
α
j
h
j
(v
j
)|≤
n

j
=1
|α

j
|·|h
j
(u
j
) − h
j
(v
j
)|
for all a =(a
1
, a
2
, , a
n
)
T
, u =(u
1
, u
2
, , u
n
)
T
, v =(v
1
, v
2

, , v
n
)
T
Î R
n
.
3 Existence and uniqueness of equilibrium point
In this section, we will proof the existence and uniqueness of equilibrium point of
model (1). For the sake of simplification, let
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˜
I
i
=
m

j=1
˜
a
ij
v
j
+ I

i
+
m
∧
j=1
T
ij
v
j
+
m
∨
j=1
H
ij
v
j
, i =1,2, , n,
˜
J
j
=
n

i
=1
˜
b
ji
u

i
+ J
j
+
n
∧
i=1
¯
T
ji
u
i
+
n
∨
i=1
¯
H
ji
u
i
, j =1,2, , m
,
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 5 of 18
then model (2) is reduced to
⎧
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙
x
i
(t )=−a
i
x
i
(t − δ
i
)+
m

j=1
a
ij
g
j
(y
j
(t )) +
m

∧
j=1
α
ij
+∞

0
K
ij
(s)g
j
(y
j
(t − s))d
s
+
m
∨
j=1
˜α
ij
+∞

0
K
ij
(s)g
j
(y
j

(t − s))ds +
˜
I
i
,
˙
y
j
(t )=−b
j
y
j
(t − θ
j
)+
n

i=1
b
ji
f
i
(u
i
(t )) +
n
∧
i=1
β
ji

+∞

0
¯
K
ji
(s)f
i
(x
i
(t − s))ds
+
n
∨
i=1
˜
β
ji
+∞

0
¯
K
ji
(s)f
i
(u
i
(t − s))ds +
˜

J
j
.
(4)
It is evident that the dynamical characteristics of model (2) are as same as of model (4).
Theorem 1 Under assumptions (H1) and (H2),system(1) has one unique equili-
brium point, if the following condition holds,
(C1) there exist vectors ξ =(ξ
1
, ξ
2
, , ξ
n
)
T
>0,h =(h
1
, h
2
, , h
m
)
T
>0and positive
number l >0such that
⎧
⎪
⎪
⎨
⎪

⎪
⎩
(λ − a
i
e
λδ
i
)ξ
i
+
m

j=1

|a
ij
| +(|α
ij
| + |˜α
ij
|)k
ij
(λ)

G
j
η
j
< 0, i =1,2, , n
,

(λ − b
j
e
λθ
j
)η
j
+
n

i
=1

|b
ji
| +(|β
ji
| + |
˜
β
ji
|)
¯
k
ji
(λ)

F
i
ξ

i
< 0. j =1,2, , m.
Proof. Let
h
(
x
1
, , x
n
, y
1
, , y
m
)
=
(
h
1
, , h
n
, h
1
, , h
m
)
T
, where
⎧
⎪
⎪

⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
h
i
= a
i
x
i
−
m

j=1
a
ij
g
j
(y
j
) −
m
∧
j=1

α
ij
k
ij
(0)g
j
(y
j
) −
m
∨
j=1
˜α
ij
k
ij
(0)g
j
(y
j
) −
˜
I
i
,
h
j
= b
j
y

j
−
n

i=1
b
ji
f
i
(x
i
) −
n
∧
i=1
β
ji
¯
k
ji
(0)f
i
(x
i
) −
n
∨
i=1
˜
β

ji
¯
k
ji
(0)f
i
(x
i
) −
˜
J
j
for i = 1, 2, , n; j = 1, 2, , m. Obviously, from assumption (H2), the equilibrium
points of model (4) are the solutions of system of equations:

h
i
=0,i =1,2, , n,
h
j
=0,j =1,2, , m
.
(5)
Define the following homotopic mapping:
H(x
1
, , x
n
, y
1

, , y
m
)=θh(x
1
, , x
n
, y
1
, , y
m
) + (1 - θ)(x
1
, , x
n
, y
1
, , y
m
)
T
, where θ
Î [0, 1]. Let H
k
(k = 1, 2, , n + m) denote the kth component of H(x
1
, , x
n
, y
1
, , y

m
),
then from assumption (H1) and Lemma 2, we have
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
|H
i
|≥[1 + θ (a
i
− 1)]|x
i
|−θ


m
j=1

|a
ij
| +(|α
ij
| + |˜α
ij
|)k
ij
(0)

G
j
|y
j
|
−θ

m
j=1

|a
ij
| +(|α
ij
| + |˜α
ij
|)k

ij
(0)

|g
j
(0)|−θ|
˜
I
i
|,
|H
n+j
|≥[1 + θ(b
j
− 1)]|y
j
|−θ

n
i=1

|b
ji
| +(|β
ji
| + |
˜
β
ji
|)

¯
k
ji
(0)

F
i
|x
i
|
−θ

n
i=1

|b
ji
| +(|β
ji
| + |
˜
β
ji
|)
¯
k
ji
(0)

|f

i
(0)|−θ|
˜
J
j
|
(6)
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 6 of 18
for i = 1, 2, , n, j = 1, 2, , m. Denote
¯
H =(|H
1
|, |H
2
|, , |H
n+m
|)
T
, z =(|x
1
|, , |x
n
|, |y
1
|, , |y
m
|)
T
,

C =diag(a
1
, , a
n
, b
1
, , b
m
), L =diag(F
1
, , F
n
, G
1
, , G
m
),
P =(|
˜
I
1
|, , |
˜
I
n
, |, |
˜
J
1
|, , |

˜
J
m
|)
T
Q =(|f
1
(0)|, , |f
n
(0)|, |g
1
(0)|, , |g
m
(0)|)
T
,
A =

|a
ij
| +(|α
ij
| + |˜α
ij
|)k
ij
(0)

n×m
, B =


|b
ji
| +(|β
ji
| + |
˜
β
ji
|)
¯
k
ji
(0)

m×n
,
T =

0 A
B 0

, ω =(ξ
1
, , ξ
n
, η
1
, , η
m

)
T
> 0.
Then, the matrix form of (6) is
¯
H ≥ [E + θ
(
C − E
)
]z − θ TLz − θ
(
P + TQ
)
=
(
1 − θ
)
z + θ [
(
C − TL
)
z −
(
P + TQ
)
]
.
Since condition (C1) holds, and k
ij
(l),

¯
k
j
i
(λ
)
are continuous on [0, δ ), when l =0in
(C1), we obtain
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
−a
i
ξ
i
+
m

j=1


|a
ij
| +(|α
ij
| + |˜α
ij
|)k
ij
(0)

G
j
η
j
< 0, i =1,2, , n
,
−b
j
η
j
+
n

i=1

|b
ji
| +(|β
ji
| + |

˜
β
ji
|)
¯
k
ji
(0)

F
i
ξ
i
< 0. j =1,2, , m.
or in matrix form,
(
−C + TL
)
ω<0
.
(7)
From Lemma 1, we know that C - TL is a nonsingular M-mat rix, so (C - TL)
-1
is a
nonnegative matrix. Let
Γ =

z =(x
1
, , x

n
, y
1
, , y
m
)
T
|z ≤ ω +(C − TL)
−1
(P + TQ)

,
then Γ is nonempty, and from (6), for any z =(x
1
, , x
n
, y
1
, , y
m
)
T
Î∂Γ, we have
¯
H ≥ (1 − θ)z + θ(C − TL)[z − (C − TL)
−1
(P + TQ)]
=
(
1 − θ

)
[ω +
(
C − TL
)
−1
(
P + TQ
)
]+θ
(
C − TL
)
ω>0, θ ∈ [0, 1]
.
Therefore, for any (x
1
, , x
n
, y
1
, ,y
m
)
T
Î ∂Γ and θ Î [0, 1], we have H ≠ 0. From
homotopy invariance theorem [39], we get
deg
(
h, Γ ,0

)
=deg
(
H, Γ ,0
)
=1
,
by topological degree theory, we know that (5) has at least one solution in Γ. That is,
model (4) has at least an equilibrium point.
Now, we show that the solution of the system of Equations (5) is unique. Assume
that
(x
∗
1
, , x
∗
n
, y
∗
1
, , y
∗
m
)
T
and
(
ˆ
x
1

, ,
ˆ
x
n
,
ˆ
y
1
, ,
ˆ
y
m
)
T
are two solutions of the system
of Equations (5), then
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 7 of 18
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
a
i
(x
∗
i
−
ˆ
x
i
)=
m

j=1
a
ij
[g
j
(y
∗
j
)+g

j
(
ˆ
y
j
)]
+

m
∧
j=1
α
ij
k
ij
(0)g
j
(y
∗
j
) −
m
∧
j=1
α
ij
k
ij
(0)g
j

(
ˆ
y
j
)

+

m
∨
j=1
˜α
ij
k
ij
(0)g
j
(y
∗
j
) −
m
∨
j=1
˜α
ij
k
ij
(0)g
j

(
ˆ
y
j
)

,
b
j
(y
∗
j
−
ˆ
y
j
)=
n

i=1
b
ji
[f
i
(x
∗
i
) − f
i
(

ˆ
x
i
)]
+

n
∧
i=1
β
ji
¯
k
ji
(0)f
i
(x
∗
i
) −
n
∧
i=1
β
ji
¯
k
ji
(0)f
i

(
ˆ
x
i
)

+

n
∨
i=1
˜
β
ji
¯
k
ji
(0)f
i
(x
∗
i
) −
n
∨
i=1
˜
β
ji
¯

k
ji
(0)f
i
(
ˆ
x
i
)

,
it follows that
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
a
i
|x

∗
i
−
ˆ
x
i
|≤
m

j=1
|a
ij
||g
j
(y
∗
j
)+g
j
(
ˆ
y
j
)|
+|
m
∧
j=1
α
ij

k
ij
(0)g
j
(y
∗
j
) −
m
∧
j=1
α
ij
k
ij
(0)g
j
(
ˆ
y
j
)|
+|
m
∨
j=1
˜α
ij
k
ij

(0)g
j
(y
∗
j
) −
m
∨
j=1
˜α
ij
k
ij
(0)g
j
(
ˆ
y
j
)|
,
b
j
|y
∗
j
−
ˆ
y
j

|≤
n

i=1
|b
ji
||f
i
(x
∗
i
) − f
i
(
ˆ
x
i
)|
+|
n
∧
i=1
β
ji
¯
k
ji
(0)f
i
(x

∗
i
) −
n
∧
i=1
β
ji
¯
k
ji
(0)f
i
(
ˆ
x
i
)|
+|
n
∨
i
=1
˜
β
ji
¯
k
ji
(0)f

i
(x
∗
i
) −
n
∨
i
=1
˜
β
ji
¯
k
ji
(0)f
i
(
ˆ
x
i
)|.
By using of Lemma 2 and hypothesis (H1), we have
⎧
⎪
⎪
⎪
⎪
⎪
⎨

⎪
⎪
⎪
⎪
⎪
⎩
a
i
|x
∗
i
−
ˆ
x
i
|−
m

j=1

|a
ij
| +(|α
ij
| + |˜α
ij
|)k
ij
(0)


G
j
|y
∗
j
−
ˆ
y
j
|≤0
,
b
j
|y
∗
j
−
ˆ
y
j
|−
n

i=1

|b
ji
| +(|β
ji
| + |

˜
β
ji
|)
¯
k
ji
(0)

F
i
|x
∗
i
−
ˆ
x
i
|≤0.
(8)
Let
Z =diag
(
|x
∗
1
−
ˆ
x
1

|, , |x
∗
n
−
ˆ
x
n
|, |y
∗
1
−
ˆ
y
1
|, , |y
∗
m
−
ˆ
y
m
|
)
, then the mat rix form of
(8) is (C -TL)Z ≤ 0. Since C - TL is a nonsingular M-matrix, (C - TL)
-1
≥ 0, thus Z ≤
0, accordingly, Z = 0, i.e.,
x
∗

i
=
ˆ
x
i
,
y
∗
j
=
ˆ
y
j
(i =1,2, , n, j =1,2, , m
)
. This shows that
model (4) has one unique equilibrium point. According to (3), this implies that system
(1) has one unique equilibrium point. The proof is completed.
Corollary 1 Under assumptions (H1) and (H2),system(1) has one unique equili-
brium point if C - TL is a nonsingular M-matrix.
Proof. Since that C - TL is a nonsingular M-matrix, from Lemma 1, there exists a
vector ω =(ξ
1
, ξ
n
, h
1
, , h
m
)

T
> 0 such that (CTL) ω >0,or(-C + TL) ω <0. It fol-
lows that
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 8 of 18
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
−a
i
ξ
i
+
m

j=1

|a
ij
| +(|α

ij
| + |˜α
ij
|)k
ij
(0)

G
j
η
j
< 0, i =1,2, , n,
−b
j
η
j
+
n

i=1

|b
ji
| +(|β
ji
| + |
˜
β
ji
|)

¯
k
ji
(0)

F
i
ξ
i
< 0, j =1,2, , m
.
From the continuity of k
ij
(l) and
¯
k
j
i
(λ
)
, it is easy to know that there exists l > 0 such
that
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪

⎪
⎪
⎪
⎪
⎩
(λ − a
i
e
λδ
i
)ξ
i
+
m

j=1

|a
ij
| +(|α
ij
| + |˜α
ij
|)k
ij
(λ)

G
j
η

j
< 0, i =1,2, , n,
(λ − b
j
e
λθ
j
)η
j
+
n

i=1

|b
ji
| +(|β
ji
| + |
˜
β
ji
|)
¯
k
ji
(λ)

F
i

ξ
i
< 0, j =1,2, , m
.
That is, condition (C1) holds. This completes the proof.
4 Exponential stability and exponential convergence rate
In this section, we will discuss the global exponential stability of system (1) and give an
estimation of exponential convergence rate.
Lemma 3 Let a < b ≤ +∞, and u(t)=(u
1
(t), , u
n
(t))
T
Î PC[[a, b), R
n
] and v(t)=(v
1
(t), , v
m
(t))
T
Î PC[[a, b), R
m
] satisfy the following integro-differential inequalities with
the initial conditions u(s) Î PC[(-∞, 0], R
n
] and v(s) Î PC[(-∞, 0], R
m
]:

⎧
⎪
⎪
⎨
⎪
⎪
⎩
D
+
u
i
(t ) ≤−r
i
u
i
(t − δ
i
)+
m

j=1
p
ij
v
j
(t )+
m

j=1
q

ij
+∞

0
|K
ij
(s)|v
j
(t − s)ds
,
D
+
v
j
(t ) ≤−
¯
r
j
v
j
(t − θ
j
)+
n

i=1
¯
p
ji
u

i
(t )+
n

i=1
¯
q
ji
+∞

0
|
¯
K
ji
(s)|u
i
(t − s)ds
(9)
for i = 1, 2, , n, j = 1, 2, , m, where r
i
>0,p
ij
>0,q
ij
>0,
¯
r
j
>

0
,
¯
p
j
i
>
0
,
¯
q
j
i
>
0
, i =
1, 2, ,n, j = 1, 2, , m. If the initial conditions satisfy

u(s) ≤ κξe
−λ(s−a)
, s ∈ (−∞, a]
,
v(s) ≤ κηe
−λ(s−a)
, s ∈ (−∞, a],
(10)
in which l >0,ξ =(ξ
1
, ξ
2

, , ξ
n
)
T
>0and h =(h
1
, h
2
, , h
m
)
T
>0satisfy
⎧
⎪
⎪
⎨
⎪
⎪
⎩
(λ − r
i
e
λδ
i
)ξ
i
+
m


j=1
(p
ij
+ q
ij
k
ij
(λ))η
j
< 0, i =1,2, , n,
(λ −
¯
r
j
e
λθ
j
)η
j
+
n

i
=1
(
¯
p
ji
+
¯

q
ji
¯
k
ji
(λ))ξ
i
< 0, j =1,2, , m
.
(11)
Then

u(t ) ≤ κξe
−λ
(
t−a
)
, t ∈ [a, b)
,
v(t) ≤ κηe
−λ(t−a)
, t ∈ [a, b).
Proof. For i Î {1, 2, , n}, j Î {1, 2, , m} and arbitrary ε > 0, set z
i
(t)=( + ε) ξ
i
e
-l
(t - a)
,

¯
z
j
(t )=(κ + ε)η
j
e
−λ(t−a
)
, we prove that
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 9 of 18

u
i
(t ) ≤ z
i
(t )=(κ + ε)ξ
i
e
−λ(t−a)
, t ∈ [a, b), i =1,2, , n,
v
j
(t ) ≤¯z
j
(t )=(κ + ε) η
j
e
−λ(t−a)
, t ∈ [a, b), j =1,2, , m

.
(12)
If this is not true, no loss of generality, suppose that there exist i
0
and t* Î [a, b)
such that
u
i
0
(t
∗
)=z
i
0
(t
∗
), D
+
u
i
0
(t
∗
) ≥˙z
i
0
(t
∗
), u
i

(t ) ≤ z
i
(t ), v
j
(t ) ≤¯z
j
(t
)
(13)
for t Î [a, t*], i = 1, 2, , n, j = 1, 2, , m.
However, from (9) and (12), we get
D
+
u
i
0
(t
∗
)
≤−r
i
0
u
i
0
(t
∗
− δ
i
0

)+
m

j=1
p
i
0
j
v
j
(t
∗
)+
m

j=1
q
i
0
j
+∞

0
|K
i
0
j
(s)|v
j
(t

∗
− s)d
s
≤−r
i
0
(κ + ε)ξ
i
0
e
−λ(t
∗
−δ
i
0
−a)
+
m

j=1
p
i
0
j
η
j
(κ + ε)η
j
e
−λ(t

∗
−a)
+
m

j=1
q
i
0
j
(κ + ε)η
j
e
−λ(t
∗
−a)
+∞

0
e
λs
|K
i
0
j
(s)|ds
=[−r
i
0
ξ

i
0
e
λδ
i
0
+
m

j
=1
(p
i
0
j
+ q
i
0
j
k
i
0
j
(λ))η
j
](κ + ε)e
−λ(t
∗
−a)
.

Since(11)holds,itfollowsthat
−r
i
0
ξ
i
0
e
λδ
i
0
+

m
j
=1
(p
i
0
j
+ q
i
0
j
k
i
0
j
(λ))η
j

< −λξ
i
0
<
0
.
Therefore, we have
D
+
u
i
0
(t
∗
) < −λξ
i
0
(κ + ε)e
−λ(t
∗
−a)
= ˙z
i
0
(t
∗
)
,
which contradicts the inequality
D

+
u
i
0
(t
∗
) ≥˙z
i
0
(t
∗
)
in (13). Thus (12) holds for all t
Î [a, b). Letting ε ® 0, we have

u
i
(t ) ≤ κξ
i
e
−λ(t−a)
, t ∈ [a, b), i =1,2, , n,
v
j
(t ) ≤ κη
j
e
−λ(t−a)
, t ∈ [a, b), j =1,2, , m
.

The proof is completed.
Remark 1. Lemma 3 is a generalization of the famous Halanay inequality.
Theorem 2 Under assumptions (H1)-(H3), if the following conditions hold,
(C1) there exist vectors ξ =(ξ
1
, ξ
2
, , ξ
n
)
T
>0,h =(h
1
, h
2
, , h
m
)
T
>0and positive
number l >0such that
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪

⎪
⎪
⎪
⎩
(λ − a
i
e
λδ
i
)ξ
i
+
m

j=1

|a
ij
| +(|α
ij
| + |˜α
ij
|)k
ij
(λ)

G
j
η
j

< 0, i =1,2, , n,
(λ − b
j
e
λθ
j
)η
j
+
n

i=1

|b
ji
| +(|β
ji
| + |
˜
β
ji
|)
¯
k
ji
(λ)

F
i
ξ

i
< 0, j =1,2, , m
;
(C2)
μ =sup
k
∈
N
{
ln μ
k
t
k
−t
k−1
} <
λ
, where
μ
k
=max
1≤i≤n,1≤
j
≤m
{1, γ
ik
, ¯γ
jk
}
, k Î N,

then system (1) has exactly one globally exponentially stable equilibrium point, and
its exponential convergence rate equals l - μ.
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 10 of 18
Proof. Since (C1) holds, from Theorem 1, we know that system (1) has one unique
equilibrium point
(x
∗
1
, , x
∗
n
, y
∗
1
, , y
∗
m
)
T
. Now, we assume that (x
1
(t), , x
n
(t), y
1
(t), ,
y
m
(t))

T
is any solution of system (1), let
¯
x
i
(t )=x
i
(t ) − x
∗
i
, i =1,2, ,n,
¯
y
j
(t )=y
j
(t ) − y
∗
j
, j = 1, 2, , m. It is easy to see that system (1) can be transformed
into the following system
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙
¯x
i
(t )=−a
i
¯
x
i
(t − δ
i

)+
m

j=1
a
ij

g
j
(
¯
y
j
(t )+y
∗
j
) − g
j
(y
∗
j
)

+
m
∧
j=1
α
ij
+∞


0
K
ij
(s)g
j
(
¯
y
j
(t − s)+y
∗
j
)ds −
m
∧
j=1
α
ij
+∞

0
K
ji
(s)g
j
(y
∗
j
)ds

+
m
∨
j=1
˜α
ij
+∞

0
K
ij
(s)g
j
(
¯
y
j
(t − s)+y
∗
j
)ds −
m
∨
j=1
˜α
ij
+∞

0
K

ij
(s)g
j
(y
∗
j
)ds
,
t = t
k
,
¯
x
i
(t
+
k
)=
˜
P
ik
(
¯
x
i
(t
−
k
)), k ∈ N
˙

¯y
j
(t )=−b
j
¯
y
j
(t − θ
j
)+
n

i=1
b
ji
(f
i
(
¯
x
i
(t )+x
∗
i
) − f
i
(x
∗
i
))

+
n
∧
i=1
β
ji
+∞

0
¯
K
ji
(s)f
i
(
¯
x
i
(t − s)+x
∗
i
)ds −
n
∧
i=1
β
ji
+∞

0

¯
K
ji
(s)f
i
(x
∗
i
)ds
+
n
∨
i=1
˜
β
ji
+∞

0
¯
K
ij
(s)f
i
(
¯
x
i
(t − s)+x
∗

i
)ds −
n
∨
i=1
˜
β
ji
+∞

0
¯
K
ij
(s)f
i
(x
∗
i
)ds,
t = t
k
,
¯
y
j
(t
+
k
)=

˜
Q
jk
(y
j
(t
−
k
)), k ∈ N ,
(14)
where
˜
P
ik
(
¯
x
i
(t )) =
¯
P
ik
(
¯
x
i
(t )+x
∗
i
) −

¯
P
ik
(x
∗
i
)
,
˜
Q
jk
(
¯
y
j
(t )) =
¯
Q
jk
(
¯
y
j
(t )+y
∗
j
) −
¯
Q
jk

(y
∗
j
)
,and
the initial conditions of (14) are

˜
φ(s)=x(s) − x
∗
= φ(s) − x
∗
, s ∈ (−∞,0]
,
˜ϕ(s)=y(s) − y
∗
= ϕ(s) − y
∗
, s ∈ (−∞,0].
From (H1) and Lemma 2, we calculate the upper right derivative along the solutions
of first equation and third equation of (14), we can obtain
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩

D
+
|
¯
x
i
(t ) |≤−a
i
|
¯
x
i
(t − δ
i
)| +
m

j=1
|a
ij
|G
j
|
¯
y
j
(t ) |
+
m


j=1
(|α
ij
| + |˜α
ij
|)G
j
+∞

0
|K
ij
(s)||
¯
y
j
(t − s)|ds
,
D
+
|
¯
y
j
(t ) |≤−b
j
|
¯
y
j

(t − θ
j
)| +
n

i=1
|b
ji
|F
i
|
¯
x
i
(t ) |
+
n

i=1
(|β
ji
| + |
˜
β
ji
|)F
i
+∞

0

|
¯
K
ji
(s)||
¯
x
i
(t − s)|ds
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 11 of 18
for i = 1, 2, , n, j = 1, 2, , m.
Let
u
i
(
t
)
= |
¯
x
i
(
t
)|
,
v
j
(t )=|
¯

y
j
(t )
|
, r
i
= a
i
, p
ij
=|a
ij
|G
j
,
q
i
j
=(|α
i
j
| + |˜α
i
j
|)G
j
,
¯
r
j

= b
j
,
¯
q
j
i
=(|β
j
i
| + |
˜
β
j
i
|)F
i
(i =1,2, , n; j =1,2, , m
)
,
¯
q
j
i
=(|β
j
i
| + |
˜
β

j
i
|)F
i
(i =1,2, , n; j =1,2, , m
)
, then we have
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
D
+
u
i
(t ) ≤−r
i
u
i

(t − δ
i
)+
m

j=1
p
ij
v
j
(t )+
m

j=1
q
ij
+∞

0
|K
ij
(s)|v
j
(t − s)ds
,
D
+
v
j
(t ) ≤−

¯
r
j
v
j
(t − θ
j
)+
n

i=1
¯
p
ji
u
i
(t )+
n

i=1
¯
q
ji
+∞

0
|
¯
K
ji

(s)|u
i
(t − s)ds
(15)
for i = 1, 2, , n, j = 1, 2, , m, and from (C1), there exist vectors ξ =(ξ
1
, ξ
2
, , ξ
n
)
T
>0,h =(h
1
, h
2
, , h
m
)
T
> 0 and positive number l > 0 such that
⎧
⎪
⎪
⎨
⎪
⎪
⎩
(λ − r
i

e
λδ
i
)ξ
i
+
m

j=1

p
ij
+ q
ij
k
ij
(λ)

G
j
η
j
< 0, i =1,2, , n,
(λ −
¯
r
j
e
λθ
j

)η
j
+
n

i
=1

¯
p
ji
+
¯
q
ji
¯
k
ji
(λ)

F
i
ξ
i
< 0, j =1,2, , m
.
(16)
Taking
||
˜

φ||+|| ˜ϕ||
min
1≤i≤n,1≤
j
≤m
{ξ
i
,η
j
}
, it is easy to prove that

u(t ) ≤ κξe
−λt
, −∞ ≤ t ≤ 0=t
0
,
v(t) ≤ κηe
−λt
, −∞ ≤ t ≤ 0=t
0
.
(17)
From Lemma 3, we obtain that

u(t ) ≤ κξe
−λt
, t
0
≤ t < t

1
,
v(t) ≤ κηe
−λt
, t
0
≤ t < t
1
.
(18)
Suppose that for l ≤ k, the inequalities

u(t ) ≤ κμ
0
μ
1
μ
l−1
ξe
−λt
, t
l−1
≤ t < t
l
,
v(t) ≤ κμ
0
μ
1
μ

l−1
ηe
−λt
, t
l−1
≤ t < t
l
.
(19)
hold, where μ
0
= 1. When l = k + 1, we note that
u
(t
k
)=|
˜
P
k
(u(t
−
k
))|≤Γ
k
u(t
−
k
) ≤ κμ
0
μ

1
μ
k−1
Γ
k
ξ lim
t→t
−
k
e
−λ
t
≤ κ
μ
0
μ
1

μ
k−1
μ
k
ξ
e
−λt
k
,
(20)
and
v(t

k
)=|
˜
Q
k
(v(t
−
k
))|≤
¯
Γ
k
v(t
−
k
) ≤ κμ
0
μ
1
μ
k−1
¯
Γ
k
η lim
t→t
−
k
e
−λ

t
≤ κμ
0
μ
1
μ
k−1
μ
k
η
e
−λt
k
.
(21)
From (20), (21) and μ
k
≥ 1, we have

u(t ) ≤ κμ
0
μ
1
μ
k−1
μ
k
ξe
−λt
, −∞ ≤ t ≤ t

k
,
v(t) ≤ κμ
0
μ
1
μ
k−1
μ
k
ηe
−λt
, −∞ ≤ t ≤ t
k
.
(22)
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 12 of 18
Combining (15),(16),(22) and Lemma 3, we obtain that

u(t ) ≤ κμ
0
μ
1
μ
k
ξe
−λt
, t
k

≤ t < t
k+1
,
v(t) ≤ κμ
0
μ
1
μ
k
ηe
−λt
, t
k
≤ t < t
k+1
.
(23)
Applying the mathematical induction, we can obtain the following inequalities

u(t ) ≤ κμ
0
μ
1
μ
k
ξe
−λt
, t ∈ [t
k
, t

k+1
), k ∈ N
,
v(t) ≤ κμ
0
μ
1
μ
k
ηe
−λt
, t ∈ [t
k
, t
k+1
), k ∈ N
.
(24)
According to (C2), we have
μ
k
≤ e
μ(t
k
−t
k−1
)
< e
λ(t
k

−t
k−1
)
, so we have
u
(t ) ≤ κe
μt
1
e
μ(t
2
−t
1
)
e
μ(t
k−1
−t
k−2
)
ξe
−λt
= κξe
μt
k−1
e
−λt
≤ κξe
−(λ−μ)t
, t ∈ [t

k−1
, t
k
)
, k ∈ N
,
and
v(t) ≤ κe
μt
1
e
μ(t
2
−t
1
)
e
μ(t
k−1
−t
k−2
)
ηe
−λt
= κηe
μt
k−1
e
−λt
≤ κηe

−(λ−μ)t
, t ∈ [t
k−1
, t
k
)
, k ∈ N
.
That is

u(t ) ≤ κξe
−(λ−μ)t
, t ∈ (−∞, t
k
), k ∈ N
,
v(t) ≤ κηe
−(λ−μ)t
, t ∈ (−∞, t
k
), k ∈ N
.
(25)
It follows that
n

i=1
|x
i
(t ) − x

∗
i
| +
m

j=1
|y
j
(t ) − y
∗
j
| =
n

i=1
u
i
(t )+
m

j=1
v
j
(t )
≤
n

i=1
κξ
i

e
−(λ−μ)t
+
m

j=1
κη
j
e
−(λ−μ)t
=

n
i=1
ξ
i
+

m
j=1
η
j
min
1≤i≤n,1≤j≤m
{ξ
i
, η
j
}
(||

˜
φ|| + || ˜ϕ||)e
−(λ−μ)
t
= M
(
||φ − x
∗
|| + ||ϕ − y
∗
||
)
e
−(λ−μ)t
,
where
M =

n
i=1
ξ
i
+

m
j=1
η
j
min
1≤i≤n,1≤

j
≤m
{ξ
i
,η
j
}
, then we have
||x(t) − x
∗
|| + ||y(t) − y
∗
|| ≤ M

||φ − x
∗
|| + ||ϕ − y
∗
||

e
−(λ−μ)t
.
The proof is completed.
Remark 2. In Theorem 2, the parameters μ
k
and μ depend on the impulsive distur-
bance of system (1), and l is actually an estimate of exponential convergence rate of
continuous system (2), whic h depends on the delay kernel functions and system para-
meters. In order to obtain more precise estimate of the exponential convergence rate

of system (1) (or system (2)), we suggest the following optimization problem:
(OP)

max λ,
s.t.(C1)holds
.
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 13 of 18
Obviously, for continuous system (2), we can immediately obtain the following
corollaries.
Corollary 2 Under assumptions (H1) and (H2), if condition (C1) holds, then system
(2) has exactly one globally exponentially stable equilibrium point, and its exponential
convergence rate equals l.
Corollary 3 Under assumptions (H1) and (H2),system(2) has exactly one globally
exponentially stable equilibrium point if C - TL is a nonsingular M-matrix.
Remark 3. Not e that Lemma 2 transforms the fuzzy AND (⋀) and the fuzzy OR (⋁)
operation into the SUM operation (∑). So above results can be applied to the following
classical impulsive BAM neural networks with time delays in the leakage terms and
distributed delays:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙
x
i
(t )=−a
i
x
i
(t − δ
i
)+
m

j=1
a
ij
g
j
(y
j
(t ))

+
m

j=1
α
ij
+∞

0
K
ij
(s)g
j
(y
j
(t − s))ds + I
i
, t = t
k
x
i
(t
+
)=x
i
(t
−
)+P
ik
(x

i
(t
−
)), t = t
k
, k ∈ N ,
˙
y
j
(t )=−b
j
y
j
(t − θ
j
)+
n

i=1
b
ji
f
i
(x
i
(t ))
+
n

i=1

β
ji
+∞

0
¯
K
ji
(s)f
i
(x
i
(t − s))ds + J
j
, t = t
k
y
j
(t
+
)=y
j
(t
−
)+Q
jk
(y
j
(t
−

)), t = t
k
, k ∈ N
(26)
for i = 1, 2, , n; j = 1, 2, , m.
For model (26), it is easy to obtain the following result:
Theorem 3 Under assumptions (H1)-(H3), if the following conditions hold,
(C1’) there exist vectors ξ =(ξ
1
, ξ
2
, , ξ
n
)
T
>0,h =(h
1
, h
2
, ,h
m
)
T
>0and positive
number l >0such that
⎧
⎪
⎪
⎪
⎪

⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
(λ − a
i
e
λδ
i
)ξ
i
+
m

j=1

|a
ij
| + |α
ij
|k
ij
(λ)

G
j

η
j
< 0, i =1,2, , n,
(λ − b
j
e
λθ
j
)η
j
+
n

i=1

|b
ji
| + |β
ji
|
¯
k
ji
(λ)

F
i
ξ
i
< 0, j =1,2, , m

;
(C2)
μ =sup
k
∈
N

ln μ
k
t
k
−t
k−1

<
λ
, where
μ
k
=max
1≤i≤n,1≤
j
≤m
{
1, γ
ik
, ¯γ
jk
}
, k Î N ,

then system (26) has exactly one globally exponentially stable equilibrium point, and
its exponential convergence rate equals l - μ.
5 An illustrative example
In order to illustrate the feasibility of our above-established criteria in the preceding
sections, we provide a concrete example. Although the selection of the coefficients and
functions in the example is somewhat artificial, the possible application of our theoreti-
cal theory is clearly expressed.
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 14 of 18
Example. Consider the following impulsive BAM FCNNs with time delays in the
leakage terms and distributed delays:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙
x
i
(t)=−a
i
x
i
(t − δ
i
)+

2

j=1
a
ij
g
j
(y
j
(t)) +
2

j=1
˜
a
ij
v
j
+ I
i
+
2
∧
j=1
α
ij
+∞

0
K

ij
(s)g
j
(y
j
(t − s))ds +
2
∨
j=1
˜α
ij
+∞

0
K
ij
(s)g
j
(y
j
(t − s))ds
+
2
∧
j=1
T
ij
v
j
+

2
∨
j=1
H
ij
v
j
, t = t
k
x
i
(t
+
)=x
i
(t
−
)+P
ik
(x
i
(t
−
)) = x
i
(t
−
) − (1+e
0.125k
)(x

i
(t
−
) − 1), t = t
k
,
˙
y
j
(t)=−b
j
y
j
(t − θ
j
)+
2

i=1
b
ji
f
i
(x
i
(t)) +
2

i=1
˜

b
ji
u
i
+ J
j
+
2
∧
i=1
β
ji
+∞

0
¯
K
ji
(s)f
i
(x
i
(t − s))ds +
2
∨
i=1
˜
β
ji
+∞


0
¯
K
ij
(s)f
i
(x
i
(t − s))ds
+
2
∧
i=1
¯
T
ji
u
i
+
2
∨
i=1
¯
H
ji
u
i
, t = t
k

y
j
(t
+
)=y
j
(t
−
)+Q
j
k
(y
j
(t
−
)) = y
j
(t
−
) − (1 + e
0.125k
)(y
j
(t
−
) − 1), t = t
k
(27)
for k Î N, i =1,2,j =1,2,t >0,t
0

=0,t
k
= t
k-1
+ 0.5k, k Î N, where
a
1
=4.5, a
2
=4.5, δ
1
=0.2, δ
2
=0.3, a
11
=
4
3
, a
12
= −
1
2
,
a
21
=
1
2
, a

22
=
2
3
,
˜
a
11
=1,
˜
a
12
= −2,
˜
a
21
= −2,
˜
a
22
=1,
I
1
=
67
12
, I
2
=
5

12
, α
11
=
1
3
, α
12
= −
1
4
, α
21
=
1
4
, α
22
=
2
3
,
˜α
11
=
1
3
, ˜α
12
=

1
4
, ˜α
21
= −
1
4
, ˜α
22
=
2
3
, T
11
=1, T
12
=0,
T
21
=0, T
22
=1, H
11
=1, H
12
=0, H
21
=0, H
22
=1,

v
1
=1, v
2
=2;
b
1
=4.5, b
2
=4.5, θ
1
=0.2, θ
2
=0.1, b
11
=
1
3
, b
12
= −
2
3
,
b
21
=
4
3
, b

22
=
1
3
,
˜
b
11
=1,
˜
b
12
=3,
˜
b
21
=2,
˜
b
22
= −2,
J
1
= −
1
2
, J
2
=
7

6
, β
11
=
1
3
, β
12
= −
1
6
, β
21
=
1
3
, β
22
=
1
3
,
˜
β
11
=
1
3
,
˜

β
12
=
1
6
,
˜
β
21
=
1
3
˜
β
22
=
1
3
,
˜
T
11
=1,
˜
T
12
=0,
˜
T
21

=0,
˜
T
22
=1,
˜
H
11
=1,
˜
H
12
=0,
˜
H
21
=0,
˜
H
22
=1,
u
1
=1, u
2
=1;
K
ij
(s)=
¯

K
ij
(s)=e
−s
, f
i
(s)=g
j
(s)=
|s+1|−|s−1|
2
, i, j =1,2.
From above parameters, we have F
1
= F
2
=1,G
1
= G
2
=1,and
(k
ij
(λ))
2×2
=(
¯
k
ji
(λ))

2×2
=

1
1−λ
1
1−λ
1
1−
λ
1
1−
λ

, Γ
k
=
¯
Γ
k
=

e
0.125k
e
0.125k

.
Solving the following optimization problem
⎧

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
max λ
0 > (λ − a
1
e
λδ
1
)ξ
1
+(|a
11
| +(|α
11
| + |˜α
11
|)k
11

(λ))G
1
η
1
+(|a
12
| +(|α
12
| + |˜α
12
|)k
12
(λ))G
2
η
2
,
0 > (λ − a
2
e
λδ
2
)ξ
1
+(|a
21
| +(|α
21
| + |˜α
21

|)k
21
(λ))G
1
η
1
+(|a
22
| +(|α
22
| + |˜α
22
|)k
22
(λ))G
2
η
2
0 > (λ − b
1
e
λθ
1
)η
1
+(|b
11
| +(|β
11
| + |

˜
β
11
|)
¯
k
11
(λ))F
1
ξ
1
+(|b
12
| +(|β
12
| + |
˜
β
12
|)
¯
k
12
(λ))F
2
ξ
2
0 > (λ − b
2
e

λθ
2
)η
2
+(|b
21
| +(|β
21
| + |
˜
β
21
|)
¯
k
21
(λ))F
1
ξ
1
+(|b
22
| +(|β
22
| + |
˜
β
22
|)
¯

k
22
(λ))F
2
ξ
2
,
λ>0, ξ =(ξ
1
, ξ
2
)
T
> 0, η =(η
1
, η
2
)
T
> 0.
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 15 of 18
We obtain that l ≈ 0.3868 > 0, ξ = ( 1082041, 1327618)
T
>0andh = (716212,
1050021)
T
> 0, so (C1) holds. From Theorem 1, we know system (27) has a unique
equilibrium point, this equilibrium point is (1, 1, 1, 1)
T

. Also,
μ
k
=max
1≤i≤2,1≤j≤2
{1, γ
ik
, ¯γ
jk
} =e
0.125k
,
μ =sup
k
∈
N
ln μ
k
t
k
− t
k−1
=
0.125k
0.5k
=0.25< 0.3868 = λ
.
That is, (C2) holds. From Theorem 2, the unique equilibrium point (1, 1, 1, 1)
T
of

system (27) is globally exponentially stable, and its exponential convergence rate is
about 0.1368. The numerical simulation is shown in Figure 1 and 2.
6 Conclusions
In this paper, a class of impulsive BAM FCNN s with ti me delays in the leakage terms
and distributed delays has been formulated and investigated. Some new criteria on the
existence, uniqueness and global exponential stability of equilibrium point for the net-
works have been derived by using M-matrix theory and the impulsive delay integro-dif-
ferential inequality. Our stability criteria are delay-dependent and impulse-dependent.
The neuronal output activation functions and the impulsive operators only need to are
Lipschitz continuous, but need not to be bounded and monotonically increasing. Some
restrictions of delay kernel functions are also removed. It is worthwhile to mention
that our technical methods a re practical, in the sense that all new stability conditions
are stated in simple algebraic forms and provided a more precise estimate of the expo-
nential convergence rate, so their verification and applications are straightforward and
0 5 10 15 20 25 30 35
−10
−5
0
5
10
15
Time: t
X
1
X
2
Figure 1 Behavior of the state variable x(t) with time impulses.
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 16 of 18
convenient. The effectiveness of our results has been demonstrated by the convenient

numerical example.
Acknowledgements
This work is supported by the Scientific Research Fund of Sichuan Provincial Education Department under Grant
09ZC057.
Authors’ contributions
ZX designed and performed all the steps of proof in this research and also wrote the paper. KL participated in the
design of the study and helped to draft and revise manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 25 May 2011 Accepted: 1 September 2011 Published: 1 September 2011
References
1. Kosko, B: Bi-directional associative memories. IEEE Trans Syst Man Cybern. 18,49–60 (1988). doi:10.1109/21.87054
2. Mathai, G, Upadhyaya, BR: Performance analysis and application of the bidirectional associative memory to industrial
spectral signatures. Proceedings of the IJCNN-1989. 1,33–37 (1989)
3. Li, C, Liao, X, Zhang, R: Delay-dependent exponential stability analysis of bi-directional associative memory neural
networks with time delay: an LMI approach. Chaos Solitons Fractals. 24, 1119–1134 (2005). doi:10.1016/j.
chaos.2004.09.052
4. Arik, S: Global asymptotic stability analysis of bidirectional associative memory neural networks with time delays. IEEE
Trans Neural Netw. 16, 580–586 (2005). doi:10.1109/TNN.2005.844910
5. Gopalsamy, K, He, X: Delay-independent stability in bi-directional associative memory with axonal signal transmission
delays. IEEE Trans Neural Netw. 5, 998–1002 (1994). doi:10.1109/72.329700
6. Mohamad, S: Global exponential stability in continuous-time and discrete-time delay bidirectional neural networks. Phys
D. 159, 233–251 (2001). doi:10.1016/S0167-2789(01)00344-X
7. Zhang, J, Yang, Y: Global stability analysis of bi-directional associative memory neural networks with time delay. Int J
Circ Theory Appl. 29, 185–196 (2001). doi:10.1002/cta.144
8. Liang, J, Cao, J: Exponential stability of continuous-time and discrete-time bidirectional associative memory networks
with delays. Chaos Solitons Fractals. 22, 773–785 (2004). doi:10.1016/j.chaos.2004.03.004
9. Feng, C, Plamondon, R: Stability analysis of bidirectional associative memory networks with time delays. IEEE Trans
Neural Netw. 14, 1560–1565 (2003). doi:10.1109/TNN.2003.820829
0 5 10 15 20 25 30 35

0.2
0.4
0.6
0.8
1
1.2
1.4
Time: t
Y
1
Y
2
Figure 2 Behavior of the state variable y(t) with time impulses.
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
/>Page 17 of 18
10. Rao, VSH, Phaneendra, BhRM: Global dynamics of bidirectional associative memory neural networks involving
transmission delays and dead zones. Neural Netw. 12, 455–465 (1999). doi:10.1016/S0893-6080(98)00134-8
11. Song, Q, Cao, J: Global exponential stability of bidirectional associative memory neural networks with distributed delays.
J Comput Appl Math. 202, 266–279 (2007). doi:10.1016/j.cam.2006.02.031
12. Gopalsamy, K: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer, Dordrecht (1992)
13. Gopalsamy, K: Leakage delays in BAM. J Math Anal Appl. 325, 1117–1132 (2007). doi:10.1016/j.jmaa.2006.02.039
14. Peng, S: Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage
terms. Nonlinear Anal Real World Appl. 11, 2141–2151 (2010). doi:10.1016/j.nonrwa.2009.06.004
15. Li, X, Fu, X, Balasubramaniam, P, Rakkiyappan, R: Existence, uniqueness and stability analysis of recurrent neural
networks with time delay in the leakage term under impulsive perturbations. Nonlinear Anal Real World Appl. 11,
4092–4108 (2010). doi:10.1016/j.nonrwa.2010.03.014
16. Li, X, Cao, J: Delay-dependent stability of neural networks of neutral type with time delay in the leakage term.
Nonlinearity. 23, 1709–1726 (2010). doi:10.1088/0951-7715/23/7/010
17. Balasubramaniam, P, Kalpana, M, Rakkiyappan, R: Global asymptotic stability of BAM fuzzy cellular neural networks with
time delay in the leakage term, discrete and unbounded distributed delays. Math Comput Model. 53, 839–853 (2011).

doi:10.1016/j.mcm.2010.10.021
18. Li, X, Rakkiyappan, R, Balasubramaniam, P: Existence and global stability analysis of equilibrium of fuzzy cellular neural
networks with time delay in the leakage term under impulsive perturbations. J Franklin Institute. 348, 135–155 (2011).
doi:10.1016/j.jfranklin.2010.10.009
19. Yang, T, Yang, L, Wu, C, Chua, L: Fuzzy cellular neural networks: theory. Proceedings of IEEE International Workshop on
Cellular Neural Networks and Applications. 181–186 (1996)
20. Yang, T, Yang, L: The global stability of fuzzy cellular neural networks. IEEE Trans Cric Syst I. 43, 880–883 (1996).
doi:10.1109/81.538999
21. Liu, Y, Tang, W: Exponential stability of fuzzy cellular neural networks with constant and time-varying delays. Phys Lett
A. 323, 224–233 (2004). doi:10.1016/j.physleta.2004.01.064
22. Yuan, K, Cao, J, Deng, J: Exponential stability and periodic solutions of fuzzy cellular neural networks with time-varying
delays. Neurocomputing. 69, 1619–1627 (2006). doi:10.1016/j.neucom.2005.05.011
23. Huang, T: Exponential stability of fuzzy cellular neural networks with distributed delay. Phys Lett A. 351,48–52 (2006).
doi:10.1016/j.physleta.2005.10.060
24. Li, K: Impulsive effect on global exponential stability of BAM fuzzy cellular neural networks with time-varying delays. Int
J Syst Sci. 41, 131–142 (2010). doi:10.1080/00207720903042921
25. Song, Q, Cao, J: Dynamical behaviors of discrete-time fuzzy cellular neural networks with variable delays and impulses. J
Franklin Institute. 345,39– 59 (2007)
26. Wang, S, Chung, KFL, Fu, D: Applying the improved fuzzy cellular neural network IFCNN to white blood cell detection.
Neurocomputing. 70, 1348–1359 (2007). doi:10.1016/j.neucom.2006.07.012
27. Li, K, Zhang, L: Global exponential stability of BAM fuzzy cellular neural networks with distributed delays and impulses. J
Appl Math Inform. 29, 211
–225 (2011)
28. Akca, H, Alassar, R, Covachev, V, Covacheva, Z, Al-Zahrani, E: Continuous-time additive Hopfield-type neural networks
with impulses. J Math Anal Appl. 290, 436–451 (2004). doi:10.1016/j.jmaa.2003.10.005
29. Xu, D, Yang, Z: Impulsive delay differential inequality and stability of neural networks. J Math Anal Appl. 305, 107–120
(2005). doi:10.1016/j.jmaa.2004.10.040
30. Yang, Z, Xu, D: Impulsive effects on stability of Cohen-Grossberg neural networks with variable delays. Appl Math
Comput. 177,63–78 (2006). doi:10.1016/j.amc.2005.10.032
31. Gopalsamy, K: Stability of artificial neural networks with impulses. Appl Math Comput. 154, 783–813 (2004). doi:10.1016/

S0096-3003(03)00750-1
32. Mohamad, S, Gopalsamy, K, Akca, H: Exponential stability of artificial neural networks with distributed delays and large
impulses. Nonlinear Anal Real World Appl. 9, 872–888 (2008). doi:10.1016/j.nonrwa.2007.01.011
33. Li, Y: Global exponential stability of BAM neural networks with delays and impulses. Chaos, Solitions Fractals. 24,
279–285 (2005)
34. Li, Y, Yang, C: Global exponential stability analysis on impulsive BAM neural networks with distributed delays. J Math
Anal Appl. 324, 1125–1139 (2006). doi:10.1016/j.jmaa.2006.01.016
35. Liu, B, Huang, L: Global exponential stability of BAM neural networks with recent-history distributed delays and
impulses. Neurocomputing. 69, 2090–2096 (2006). doi:10.1016/j.neucom.2005.09.014
36. Huang, Z, Luo, X, Yang, Q: Global asymptotic stability analysis of bidirectional associative memory neural networks with
distributed delays and impulse. Chaos, Solitons Fractals. 34, 878–885 (2007). doi:10.1016/j.chaos.2006.03.112
37. Zhou, Q: Global exponential stability of BAM neural networks with distributed delays and impulses. Nonlinear Anal Real
World Appl. 10, 144–153 (2009). doi:10.1016/j.nonrwa.2007.08.019
38. Horn, RA, Johnson, CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
39. Cronin, J: Fixed Points and Topological Degree in Nonlinear Analysis. American Mathematical Society, Providence.
(1964)
doi:10.1186/1029-242X-2011-43
Cite this article as: Zhang and Li: Integro-differential inequality and stability of BAM FCNNs with time delays in
the leakage terms and distributed delays. Journal of Inequalities and Applications 2011 2011:43.
Zhang and Li Journal of Inequalities and Applications 2011, 2011:43
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