Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 689845, 12 pages
doi:10.1155/2012/689845
Research Article
Fixed Point and Asymptotic Analysis of
Cellular Neural Networks
Xianghong Lai1 and Yutian Zhang2
1
School of Economics & Management, Nanjing University of Information Science & Technology,
Nanjing 210044, China
2
School of Mathematics & Statistics, Nanjing University of Information Science and Technology,
Nanjing 210044, China
Correspondence should be addressed to Yutian Zhang, ytzhang81@163.com
Received 24 April 2012; Revised 16 July 2012; Accepted 2 August 2012
Academic Editor: Naseer Shahzad
Copyright q 2012 X. Lai and Y. Zhang. 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.
We firstly employ the fixed point theory to study the stability of cellular neural networks
without delays and with time-varying delays. Some novel and concise sufficient conditions are
given to ensure the existence and uniqueness of solution and the asymptotic stability of trivial
equilibrium at the same time. Moreover, these conditions are easily checked and do not require the
differentiability of delays.
1. Introduction
Cellular neural networks CNNs were firstly proposed by Chua and Yang in 1988 1, 2 and
have become a research focus for their numerous successful applications in various fields
such as optimization, linear, and nonlinear programming, associative memory, pattern recognition, and computer vision. Owing to the finite switching speed of neurons and amplifiers
in the implementation of neural networks, it turns out that the time delays are inevitable
and therefore the model of delayed cellular neural networks DCNNs is of greater realistic
significance. Research on the dynamic behaviors of CNNs and DCNNs has received much
attention, and nowadays there have been a large number of achievements reported 3–5.
In fact, besides delay effects, stochastic and impulsive as well as diffusing effects are
also likely to exist in the neural networks. As a result, they have formed complex CNNs
including impulsive delayed reaction-diffusion CNNs, stochastic delayed reaction-diffusion
CNNs, and so forth. One can refer to 6–11 for the relevant researches. Synthesizing the
existing publications about complex CNNs, we find that Lyapunov method is the primary
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Journal of Applied Mathematics
technique. However, we also notice that there exist lots of difficulties in the applications of
corresponding results to practical problems and so it does seem that new methods are needed
to address those difficulties.
Encouragingly, Burton and other authors have recently applied the fixed point theory
to investigate the stability of deterministic systems and obtained more applicable results, for
example, see the monograph 12 and the papers 13–24. Furthermore, there has been found
that the fixed point theory is also effective to the stability analysis of stochastic delayed
differential equations, see 25–31. Particularly, in 26–28, Luo used the fixed point theory to
study the exponential stability of mild solutions of stochastic partial differential equations
with bounded delays and with infinite delays. In 29, 30, Sakthivel and Luo used the
fixed point theory to investigate the asymptotic stability in pth moment of mild solutions to
nonlinear impulsive stochastic partial differential equations with bounded delays and with
infinite delays. In 31, Luo used the fixed point theory to study the exponential stability of
stochastic Volterra-Levin equations. With these motivations, we wonder if we can use the
fixed point theory to study the stability of complex neural networks, thus obtaining more
applicable results.
In the present paper, we aim to discuss the asymptotic stability of CNNs and
DCNNs. Our method is based on the contraction mapping theory, which is different from
the usual method of Lyapunov theory. Some new and easily checked algebraic criteria are
presented ensuring the existence and uniqueness of solution and the asymptotic stability
of trivial equilibrium at the same time. These sufficient conditions do not require even the
differentiability of delays, let alone the monotone decreasing behavior of delays.
2. Preliminaries
Let Rn denote the n-dimensional Euclidean space and · represent the Euclidean norm.
N {1, 2, . . . , n}. R 0, ∞. CX, Y corresponds to the space of continuous mappings
from the topological space X to the topological space Y .
In this paper, we consider the cellular neural network described by
n
dxi t
−ai xi t bij fj xj t ,
dt
j1
t ≥ 0,
2.1
2.2
xi 0 x0i ,
and the following cellular neural network with time-varying delays as
n
n
dxi t
−ai xi t bij fj xj t cij gj xj t − τj t ,
dt
j1
j1
xi s ϕi s,
−τ ≤ s ≤ 0,
t≥0
2.3
2.4
where i ∈ N and n is the number of neurons in the neural network. xi t corresponds to the
state of the ith neuron at time t. x0 x01 , . . . , x0n T ∈ Rn . fj xj t denotes the activation
function of the jth neuron at time t and gj xj t − τj t is the activation function of the jth
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3
neuron at time t − τj t. The constant bij represents the connection weight of the jth neuron
on the ith neuron at time t. The constant ai > 0 represents the rate with which the ith
neuron will reset its potential to the resting state when disconnected from the network and
external inputs. The constant cij represents the connection strength of the jth neuron on the
ith neuron at time t − τj t, where τj t corresponds to the transmission delay along the
axon of the jth neuron and satisfies 0 ≤ τj t ≤ τ τ is a constant. fj ·, gj · ∈ CR, R.
ϕs ϕ1 s, . . . , ϕn sT ∈ Rn and ϕi s ∈ C−τ, 0, R. Denote |ϕ| sups∈−τ,0 ϕs.
Throughout this paper, we always assume that fj 0 gj 0 0 for j ∈ N and
therefore 2.1 and 2.3 admit a trivial equilibrium x 0.
Denote by xt; 0, x0 x1 t; 0, x01 , . . . , xn t; 0, x0n T ∈ Rn the solution of 2.1 with the
initial condition 2.2 and denote by xt; s, ϕ x1 t; s, ϕ1 , . . . , xn t; s, ϕn T ∈ Rn the solution
of 2.3 with the initial condition 2.4.
Definition 2.1 see 32. The trivial equilibrium x 0 of 2.1 is said to be stable if for any
ε > 0, there exists δ > 0 such that for any initial condition x0 satisfying x0 < δ,
xt; 0, x0 < ε,
t ≥ 0.
2.5
Definition 2.2 see 32. The trivial equilibrium x 0 of 2.1 is said to be asymptotically
stable if it is stable and for any x0 ∈ Rn ,
lim xt; 0, x0 0.
t→∞
2.6
Definition 2.3 see 32. The trivial equilibrium x 0 of 2.3 is said to be stable if for any
ε > 0, there exists δ > 0 such that for any initial condition ϕs ∈ C−τ, 0, Rn satisfying
|ϕ| < δ,
x t; s, ϕ < ε,
t ≥ 0.
2.7
Definition 2.4 see 32. The trivial equilibrium x 0 of 2.3 is said to be asymptotically
stable if it is stable and for any initial condition ϕs ∈ C−τ, 0, Rn ,
lim x t; s, ϕ 0.
t→∞
2.8
The consideration of this paper is based on the following fixed point theorem.
Lemma 2.5 see 33. Let Υ be a contraction operator on a complete metric space Θ, then there exists
a unique point ζ ∈ Θ for which Υζ ζ.
3. Asymptotic Stability of Cellular Neural Networks
In this section, we will simultaneously consider the existence and uniqueness of solution
to 2.1-2.2 and the asymptotic stability of trivial equilibrium x 0 of 2.1 by means
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Journal of Applied Mathematics
of the contraction mapping principle. Before proceeding, we firstly introduce the following
assumption:
A1 There exist nonnegative constants lj such that for η, υ ∈ R,
fj η − fj υ ≤ lj η − υ,
j ∈ N.
3.1
Let S S1 × S2 × · · · × Sn , where Si i ∈ N is the space consisting of continuous
functions φi t : R → R such that φi 0 x0i and φi t → 0 as t → ∞, here x0i is the same
as defined in Section 2. Also S is a complete metric space when it is equipped with a metric
defined by
n qi t − hi t,
d qt, ht sup
3.2
t≥0 i1
where qt q1 t, . . . , qn t ∈ S and ht h1 t, . . . , hn t ∈ S.
Theorem 3.1. Assume the condition (A1) holds. If the following inequalities hold
n 1
max bij lj < 1,
ai j
i1
where λi 1/ai n
j1
1
max{λi } < √ ,
i∈N
n
3.3
|bij lj |, then the trivial equilibrium x 0 of 2.1 is asymptotically stable.
Proof. Multiplying both sides of 2.1 with eai t gives
deai t xi t eai t dxi t ai xi teai t dt eai t
n
bij fj xj t dt,
t ≥ 0, i ∈ N,
3.4
j1
which yields after integrating from 0 to t as
xi t x0i e−ai t e−ai t
t
eai s
0
n
bij fj xj s ds,
t ≥ 0, i ∈ N.
3.5
j1
Now, for any yt y1 t, . . . , yn t ∈ S, we define the following operator Φ acting on S as
Φyt Φ y1 t, . . . , Φ yn t ,
t ≥ 0,
3.6
where
Φ yi t x0i e−ai t e−ai t
t
0
eai s
n
j1
bij fj yj s ds,
i ∈ N.
3.7
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5
The following proof is based on the contraction mapping principle, which can be
divided into two steps as follows.
Step 1. We need to prove ΦS ⊂ S. Recalling the construction of S, we know that
it is necessary to show the continuity of Φ on 0, ∞ and Φyi t|t0 x0i as well as
limt → ∞ Φyi t 0 for i ∈ N.
From 3.7, it is easy to see Φyi t|t0 x0i . Moreover, for a fixed time t1 ≥ 0, we have
Φ yi t1 r − Φ yi t1 x0i e−ai t1 r
−ai t1
− x0i e
−ai t1 r
e
t1 r
eai s
0
−ai t1
−e
t1
0
eai s
n
bij fj yj s ds
j1
3.8
n
bij fj yj s ds.
j1
It is not difficult to see that Φyi t1 r − Φyi t1 → 0 as r → 0 which implies Φ is
continuous on 0, ∞.
Next we shall prove limt → ∞ Φyi t 0 for yi t ∈ Si . Since yj t ∈ Sj , we get
limt → ∞ yj t 0. Then for any ε > 0, there exists a Tj > 0 such that s ≥ Tj implies |yj s| < ε.
Choose T ∗ maxj∈N {Tj }. It is then derived form A1 that
e−ai t
t
eai s
0
n
bij fj yj s ds
j1
≤ e−ai t
t
eai s
0
−ai t
e
T∗
0
−ai t
≤e
n
j1
−ai t
≤e
n
j1
n bij lj yj s ds
j1
t
n n −ai t
bij lj yj s ds
bij lj yj s ds e
e
eai s
ai s
T∗
j1
j1
3.9
T ∗
n t a s
a
s
bij lj bij lj sup yj s
e i ds εe−ai t
e i ds
s∈0,T ∗ 0
j1
T∗
T ∗
n ε
ai s
bij lj .
bij lj sup yj s
e ds ai j1
0
s∈0,T ∗ t
As ai > 0, we obtain e−ai t 0 eai s nj1 bij fj yj sds → 0 as t → ∞. So limt → ∞ Φyi t 0 for
i ∈ N. We therefore conclude that ΦS ⊂ S.
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Journal of Applied Mathematics
Step 2. We need to prove Φ is contractive. For any y y1 t, . . . , yn t ∈ S and z z1 t, . . . , zn t ∈ S, we compute
sup
n Φ yi t − Φzi t
t∈0,T i1
≤ sup
⎧
n ⎨
e−ai t
t∈0,T i1
≤ sup
⎧
n ⎨
e−ai t
t∈0,T i1
≤ sup
⎩
⎩
⎫
n ⎬
bij fj yj s − fj zj s ds
eai s
⎭
0
j1
t
⎫
n ⎬
bij lj yj s − zj s ds
eai s
⎭
0
j1
t
3.10
⎫
⎧
⎫
n ⎨
⎬ −a t t a s ⎬
yj s − zj s e i
max bij lj sup
e i ds
⎭
⎩ j∈N
⎭
0
s∈0,T ⎩ j1
⎧
n ⎨
t∈0,T i1
⎫
⎧
n n ⎨
⎬
1
yj s − zj s .
max bij lj ≤
sup
⎭
ai j∈N
s∈0,T ⎩ j1
i1
n
maxj∈N {|bij lj |}} < 1, Φ is a contraction mapping.
Therefore, by the contraction mapping principle, we see there must exist a unique
fixed point u· of Φ in S which means uT · is the solution of 2.1-2.2 and uT · → 0 as
t → ∞.
To obtain the asymptotic stability, we still need to prove that the trivial equilibrium
x 0 of 2.1 is stable. For any ε > 0, from the conditions of Theorem 3.1, we can find δ
√
satisfying 0 < δ < ε such that δ maxi∈N {λi }ε ≤ ε/ n.
Let x0 < δ. According to what have been discussed above, we know that there must
exist a unique solution xt; 0, x0 x1 t; 0, x01 , . . . , xn t; 0, x0n T ∈ Rn to 2.1-2.2, and
As
i1 {1/ai xi t Φxi t J1 t J2 t,
t ≥ 0,
3.11
t
where J1 t x0i e−ai t , J2 t e−ai t 0 eai s nj1 bij fj xj sds.
Suppose there exists t∗ > 0 such that xt∗ ; 0, x0 ε and xt; 0, x0 < ε as 0 ≤ t < t∗ .
It follows from 3.11 that |xi t∗ | ≤ |J1 t∗ | |J2 t∗ |.
∗
∗ t∗
As |J1 t∗ | |x0i e−ai t | ≤ δ and |J2 t∗ | ≤ e−ai t 0 eai s nj1 |bij lj xj s|ds <
n
ε/ai j1 |bij lj |, we obtain |xi t∗ | < δ λi ε. Hence
xt∗ ; 0, x0 2 2
n n |xi t∗ |2 <
|δ λi ε|2 ≤ nδ max{λi }ε ≤ ε2 .
i1
i1
i∈N
3.12
This contradicts to the assumption of xt∗ ; 0, x0 ε. Therefore, xt; 0, x0 < ε holds
for all t ≥ 0. This completes the proof.
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7
4. Asymptotic Stability of Delayed Cellular Neural Networks
In this section, we will simultaneously consider the existence and uniqueness of solution to
2.3-2.4 and the asymptotic stability of trivial equilibrium x 0 of 2.3 by means of the
contraction mapping principle. Before proceeding, we give the assumption as follows.
A2 There exist nonnegative constants kj such that for η, υ ∈ R,
gj η − gj υ ≤ kj η − υ,
j ∈ N.
4.1
Let H H1 ×· · ·×Hn , where Hi i ∈ N is the space consisting of continuous functions
φi t : −τ, ∞ → R such that φi s ϕi s on s ∈ −τ, 0 and φi t → 0 as t → ∞, here ϕi s
is the same as defined in Section 2. Also H is a complete metric space when it is equipped
with a metric defined by
n qi t − hi t,
d qt, ht sup
t≥−τ i1
4.2
where qt q1 t, . . . , qn t ∈ H and ht h1 t, . . . , hn t ∈ H.
Theorem 4.1. Assume the conditions (A1)-(A2) hold. If the following inequalities hold
n 1
< 1,
max bij lj max cij kj
j∈N
ai j∈N
i1
1
max λi ∗ < √
i∈N
n
4.3
where λi ∗ 1/ai nj1 |bij lj | 1/ai nj1 |cij kj |, then the trivial equilibrium x 0 of 2.3 is
asymptotically stable.
Proof. Multiplying both sides of 2.3 with eai t gives
deai t xi t eai t dxi t ai xi teai t dt
⎧
⎫
n
n
⎨
⎬
dt,
eai t
bij fj xj t cij gj xj t − τj t
⎩ j1
⎭
j1
4.4
which yields after integrating from 0 to t as
−ai t
xi t ϕi 0e
−ai t
e
⎧
n
⎨
⎫
n
⎬
e
b f x s cij gj xj s − τj s
ds.
⎩ j1 ij j j
⎭
0
j1
t
ai s
4.5
Now for any yt y1 t, . . . , yn t ∈ H, we define the following operator π acting on H
πyt π y1 t, . . . , π yn t ,
4.6
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Journal of Applied Mathematics
where
π yi t ϕi 0e−ai t e−ai t
⎧
n
⎨
⎫
n
⎬
ds,
eai s
bij fj yj s cij gj yj s − τj s
⎩ j1
⎭
0
j1
t
t ≥ 0,
4.7
and πyi s ϕi s on s ∈ −τ, 0 for i ∈ N.
Similar to the proof of Theorem 3.1, we shall apply the contraction mapping principle
to prove Theorem 4.1. The subsequent proof can be divided into two steps.
Step 1. We need prove πH ⊂ H. To prove πH ⊂ H, it is necessary to show the continuity
of π on −τ, ∞ and limt → ∞ πyi t 0 for yi t ∈ Hi and i ∈ N. In light of 4.7, we have,
for a fixed time t1 ≥ 0,
π yi t1 r − π yi t1 I1 I2 I3,
4.8
where
I1 ϕi 0 e−ai t1 r − ϕi 0 e−ai t1 ,
t1 r
t1
n
n
eai s bij fj yj s ds − e−ai t1
eai s bij fj yj s ds,
I2 e−ai t1 r
0
−ai t1 r
I3 e
0
j1
t1 r
j1
4.9
t1
n
n
−ai t1
e
cij gj yj s − τj s ds − e
eai s cij gj yj s − τj s ds.
ai s
0
0
j1
j1
It is easy to see that limr → 0 {πyi t1 r − πyi t1 } 0. Thus, π is continuous on 0, ∞.
Noting ϕi s ∈ C−τ, 0, R and πyi 0 ϕi 0, we obtain π is indeed continuous on
−τ, ∞.
Next, we will prove limt → ∞ πyi t 0 for yi t ∈ Hi . As we did in Section 3, we
t
know limt → ∞ e−ai t 0 and e−ai t 0 eai s nj1 bij fj yj sds → 0 as t → ∞. In what follows, we
t
n
will show e−ai t 0 eai s
j1 cij gj yj s − τj sds → 0 as t → ∞. In fact, since yj t ∈ Hj , we
have limt → ∞ yj t 0. Then for any ε > 0, there exists a Tj > 0 such that s ≥ Tj − τ implies
|yj s| < ε. Select T maxj∈N {Tj }. It is then derived from A2 that
e−ai t
t
eai s
0
n
cij gj yj s − τj s ds
j1
−ai t
≤e
t
eai s
0
−ai t
e
T
j1
eai s
0
e−ai t
n cij kj yj s − τj s ds
n cij kj yj s − τj s ds
j1
t
T
eai s
n cij kj yj s − τj s ds
j1
Journal of Applied Mathematics
⎧
⎫
n ⎨
n ⎬ T
t a s
−ai t
a
s
cij kj sup yj s
cij kj ≤e
e i ds εe−ai t
e i ds
⎩
⎭ 0
T
j1
j1
s∈−τ,T ⎫
⎧
n n ⎨
⎬ T
ε
−ai t
a
s
cij kj sup yj s
cij kj .
e i ds ≤e
⎭ 0
⎩
ai j1
j1
s∈−τ,T 9
4.10
t
As limt → ∞ e−ai t 0, we obtain e−ai t 0 eai s nj1 cij gj yj s − τj sds → 0 as t → ∞, which
leads to limt → ∞ πyi t 0 for yi t ∈ Hi and i ∈ N. We therefore conclude πH ⊂ H.
Step 2. We need to prove π is contractive. For any y y1 t, . . . , yn t ∈ H and z z1 t, . . . , zn t ∈ H, we estimate
n π yi t − πzi t
i1
≤
⎧
n ⎨
e−ai t
i1
⎩
⎧
n ⎨
e−ai t
i1
≤
e−ai t
i1
⎩
⎧
n ⎨
≤
⎫
n ⎬
bij lj yj s − zj s ds
eai s
⎭
0
j1
t
e−ai t
i1
⎫
n ⎬
cij gj yj s − τj s − gj zj s − τj s ds
eai s
⎭
0
j1
t
⎩
⎧
n ⎨
⎫
n ⎬
bij fj yj s − fj zj s ds
eai s
⎭
0
j1
t
⎩
⎫
n ⎬
cij kj yj s − τj s − zj s − τj s ds
eai s
⎭
0
j1
t
⎫
⎫
⎧
n ⎨
⎬ −a t t a s ⎬
y s − zj s e i
e i ds
maxb l sup
⎭
⎭
⎩ j∈N ij j s∈0,t⎩ j1 j
0
⎧
n ⎨
i1
⎫
⎧
⎫
n ⎨
⎬ −a t t a s ⎬
y s − zj s e i
maxc k sup
e i ds
⎭
⎩ j∈N ij j s∈−τ,t⎩ j1 j
⎭
0
⎧
n ⎨
i1
⎫
⎧
n n ⎬
⎨
1
yj s − zj s
≤
maxbij lj sup
⎭
ai j∈N
s∈0,t⎩ j1
i1
⎫
⎧
n n ⎨
⎬
1
yj s − zj s .
sup
maxcij kj ⎭
⎩
j∈N
a
i
s∈−τ,t
i1
j1
4.11
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Journal of Applied Mathematics
Hence,
sup
n π yi t − πzi t
t∈−τ,T i1
≤
⎧
⎫
n n n ⎨
⎬
1
1
yj s − zj s .
sup
maxbij lj maxcij kj ⎩
⎭
j∈N
j∈N
a
a
i
i
s∈−τ,T
i1
i1
j1
4.12
As ni1 {1/ai maxj∈N |bij lj | maxj∈N |cij kj |} < 1, π is a contraction mapping and
hence there exists a unique fixed point u· of π in H which means uT · is the solution of
2.3-2.4 and uT · → 0 as t → ∞.
To obtain the asymptotic stability, we still need to prove that the trivial equilibrium
of 2.3 is stable. For any ε > 0, from the conditions of Theorem 4.1, we can find δ satisfying
√
0 < δ < ε such that δ maxi∈N {λ∗i }ε ≤ ε/ n.
Let |ϕ| < δ. According to what have been discussed above, we know that there exists a
unique solution xt; s, ϕ x1 t; s, ϕ1 , . . . , xn t; s, ϕn T ∈ Rn to 2.3-2.4, and
xi t πxi t J1 t J2 t J3 t,
t ≥ 0,
4.13
where
J1 t x0i e−ai t ,
J2 t e−ai t
t
0
J3 e−ai t
t
eai s
0
n
eai s
n
bij fj xj s ds,
j1
4.14
cij gj xj s − τj s ds.
j1
Suppose there exists t∗ > 0 such that xt∗ ; s, ϕ ε and xt; s, ϕ < ε as 0 ≤ t < t∗ . It
follows from 4.13 that |xi t∗ | ≤ |J1 t∗ | |J2 t∗ | |J3 t∗ |.
∗
As |J1 t∗ | |x0i e−ai t | ≤ δ, |J2 t∗ | < ε/ai nj1 |bij lj | and
∗
|J3 t∗ | ≤ e−ai t
t∗
eai s
0
n n cij kj xj s − τj s ds < ε
cij kj ,
ai j1
j1
4.15
we obtain |xi t∗ | < δ λi ∗ ε. Hence
n n ∗
∗ 2
∗ 2
∗ 2
x t ; s, ϕ 2 δ λi ε
≤ nδ max λi ε ≤ ε2 .
|xi t | <
i1
i1
i∈N
4.16
This contradicts to the assumption of xt∗ ; s, ϕ ε. Therefore, xt; s, ϕ < ε holds
for all t ≥ 0. This completes the proof.
Remark 4.2. In Theorems 3.1 and 4.1, we use the contraction mapping principle to study the
existence and uniqueness of solution and the asymptotic stability of trivial equilibrium at the
same time, while Lyapunov method fails to do this.
Journal of Applied Mathematics
11
Remark 4.3. The provided sufficient conditions in Theorem 4.1 do not require even the
differentiability of delays, let alone the monotone decreasing behavior of delays which is
necessary in some relevant works.
5. Example
Consider the following two-dimensional cellular neural network with time-varying delays
2
2
dxi t
−ai xi t bij fj xj t cij gj xj t − τj t ,
dt
j1
j1
5.1
with the initial conditions x1 s coss, x2 s sins on −τ ≤ s ≤ 0, where a1 a2 3, b11 0, b12 1/7, b21 −1/7, b22 −1/7, c11 3/7, c12 2/7, c21 0, c22 1/7,
fj s gj s |s 1| − |s − 1|/2, τj t is bounded by τ.
It is easily to know that lj kj 1 for j 1, 2. Compute
2 1
< 1,
maxbij lj maxcij kj j1,2
ai j1,2
i1
⎫
⎧
n n ⎨1
1
⎬
bij lj cij kj < √1 . 5.2
max
⎭
⎩
i∈N
ai j1
ai j1
2
From Theorem 4.1, we conclude that the trivial equilibrium x 0 of this two-dimensional
cellular neural network is asymptotically stable.
Acknowledgments
This work is supported by National Natural Science Foundation of China under Grant
60904028 and 71171116.
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