An. Şt. Univ. Ovidius Constanţa
Vol. 18(2), 2010, 295–314
GENERAL RESULTS ON THE EXISTENCE
AND GLOBAL EXPONENTIAL STABILITY
OF PERIODIC SOLUTIONS FOR
GENERALIZED SHUNTING INHIBITORY
CELLULAR NEURAL NETWORKS WITH
DELAYS
Xinsong Yang, Chuangxia Huang
Abstract
In this paper, with the help of the Leray-Schauder fixed point theorem, differential inequality techniques and suitable Lyapunov functional,
several novel sufficient conditions on the existence and global exponential stability of periodic solutions for the generalized shunting Inhibitory
cellular neural networks are developed, which improve some published
results. Particularly, the precise convergence rate index is also obtained.
One example with its numerical simulation is employed to illustrate the
generalized results.
1
Introduction and preliminaries
In recent years, the shunting inhibitory cellular neural networks (SICNNs) have
been extensively studied and found many important applications in different
Key Words: Shunting inhibitory cellular neural networks; Global exponential stability;
Periodic solution
2010 Mathematics Subject Classification: 34C25; 34K13; 34K25
The corresponding author: Xinsong Yang
This work was supported in part by the Foundation of Chinese Society for Electrical
Engineering (2008), the Excellent Youth Foundation of Educational Committee of Hunan
Provincial under Grant No. 10B002, the Scientific Research Fund of Hunan Provincial Science and Technology Department of China under Grant No. 2009FJ3103 and the Scientific
Research Fund of Yunnan Province.
Received: August, 2009
Accepted: January, 2010
295
296
Xinsong Yang, Chuangxia Huang
areas such as psychophysics, speech, perception, robotics, adaptive pattern
recognition, vision, and image processing. To date, many interesting results
on stability of SICNNs have been obtained. In particular, some results on the
existence and exponential stability of (almost) periodic solutions for SICNNs
with delays have been reported in [1, 3, 4, 5, 6, 7, 8]. However, SICNNs in [1,
3, 4, 5, 6, 7, 8] are all with constant coefficients. Non-autonomous phenomena
often occur in many realistic systems, hence, it is of prime importance to
study the existence and exponential stability of periodic solutions and almost
periodic solutions of SICNNs with variable coefficients. In this paper, we
consider the following general SICNNs with delays, which includes SICNNs in
[1, 3, 4, 6] as special cases.
x0ij (t) =
−aij (t, xij (t)) +
+
X
X
B kl ∈N
kl
Bij
(t)fij (t, xkl (t))xij (t)
r (i,j)
kl
Cij
(t)gij (t, xkl (t
− τkl (t)))xij (t) + Iij (t), (1.1)
C kl ∈Nr (i,j)
where i = 1, 2, · · · , n, j = 1, 2, · · · , m. τij (t) represents axonal signal transmission delays and continuous with 0 ≤ τkl (t) ≤ τ ; Cij (t) denotes the cell at the
(i, j) position of the lattice at the t, the r-neighborhood Nr (i, j) of Cij (t) is
Nr (i, j) = {C kl (t) : max(|k − i|, |l − j|) ≤ r, 1 ≤ k ≤ m, 1 ≤ l ≤ n}.
xij (t) is the activity of the cell Cij (t), Iij (t) is the external input to Cij (t) ,
kl
(t) is
aij (t, xij (t)) represents the passive decay function of the cell activity; Cij
the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell Cij (t), the activity function fij (t, ·) is a continuous function representing the output or firing rate of the cell C kl (t); ϕij (t) is the
initial function, and is assumed to be bounded and continuous on [−τ, 0].
kl
(t), fij (t, ·), Iij (t), ϕij (t) are all continuous periodic functions.
aij (t), Cij
For convenience, we introduce the notations: τ = max(i,j) {τij (t)|t ∈ [0, ω]},
I ij = maxt∈[0,ω] |Iij (t)|, µij = mint∈[0,ω] µij (t),
x = (x11 , x12 , · · · , x1m , · · · , xn1 , xn2 , · · · , xnm )T
be a column vector, in which the symbol T denotes the transpose of a vector.
The initial condition φ = (φ11 , · · · , φ1m , · · · , φn1 , · · · , φnm )T of (1.1) is of
the form
xij (s) = φij (s), s ∈ (−τ, 0],
where φij (s), i = 1, 2, · · · , n, j = 1, · · · , m, are continuous ω-periodic solutions.
GENERAL RESULTS ON THE EXISTENCE AND GLOBAL EXPONENTIAL
STABILITY OF PERIODIC SOLUTIONS FOR GENERALIZED SHUNTING
INHIBITORY CELLULAR NEURAL NETWORKS WITH DELAYS
297
Definition 1.1. Let x∗ (t) be an ω-periodic solution of (1.1) with initial value
φ∗ . If there exist constants α > 0 and P ≥ 1 such that for every solution x(t)
of (1.1) with initial value φ,
|xij (t) − x∗ij (t)| ≤ P kφ − φ∗ ke−αt ,
∀t > 0, i = 1, 2, · · · , n, j = 1, 2, · · · , m,
where kφ − φ∗ k = max(i,j) sup−τ ≤s≤0 {|φij (s) − φ∗ij (s)|}. Then x∗ (t) is said to
be globally exponentially stable.
Lemma 1.1(Leray-Schauder). Let E be a Banach space, and let the operator
A : E → E be completely continuous. If the set {kxk|x ∈ E, x = λAx, 0 <
λ < 1} is bounded, then A has a fixed point in T , where
T = {x|x ∈ E, kxk ≤ R}, R = sup{kxk|x = λAx, 0 < λ < 1}.
Lemma 1.2 [2]. For any x, y ≥ 0, q > 0.5, the inequality
x2q−1 y ≤
1
2q − 1 2q
x + y 2q
2q
2q
holds.
Obviously, when q = 0.5, the above inequality also holds. Hence, for any
x, y ≥ 0, q ≥ 1, we have
xq−1 y ≤
q−1 q 1 q
x + y .
q
q
(1.2)
The main purpose of this paper is to obtain sufficient conditions for the
existence and global exponential stability of periodic solutions for (1.1). The
main methods used in this paper are Leray-Schauder’s fixed point theorem,
differential inequality techniques and Lyapunov functional. The results of this
paper generalize and complement some published results. One example is
employed to illustrate our feasible results.
Throughout this paper, we assume that
kl
kl
(H1 ) Bij
(t), Cij
(t), τij (t) ≥ 0, Iij (t) are continuous ω-periodic functions. ω >
0 is a constant, i = 1, 2, · · · , n, j = 1, 2, · · · , m;
(H2 ) aij (t, u) ∈ C(R2 , R) are ω-periodic about the first argument, aij (t, 0) = 0
and there are positive continuous ω-periodic functions µij (t) such that
∂aij (t,u)
∂a (t,u)
≥ µij (t), ij∂u
are bounded, i = 1, 2, · · · , n, j = 1, 2, · · · , m;
∂u
(H3 ) fij (t, u), gij (t, u) ∈ C(R2 , R) are ω-periodic about the first argument.
There are continuous ω-periodic solutions δij (t) and γij (t) such that
δij (t) = supu∈R |fij (t, u)|, γij (t) = supu∈R |gij (t, u)|, i = 1, 2, · · · , n,
j = 1, 2, · · · , m;
298
Xinsong Yang, Chuangxia Huang
½
(H4 ) max(i,j) sup0≤t≤ω
δij (t)
P
B kl ∈Nr (i,j)
kl
|Bij
(t)|+γij (t)
µij (t)
P
C kl ∈Nr (i,j)
kl
|Cij
(t)|
¾
=θ
and θ < 1;
(H5 ) There are non-negative continuous ω-periodic solutions αij (t) and βij (t)
such that
¯
¯
¯
¯
¯ f (t,u)−f (t,v) ¯
¯
¯
¯, βij (t) = supu6=v ¯ gij (t,u)−gij (t,v) ¯ for
αij (t) = supu6=v ¯¯ ij u−vij
u−v
¯
¯
¯
all u, v ∈ R, u 6= v, i = 1, 2, · · · , n j = 1, 2, · · · , m.
The organization of this paper is as follows. In Section 2, we study the
existence of periodic solutions of system (1.1) by using the Leray-Schauder’s
fixed point theorem. In Section 3, by constructing Lyapunov functional, we
shall derive new sufficient conditions for the global exponential stability of the
periodic solution of system (1.1). Moreover, we compare our results with some
of previously know results. At last, an example is employed to illustrate the
feasible results of this paper.
2
Existence of periodic solutions
Let ξij , i = 1, 2, · · · , n, j = 1, 2, · · · , m be positive constants. Make the change
of variables
xij = ξij yij (t), i = 1, 2, · · · , n, j = 1, 2, · · · , m,
(2.1)
then (1.1) can be reformulated as
−1
0
yij
(t) = −ξij
aij (t, ξij yij (t)) +
+
X
B kl ∈N
X
kl
Bij
(t)fij (t, ξkl ykl (t))yij (t)
r (i,j)
kl
Cij
(t)gij (t, ξkl ykl (t
−1
− τkl (t)))yij (t) + ξij
Iij (t).
(2.2)
C kl ∈Nr (i,j)
System (2.2) can be rewritten as
0
yij
(t) = −dij (t, yij (t))yij (t) +
+
X
X
B kl ∈N
kl
Bij
(t)fij (t, ξkl ykl (t))yij (t)
r (i,j)
kl
Cij
(t)gij (t, ξkl ykl (t
−1
− τkl (t)))yij (t) + ξij
Iij (t),
(2.3)
C kl ∈Nr (i,j)
. ∂a (t,u)
where dij (t, yij (t)) = ij∂u |u=dij , dij is between 0 and ξij yij (t), dij ∈ R.
By (H2 ), we obtain aij (t, ξij yij ) is strictly monotone increasing about yij .
Hence, dij (t, yij (t)) is unique for any yij (t). Obviously, dij (t, yij (t)) is continuous ω-periodic about the first argument and dij (t, yij (t)) ≥ µij (t).
GENERAL RESULTS ON THE EXISTENCE AND GLOBAL EXPONENTIAL
STABILITY OF PERIODIC SOLUTIONS FOR GENERALIZED SHUNTING
INHIBITORY CELLULAR NEURAL NETWORKS WITH DELAYS
299
Lemma 2.1. Suppose that (H1 )-(H3 ) hold and let x(t) be an ω-periodic
solution of (1.1). Then,
·
Z ω
X
y
kl
yij (t) =
Hij
(t, s)
Bij
(s)fij (s, ξkl ykl (s))yij (s)
0
B kl ∈Nr (i,j)
X
+
kl
Cij
(s)gij (s, ξkl ykl (s
− τkl (s)))yij (s) +
−1
ξij
Iij (s)
¸
ds,
C kl ∈Nr (i,j)
t ∈ [0, ω], i = 1, 2, · · · , n, j = 1, 2, · · · , m,
(2.4)
where,
y
Hij
(t, s) =


Rt
e− sR dij (v,yij (v))dv
,
0≤s≤t
ω
0 dij (v,yij (v))dv
1−eR−
R
−( 0ω dij (v,yij (v))dv− ts dij (v,yij (v))dv)
e
Rω
,
1−e− 0 dij (v,yij (v))dv

≤ ω,
0 ≤ t < s ≤ ω.
Proof. From the system (2.3), we have
µ
¶0 ·
Rt
dij (s,yij (s))ds
0
yij (t)e
=
X
B kl ∈N
X
kl
Bij
(t)fij (t, ξkl ykl (t))yij (t) +
r (i,j)
¸ R
t
−1
kl
Cij
(t)gij (t, ξkl ykl (t − τkl (t)))yij (t) + ξij
Iij (t) e 0 dij (s,yij (s))ds .(2.5)
C kl ∈Nr (i,j)
Integrating (2.5) from 0 to t, we have
Z t·
R
− 0t dij (s,yij (s))ds
yij (t) = e
yij (0) +
0
X
X
kl
Bij
(s)fij (s, ξkl ykl (s))yij (s) +
B kl ∈Nr (i,j)
¸ R
t
−1
kl
(s)gij (s, ξkl ykl (s − τkl (s)))yij (s) + ξij
Cij
Iij (s) e− s dij (v,yij (v))dv ds.(2.6)
C kl ∈Nr (i,j)
From xij (ω) = xij (0) and (2.1), we have yij (ω) = yij (0). By (2.6), we obtain
−
1−e
X
C kl ∈Nr (i,j)
Rω
0
1
dij (s,yij (s))ds
Z
ω
0
·
yij (0) =
X
B kl ∈Nr (i,j)
kl
Bij
(s)fij (s, ξkl ykl (s))yij (s) +
¸ R
ω
−1
kl
Cij
(s)gij (s, ξkl ykl (s − τkl (s)))yij (s) + ξij
Iij (s) e− s dij (v,yij (v))dv ds.(2.7)
300
Xinsong Yang, Chuangxia Huang
Substituting (2.7) into (2.6), we obtain
yij (t) =
e−
1−e
Rt
dij (s,yij (s))ds
R
− 0ω dij (v,yij (v))dv
0
X
C kl ∈N
X
0
X
kl
Bij
(s)fij (s, ξkl ykl (s))yij (s) +
B kl ∈Nr (i,j)
− τkl (s)))yij (s) +
¸
−1
ξij
Iij (s)
C kl ∈N
e−
Rω
dij (v,yij (v))dv
s
ds
kl
(s)fij (s, ξkl ykl (s))yij (s) +
Bij
B kl ∈Nr (i,j)
X
0
0
·
r (i,j)
Z t·
Z
ω
kl
Cij
(s)gij (s, ξkl ykl (s
+
=
Z
¸
kl
Cij
(s)gij (s, ξkl ykl (s
− τkl (s)))yij (s) +
−1
ξij
Iij (s)
e−
Rt
s
dij (v,yij (v))dv
r (i,j)
ω
·
y
Hij
(t, s)
X
kl
Bij
(s)fij (s, ξkl ykl (s))yij (s) +
B kl ∈Nr (i,j)
X
¸
kl
Cij
(s)gij (s, ξkl ykl (s
− τkl (s)))yij (s) +
−1
ξij
Iij (s)
ds.
C kl ∈Nr (i,j)
This completes the proof.
In order to use Lemma 1.1, we take X = {y|y ∈ C([0, ω], Rnm )}. Then X
is a Banach space with the norm
kyk = max{|yij |0 }, |yij |0 = sup |yij (t)|, i = 1, · · · , n, j = 1, · · · , m.
(i,j)
0≤t≤ω
Take a mapping Φ : X → X by setting
(Φy)(t) = ((Φy)11 (t), (Φy)12 (t), · · · , (Φy)nm (t))T ,
where
Z
(Φy)ij (t) =
0
ω
·
y
Hij
(t, s)
X
B kl ∈N
X
kl
Bij
(s)fij (s, ξkl ykl (s))yij (s) +
r (i,j)
¸
−1
kl
(s)gij (s, ξkl ykl (s − τkl (s)))yij (s) + ξij
Cij
Iij (s) ds,
C kl ∈Nr (i,j)
i = 1, 2, · · · , n, j = 1, 2, · · · , m.
It is easy to know the fact that the existence of ω-periodic solution of (1.1)
is equivalent to the existence of fixed point of the mapping Φ in X.
ds
GENERAL RESULTS ON THE EXISTENCE AND GLOBAL EXPONENTIAL
STABILITY OF PERIODIC SOLUTIONS FOR GENERALIZED SHUNTING
INHIBITORY CELLULAR NEURAL NETWORKS WITH DELAYS
301
Lemma 2.2. Suppose that (H1 )-(H4 ) hold. Then Φ : X → X is completely
continuous.
Proof. Under our assumptions, it is clear that the operator Φ is continuous.
Next, we show that Φ is compact.
Since µij (t), i = 1, 2, · · · , n, j = 1, 2, · · · , m are positive ω-periodic solutions, µij > 0. For any constant D > 0, let Ω = {y|y ∈ X, kyk < D}. For any
y ∈ Ω, we have
kΦyk
=
½¯ Z
¯
¯
¯
0≤t≤ω
ω
max sup
(i,j)
+
0
·
½Z
ω
max sup
(i,j) 0≤t≤ω
B kl ∈Nr (i,j)
0
·µ
y
(t, s)
Hij
δij (s)
X
kl
|Bij
(s)|
B kl ∈Nr (i,j)
¶
¸ ¾
−1
kl
|Cij
(s)| yij (s) + ξij
Iij (s) ds
X
+γij (s)
kl
Bij
(s)fij (s, ξkl ykl (s))yij (s)
¸ ¯¾
¯
−1
kl
Iij (s) ds¯¯
Cij
(s)gij (s, ξkl ykl (s − τkl (s)))yij (s) + ξij
X
C kl ∈Nr (i,j)
≤
X
y
Hij
(t, s)
C kl ∈Nr (i,j)
¾
¾
½
I ij
y
Hij
(t, s)µij (s)θds kyk + max
ξij µij
(i,j) 0≤t≤ω
(i,j)
0
½
¾
I ij
θD + max
,
ξij µij
(i,j)
½Z
≤
<
ω
max sup
this implies that Φ(Ω) is uniformly bounded, where,
½Z t R
t
1
y
Rω
e− s dij (v,yij (v))dv µij (s)ds
Hij
(t, s)µij (s)ds =
− 0 dij (v,yij (v))dv
1−e
0
¾0
Z ω R
Rω
s
− 0 dij (v,yij (v))dv
d
(v,y
(v)dv)
ij
ij
+e
et
µij (s)ds
t
½Z t R
t
1
Rω
e− s µij (v)dv µij (s)ds
1 − e− 0 dij (v,yij (v))dv
0
¾
Z ω R
R
s
− 0ω dij (v,yij (v))dv
e t dij (v,yij (v))dv dij (s, yij (s))ds
+e
t
½
³ Rω
´¾
R
Rω
− 0t µij (s)dv
1−e
+ e− 0 dij (v,yij (v))dv e t dij (v,yij (v)dv) − 1
Z
≤
=
ω
1 − e−
Rω
0
dij (v,yij (v))dv
302
Xinsong Yang, Chuangxia Huang
½
1 − e−
Rt
0
µij (s)dv
=
+ e−
Rt
0
dij (v,yij (v))dv
1 − e−
Rω
0
− e−
Rω
0
¾
dij (v,yij (v))dv
≤ 1.
dij (v,yij (v))dv
By (H2 ), there exists a constant M > 0 such that
|dij (t, yij (t))| ≤ M,
for t ∈ [0, ω] × Ω, i = 1, 2, · · · , n, j = 1, 2, · · · , m.
In view of the definition of Φ, we have
µZ ω
·
X
d
y
(Φy)0ij (t) =
Hij
(t, s)
dt
0
kl
B
X
kl
Bij
(s)fij (s, ξkl ykl (s))yij (s) +
∈Nr (i,j)
C kl ∈Nr (i,j)
=
− τkl (s)))yij (s) +
X
−dij (t, yij (t))(Φy)ij (t) +
−1
ξij
Iij (s)
¶
ds
kl
(s)fij (t, ξkl ykl (t))yij (t) +
Bij
B kl ∈Nr (i,j)
X
C kl ∈N
¸
kl
Cij
(s)gij (s, ξkl ykl (s
−1
kl
Cij
(t)gij (t, ξkl ykl (t − τkl (t)))yij (t) + ξij
Iij (t).
r (i,j)
Hence,
µ
½
¾¶
¯
¯
I ij
¯(Φy)0ij (t)¯ ≤ M θD + max
+
ξij µij
(i,j)
½
¾
X
X
I ij
kl
kl
max
B ij δ ij D +
C ij γ ij D +
,
ξij
(i,j)
kl
kl
B
kl
∈Nr (i,j)
C
∈Nr (i,j)
kl
kl
kl
where B ij = supt∈[0,ω] |Bij
(t)|, δ ij = supt∈[0,ω] δij (t) C ij = supt∈[0,ω] |Cij
(t)|,
γ ij = supt∈[0,ω] γij (t). So, Φ(Ω) ⊆ X is a family of uniformly bounded and
equi-continuous subsets. By using the Arzela-Ascoli Theorem, Φ : X → X is
compact. Therefore, Φ : X → X is completely continuous. This completes
the proof.
Theorem 2.1. Suppose that (H1 )-(H4 ) hold. Let ξij , i = 1, 2, · · · , n, j =
1, 2, · · · , m be positive constants. Then the system (1.1) has an ω-periodic
.
e=
solution x∗ (t) with kx∗ k ≤ max(i,j) {ξij }R
R0 , where
½
¾
I
max(i,j) ξijij
µ
ij
e=
R
.
1−θ
GENERAL RESULTS ON THE EXISTENCE AND GLOBAL EXPONENTIAL
STABILITY OF PERIODIC SOLUTIONS FOR GENERALIZED SHUNTING
INHIBITORY CELLULAR NEURAL NETWORKS WITH DELAYS
303
Proof. Let y ∈ X, t ∈ [0, ω]. We consider the operator equation
y = λΦy, λ ∈ (0, 1).
(2.8)
If y is a solution of (2.8), for t ∈ [0, ω], we obtain
½
¾
I ij
kyk ≤ kΦyk ≤ θkyk + max
.
ξij µij
(i,j)
This and (H4 ) imply that
e
kyk ≤ R.
e
In view of Lemma 1.1, we obtain that Φ has a fixed point y ∗ (t) with ky ∗ k ≤ R.
Hence, system (2.3) has one ω-periodic solution
∗
∗
∗
∗
∗
∗
, y12
, · · · , y1m
, · · · , yn1
, yn2
, · · · , ynm
)T
y ∗ (t) = (y11
e It follows from (2.1) that
with ky ∗ k ≤ R.
x∗ (t) = (x∗11 (t), x∗12 (t), · · · , x∗1m (t), · · · , x∗n1 (t), x∗n2 (t), · · · , x∗nm (t))T =
∗
∗
∗
∗
∗
∗
= (ξ11 y11
, ξ12 y12
, · · · , ξ1m y1m
, · · · , ξn1 yn1
, ξn2 yn2
, · · · , ξnm ynm
)T
is one ω-periodic solution of (1.1) with
.
e=
kx∗ k ≤ max{ξij }R
R0 .
(i,j)
This completes the proof.
3
Global exponential stability of periodic solution
In this Section, we shall construct a suitable Lyapunov functional to derive
sufficient conditions ensuring that (1.1) has a unique ω-periodic solution and
all solutions of (1.1) exponentially converge to its unique ω-periodic solution.
Theorem 3.1. Assume (H1 )-(H3 ), (H5 ) and (H6 ); where (H6 ) is
(H6 ) There are positive constants p ≥ 1 and ξij , i = 1, 2, · · · , n, j = 1, 2, · · · , m,
such that
½ kl
¾
kl
pAij (t)ξij + R0 (p − 1)Dij
(t)ξij + R0 Fijkl (ξ, p, t)
max sup
< 1,
pµij (t)ξij
(i,j) 0≤t≤ω
304
Xinsong Yang, Chuangxia Huang
where,
Akl
ij (t) = δij (t)
X
B kl ∈N
X
kl
Dij
(t) =
r (i,j)
X
=
X
kl
|Bij
(t)| +
B kl ∈Nr (i,j)
Fijkl (ξ, p, t)
X
kl
|Bij
(t)| + γij (t)
C kl ∈N
r (i,j)
kl
|Cij
(t)|,
C kl ∈Nr (i,j)
p
kl
|Bij (t)|ξkl αij
(t) +
B kl ∈Nr (i,j)
kl
|Cij
(t)|,
X
p
kl
|Cij
(t)|ξkl βij
(t).
C kl ∈Nr (i,j)
Then the system (1.1) has exactly one ω-periodic solution, which is globally
exponentially stable.
Proof. Obviously that (H6 ) implies (H4 ). By Theorem 2.1, there exists an
ω-periodic solution x∗ (t) of (1.2) with initial value
φ∗ (t) = (φ∗11 (t), · · · , φ∗1m (t), · · · , φ∗n1 (t), · · · , φ∗nm (t))T
and kx∗ k ≤ R0 . Suppose that x(t) is an arbitrary solution of system (1.1)
with initial value φ(t) = (φ11 (t), · · · , φ1m (t), · · · , φn1 (t), · · · , φnm (t))T . Set
z(t) = (z11 (t), · · · , z1m (t), · · · , zn1 (t), · · · , znm (t))T = x(t) − x∗ (t). Then,
from system (1.1) we have
X
+
+
kl
(t)[fij (t, xkl (t))xij (t) − fij (t, x∗kl (t))x∗ij (t)]
Bij
B kl ∈Nr (i,j)
X
C kl ∈N
0
(t) = −[aij (t, xij (t)) − aij (t, x∗ij (t))]
zij
kl
Cij
(t)[gij (t, xkl (t − τkl (t)))xij (t) − gij (t, x∗kl (t − τkl (t)))x∗ij (t)].(3.1)
r (i,j)
From (H6 ) we have
kl
kl
−pµij (t)ξij + pAkl
ij (t)ξij + R0 (p − 1)Dij (t)ξij + R0 Fij (ξ, p, t) < 0,
i = 1, 2, · · · , n, j = 1, 2, · · · , m.
(3.2)
Set
hij (λ) =
kl
λξij − pµij (t)ξij + pAkl
ij (t)ξij + R0 (p − 1)Dij (t)ξij
µ
¶
X
X
p
p
kl
kl
+R0
|Bij
(t)|ξkl αij
(t) +
|Cij
(t)|ξkl βij
(t)eλτ .
B kl ∈Nr (i,j)
C kl ∈Nr (i,j)
Clearly, hij (λ), i = 1, 2, · · · , n, j = 1, 2, · · · , m, are continuous functions on R.
Since hij (0) < 0,
dhij (λ)
= ξij + λR0
dλ
X
C kl ∈Nr (i,j)
p
kl
|Cij
(t)|βij
(t)eλτ ξkl > 0,
GENERAL RESULTS ON THE EXISTENCE AND GLOBAL EXPONENTIAL
STABILITY OF PERIODIC SOLUTIONS FOR GENERALIZED SHUNTING
INHIBITORY CELLULAR NEURAL NETWORKS WITH DELAYS
305
and hij (+∞) = +∞, hence hij (λ), i = 1, 2, · · · , n, j = 1, 2, · · · , m, are strictly
monotone increasing functions. Therefore, for any i ∈ {1, 2, · · · , n}, j ∈
{1, 2, · · · , m} and t ≥ 0, there is unique λ(t) such that
kl
λ(t)ξij − pµij (t)ξij + pAkl
ij (t)ξij + R0 (p − 1)Dij (t)ξij
µ
¶
X
X
p
p
kl
kl
λ(t)τ
+R0
|Bij (t)|ξkl αij (t) +
|Cij (t)|ξkl βij (t)e
= 0.
B kl ∈Nr (i,j)
Let
C kl ∈Nr (i,j)
(
λ∗ij
µ
+R0
¯
¯
kl
= inf λ(t)¯¯λ(t)ξij − pµij (t)ξij + pAkl
ij (t)ξij + R0 (p − 1)Dij (t)ξij
t≥0
¶
¾
X
X
p
p
kl
kl
λ(t)τ
|Bij (t)|ξkl αij (t) +
|Cij (t)|ξkl βij (t)e
=0 .
B kl ∈Nr (i,j)
C kl ∈Nr (i,j)
Obviously, λ∗ij ≥ 0, i = 1, 2, · · · , n, j = 1, 2, · · · , m. Now, we shall prove that
λ∗ij > 0. Suppose this is not true. From (3.2), there exists a positive constant
η such that
½
¾
kl
kl
pµij (t0 )ξij − pAkl
ij (t0 )ξij − R0 (p − 1)Dij (t0 )ξij − R0 Fij (ξ, p, t0 )
P
inf
≥ η.
p
kl (t)|ξ
t≥0,(i,j)
ξij + 1.5τ R0 βij
(t) C kl ∈Nr (i,j) |Cij
kl
Pick small ε > 0, then there exists t0 ≥ 0 such that
0 < λ∗ij (t0 ) < ε < η.
Let us recall the inequality ex < 1 + 1.5x for sufficiently small x > 0. Then
we obtain
0 =
kl
λ(t0 )ξij − pµij (t0 )ξij + pAkl
ij (t0 )ξij + R0 (p − 1)Dij (t0 )ξij
µ
¶
X
X
p
p
kl
kl
+R0
|Bij
(t0 )|ξkl αij
(t0 ) +
|Cij
(t0 )|ξkl βij
(t0 )eλ(t0 )τ
B kl ∈Nr (i,j)
C kl ∈Nr (i,j)
kl
(t0 )ξij
< εξij − pµij (t0 )ξij +
+ R0 (p − 1)Dij
µ
¶
X
X
p
p
kl
kl
+R0
|Bij
(t0 )|ξkl αij
(t0 ) +
|Cij
(t0 )|ξkl βij
(t0 )eετ
pAkl
ij (t0 )ξij
B kl ∈Nr (i,j)
C kl ∈Nr (i,j)
kl
< ηξij − pµij (t0 )ξij +
+ R0 (p − 1)Dij
(t0 )ξij
¶
µ
X
X
p
p
kl
kl
|Bij (t0 )|ξkl αij (t0 ) +
|Cij (t0 )|ξkl βij (t0 )(1 + 1.5ητ )
+R0
pAkl
ij (t0 )ξij
B kl ∈Nr (i,j)
=
C kl ∈Nr (i,j)
kl
kl
−pµij (t0 )ξij + pAkl
ij (t0 )ξij + R0 (p − 1)Dij (t0 )ξij + R0 Fij (ξ, p, t0 )
306
Xinsong Yang, Chuangxia Huang
µ
p
+η ξij + 1.5τ R0 βij
(t0 )
¶
X
kl
|Cij
(t0 )|ξkl
C kl ∈Nr (i,j)
kl
kl
−pµij (t0 )ξij + pAkl
ij (t0 )ξij + R0 (p − 1)Dij (t0 )ξij + R0 Fij (ξ, p, t0 )
kl
kl
pµij (t0 )ξij − pAkl
ij (t0 )ξij − R0 (p − 1)Dij (t0 )ξij − R0 Fij (ξ, p, t0 )
P
+
p
kl
ξij + 1.5τ R0 βij (t0 ) C kl ∈Nr (i,j) |Cij (t0 )|ξkl
≤
µ
p
ξij + 1.5τ R0 βij
(t0 )
×
¶
X
kl
(t0 )|ξkl
|Cij
= 0,
C kl ∈Nr (i,j)
which is a contradiction, and hence λ∗ij > 0, i = 1, 2, · · · , n, j = 1, 2, · · · , m.
Let ² = min(i,j) {λ∗ij }. Obviously,
kl
hij (²) = ²ξij − pµij (t)ξij + pAkl
ij (t)ξij + R0 (p − 1)Dij (t)ξij
¶
X
X
p
p
kl
kl
²τ
≤ 0,
|Bij (t)|ξkl αij (t) +
|Cij (t)|ξkl βij (t)e
µ
+R0
B kl ∈Nr (i,j)
C kl ∈Nr (i,j)
i = 1, 2, · · · , n, j = 1, 2, · · · , m. (3.3)
We choose a constant d > 1 such that
pdξij e−²t ≥ 1, for t ∈ (−τ, 0], i = 1, 2, · · · , n, j = 1, 2, · · · , m.
It is obvious that
|zij (t)|p ≤ kφ − φ∗ kp ≤ pdξij kφ − φ∗ kp e−²t , for t ∈ (−τ, 0] and i = 1, 2, · · · , n,
j = 1, 2, · · · , m, where kφ − φ∗ k is defined as that in Definition 1.1.
Define a Lyapunov functional V (t) = (V11 (t), · · · , V1m (t), · · · , Vn1 (t), · · · , Vnm (t), )T
by Vij (t) = p1 e²t |zij (t)|p , i = 1, 2, · · · , n, j = 1, 2, · · · , m. In view of (1.2) and
(3.1), we obtain
½
d+ Vij (t)
= |zij (t)|p−1 e²t sgn zij − [aij (t, xij (t)) − aij (t, x∗ij (t))]
dt
X
kl
+
Bij
(t)[fij (t, xkl (t))xij (t) − fij (t, x∗kl (t))x∗ij (t)]
B kl ∈Nr (i,j)
+
X
kl
(t)[gij (t, xkl (t
Cij
¾
− τkl (t)))xij (t) −
gij (t, x∗kl (t
−
τkl (t)))x∗ij (t)]
C kl ∈Nr (i,j)
²
+ e²t |zij (t)|p
p
½
≤ |zij (t)|p−1 e²t
− µij (t)|zij (t)| +
X
B kl ∈Nr (i,j)
£
kl
|Bij
(t)| |fij (t, xkl (t))||zij (t)|
GENERAL RESULTS ON THE EXISTENCE AND GLOBAL EXPONENTIAL
STABILITY OF PERIODIC SOLUTIONS FOR GENERALIZED SHUNTING
INHIBITORY CELLULAR NEURAL NETWORKS WITH DELAYS
X
+
¤
+|fij (t, xkl (t)) − fij (t, x∗kl (t))||x∗ij (t)|
£
kl
|Cij
(t)| |gij (t, xkl (t − τkl (t)))||zij (t)|
C kl ∈Nr (i,j)
+|gij (t, xkl (t − τkl (t))) − gij (t, x∗kl (t − τkl (t)))||x∗ij (t)|
½µ
≤ e²t
²
− µij (t) + δij (t)
p
X
B kl ∈N
X
+R0
X
+R0
² − pµij (t) +
p
kl
|Bij
(t)|αij
(t)Vkl (t) +
B kl ∈Nr (i,j)
We claim that
1
Vij (t) = e²t |zij (t)|p ≤ dξij kφ−φ∗ kp ,
p
¾
kl
|Cij
(t)||zij (t)|p−1 βij (t)|zkl (t − τkl (t))|
C kl ∈Nr (i,j)
≤
kl
|Bij
(t)||zij (t)|p−1 αij (t)|zkl (t)|
r (i,j)
µ
µ
²
+ e²t |zij (t)|p
p
¶
kl
|Cij
(t)| |zij (t)|p
C kl ∈Nr (i,j)
X
+R0
¾
¤
X
kl
|Bij
(t)| + γij (t)
B kl ∈Nr (i,j)
307
¶
+ R0 (p −
Vij (t) +
¶
p
kl
²τ
|Cij (t)|βij (t)e Vkl (t − τkl (t)) .(3.4)
pAkl
ij (t)
X
kl
1)Dij
(t)
C kl ∈Nr (i,j)
i = 1, 2, · · · , n, j = 1, 2, · · · , m, for all t > 0.
(3.5)
Contrarily, there must exists some some i ∈ {1, 2, · · · , n}, j ∈ {1, 2, · · · , m}
and e
t > 0 such that
d+ Vij (e
t)
Vij (e
t) = dξij kφ − φ∗ kp ,
> 0 and Vij (t) ≤ dξij kφ − φ∗ kp ,
(3.6)
dt
∀t ∈ (−τ, e
t], i = 1, 2, · · · , n, j = 1, 2, · · · , m. Together with (3.4) and (3.6), we
obtain
½
d+ Vij (e
t)
∗ p
kl
0<
≤ dkφ − φ k ²ξij − pµij (t)ξij + pAkl
ij (t)ξij + R0 (p − 1)Dij (t)ξij
dt
µ
¶¾
X
X
p
p
kl
kl
²τ
+R0
|Bij (t)|ξkl αij (t) +
|Cij (t)|ξkl βij (t)e
.
B kl ∈Nr (i,j)
C kl ∈Nr (i,j)
Hence,
0
<
kl
²ξij − pµij (t)ξij + pAkl
ij (t)ξij + R0 (p − 1)Dij (t)ξij
¶
µ
X
X
p
p
kl
kl
|Cij
(t)|ξkl βij
(t)e²τ ,
+R0
|Bij
(t)|ξkl αij
(t) +
B kl ∈Nr (i,j)
C kl ∈Nr (i,j)
308
Xinsong Yang, Chuangxia Huang
which contradicts (3.3). Hence, (3.5) holds. It follows that
²
1
|xij (t) − x∗ij (t)| = |zij (t)| ≤ (pdξij ) p kφ − φ∗ ke− p t ,
∀t > 0, i = 1, 2, · · · , n,
1
j = 1, 2, · · · , m. Let M = max(i,j) {(pdξij ) p + 1}. Then, we have
²
|xij (t) − x∗ij (t)| ≤ M kφ − φ∗ ke− p t ,
∀t > 0, i = 1, 2, · · · , n, j = 1, 2, · · · , m.
In view of Definition 1.1, the ω-periodic solution x∗ (t) of the system (1.1) is
globally exponentially stable. This completes the proof.
Taking p = 1 in Theorem 3.1, we have the following corollary.
Corollary 3.1. Assume (H1 )-(H3 ), (H5 ) and (H6 ) hold, where
(H06 ) There are positive constants ξij , i = 1, 2, · · · , n, j = 1, 2, · · · , m, such
that
½ kl
¾
kl
Aij (t)ξij + R0 Eij
(ξ, t)
max sup
< 1,
µij (t)ξij
(i,j) 0≤t≤ω
X
X
kl
kl
kl
where, Eij
(ξ, t) =
|Bij
(t)|ξkl αij (t)+
|Cij
(t)|ξkl βij (t).
B kl ∈Nr (i,j)
C kl ∈Nr (i,j)
Then the system (1.1) has exactly one ω-periodic solution, which is globally
exponentially stable.
Remark 1. In [4], the authors studied the existence and exponential stability
of the following SICNNs with delays and variable coefficients
X
kl
x0ij (t) = −hij (t)xij (t) −
Jij
(t)eij (xkl (t))xij (t)
−
X
J kl ∈Nr (i,j)
Wijkl (t)qij (xkl (t − τkl (t)))xij (t) + Iij (t), (3.7)
W kl ∈Nr (i,j)
kl
where hij (t) > 0, Jij
(t) ≥ 0, Wijkl (t) ≥ 0 and Iij (t) are all continuous ωperiodic solutions.
Theorem ∗ (Li[4]). Assume that the following conditions are satisfied:
(F1 ) eij , qij ∈ C(R, R) are bounded on R, i = 1, 2, · · · , n, j = 1, 2, · · · , m;
(F2 ) there exist positive numbers µij and νij such that |eij (x) − eij (y)| ≤
µij |x − y|, |qij (x) − eij (y)| ≤ νij |x − y| for all x, y ∈ R, i = 1, 2, · · · , n,
j = 1, 2, · · · , m;
GENERAL RESULTS ON THE EXISTENCE AND GLOBAL EXPONENTIAL
STABILITY OF PERIODIC SOLUTIONS FOR GENERALIZED SHUNTING
INHIBITORY CELLULAR NEURAL NETWORKS WITH DELAYS
309
(F3 ) for each i = 1, 2, · · · , n, j = 1, 2, · · · , m
hij − Mij
·
I ij
−
µij
Γij
X
kl
J ij − Nij
J kl ∈Nr (i,j)
X
J kl ∈Nr (i,j)
X
kl
W ij
W kl ∈Nr (i,j)
kl
J ij
+ νij
X
kl
W ij
¸
> 0.
W kl ∈Nr (i,j)
Then the system (3.7) has a unique ω-periodic solution which is globally exponentially stable, where
kl
kl
kl
(t)}, W ij = maxt∈[0,ω] {Wijkl (t)},
hij = mint∈[0,ω] {hij (t)}, J ij = maxt∈[0,ω] {Jij
I ij = maxt∈[0,ω] {|Iij (t)|}, Mij = supu∈R {|eij (u)|}, Nij = supu∈R {|qij (u)|}.
One can easily sees that (F1 )-(F3 ) are special cases of (H3 ), (H5 ) and (H06 ),
respectively. Taking ξij = 1, i = 1, 2, · · · , n, j = 1, 2, · · · , m in Corollary 3.1,
it is obvious that Theorem 3.2 is direct corollary of Corollary 3.1.
Remark 2. It is worth noting that our results Theorem 2.1, Theorem 3.1 and
Corollary 3.1 are also applicable to SCINNs with distributed delays [5, 7, 8].
On the other hand, the results of this paper are also applicable to studying
the existence and exponential stability of almost periodic solution for SCINNs
with time-varying coefficients and delays. Consider the following SCINNs with
time-varying coefficients and delays:

 x0ij (t) =

xij (t) =
−e
aij (t)xij (t) −
X
ekl
C
kl
eij
C
(t)e
gij (t, xkl (t − τekl (t)))xij (t) + Ieij (t),
(3.8)
ϕij (t), t ∈ [0, ω] i = 1, 2, · · · , n, j = 1, 2, · · · , m.
e kl (t) ≥ 0, geij (t, ·), τeij (t) ≥ 0, ϕij (t), Ieij (t) are all
where t ≥ 0, e
aij (t) > 0, C
ij
continuous almost periodic functions.
We further assume that
(F01 ) geij (t, u) ∈ C(R2 , R) are almost periodic about the first argument. There
are continuous almost periodic solutions γ
eij (t) such that
γ
eij (t) = supu∈R |e
gij (t, u)|, i = 1, 2, · · · , n, j = 1, 2, · · · , m;
(F02 ) there are non-negative continuous almost periodic solutions βeij (t) such
that
¯
¯
¯ ge (t,u)−eg (t,v) ¯
¯, for all u, v ∈ R, u 6= v, i = 1, 2, · · · , n
βeij (t) = supu6=v ¯¯ ij u−vij
¯
j = 1, 2, · · · , m;
310
Xinsong Yang, Chuangxia Huang
(F03 ) There are positive constants p ≥ 1 and ξij , i = 1, 2, · · · , n, j = 1, 2, · · · , m,
such that
½P
e0 (p − 1)ξij + R
e0 ξkl βep (t)) ¾
e kl
γij (t)ξij + R
ekl ∈Nr (i,j) Cij (t)(e
ij
C
max sup
< 1,
e
aij (t)ξij
(i,j) t∈R
½
¾Á
I ij
e
e
e
where R0 = max(i,j) {ξij }R, R = max(i,j) ξij ea
(1 − θ), I ij =
ij
aij (t),
maxt∈R |Iij (t)|, e
a = mint∈R e
½ij
¾
P
ekl (t)|
γ
eij (t) Ce kl ∈N (i,j) |C
ij
r
max(i,j) supt∈R
= θ < 1.
e
aij (t)
Let ξij , i = 1, 2, · · · , n, j = 1, 2, · · · , m be positive constants, ψ(t) be continuous almost periodic solution. Make the change of variables as that of (1.2).
We have the unique almost periodic solution from system (3.8)
½Z
y ψ(t) (t) =
·
X
−
t
e−
Rt
s
dij (u,ψij (u))du
−
−∞
¸ ¾
−1
kl
Cij
(s)fij (s, ξkl ψkl (s − τkl (s)))ψij (s) + ξij
Iij (s) ds .
C kl ∈Nr (i,j)
Similar to the proof of Theorem 2.1 and Theorem 3.1 of this paper, we
obtain the following theorem.
Theorem 3.2. Assume (F01 )-(F03 ) hold. Then the system (3.8) has exactly
one almost periodic solution, which is globally exponentially stable.
Take p = 1 in Theorem 3.3 and obtain the following corollary:
Corollary 3.2. Assume (F01 ), (F02 ) hold and
(F003 ) There are positive constants ξij , i = 1, 2, · · · , n, j = 1, 2, · · · , m, such
that
½P
e0 ξkl βeij (t)) ¾
e kl
γij (t)ξij + R
e kl ∈Nr (i,j) Cij (t)(e
C
max sup
< 1,
e
aij (t)ξij
(i,j) t∈R
e0 is the same as that in (F0 ).
where R
3
Then the system (3.8) has exactly one almost periodic solution, which is globally exponentially stable.
Remark 3. From (3.3), it is obvious that the convergent index ² is more
precise than the estimation of convergent index in [1, 3, 4, 5, 6, 7, 8].
GENERAL RESULTS ON THE EXISTENCE AND GLOBAL EXPONENTIAL
STABILITY OF PERIODIC SOLUTIONS FOR GENERALIZED SHUNTING
INHIBITORY CELLULAR NEURAL NETWORKS WITH DELAYS
4
311
Application
In this Section, we give an example to illustrate the obtained results. Consider the following generalized SICNNs with time-varying delays and coefficients
X
kl
x0ij (t) = −aij (t, xij (t)) +
Cij
(t)gij (t, xkl (t − τkl (t)))xij (t) + Iij (t),(4.1)
C kl ∈Nr (i,j)
where r = 1. Take a11 (t, x) = a13 (t, x) = a21 (t, x) = a32 (t, x) = 4x + sin x +
x sin t, a12 (t, x) = a23 (t, x) = a31 (t, x) = a33 (t, x) = 5x − sin x + x cos t,
a22 (t, x) = 3x + cos x − x sin t, gij (t, x) = 0.1 sin x, τij (t) = (cos t)2 , and

 

C11 (t) C12 (t) C13 (t)
0.2 cos t 0.4 sin t 0.3 cos t
 C21 (t) C22 (t) C23 (t)  =  0.6 cos t
0
0.5 sin t  ,
C31 (t) C32 (t) C33 (t)
0.5 sin t 0.6 cos t 0.5 sin t

 

1.5 cos t 2 sin t 2 cos t
I11 (t) I12 (t) I13 (t)
 I21 (t) I22 (t) I23 (t)  =  3.5 cos t 4 cos t 2 sin t  .
5 sin t 3 cos t 4 sin t
I31 (t) I32 (t) I33 (t)
Then ω = 2π, µ11 (t) = µ13 (t) = µ21 (t) = µ32 (t) = 3 + sin t, µ12 (t) =
µ23 (t) = µ31 (t) = µ33 (t) = 4 + cos t, µ22 (t) =
= βij (t) = 0.1,
P2 − sin t, γij (t)
P
kl
kl
(t)
= 1.1| cos t|+
C
(t)|
=
0.8|
cos
t|+0.4|
sin
t|,
|C
C kl ∈Nr (1,2) 12
C kl ∈Nr (1,1)
P
P 11
kl
kl
(t)| =
0.9| sin t|, C kl ∈Nr (1,3) |C13 (t)| = 0.3| cos t|+0.9| sin t|, C kl ∈Nr (2,1) |C21
P
P
kl
1.4| cos t|+0.9| sin t|, C kl ∈Nr (2,2) |C22 (t)| = 1.7| cos t|+1.9| sin t|, C kl ∈Nr (2,3)
P
kl
kl
(t)| = 1.2| cos t| + 0.5| sin t|,
(t)| = 0.9| cos t| + 1.4| sin t|, C kl ∈Nr (3,1) |C31
|C23
P
P
kl
kl
(t)|
=
1.2|
cos
t|+1.5|
sin
t|,
|C
32
C kl ∈Nr (3,3) |C33 (t)| = 0.6| cos t|+
C kl ∈Nr (3,2)
| sin t|.
Take ξij = 1, i, j = 1, 2, 3. Computing by MATLAB, we have θ ≈
0.22358 < 1, R0 ≈ 2.1466, and κ ≈ 0.7035 < 1. It’s easy to check that all
the conditions in Corollary 3.1 are hold. Therefore, system (4.1) has a unique
2π-periodic solution x∗ (t) with kx∗ k ≤ R0 , which is globally exponentially
stable. The numerical simulations with the following initial conditions:
(x11 (s), x12 (s), x13 (s), x21 (s), x22 (s), x23 (s), x31 (s), x32 (s), x33 (s))T
= (1, 2, 3, 4, 5, 6, 7, 8, 9)T
and
(x11 (s), x12 (s), x13 (s), x21 (s), x22 (s), x23 (s), x31 (s), x32 (s), x33 (s))T
= (−1, −2, −3, −4, −5, −6, −7, −8, −9)T
312
Xinsong Yang, Chuangxia Huang
for s ∈ [−1, 0]. For the trajectories of xij (t), i, j = 1, 2, 3, please see the below
Figure.
10
8
6
4
solution x
2
0
−2
−4
−6
−8
−10
0
1
2
3
4
5
time t
6
7
8
9
10
Fig. : Trajectories of xij (t), i, j = 1, 2, 3 for system (4.1).
Remark 4. The system (4.1) is a simple generalized SICNN with delays
and time-varying coefficients. Obviously, aij (t, x), i, j = 1, 2, 3, are non-linear
about x, hence none of the the results in [1, 3, 4, 5, 6, 7, 8] and references
cited therein can be applied to (4.1). Moreover, the periodic solution x∗ (t)
satisfies kx∗ k ≤ R0 , which has nothing to do with the initial value of (4.1).
Hence, the results of this paper generalize and complement some previously
known results.
References
[1] A. Chen, J. Cao, Almost periodic solution of shunting inhibitory CNNs
with delays, Phys Lett., A 298 (2002), 161-170.
[2] S. Guo, L. Huang, Exponential stability and periodic solutions of neural
networks with continuously distributed delays, Phys. Rev., E 67 (2003),
1902-1909.
[3] X. Huang, J. Cao, Almost periodic solution of shunting inhibitory cellular
neural networks with time-varying delay, Phys Lett, A 314 (2003), 222231.
GENERAL RESULTS ON THE EXISTENCE AND GLOBAL EXPONENTIAL
STABILITY OF PERIODIC SOLUTIONS FOR GENERALIZED SHUNTING
INHIBITORY CELLULAR NEURAL NETWORKS WITH DELAYS
313
[4] Y. Li, C. Liu, L. Zh Global exponential stability of periodic solution of
shunting inhibitory CNNs with delays, Phys. Lett., A 337 (2005), 46-54.
[5] Y. Li, et al. Exponential convergence behavior of shunting inhibitory cellular neural networks with time-varying coefficients, J. Comput. Appl.
Math., 216 (2008), 164-169.
[6] B. Liu, L. Huang, Existence and stability of almost periodic solutions
for shunting inhibitory cellular neural networks with time-varying delays,
Chaos, Solitons & Fractals, 31 (2007), 211-217.
[7] Y. Liu, et al. On the almost periodic solution of generalized shunting
inhibitory cellular neural networks with continuously distributed delays,
Phys. Lett., A 360 (2006), 122-130.
[8] H. Meng, Y. Li, New convergence behavior of shunting inhibitory cellular neural networks with time-varying coefficients, Applied Mathematics
Letters, 21 (7) (2008), 717-721.
Department of Mathematics
Honghe University
Mengzi Yunnan 661100, China,
Email: xinsongyang@163.com
Department of Applied Mathematics,
College of Mathematics and Computing Science,
Changsha University of Science and Technology,
Changsha, Hunan 410076, China
Email: cxiahuang@126.com
314
Xinsong Yang, Chuangxia Huang