Electronic Journal of Differential Equations, Vol. 2006(2006), No. 19, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
EXISTENCE AND STABILITY OF ALMOST PERIODIC
SOLUTIONS FOR SHUNTING INHIBITORY CELLULAR
NEURAL NETWORKS WITH CONTINUOUSLY
DISTRIBUTED DELAYS
QIYUAN ZHOU, BING XIAO, YUEHUA YU
Abstract. In this paper, we consider shunting inhibitory cellular neural networks (SICNNs) with continuously distributed delays. Sufficient conditions
for the existence and local exponential stability of almost periodic solutions
are established using a fixed point theorem, Lyapunov functional method, and
differential inequality techniques. We illustrate our results with an example
for which our conditions are satisfied, but not the conditions in [4, 6, 8].
1. Introduction
Consider the shunting inhibitory cellular neural networks (SICNNs) with continuously distributed delays
Z ∞
X
kl
x0ij (t) = −aij (t)xij (t) −
Cij
f(
Kij (u)xkl (t − u)du)xij (t) + Lij (t),
0
Ckl ∈Nr (i,j)
(1.1)
where i = 1, 2, . . . , m, j = 1, 2, . . . , n, Cij denote the cell at the (i, j) position of
the lattice, the r-neighborhood Nr (i, j) of Cij is
Nr (i, j) = {Ckl : max(|k − i|, |l − j|) ≤ r, 1 ≤ k ≤ m, 1 ≤ l ≤ n}.
xij is the activity of the cell Cij , Lij (t) is the external input to Cij , aij (t) > 0
kl
represent the passive decay rate of the cell activity, Cij
≥ 0 is the connection or
coupling strength of postsynaptic activity of the cell transmitted to the cell Cij , and
the activity function f is a continuous function representing the output or firing
rate of the cell Ckl .
Since Bouzerdout and Pinter in [1, 2, 3] described SICNNs as a new cellular
neural networks(CNNs), SICNNs have been extensively applied in psychophysics,
speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Hence, they have been the object of intensive analysis by numerous authors
2000 Mathematics Subject Classification. 34C25, s4K13, 34K25.
Key words and phrases. Shunting inhibitory cellular neural networks; almost periodic solution;
exponential stability; continuously distributed delays.
c
2006
Texas State University - San Marcos.
Submitted December 3, 2005. Published February 7, 2006.
Supported by the NNSF of China and by grant 05JJ40009 from Hunan Provincial
Natural Science Foundation of China.
1
2
Q. ZHOU, B. XIAO, Y. YU
EJDE-2006/19
in recent years. In particular, there have been extensive results on the problem of
the existence and stability of periodic and almost periodic solutions of SICNNs with
constant time delays and time-varying delays in the literature. We refer the reader
to [4, 6, 8] and the references cited therein. Moreover, in the above-mentioned
literature, we observe that the assumption
(T0) there exists a nonnegative constant Mf such that Mf = supx∈R |f (x)|
has been considered as fundamental for the considered existence and stability of
periodic and almost periodic solutions of SICNNS. However, to the best of our
knowledge, few authors have considered SICNNS without the assumptions (T0).
Thus, it is worth while to continue to investigate the existence and stability of
almost periodic solutions of SICNNS.
The main purpose of this paper is to obtain some sufficient conditions for the existence and stability and local exponential stability of the almost periodic solutions
for system (1.1). By applying fixed point theorem, Lyapunov functional method
and differential inequality techniques, we derive some new sufficient conditions ensuring the existence and local exponential stability of the almost periodic solution
of system (1.1), which are new and they complement previously known results. In
particular, we do not need the assumption (T0). Moreover, an example is also
provided to illustrate the effectiveness of the new results.
Throughout this paper, we set
{xij (t)} = (x11 (t), . . . , x1n (t), . . . , xi1 (t), . . . , xin (t), . . . , xm1 (t), . . . , xmn (t)).
For all x = {xij (t)} ∈ Rm×n , we define the norm kxk = max(i,j) {|xij (t)|}. Set
B = ϕ : ϕ = {ϕij (t)} = (ϕ11 (t), . . . , ϕ1n (t), . . . ,
ϕi1 (t), . . . , ϕin (t), . . . , ϕm1 (t), . . . , ϕmn (t)) ,
where ϕ is an almost periodic function on R. For all ϕ ∈ B, we define induced
module kϕkB = supt∈R kϕ(t)k, then B is a Banach space.
The initial conditions associated with system (1.1) are
xij (s) = ϕij (s),
s ∈ (−∞, 0], i = 1, 2, . . . , m, j = 1, 2, . . . , n,
(1.2)
where ϕij (·) denotes real-valued bounded continuous function defined on (−∞, 0].
We also assume that the following conditions
(T1) For i ∈ {1, 2, . . . , m}, j ∈ {1, 2, . . . , n}, the delay kernels Kij : [0, ∞) → R
are continuous, integrable and there exist nonnegative constants kij such
that
Z ∞
|Kij (s)|ds ≤ kij .
0
(T2) For each i ∈ {1, 2, . . . , m}, j ∈ {1, 2, . . . , n}, Lij (t) and aij (t) are almost
periodic functions on R, let L+
ij = supt∈R |Lij (t)|, 0 < aij = inf t∈R aij (t).
(T3) f (0) = 0, f : R → R is Lipschitz with Lipschitz constant µ, i.e.,
|f (u) − f (v)| ≤ µ|u − v|,
for all u, v ∈ R.
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EXISTENCE AND STABILITY
3
(T4) there exist nonnegative constants L, q and δ such that
P
kl o
n µkij
L+
Ckl ∈Nr (i,j) Cij
ij
L = max{
< 1,
}, δ = max
aij
(i,j) aij
(i,j)
L
L
≤ 1, q = 2δ
< 1.
(1 − δ)
(1 − δ)
(T5) For i ∈ {1, 2, . . . , m}, j ∈ {1, 2, . . . , n}, there exists a constant λ0 > 0 such
that
Z ∞
|Kij (s)|eλ0 s ds < +∞.
0
Definition. (see [5, 6]) Let u(t) : R → Rn be continuous in t. u(t) is said to be
almost periodic on R if, for each ε > 0, the set T (u, ε) = {δ : |u(t + δ) − u(t)| <
ε, ∀t ∈ R} is relatively dense; i.e., for all ε > 0, it is possible to find a real number
l = l(ε) > 0, so that for any interval of length l(ε), there exists a number δ = δ(ε)
in this interval such that |u(t + δ) − u(t)| < ε, for all t ∈ R.
The remaining part of this paper is organized as follows. In Section 2, we shall derive new sufficient conditions for checking the existence of almost periodic solutions.
In Section 3, we present some new sufficient conditions for the local exponential
stability of the almost periodic solution of (1.1). In Section 4, we shall give an
example to illustrate our results obtained in previous sections.
2. Existence of Almost Periodic Solutions
Theorem 2.1. Under conditions (T1)–(T4)there exists a unique almost periodic
δL
solution of (1.1) in the region B ∗ = {ϕ : ϕ ∈ B, kϕ − ϕ0 kB ≤ 1−δ
}, where
Z
n t
o
Rt
ϕ0 (t) =
e− s aij (u)du )Lij (s)ds
=
Z
−∞
t
e−
−∞
Z t
Rt
a1j (u)du
s
Z
t
L11 (s)ds, . . . ,
e−
Rt
s
aij (u)du
Lij (s)ds,
−∞
e−
...,
Rt
s
amn (u)du
Lmn (s)ds .
−∞
Proof. For ϕ ∈ B, we consider the almost periodic solution xϕ (t) of nonlinear
almost periodic differential equation
Z ∞
X
dxij
kl
= −aij (t)xij (t) −
Cij
f(
Kij (u)ϕkl (t − u)du)ϕij (t) + Lij (t),
dt
0
Ckl ∈Nr (i,j)
(2.1)
Because ϕij (t), Lij (t), i = 1, 2, . . . , m, j = 1, 2, . . . , n, are almost periodic functions.
By [7, P. 90-120], (2.1) has a unique almost periodic solution
nZ t
h
Rt
X
kl
xϕ (t) =
e− s aij (u)du −
Cij
−∞
Z
×f
Ckl ∈Nr (i,j)
∞
i o
Kij (u)ϕkl (s − u)du ϕij (s) + Lij (s) ds .
0
Now, we define a mapping T : B → B by setting
T (ϕ)(t) = xϕ (t),
∀ϕ ∈ B.
(2.2)
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Q. ZHOU, B. XIAO, Y. YU
EJDE-2006/19
δL
Since B ∗ = {ϕ : ϕ ∈ B, kϕ − ϕ0 kB ≤ 1−δ
}, it is easy to see that B ∗ is a closed
convex subset of B. According to the definition of the norm of Banach space B,
we have
kϕ0 kB = sup max
t∈R (i,j)
≤ sup max
nZ
t
Lij (s)e−
−∞
+
Lij aij
t∈R (i,j)
Rt
s
aij (u)du
o
ds
(2.3)
L+
ij = max
= L.
aij
(i,j)
Therefore, for all ϕ ∈ B ∗ , we have
kϕkB ≤ kϕ − ϕ0 kB + kϕ0 kB ≤
L
δL
+L=
.
1−δ
1−δ
(2.4)
In view of (T3), we have
|f (u)| = |f (u) − f (0)| ≤ µ|u|,
∀u ∈ R.
(2.5)
Now, we prove that the mapping T is a self-mapping from B ∗ to B ∗ . In fact, for
L
all ϕ ∈ B ∗ , together with (2.4), (2.5) and 1−δ
≤ 1, we obtain
kT ϕ − ϕ0 kB
n Z
= sup max |
t∈R (i,j)
t
−
e
s
t
e−
≤ sup max{
t∈R (i,j)
t∈R (i,j)
µkij
Rt
s
aij (u)du
|Kij (u)|du
0
X
kl
Cij
Z
(i,j)
= δkϕk2B ≤ δ(
Ckl ∈Nr (i,j)
aij
kl
Cij
e−
X
kl
Cij
|ϕij (s)|ds}
Ckl ∈Nr (i,j)
Rt
s
aij (u)du
dskϕk2B
o
−∞
Ckl ∈Nr (i,j)
P
t
o
Kij (u)ϕkl (s − u)du)ϕij (s)ds|
0
∞
µkϕkB
−∞
n
≤ sup max µkij
kl
Cij
f(
Ckl ∈Nr (i,j)
Z
∞
Z
X
aij (u)du
−∞
Z
≤ max
Rt
kϕk2B
L 2
L
) ≤δ
,
1−δ
1−δ
which implies T (ϕ)(t) ∈ B ∗ . So, the mapping T is a mapping from B ∗ to B ∗ . Next,
we prove that the mapping T is a contraction mapping of the B ∗ . In fact, for all
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EXISTENCE AND STABILITY
5
ϕ, ψ ∈ B ∗ , we have
kT (ϕ) − T (ψ)kB
= sup kT (ϕ)(t) − T (ψ)(t)k
t∈R
Z
t
e−
= sup max{|
t∈R (i,j)
Rt
s
−∞
Z
kl
Cij
(f (
X
aij (u)du
∞
Kij (u)ϕkl (s − u)du)
0
Ckl ∈Nr (i,j)
Z ∞
× ϕij (s) − f (
Kij (u)ψkl (s − u)du)ψij (s))ds|}
0
Z t
Z ∞
R
X
− st aij (u)du
kl
≤ sup max{
e
Cij |f (
Kij (u)ϕkl (s − u)du)ϕij (s)
t∈R (i,j)
−∞
0
Ckl ∈Nr (i,j)
Z ∞
Z ∞
Kij (u)ψkl (s − u)du)ϕij (s) + f (
Kij (u)ψkl (s − u)du)ϕij (s)
− f(
0
Z0 ∞
− f(
Kij (u)ψkl (s − u)du)ψij (s)|ds} .
0
In view of condition (T3), (2.4), (2.5) and the above inequality, we have
kT (ϕ) − T (ψ)kB
Z t
Rt
≤ sup max{
e− s aij (u)du
t∈R (i,j)
−∞
X
Z
kl
Cij
[|f (
∞
Kij (u)ϕkl (s − u)du)
0
Ckl ∈Nr (i,j)
Z ∞
− f(
Kij (u)ψkl (s − u)du)||ϕij (s)|
Z0 ∞
+ |f (
Kij (u)ψkl (s − u)du)||ϕij (s) − ψij (s)|]ds}
0
Z t
Z ∞
R
X
− st aij (u)du
kl
≤ sup max{
e
Cij [
|Kij (u)|µ|ϕkl (s − u)
t∈R (i,j)
−∞
Z
Ckl ∈Nr (i,j)
∞
− ψkl (s − u)|du|ϕij (s)| +
0
Z t
Rt
≤ sup max{
e− s aij (u)du
t∈R (i,j)
≤ max{
µkij
−∞
|Kij (u)|µ|ψkl (s − u)|du|ϕij (s) − ψij (s)|]ds}
X
kl
Cij
kij µ[kϕkB + kψkB ]kϕ − ψkB ds}
Ckl ∈Nr (i,j)
kl
Ckl ∈Nr (i,j) Cij
P
aij
L
= 2δ
kϕ − ψkB ,
(1 − δ)
(i,j)
0
2L
}kϕ − ψkB
(1 − δ)
i.e.
kT (ϕ) − T (ψ)kB ≤ qkϕ − ψkB .
L
Note that q = 2δ (1−δ)
< 1, it is clear that the mapping T is a contraction. Therefore
the mapping T possesses a unique fixed point ϕ∗ ∈ B ∗ , T ϕ∗ = ϕ∗ . By (2.1), ϕ∗
satisfies (1.1). So ϕ∗ is an almost periodic solution of system (1.1) in B ∗ . The proof
is complete.
6
Q. ZHOU, B. XIAO, Y. YU
EJDE-2006/19
3. Stability of the almost periodic solution
In this section, we establish some results for the uniqueness and local exponential
stability of the almost periodic solution of system (1.1) in the region B ∗ .
L
) < 1 and suppose that conditions (T1)–(T5) hold.
Theorem 3.1. Let δ(1 + 2 1−δ
Then (1.1) has exactly one almost periodic solution ϕ∗ (t) = {x∗ij (t)} = {ϕ∗ij (t)} in
the region B ∗ . Moreover, ϕ∗ (t) is locally exponentially stable, and the domain of
attraction of ϕ∗ (t) is the set
G1 (ϕ∗ ) = {ϕ : ϕ ∈ C((−∞, 0]; Rmn ), kϕ−ϕ∗ k =
sup
max |ϕij (s)−ϕ∗ij (s)| < 1};
−∞≤s≤0 (i,j)
namely, there exist constants λ > 0 and M > 1 such that for every solution Z(t) =
{xij (t)} of system (1.1) with any initial value ϕ = {ϕij (t)} ∈ G1 (ϕ∗ ),
|xij (t) − x∗ij (t)| ≤ M kϕ − ϕ∗ ke−λt ,
for all t > 0, i = 1, 2, . . . , m, j = 1, 2, . . . , n.
Proof. From Theorem 2.1, system (1.1) has exactly one almost periodic solution
ϕ∗ (t) = {x∗ij (t)} = {ϕ∗ij (t)} in the region B ∗ . Let Z(t) = {xij (t)} be an arbitrary
solution of system (1.1) with initial value ϕ = {ϕij (t)} ∈ G1 (ϕ∗ ). Set y(t) =
{yij (t)} = {xij (t) − x∗ij (t)} = Z(t) − ϕ∗ (t). Then
Kij (u)xkl (t − u)du)xij (t)
0
Ckl ∈Nr (i,j)
Z
− f(
∞
Z
kl
Cij
[f (
X
0
yij
(t) = −aij (t)yij (t) −
(3.1)
∞
Kij (u)x∗kl (t
−
u)du)x∗ij (t)],
0
L
for i = 1, 2, . . . , m, j = 1, 2, . . . , n. Since δ < 1, δ(1 + 2 1−δ
) < 1, we can easily
obtain
X
aij > µkij
Ckl ∈Nr (i,j)
+ µkij
X
kl
Cij
+ µkij
Ckl ∈Nr (i,j)
X
Ckl ∈Nr (i,j)
kl
Cij
L
1−δ
kl L
Cij
,
1−δ
where i = 1, 2, . . . , m, j = 1, 2, . . . , n. Set
Γij (ω) = ω − aij +
X
kl
Cij
[µ
Ckl ∈Nr (i,j)
Z
+µ
0
∞
|Kij (u)|eωu du
Z
∞
0
L
],
1−δ
|Kij (u)|eωu du + µkij
L
1−δ
EJDE-2006/19
EXISTENCE AND STABILITY
7
where i = 1, 2, . . . , m, j = 1, 2, . . . , n. Clearly, Γij (ω), i = 1, 2, . . . , m, j =
1, 2, . . . , n, are continuous functions on [0, λ0 ]. Since
Z ∞
X
L
kl
Γij (0) = −aij +
Cij [µ
|Kij (u)|du + µkij
1
−
δ
0
Ckl ∈Nr (i,j)
Z ∞
L
+µ
|Kij (u)|du
]
1−δ
0
X
X
kl
kl L
≤ −aij + µkij
Cij
+ µkij
Cij
1−δ
Ckl ∈Nr (i,j)
+ µkij
X
Ckl ∈Nr (i,j)
Ckl ∈Nr (i,j)
kl L
< 0,
Cij
1−δ
where i = 1, 2, . . . , m, j = 1, 2, . . . , n. It follows that we can choose a positive
constant λ ∈ [0, λ0 ] such that
Z ∞
X
L
kl
Γij (λ) = (λ − aij ) +
Cij
[µ
|Kij (u)|eλu du + µkij
1
−
δ
0
Ckl ∈Nr (i,j)
(3.2)
Z ∞
L
λu
+µ
] < 0,
|Kij (u)|e du
1−δ
0
where i = 1, 2, . . . , m, j = 1, 2, . . . , n. We consider the Lyapunov functional
Vij (t) = |yij (t)|eλt ,
i = 1, 2, . . . , m, j = 1, 2, . . . , n.
(3.3)
Calculating the upper right derivative of Vij (t) along the solution y(t) = {yij (t)}
of system (3.1) with the initial value ϕ̄ = ϕ − ϕ∗ , we have
D+ (Vij (t))
∞
Z
kl
Cij
|f (
X
≤ −aij |yij (t)|eλt +
Kij (u)xkl (t − u)du)xij (t)
0
Ckl ∈Nr (i,j)
∞
Z
− f(
Kij (u)x∗kl (t − u)du)x∗ij (t)|eλt + λ|yij (t)|eλt
Z ∞
X
kl
= (λ − aij )|yij (t)|eλt +
Cij
|f (
Kij (u)xkl (t − u)du)yij (t)
0
0
Ckl ∈Nr (i,j)
Z
+ [f (
∞
(3.4)
Z
Kij (u)xkl (t − u)du) − f (
0
∞
Kij (u)x∗kl (t − u)du)]x∗ij (t)|eλt ,
0
where i = 1, 2, . . . , m, j = 1, 2, . . . , n. Let
kϕ − ϕ∗ k =
sup
max |ϕij (s) − ϕ∗ij (s)| > 0.
−∞≤s≤0 (i,j)
Since kϕ − ϕ∗ k < 1, we can choose a positive constant M > 1 such that
M kϕ − ϕ∗ k < 1,
(M kϕ − ϕ∗ k)2 < M kϕ − ϕ∗ k.
(3.5)
It follows from (3.3) that
Vij (t) = |yij (t)|eλt < M kϕ − ϕ∗ k,
for all t ∈ (−∞, 0], i = 1, 2, . . . , m, j = 1, 2, . . . , n. We claim that
Vij (t) = |yij (t)|eλt < M kϕ − ϕ∗ k,
(3.6)
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Q. ZHOU, B. XIAO, Y. YU
EJDE-2006/19
for all t > 0, i = 1, 2, . . . , m, j = 1, 2, . . . , n. Contrarily, there must exist ij ∈
{11, 12, . . . , 1n, . . . , m1, m2, . . . , mn} and tij > 0 such that
Vij (tij ) = M kϕ − ϕ∗ k,
Vij (t) < M kϕ − ϕ∗ k, ∀t ∈ (−∞, tij ),
(3.7)
where ij ∈ {11, 12, . . . , 1n, . . . , m1, m2, . . . , mn}. It follows from (3.7) that
Vij (tij ) − M kϕ − ϕ∗ k = 0,
Vij (t) − M kϕ − ϕ∗ k < 0,
(3.8)
for all t ∈ (−∞, tij ), where ij ∈ {11, 12, . . . , 1n, . . . , m1, m2, . . . , mn}. From (2.4),
(2.5), (3.4) , (3.5) and (3.8), we obtain
0 ≤ D+ (Vij (tij ) − M kϕ − ϕ∗ k)
= D+ (Vij (tij ))
≤ (λ − aij )|yij (tij )|eλtij +
Z
kl
Cij
|f (
X
∞
Z
Kij (u)xkl (tij − u)du) − f (
0
∞
Kij (u)x∗kl (tij − u)du)]x∗ij (tij )|eλtij
0
≤ (λ − aij )M kϕ − ϕ∗ k +
X
Ckl ∈Nr (i,j)
Z
Kij (u)xkl (tij − u)du)yij (tij )
0
Ckl ∈Nr (i,j)
Z
+ [f (
∞
kl
Cij
[µ
∞
Z
|Kij (u)||xkl (tij − u)|du|yij (tij )|eλtij
0
∞
|Kij (u)|eλu µ|ykl (tij − u)|eλ(tij −u) du|x∗ij (tij )|]
0
Z ∞
X
kl
≤ (λ − aij )M kϕ − ϕ∗ k +
Cij
[µ
|Kij (u)|(|xkl (tij − u) − x∗ij (tij − u)|
+
Ckl ∈Nr (i,j)
0
+ |x∗ij (tij − u)|)du|yij (tij )|eλtij
Z ∞
+
|Kij (u)|eλu µ|ykl (tij − u)|eλ(tij −u) du|x∗ij (tij )|]
0
Z ∞
X
kl
≤ (λ − aij )M kϕ − ϕ∗ k +
Cij
[µ
|Kij (u)|eλu |ykl (tij − u)|eλ(tij −u) du
Ckl ∈Nr (i,j)
Z ∞
0
× |yij (tij )|eλtij e−λtij + µ
|Kij (u)||x∗ij (tij − u)|du|yij (tij )|eλtij
0
Z ∞
+
|Kij (u)|eλu µ|ykl (tij − u)|eλ(tij −u) du|x∗ij (tij )|]
0
Z ∞
X
kl
(λ − aij )M kϕ − ϕ∗ k +
Cij
[µ
|Kij (u)|eλu du(M kϕ − ϕ∗ k)2 e−λtij
Ckl ∈Nr (i,j)
0
Z ∞
L
L
∗
M kϕ − ϕ k + µ
|Kij (u)|eλu du
M kϕ − ϕ∗ k]
+µ
|Kij (u)|du
1
−
δ
1
−
δ
0
0
Z ∞
X
L
kl
{(λ − aij ) +
Cij
[µ
|Kij (u)|eλu du + µkij
1
−
δ
0
Ckl ∈Nr (i,j)
Z ∞
L
]}M kϕ − ϕ∗ k.
+µ
|Kij (u)|eλu du
1
−
δ
0
Z
∞
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EXISTENCE AND STABILITY
9
Therefore,
X
0 ≤ (λ − aij ) +
kl
Cij
[µ
+µ
∞
|Kij (u)|eλu du
0
|Kij (u)|eλu du + µkij
0
Ckl ∈Nr (i,j)
Z
∞
Z
L
1−δ
L
],
1−δ
which contradicts (3.2). Hence, (3.6) holds. It follows that
|xij (t) − x∗ij (t)| = |yij (t)| < M kϕ − ϕ∗ ke−λt ,
for t > 0, i = 1, 2, . . . , m, j = 1, 2, . . . , n. This completes the proof.
4. Example
To illustrate the results obtained in previous sections we present the following
example. Consider the shunting inhibitory cellular neural network with delays
Z ∞
X
dxij
kl
= −aij (t)xij −
Cij f (
Kij (u)xkl (t − u)du)xij + Lij (t),
dt
0
Ckl ∈Nr (i,j)
where i = 1, 2, 3,

a11 (t)
a21 (t)
a31 (t)
j = 1, 2, 3,
 

1 + | sin t| 1 + | sin t| 3 + | sin t|
a12 (t) a13 (t)
a22 (t) a23 (t) = 3 + | sin t| 1 + | sin t| 3 + | sin t| ,
3 + | sin t| 1 + | sin t| 3 + | sin t|
a32 (t) a33 (t)


 
C11 C12 C13
0.1 0.2 0.1
C21 C22 C23  = 0.2 0 0.2 ,
C31 C32 C33
0.1 0.2 0.1

 

L11 L12 L13
0.5 sin t 0.5 cos t 0.2 sin t
L21 L22 L23  = 0.4 cos t 0.2 sin t 0.3 sin t  .
L31 L32 L33
0.4 cos t 0.6 sin t 0.2 cos t
Set r = 1, Kij (u) = (sin u)e−u , i = 1, 2, 3, j = 1, 2, 3, and f (x) =
P
kl
µ = 0.1, Ckl ∈N1 (1,1) C11
= 0.5,
X
X
kl
C12
= 0.8 ,
Ckl ∈N1 (1,2)
X
Ckl ∈N1 (1,3)
kl
C21
kl
C22
= 1.2 ,
Ckl ∈N1 (2,2)
kl
C23
X
= 0.8 ,
Ckl ∈N1 (2,3)
X
X
= 0.8 ,
Ckl ∈N1 (2,1)
X
kl
C13
= 0.5 ,
kl
C31
= 0.5 ,
Ckl ∈N1 (3,1)
kl
C32
X
= 0.8 ,
Ckl ∈N1 (3,2)
kl
C33
= 0.5 ,
Ckl ∈N1 (3,3)
X
X
(i,j) Ckl ∈N1 (i,j)
kl
Cij
= 6.4 ,
1
10 x.
Clearly
10
Q. ZHOU, B. XIAO, Y. YU
EJDE-2006/19
kij = 1, i = 1, 2, 3, j = 1, 2, 3,
δ = max
(i,j)
µ
P
Ckl ∈N1 (i,j)
kl
Cij
aij
= 0.12 < 1,
L+
L
0.6
ij = 0.6,
=
< 1,
aij
(1 − δ)
1 − 0.12
(i,j)
L
0.6
q = 2δ
= 2 × 0.12 ×
< 1,
(1 − δ)
1 − 0.12
L
0.6
δ(1 + 2
) = 0.12(1 + 2 × 0.12 ×
) < 1.
1−δ
1 − 0.12
By theorem 3.1, the system (4) has a unique almost periodic solution ϕ∗ (t) in the
region kϕ − ϕ0 kB ≤ 0.08128. Moreover, ϕ∗ (t) is locally exponentially stable, the
domain of attraction of ϕ∗ (t) is the set G1 (ϕ∗ ).
We remark that System (4) is a very simple form of SICNNs, and that it does
not not satisfy the condition (T0). Therefore, the results in [4, 6, 8] can not be
applied to this system. This implies that the results of this paper are essentially
new.
L = max
Conclusion. The shunting inhibitory cellular neural networks with continuously
distributed delays have been studied. Some sufficient conditions for the existence
and local exponential stability of almost periodic solutions have been established.
The obtained results are new and complement previously known results. Moreover,
an example is given to illustrate our results.
References
[1] A. Bouzerdoum and R. B. Pinter, ”Shunting Inhibitory Cellular Neural Networks: Derivation
and Stability Analysis.” IEEE Trans. Circuits Syst. 1-Fundamental Theory and Applications,
vol. 40 (1993), 215-221.
[2] A. Bouzerdoum and R. B. Pinter, ”Analysis and analog implementation of directionally sensitive shunting inhibitory Cellular Neural Networks.” Visual Information Processing: From
neurons to Chips, vol. SPIE-1473 (1991), 29-38.
[3] A. Bouzerdoum and R. B. Pinter, ”Nonlinear lateral inhibition applied to motion detection
in the fly visual system.” Nonlinear Vision, R. B. Pinter and B. Nabet, Eds. Boca Raton,
FL: CRC Press, 1992, pp. 423-450.
[4] A. Chen and J. Cao L. Huang, Almost periodic solution of shunting inhibitory CNNs with
delays, Physics Letters A, 298 (2002), 161-170.
[5] A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377,
Springer, Berlin, 1974.
[6] X. Huang and J. Cao, Almost periodic solutions of inhibitory cellular neural networks with
with time-vary delays, Physics Letters A, 314 (2003), 222-231.
[7] C. Y. He, Almost periodic differential equation, Higher Education Publishing House, Beijing,
1992. [In Chinese]
[8] Y. Li, C. Liu and L. Zhu, Global exponential stability of periodic solution of shunting inhibitory CNNs with delays, Physics Letters A, Vol. 337 (2005), 46-54.
Department of Mathematics, Hunan University of Arts and Science, Changde, Hunan
415000, China
E-mail address, Q. Zhou: zhouqiyuan65@yahoo.com.cn
E-mail address, B. Xiao: changde1218@yahoo.com.cn
E-mail address, Y. Yu: jinli127@yahoo.com.cn