Directory UMM :Journals:Journal_of_mathematics:EJQTDE:
Electronic Journal of Qualitative Theory of Differential Equations
2011, No. 30, 1-12; http://www.math.u-szeged.hu/ejqtde/
Existence of almost periodic solution for SICNN with a neutral delay
Zheng Fang1
Yongqing Yang2
School of Science, Jiangnan University, Wuxi, 214122, PR China
Abstract
In this paper, a kind of shunting inhibitory cellular neural network with
a neutral delay was considered. By using the Banach fixed point theorem,
we established a result about the existence and uniqueness of the almost
periodic solution for the shunting inhibitory cellular neural network.
Keywords: shunting inhibitory cellular neural network, neutral delay,
almost periodic solution, fixed point theorem.
1. Introduction
Shunting inhibitory cellular neural network (SICNN) is a kind of very important
model and has been investigated by many authors (see [1, 2, 3, 4] and the reference
therein) due to its wide applications in practical fields such as robotics, adaptive pattern
recognition and image processing. In [1], Ding studied the following SICNN
x′ij = −aij xij −
X
Cijkl f [xkl (t − τ (t))]xij (t) + Lij (t),
Ckl ∈Nr (i,j)
Most of the existing SICNN models are concerns with the delays in the state.
However, it is not enough for it can not describe the phenomenon precisely. It is
natural and useful to consider the model with a neutral delay, it means that the system
describing the model depends on not only the past information of the state but also
the information of the derivative of the state, i.e., the decay rate of the cells. This kind
of model is described by a differential equation with a neutral delay. The neutral type
differential equations have many applications, for more details we refer to [6]. Some
authors have considered the Hopfield neural networks with neutral delays, see [7, 8].
To the best of our knowledge, there is few consideration about the shunting inhibitory
cellular neural network with a neutral delay. In this paper, we consider the following
1
2
e-mail: [email protected], corresponding author
e-mail: [email protected]
EJQTDE, 2011 No. 30, p. 1
shunting inhibitory cellular neural network with a neutral delay:
X
x′ij = −aij (t)xij −
Cijkl (t)f [xkl (t − τ (t))]xij (t)
Ckl ∈Nr (i,j)
−
X
Dijkl (t)g[x′kl (t − σ(t))]xij (t) + Iij (t).
(1.1)
Ckl ∈Ns (i,j)
In this model, Cij represents the cell at the (i, j) position of the lattice, the
r−neighborhood Nr (i, j) of Cij is defined as follows
n
o
Nr (i, j) = Ckl : max{| k − i |, | l − j |} ≤ r, 1 ≤ k ≤ m, 1 ≤ l ≤ n ,
xij (t) describes the state of the cell Cij , the coefficient aij (t) > 0 is the passive decay
rate of the cell activity, Cijkl (t), Dijkl (t) are connection weights or coupling strength of
postsynaptic activity of the cell Ckl transmitted to the cell Cij , and f, g are continuous
activity functions, representing the output or firing rate of the cell Ckl , and τ (t), σ(t)
correspond to the transmission delays.
In the following, we give some basic knowledge about the almost periodic functions
and almost periodic solutions of differential equations, please refer to [9, 10] for more
details.
Definition 1.1 (See [10]) Let u : R → Rn be continuous in t. u is said to be almost
periodic on R if, for any ǫ > 0, the set T (u, ǫ) = {δ :| u(t + δ) − u(t) |< ǫ, ∀ t ∈ R} is
relatively dense, i.e., for ∀ǫ > 0, it is possible to find a real number l = l(ǫ) > 0, for
any interval with length l(ǫ), there exists a number δ = δ(ǫ) in this interval such that
| u(t + δ) − u(t) |< ǫ, for all t ∈ R.
Definition 1.2 ([9, 10]) If u : R → Rn is continuously differentiable in t, u(t) and
u′ (t) are almost periodic on R, then u(t) is said to be a continuously differentiable
almost periodic function.
Let AP (R, Rm×n ) and AP 1 (R, Rm×n ) be the set of continuous almost periodic functions, and continuously differentiable almost periodic functions from R to Rm×n , respectively. For each ϕ ∈ AP 1 (R, Rm×n ), define
kϕk0 = sup max{| ϕi,j |},
t∈R
i,j
kϕk = max{kϕk0 , kϕ′ k0 }.
It is easy to check that (AP (R, Rm×n ), k · k0 ) and (AP 1 (R, Rm×n ), k · k) are all Banach
spaces.
Definition 1.3 ([9, 10]) Let x ∈ Rn and Q(t) be an n × n continuous matrix
defined on R. The linear system
x′ (t) = Q(t)x(t)
(1.2)
EJQTDE, 2011 No. 30, p. 2
is said to admit an exponential dichotomy on R if there exist positive constants k, α,
projection P and the fundamental solution matrix X(t) of (1.2) satisfying
k X(t)P X −1 (s) k< ke−α(t−s) ,
k X(t)(I − P )X −1(s) k< ke−α(s−t) ,
t ≥ s,
t ≤ s.
Lemma 1.1 ([9, 10]) If the linear system (1.2) admits an exponential dichotomy,
then almost periodic system
x′ (t) = Q(t)x(t) + g(t)
(1.3)
has a unique almost periodic solution x(t), and
Z t
Z +∞
−1
x(t) =
X(t)P X (s)ds −
X(t)(I − P )X(s)ds.
(1.4)
t
−∞
Lemma 1.2 ([9, 10]) Let c(t) be an almost periodic function on R and
Z
1 t+T
ci (s)ds > 0, i = 1, 2, · · · n.
M[ci ] = lim
T →+∞ T t
Then the linear system
x′ (t) = diag(−c1 (t), −c2 (t), · · · , cn (t))x(t)
admits an exponential dichotomy on R.
2. Main results
Firstly, We give some assumptions.
(H1 ) aij (t), Cijkl (t), Dijkl (t), Iij (t), τ (t), σ(t) are all almost periodic functions,
i = 1, 2, · · · , n, j = 1, 2, · · · , m;
(H2 ) the activity functions f and g are Lipschtiz functions, i.e., there exist L > 0,
l > 0 such that
|f (x) − f (y)| ≤ L|x − y|, ∀x, y ∈ R,
|g(x) − g(y)| ≤ l|x − y|,
∀x, y ∈ R;
(H3 ) sup |Cijkl (t)| = Cijkl < +∞, sup |Dijkl (t)| = Dijkl < +∞, sup |aij (t)| = a+
ij <
t∈R
t∈R
t∈R
+∞, aij (u) ≥ aij∗ > 0.
EJQTDE, 2011 No. 30, p. 3
Let
ϕ0 =
n Zt
e−
Rt
s
aij (u)du
−∞
o
Iij (s)ds , kϕ0 k = R0
θ1 = max
n
X
Ckl ∈Nr (i,j)
Cijkl 2LR + |f (0)| +
X
Ckl ∈Ns (i,j)
o
Dijkl 2lR + |g(0)| ;
θ2 = max
n
X
Cijkl 4LR + |f (0)| +
X
o
Dijkl 4lR + |g(0)| .
i,j
i,j
Ckl ∈Nr (i,j)
Ckl ∈Ns (i,j)
where R is a constant with R ≥ R0 .
Theorem 2.1 If (H1 ), (H2 ), (H3 ) and the following conditions are satisfied
(1) R0 ≤ R < +∞;
(2) θ1 · max
i,j
n
1
,1 +
aij∗
(3) θ = θ2 · max
i,j
n
a+
ij
aij∗
o
1
,1 +
aij∗
≤ 21 ;
a+
ij
aij∗
o
< 1,
then Eq.(1.1) has a unique almost periodic solution.
Proof. For any given ϕ = {ϕij } ∈ AP 1 (R, Rm×n ), we consider the almost periodic
solution of the following differential equation
X
Cijkl (t)f (ϕkl (t − τ (t)))ϕij (t)
x′ij = −aij (t)xij −
Ckl ∈Nr (i,j)
X
−
Dijkl (t)g(ϕ′kl (t − σ(t)))ϕij (t) + Iij (t).
(2.1)
Ckl ∈Ns (i,j)
Since ϕij (t), aij (t), Cijkl (t), Dijkl (t), τ (t), σ(t) and Iij (t) are all almost periodic functions, and M[aij ] > 0, according to Lemma 1 and lemma 2, we know that Eq.(2.1) has
a unique almost periodic solution xϕ = {xϕij }, which can be expressed as follows
xϕij =
Zt
−∞
e−
Rt
s
aij (u)du
h
−
X
Cijkl (s)f (ϕkl (s − τ (s)))ϕij (s)
X
i
Dijkl (s)g(ϕ′kl (s − σ(s)))ϕij (s) + Iij (s) ds.
Ckl ∈Nr (i,j)
−
Ckl ∈Ns (i,j)
We define a nonlinear operator on AP 1 (R, Rm×n ) as follows
T (ϕ)(t) = xϕ (t),
∀ϕ ∈ AP 1 (R, Rm×n ).
EJQTDE, 2011 No. 30, p. 4
It is obvious that the fixed point of T is a solution of Eq.(1.1). In the following we will
show that T is a contract mapping, thus the Banach fixed point theorem assert that T
has a fixed point.
Let E be defined as follows
E = {ϕ ∈ AP 1 (R, Rm×n )| kϕ − ϕ0 k ≤ R}.
Firstly, we show that T (E) ⊆ E. For each ϕ ∈ E, we have kϕ − ϕ0 k ≤ R,
kϕk ≤ kϕ − ϕ0 k + kϕ0 k ≤ R + R ≤ 2R. Thus
kT ϕ − ϕ0 k0
h
n Z t
Rt
= sup max
e− s aij (u)du −
t∈R
i,j
−∞
X
−
Ckl ∈Ns (i,j)
≤ sup max
t∈R
i,j
n
Zt
e−
Rt
s
−∞
−
−
Ckl ∈Ns (i,j)
≤ sup max
t∈R
i,j
n
Zt
e−
Rt
s
aij (u)du
−∞
h
X
+
≤ sup max
t∈R
i,j
n
e−
Rt
s
aij (u)du
−∞
X
+
Ckl ∈Ns (i,j)
≤ sup max
t∈R
i,j
n
Zt
e−
Rt
s
aij (u)du
≤ sup max
t∈R
i,j
n
−∞
−
e
Rt
s
Cijkl (s)f (ϕkl(s − τ (s)))ϕij (s)
Ckl ∈Nr (i,j)
o
Dijkl (s)g(ϕ′kl(s − σ(s)))ϕij (s) ds
|Cijkl (s)||f (ϕkl (s − τ (s)))||ϕij (s)|
Ckl ∈Nr (i,j)
i o
|Dijkl (s)||g(ϕ′kl(s − σ(s)))||ϕij (s)| ds
X
Ckl ∈Nr (i,j)
Cijkl |f (ϕkl(s − τ (s))) − f (0)| + |f (0)|
h
X
Cijkl (L|ϕkl (s − τ (s))| + |f (0)|)
Ckl ∈Nr (i,j)
X
i o
Dijkl (l|ϕ′kl (s − σ(s))| + |g(0)|) ds 2R
aij (u)du
h
Ckl ∈Ns (i,j)
Zt
X
i o
Dijkl |g(ϕ′kl(s − σ(s)) − g(0)| + |g(0)|) ds 2R
−∞
+
h
Ckl ∈Nr (i,j)
X
Ckl ∈Ns (i,j)
Zt
Cijkl (s)f (ϕkl(s − τ (s)))ϕij (s)
i o
Dijkl (s)g(ϕ′kl (s − σ(s)))ϕij (s) ds
aij (u)du
X
X
X
Cijkl (2LR + |f (0)|)
Ckl ∈Nr (i,j)
EJQTDE, 2011 No. 30, p. 5
i o
Dijkl (2lR + |g(0)|) ds 2R
X
+
Ckl ∈Ns (i,j)
≤ sup max
t∈R
i,j
n
Zt
−aij∗ (t−s)
e
−∞
o
ds 2Rθ1
n 1 o
≤ 2Rθ1 max
i,j
aij∗
≤ R
(2.2)
and
=
k(T ϕ − ϕ0 )′ k0
Z t
Rt
sup max
−aij (t)e− s aij (u)du ·
t∈R i,j
−∞
h
X
Cijkl (s)f (ϕkl (s − τ (s)))
−
Ckl ∈Nr (i,j)
X
−
Dijkl (s)g(ϕ′kl(s
Ckl ∈Ns (i,j)
h
+
X
−
Cijkl (t)f (ϕkl (t − τ (t)))
Ckl ∈Nr (i,j)
X
−
Dijkl (t)g(ϕ′kl (t
Ckl ∈Ns (i,j)
=
t
Z
sup max
t∈R
i,j
aij (t)e−
Rt
s
aij (u)du
h
X
·
Cijkl (s)f (ϕkl (s − τ (s)))
Ckl ∈Nr (i,j)
X
Ckl ∈Ns (i,j)
−
h
X
i
Dijkl (s)g(ϕ′kl(s − σ(s))) ϕij (s)ds
Cijkl (t)f (ϕkl (t − τ (t)))
Ckl ∈Nr (i,j)
X
+
Dijkl (t)g(ϕ′kl(t
Ckl ∈Ns (i,j)
sup max
t∈R
i,j
i
− σ(t))) ϕij (t)
−∞
+
≤
i
− σ(s))) ϕij (s)ds
Z
t
|aij (t)|e−
−∞
h X
Rt
s
aij (u)du
i
− σ(t))) ϕij (t)
·
|Cijkl (s)| |f (ϕkl (s − τ (s)))|
Ckl ∈Nr (i,j)
+
X
Ckl ∈Ns (i,j)
i
|Dijkl (s)| |g(ϕ′kl (s − σ(s)))| |ϕij (s)|ds
EJQTDE, 2011 No. 30, p. 6
+
h
X
|Cijkl (t)| |f (ϕkl (t − τ (t)))|
Ckl ∈Nr (i,j)
X
+
|Dijkl (t)|
|g(ϕ′kl (t
Ckl ∈Ns (i,j)
≤
sup max
t∈R
i,j
Z
i
− σ(t)))| |ϕij (t)|
t
−aij∗ (t−s)
a+
ds ·
ij e
−∞
h X
Cijkl 2LR + |f (0)|
Ckl ∈Nr (i,j)
X
+
Ckl ∈Ns (i,j)
+
h
X
Ckl ∈Nr (i,j)
+
Cijkl 2LR + |f (0)|
X
Ckl ∈Ns (i,j)
≤
n
2Rθ1 max 1 +
≤
R.
i,j
a+
ij
aij∗
i
Dijkl 2lR + |g(0)|
Dijkl
i
2lR + |g(0)|
· 2R
o
(2.3)
From (2.2) and (2.3), we have kT ϕ − ϕ0 k ≤ R, thus T (E) ⊆ E.
Let ϕ, ψ ∈ E, denote by
I1 (s) =
Cijkl (s)
X
Dijkl (s)
Ckl ∈Nr (i,j)
I2 (s) =
h
i
f ϕkl (s − τ (s)) ϕij (s) − f ψkl (s − σ(s)) ψij (s) ,
X
Ckl ∈Ns (i,j)
h
i
′
′
g ϕkl (s − σ(s)) ϕij (s) − g ψkl (s − σ(s)) ψij (s) .
We have
|I1 (s)|
X
=
Ckl ∈Nr (i,j)
≤
X
Ckl ∈Nr (i,j)
=
X
Ckl ∈Nr (i,j)
h
i
Cijkl (s) f ϕkl (s − τ (s)) ϕij (s) − f ψkl (s − σ(s)) ψij (s)
h
Cijkl f ϕkl (s − τ (s)) ϕij (s) − f ϕkl (s − τ (s)) ψij (s)
i
+ f ϕkl (s − τ (s)) ψij (s) − f ψkl (s − σ(s)) ψij (s)
h
Cijkl f ϕkl (s − τ (s)) · ϕij (s) − ψij (s)
EJQTDE, 2011 No. 30, p. 7
i
+ f ϕkl (s − τ (s)) − f ψkl (s − σ(s)) · ψij (s)
h
X
kl
Cij f ϕkl (s − τ (s)) · ϕij (s) − ψij (s)
≤
Ckl ∈Nr (i,j)
X
≤
Ckl ∈Nr (i,j)
i
+L · ϕkl (s − τ (s)) − ψkl (s − σ(s)) · ψij (s)
h
kl
Cij f ϕkl (s − τ (s)) − f (0) + f (0) ·
ϕij (s) − ψij (s)
X
≤
Ckl ∈Nr (i,j)
X
≤
Ckl ∈Nr (i,j)
i
+L · ϕkl (s − τ (s)) − ψkl (s − σ(s)) · ψij (s)
h
Cijkl 2LR + |f (0)| · ϕij (s) − ψij (s)
i
+2LR · ϕkl (s − τ (s)) − ψkl (s − σ(s))
kl
Cij 4LR + |f (0)| · kϕ − ψk
Similarly,
|I2 (s)|
X
=
Dijkl (s)
Ckl ∈Ns (i,j)
X
≤
Dijkl
Ckl ∈Ns (i,j)
h
i
′
′
g ϕkl (s − τ (s)) ϕij (s) − g ψkl (s − σ(s)) ψij (s)
4lR + |g(0)| · kϕ − ψk
kT ϕ − T ψk0
Z t
Rt
= sup max
e− s aij (u)du ·
i,j
t∈R
−∞
h
X
−
−
Ckl ∈Nr (i,j)
X
Cijkl (s) [f (ϕkl (s − τ (s)))ϕij (s) − f (ψkl (s − σ(s)))ψij (s)
Dijkl (s)
Ckl ∈Ns (i,j)
≤ sup max
t∈R
i,j
n Zt
i
′
′
g(ϕkl (s − σ(s)))ϕij (s) − g(ψkl (s − σ(s)))ψij (s) ds
−
Rt
s
aij (u)du
−
Rt
aij (u)du
e
−∞
≤ sup max
t∈R
i,j
n Zt
−∞
e
s
o
|I1 (s)| + |I2 (s)| ds
ds
h
X
Ckl ∈Nr (i,j)
Cijkl
4LR + |f (0)|
EJQTDE, 2011 No. 30, p. 8
+
io
Dijkl 4lR + |g(0)|
· kϕ − ψk
X
Ckl ∈Ns (i,j)
≤ sup max
t∈R
i,j
n Zt
−aij∗ (t−s)
e
o
ds · θ2 · kϕ − ψk
−∞
≤ θ2 · max{
i,j
1
} · kϕ − ψk
aij∗
(2.4)
And
k(T ϕ − T ψ)′ k0
Z t
Rt
= sup max
−aij (t)e− s aij (u)du ·
t∈R i,j
−∞
nh
X
−
Cijkl (s)f (ϕkl (s − τ (s)))
Ckl ∈Nr (i,j)
X
−
Dijkl (s)g(ϕ′kl(s
Ckl ∈Ns (i,j)
+
h
X
Cijkl (s)f (ψkl (s − τ (s)))
Ckl ∈Nr (i,j)
X
+
Ckl ∈Ns (i,j)
+
nh
−
X
i
o
′
Dijkl (s)g(ψkl
(s − σ(s))) ψij (s) ds
Cijkl (t)f (ϕkl (t − τ (t)))
Ckl ∈Nr (i,j)
X
−
Ckl ∈Ns (i,j)
+
h
X
Ckl ∈Nr (i,j)
t∈R
i,j
t
−
|aij (t)|e
−∞
Rt
s
i
o
− σ(t))) ψij (t)
X
′
Dijkl (t)g(ψkl
(t
aij (u)du
|I1 (s)| + |I2 (s)| ds
Ckl ∈Ns (i,j)
≤ sup max
i
Dijkl (t)g(ϕ′kl(t − σ(t))) ϕij (t)
Cijkl (t)f (ψkl (t − τ (t)))
+
nZ
i
− σ(s))) ϕij (s)
o
+ |I1 (t)| + |I2 (t)|
nZ t
o
Rt
≤ sup max
|aij (t)|e− s aij (u)du ds + 1 · θ2 · kϕ − ψk
t∈R i,j
−∞
nZ t
o
+ −aij∗ (t−s)
≤ sup max
aij e
ds + 1 · θ2 · kϕ − ψk
t∈R
i,j
−∞
EJQTDE, 2011 No. 30, p. 9
o
n
a+
ij
· kϕ − ψk
≤ θ2 · max 1 +
i,j
aij∗
(2.5)
From (2.4) and (2.5), we have kT ϕ − T ψk ≤ θ2 · max
i,j
n
1
aij∗
, 1+
a+
ij
aij∗
o
· kϕ − ψk =
θ · kϕ − ψk. According to the condition of this theorem we know θ < 1, therefore T
has a unique fixed point.
Example We consider the following SICCN with neutral a delay:
x′ij = −aij (t)xij −
X
Cijkl (t)f [xkl (t − τ (t))]xij (t)
X
Dijkl (t)g[x′kl (t − σ(t))]xij (t) + Iij (t),
Ckl ∈Nr (ij)
+
(2.6)
Ckl ∈Ns (i,j)
where i = 1, 2, 3, j = 1, 2, 3, τ (t) = cos2 t, σ(t) = sin 2t, f (x) = 54 sin x, g(x) = 34 |x|,
a11 (t) a12 (t) a13 (t)
5 + | sin t| 5 + | sin 2t| 9 + | sin t|
a21 (t) a22 (t) a23 (t) = 6 + | cos t| 6 + | sin t| 7 + | cos t|
a31 (t) a32 (t) a33 (t)
8 + | cos t| 8 + | sin t| 5 + | sin 2t|
c11 (t) c12 (t) c13 (t)
0.004| sin 3t| 0.002| sin 3t| 0.001| sin 3t|
c21 (t) c22 (t) c23 (t) = 0.002| sin 3t| 0.001| sin 3t| 0.001| sin 3t|
c31 (t) c32 (t) c33 (t)
0.001| sin 3t| 0.002| sin 3t| 0.001| sin 3t|
d11 (t) d12 (t) d13 (t)
0.001| cos 2t| 0.001| cos 2t| 0.002| cos 2t|
d21 (t) d22 (t) d23 (t) = 0.001| cos 2t| 0.002| cos 2t| 0.003| cos 2t|
d31 (t) d32 (t) d33 (t)
0.002| cos 2t| 0.002| cos 2t| 0.001| cos 2t|
I11 (t) I12 (t) I13 (t)
sin t sin t cos t
I21 (t) I22 (t) I23 (t) = sin t cos t cos t
I31 (t) I32 (t) I33 (t)
cos t cos t cos t
In the following, we will check that all assumptions of the theorem are satisfied. By
computing,we have
+
+
a+
a
a
6
6
10
11
12
13
+
+
a
8 ,
a21 a+
22
23 = 7 7
+
+
a+
9 9 6
31 a32 a33
a11∗ a12∗ a13∗
5 5 9
a21∗ a22∗ a23∗ = 6 6 7
a31∗ a32∗ a33∗
8 8 5
EJQTDE, 2011 No. 30, p. 10
c11
c21
c31
d11
d21
d31
c12
c22
c32
d12
d22
d32
0.004
c13
c23 = 0.002
c33
0.001
0.001
d13
d23 = 0.001
d33
0.002
0.002
0.001
0.002
0.001
0.002
0.002
0.001
0.001 ,
0.001
0.002
0.003 .
0.001
Note that f and g are Lipschtz functions with f (0) = g(0) = 0, the Lipschtz
constants of f and g, L, l, are less than 1, we take L = l = 1.
X
ckl +
X
ckl +
X
ckl +
X
ckl +
X
ckl +
dkl = 0.014,
X
dkl = 0.013,
ckl ∈N1 (1,3)
X
dkl = 0.030,
X
dkl = 0.013,
X
dkl = 0.013,
P
ckl +
P
ckl +
P
ckl +
ckl ∈N1 (3,2)
ckl ∈N1 (3,1)
P
dkl = 0.021,
P
dkl = 0.021,
P
dkl = 0.019,
P
dkl = 0.019,
ckl ∈N1 (1,2)
ckl ∈N1 (2,1)
ckl ∈N1 (2,3)
ckl ∈N1 (2,2)
ckl ∈N1 (3,1)
ckl +
ckl ∈N1 (2,1)
ckl ∈N1 (1,3)
ckl ∈N1 (2,2)
P
ckl ∈N1 (1,2)
ckl ∈N1 (1,1)
ckl ∈N1 (1,1)
ckl ∈N1 (3,3)
X
ckl ∈N1 (2,3)
ckl ∈N1 (3,2)
ckl ∈N1 (3,3)
n
1
,
we
choose
R
=
3.
Obviously
max
,1 +
From computing we know kϕ0 k ≤ 11
5
aij∗
i,j
o
a+
a+
ij
ij
11
1
=
.
θ
=
0.18,
θ
=
0.36,
θ
·
max
{
,
1
+
} = 0.396 ≤ 12 , θ = θ2 ·
1
2
1
aij∗
5
aij∗
aij∗
i,j
n
o
a+
ij
1
max aij∗
, 1 + aij∗
= 0.792 < 1. So all the conditions of theorem 2.1 are satisfied,
i,j
hence by the theorem 2.1, Eq.(1.1) has a unique almost periodic solution.
Acknowledgements
The authors would like to thank the referees for their valuable suggestions and
comments.
The first author was supported by Fund for Young Teachers of Jiangnan University
(2008LQN010) and Program for Innovative Research Team of Jiangnan University.
The second author was supported by the National Natural Science Foundation of
China under Grant 60875036, the Key Research Foundation of Science and Technology
of the Ministry of Education of China under Grant 108067, and Program for Innovative
Research Team of Jiangnan University.
EJQTDE, 2011 No. 30, p. 11
References
[1] H. S. Ding, J. Liang, T. J. Xiao, Existence of almost periodic solutions for SICNNs
with time-varying delays, Physics Letters A, 372 (2008) 5411-5416.
[2] Y. Xia , J. Cao, Zhenkun Huang, Existence and exponential stability of almost
periodic solution for shunting inhibitory cellular neural networks with impulses,
Chaos, Solitons and Fractals, 34 (2007) 1599-1607.
[3] L. Wang, Existence and global attractivity of almost periodic solutions for delayed
high-ordered neural networks, Neurocomputing, 73 (2010) 802-808.
[4] L. Chen, H. Zhao, Global stability of almost periodic solution of shunting inhibitory cellular neural networks with variable coefficients, Chaos, Solitons and
Fractals, 35 (2008) 351-357.
[5] C. Ou, Almost periodic solutions for shunting inhibitory cellular neural networks,
Nonlinear Analysis: Real World Applications, 10 (2009) 2652-2658.
[6] J. Hale, Theory of functional differential equations, Springer-Verlag, New York,
1977.
[7] B. Xiao, Existence and uniqueness of almost periodic solutions for a class of Hopfield neural networks with neutral delays, Applied Mathematics Letters, 22 (2009)
528-533.
[8] C. Bai, Existence and stability of almost periodic solutions of Hopfield neural
networks with continuously distributed delays, Nonlinear Analysis, 71 (2009) 58505859.
[9] C. Y. He, Almost periodic differential equation, Higher Education Publishing
House, Beijing, 1992 (in Chinese).
[10] A. M. Fink, Almost periodic differential equation, in: Lecture Notes of Mathematics, Vol. 377, Springer, Berlin, 1974.
(Received July 1, 2010)
EJQTDE, 2011 No. 30, p. 12
2011, No. 30, 1-12; http://www.math.u-szeged.hu/ejqtde/
Existence of almost periodic solution for SICNN with a neutral delay
Zheng Fang1
Yongqing Yang2
School of Science, Jiangnan University, Wuxi, 214122, PR China
Abstract
In this paper, a kind of shunting inhibitory cellular neural network with
a neutral delay was considered. By using the Banach fixed point theorem,
we established a result about the existence and uniqueness of the almost
periodic solution for the shunting inhibitory cellular neural network.
Keywords: shunting inhibitory cellular neural network, neutral delay,
almost periodic solution, fixed point theorem.
1. Introduction
Shunting inhibitory cellular neural network (SICNN) is a kind of very important
model and has been investigated by many authors (see [1, 2, 3, 4] and the reference
therein) due to its wide applications in practical fields such as robotics, adaptive pattern
recognition and image processing. In [1], Ding studied the following SICNN
x′ij = −aij xij −
X
Cijkl f [xkl (t − τ (t))]xij (t) + Lij (t),
Ckl ∈Nr (i,j)
Most of the existing SICNN models are concerns with the delays in the state.
However, it is not enough for it can not describe the phenomenon precisely. It is
natural and useful to consider the model with a neutral delay, it means that the system
describing the model depends on not only the past information of the state but also
the information of the derivative of the state, i.e., the decay rate of the cells. This kind
of model is described by a differential equation with a neutral delay. The neutral type
differential equations have many applications, for more details we refer to [6]. Some
authors have considered the Hopfield neural networks with neutral delays, see [7, 8].
To the best of our knowledge, there is few consideration about the shunting inhibitory
cellular neural network with a neutral delay. In this paper, we consider the following
1
2
e-mail: [email protected], corresponding author
e-mail: [email protected]
EJQTDE, 2011 No. 30, p. 1
shunting inhibitory cellular neural network with a neutral delay:
X
x′ij = −aij (t)xij −
Cijkl (t)f [xkl (t − τ (t))]xij (t)
Ckl ∈Nr (i,j)
−
X
Dijkl (t)g[x′kl (t − σ(t))]xij (t) + Iij (t).
(1.1)
Ckl ∈Ns (i,j)
In this model, Cij represents the cell at the (i, j) position of the lattice, the
r−neighborhood Nr (i, j) of Cij is defined as follows
n
o
Nr (i, j) = Ckl : max{| k − i |, | l − j |} ≤ r, 1 ≤ k ≤ m, 1 ≤ l ≤ n ,
xij (t) describes the state of the cell Cij , the coefficient aij (t) > 0 is the passive decay
rate of the cell activity, Cijkl (t), Dijkl (t) are connection weights or coupling strength of
postsynaptic activity of the cell Ckl transmitted to the cell Cij , and f, g are continuous
activity functions, representing the output or firing rate of the cell Ckl , and τ (t), σ(t)
correspond to the transmission delays.
In the following, we give some basic knowledge about the almost periodic functions
and almost periodic solutions of differential equations, please refer to [9, 10] for more
details.
Definition 1.1 (See [10]) Let u : R → Rn be continuous in t. u is said to be almost
periodic on R if, for any ǫ > 0, the set T (u, ǫ) = {δ :| u(t + δ) − u(t) |< ǫ, ∀ t ∈ R} is
relatively dense, i.e., for ∀ǫ > 0, it is possible to find a real number l = l(ǫ) > 0, for
any interval with length l(ǫ), there exists a number δ = δ(ǫ) in this interval such that
| u(t + δ) − u(t) |< ǫ, for all t ∈ R.
Definition 1.2 ([9, 10]) If u : R → Rn is continuously differentiable in t, u(t) and
u′ (t) are almost periodic on R, then u(t) is said to be a continuously differentiable
almost periodic function.
Let AP (R, Rm×n ) and AP 1 (R, Rm×n ) be the set of continuous almost periodic functions, and continuously differentiable almost periodic functions from R to Rm×n , respectively. For each ϕ ∈ AP 1 (R, Rm×n ), define
kϕk0 = sup max{| ϕi,j |},
t∈R
i,j
kϕk = max{kϕk0 , kϕ′ k0 }.
It is easy to check that (AP (R, Rm×n ), k · k0 ) and (AP 1 (R, Rm×n ), k · k) are all Banach
spaces.
Definition 1.3 ([9, 10]) Let x ∈ Rn and Q(t) be an n × n continuous matrix
defined on R. The linear system
x′ (t) = Q(t)x(t)
(1.2)
EJQTDE, 2011 No. 30, p. 2
is said to admit an exponential dichotomy on R if there exist positive constants k, α,
projection P and the fundamental solution matrix X(t) of (1.2) satisfying
k X(t)P X −1 (s) k< ke−α(t−s) ,
k X(t)(I − P )X −1(s) k< ke−α(s−t) ,
t ≥ s,
t ≤ s.
Lemma 1.1 ([9, 10]) If the linear system (1.2) admits an exponential dichotomy,
then almost periodic system
x′ (t) = Q(t)x(t) + g(t)
(1.3)
has a unique almost periodic solution x(t), and
Z t
Z +∞
−1
x(t) =
X(t)P X (s)ds −
X(t)(I − P )X(s)ds.
(1.4)
t
−∞
Lemma 1.2 ([9, 10]) Let c(t) be an almost periodic function on R and
Z
1 t+T
ci (s)ds > 0, i = 1, 2, · · · n.
M[ci ] = lim
T →+∞ T t
Then the linear system
x′ (t) = diag(−c1 (t), −c2 (t), · · · , cn (t))x(t)
admits an exponential dichotomy on R.
2. Main results
Firstly, We give some assumptions.
(H1 ) aij (t), Cijkl (t), Dijkl (t), Iij (t), τ (t), σ(t) are all almost periodic functions,
i = 1, 2, · · · , n, j = 1, 2, · · · , m;
(H2 ) the activity functions f and g are Lipschtiz functions, i.e., there exist L > 0,
l > 0 such that
|f (x) − f (y)| ≤ L|x − y|, ∀x, y ∈ R,
|g(x) − g(y)| ≤ l|x − y|,
∀x, y ∈ R;
(H3 ) sup |Cijkl (t)| = Cijkl < +∞, sup |Dijkl (t)| = Dijkl < +∞, sup |aij (t)| = a+
ij <
t∈R
t∈R
t∈R
+∞, aij (u) ≥ aij∗ > 0.
EJQTDE, 2011 No. 30, p. 3
Let
ϕ0 =
n Zt
e−
Rt
s
aij (u)du
−∞
o
Iij (s)ds , kϕ0 k = R0
θ1 = max
n
X
Ckl ∈Nr (i,j)
Cijkl 2LR + |f (0)| +
X
Ckl ∈Ns (i,j)
o
Dijkl 2lR + |g(0)| ;
θ2 = max
n
X
Cijkl 4LR + |f (0)| +
X
o
Dijkl 4lR + |g(0)| .
i,j
i,j
Ckl ∈Nr (i,j)
Ckl ∈Ns (i,j)
where R is a constant with R ≥ R0 .
Theorem 2.1 If (H1 ), (H2 ), (H3 ) and the following conditions are satisfied
(1) R0 ≤ R < +∞;
(2) θ1 · max
i,j
n
1
,1 +
aij∗
(3) θ = θ2 · max
i,j
n
a+
ij
aij∗
o
1
,1 +
aij∗
≤ 21 ;
a+
ij
aij∗
o
< 1,
then Eq.(1.1) has a unique almost periodic solution.
Proof. For any given ϕ = {ϕij } ∈ AP 1 (R, Rm×n ), we consider the almost periodic
solution of the following differential equation
X
Cijkl (t)f (ϕkl (t − τ (t)))ϕij (t)
x′ij = −aij (t)xij −
Ckl ∈Nr (i,j)
X
−
Dijkl (t)g(ϕ′kl (t − σ(t)))ϕij (t) + Iij (t).
(2.1)
Ckl ∈Ns (i,j)
Since ϕij (t), aij (t), Cijkl (t), Dijkl (t), τ (t), σ(t) and Iij (t) are all almost periodic functions, and M[aij ] > 0, according to Lemma 1 and lemma 2, we know that Eq.(2.1) has
a unique almost periodic solution xϕ = {xϕij }, which can be expressed as follows
xϕij =
Zt
−∞
e−
Rt
s
aij (u)du
h
−
X
Cijkl (s)f (ϕkl (s − τ (s)))ϕij (s)
X
i
Dijkl (s)g(ϕ′kl (s − σ(s)))ϕij (s) + Iij (s) ds.
Ckl ∈Nr (i,j)
−
Ckl ∈Ns (i,j)
We define a nonlinear operator on AP 1 (R, Rm×n ) as follows
T (ϕ)(t) = xϕ (t),
∀ϕ ∈ AP 1 (R, Rm×n ).
EJQTDE, 2011 No. 30, p. 4
It is obvious that the fixed point of T is a solution of Eq.(1.1). In the following we will
show that T is a contract mapping, thus the Banach fixed point theorem assert that T
has a fixed point.
Let E be defined as follows
E = {ϕ ∈ AP 1 (R, Rm×n )| kϕ − ϕ0 k ≤ R}.
Firstly, we show that T (E) ⊆ E. For each ϕ ∈ E, we have kϕ − ϕ0 k ≤ R,
kϕk ≤ kϕ − ϕ0 k + kϕ0 k ≤ R + R ≤ 2R. Thus
kT ϕ − ϕ0 k0
h
n Z t
Rt
= sup max
e− s aij (u)du −
t∈R
i,j
−∞
X
−
Ckl ∈Ns (i,j)
≤ sup max
t∈R
i,j
n
Zt
e−
Rt
s
−∞
−
−
Ckl ∈Ns (i,j)
≤ sup max
t∈R
i,j
n
Zt
e−
Rt
s
aij (u)du
−∞
h
X
+
≤ sup max
t∈R
i,j
n
e−
Rt
s
aij (u)du
−∞
X
+
Ckl ∈Ns (i,j)
≤ sup max
t∈R
i,j
n
Zt
e−
Rt
s
aij (u)du
≤ sup max
t∈R
i,j
n
−∞
−
e
Rt
s
Cijkl (s)f (ϕkl(s − τ (s)))ϕij (s)
Ckl ∈Nr (i,j)
o
Dijkl (s)g(ϕ′kl(s − σ(s)))ϕij (s) ds
|Cijkl (s)||f (ϕkl (s − τ (s)))||ϕij (s)|
Ckl ∈Nr (i,j)
i o
|Dijkl (s)||g(ϕ′kl(s − σ(s)))||ϕij (s)| ds
X
Ckl ∈Nr (i,j)
Cijkl |f (ϕkl(s − τ (s))) − f (0)| + |f (0)|
h
X
Cijkl (L|ϕkl (s − τ (s))| + |f (0)|)
Ckl ∈Nr (i,j)
X
i o
Dijkl (l|ϕ′kl (s − σ(s))| + |g(0)|) ds 2R
aij (u)du
h
Ckl ∈Ns (i,j)
Zt
X
i o
Dijkl |g(ϕ′kl(s − σ(s)) − g(0)| + |g(0)|) ds 2R
−∞
+
h
Ckl ∈Nr (i,j)
X
Ckl ∈Ns (i,j)
Zt
Cijkl (s)f (ϕkl(s − τ (s)))ϕij (s)
i o
Dijkl (s)g(ϕ′kl (s − σ(s)))ϕij (s) ds
aij (u)du
X
X
X
Cijkl (2LR + |f (0)|)
Ckl ∈Nr (i,j)
EJQTDE, 2011 No. 30, p. 5
i o
Dijkl (2lR + |g(0)|) ds 2R
X
+
Ckl ∈Ns (i,j)
≤ sup max
t∈R
i,j
n
Zt
−aij∗ (t−s)
e
−∞
o
ds 2Rθ1
n 1 o
≤ 2Rθ1 max
i,j
aij∗
≤ R
(2.2)
and
=
k(T ϕ − ϕ0 )′ k0
Z t
Rt
sup max
−aij (t)e− s aij (u)du ·
t∈R i,j
−∞
h
X
Cijkl (s)f (ϕkl (s − τ (s)))
−
Ckl ∈Nr (i,j)
X
−
Dijkl (s)g(ϕ′kl(s
Ckl ∈Ns (i,j)
h
+
X
−
Cijkl (t)f (ϕkl (t − τ (t)))
Ckl ∈Nr (i,j)
X
−
Dijkl (t)g(ϕ′kl (t
Ckl ∈Ns (i,j)
=
t
Z
sup max
t∈R
i,j
aij (t)e−
Rt
s
aij (u)du
h
X
·
Cijkl (s)f (ϕkl (s − τ (s)))
Ckl ∈Nr (i,j)
X
Ckl ∈Ns (i,j)
−
h
X
i
Dijkl (s)g(ϕ′kl(s − σ(s))) ϕij (s)ds
Cijkl (t)f (ϕkl (t − τ (t)))
Ckl ∈Nr (i,j)
X
+
Dijkl (t)g(ϕ′kl(t
Ckl ∈Ns (i,j)
sup max
t∈R
i,j
i
− σ(t))) ϕij (t)
−∞
+
≤
i
− σ(s))) ϕij (s)ds
Z
t
|aij (t)|e−
−∞
h X
Rt
s
aij (u)du
i
− σ(t))) ϕij (t)
·
|Cijkl (s)| |f (ϕkl (s − τ (s)))|
Ckl ∈Nr (i,j)
+
X
Ckl ∈Ns (i,j)
i
|Dijkl (s)| |g(ϕ′kl (s − σ(s)))| |ϕij (s)|ds
EJQTDE, 2011 No. 30, p. 6
+
h
X
|Cijkl (t)| |f (ϕkl (t − τ (t)))|
Ckl ∈Nr (i,j)
X
+
|Dijkl (t)|
|g(ϕ′kl (t
Ckl ∈Ns (i,j)
≤
sup max
t∈R
i,j
Z
i
− σ(t)))| |ϕij (t)|
t
−aij∗ (t−s)
a+
ds ·
ij e
−∞
h X
Cijkl 2LR + |f (0)|
Ckl ∈Nr (i,j)
X
+
Ckl ∈Ns (i,j)
+
h
X
Ckl ∈Nr (i,j)
+
Cijkl 2LR + |f (0)|
X
Ckl ∈Ns (i,j)
≤
n
2Rθ1 max 1 +
≤
R.
i,j
a+
ij
aij∗
i
Dijkl 2lR + |g(0)|
Dijkl
i
2lR + |g(0)|
· 2R
o
(2.3)
From (2.2) and (2.3), we have kT ϕ − ϕ0 k ≤ R, thus T (E) ⊆ E.
Let ϕ, ψ ∈ E, denote by
I1 (s) =
Cijkl (s)
X
Dijkl (s)
Ckl ∈Nr (i,j)
I2 (s) =
h
i
f ϕkl (s − τ (s)) ϕij (s) − f ψkl (s − σ(s)) ψij (s) ,
X
Ckl ∈Ns (i,j)
h
i
′
′
g ϕkl (s − σ(s)) ϕij (s) − g ψkl (s − σ(s)) ψij (s) .
We have
|I1 (s)|
X
=
Ckl ∈Nr (i,j)
≤
X
Ckl ∈Nr (i,j)
=
X
Ckl ∈Nr (i,j)
h
i
Cijkl (s) f ϕkl (s − τ (s)) ϕij (s) − f ψkl (s − σ(s)) ψij (s)
h
Cijkl f ϕkl (s − τ (s)) ϕij (s) − f ϕkl (s − τ (s)) ψij (s)
i
+ f ϕkl (s − τ (s)) ψij (s) − f ψkl (s − σ(s)) ψij (s)
h
Cijkl f ϕkl (s − τ (s)) · ϕij (s) − ψij (s)
EJQTDE, 2011 No. 30, p. 7
i
+ f ϕkl (s − τ (s)) − f ψkl (s − σ(s)) · ψij (s)
h
X
kl
Cij f ϕkl (s − τ (s)) · ϕij (s) − ψij (s)
≤
Ckl ∈Nr (i,j)
X
≤
Ckl ∈Nr (i,j)
i
+L · ϕkl (s − τ (s)) − ψkl (s − σ(s)) · ψij (s)
h
kl
Cij f ϕkl (s − τ (s)) − f (0) + f (0) ·
ϕij (s) − ψij (s)
X
≤
Ckl ∈Nr (i,j)
X
≤
Ckl ∈Nr (i,j)
i
+L · ϕkl (s − τ (s)) − ψkl (s − σ(s)) · ψij (s)
h
Cijkl 2LR + |f (0)| · ϕij (s) − ψij (s)
i
+2LR · ϕkl (s − τ (s)) − ψkl (s − σ(s))
kl
Cij 4LR + |f (0)| · kϕ − ψk
Similarly,
|I2 (s)|
X
=
Dijkl (s)
Ckl ∈Ns (i,j)
X
≤
Dijkl
Ckl ∈Ns (i,j)
h
i
′
′
g ϕkl (s − τ (s)) ϕij (s) − g ψkl (s − σ(s)) ψij (s)
4lR + |g(0)| · kϕ − ψk
kT ϕ − T ψk0
Z t
Rt
= sup max
e− s aij (u)du ·
i,j
t∈R
−∞
h
X
−
−
Ckl ∈Nr (i,j)
X
Cijkl (s) [f (ϕkl (s − τ (s)))ϕij (s) − f (ψkl (s − σ(s)))ψij (s)
Dijkl (s)
Ckl ∈Ns (i,j)
≤ sup max
t∈R
i,j
n Zt
i
′
′
g(ϕkl (s − σ(s)))ϕij (s) − g(ψkl (s − σ(s)))ψij (s) ds
−
Rt
s
aij (u)du
−
Rt
aij (u)du
e
−∞
≤ sup max
t∈R
i,j
n Zt
−∞
e
s
o
|I1 (s)| + |I2 (s)| ds
ds
h
X
Ckl ∈Nr (i,j)
Cijkl
4LR + |f (0)|
EJQTDE, 2011 No. 30, p. 8
+
io
Dijkl 4lR + |g(0)|
· kϕ − ψk
X
Ckl ∈Ns (i,j)
≤ sup max
t∈R
i,j
n Zt
−aij∗ (t−s)
e
o
ds · θ2 · kϕ − ψk
−∞
≤ θ2 · max{
i,j
1
} · kϕ − ψk
aij∗
(2.4)
And
k(T ϕ − T ψ)′ k0
Z t
Rt
= sup max
−aij (t)e− s aij (u)du ·
t∈R i,j
−∞
nh
X
−
Cijkl (s)f (ϕkl (s − τ (s)))
Ckl ∈Nr (i,j)
X
−
Dijkl (s)g(ϕ′kl(s
Ckl ∈Ns (i,j)
+
h
X
Cijkl (s)f (ψkl (s − τ (s)))
Ckl ∈Nr (i,j)
X
+
Ckl ∈Ns (i,j)
+
nh
−
X
i
o
′
Dijkl (s)g(ψkl
(s − σ(s))) ψij (s) ds
Cijkl (t)f (ϕkl (t − τ (t)))
Ckl ∈Nr (i,j)
X
−
Ckl ∈Ns (i,j)
+
h
X
Ckl ∈Nr (i,j)
t∈R
i,j
t
−
|aij (t)|e
−∞
Rt
s
i
o
− σ(t))) ψij (t)
X
′
Dijkl (t)g(ψkl
(t
aij (u)du
|I1 (s)| + |I2 (s)| ds
Ckl ∈Ns (i,j)
≤ sup max
i
Dijkl (t)g(ϕ′kl(t − σ(t))) ϕij (t)
Cijkl (t)f (ψkl (t − τ (t)))
+
nZ
i
− σ(s))) ϕij (s)
o
+ |I1 (t)| + |I2 (t)|
nZ t
o
Rt
≤ sup max
|aij (t)|e− s aij (u)du ds + 1 · θ2 · kϕ − ψk
t∈R i,j
−∞
nZ t
o
+ −aij∗ (t−s)
≤ sup max
aij e
ds + 1 · θ2 · kϕ − ψk
t∈R
i,j
−∞
EJQTDE, 2011 No. 30, p. 9
o
n
a+
ij
· kϕ − ψk
≤ θ2 · max 1 +
i,j
aij∗
(2.5)
From (2.4) and (2.5), we have kT ϕ − T ψk ≤ θ2 · max
i,j
n
1
aij∗
, 1+
a+
ij
aij∗
o
· kϕ − ψk =
θ · kϕ − ψk. According to the condition of this theorem we know θ < 1, therefore T
has a unique fixed point.
Example We consider the following SICCN with neutral a delay:
x′ij = −aij (t)xij −
X
Cijkl (t)f [xkl (t − τ (t))]xij (t)
X
Dijkl (t)g[x′kl (t − σ(t))]xij (t) + Iij (t),
Ckl ∈Nr (ij)
+
(2.6)
Ckl ∈Ns (i,j)
where i = 1, 2, 3, j = 1, 2, 3, τ (t) = cos2 t, σ(t) = sin 2t, f (x) = 54 sin x, g(x) = 34 |x|,
a11 (t) a12 (t) a13 (t)
5 + | sin t| 5 + | sin 2t| 9 + | sin t|
a21 (t) a22 (t) a23 (t) = 6 + | cos t| 6 + | sin t| 7 + | cos t|
a31 (t) a32 (t) a33 (t)
8 + | cos t| 8 + | sin t| 5 + | sin 2t|
c11 (t) c12 (t) c13 (t)
0.004| sin 3t| 0.002| sin 3t| 0.001| sin 3t|
c21 (t) c22 (t) c23 (t) = 0.002| sin 3t| 0.001| sin 3t| 0.001| sin 3t|
c31 (t) c32 (t) c33 (t)
0.001| sin 3t| 0.002| sin 3t| 0.001| sin 3t|
d11 (t) d12 (t) d13 (t)
0.001| cos 2t| 0.001| cos 2t| 0.002| cos 2t|
d21 (t) d22 (t) d23 (t) = 0.001| cos 2t| 0.002| cos 2t| 0.003| cos 2t|
d31 (t) d32 (t) d33 (t)
0.002| cos 2t| 0.002| cos 2t| 0.001| cos 2t|
I11 (t) I12 (t) I13 (t)
sin t sin t cos t
I21 (t) I22 (t) I23 (t) = sin t cos t cos t
I31 (t) I32 (t) I33 (t)
cos t cos t cos t
In the following, we will check that all assumptions of the theorem are satisfied. By
computing,we have
+
+
a+
a
a
6
6
10
11
12
13
+
+
a
8 ,
a21 a+
22
23 = 7 7
+
+
a+
9 9 6
31 a32 a33
a11∗ a12∗ a13∗
5 5 9
a21∗ a22∗ a23∗ = 6 6 7
a31∗ a32∗ a33∗
8 8 5
EJQTDE, 2011 No. 30, p. 10
c11
c21
c31
d11
d21
d31
c12
c22
c32
d12
d22
d32
0.004
c13
c23 = 0.002
c33
0.001
0.001
d13
d23 = 0.001
d33
0.002
0.002
0.001
0.002
0.001
0.002
0.002
0.001
0.001 ,
0.001
0.002
0.003 .
0.001
Note that f and g are Lipschtz functions with f (0) = g(0) = 0, the Lipschtz
constants of f and g, L, l, are less than 1, we take L = l = 1.
X
ckl +
X
ckl +
X
ckl +
X
ckl +
X
ckl +
dkl = 0.014,
X
dkl = 0.013,
ckl ∈N1 (1,3)
X
dkl = 0.030,
X
dkl = 0.013,
X
dkl = 0.013,
P
ckl +
P
ckl +
P
ckl +
ckl ∈N1 (3,2)
ckl ∈N1 (3,1)
P
dkl = 0.021,
P
dkl = 0.021,
P
dkl = 0.019,
P
dkl = 0.019,
ckl ∈N1 (1,2)
ckl ∈N1 (2,1)
ckl ∈N1 (2,3)
ckl ∈N1 (2,2)
ckl ∈N1 (3,1)
ckl +
ckl ∈N1 (2,1)
ckl ∈N1 (1,3)
ckl ∈N1 (2,2)
P
ckl ∈N1 (1,2)
ckl ∈N1 (1,1)
ckl ∈N1 (1,1)
ckl ∈N1 (3,3)
X
ckl ∈N1 (2,3)
ckl ∈N1 (3,2)
ckl ∈N1 (3,3)
n
1
,
we
choose
R
=
3.
Obviously
max
,1 +
From computing we know kϕ0 k ≤ 11
5
aij∗
i,j
o
a+
a+
ij
ij
11
1
=
.
θ
=
0.18,
θ
=
0.36,
θ
·
max
{
,
1
+
} = 0.396 ≤ 12 , θ = θ2 ·
1
2
1
aij∗
5
aij∗
aij∗
i,j
n
o
a+
ij
1
max aij∗
, 1 + aij∗
= 0.792 < 1. So all the conditions of theorem 2.1 are satisfied,
i,j
hence by the theorem 2.1, Eq.(1.1) has a unique almost periodic solution.
Acknowledgements
The authors would like to thank the referees for their valuable suggestions and
comments.
The first author was supported by Fund for Young Teachers of Jiangnan University
(2008LQN010) and Program for Innovative Research Team of Jiangnan University.
The second author was supported by the National Natural Science Foundation of
China under Grant 60875036, the Key Research Foundation of Science and Technology
of the Ministry of Education of China under Grant 108067, and Program for Innovative
Research Team of Jiangnan University.
EJQTDE, 2011 No. 30, p. 11
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(Received July 1, 2010)
EJQTDE, 2011 No. 30, p. 12