# Global Exponential Stability of a Unique Almost Periodic Solution for Neutral-Type Cellular Neural Networks with Distributed Delays

Authors: Wenquan Wu
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

This paper is concerned with the problem of the existence, uniqueness, and global exponential stability for neutral-type cellular neural networks with distributed delays. Based on fixed point theory and Lyapunov functional, several sufficient conditions are established for the existence, uniqueness, and global exponential stability of almost periodic solution for the above system. Finally, a simple example is given to illustrate the feasibility and effectiveness of our main results.


---

## Body

## 1. Introduction

It is well known that the delayed cellular neural networks (DCNNs) have been successfully applied to many practical problems, such as signal and image processing, pattern recognition, and optimization. Hence, they have been investigated extensively by many researchers in recent years. In particular, the stability and the existence of almost periodic solutions are two important properties, which have close relation to the applications of neural networks, so they have been widely studied. Therefore, there have been extensive results on the existence and stability of periodic solutions and almost periodic solutions of delayed cellular neural networks in the literature. We refer the reader to [1–12] and the references cited therein for more details.As pointed out in [13], in the biochemistry experiments, neural information may transfer across chemical reactivity, which results in a neutral-type process. Moreover, in many cases the existing neural network models cannot characterize the properties of a neural reaction process precisely. To solve this problem, it is naturally introducing the neutral networks. To the best of our knowledge, however, the problem of global exponential stability of almost periodic solution for neutral-type cellular neural networks has not been fully investigated in the literature.In this paper, we consider the following neutral-type cellular neural networks with distributed delays:(1)
x
˙
i
(
t
)
=
-
a
i
(
t
)
x
i
(
t
)
+
∑
j
=
1
n
‍
b
i
j
(
t
)
f
j
(
x
j
(
t
)
)
+
∑
j
=
1
n
‍
c
i
j
(
t
)
g
j
(
∫
0
∞
‍
w
j
(
s
)
x
j
(
t
-
s
)
d
s
)
+
∑
j
=
1
n
‍
d
i
j
(
t
)
h
j
(
∫
0
∞
‍
v
j
(
s
)
x
˙
j
(
t
-
s
)
d
s
)
+
I
i
(
t
)
,
where n denotes the number of units in a neural network, x
i
(
t
) corresponds to the state of the ith unit at time t, a
i
>
0 represents the passive decay rates at time t, f
j, g
j, and h
j are the neuronal output signal functions, a
i
j
(
t
), b
i
j
(
t
), c
i
j
(
t
), and d
i
j
(
t
) denote the strength of the jth unit on the ith unit at time t, I
i
(
t
) denotes the external inputs at time t, and a
i, a
i
j, b
i
j, c
i
j, d
i
j, and I
i are all almost periodic functions, i
,
j
=
1,2
,
…
,
n. The kernel functions w
j
,
v
j
:
[
0
,
+
∞
)
→
[
0
,
+
∞
) are continuous with ∫
0
∞
‍
w
j
(
s
)
d
s
=
∫
0
∞
‍
v
j
(
s
)
d
s
=
1 and satisfy
(2)
∫
0
∞
‍
e
ξ
s
w
j
(
s
)
d
s
=
p
j
(
ξ
)
,
∫
0
∞
‍
e
ξ
s
v
j
(
s
)
d
s
=
q
j
(
ξ
)
,
where p
j and q
j are continuous functions in [
0
,
δ
]
(
δ
>
0
) and p
j
(
0
)
=
q
j
(
0
)
=
1, j
=
1,2
,
…
,
n.LetC
(
𝕏
,
𝕐
) and C
1
(
𝕏
,
𝕐
) be the space of continuous functions and continuously differential functions which map 𝕏 into 𝕐, respectively. Especially, C
(
𝕏
)
:
=
C
(
𝕏
,
𝕏
), C
1
(
𝕏
)
:
=
C
1
(
𝕏
,
𝕏
). For any bounded function f
∈
C
(
ℝ
), f
+
=
sup
⁡
s
∈
ℝ
|
f
(
s
)
|, f
-
=
inf
⁡
s
∈
ℝ
|
f
(
s
)
|.The initial conditions associated with system (1) are of the form
(3)
x
i
(
s
)
=
x
i
*
(
s
)
,
∀
s
∈
(
-
∞
,
0
]
,
x
i
*
∈
C
1
(
(
-
∞
,
0
]
,
ℝ
)
,
i
=
1,2
,
…
,
n
.We list some assumptions which will be used in this paper.(H
1)
min
⁡
1
≤
i
≤
n
{
a
i
-
}
>
0.
(H
2)
There exist some positive constantsL
j
f, L
j
g, and L
j
h such that
(4)
|
f
j
(
u
)
-
f
j
(
v
)
|
≤
L
j
f
|
u
-
v
|
,
|
g
j
(
u
)
-
g
j
(
v
)
|
≤
L
j
g
|
u
-
v
|
,
|
h
j
(
u
)
-
h
j
(
v
)
|
≤
L
j
h
|
u
-
v
|
,
for  all u
,
v
∈
ℝ,  j
=
1
,
…
,
n.
(H
3)
θ
=
max
⁡
{
θ
1
,
θ
2
}
<
1, where
(5)
θ
1
=
max
⁡
1
≤
i
≤
n
{
(
a
i
-
)
-
1
∑
j
=
1
n
‍
(
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
+
d
i
j
+
L
j
h
)
}
,
θ
2
=
max
⁡
1
≤
i
≤
n
{
[
1
+
a
i
u
a
i
l
]
∑
j
=
1
n
‍
(
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
+
d
i
j
+
L
j
h
)
}
.The organization of this paper is as follows. In Section2, we give some basic definitions and necessary lemmas which will be used in later sections. In Sections 3 and 4, by using a fixed point theorem and constructing suitable Lyapunov functional, we obtain some sufficient conditions ensuring existence, uniqueness, and global exponential stability of almost periodic solution of system (1). Finally, an example is given to illustrate that our results are feasible.


## 2. Preliminaries

Now, let us state the following definitions and lemmas, which will be useful in proving our main result.Definition 1 (see [14]).
x
∈
C
(
ℝ
) is called almost periodic, if for any ϵ
>
0 it is possible to find a real number l
=
l
(
ϵ
)
>
0 and for any interval with length l
(
ϵ
) there exists a number τ
=
τ
(
ϵ
) in this interval such that |
x
(
t
+
τ
)
-
x
(
t
)
|
<
ϵ, ∀
t
∈
ℝ. The collection of those functions is denoted by A
P
(
ℝ
).Definition 2 (see [14]).
Lety
∈
C
(
ℝ
,
ℝ
n
) and let P
(
t
) be a n
×
n continuous matrix defined on ℝ. The linear system
(6)
y
˙
(
t
)
=
P
(
t
)
y
(
t
)
is said to be an exponential dichotomy on ℝ if there exist constants k
,
λ
>
0, projection S, and the fundamental matrix Y
(
t
) satisfying
(7)
∥
Y
(
t
)
S
Y
-
1
(
s
)
∥
≤
k
e
-
λ
(
t
-
s
)
,
∀
t
≥
s
,
∥
Y
(
t
)
(
I
-
S
)
Y
-
1
(
s
)
∥
≤
k
e
-
λ
(
s
-
t
)
,
∀
t
≤
s
.Lemma 3 (see [14]).
If the linear systemy
˙
(
t
)
=
P
(
t
)
y
(
t
) has an exponential dichotomy, then almost periodic system
(8)
y
˙
(
t
)
=
P
(
t
)
y
(
t
)
+
g
(
t
)
has a unique almost periodic solution y
(
t
) which can be expressed as follows:
(9)
y
(
t
)
=
∫
-
∞
t
‍
Y
(
t
)
S
Y
-
1
(
s
)
g
(
s
)
d
s
-
∫
t
∞
‍
Y
(
t
)
(
I
-
S
)
Y
-
1
(
s
)
g
(
s
)
d
s
.Lemma 4 (see [14]).
Leta
i
(
t
) be an almost periodic function and
(10)
M
[
a
i
]
=
lim
⁡
T
→
∞
1
T
∫
t
t
+
T
‍
a
i
(
s
)
d
s
>
0
,
i
=
1,2
,
…
,
n
.
Then the linear systemy
˙
(
t
)
=
-
A
(
t
)
y
(
t
) admits an exponential dichotomy, where A
(
t
)
=
diag
⁡
{
a
1
(
t
)
,
a
2
(
t
)
,
…
,
a
n
(
t
)
}.Definition 5.
The almost periodic solutionz
=
(
x
1
,
…
,
x
n
)
T of system (1) with the initial value z
*
=
(
x
1
*
,
…
,
x
n
*
)
T is said to be globally exponentially stable, if there exist constants ω
>
0 and M
≥
1, for any solution ϕ
=
(
φ
1
,
…
,
φ
n
)
T of system (1) with initial value ϕ
*
=
(
φ
1
*
,
…
,
φ
n
*
)
T such that
(11)
∑
i
=
1
n
‍
|
x
i
(
t
)
-
φ
i
(
t
)
|
≤
M
∥
z
*
-
ϕ
*
∥
∞
e
-
ω
t
,
∀
t
>
0
,
where
(12)
∥
z
*
-
ϕ
*
∥
∞
:
=
max
⁡
1
≤
i
≤
n
sup
⁡
s
∈
(
-
∞
,
0
]
{
|
x
i
*
(
s
)
-
φ
i
*
(
s
)
|
+
|
x
˙
i
*
(
s
)
-
φ
˙
i
*
(
s
)
|
}
.Lemma 6 (see [15, 16]).
Let(
𝕏
,
∥
·
∥
) be a Banach space. Assume that 0
∈
Ω is an open bounded subset of 𝕏; T
:
Ω
¯
→
𝕏 is completely continuous satisfying
(13)
∥
T
x
∥
≤
∥
x
∥
,
∀
x
∈
∂
Ω
;
then T has a fixed point in Ω
¯.Let𝔼
=
{
x
∈
A
P
(
ℝ
)
∩
C
1
(
ℝ
)
:
x
˙
∈
A
P
(
ℝ
)
} and
(14)
𝕏
=
{
z
=
(
x
1
,
…
,
x
n
)
T
:
x
i
∈
𝔼
,
i
=
1
,
…
,
n
}
with the norm
(15)
∥
z
∥
=
max
⁡
{
∥
z
∥
0
,
∥
z
˙
∥
0
}
=
max
⁡
{
max
⁡
1
≤
i
≤
n
{
|
x
i
|
0
}
,
max
⁡
1
≤
i
≤
n
{
|
x
˙
i
|
0
}
}
,
where ∥
z
∥
0
=
max
⁡
1
≤
i
≤
n
⁡
{
|
x
i
|
0
},  ∥
z
˙
∥
0
=
max
⁡
1
≤
i
≤
n
⁡
{
|
x
˙
i
|
0
},  |
f
|
0
=
sup
⁡
s
∈
ℝ
⁡
|
f
(
s
)
|,  ∀
f
∈
A
P
(
ℝ
). Then 𝕏 is a Banach space with the norm ∥
·
∥.Letρ
=
max
⁡
{
ρ
1
,
ρ
2
}, where
(16)
ρ
1
=
max
⁡
1
≤
i
≤
n
{
(
a
i
-
)
-
1
[
∑
j
=
1
n
‍
b
i
j
+
f
j
(
0
)
+
∑
j
=
1
n
‍
c
i
j
+
g
j
(
0
)
+
∑
j
=
1
n
‍
d
i
j
+
h
j
(
0
)
+
I
i
+
]
}
,
(17)
ρ
2
=
max
⁡
1
≤
i
≤
n
{
[
1
+
a
i
u
a
i
l
]
[
∑
j
=
1
n
‍
b
i
j
+
f
j
(
0
)
+
∑
j
=
1
n
‍
c
i
j
+
g
j
(
0
)
+
∑
j
=
1
n
‍
d
i
j
+
h
j
(
0
)
+
I
i
+
]
}
.Define an open bounded subsetΩ in 𝕏 by
(18)
Ω
=
{
z
∈
𝕏
:
∥
z
∥
<
r
0
=
ρ
1
-
θ
}
.By Lemmas3 and 4, system (1) has a unique almost periodic solution z
ϕ
=
(
x
1
φ
1
,
…
,
x
n
φ
n
)
T which can be expressed as follows:
(19)
x
i
φ
i
=
∫
-
∞
t
‍
e
-
∫
s
t
‍
a
i
(
u
)
d
u
F
i
(
s
,
ϕ
(
s
)
)
d
s
,
where ϕ
=
(
φ
1
,
…
,
φ
n
)
T,
(20)
F
i
(
t
,
ϕ
(
t
)
)
=
∑
j
=
1
n
‍
b
i
j
(
t
)
f
j
(
φ
j
(
t
)
)
+
∑
j
=
1
n
‍
c
i
j
(
t
)
g
j
(
∫
0
∞
‍
w
i
j
(
s
)
φ
j
(
t
-
s
)
d
s
)
+
∑
j
=
1
n
‍
d
i
j
(
t
)
h
j
(
∫
0
∞
‍
v
i
j
(
s
)
φ
˙
j
(
t
-
s
)
d
s
)
+
I
i
(
t
)
,
where i
=
1,2
,
…
,
n, ∀
t
∈
ℝ.Let the mapT
:
𝕏
→
𝕏 be defined by
(21)
T
(
ϕ
)
=
(
Φ
1
(
ϕ
)
,
…
,
Φ
n
(
ϕ
)
)
T
=
(
x
1
φ
1
,
…
,
x
n
φ
n
)
T
,
aaaaaaaaaaaaaaaaaaaaaa
∀
ϕ
=
(
φ
1
,
…
,
φ
n
)
T
∈
𝕏
.Lemma 7.
T
:
Ω
¯
→
𝕏 is completely continuous.Proof.
First, we show thatT
:
Ω
¯
→
𝕏 is continuous. Assume that ϕ
1
=
(
φ
1
1
,
…
,
φ
n
1
)
T
,
ϕ
2
=
(
φ
1
2
,
…
,
φ
n
2
)
T
∈
Ω; then
(22)
|
F
i
(
t
,
ϕ
1
)
-
F
i
(
t
,
ϕ
2
)
|
=
|
∑
j
=
1
n
‍
b
i
j
(
t
)
[
f
j
(
φ
j
1
(
t
)
)
-
f
j
(
φ
j
2
(
t
)
)
]
+
∑
j
=
1
n
‍
c
i
j
(
t
)
[
g
j
(
∫
0
∞
‍
w
i
j
(
s
)
φ
j
1
(
t
-
s
)
d
s
)
-
g
j
(
∫
0
∞
‍
w
i
j
(
s
)
φ
j
2
(
t
-
s
)
d
s
)
]
+
∑
j
=
1
n
‍
d
i
j
(
t
)
[
h
j
(
∫
0
∞
‍
v
i
j
(
s
)
φ
˙
j
1
(
t
-
s
)
d
s
)
-
h
j
(
∫
0
∞
‍
v
i
j
(
s
)
φ
˙
j
2
(
t
-
s
)
d
s
)
]
∑
j
=
1
n
‍
|
≤
∑
j
=
1
n
‍
(
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
)
|
φ
j
1
-
φ
j
2
|
0
+
∑
j
=
1
n
‍
d
i
j
+
L
j
h
|
φ
˙
j
1
-
φ
˙
j
2
|
0
≤
max
⁡
{
∑
j
=
1
n
‍
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
,
∑
j
=
1
n
‍
d
i
j
+
L
j
h
}
∥
ϕ
1
-
ϕ
2
∥
,
which implies that
(23)
|
Φ
i
(
ϕ
1
)
-
Φ
i
(
ϕ
2
)
|
≤
∫
-
∞
t
‍
e
-
∫
s
t
‍
a
i
(
u
)
d
u
|
F
i
(
s
,
ϕ
1
(
s
)
)
-
F
i
(
s
,
ϕ
2
(
s
)
)
|
d
s
≤
(
a
i
-
)
-
1
max
⁡
{
∑
j
=
1
n
‍
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
,
∑
j
=
1
n
‍
d
i
j
+
L
j
h
}
∥
ϕ
1
-
ϕ
2
∥
≤
∑
i
=
1
n
‍
(
a
i
-
)
-
1
max
⁡
{
∑
j
=
1
n
‍
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
,
∑
j
=
1
n
‍
d
i
j
+
L
j
h
}
∥
ϕ
1
-
ϕ
2
∥
∶
=
l
1
∥
ϕ
1
-
ϕ
2
∥
,
where l
1
=
∑
i
=
1
n
‍
(
a
i
-
)
-
1
max
⁡
⁡
{
∑
j
=
1
n
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
,
∑
j
=
1
n
d
i
j
+
L
j
h
},  i
=
1
,
…
,
n. Since
(24)
|
Φ
˙
i
(
ϕ
1
)
-
Φ
˙
i
(
ϕ
2
)
|
=
|
[
-
a
i
(
t
)
Φ
i
(
ϕ
1
)
+
F
i
(
t
,
ϕ
1
)
]
-
[
a
i
(
t
)
Φ
i
(
ϕ
2
)
+
F
i
(
t
,
ϕ
2
)
]
|
≤
a
i
+
|
Φ
i
(
ϕ
1
)
-
Φ
i
(
ϕ
2
)
|
+
|
F
i
(
t
,
ϕ
1
)
-
F
i
(
t
,
ϕ
2
)
|
≤
(
1
+
a
i
+
a
i
-
)
max
⁡
{
∑
j
=
1
n
‍
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
,
∑
j
=
1
n
‍
d
i
j
+
L
j
h
}
∥
ϕ
1
-
ϕ
2
∥
∶
=
l
2
∥
ϕ
1
-
ϕ
2
∥
,
where l
2
=
(
1
+
(
a
i
+
/
a
i
-
)
)
max
⁡
{
∑
j
=
1
n
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
,
∑
j
=
1
n
d
i
j
+
L
j
h
},  i
=
1
,
…
,
n, so
(25)
∥
T
(
ϕ
1
)
-
T
(
ϕ
2
)
∥
≤
∑
i
=
1
2
‍
l
i
∥
ϕ
1
-
ϕ
2
∥
.
For arbitraryϵ
>
0, taking δ
=
(
ϵ
/
∑
i
=
1
2
‍
l
i
), when ∥
ϕ
1
-
ϕ
2
∥
<
δ, one has
(26)
∥
T
(
ϕ
1
)
-
T
(
ϕ
2
)
∥
<
ϵ
,
which implies that T
:
Ω
¯
→
𝕏 is continuous.
Next, we show thatT maps bounded set onto itself. Assume c is a positive constant and ϕ
∈
Ω
¯
c
=
{
z
∈
Ω
:
∥
z
∥
≤
c
}. By the almost periodicity of system (1), there exists a constant C
>
0 such that
(27)
max
⁡
1
≤
i
≤
n
,
1
≤
j
≤
m
max
⁡
s
∈
ℝ
,
ϕ
∈
Ω
¯
c
{
F
i
(
s
,
0
)
,
F
i
(
s
,
ϕ
)
}
<
C
.
From the above analysis, we get that(28)
|
Φ
i
(
ϕ
)
|
=
∫
-
∞
t
‍
e
-
∫
s
t
‍
a
i
(
u
)
d
u
|
F
i
(
s
,
ϕ
(
s
)
)
-
F
i
(
s
,
0
)
|
d
s
+
∫
-
∞
t
‍
e
-
∫
s
t
‍
a
i
(
u
)
d
u
F
i
(
s
,
0
)
d
s
≤
l
1
∥
ϕ
∥
+
C
a
i
-
,
i
=
1
,
…
,
n
,
|
Φ
˙
i
(
ϕ
)
|
=
|
-
a
i
(
t
)
Φ
i
(
ϕ
(
t
)
)
+
F
i
(
t
,
ϕ
)
|
≤
a
i
+
[
l
1
∥
ϕ
∥
+
C
a
i
-
]
+
C
,
i
=
1
,
…
,
n
,
which imply that T
Ω
¯
c is uniformly bounded. In addition, notice that
(29)
|
Φ
i
(
ϕ
(
t
1
)
)
-
Φ
i
(
ϕ
(
t
2
)
)
|
=
|
∫
t
2
t
1
‍
e
-
∫
s
t
‍
a
i
(
u
)
d
u
F
i
(
s
,
ϕ
(
s
)
)
d
s
|
≤
C
|
t
1
-
t
2
|
,
i
=
1
,
…
,
n
.
Using Arzela-Ascoli theorem, we obtain thatT
Ω
¯
c is relatively compact. Hence, T is completely continuous. The proof of this lemma is complete.


## 3. Existence of Almost Periodic Solution

In this section, we study the existence of almost periodic solutions of system (1).Theorem 8.
Assume that (H
1)–(H
3) hold; then system (1) admits at least one almost periodic solution.Proof.
Consider the following nonlinear operator:(30)
T
(
ϕ
)
=
(
Φ
1
(
ϕ
)
,
…
,
Φ
n
(
ϕ
)
)
T
,
∀
ϕ
=
(
φ
1
,
…
,
φ
n
)
T
∈
Ω
¯
.
For∀
ϕ
=
(
φ
1
,
…
,
φ
n
,
ψ
1
,
…
,
ψ
m
)
T
∈
∂
Ω we have ∥
ϕ
∥
=
r
0. From Lemma 7, T
:
Ω
¯
→
𝕏 is completely continuous. Similar to the arguments as that in Lemma 7, it follows from the definition of F
i and G
j that
(31)
|
F
i
|
0
=
sup
⁡
s
∈
ℝ
|
F
i
(
s
,
ϕ
)
|
≤
sup
⁡
s
∈
ℝ
|
F
i
(
s
,
ϕ
)
-
F
i
(
s
,
0
)
|
+
sup
⁡
s
∈
ℝ
|
F
i
(
s
,
0
)
|
≤
∑
j
=
1
n
‍
(
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
)
|
φ
j
|
0
+
∑
j
=
1
n
‍
d
i
j
+
L
j
h
|
φ
˙
j
|
0
+
∑
j
=
1
n
‍
b
i
j
+
f
j
(
0
)
+
∑
j
=
1
n
‍
c
i
j
+
g
j
(
0
)
+
∑
j
=
1
n
‍
d
i
j
+
h
j
(
0
)
+
I
i
+
,
which yields that
(32)
|
Φ
i
(
ϕ
)
|
0
=
sup
⁡
t
∈
ℝ
|
Φ
i
(
ϕ
(
t
)
)
|
≤
sup
⁡
t
∈
ℝ
|
∫
-
∞
t
‍
e
-
∫
s
t
‍
a
i
(
u
)
d
u
F
i
(
s
,
ϕ
(
s
)
)
d
s
|
≤
(
a
i
-
)
-
1
∑
j
=
1
n
‍
(
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
+
d
i
j
+
L
j
h
)
∥
ϕ
∥
+
(
a
i
-
)
-
1
[
∑
j
=
1
n
‍
b
i
j
+
f
j
(
0
)
+
∑
j
=
1
n
‍
c
i
j
+
g
j
(
0
)
+
∑
j
=
1
n
‍
d
i
j
+
h
j
(
0
)
+
I
i
+
]
,
|
Φ
˙
i
(
ϕ
)
|
0
=
sup
⁡
t
∈
ℝ
|
Φ
˙
i
(
ϕ
(
t
)
)
|
=
sup
⁡
t
∈
ℝ
|
-
a
i
(
t
)
Φ
i
(
ϕ
)
+
F
i
(
t
,
ϕ
)
|
≤
[
1
+
a
i
u
a
i
l
]
∑
j
=
1
n
‍
(
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
+
d
i
j
+
L
j
h
)
∥
ϕ
∥
+
[
1
+
a
i
u
a
i
l
]
[
∑
j
=
1
n
‍
b
i
j
+
f
j
(
0
)
+
∑
j
=
1
n
‍
c
i
j
+
g
j
(
0
)
+
∑
j
=
1
n
‍
d
i
j
+
h
j
(
0
)
+
I
i
+
]
,
where i
=
1,2
,
…
,
n. Hence,
(33)
∥
T
(
ϕ
)
∥
0
=
max
⁡
1
≤
i
≤
n
{
|
Φ
i
(
ϕ
)
|
0
}
≤
max
⁡
1
≤
i
≤
n
{
(
a
i
-
)
-
1
∑
j
=
1
n
‍
(
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
+
d
i
j
+
L
j
h
)
}
∥
ϕ
∥
+
max
⁡
1
≤
i
≤
n
{
(
a
i
-
)
-
1
[
∑
j
=
1
n
‍
b
i
j
+
f
j
(
0
)
+
∑
j
=
1
n
‍
c
i
j
+
g
j
(
0
)
+
∑
j
=
1
n
‍
d
i
j
+
h
j
(
0
)
+
I
i
+
∑
j
=
1
n
‍
c
i
j
+
g
j
(
0
)
∑
j
=
1
n
‍
b
i
j
+
f
j
(
0
)
]
}
=
θ
1
r
0
+
ρ
1
,
∥
T
˙
(
ϕ
)
∥
0
=
max
⁡
1
≤
i
≤
n
{
|
Φ
˙
i
(
ϕ
)
|
0
}
≤
max
⁡
1
≤
i
≤
n
{
[
1
+
a
i
u
a
i
l
]
∑
j
=
1
n
‍
(
b
i
j
+
L
j
f
+
c
i
j
+
L
j
g
+
d
i
j
+
L
j
h
)
}
∥
ϕ
∥
+
max
⁡
1
≤
i
≤
n
{
[
1
+
a
i
u
a
i
l
]
[
∑
j
=
1
n
‍
b
i
j
+
f
j
(
0
)
+
∑
j
=
1
n
‍
c
i
j
+
g
j
(
0
)
+
∑
j
=
1
n
‍
d
i
j
+
h
j
(
0
)
+
I
i
+
+
∑
j
=
1
n
‍
c
i
j
+
g
j
(
0
)
∑
j
=
1
n
‍
b
i
j
+
f
j
(
0
)
]
}
=
θ
2
r
0
+
ρ
2
.
So(34)
∥
T
(
ϕ
)
∥
=
max
⁡
{
∥
T
(
ϕ
)
∥
0
,
∥
T
˙
(
ϕ
)
∥
0
}
≤
max
⁡
{
θ
1
,
θ
2
}
∥
ϕ
∥
+
max
⁡
{
ρ
1
,
ρ
2
}
=
θ
r
0
+
ρ
≤
ρ
1
-
θ
=
r
0
=
∥
ϕ
∥
.
By Lemma6, there exists at least a fixed point ϕ
0
∈
Ω
¯ satisfying T
(
ϕ
0
)
=
ϕ
0, which implies system (1) has at least one almost periodic solution. This completes the proof.


## 4. Globally Exponential Stability of Almost Periodic Solution

Theorem 9.
Assume that(
H
1
)
–
(
H
3
) hold; suppose further the following.
(
H
4
) There exists a positive constant λ
<
1 such that
(35)
sup
⁡
s
∈
ℝ
{
∑
j
=
1
n
‍
[
∫
0
∞
‍
w
i
(
μ
)
|
c
j
i
(
s
+
μ
)
|
d
μ
|
b
j
i
(
s
)
|
L
i
f
+
(
1
+
λ
)
L
i
g
×
∫
0
∞
‍
w
i
(
μ
)
|
c
j
i
(
s
+
μ
)
|
d
μ
]
-
(
1
-
λ
)
a
i
(
s
)
∑
j
=
1
n
‍
}
<
0
,
sup
⁡
s
∈
ℝ
{
[
1
+
λ
]
∑
j
=
1
n
‍
|
d
j
i
(
s
)
|
L
i
h
-
λ
}
<
0
,
i
=
1,2
,
…
,
n
.
Then system (1) has a unique almost periodic solution, which is globally exponentially stable.Proof.
It follows from Theorem8 that system (1) has at least one almost periodic solution ϕ
=
(
φ
1
,
…
,
φ
n
)
T with initial value ϕ
*
=
(
φ
1
*
,
…
,
φ
n
*
)
T. We next show that the almost periodic solution ϕ is globally exponentially stable.
Make a transformation for system (1): x
i
=
u
i
-
φ
i, x
i
*
=
u
i
*
-
φ
i
*, i
=
1,2
,
…
,
n, where z
=
(
u
1
,
…
,
u
n
)
T is arbitrary solution of system (1) with initial value z
*
=
(
u
1
*
,
…
,
u
n
*
)
T.
By (2) and (
H
4
), there exist small enough positive constants ω
≤
δ and ϵ such that
(36)
∫
0
∞
‍
v
i
(
ν
)
e
ω
ν
d
ν
≤
∫
0
∞
‍
v
i
(
ν
)
(
1
+
ϵ
)
d
ν
=
(
1
+
ϵ
)
,
∫
0
∞
‍
w
j
(
μ
)
|
c
i
j
(
s
+
μ
)
|
e
ω
μ
d
μ
≤
(
1
+
ϵ
)
∫
0
∞
‍
w
j
(
μ
)
|
c
i
j
(
s
+
μ
)
|
d
μ
,
ω
+
∑
j
=
1
n
‍
(
∫
0
∞
‍
w
i
(
μ
)
|
c
j
i
(
s
+
μ
)
|
d
μ
|
b
j
i
(
s
)
|
L
i
f
+
[
1
+
λ
(
1
+
ϵ
)
]
(
1
+
ϵ
)
L
i
g
×
∫
0
∞
‍
w
i
(
μ
)
|
c
j
i
(
s
+
μ
)
|
d
μ
)
≤
[
1
-
λ
(
1
+
ϵ
)
]
a
i
(
s
)
,
[
1
+
λ
(
1
+
ϵ
)
]
∑
j
=
1
n
‍
|
d
j
i
(
s
)
|
L
i
h
-
λ
≤
0
,
where i
,
j
=
1,2
,
…
,
n, s
∈
ℝ.
Define(37)
V
1
(
t
)
=
∑
i
=
1
n
‍
e
ω
t
|
x
i
(
t
)
|
.
In view of system (1), we have
(38)
D
+
V
1
(
t
)
≤
ω
∑
i
=
1
n
‍
e
ω
t
|
x
i
(
t
)
|
+
∑
i
=
1
n
‍
e
ω
t
[
-
a
i
(
t
)
|
x
i
(
t
)
|
+
∑
j
=
1
n
‍
|
b
i
j
(
t
)
|
L
j
f
|
x
j
(
t
)
|
+
∑
j
=
1
n
‍
|
c
i
j
(
t
)
|
L
j
g
∫
0
∞
‍
w
j
(
μ
)
|
x
j
(
t
-
μ
)
|
d
μ
+
∑
j
=
1
n
‍
|
d
i
j
(
t
)
|
L
j
h
∫
0
∞
‍
v
j
(
ν
)
|
x
˙
j
(
t
-
ν
)
|
d
ν
]
.
Let(39)
V
2
(
t
)
=
λ
∑
j
=
1
n
‍
∫
0
∞
‍
v
j
(
ν
)
∫
t
-
ν
t
‍
e
ω
(
r
+
ν
)
|
x
˙
j
(
r
)
|
d
r
d
ν
,
V
3
(
t
)
=
∑
i
=
1
n
‍
∑
j
=
1
n
‍
[
1
+
λ
(
1
+
ϵ
)
]
L
j
g
×
∫
0
∞
‍
w
j
(
μ
)
∫
t
-
μ
t
‍
|
c
i
j
(
r
+
μ
)
|
e
ω
(
r
+
μ
)
|
x
j
(
r
)
|
d
r
d
μ
.
So(40)
D
+
V
2
(
t
)
≤
λ
∑
j
=
1
n
‍
e
ω
t
∫
0
∞
‍
v
j
(
ν
)
e
ω
ν
d
ν
|
x
˙
j
(
t
)
|
-
λ
∑
j
=
1
n
‍
e
ω
t
∫
0
∞
‍
v
j
(
ν
)
|
x
˙
j
(
t
-
ν
)
|
d
ν
≤
λ
∑
i
=
1
n
‍
e
ω
t
(
1
+
ϵ
)
[
∑
j
=
1
n
‍
a
i
(
t
)
|
x
i
(
t
)
|
+
∑
j
=
1
n
‍
|
b
i
j
(
t
)
|
L
j
f
|
x
j
(
t
)
|
+
∑
j
=
1
n
‍
|
c
i
j
(
t
)
|
L
j
g
×
∫
0
∞
‍
w
j
(
μ
)
|
x
j
(
t
-
μ
)
|
d
μ
∑
j
=
1
n
‍
]
-
λ
∑
j
=
1
n
‍
e
ω
t
[
1
-
∑
i
=
1
n
‍
(
1
+
ϵ
)
|
d
i
j
(
t
)
|
L
j
h
]
×
∫
0
∞
‍
v
j
(
ν
)
|
x
˙
j
(
t
-
ν
)
|
d
ν
,
(41)
D
+
V
3
(
t
)
≤
∑
i
=
1
n
‍
∑
j
=
1
n
‍
[
1
+
λ
(
1
+
ϵ
)
]
e
ω
t
L
j
g
×
∫
0
∞
‍
w
j
(
μ
)
|
c
i
j
(
t
+
μ
)
|
e
ω
μ
d
μ
|
x
j
(
t
)
|
-
∑
i
=
1
n
‍
∑
j
=
1
n
‍
[
1
+
λ
(
1
+
ϵ
)
]
e
ω
t
|
c
i
j
(
t
)
|
L
j
g
×
∫
0
∞
‍
w
j
(
μ
)
|
x
j
(
t
-
μ
)
|
d
μ
≤
∑
i
=
1
n
‍
∑
j
=
1
n
‍
[
1
+
λ
(
1
+
ϵ
)
]
(
1
+
ϵ
)
e
ω
t
L
j
g
×
∫
0
∞
‍
w
j
(
μ
)
|
c
i
j
(
t
+
μ
)
|
d
μ
|
x
j
(
t
)
|
-
∑
i
=
1
n
‍
∑
j
=
1
n
‍
[
1
+
λ
(
1
+
ϵ
)
]
e
ω
t
|
c
i
j
(
t
)
|
L
j
g
×
∫
0
∞
‍
w
j
(
μ
)
|
x
j
(
t
-
μ
)
|
d
μ
.
LetV
(
t
)
=
∑
q
=
1
3
‍
V
q. From (38)–(41), it follows that
(42)
D
+
V
(
t
)
≤
e
ω
t
∑
i
=
1
n
‍
{
∫
0
∞
‍
w
i
(
μ
)
|
c
j
i
(
t
+
μ
)
|
ω
-
[
1
-
λ
(
1
+
ϵ
)
]
a
i
(
t
)
+
∑
j
=
1
n
‍
|
b
j
i
(
t
)
|
L
i
f
+
∑
j
=
1
n
‍
[
1
+
λ
(
1
+
ϵ
)
]
(
1
+
ϵ
)
L
i
g
×
∫
0
∞
‍
w
i
(
μ
)
|
c
j
i
(
t
+
μ
)
|
d
μ
}
|
x
i
(
t
)
|
+
e
ω
t
∑
j
=
1
n
‍
{
[
1
+
λ
(
1
+
ϵ
)
]
∑
i
=
1
n
‍
|
d
i
j
(
t
)
|
L
j
h
-
λ
}
×
∫
0
∞
‍
v
j
(
ν
)
|
x
˙
j
(
t
-
ν
)
|
d
ν
≤
0
,
which implies that V
(
t
)
≤
V
(
0
), ∀
t
>
0. Obviously,
(43)
∑
i
=
1
n
‍
e
ω
t
|
x
i
(
t
)
|
≤
V
(
t
)
.
On the other hand, we have(44)
V
(
0
)
=
∑
i
=
1
n
‍
|
x
i
(
0
)
|
+
λ
∑
j
=
1
n
‍
∫
0
∞
‍
v
j
(
ν
)
∫
-
ν
0
‍
e
ω
(
r
+
ν
)
|
x
˙
j
(
r
)
|
d
r
d
ν
+
∑
i
=
1
n
‍
∑
j
=
1
n
‍
[
1
+
λ
(
1
+
ϵ
)
]
L
j
g
×
∫
0
∞
‍
w
j
(
μ
)
∫
-
μ
0
‍
|
c
i
j
(
r
+
μ
)
|
e
ω
(
r
+
μ
)
|
x
j
(
r
)
|
d
r
d
μ
≤
{
n
+
2
n
ω
+
∑
j
=
1
m
‍
∑
i
=
1
n
‍
6
c
i
j
+
L
j
g
ω
}
∥
z
*
-
ϕ
*
∥
,
which implies from (43) that
(45)
∑
i
=
1
n
‍
|
x
i
(
t
)
|
≤
M
∥
z
*
-
ϕ
*
∥
e
-
ω
t
,
∀
t
>
0
,
where
(46)
M
:
=
n
+
2
n
ω
+
∑
j
=
1
m
‍
∑
i
=
1
n
‍
6
c
i
j
+
L
j
g
ω
.
Thus, the almost periodic solution of system (1) is globally exponentially stable. The globally exponential stability implies that the almost periodic solution is unique. This completes the proof.


## 5. An Example

Example 1.
Consider the following neutral-type cellular neural networks with distributed delays:(47)
x
˙
i
(
t
)
=
-
x
i
(
t
)
+
sin
(
2
t
)
×
∑
j
=
1
2
‍
g
j
(
∫
0
∞
‍
e
-
s
x
j
(
t
-
s
)
d
s
)
+
0.1
[
2
+
cos
⁡
2
(
3
t
)
]
∑
j
=
1
2
‍
h
j
(
∫
0
∞
‍
e
-
s
x
˙
j
(
t
-
s
)
d
s
)
+
sin
(
2
t
)
,
i
=
1,2
,
where g
1
(
s
)
=
g
2
(
s
)
=
0.1
s, h
1
(
s
)
=
0.1
sin
(
s
), and h
2
(
s
)
=
0.1
cos
⁡
(
s
), ∀
s
∈
ℝ. Then system (47) has a unique almost periodic solution, which is globally exponentially stable.Proof.
Corresponding to system (1), a
i
-
=
1
,
L
j
g
=
L
j
h
=
0.1,  c
i
j
+
=
1
,
d
i
j
+
=
0.3,  w
j
(
s
)
=
v
j
(
s
)
=
e
-
s,  i
,
j
=
1,2. Taking λ
=
0.5, it is easy to verify that (
H
1
)
–
(
H
4
) hold and the results follow from Theorems 8 and 9 (see Figures 1, 2, and 3). This completes the proof.Figure 1
Almost periodicity of state variablesx
1
(
t
) and x
2
(
t
).Figure 2
Global exponential stability of state variablex
1
(
t
).Figure 3
Global exponential stability of state variablex
2
(
t
).Remark 10.
One can observe that all the results in [9–12] and the references therein cannot be applicable to system (1) to obtain the existence and exponential stability of the almost periodic solutions. This implies that the results of this paper are essentially new.


## 6. Conclusion

This paper is concerned with the neutral cellular neural networks with distributed delays. By using fixed point theory and constructing suitable Lyapunov functional, some new sufficient conditions are obtained for the existence and global exponential stability of almost periodic solution of the system. Conditions(
H
3
) and (
H
4
) in Theorems 8 and 9 indicate that the neutral terms and the distributed delays are harm for the existence and global exponential stability of almost periodic solution of the neutral-type system. The method used in this paper provides a possible method to study the existence and global exponential stability of almost periodic solution of other neutral neural networks.



---

*Source: 281016-2014-05-13.xml*