# Feedback Control in a General Almost Periodic Discrete System of Plankton Allelopathy

Authors: Wenshuang Yin
Journal: The Scientific World Journal
 (2014)

---

## Abstract

We study the properties of almost periodic solutions for a general discrete system of plankton allelopathy with feedback controls and establish a theorem on the uniformly asymptotic stability of almost periodic solutions.


---

## Body

## 1. Introduction

Allelopathy is first used by Hans (see [1]) in 1937, which is a biological phenomenon that is characteristic of any process involving secondary metabolites produced by some plants, algae, bacteria, and fungi which influences the growth and development of biological systems. There are many investigations which have been conducted to study the toxic effect on plankton allelopathy (see [2–9]). In this paper, we investigate the following system with feedback controls:
(1)
x
i
(
n
+
1
)
=
x
i
(
n
)
exp
⁡
⁡
[
r
i
(
n
)
-
a
i
i
(
n
)
x
i
(
n
)
-
a
i
j
(
n
)
x
j
(
n
)
ggggggggg
-
b
i
(
n
)
x
i
(
n
)
x
j
(
n
)
-
c
i
(
n
)
u
i
(
n
)
]
,
Δ
u
i
(
n
)
=
-
d
i
(
n
)
u
i
(
n
)
+
e
i
(
n
)
x
i
(
n
)
,
i
,
j
=
1,2
;
i
≠
j
.
Here x
i
(
n
) represent the densities of species x
i at the nth generation, r
i
(
n
) are the intrinsic growth rates of species x
i at the nth generation, a
i
i
(
n
) measure the intraspecific effects of the generation of species x
i on their own population, and a
i
j
(
n
) stand for the interspecific effects of the nth generation of species x
i on species x
j; b
i
(
n
) are the rates of toxic inhibition of species x
i by species x
j and vice versa at the nth generation, Δ is the first-order forward difference operator Δ
u
i
(
n
)
=
u
i
(
n
+
1
)
-
u
i
(
n
), and u
i
(
n
) are the feedback control variables (
i
,
j
=
1,2; i
≠
j
). Meanwhile, {
r
i
(
n
)
}, {
a
i
j
(
n
)
}, {
b
i
(
n
)
}, {
c
i
(
n
)
}, {
d
i
(
n
)
}, and {
e
i
(
n
)
} are bounded nonnegative sequences such that
(2)
0
<
r
i
l
≤
r
i
(
n
)
≤
r
i
u
,
0
<
a
i
j
l
≤
a
i
j
(
n
)
≤
a
i
j
u
,
0
<
b
i
l
≤
b
i
(
n
)
≤
b
i
u
,
0
<
c
i
l
≤
c
i
(
n
)
≤
c
i
u
,
0
<
e
i
l
≤
e
i
(
n
)
≤
e
i
u
,
0
<
d
i
l
≤
d
i
(
n
)
≤
d
i
u
<
1
,
for i
,
j
=
1,2. Here, we use the notations f
u
=
su
p
n
∈
ℤ
+
f
(
n
) andf
l
=
in
f
n
∈
ℤ
+
f
(
n
) for any bounded sequence {
f
(
n
)
}.  For the simplicity and convenience of exposition, throughout this paper we let ℤ, ℤ
+, and ℝ
+ denote the sets of all integers, and nonnegative integers, nonnegative real numbers, respectively.From the point of view of biology, we only consider{
x
1
(
n
)
,
x
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
} are positive; that is,
(3)
x
i
(
n
)
>
0
,
u
i
(
n
)
>
0
,
i
=
1,2
.For convenience, we give the following definition of persistence of system (1).Definition 1.
System (1) is said to be persistent if there exist positive constants x
i
*, u
i
*, x
i
*, and u
i
* which are independent of the solutions of system (1), such that any positive solution {
x
1
(
n
)
,
x
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
} of system (1) satisfies
(4)
x
i
*
≤
liminf
⁡
n
→
+
∞
x
i
(
n
)
≤
limsup
⁡
n
→
+
∞
x
i
(
n
)
≤
x
i
*
,
u
i
*
≤
liminf
⁡
n
→
+
∞
u
i
(
n
)
≤
limsup
⁡
n
→
+
∞
u
i
(
n
)
≤
u
i
*
,
for i
=
1,2.The rest of this paper is organized as follows. We first give some preliminaries in Section2. Next, we study the persistent property of (1) in Section 3 and the almost periodic property in Section 4. To the best of our knowledge, no work has been done for the discrete system (1) with feedback controls.


## 2. Preliminaries

In this section, we first introduce some definitions. For our purpose, we introduce the following notions with their properties. For convenience,n, m, l, s, h, i, j, k, τ, δ, and r in the sequel always denote integers, and the relevant intervals and inequalities are discrete ones. This section, devoted to investigate the persistent property of system (1). To do so, we need to make some preparations.We first introduce some definitions.Definition 2 (see [10]).
A sequencex
:
ℤ
→
ℝ
k is called an almost periodic sequence if the ε-translation set of x
(5)
E
{
ε
,
x
}
:
=
{
τ
∈
ℤ
:
|
x
(
n
+
τ
)
-
x
(
n
)
|
<
ε
∀
n
∈
ℤ
}
is a relatively dense set in ℤ for all ε
>
0; that is, for any given ε
>
0, there exists an integer l
(
ε
)
>
0 such that each discrete interval of length l
(
ε
) contains a τ
=
τ
(
ε
)
∈
E
{
ε
,
x
} such that
(6)
|
x
(
n
+
τ
)
-
x
(
n
)
|
<
ε
for all n
∈
ℤ. τ is called the ε-translation number of x
(
n
).Definition 3 (see [10]).
Letf
:
ℤ
×
𝔻
→
ℝ
k,  where 𝔻 is an open set in ℝ
k. f
(
n
,
x
) is said to be almost periodic in n uniformly for x
∈
𝔻 or uniformly almost periodic for short if, for any ε
>
0 and any compact set S in 𝔻, there exists a positive integer l
(
ε
,
S
) such that any interval of length l
(
ε
,
S
) contains a τ for which
(7)
|
f
(
n
+
τ
,
x
)
-
f
(
n
,
x
)
|
<
ε
for all n
∈
ℤ and all x
∈
S. τ is called the ε-translation number of f
(
x
,
n
).Next, we will introduce the following Lemmas.Lemma 4 (see [10]).
{
x
(
n
)
} is an almost periodic sequence if and only if for any sequence {
h
k
}
⊂
{
h
k
′
} such that x
(
n
+
h
k
) converges uniformly on n
∈
ℤ as k
→
∞. Furthermore, the limit sequence is also an almost periodic sequence.Lemma 5 (see [11]).
Assume that{
x
(
n
)
} satisfies x
(
n
)
>
0 and
(8)
x
(
n
+
1
)
≤
x
(
n
)
exp
⁡
[
a
(
n
)
-
b
(
n
)
x
(
n
)
]
for n
∈
ℤ
+, where a
(
n
) and b
(
n
) are nonnegative sequences bounded above and below by positive constants. Then
(9)
limsup
⁡
n
→
+
∞
⁡
x
(
n
)
≤
1
b
l
exp
⁡
(
a
u
-
1
)
.Lemma 6 (see [11]).
Assume that{
x
(
n
)
} satisfies
(10)
x
(
n
+
1
)
≥
x
(
n
)
exp
⁡
[
a
(
n
)
-
b
(
n
)
x
(
n
)
]
,
n
≥
n
0
,
and limsu
p
n
→
+
∞
x
(
n
)
≤
x
*
,
b
u
x
*
>
a
l
,
x
(
n
0
)
>
0, where a
(
n
) and b
(
n
) are nonnegative sequences bounded above and below by positive constants and n
0
∈
ℤ
+. Then
(11)
liminf
⁡
n
→
+
∞
⁡
x
(
n
)
≥
a
l
b
u
exp
⁡
(
a
l
-
b
u
x
*
)
.


## 3. Persistence

In this section, we will establish sufficient conditions for the persistence of system (1).Proposition 7.
Assume that (2) and (3) hold; any solution {
x
1
(
n
)
,
x
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
} of system (1) satisfies
(12)
limsup
⁡
n
→
+
∞
⁡
x
i
(
n
)
≤
x
i
*
,
limsup
⁡
n
→
+
∞
⁡
u
i
(
n
)
≤
u
i
*
,
i
=
1,2
,
where
(13)
x
i
*
=
def
exp
⁡
(
r
i
u
-
1
)
a
i
i
l
,
u
i
*
=
def
f
i
u
exp
⁡
(
r
i
u
-
1
)
a
i
i
l
e
i
l
.Proof.
Let{
x
1
(
n
)
,
x
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
} be any positive solution of system (1) with initial condition (3). By assumption (2), it follows from the first and second equations of system (1) that
(14)
x
i
(
n
+
1
)
≤
x
i
(
n
)
exp
⁡
[
r
i
(
n
)
-
a
i
i
(
n
)
x
i
(
n
)
]
,
i
=
1,2
.
Applying Lemma 5, one obtains
(15)
limsup
⁡
n
→
+
∞
⁡
x
i
(
n
)
≤
exp
⁡
(
r
i
u
-
1
)
a
i
i
l
=
x
i
*
,
i
=
1,2
.
Thus, for any ε
>
0 small enough, it follows from (15) that there exists a large enough integer n
1 such that
(16)
x
i
(
n
)
≤
x
i
*
+
ε
for
n
≥
n
1
,
i
=
1,2
.
Then the third and fourth equations of system (1) lead to
(17)
u
i
(
n
)
=
∏
p
=
0
n
-
1
(
1
-
d
i
(
p
)
)
[
u
i
(
0
)
+
∑
p
=
0
n
-
1
e
i
(
p
)
x
i
(
p
)
∏
q
=
0
n
-
1
1
-
d
i
(
q
)
]
≤
(
1
-
d
i
l
)
n
(
u
i
(
0
)
+
v
i
)
+
e
i
u
(
x
i
*
+
ε
)
∑
p
=
n
0
n
-
1
∏
q
=
p
+
1
n
-
1
(
1
-
e
i
(
q
)
)
≤
(
1
-
d
i
l
)
n
(
u
1
(
0
)
+
v
i
)
+
e
i
u
(
x
i
*
+
ε
)
∑
p
=
n
0
n
-
1
(
1
-
e
i
l
)
n
-
p
-
1
,
where v
i
=
∑
p
=
0
n
-
1
(
e
i
(
p
)
x
i
(
p
)
/
∏
q
=
0
n
-
1
1
-
d
i
(
q
)
), i
=
1,2. Since 0
<
d
1
l
<
1, we can find a  g
∈
ℤ
+ such that 1
-
d
1
l
=
e
-
g; by Stolz’s theorem, we have
(18)
∑
p
=
n
0
n
-
1
(
1
-
e
i
l
)
n
-
p
-
1
=
∑
p
=
n
0
n
-
1
e
g
(
p
+
1
)
e
g
n
⟶
1
1
-
e
-
g
=
1
d
1
l
,
(
n
⟶
∞
)
.
Thus,
(19)
limsup
⁡
n
→
+
∞
⁡
u
i
(
n
)
≤
(
x
i
*
+
ε
)
f
i
u
e
i
l
.
Letting ε
→
0 and substituting (16) in the above inequality lead to
(20)
limsup
⁡
n
→
+
∞
u
i
(
n
)
≤
f
i
u
exp
⁡
(
r
i
u
-
1
)
a
i
i
l
e
i
l
=
u
i
*
,
i
=
1,2
.Proposition 8.
Assume that  (2) and (3); furthermore
(21)
r
i
l
≥
a
i
j
u
x
j
*
+
c
i
u
u
i
*
,
a
i
i
u
+
c
i
u
u
i
*
+
(
b
i
u
+
a
i
j
u
)
x
j
*
≥
r
i
l
,
for i
,
j
=
1,2, i
≠
j, are satisfied, where u
i
* and x
i
* are the same as those in Proposition 7. Then
(22)
liminf
⁡
n
→
+
∞
x
i
(
n
)
≥
x
i
*
,
liminf
⁡
n
→
+
∞
u
i
(
n
)
≥
u
i
*
,
where
(23)
u
i
*
=
def
x
i
*
f
i
l
e
i
u
,
x
i
*
=
def
r
i
l
-
a
i
j
u
x
j
*
-
c
i
u
x
i
*
a
i
i
u
+
b
i
u
x
j
*
×
exp
⁡
{
(
r
i
l
-
a
i
j
u
x
j
*
-
c
i
u
u
i
*
)
-
(
a
i
i
u
+
b
i
u
x
j
*
)
x
i
*
}
.Proof.
Forε
>
0 and n
>
n
2, according to Proposition 7, there exists a n
2
∈
ℤ
+ such that
(24)
x
i
(
n
)
≤
x
i
*
+
ε
,
u
i
(
n
)
≤
u
i
*
+
ε
.
By Lemma 6, associating (21) with (1), one has
(25)
liminf
⁡
n
→
+
∞
x
i
(
n
)
≥
r
i
l
-
a
i
j
u
(
x
j
*
+
ε
)
-
c
i
u
(
x
i
*
+
ε
)
a
i
i
u
+
b
i
u
(
x
j
*
+
ε
)
×
exp
⁡
⁡
{
r
i
l
-
a
i
j
u
(
x
j
*
+
ε
)
-
c
i
u
(
u
i
*
+
ε
)
f
f
f
f
-
[
a
i
i
u
+
b
i
u
(
x
j
*
+
ε
)
]
x
i
*
}
.
Letting ε
→
0 in the above inequality leads to
(26)
liminf
⁡
n
→
+
∞
x
i
(
n
)
≥
r
i
l
-
a
i
j
u
x
j
*
-
c
i
u
x
i
*
a
i
i
u
+
b
i
u
x
j
*
×
exp
⁡
{
(
r
i
l
-
a
i
j
u
x
j
*
-
c
i
u
u
i
*
)
f
f
f
f
f
-
(
a
i
i
u
+
b
i
u
x
j
*
)
x
i
*
}
=
x
i
*
.
Also from (26), for each ε
>
0, there exists a large enough integer n
0 such that
(27)
x
i
(
n
)
≥
x
i
*
-
ε
for
n
≥
n
0
,
i
=
1,2
.
By the third and fourth equations of system (1), we can get that
(28)
u
i
(
n
)
=
∏
p
=
0
n
-
1
(
1
-
d
i
(
p
)
)
[
u
i
(
0
)
+
∑
i
=
0
n
-
1
e
i
(
p
)
x
i
(
p
)
∏
q
=
0
p
1
-
d
i
(
q
)
]
≥
(
1
-
d
i
u
)
n
(
u
i
(
0
)
+
v
1
)
+
e
i
u
(
x
i
*
-
ε
)
×
∑
p
=
n
0
n
-
1
∏
q
=
p
+
1
n
-
1
(
1
-
e
i
(
j
)
)
≥
(
1
-
d
i
u
)
n
(
u
i
(
0
)
+
v
1
)
+
e
i
u
(
x
i
*
-
ε
)
∑
p
=
n
0
n
-
1
(
1
-
e
i
u
)
n
-
p
-
1
,
where v
i
=
∑
p
=
0
n
-
1
(
e
i
(
p
)
x
i
(
p
)
/
∏
q
=
0
n
-
1
1
-
d
i
(
q
)
), i
=
1,2. As 0
<
d
i
u
<
1, similar  to the analysis in the proof of Proposition 7, we have
(29)
liminf
⁡
n
→
+
∞
⁡
u
i
(
n
)
≥
x
i
*
f
i
l
e
i
u
=
u
i
*
,
i
=
1,2
.Now, we are in a position to state Theorem9 whose proof is a direct consequence of Propositions 7 and 8.Theorem 9.
If the inequalities in (2), (3), and (21) hold, then system (1) is persistent.


## 4. Almost Periodic Solution

Based on Theorem9, in Section 3, we discuss the almost periodic property for system (1).Consider the following almost periodic difference system:(30)
x
(
n
+
1
)
=
f
(
n
,
x
(
n
)
)
,
n
∈
ℤ
+
,
where f
:
ℤ
×
S
B
→
ℝ
k, S
B
=
{
x
∈
ℝ
k
:
∥
x
∥
<
B
}, and f
(
n
,
x
) is almost periodic in n uniformly for x
∈
S
B and is continuous in x. The product system of system (30) is as follows:
(31)
x
(
n
+
1
)
=
f
(
n
,
x
(
n
)
)
,
y
(
n
+
1
)
=
f
(
n
,
y
(
n
)
)
.Lemma 10 (see [2]).
Suppose that there exists a Lyapunov functionV
(
n
,
x
,
y
) defined for n
∈
ℕ, ∥
x
∥
<
B, ∥
y
∥
<
B satisfying that(i)
a
(
∥
x
-
y
∥
)
≤
V
(
n
,
x
,
y
)
≤
b
(
∥
x
-
y
∥
), where a
,
b
∈
K, with K
=
{
a
∈
C
(
ℝ
+
,
ℝ
+
)
:
a
(
0
)
=
0
}, and a is increasing;
(ii)
|
V
(
n
,
x
1
,
y
1
)
-
V
(
n
,
x
2
,
y
2
)
|
≤
L
(
∥
x
1
-
x
2
∥
+
∥
y
1
-
y
2
∥
), where L
>
0 is a constant;
(iii)
Δ
V
(
10
)
(
n
,
x
,
y
)
≤
-
a
V
(
n
,
x
,
y
), where 0
<
a
<
1 is a constant and Δ
V
(
10
)
(
n
,
x
,
y
)
=
V
(
n
+
1
,
f
(
n
,
x
)
,
f
(
n
,
y
)
)
-
V
(
n
,
x
,
y
).Moreover, if there exists a solution φ
(
n
) of system (30) such that ∥
φ
(
n
)
∥
≤
B
*
<
B for n
∈
ℤ
+, then there exists a unique uniformly asymptotically stable almost periodic solution p
(
n
) of system (30) which is bounded by B
*. In particular, if f
(
n
,
x
) is periodic of period ω, then there exists a unique uniformly asymptotically stable periodic solution of (30) of period ω.According to Lemma10, we first prove that there exists a bounded solution of system (1) and then construct an adaptive Lyapunov functional for system (1). We denote by Ω the set of all solutions X
(
n
)
=
{
x
1
(
n
)
,
x
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
} of system (1) satisfying x
i
*
≤
x
i
(
n
)
≤
x
i
*, u
i
*
≤
u
i
(
n
)
≤
u
i
*.Lemma 11.
Assume that (2), (3), and the conditions of Theorem 9 hold; then Ω
≠
Ø.Proof.
It is now possible to show by an inductive argument that the system (1) leads to
(32)
x
i
(
n
)
=
x
i
(
0
)
exp
⁡
∑
l
=
0
n
-
1
[
r
i
(
l
)
-
a
i
i
(
l
)
x
i
(
l
)
-
a
i
j
(
l
)
x
j
(
l
)
f
f
f
f
f
f
f
f
-
b
i
(
l
)
x
i
(
l
)
x
j
(
l
)
-
c
i
(
l
)
u
i
(
l
)
]
,
u
i
(
n
)
=
u
i
(
0
)
-
∑
l
=
0
n
-
1
{
d
i
(
l
)
u
i
(
l
)
-
e
i
(
l
)
x
i
(
l
)
}
,
for i
,
j
=
1,2, i
≠
j. From Theorem 9, for any solution X
(
n
)
=
{
x
1
(
n
)
,
x
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
} of system (1) with initial condition (2) satisfies Definition 1.  Hence, for any ε
>
0, there exists a n
0; if n
0 is sufficiently large, we have
(33)
x
i
*
-
ε
≤
x
i
(
n
)
≤
x
i
*
+
ε
,
u
i
*
-
ε
≤
u
i
(
n
)
≤
u
i
*
+
ε
,
x
i
*
-
ε
≤
x
i
(
n
)
≤
x
i
*
+
ε
,
u
i
*
-
ε
≤
u
i
(
n
)
i
=
1,2
.
Let {
τ
α
} be any integer valued sequence such that τ
α
→
∞ as α
→
∞; we claim that there exists a subsequence of {
τ
α
}, and we still denote it by {
τ
α
}, such that x
i
(
n
+
τ
α
)
→
x
i
*
(
n
) uniformly in n on any finite subset B of ℤ as α
→
∞, where
(34)
B
=
{
a
1
,
a
2
,
…
,
a
m
}
,
a
h
∈
ℤ
,
h
=
(
1,2
,
…
,
m
)
,
and m is a finite number.
In fact, for any finite subsetB
⊂
ℤ, when α is large enough, τ
α
+
a
α
>
n
0, h
=
1,2
,
…
,
m. So
(35)
x
i
*
-
ε
≤
x
i
(
n
+
τ
α
)
≤
x
i
*
+
ε
,
u
i
*
-
ε
≤
u
i
(
n
+
τ
α
)
≤
u
i
*
+
ε
.
That is, {
x
i
(
k
+
τ
α
)
,
u
i
(
k
+
τ
α
)
} are uniformly bounded for large enough α.
Now, fora
1
∈
B, we can choose a subsequence τ
α
(
1
) of {
τ
α
} such that {
x
i
(
a
1
+
τ
α
(
1
)
)
,
u
i
(
a
1
+
τ
α
(
1
)
)
} uniformly converges on ℤ
+ for α large enough.
Similarly, fora
2
∈
B, we can choose a subsequence τ
α
(
2
) of {
τ
α
} such that {
x
i
(
a
2
+
τ
α
(
2
)
)
,
u
i
(
a
2
+
τ
α
(
2
)
)
} uniformly converges on ℤ
+ for α  large enough.
Repeating this procedure, fora
m
∈
B, we can choose a subsequence {
τ
α
(
m
-
1
)
} of {
τ
α
(
m
)
} such that {
x
i
(
a
m
+
τ
α
(
m
)
)
,
u
i
(
a
m
+
τ
α
(
m
)
)
}   uniformly converges on ℤ
+ for α large enough.
Now pick the sequence{
τ
α
(
m
)
} which is a subsequence of {
τ
α
}; we still denote it by {
τ
α
}; then for all k
∈
B, we have
(36)
{
x
i
(
n
+
τ
α
)
⟶
x
i
*
,
u
i
(
n
+
τ
α
)
⟶
u
i
*
}
uniformly in k
∈
B as α
→
∞.
By the arbitrary ofB, the conclusion is valid.Since{
r
i
(
n
)
}, {
a
i
j
(
n
)
}, {
b
i
(
n
)
}, {
c
i
(
n
)
}, {
d
i
(
n
)
}, and {
e
i
(
n
)
} are almost periodic sequence, for the above sequence τ
α
,
τ
α
→
∞ as α
→
∞, there exists a subsequence which we still denote it by {
τ
α
} (if necessary, we take subsequence), such that
(37)
r
i
(
n
+
τ
α
)
⟶
r
i
(
n
)
,
a
i
j
(
n
+
τ
α
)
⟶
r
i
(
n
)
,
b
i
(
n
+
τ
α
)
⟶
b
i
(
n
)
,
c
i
(
n
+
τ
α
)
⟶
c
i
(
n
)
,
d
i
(
n
+
τ
α
)
⟶
d
i
(
n
)
,
e
i
(
n
+
τ
α
)
⟶
e
i
(
n
)
,
as α
→
∞ uniformly on ℤ
+.For anyδ
∈
ℤ, we can assume that τ
α
+
δ
≥
n
0 for δ large enough. Letting  n
≥
0 and n
∈
ℤ, by an inductive argument of (1) from τ
α
+
δ to n
+
τ
α
+
δ, leads to
(38)
x
i
(
n
+
τ
α
+
δ
)
=
x
i
(
τ
α
+
δ
)
×
exp
⁡
∑
l
=
τ
α
+
δ
n
+
τ
α
+
δ
-
1
[
r
i
(
l
)
-
a
i
i
(
l
)
x
i
(
l
)
-
a
i
j
(
l
)
x
j
(
l
)
f
f
f
f
f
f
f
f
f
f
-
b
i
(
l
)
x
i
(
l
)
x
j
(
l
)
-
c
i
(
l
)
u
i
(
l
)
]
,
u
i
(
n
+
τ
α
+
δ
)
=
u
i
(
τ
α
+
δ
)
-
∑
l
=
τ
α
n
+
τ
α
+
δ
-
1
{
d
i
(
l
)
u
i
(
l
)
-
e
i
(
l
)
x
i
(
l
)
}
.
Then, for i
,
j
=
1,2, i
≠
j, we have
(39)
x
i
(
n
+
τ
α
+
δ
)
=
x
i
(
τ
α
+
δ
)
exp
⁡
∑
l
=
δ
n
+
δ
-
1
[
r
i
(
l
+
τ
α
)
-
a
i
i
(
l
+
τ
α
)
x
i
(
l
+
τ
α
)
f
f
f
f
f
d
d
d
d
d
d
d
f
11
-
a
i
j
(
l
+
τ
α
)
x
j
(
l
+
τ
α
)
f
f
f
f
f
d
d
d
d
d
d
d
f
11
-
b
i
(
l
+
τ
α
)
x
i
(
l
+
τ
α
)
x
j
(
l
+
τ
α
)
f
f
f
f
f
f
d
d
d
d
d
d
f
11
-
c
i
(
l
+
τ
α
)
u
i
(
l
+
τ
α
)
]
,
u
i
(
n
+
τ
α
+
δ
)
=
u
i
(
τ
α
+
δ
)
-
∑
l
=
δ
n
+
δ
-
1
{
d
i
(
l
+
τ
α
)
u
i
(
l
+
τ
α
)
f
f
f
f
f
f
f
f
f
f
f
f
2
-
e
i
(
l
+
τ
α
)
x
i
(
l
+
τ
α
)
}
.
Let α
→
∞; for any k
≥
0,
(40)
x
i
*
(
n
+
δ
)
=
x
i
(
δ
)
exp
⁡
∑
l
=
δ
n
+
δ
-
1
[
r
i
(
l
)
-
a
i
i
(
l
)
x
i
(
l
)
-
a
i
j
(
l
)
x
j
(
l
)
f
f
g
g
g
g
g
f
f
f
f
i
1
-
b
i
(
l
)
x
i
(
l
)
x
j
(
l
)
-
c
i
(
l
)
u
i
(
l
)
]
,
u
i
*
(
n
+
δ
)
=
u
i
(
δ
)
-
∑
l
=
δ
n
+
δ
-
1
{
d
i
(
l
)
u
i
(
l
)
-
e
i
(
l
)
x
i
(
l
)
}
.By the arbitrariness ofδ
,
X
*
(
n
)
=
{
x
1
*
(
n
)
,
x
2
*
(
n
)
,
u
1
*
(
n
)
,
u
2
*
(
n
)
} is a solution of system (1) on ℤ
+. It is clear that 0
≤
x
i
*
≤
x
i
(
n
)
≤
x
i
*; 0
≤
u
i
*
≤
u
i
(
n
)
≤
u
i
*, n
∈
ℤ
+.SoΩ
≠
Φ. Lemma 10 is valid.Before stating Theorem12, for the sake of convenience, we set
(41)
r
i
j
=
2
(
a
i
i
l
+
b
i
l
x
i
*
)
-
[
(
a
i
i
u
+
b
i
u
x
j
*
)
2
+
e
i
2
u
]
x
i
*
2
-
[
a
j
i
u
+
b
j
u
x
j
*
]
2
x
i
*
2
+
(
a
i
j
u
+
b
i
u
x
i
*
)
+
(
a
j
i
u
+
b
j
u
x
j
*
)
-
(
a
i
i
u
+
b
i
u
x
j
*
)
(
a
i
j
u
+
b
i
u
x
i
*
)
x
i
*
x
j
*
-
(
a
j
j
u
+
b
j
u
x
i
*
)
(
a
j
i
u
+
b
j
u
x
j
*
)
x
i
*
x
j
*
,
+
c
i
l
(
a
i
i
l
+
b
i
l
x
j
*
)
x
i
*
-
c
i
u
-
(
1
-
d
i
L
)
e
i
u
x
i
*
-
c
j
u
(
a
j
i
u
+
b
i
u
x
i
*
)
x
i
*
,
r
i
j
*
=
-
c
i
u
-
(
1
-
d
i
l
)
e
i
u
x
i
*
+
c
i
l
(
a
i
i
l
+
b
i
l
x
j
*
)
x
i
*
-
c
i
u
(
a
i
j
u
+
b
j
u
x
i
*
)
x
j
*
-
d
i
2
u
+
2
d
i
l
-
c
i
u
.Theorem 12.
Suppose that the conditions of Lemma10 are satisfied; moreover, 0
<
β
<
1, where
(42)
β
=
min
⁡
{
r
i
j
,
r
i
j
*
}
,
for
i
,
j
=
1,2
,
i
≠
j
.
Then there exists a unique uniformly asymptotically stable almost periodic solution X
(
n
)
=
{
x
1
(
n
)
,
x
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
} of system (1) which is bounded by Ω for all n
∈
ℤ
+.Proof.
Letx
i
(
n
)
=
exp
⁡
p
i
(
n
); from system (1), we have
(43)
p
i
(
n
+
1
)
=
p
i
(
n
)
+
r
i
(
n
)
-
a
i
i
(
n
)
exp
⁡
p
i
(
n
)
-
a
i
j
(
n
)
exp
⁡
p
j
(
n
)
-
b
i
(
n
)
exp
⁡
p
i
(
n
)
exp
⁡
p
j
(
n
)
-
c
i
(
n
)
u
i
(
n
)
,
Δ
u
i
(
n
)
=
-
d
i
(
n
)
u
i
(
n
)
+
e
i
(
n
)
exp
⁡
p
i
(
n
)
,
where i
,
j
=
1,2, i
≠
j. From Lemma 10, it shows that system (43) has a bounded solution Y
(
n
)
=
{
p
1
(
n
)
,
p
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
} satisfying
(44)
ln
⁡
x
i
*
≤
p
i
(
n
)
<
ln
⁡
x
i
*
,
u
i
*
≤
u
i
(
n
)
≤
u
i
*
,
ln
⁡
x
i
*
≤
p
i
(
n
)
<
ln
⁡
x
i
*
,
i
=
1,2
,
n
∈
ℤ
+
.
Thus,
(45)
|
p
i
(
n
)
|
≤
A
i
,
|
u
i
(
n
)
|
≤
B
i
,
A
i
=
max
⁡
⁡
{
|
ln
⁡
x
i
*
|
,
|
ln
⁡
x
i
*
|
}
,
B
i
=
max
⁡
⁡
{
u
i
*
,
u
i
*
}
,
i
=
1,2
.
For (
X
,
U
)
∈
ℝ
2
+
2, we define a norm
(46)
∥
(
X
,
U
)
∥
=
∑
i
=
1
2
|
x
i
|
+
∑
i
=
1
2
|
u
i
|
.
For the product form of system (43)
(47)
p
i
(
n
+
1
)
=
p
i
(
n
)
+
r
i
(
n
)
-
a
i
i
(
n
)
exp
⁡
p
i
(
n
)
-
a
i
j
(
n
)
exp
⁡
p
j
(
n
)
-
b
i
(
n
)
exp
⁡
p
i
(
n
)
exp
⁡
p
j
(
n
)
-
c
i
(
n
)
u
i
(
n
)
,
Δ
u
i
(
n
)
=
-
d
i
(
n
)
u
i
(
n
)
+
e
i
(
n
)
exp
⁡
p
i
(
n
)
,
q
i
(
n
+
1
)
=
q
i
(
n
)
+
r
i
(
n
)
-
a
i
i
(
n
)
exp
⁡
q
i
(
n
)
-
a
i
j
(
n
)
exp
⁡
q
j
(
n
)
-
b
i
(
n
)
exp
⁡
q
i
(
n
)
exp
⁡
q
j
(
n
)
-
c
i
(
n
)
v
i
(
n
)
,
Δ
v
i
(
n
)
=
-
d
i
(
n
)
v
i
(
n
)
+
e
i
(
n
)
exp
⁡
q
i
(
n
)
.
Let Z
=
{
p
1
(
n
)
,
p
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
} and W
=
{
q
1
(
n
)
,
q
2
(
n
)
,
v
1
(
n
)
,
v
2
(
n
)
} be any two solutions of system (30) defined on ℤ
+
×
S
*
×
S
*; then ∥
Z
∥
≤
B and ∥
W
∥
≤
B, where B
=
∑
i
=
1
2
{
A
i
+
B
i
}, and
(48)
S
*
=
{
Y
(
n
)
∣
ln
⁡
x
i
*
≤
p
i
(
n
)
≤
ln
⁡
x
i
*
,
d
u
i
*
≤
u
i
(
n
)
≤
u
i
*
,
i
=
1,2
}
.
For convenience, we denote
(49)
M
i
(
n
)
=
p
i
(
n
)
-
q
i
(
n
)
,
N
i
(
n
)
=
u
i
(
n
)
-
v
i
(
n
)
,
M
~
i
(
n
)
=
p
~
i
(
n
)
-
q
~
i
(
n
)
,
N
~
i
(
n
)
=
u
~
i
(
n
)
-
v
~
i
(
n
)
.
Let
(50)
V
(
n
,
Z
,
W
)
=
∑
i
=
1
2
M
i
2
(
n
)
+
N
i
2
(
n
)
be a function on ℤ
+
×
S
*
×
S
*.
Then(51)
∥
Z
-
W
∥
=
∑
i
=
1
2
{
|
M
i
(
n
)
|
+
|
N
i
(
n
)
|
}
,
∥
Z
-
W
∥
*
=
{
∑
i
=
1
2
{
M
i
2
(
n
)
+
N
i
2
(
n
)
}
}
1
/
2
are equivalent. It is easy to see that
(52)
(
C
1
∥
Z
-
W
∥
)
2
≤
V
(
n
,
Z
,
W
)
≤
(
C
2
∥
Z
-
W
∥
)
2
.
Let a
∈
C
(
ℝ
+
,
ℝ
+
), a
(
x
)
=
C
1
2
x
2, b
∈
C
(
ℝ
+
,
ℝ
+
), and b
(
x
)
=
C
2
2
x
2; thus condition (i) in Lemma 10 is satisfied.
In addition,(53)
|
V
(
n
,
Z
,
W
)
-
V
(
n
,
Z
~
,
W
~
)
|
=
|
∑
i
=
1
2
(
M
i
2
+
N
i
2
)
-
∑
i
=
1
2
(
M
~
i
2
+
N
~
i
2
)
|
≤
∑
i
=
1
2
{
|
M
i
(
n
)
+
M
~
i
(
n
)
|
·
|
M
i
(
n
)
-
M
~
i
(
n
)
|
}
+
∑
i
=
1
2
{
|
N
i
(
n
)
+
N
~
i
(
n
)
|
·
|
N
i
(
n
)
-
N
~
i
(
n
)
|
}
≤
L
{
∑
i
=
1
2
(
|
M
i
(
n
)
|
+
|
M
~
i
(
n
)
|
)
+
∑
i
=
1
2
(
|
N
i
(
n
)
|
+
|
N
~
i
(
n
)
|
)
}
=
L
{
∥
Z
-
Z
~
∥
+
∥
W
-
W
~
∥
}
,
where L
=
4
max
⁡
{
A
i
,
B
i
}, i
=
1,2. And  Lemma 10 (ii) is held.
Finally, calculating theΔ
V along the solutions of (47) leads to
(54)
Δ
V
(
12
)
(
n
)
=
∑
i
=
1
2
{
M
i
+
1
2
(
n
)
+
N
i
+
1
2
(
n
)
}
-
∑
i
=
1
2
{
M
i
2
(
n
)
+
N
i
2
(
n
)
}
=
∑
i
=
1
2
[
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
2
+
e
i
2
(
n
)
]
(
e
p
i
(
n
)
-
e
q
i
(
n
)
)
2
+
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
i
(
n
)
]
2
(
e
p
j
(
n
)
-
e
q
j
(
n
)
)
2
+
2
[
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
]
·
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
i
(
n
)
]
(
e
p
i
(
n
)
-
e
q
i
(
n
)
)
×
(
e
p
j
(
n
)
-
e
q
j
(
n
)
)
-
2
M
i
(
n
)
·
[
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
]
(
e
p
i
(
n
)
-
e
q
i
(
n
)
)
-
2
M
i
(
n
)
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
j
(
n
)
]
·
(
e
p
j
(
n
)
-
e
q
j
(
n
)
)
+
2
c
i
M
i
(
n
)
N
i
(
n
)
+
2
[
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
(
1
-
d
i
(
n
)
)
e
i
(
n
)
+
c
i
(
n
)
f
f
f
f
f
f
·
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
]
N
i
(
n
)
(
e
p
i
(
n
)
-
e
q
i
(
n
)
)
+
2
c
i
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
i
(
n
)
]
·
N
i
(
n
)
(
e
p
j
(
n
)
-
e
q
j
(
n
)
)
+
[
d
i
2
(
n
)
-
2
d
i
(
n
)
+
c
i
(
n
)
]
2
N
i
2
(
n
)
.
By the mean value theorem we get e
p
i
(
n
)
-
e
q
i
(
n
)
=
ξ
(
n
)
M
i
(
n
), where ξ
(
n
) lies between e
p
i
(
n
) and e
q
i
(
n
).
From (49), (54), one has
(55)
Δ
V
(
12
)
(
n
)
=
∑
i
=
1
2
{
[
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
2
+
e
i
2
(
n
)
]
f
f
f
f
f
f
f
×
ξ
i
2
[
(
p
i
(
n
)
-
q
i
(
n
)
]
2
f
f
f
f
f
f
f
+
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
i
(
n
)
]
2
ξ
j
2
[
p
j
(
n
)
-
q
j
(
n
)
]
2
f
f
f
f
f
f
f
+
2
[
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
]
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
i
(
n
)
]
f
f
f
f
f
f
f
·
ξ
i
ξ
j
[
p
i
(
n
)
-
q
i
(
n
)
]
[
p
j
(
n
)
-
q
j
(
n
)
]
f
f
f
f
f
f
f
-
2
[
p
i
(
n
)
-
q
i
(
n
)
]
[
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
]
f
f
f
f
f
f
f
×
ξ
i
[
p
i
(
n
)
-
q
i
(
n
)
]
f
f
f
f
f
f
f
+
2
[
p
i
(
n
)
-
q
i
(
n
)
]
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
j
(
n
)
]
f
f
f
f
f
f
f
×
ξ
j
[
p
j
(
n
)
-
q
j
(
n
)
]
f
f
f
f
f
f
f
+
2
c
i
[
p
i
(
n
)
-
q
i
(
n
)
]
[
u
i
(
n
)
-
v
i
(
n
)
]
f
f
f
f
f
f
f
+
2
[
(
1
-
d
i
(
n
)
)
e
i
(
n
)
+
c
i
(
n
)
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
]
f
f
f
f
f
f
f
·
(
u
i
(
n
)
-
v
i
(
n
)
)
ξ
i
[
p
i
(
n
)
-
q
i
(
n
)
]
f
f
f
f
f
f
f
+
2
c
i
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
i
(
n
)
]
(
u
i
(
n
)
-
v
i
(
n
)
)
f
f
f
f
f
f
f
×
ξ
j
[
p
j
(
n
)
-
q
j
(
n
)
]
f
f
f
f
f
f
f
+
[
d
i
2
(
n
)
-
2
d
i
(
n
)
+
c
i
(
n
)
]
2
(
u
i
(
n
)
-
v
i
(
n
)
)
2
[
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
2
+
e
i
2
(
n
)
]
}
=
∑
i
=
1
2
{
{
[
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
2
+
e
i
2
(
n
)
]
ξ
i
2
f
f
f
f
f
f
f
f
-
2
[
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
ξ
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
2
]
}
f
f
f
f
f
f
f
·
M
i
2
(
n
)
+
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
i
(
n
)
]
2
ξ
j
2
M
j
2
(
n
)
f
f
f
f
f
f
f
+
(
2
[
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
]
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
i
(
n
)
]
ξ
i
ξ
j
f
f
f
f
f
f
f
f
f
f
-
2
[
a
i
j
(
n
)
+
b
i
(
n
)
·
e
q
j
(
n
)
]
ξ
j
)
M
i
(
n
)
M
j
(
n
)
f
f
f
f
f
f
f
+
[
d
i
2
(
n
)
-
2
d
i
(
n
)
+
c
i
(
n
)
]
2
N
i
2
(
n
)
f
f
f
f
f
f
f
+
2
[
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
c
i
+
[
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
(
1
-
d
i
(
n
)
)
e
i
(
n
)
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
+
c
i
(
n
)
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
]
ξ
i
]
f
f
f
f
f
f
f
×
M
i
(
n
)
N
i
(
n
)
+
2
c
i
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
i
(
n
)
]
f
f
f
f
f
f
f
×
N
i
(
n
)
ξ
j
[
p
j
(
n
)
-
q
j
(
n
)
]
[
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
2
+
e
i
2
(
n
)
]
}
.
We get for j
=
1,2 that
(56)
Δ
V
(
12
)
(
n
)
≤
∑
i
=
1
2
{
V
1
i
j
+
V
2
i
j
+
V
3
i
j
+
V
4
i
j
+
V
5
i
j
+
V
6
i
j
}
,
where
(57)
V
1
i
j
=
{
[
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
2
+
e
i
2
(
n
)
]
ξ
i
2
f
f
f
f
-
2
[
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
ξ
i
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
2
]
}
M
i
2
(
n
)
≤
{
(
a
i
i
u
+
b
i
u
x
j
*
)
2
+
e
i
2
u
)
x
i
*
2
-
2
(
a
i
i
l
+
b
i
l
x
i
*
)
x
i
*
}
M
i
2
(
n
)
;
V
2
i
j
=
{
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
i
(
n
)
]
2
ξ
j
2
}
M
j
2
(
n
)
,
≤
{
[
a
i
j
u
+
b
i
u
x
i
*
]
2
x
j
*
2
}
M
j
2
(
n
)
;
V
3
i
j
=
{
2
[
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
]
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
i
(
n
)
]
ξ
i
ξ
j
f
f
f
-
2
[
a
i
j
(
n
)
+
b
i
(
n
)
e
q
j
(
n
)
]
ξ
j
}
M
i
(
n
)
M
j
(
n
)
≤
{
(
a
i
i
u
+
b
i
u
x
j
*
)
(
a
i
j
u
+
b
i
u
x
i
*
)
x
i
*
x
j
*
-
(
a
i
j
u
+
b
i
u
x
i
*
)
}
×
[
M
i
2
(
n
)
+
M
j
2
(
n
)
]
;
V
4
i
j
=
{
[
(
1
-
d
i
l
)
e
i
u
+
c
i
l
(
a
i
i
l
+
b
i
l
x
j
*
)
]
x
i
*
2
c
i
+
2
[
c
i
(
n
)
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
(
1
-
d
i
(
n
)
)
e
i
(
n
)
f
f
f
f
f
f
f
f
f
f
f
+
c
i
(
n
)
(
a
i
i
(
n
)
+
b
i
(
n
)
e
p
j
(
n
)
)
]
ξ
i
}
×
M
i
(
n
)
N
i
(
n
)
≤
{
c
i
u
+
[
(
1
-
d
i
l
)
e
i
u
+
c
i
l
(
a
i
i
l
+
b
i
l
x
j
*
)
]
x
i
*
}
×
[
M
i
2
(
n
)
+
N
i
2
(
n
)
]
;
V
5
i
j
=
{
2
c
i
[
a
i
j
(
n
)
+
b
j
(
n
)
e
q
i
(
n
)
]
ξ
j
}
M
j
(
n
)
N
i
(
n
)
≤
{
c
i
u
(
a
i
j
u
+
b
j
u
x
i
*
)
x
j
*
}
[
M
j
2
(
n
)
+
N
i
2
(
n
)
]
;
V
6
i
j
=
{
[
d
i
2
(
n
)
-
2
d
i
(
n
)
+
c
i
(
n
)
]
2
}
N
i
2
(
n
)
≤
{
d
i
2
u
-
2
d
i
l
+
c
i
u
}
N
i
2
(
n
)
.
Hence,
(58)
Δ
V
(
12
)
(
n
)
≤
∑
i
=
1
2
{
[
[
(
a
i
i
u
+
b
i
u
x
j
*
)
2
+
e
i
2
u
]
x
i
*
2
-
2
(
a
i
i
l
+
b
i
l
x
i
*
)
x
i
*
]
f
f
f
f
f
f
·
[
(
p
i
(
n
)
-
q
i
(
n
)
]
2
+
[
[
a
i
j
u
+
b
i
u
x
i
*
]
2
x
j
*
2
]
M
j
2
(
n
)
f
f
f
f
f
f
+
[
(
a
i
i
u
+
b
i
u
x
j
*
)
(
a
i
j
u
+
b
i
u
x
i
*
)
x
i
*
x
j
*
-
(
a
i
j
u
+
b
i
u
x
i
*
)
]
f
f
f
f
f
f
×
M
i
2
(
n
)
M
j
2
(
n
)
f
f
f
f
f
f
+
[
c
i
u
+
[
(
1
-
d
i
l
)
e
i
u
-
c
i
l
(
a
i
i
l
+
b
i
l
x
j
*
)
]
x
i
*
]
d
d
d
d
d
×
M
i
2
(
n
)
N
i
2
(
n
)
f
f
f
f
f
f
+
[
c
i
u
(
a
i
j
u
+
b
i
u
x
i
*
)
x
j
*
]
[
M
j
2
(
n
)
+
N
i
2
(
n
)
]
f
f
f
f
f
f
+
[
d
i
2
u
-
2
d
i
l
+
c
i
u
]
N
i
2
(
n
)
[
[
(
a
i
i
u
+
b
i
u
x
j
*
)
2
+
e
i
2
u
]
x
i
*
2
-
2
(
a
i
i
l
+
b
i
l
x
i
*
)
x
i
*
]
}
=
-
∑
i
=
1
2
{
[
2
(
a
i
i
l
+
b
i
l
x
i
*
)
-
[
(
a
i
i
u
+
b
i
u
x
j
*
)
2
+
e
i
2
u
]
x
i
*
2
d
d
d
d
d
d
d
-
[
a
j
i
u
+
b
j
u
x
j
*
]
2
x
i
*
2
+
(
a
i
j
u
+
b
i
u
x
i
*
)
+
(
a
j
i
u
+
b
j
u
x
j
*
)
d
d
d
d
d
d
d
-
(
a
i
i
u
+
b
i
u
x
j
*
)
(
a
i
j
u
+
b
i
u
x
i
*
)
x
i
*
x
j
*
d
d
d
d
d
d
d
-
(
a
j
j
u
+
b
j
u
x
i
*
)
(
a
j
i
u
+
b
j
u
x
j
*
)
x
i
*
x
j
*
d
d
d
d
d
d
d
+
c
i
l
(
a
i
i
l
+
b
i
l
x
j
*
)
x
i
*
-
c
i
u
-
(
1
-
d
i
l
)
e
i
u
x
i
*
dddddddd
-
c
j
u
(
a
j
i
u
+
b
i
u
x
i
*
)
x
i
*
[
(
a
i
i
u
+
b
i
u
x
j
*
)
2
+
e
i
2
u
]
]
·
M
i
2
(
n
)
d
d
d
d
d
d
d
+
[
-
c
i
u
-
(
1
-
d
i
l
)
e
i
u
x
i
*
+
c
i
l
(
a
i
i
l
+
b
i
l
x
j
*
)
x
i
*
d
d
d
d
d
d
d
d
d
d
-
c
i
u
(
a
i
j
u
+
b
i
u
x
i
*
)
x
j
*
-
d
i
2
u
+
2
d
i
l
-
c
i
u
]
N
i
2
(
n
)
[
(
a
i
i
u
+
b
i
u
x
j
*
)
2
+
e
i
2
u
]
}
≤
-
∑
i
=
1
2
[
r
i
j
M
i
2
(
n
)
+
r
i
j
*
N
i
2
(
n
)
]
≤
-
β
∑
i
=
1
2
[
M
i
2
(
n
)
+
N
i
2
(
n
)
]
≤
-
β
V
(
n
)
,
where β
=
min
⁡
{
r
i
j
,
r
i
j
*
}, i
,
j
=
1,2, i
≠
j. That is, there exists a positive constant 0
<
β
<
1 such that
(59)
Δ
V
(
12
)
(
n
)
≤
-
β
V
(
n
)
.
From 0
<
β
<
1, condition (iii) of Lemma 10 is satisfied. So, from Lemma 10, there exists a uniqueness uniformly asymptotically stable almost periodic solution X
(
n
)
=
{
x
1
(
n
)
,
x
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
} of system (43) which is bounded by S
* for all n
∈
ℤ
+, which means that there exists a uniqueness uniformly asymptotically stable almost periodic solution  X
(
n
)
=
{
x
1
(
n
)
,
x
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
} of system (1) which is bounded by Ω for all n
∈
ℤ
+. This completes the proof.



---

*Source: 824801-2014-01-23.xml*