Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 182158, 9 pages
http://dx.doi.org/10.1155/2013/182158
Research Article
Uniformly Asymptotic Stability of Positive Almost Periodic
Solutions for a Discrete Competitive System
Qinglong Wang and Zhijun Liu
Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China
Correspondence should be addressed to Zhijun Liu; zhijun liu47@hotmail.com
Received 11 February 2013; Accepted 26 April 2013
Academic Editor: Yansheng Liu
Copyright © 2013 Q. Wang and Z. Liu. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is devoted to the study of almost periodic solutions of a discrete two-species competitive system. With the help of the
methods of the Lyapunov function, some analysis techniques, and preliminary lemmas, we establish a criterion for the existence,
uniqueness, and uniformly asymptotic stability of positive almost periodic solution of the system. Numerical simulations are
presented to illustrate the analytical results.
1. Introduction
In recent years, many works have been done for the difference
system (see [1–14] and the references cited therein) since the
discrete time models governed by the difference equation are
more appropriate than the continuous ones when the populations have a short life expectancy, nonoverlapping generations in the real world. In particular, Qin et al. [1] introduced
the following discrete Lotka-Volterra competitive system:
𝑥1 (𝑛 + 1)
= 𝑥1 (𝑛) exp [𝑟1 (𝑛) − 𝑎1 (𝑛) 𝑥1 (𝑛) −
𝑐2 (𝑛) 𝑥2 (𝑛)
],
1 + 𝑥2 (𝑛)
𝑥2 (𝑛 + 1)
= 𝑥2 (𝑛) exp [𝑟2 (𝑛) − 𝑎2 (𝑛) 𝑥2 (𝑛) −
𝑐1 (𝑛) 𝑥1 (𝑛)
],
1 + 𝑥1 (𝑛)
𝑛 = 0, 1, 2 . . . ,
investigated the permanence and global asymptotic stability
of positive periodic solutions of system (1).
Notice that the investigation of almost periodic solutions
for difference equations is one of most important topics in
the qualitative theory of difference equations due to the
applications in biology, ecology, neural network, and so forth
(see [10–14] in detail), and few work has been done previously
on an almost periodic version which is corresponding to periodic system (1). In this paper, we will further investigate the
existence, uniqueness, and uniformly asymptotic stability of
positive almost periodic solution of the above almost periodic
version. To this end, we assume that the coefficients of system
(1) {𝑟𝑖 (𝑛)}, {𝑎𝑖 (𝑛)} and {𝑐𝑖 (𝑛)} are bounded nonnegative almost
periodic sequences.
For the sake of simplicity and convenience in the following discussion, the notations below will be used throughout
this paper:
𝑓𝑈 = sup {𝑓 (𝑛)} ,
𝑛∈Z+
(1)
where 𝑥𝑖 (0) > 0, 𝑥𝑖 (𝑛) stand for the densities of species 𝑥𝑖
at the 𝑛th generation, 𝑟𝑖 (𝑛) represent the natural growth rates
of species 𝑥𝑖 at the 𝑛th generation, 𝑎𝑖 (𝑛) are the intraspecific
effects of the 𝑛th generation of species 𝑥𝑖 on own population, and 𝑐𝑖 (𝑛) measure the interspecific effects of the 𝑛th
generation of species 𝑥𝑖 on species 𝑥𝑗 (𝑖, 𝑗 = 1, 2; 𝑖 ≠ 𝑗). They
𝑓𝐿 = inf+ {𝑓 (𝑛)} ,
𝑛∈Z
(2)
where {𝑓(𝑛)} is a bounded sequence and Z+ = {0, 1, 2, . . .}.
The remaining part of this paper is organized as follows.
In the next section, we introduce some notations, definitions,
and lemmas which are available for our main results. In
Section 3, sufficient conditions for the existence, uniqueness,
and uniformly asymptotic stability of positive almost periodic
solution of system (1) are given. Numerical simulations are
Journal of Applied Mathematics
1.25
1.25
1.2
1.2
𝑥1∗ (𝑛)
𝑥1∗ (𝑛)
2
1.15
1.1
1.05
1.15
1.1
0
20
40
60
80
1.05
700
100
720
740
1.25
1.25
1.2
1.2
1.15
1.15
1.1
1.1
1.05
1.05
0
20
780
800
760
780
800
(b)
𝑥2∗ (𝑛)
𝑥2∗ (𝑛)
(a)
1
760
𝑛
𝑛
40
60
80
100
1
700
720
740
𝑛
𝑛
(c)
(d)
𝑥1∗ (𝑛)
Figure 1: Positive almost periodic solution of system (13). (a), (c) Time-series
and
with initial values 𝑥1∗ (0) = 1.19, 𝑥2∗ (0) = 1.18
∗
∗
for 𝑛 ∈ [0, 100], respectively. (b), (d) Time-series 𝑥1 (𝑛) and 𝑥2 (𝑛) with the above initial values for 𝑛 ∈ [700, 800], respectively.
carried out to substantiate the above analytical results in
Section 4. Finally, we give some proofs of theorems in the
appendices for convenience in reading this paper.
𝑥2∗ (𝑛)
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
|𝑥 (𝑛 + 𝜏) − 𝑥 (𝑛)| < 𝜀,
2. Preliminaries
∀𝑛 ∈ Z.
In this section, we will need some preparations and give some
notations, definitions, and lemmas which will be useful for
our main results.
Denote by R, R+ , Z, and Z+ the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers,
respectively. R2 and R𝑘 denote the cone of 2-dimensional and
𝑘-dimensional real Euclidean space, respectively.
𝜏 is called the 𝜀-translation number of 𝑥(𝑛).
Definition 1 (see [13]). A sequence 𝑥 : Z → R𝑘 is called an
almost periodic sequence if the following 𝜀-translation set of
𝑥
󵄨
󵄨󵄨
󵄨󵄨𝑓 (𝑛 + 𝜏, 𝑥) − 𝑓 (𝑛, 𝑥)󵄨󵄨󵄨 < 𝜀,
𝐸 {𝜀, 𝑥} := {𝜏 ∈ Z : |𝑥 (𝑛 + 𝜏) − 𝑥 (𝑛)| < 𝜀, ∀𝑛 ∈ Z}
(3)
(4)
Definition 2 (see [13]). Let 𝑓 : Z × D → R𝑘 , where D is
an open set in R𝑘 . 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 an integer 𝜏 for which
(5)
for all 𝑛 ∈ Z and all 𝑥 ∈ S. 𝜏 is called the 𝜀-translation number
of 𝑓(𝑛, 𝑥).
3
1.22
1.22
1.2
1.2
1.18
1.18
1.16
1.16
1.14
1.14
𝑥2∗ (𝑛)
𝑥2∗ (𝑛)
Journal of Applied Mathematics
1.12
1.12
1.1
1.1
1.08
1.08
1.06
1.06
1.04
1.08
1.1
1.12
1.14
1.16
𝑥1∗ (𝑛)
1.18
1.2
1.04
1.08
1.22
1.1
1.12
1.25
1.25
1.2
1.2
1.15
1.15
1.1
1.2
1.22
1.1
1.05
1.05
1
1.25
1.18
(b)
𝑥2∗ (𝑛)
𝑥2∗ (𝑛)
(a)
1.14
1.16
𝑥1∗ (𝑛)
1
1.25
1.2
1.15
𝑥1∗(
𝑛)
1.1
1.05 0
20
40
60
80
100
1.2
1.15
𝑥∗
1 (𝑛
)
𝑛
(c)
1.1
1.05 700
720
740
760
780
800
𝑛
(d)
Figure 2: Phase portrait. (a), (b) 2-dimensional phase portrait of almost periodic system (13). Time-series 𝑥1∗ (𝑛) and 𝑥2∗ (𝑛) with initial values
𝑥1∗ (0) = 1.19, 𝑥2∗ (0) = 1.18 for 𝑛 ∈ [0, 100] and 𝑛 ∈ [700, 800], respectively. (c), (d) 3-dimensional phase portrait of almost periodic system
(13). Time-series 𝑥1∗ (𝑛) and 𝑥2∗ (𝑛) with the above initial values for 𝑛 ∈ [0, 100] and 𝑛 ∈ [700, 800], respectively.
Lemma 3 (see [13]). {𝑥(𝑛)} is an almost periodic sequence if
and only if for any sequence {ℎ𝑘󸀠 } ⊂ Z there exists a subsequence
{ℎ𝑘 } ⊂ {ℎ𝑘󸀠 } such that 𝑥(𝑛 + ℎ𝑘 ) converges uniformly on 𝑛 ∈ Z
as 𝑘 → ∞. Furthermore, the limit sequence is also an almost
periodic sequence.
Consider the following almost periodic difference system:
𝑥 (𝑛 + 1) = 𝑓 (𝑛, 𝑥 (𝑛)) ,
+
𝑘
𝑛 ∈ Z+ ,
(6)
𝑘
where 𝑓 : Z × S𝐵 → R , S𝐵 = {𝑥 ∈ R : ‖𝑥‖ < 𝐵}, and
𝑓(𝑛, 𝑥) is almost periodic in 𝑛 uniformly for 𝑥 ∈ S𝐵 and is
continuous in 𝑥. The product system of (6) is the following
system:
𝑥 (𝑛 + 1) = 𝑓 (𝑛, 𝑥 (𝑛)) ,
𝑦 (𝑛 + 1) = 𝑓 (𝑛, 𝑦 (𝑛)) , (7)
and Zhang [14] obtained the following lemma.
Lemma 4 (see [14]). Suppose that there exists a Lyapunov
function 𝑉(𝑛, 𝑥, 𝑦) defined for 𝑛 ∈ Z+ , ‖𝑥‖ < 𝐵, ‖𝑦‖ < 𝐵
satisfying the following conditions:
(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, 𝑥1 , 𝑦1 ) − 𝑉(𝑛, 𝑥2 , 𝑦2 )| ≤ 𝐿(‖𝑥1 − 𝑥2 ‖ + ‖𝑦1 − 𝑦2 ‖),
where 𝐿 > 0 is a constant;
(iii) Δ𝑉(2.2) (𝑛, 𝑥, 𝑦) ≤ −𝛽𝑉(𝑛, 𝑥, 𝑦), where 0 < 𝛽 < 1 is a
constant and
Δ𝑉(2.2) (𝑛, 𝑥, 𝑦) = 𝑉 (𝑛 + 1, 𝑓 (𝑛, 𝑥) , 𝑓 (𝑛, 𝑦)) − 𝑉 (𝑛, 𝑥, 𝑦) .
(8)
Moreover, if there exists a solution 𝜑(𝑛) of system (6) such that
‖𝜑(𝑛)‖ ≤ 𝐵∗ < 𝐵 for 𝑛 ∈ Z+ , then there exists a unique uniformly asymptotically stable almost periodic solution 𝑝(𝑛) of
system (6) which satisfies ‖𝑝(𝑛)‖ ≤ 𝐵∗ . In particular, if 𝑓(𝑛, 𝑥)
Journal of Applied Mathematics
1.25
1.25
1.2
1.2
𝑥1∗ (𝑛), 𝑥1 (𝑛)
𝑥1∗ (𝑛), 𝑥1 (𝑛)
4
1.15
1.1
1.1
1.05
1.15
0
20
40
80
60
1.05
700
100
720
740
(a)
800
760
780
800
(b)
1.25
1.25
1.2
1.2
1.15
𝑥2∗ (𝑛), 𝑥2 (𝑛)
𝑥2∗ (𝑛), 𝑥2 (𝑛)
780
𝑥1∗ (𝑛)
𝑥1 (𝑛)
𝑥1∗ (𝑛)
𝑥1 (𝑛)
1.1
1.05
1
760
𝑛
𝑛
1.15
1.1
1.05
0
20
40
80
60
100
1
700
720
740
𝑛
𝑛
𝑥2∗ (𝑛)
𝑥2 (𝑛)
𝑥2∗ (𝑛)
𝑥2 (𝑛)
(c)
(d)
𝑥1∗ (𝑛)
Figure 3: Uniformly asymptotic stability. (a), (c) Time-series
𝑥2 (𝑛) with initial values 𝑥1 (0) = 1.06, 𝑥2 (0) = 1.03 for 𝑛 ∈ [0, 100],
above initial values for 𝑛 ∈ [700, 800], respectively.
is periodic of period 𝜔, then there exists a unique uniformly
asymptotically stable periodic solution of system (6) of periodic
𝜔.
Lemma 5 (see [1]). Any positive solution (𝑥1 (𝑛), 𝑥2 (𝑛)) of system (1) satisfies
lim sup 𝑥𝑖 (𝑛) ≤
𝑛 → +∞
𝑈
def [exp (𝑟𝑖
𝑀𝑖 =
𝑎𝑖𝐿
− 1)]
𝑟2𝐿 > 𝑐1𝑈.
and
with initial values 𝑥1∗ (0) = 1.19, 𝑥2∗ (0) = 1.18
respectively. (b), (d) Time-series 𝑥1∗ (𝑛), 𝑥2∗ (𝑛), 𝑥1 (𝑛), and
and 𝑥1 (𝑛) and
𝑥2 (𝑛) with the
Then, any positive solution (𝑥1 (𝑛), 𝑥2 (𝑛)) of system (1) satisfies
def
lim inf 𝑥𝑖 (𝑛) ≥ 𝑚𝑖 =
𝑛 → +∞
𝑟𝑖𝐿 − 𝑐𝑗𝑈
𝑎𝑖𝑈
exp (𝑟𝑖𝐿 − 𝑎𝑖𝑈𝑀𝑖 − 𝑐𝑗𝑈) ,
𝑖 ≠ 𝑗; 𝑖, 𝑗 = 1, 2.
(11)
,
𝑖 = 1, 2.
(9)
Lemma 6 (see [1]). Suppose that system (1) satisfies the following assumptions:
𝑟1𝐿 > 𝑐2𝑈,
𝑥2∗ (𝑛)
(10)
3. Main Result
From (9) and (11), we denote by Ω the set of all solutions
(𝑥1 (𝑛), 𝑥2 (𝑛)) of system (1) satisfying 𝑚𝑖 ≤ 𝑥𝑖 (𝑛) ≤ 𝑀𝑖 , 𝑖 =
1, 2 for all 𝑛 ∈ Z+ . According to Lemma 4, we first prove that
Journal of Applied Mathematics
5
there is a bounded solution of system (1), and then structure
a suitable Lyapunov function for system (1).
Theorem 7. If the assumptions in (10) hold, then Ω ≠ Φ.
A computation shows that
𝑟1𝐿 − 𝑐2𝑈 = 1.1530 > 0,
Theorem 8. If the assumptions in (10) are satisfied, furthermore, 0 < 𝛽 < 1, where 𝛽 = min{𝑠1 , 𝑠2 }, and
−
2
(1 + 𝑚2 )
2
(1 + 𝑚1 )
𝑠2 = 2𝑎2𝐿 𝑚2 − 𝑎2𝑈2 𝑀22 −
−
(1 + 𝑎1𝑈𝑀1 ) 𝑐2𝑈𝑀2
(1 + 𝑎2𝑈𝑀2 ) 𝑐1𝑈𝑀1
−
𝑐1𝑈2 𝑀12
4
(1 + 𝑚1 )
,
(1 + 𝑎2𝑈𝑀2 ) 𝑐1𝑈𝑀1
2
(1 + 𝑚2 )
(12)
2
(1 + 𝑚1 )
(1 + 𝑎1𝑈𝑀1 ) 𝑐2𝑈𝑀2
−
𝑐2𝑈2 𝑀22
(14)
and moreover, we have
The proof of Theorem 7 is given in Appendix A.
𝑠1 = 2𝑎1𝐿 𝑚1 − 𝑎1𝑈2 𝑀12 −
𝑟2𝐿 − 𝑐1𝑈 = 1.0900 > 0,
𝑠1 = 0.3515,
𝑠2 = 0.2502,
(15)
that is, 0 < 𝛽 = min{𝑠1 , 𝑠2 } = 0.2502 < 1. It is easy to see
that the assumptions of Theorem 8 are satisfied. Hence, in
system (13) there exists a unique uniformly asymptotically
stable positive almost periodic solution. From Figure 1, it is
easy to see that there exists a positive almost periodic solution
(𝑥1∗ (𝑡), 𝑥2∗ (𝑡)), and the 2-dimensional and 3-dimensional
phase portraits of almost periodic system (13) are revealed
in Figure 2, respectively. Figure 3 shows that any positive
solution (𝑥1 (𝑛), 𝑥2 (𝑛)) tends to the almost periodic solution
(𝑥1∗ (𝑛), 𝑥2∗ (𝑛)).
Appendices
4
(1 + 𝑚2 )
A. Proof of Theorem 7
,
then there exists a unique uniformly asymptotically stable
almost periodic solution of system (1) which is bounded by Ω
for all 𝑛 ∈ Z+ .
The proof of Theorem 8 is given in Appendix B.
Clearly, by an inductive argument we have from system (1)
that
𝑛−1
𝑥1 (𝑛) = 𝑥1 (0) exp ∑ [𝑟1 (𝑙) − 𝑎1 (𝑙) 𝑥1 (𝑙) −
𝑙=0
𝑛−1
𝑥2 (𝑛) = 𝑥2 (0) exp ∑ [𝑟2 (𝑙) − 𝑎2 (𝑙) 𝑥2 (𝑙) −
𝑙=0
4. Numerical Simulations
In this section, we give the following example to check the
feasibility of the assumptions of Theorem 8.
Example 9. Consider the following discrete system:
𝑐1 (𝑙) 𝑥1 (𝑙)
].
1 + 𝑥1 (𝑙)
(A.1)
According to Lemmas 5 and 6, for any solution (𝑥1 (𝑛), 𝑥2 (𝑛))
of system (1) and an arbitrarily small constant 𝜀 > 0, there
exists 𝑛0 sufficiently large such that
𝑚1 − 𝜀 ≤ 𝑥1 (𝑛) ≤ 𝑀1 + 𝜀,
𝑥1 (𝑛 + 1)
𝑐2 (𝑙) 𝑥2 (𝑙)
],
1 + 𝑥2 (𝑙)
𝑚2 − 𝜀 ≤ 𝑥2 (𝑛) ≤ 𝑀2 + 𝜀,
∀𝑛 ≥ 𝑛0 .
= 𝑥1 (𝑛) exp [1.20 − 0.02 sin (√2𝑛𝜋)
(A.2)
− (1.05 + 0.01 sin (√2𝑛𝜋)) 𝑥1 (𝑛)
−
(0.025 + 0.002 cos (√2𝑛𝜋)) 𝑥2 (𝑛)
1 + 𝑥2 (𝑛)
],
𝑥2 (𝑛 + 1)
= 𝑥2 (𝑛) exp [1.15 − 0.02 cos (√2𝑛𝜋)
− (1.02 + 0.02 cos (√2𝑛𝜋)) 𝑥2 (𝑛)
−
(0.035 + 0.005 sin (√3𝑛𝜋)) 𝑥1 (𝑛)
1 + 𝑥1 (𝑛)
].
(13)
Set {𝜏𝑘 } be any positive integer sequence such that 𝜏𝑘 → +∞
as 𝑘 → +∞, we can show that there exists a subsequence of
{𝜏𝑘 } still denoted by {𝜏𝑘 }, such that 𝑥𝑖 (𝑛+𝜏𝑘 ) → 𝑥𝑖∗ (𝑛), 𝑖 = 1, 2
uniformly in 𝑛 on any finite subset 𝐶 of Z+ as 𝑘 → +∞,
where 𝐶 = {𝑎1 , 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑚 }, 𝑎ℎ ∈ Z+ (ℎ = 1, 2 ⋅ ⋅ ⋅ 𝑚), and 𝑚 is a
finite number.
As a matter of fact, for any finite subset 𝐶 ⊂ Z+ , 𝜏𝑘 + 𝑎ℎ >
𝑛0 , ℎ = 1, 2 ⋅ ⋅ ⋅ 𝑚, when 𝑘 is large enough. Therefore, 𝑚𝑖 − 𝜀 ≤
𝑥𝑖 (𝑛 + 𝜏𝑘 ) ≤ 𝑀𝑖 + 𝜀, 𝑖 = 1, 2; that is, {𝑥𝑖 (𝑛 + 𝜏𝑘 )} are uniformly
bounded for 𝑘 large enough.
Now, for 𝑎1 ∈ 𝐶, we can choose a subsequence {𝜏𝑘(1) } of
{𝜏𝑘 } such that {𝑥1 (𝑎1 +𝜏𝑘(1) )} and {𝑥2 (𝑎1 +𝜏𝑘(1) )} uniformly converge on Z+ for 𝑘 large enough.
Analogously, for 𝑎2 ∈ 𝐶, we can also choose a subsequence {𝜏𝑘(2) } of {𝜏𝑘(1) } such that {𝑥1 (𝑎2 + 𝜏𝑘(2) )} and {𝑥2 (𝑎2 +
𝜏𝑘(2) )} uniformly converge on Z+ for 𝑘 large enough.
6
Journal of Applied Mathematics
Repeating the above process, for 𝑎𝑚 ∈ 𝐶, we get a subsequence {𝜏𝑘(𝑚) } of {𝜏𝑘(𝑚−1) } such that {𝑥1 (𝑎𝑚 +𝜏𝑘(𝑚) )} and {𝑥2 (𝑎𝑚 +
𝜏𝑘(𝑚) )} uniformly converge on Z+ for 𝑘 large enough.
Now, we choose the sequence {𝜏𝑘(𝑚) } which is a subsequence of {𝜏𝑘 } denoted by {𝜏𝑘 }; then, for all 𝑛 ∈ 𝐶, we obtain
that 𝑥𝑖 (𝑛 + 𝜏𝑘 ) → 𝑥𝑖∗ (𝑛), 𝑖 = 1, 2 uniformly in 𝑛 ∈ 𝐶 as
𝑘 → +∞. Hence, the conclusion is valid by the arbitrary of
𝐶.
Recall the almost periodicity of {𝑟𝑖 (𝑛)}, {𝑎𝑖 (𝑛)} and {𝑐𝑖 (𝑛)},
𝑖 = 1, 2, for the above sequence {𝜏𝑘 }, 𝜏𝑘 → +∞ as 𝑘 → +∞,
there exists a subsequence denoted by {𝜏𝑘 } such that
𝑟𝑖 (𝑛 + 𝜏𝑘 ) 󳨀→ 𝑟𝑖 (𝑛) ,
𝑎𝑖 (𝑛 + 𝜏𝑘 ) 󳨀→ 𝑎𝑖 (𝑛) ,
(A.3)
Let 𝑘 → +∞, we have
𝑥1∗ (𝑛 + 𝛼)
𝑛+𝛼−1
= 𝑥1∗ (𝛼) exp ∑ [𝑟1 (𝑙) − 𝑎1 (𝑙) 𝑥1∗ (𝑙) −
𝑙=𝛼
𝑐2 (𝑙) 𝑥2∗ (𝑙)
],
1 + 𝑥2∗ (𝑙)
𝑥2∗ (𝑛 + 𝛼)
𝑛+𝛼−1
= 𝑥2∗ (𝛼) exp ∑ [𝑟2 (𝑙) − 𝑎2 (𝑙) 𝑥2∗ (𝑙) −
𝑙=𝛼
𝑐1 (𝑙) 𝑥1∗ (𝑙)
].
1 + 𝑥1∗ (𝑙)
(A.6)
Since 𝛼 is arbitrary, we know that (𝑥1∗ (𝑛), 𝑥2∗ (𝑛)) is a solution
of system (1) on Z+ , and
𝑐𝑖 (𝑛 + 𝜏𝑘 ) 󳨀→ 𝑐𝑖 (𝑛) ,
0 < 𝑚1 − 𝜀 ≤ 𝑥1∗ (𝑛) ≤ 𝑀1 + 𝜀,
as 𝑘 → +∞ uniformly on Z+ .
For any 𝛼 ∈ Z+ , we can assume that 𝜏𝑘 + 𝛼 ≥ 𝑛0 for 𝑘 large
enough. Let 𝑛 ∈ Z+ , by an inductive argument of system (1)
from 𝜏𝑘 + 𝛼 to 𝑛 + 𝜏𝑘 + 𝛼, we obtain
𝑥1 (𝑛 + 𝜏𝑘 + 𝛼)
0 < 𝑚2 − 𝜀 ≤ 𝑥2∗ (𝑛) ≤ 𝑀2 + 𝜀,
𝑛+𝜏𝑘 +𝛼−1
𝑙=𝜏𝑘 +𝛼
0 < 𝑚2 ≤ 𝑥2∗ (𝑛) ≤ 𝑀2 ,
𝑐2 (𝑙) 𝑥2 (𝑙)
],
1 + 𝑥2 (𝑙)
𝑥2 (𝑛 + 𝜏𝑘 + 𝛼)
(A.7)
Notice that 𝜀 is an arbitrarily small positive constant; it follows
that
0 < 𝑚1 ≤ 𝑥1∗ (𝑛) ≤ 𝑀1 ,
= 𝑥1 (𝜏𝑘 +𝛼) exp ∑ [𝑟1 (𝑙)−𝑎1 (𝑙) 𝑥1 (𝑙)−
∀𝑛 ∈ Z+ .
∀𝑛 ∈ Z+ .
(A.8)
Thus, Ω ≠ Φ. This completes the proof.
𝑛+𝜏𝑘 +𝛼−1
𝑐 (𝑙) 𝑥1 (𝑙)
= 𝑥2 (𝜏𝑘 +𝛼) exp ∑ [𝑟2 (𝑙)−𝑎2 (𝑙) 𝑥2 (𝑙)− 1
].
1 + 𝑥1 (𝑙)
𝑙=𝜏 +𝛼
𝑘
(A.4)
B. Proof of Theorem 8
Denote 𝑝1 (𝑛) = ln 𝑥1 (𝑛), 𝑝2 (𝑛) = ln 𝑥2 (𝑛). It follows from
system (1) that
Thus, it derives that
𝑥1 (𝑛 + 𝜏𝑘 + 𝛼)
𝑛+𝛼−1
= 𝑥1 (𝜏𝑘 + 𝛼) exp ∑ [𝑟1 (𝑙 + 𝜏𝑘 ) − 𝑎1 (𝑙 + 𝜏𝑘 ) 𝑥1 (𝑙 + 𝜏𝑘 )
𝑝1 (𝑛 + 1) = 𝑝1 (𝑛) + 𝑟1 (𝑛) − 𝑎1 (𝑛) 𝑒𝑝1 (𝑛) −
𝑐2 (𝑛) 𝑒𝑝2 (𝑛)
,
1 + 𝑒𝑝2 (𝑛)
𝑝2 (𝑛 + 1) = 𝑝2 (𝑛) + 𝑟2 (𝑛) − 𝑎2 (𝑛) 𝑒𝑝2 (𝑛) −
𝑐1 (𝑛) 𝑒𝑝1 (𝑛)
.
1 + 𝑒𝑝1 (𝑛)
(B.1)
𝑙=𝛼
−
𝑐2 (𝑙 + 𝜏𝑘 ) 𝑥2 (𝑙 + 𝜏𝑘 )
],
1 + 𝑥2 (𝑙 + 𝜏𝑘 )
According to Theorem 7, we can see that the system (B.1) has
a bounded solution (𝑝1 (𝑛), 𝑝2 (𝑛)) satisfying
𝑥2 (𝑛 + 𝜏𝑘 + 𝛼)
𝑛+𝛼−1
= 𝑥2 (𝜏𝑘 + 𝛼) exp ∑ [𝑟2 (𝑙 + 𝜏𝑘 ) − 𝑎2 (𝑙 + 𝜏𝑘 ) 𝑥2 (𝑙 + 𝜏𝑘 )
𝑙=𝛼
−
𝑐1 (𝑙 + 𝜏𝑘 ) 𝑥1 (𝑙 + 𝜏𝑘 )
].
1 + 𝑥1 (𝑙 + 𝜏𝑘 )
ln 𝑚1 ≤ 𝑝1 (𝑛) ≤ ln 𝑀1 ,
ln 𝑚2 ≤ 𝑝2 (𝑛) ≤ ln 𝑀2 ,
(A.5)
𝑛 ∈ Z+ .
(B.2)
Thus, |𝑝1 (𝑛)| ≤ 𝐴, |𝑝2 (𝑛)| ≤ 𝐵, where 𝐴 = max{| ln 𝑚1 |,
| ln 𝑀1 |}, 𝐵 = max{| ln 𝑚2 |, | ln 𝑀2 |}. Define the norm
Journal of Applied Mathematics
7
󵄨 󵄨
󵄨 󵄨
󵄨 󵄨
󵄨
󵄨
≤ (󵄨󵄨󵄨𝑝1 (𝑛)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑞1 (𝑛)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑝̃1 (𝑛)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑞̃1 (𝑛)󵄨󵄨󵄨)
‖(𝑝1 (𝑛), 𝑝2 (𝑛))‖ = |𝑝1 (𝑛)| + |𝑝2 (𝑛)|, where (𝑝1 (𝑛), 𝑝2 (𝑛)) ∈
R2 . Consider the product system of system (B.1) as follow:
𝑝1 (𝑛 + 1) = 𝑝1 (𝑛) + 𝑟1 (𝑛) − 𝑎1 (𝑛) 𝑒𝑝1 (𝑛) −
󵄨
󵄨 󵄨
󵄨
⋅ (󵄨󵄨󵄨𝑝1 (𝑛) − 𝑝̃1 (𝑛)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑞1 (𝑛) − 𝑞̃1 (𝑛)󵄨󵄨󵄨)
𝑐2 (𝑛) 𝑒𝑝2 (𝑛)
,
1 + 𝑒𝑝2 (𝑛)
󵄨
󵄨 󵄨
󵄨 󵄨
󵄨 󵄨
󵄨
+ (󵄨󵄨󵄨𝑝2 (𝑛)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑞2 (𝑛)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑝̃2 (𝑛)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑞̃2 (𝑛)󵄨󵄨󵄨)
󵄨
󵄨 󵄨
󵄨
⋅ (󵄨󵄨󵄨𝑝2 (𝑛) − 𝑝̃2 (𝑛)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑞2 (𝑛) − 𝑞̃2 (𝑛)󵄨󵄨󵄨)
𝑐 (𝑛) 𝑒𝑝1 (𝑛)
𝑝2 (𝑛 + 1) = 𝑝2 (𝑛) + 𝑟2 (𝑛) − 𝑎2 (𝑛) 𝑒𝑝2 (𝑛) − 1
,
1 + 𝑒𝑝1 (𝑛)
𝑞1 (𝑛 + 1) = 𝑞1 (𝑛) + 𝑟1 (𝑛) − 𝑎1 (𝑛) 𝑒𝑞1 (𝑛) −
𝑐2 (𝑛) 𝑒𝑞2 (𝑛)
,
1 + 𝑒𝑞2 (𝑛)
𝑞2 (𝑛 + 1) = 𝑞2 (𝑛) + 𝑟2 (𝑛) − 𝑎2 (𝑛) 𝑒𝑞2 (𝑛) −
𝑐1 (𝑛) 𝑒𝑞1 (𝑛)
.
1 + 𝑒𝑞1 (𝑛)
(B.3)
We assume that 𝑌 = (𝑝1 (𝑛), 𝑝2 (𝑛)), 𝑊 = (𝑞1 (𝑛), 𝑞2 (𝑛)) are
any two solutions of system (B.1) defined on S; then, ‖𝑌‖ ≤ 𝐷,
‖𝑊‖ ≤ 𝐷, where 𝐷 = 𝐴 + 𝐵, and S = {(𝑝1 (𝑛), 𝑝2 (𝑛)) | ln 𝑚𝑖 ≤
𝑝𝑖 (𝑛) ≤ ln 𝑀𝑖 , 𝑖 = 1, 2, 𝑛 ∈ Z+ }.
Let us construct a Lyapunov function defined on Z+ × S ×
S as follows:
2
2
𝑉 (𝑛, 𝑌, 𝑊) = (𝑝1 (𝑛) − 𝑞1 (𝑛)) + (𝑝2 (𝑛) − 𝑞2 (𝑛)) .
(B.4)
It is obvious that the norm ‖𝑌−𝑊‖ = |𝑝1 (𝑛)−𝑞1 (𝑛)|+|𝑝2 (𝑛)−
𝑞2 (𝑛)| is equivalent to ‖𝑌 − 𝑊‖∗ = [(𝑝1 (𝑛)−𝑞1 (𝑛))2 +(𝑝2 (𝑛)−
𝑞2 (𝑛))2 ]1/2 ; that is, there are two constants 𝐶1 > 0, 𝐶2 > 0,
such that
𝐶1 ‖𝑌 − 𝑊‖ ≤ ‖𝑌 − 𝑊‖∗ ≤ 𝐶2 ‖𝑌 − 𝑊‖ ,
(B.5)
then,
2
2
(𝐶1 ‖𝑌 − 𝑊‖) ≤ 𝑉 (𝑛, 𝑌, 𝑊) ≤ (𝐶2 ‖𝑌 − 𝑊‖) .
(B.6)
Let 𝑎, 𝑏 ∈ 𝐶(R+ , R+ ), 𝑎(𝑥) = 𝐶12 𝑥2 , 𝑏(𝑥) = 𝐶22 𝑥2 ; then,
condition (i) of Lemma 4 is satisfied.
̃ 𝑊)
̃ ∈ Z+ × S × S, we
Moreover, for any (𝑛, 𝑌, 𝑊), (𝑛, 𝑌,
have
󵄨󵄨
̃ 𝑊)
̃ 󵄨󵄨󵄨󵄨
󵄨󵄨𝑉 (𝑛, 𝑌, 𝑊) − 𝑉 (𝑛, 𝑌,
󵄨
󵄨
󵄨󵄨
2
2
= 󵄨󵄨󵄨(𝑝1 (𝑛) − 𝑞1 (𝑛)) + (𝑝2 (𝑛) − 𝑞2 (𝑛))
2
2󵄨
−(𝑝̃1 (𝑛) − 𝑞̃1 (𝑛)) − (𝑝̃2 (𝑛) − 𝑞̃2 (𝑛)) 󵄨󵄨󵄨󵄨
󵄨
2
2󵄨
≤ 󵄨󵄨󵄨󵄨(𝑝1 (𝑛) − 𝑞1 (𝑛)) − (𝑝̃1 (𝑛) − 𝑞̃1 (𝑛)) 󵄨󵄨󵄨󵄨
󵄨
2
2󵄨
+ 󵄨󵄨󵄨󵄨(𝑝2 (𝑛) − 𝑞2 (𝑛)) − (𝑝̃2 (𝑛) − 𝑞̃2 (𝑛)) 󵄨󵄨󵄨󵄨
󵄨
󵄨
= 󵄨󵄨󵄨(𝑝1 (𝑛) − 𝑞1 (𝑛)) + (𝑝̃1 (𝑛) − 𝑞̃1 (𝑛))󵄨󵄨󵄨
󵄨
󵄨
⋅ 󵄨󵄨󵄨(𝑝1 (𝑛) − 𝑞1 (𝑛)) − (𝑝̃1 (𝑛) − 𝑞̃1 (𝑛))󵄨󵄨󵄨
󵄨
󵄨
+ 󵄨󵄨󵄨(𝑝2 (𝑛) − 𝑞2 (𝑛)) + (𝑝̃2 (𝑛) − 𝑞̃2 (𝑛))󵄨󵄨󵄨
󵄨
󵄨
⋅ 󵄨󵄨󵄨(𝑝2 (𝑛) − 𝑞2 (𝑛)) − (𝑝̃2 (𝑛) − 𝑞̃2 (𝑛))󵄨󵄨󵄨
󵄨
󵄨 󵄨
󵄨
≤ 𝐿 {󵄨󵄨󵄨𝑝1 (𝑛) − 𝑝̃1 (𝑛)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑝2 (𝑛) − 𝑝̃2 (𝑛)󵄨󵄨󵄨
󵄨
󵄨 󵄨
󵄨
+ 󵄨󵄨󵄨𝑞1 (𝑛) − 𝑞̃1 (𝑛)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑞2 (𝑛) − 𝑞̃2 (𝑛)󵄨󵄨󵄨}
󵄩
̃󵄩󵄩󵄩󵄩} ,
̃ 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑊 − 𝑊
= 𝐿 {󵄩󵄩󵄩󵄩𝑌 − 𝑌
󵄩
󵄩 󵄩
(B.7)
̃ = (𝑝̃1 (𝑛), 𝑝̃2 (𝑛)), 𝑊
̃ = (̃
where 𝑌
𝑞1 (𝑛), 𝑞̃2 (𝑛)), and 𝐿 =
4 max{𝐴, 𝐵}. Thus, condition (ii) of Lemma 4 is satisfied.
Finally, calculating the Δ𝑉(𝑛) of 𝑉(𝑛) along the solutions
of system (B.3), we have
Δ𝑉(𝐵.3) (𝑛)
= 𝑉 (𝑛 + 1) − 𝑉 (𝑛)
2
= (𝑝1 (𝑛 + 1) − 𝑞1 (𝑛 + 1))
2
+ (𝑝2 (𝑛 + 1) − 𝑞2 (𝑛 + 1))
2
2
− (𝑝1 (𝑛) − 𝑞1 (𝑛)) − (𝑝2 (𝑛) − 𝑞2 (𝑛))
2
2
= [(𝑝1 (𝑛 + 1) − 𝑞1 (𝑛 + 1)) − (𝑝1 (𝑛) − 𝑞1 (𝑛)) ]
2
2
+ [(𝑝2 (𝑛 + 1) − 𝑞2 (𝑛 + 1)) − (𝑝2 (𝑛) − 𝑞2 (𝑛)) ]
= [ (𝑝1 (𝑛) − 𝑞1 (𝑛)) − 𝑎1 (𝑛) (𝑒𝑝1 (𝑛) − 𝑒𝑞1 (𝑛) )
2
−𝑐2 (𝑛) (
𝑒𝑞2 (𝑛)
𝑒𝑝2 (𝑛)
2
−
)] −(𝑝1 (𝑛)−𝑞1 (𝑛))
𝑝
(𝑛)
1+𝑒 2
1 + 𝑒𝑞2 (𝑛)
+ [ (𝑝2 (𝑛) − 𝑞2 (𝑛)) − 𝑎2 (𝑛) (𝑒𝑝2 (𝑛) − 𝑒𝑞2 (𝑛) )
2
−𝑐1 (𝑛) (
𝑒𝑞1 (𝑛)
𝑒𝑝1 (𝑛)
2
−
)] −(𝑝2 (𝑛)−𝑞2 (𝑛))
𝑝
(𝑛)
1+𝑒 1
1 + 𝑒𝑞1 (𝑛)
= −2𝑎1 (𝑛) (𝑝1 (𝑛) − 𝑞1 (𝑛)) (𝑒𝑝1 (𝑛) − 𝑒𝑞1 (𝑛) )
− 2𝑐2 (𝑛) (𝑝1 (𝑛) − 𝑞1 (𝑛)) (
+ 2𝑎1 (𝑛) 𝑐2 (𝑛) (
𝑒𝑞2 (𝑛)
𝑒𝑝2 (𝑛)
−
) (𝑒𝑝1 (𝑛) − 𝑒𝑞1 (𝑛) )
1 + 𝑒𝑝2 (𝑛) 1 + 𝑒𝑞2 (𝑛)
+ 𝑎12 (𝑛) (𝑒𝑝1 (𝑛) − 𝑒𝑞1 (𝑛) )
+ 𝑐22 (𝑛) (
𝑒𝑝2 (𝑛)
𝑒𝑞2 (𝑛)
−
)
1 + 𝑒𝑝2 (𝑛) 1 + 𝑒𝑞2 (𝑛)
2
2
𝑒𝑝2 (𝑛)
𝑒𝑞2 (𝑛)
−
)
1 + 𝑒𝑝2 (𝑛) 1 + 𝑒𝑞2 (𝑛)
8
Journal of Applied Mathematics
− 2𝑎2 (𝑛) (𝑝2 (𝑛) − 𝑞2 (𝑛)) (𝑒𝑝2 (𝑛) − 𝑒𝑞2 (𝑛) )
− 2𝑐1 (𝑛) (𝑝2 (𝑛) − 𝑞2 (𝑛)) (
+ 2𝑎2 (𝑛) 𝑐1 (𝑛) (
𝑝1 (𝑛)
𝑞1 (𝑛)
𝑒
𝑒
−
)
𝑝
(𝑛)
1
1+𝑒
1 + 𝑒𝑞1 (𝑛)
2
𝑒𝑞1 (𝑛)
𝑒𝑝1 (𝑛)
−
) (𝑒𝑝2 (𝑛) − 𝑒𝑞2 (𝑛) )
1 + 𝑒𝑝1 (𝑛) 1 + 𝑒𝑞1 (𝑛)
2
2
+ 2[
𝑎2 (𝑛) 𝑐1 (𝑛) 𝜉2 (𝑛) 𝜂1 (𝑛)
2
(1 + 𝜂1 (𝑛))
+
By the mean value theorem, it derives that
𝑒𝑝𝑖 (𝑛) − 𝑒𝑞𝑖 (𝑛) = 𝜉𝑖 (𝑛) (𝑝𝑖 (𝑛) − 𝑞𝑖 (𝑛)) ,
𝜂𝑖 (𝑛)
𝑒𝑝𝑖 (𝑛)
𝑒𝑞𝑖 (𝑛)
−
=
(𝑝𝑖 (𝑛) − 𝑞𝑖 (𝑛)) ,
2
𝑝
(𝑛)
𝑞
(𝑛)
𝑖
𝑖
1+𝑒
1+𝑒
(1 + 𝜂𝑖 (𝑛))
(B.9)
𝑖 = 1, 2, where 𝜉𝑖 (𝑛) and 𝜂𝑖 (𝑛) lie between 𝑒𝑝𝑖 (𝑛) and 𝑒𝑞𝑖 (𝑛) ,
respectively. Substituting (B.9) into (B.8), we get
2
= −2𝑎1 (𝑛) 𝜉1 (𝑛) (𝑝1 (𝑛) − 𝑞1 (𝑛))
−
4
2𝑐2 (𝑛) 𝜂2 (𝑛)
(1 + 𝜂2 (𝑛))
2
2
(𝑝2 (𝑛) − 𝑞2 (𝑛))
(𝑝1 (𝑛) − 𝑞1 (𝑛)) (𝑝2 (𝑛) − 𝑞2 (𝑛))
2𝑎1 (𝑛) 𝑐2 (𝑛) 𝜉1 (𝑛) 𝜂2 (𝑛)
2
(1 + 𝜂2 (𝑛))
(𝑝1 (𝑛) − 𝑞1 (𝑛))
− 2𝑎2 (𝑛) 𝜉2 (𝑛) (𝑝2 (𝑛) − 𝑞2 (𝑛))
+
−
4
(1 + 𝜂1 (𝑛))
2𝑐1 (𝑛) 𝜂1 (𝑛)
(1 + 𝜂1 (𝑛))
2
2
𝑎1 (𝑛) 𝑐2 (𝑛) 𝜉1 (𝑛) 𝜂2 (𝑛)
(1 + 𝜂2 (𝑛))
−
2
2
𝑐2 (𝑛) 𝜂2 (𝑛)
2
(1 + 𝜂2 (𝑛))
]
≤ [−2𝑎1𝐿 𝑚1 + 𝑎1𝑈2 𝑀12 +
𝑐1𝑈2 𝑀12
(1 + 𝑚1 )
+ [−2𝑎2𝐿 𝑚2 + 𝑎2𝑈2 𝑀22 +
+[
𝑎2𝑈𝑐1𝑈𝑀1 𝑀2
2
(1 + 𝑚1 )
𝑐2𝑈2 𝑀22
2
4
(1 + 𝑚2 )
𝑐1𝑈𝑀1
+
2
] (𝑝1 (𝑛) − 𝑞1 (𝑛))
4
+
2
(1 + 𝑚1 )
] (𝑝2 (𝑛) − 𝑞2 (𝑛))
𝑎1𝑈𝑐2𝑈𝑀1 𝑀2
+
2
(1 + 𝑚2 )
2
𝑐2𝑈𝑀2
2
(1 + 𝑚2 )
]
2
× {(𝑝1 (𝑛) − 𝑞1 (𝑛)) + (𝑝2 (𝑛) − 𝑞2 (𝑛)) }
= − [2𝑎1𝐿 𝑚1 − 𝑎1𝑈2 𝑀12 −
−
𝑐1𝑈𝑀1
(1 + 𝑚1 )
2
𝑐1𝑈2 𝑀12
𝑎1𝑈𝑐2𝑈𝑀1 𝑀2
−
4
(1 + 𝑚1 )
−
2
(1 + 𝑚2 )
−
𝑎2𝑈𝑐1𝑈𝑀1 𝑀2
2
(1 + 𝑚1 )
𝑐2𝑈𝑀2
]
2
(1 + 𝑚2 )
− [2𝑎2𝐿 𝑚2 − 𝑎2𝑈2 𝑀22 −
−
𝑐1𝑈𝑀1
2
(1 + 𝑚1 )
−
𝑐2𝑈2 𝑀22
4
(1 + 𝑚2 )
𝑎1𝑈𝑐2𝑈𝑀1 𝑀2
2
(1 + 𝑚2 )
−
−
𝑎2𝑈𝑐1𝑈𝑀1 𝑀2
2
(1 + 𝑚1 )
𝑐2𝑈𝑀2
2
(1 + 𝑚2 )
]
2
2
(𝑝1 (𝑛) − 𝑞1 (𝑛))
(𝑝2 (𝑛) − 𝑞2 (𝑛)) (𝑝1 (𝑛) − 𝑞1 (𝑛))
2
+ 𝑎22 (𝑛) 𝜉22 (𝑛) (𝑝2 (𝑛) − 𝑞2 (𝑛))
+
(1 + 𝜂1 (𝑛))
× (𝑝1 (𝑛) − 𝑞1 (𝑛))
× (𝑝2 (𝑛) − 𝑞2 (𝑛))
𝑐12 (𝑛) 𝜂12 (𝑛)
𝑐1 (𝑛) 𝜂1 (𝑛)
−
2
2
+ 𝑎12 (𝑛) 𝜉12 (𝑛) (𝑝1 (𝑛) − 𝑞1 (𝑛))
+
]
× (𝑝1 (𝑛) − 𝑞1 (𝑛)) (𝑝2 (𝑛) − 𝑞2 (𝑛))
Δ𝑉(𝐵.3) (𝑛)
(1 + 𝜂2 (𝑛))
4
(1 + 𝜂2 (𝑛))
2
(B.8)
+
𝑐22 (𝑛) 𝜂22 (𝑛)
+ [−2𝑎2 (𝑛) 𝜉2 (𝑛) + 𝑎22 (𝑛) 𝜉22 (𝑛) +
× (𝑝2 (𝑛) − 𝑞2 (𝑛))
𝑒𝑝1 (𝑛)
𝑒𝑞1 (𝑛)
−
).
𝑝
(𝑛)
1+𝑒 1
1 + 𝑒𝑞1 (𝑛)
𝑐22 (𝑛) 𝜂22 (𝑛)
]
4
(1 + 𝜂1 (𝑛))
× (𝑝1 (𝑛) − 𝑞1 (𝑛))
+ 𝑎22 (𝑛) (𝑒𝑝2 (𝑛) − 𝑒𝑞2 (𝑛) )
+ 𝑐12 (𝑛) (
𝑐12 (𝑛) 𝜂12 (𝑛)
= [−2𝑎1 (𝑛) 𝜉1 (𝑛) + 𝑎12 (𝑛) 𝜉12 (𝑛) +
2𝑎2 (𝑛) 𝑐1 (𝑛) 𝜉2 (𝑛) 𝜂1 (𝑛)
2
(1 + 𝜂1 (𝑛))
× (𝑝1 (𝑛) − 𝑞1 (𝑛))
(𝑝2 (𝑛) − 𝑞2 (𝑛))
× (𝑝2 (𝑛) − 𝑞2 (𝑛))
2
2
= − {𝑠1 (𝑝1 (𝑛) − 𝑞1 (𝑛)) + 𝑠2 (𝑝2 (𝑛) − 𝑞2 (𝑛)) }
2
2
≤ −𝛽 {(𝑝1 (𝑛) − 𝑞1 (𝑛)) + (𝑝2 (𝑛) − 𝑞2 (𝑛)) }
= −𝛽𝑉 (𝑛) ,
(B.10)
where 𝛽 = min{𝑠1 , 𝑠2 }. By the conditions of Theorem 8, we
have 0 < 𝛽 < 1, and hence, condition (iii) of Lemma 4 is
satisfied. So, it follows from Lemma 4 that there exists a unique uniformly asymptotically stable almost periodic solution
Journal of Applied Mathematics
(𝑝1∗ (𝑛), 𝑝2∗ (𝑛)) of system (B.1) which is bounded by S for all
𝑛 ∈ Z+ ; that is, there exists a unique uniformly asymptotically
stable almost periodic solution (𝑥1∗ (𝑛), 𝑥2∗ (𝑛)) of system (1)
which is bounded by Ω for all 𝑛 ∈ Z+ . This completed the
proof.
Acknowledgments
This work is supported by the National Natural Science
Foundation of China (no. 11261017), the Key Project of Chinese Ministry of Education (nos. 210134, 212111), the Project
of Key Laboratory of Biological Resources Protection and
Utilization of Hubei Province, and the Key Subject of Hubei
Province (Forestry).
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