Research Article Uniformly Asymptotic Stability of Positive Almost Periodic

advertisement
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 [email protected]
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:
π‘₯1 (𝑛 + 1)
= π‘₯1 (𝑛) exp [π‘Ÿ1 (𝑛) − π‘Ž1 (𝑛) π‘₯1 (𝑛) −
𝑐2 (𝑛) π‘₯2 (𝑛)
],
1 + π‘₯2 (𝑛)
π‘₯2 (𝑛 + 1)
= π‘₯2 (𝑛) exp [π‘Ÿ2 (𝑛) − π‘Ž2 (𝑛) π‘₯2 (𝑛) −
𝑐1 (𝑛) π‘₯1 (𝑛)
],
1 + π‘₯1 (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) {π‘Ÿπ‘– (𝑛)}, {π‘Žπ‘– (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:
π‘“π‘ˆ = sup {𝑓 (𝑛)} ,
𝑛∈Z+
(1)
where π‘₯𝑖 (0) > 0, π‘₯𝑖 (𝑛) stand for the densities of species π‘₯𝑖
at the 𝑛th generation, π‘Ÿπ‘– (𝑛) represent the natural growth rates
of species π‘₯𝑖 at the 𝑛th generation, π‘Žπ‘– (𝑛) are the intraspecific
effects of the 𝑛th generation of species π‘₯𝑖 on own population, and 𝑐𝑖 (𝑛) measure the interspecific effects of the 𝑛th
generation of species π‘₯𝑖 on species π‘₯𝑗 (𝑖, 𝑗 = 1, 2; 𝑖 =ΜΈ 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
π‘₯1∗ (𝑛)
π‘₯1∗ (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)
π‘₯2∗ (𝑛)
π‘₯2∗ (𝑛)
(a)
1
760
𝑛
𝑛
40
60
80
100
1
700
720
740
𝑛
𝑛
(c)
(d)
π‘₯1∗ (𝑛)
Figure 1: Positive almost periodic solution of system (13). (a), (c) Time-series
and
with initial values π‘₯1∗ (0) = 1.19, π‘₯2∗ (0) = 1.18
∗
∗
for 𝑛 ∈ [0, 100], respectively. (b), (d) Time-series π‘₯1 (𝑛) and π‘₯2 (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.
π‘₯2∗ (𝑛)
is a relatively dense set in Z for all πœ€ > 0; that is, for any given
πœ€ > 0, there exists an integer 𝑙(πœ€) > 0 such that each discrete
interval of length 𝑙(πœ€) contains a 𝜏 = 𝜏(πœ€) ∈ 𝐸{πœ€, π‘₯} such that
|π‘₯ (𝑛 + 𝜏) − π‘₯ (𝑛)| < πœ€,
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 Rπ‘˜ denote the cone of 2-dimensional and
π‘˜-dimensional real Euclidean space, respectively.
𝜏 is called the πœ€-translation number of π‘₯(𝑛).
Definition 1 (see [13]). A sequence π‘₯ : Z → Rπ‘˜ is called an
almost periodic sequence if the following πœ€-translation set of
π‘₯
󡄨
󡄨󡄨
󡄨󡄨𝑓 (𝑛 + 𝜏, π‘₯) − 𝑓 (𝑛, π‘₯)󡄨󡄨󡄨 < πœ€,
𝐸 {πœ€, π‘₯} := {𝜏 ∈ Z : |π‘₯ (𝑛 + 𝜏) − π‘₯ (𝑛)| < πœ€, ∀𝑛 ∈ Z}
(3)
(4)
Definition 2 (see [13]). Let 𝑓 : Z × D → Rπ‘˜ , where D is
an open set in Rπ‘˜ . 𝑓(𝑛, π‘₯) is said to be almost periodic in 𝑛
uniformly for π‘₯ ∈ D, or uniformly almost periodic for short,
if for any πœ€ > 0 and any compact set S in D there exists a
positive integer 𝑙(πœ€, S) such that any interval of length 𝑙(πœ€, S)
contains an integer 𝜏 for which
(5)
for all 𝑛 ∈ Z and all π‘₯ ∈ S. 𝜏 is called the πœ€-translation number
of 𝑓(𝑛, π‘₯).
3
1.22
1.22
1.2
1.2
1.18
1.18
1.16
1.16
1.14
1.14
π‘₯2∗ (𝑛)
π‘₯2∗ (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
π‘₯1∗ (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)
π‘₯2∗ (𝑛)
π‘₯2∗ (𝑛)
(a)
1.14
1.16
π‘₯1∗ (𝑛)
1
1.25
1.2
1.15
π‘₯1∗(
𝑛)
1.1
1.05 0
20
40
60
80
100
1.2
1.15
π‘₯∗
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 π‘₯1∗ (𝑛) and π‘₯2∗ (𝑛) with initial values
π‘₯1∗ (0) = 1.19, π‘₯2∗ (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 π‘₯1∗ (𝑛) and π‘₯2∗ (𝑛) with the above initial values for 𝑛 ∈ [0, 100] and 𝑛 ∈ [700, 800], respectively.
Lemma 3 (see [13]). {π‘₯(𝑛)} is an almost periodic sequence if
and only if for any sequence {β„Žπ‘˜σΈ€  } ⊂ Z there exists a subsequence
{β„Žπ‘˜ } ⊂ {β„Žπ‘˜σΈ€  } such that π‘₯(𝑛 + β„Žπ‘˜ ) converges uniformly on 𝑛 ∈ Z
as π‘˜ → ∞. Furthermore, the limit sequence is also an almost
periodic sequence.
Consider the following almost periodic difference system:
π‘₯ (𝑛 + 1) = 𝑓 (𝑛, π‘₯ (𝑛)) ,
+
π‘˜
𝑛 ∈ Z+ ,
(6)
π‘˜
where 𝑓 : Z × S𝐡 → R , S𝐡 = {π‘₯ ∈ R : β€–π‘₯β€– < 𝐡}, and
𝑓(𝑛, π‘₯) is almost periodic in 𝑛 uniformly for π‘₯ ∈ S𝐡 and is
continuous in π‘₯. The product system of (6) is the following
system:
π‘₯ (𝑛 + 1) = 𝑓 (𝑛, π‘₯ (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, π‘₯, 𝑦) defined for 𝑛 ∈ Z+ , β€–π‘₯β€– < 𝐡, ‖𝑦‖ < 𝐡
satisfying the following conditions:
(i) π‘Ž(β€–π‘₯ − 𝑦‖) ≤ 𝑉(𝑛, π‘₯, 𝑦) ≤ 𝑏(β€–π‘₯ − 𝑦‖), where π‘Ž, 𝑏 ∈
𝐾 with 𝐾 = {π‘Ž ∈ 𝐢(R+ , R+ ) : π‘Ž(0) = 0 and π‘Ž is
increasing};
(ii) |𝑉(𝑛, π‘₯1 , 𝑦1 ) − 𝑉(𝑛, π‘₯2 , 𝑦2 )| ≤ 𝐿(β€–π‘₯1 − π‘₯2 β€– + ‖𝑦1 − 𝑦2 β€–),
where 𝐿 > 0 is a constant;
(iii) Δ𝑉(2.2) (𝑛, π‘₯, 𝑦) ≤ −𝛽𝑉(𝑛, π‘₯, 𝑦), where 0 < 𝛽 < 1 is a
constant and
Δ𝑉(2.2) (𝑛, π‘₯, 𝑦) = 𝑉 (𝑛 + 1, 𝑓 (𝑛, π‘₯) , 𝑓 (𝑛, 𝑦)) − 𝑉 (𝑛, π‘₯, 𝑦) .
(8)
Moreover, if there exists a solution πœ‘(𝑛) of system (6) such that
β€–πœ‘(𝑛)β€– ≤ 𝐡∗ < 𝐡 for 𝑛 ∈ Z+ , then there exists a unique uniformly asymptotically stable almost periodic solution 𝑝(𝑛) of
system (6) which satisfies ‖𝑝(𝑛)β€– ≤ 𝐡∗ . In particular, if 𝑓(𝑛, π‘₯)
Journal of Applied Mathematics
1.25
1.25
1.2
1.2
π‘₯1∗ (𝑛), π‘₯1 (𝑛)
π‘₯1∗ (𝑛), π‘₯1 (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
π‘₯2∗ (𝑛), π‘₯2 (𝑛)
π‘₯2∗ (𝑛), π‘₯2 (𝑛)
780
π‘₯1∗ (𝑛)
π‘₯1 (𝑛)
π‘₯1∗ (𝑛)
π‘₯1 (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
π‘₯2∗ (𝑛)
π‘₯2 (𝑛)
π‘₯2∗ (𝑛)
π‘₯2 (𝑛)
(c)
(d)
π‘₯1∗ (𝑛)
Figure 3: Uniformly asymptotic stability. (a), (c) Time-series
π‘₯2 (𝑛) with initial values π‘₯1 (0) = 1.06, π‘₯2 (0) = 1.03 for 𝑛 ∈ [0, 100],
above initial values for 𝑛 ∈ [700, 800], respectively.
is periodic of period πœ”, then there exists a unique uniformly
asymptotically stable periodic solution of system (6) of periodic
πœ”.
Lemma 5 (see [1]). Any positive solution (π‘₯1 (𝑛), π‘₯2 (𝑛)) of system (1) satisfies
lim sup π‘₯𝑖 (𝑛) ≤
𝑛 → +∞
π‘ˆ
def [exp (π‘Ÿπ‘–
𝑀𝑖 =
π‘Žπ‘–πΏ
− 1)]
π‘Ÿ2𝐿 > 𝑐1π‘ˆ.
and
with initial values π‘₯1∗ (0) = 1.19, π‘₯2∗ (0) = 1.18
respectively. (b), (d) Time-series π‘₯1∗ (𝑛), π‘₯2∗ (𝑛), π‘₯1 (𝑛), and
and π‘₯1 (𝑛) and
π‘₯2 (𝑛) with the
Then, any positive solution (π‘₯1 (𝑛), π‘₯2 (𝑛)) of system (1) satisfies
def
lim inf π‘₯𝑖 (𝑛) ≥ π‘šπ‘– =
𝑛 → +∞
π‘Ÿπ‘–πΏ − π‘π‘—π‘ˆ
π‘Žπ‘–π‘ˆ
exp (π‘Ÿπ‘–πΏ − π‘Žπ‘–π‘ˆπ‘€π‘– − π‘π‘—π‘ˆ) ,
𝑖 =ΜΈ 𝑗; 𝑖, 𝑗 = 1, 2.
(11)
,
𝑖 = 1, 2.
(9)
Lemma 6 (see [1]). Suppose that system (1) satisfies the following assumptions:
π‘Ÿ1𝐿 > 𝑐2π‘ˆ,
π‘₯2∗ (𝑛)
(10)
3. Main Result
From (9) and (11), we denote by Ω the set of all solutions
(π‘₯1 (𝑛), π‘₯2 (𝑛)) of system (1) satisfying π‘šπ‘– ≤ π‘₯𝑖 (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 =ΜΈ Φ.
A computation shows that
π‘Ÿ1𝐿 − 𝑐2π‘ˆ = 1.1530 > 0,
Theorem 8. If the assumptions in (10) are satisfied, furthermore, 0 < 𝛽 < 1, where 𝛽 = min{𝑠1 , 𝑠2 }, and
−
2
(1 + π‘š2 )
2
(1 + π‘š1 )
𝑠2 = 2π‘Ž2𝐿 π‘š2 − π‘Ž2π‘ˆ2 𝑀22 −
−
(1 + π‘Ž1π‘ˆπ‘€1 ) 𝑐2π‘ˆπ‘€2
(1 + π‘Ž2π‘ˆπ‘€2 ) 𝑐1π‘ˆπ‘€1
−
𝑐1π‘ˆ2 𝑀12
4
(1 + π‘š1 )
,
(1 + π‘Ž2π‘ˆπ‘€2 ) 𝑐1π‘ˆπ‘€1
2
(1 + π‘š2 )
(12)
2
(1 + π‘š1 )
(1 + π‘Ž1π‘ˆπ‘€1 ) 𝑐2π‘ˆπ‘€2
−
𝑐2π‘ˆ2 𝑀22
(14)
and moreover, we have
The proof of Theorem 7 is given in Appendix A.
𝑠1 = 2π‘Ž1𝐿 π‘š1 − π‘Ž1π‘ˆ2 𝑀12 −
π‘Ÿ2𝐿 − 𝑐1π‘ˆ = 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
(π‘₯1∗ (𝑑), π‘₯2∗ (𝑑)), 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 (π‘₯1 (𝑛), π‘₯2 (𝑛)) tends to the almost periodic solution
(π‘₯1∗ (𝑛), π‘₯2∗ (𝑛)).
Appendices
4
(1 + π‘š2 )
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
π‘₯1 (𝑛) = π‘₯1 (0) exp ∑ [π‘Ÿ1 (𝑙) − π‘Ž1 (𝑙) π‘₯1 (𝑙) −
𝑙=0
𝑛−1
π‘₯2 (𝑛) = π‘₯2 (0) exp ∑ [π‘Ÿ2 (𝑙) − π‘Ž2 (𝑙) π‘₯2 (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) π‘₯1 (𝑙)
].
1 + π‘₯1 (𝑙)
(A.1)
According to Lemmas 5 and 6, for any solution (π‘₯1 (𝑛), π‘₯2 (𝑛))
of system (1) and an arbitrarily small constant πœ€ > 0, there
exists 𝑛0 sufficiently large such that
π‘š1 − πœ€ ≤ π‘₯1 (𝑛) ≤ 𝑀1 + πœ€,
π‘₯1 (𝑛 + 1)
𝑐2 (𝑙) π‘₯2 (𝑙)
],
1 + π‘₯2 (𝑙)
π‘š2 − πœ€ ≤ π‘₯2 (𝑛) ≤ 𝑀2 + πœ€,
∀𝑛 ≥ 𝑛0 .
= π‘₯1 (𝑛) exp [1.20 − 0.02 sin (√2π‘›πœ‹)
(A.2)
− (1.05 + 0.01 sin (√2π‘›πœ‹)) π‘₯1 (𝑛)
−
(0.025 + 0.002 cos (√2π‘›πœ‹)) π‘₯2 (𝑛)
1 + π‘₯2 (𝑛)
],
π‘₯2 (𝑛 + 1)
= π‘₯2 (𝑛) exp [1.15 − 0.02 cos (√2π‘›πœ‹)
− (1.02 + 0.02 cos (√2π‘›πœ‹)) π‘₯2 (𝑛)
−
(0.035 + 0.005 sin (√3π‘›πœ‹)) π‘₯1 (𝑛)
1 + π‘₯1 (𝑛)
].
(13)
Set {πœπ‘˜ } be any positive integer sequence such that πœπ‘˜ → +∞
as π‘˜ → +∞, we can show that there exists a subsequence of
{πœπ‘˜ } still denoted by {πœπ‘˜ }, such that π‘₯𝑖 (𝑛+πœπ‘˜ ) → π‘₯𝑖∗ (𝑛), 𝑖 = 1, 2
uniformly in 𝑛 on any finite subset 𝐢 of Z+ as π‘˜ → +∞,
where 𝐢 = {π‘Ž1 , π‘Ž2 ⋅ ⋅ ⋅ π‘Žπ‘š }, π‘Žβ„Ž ∈ Z+ (β„Ž = 1, 2 ⋅ ⋅ ⋅ π‘š), and π‘š is a
finite number.
As a matter of fact, for any finite subset 𝐢 ⊂ Z+ , πœπ‘˜ + π‘Žβ„Ž >
𝑛0 , β„Ž = 1, 2 ⋅ ⋅ ⋅ π‘š, when π‘˜ is large enough. Therefore, π‘šπ‘– − πœ€ ≤
π‘₯𝑖 (𝑛 + πœπ‘˜ ) ≤ 𝑀𝑖 + πœ€, 𝑖 = 1, 2; that is, {π‘₯𝑖 (𝑛 + πœπ‘˜ )} are uniformly
bounded for π‘˜ large enough.
Now, for π‘Ž1 ∈ 𝐢, we can choose a subsequence {πœπ‘˜(1) } of
{πœπ‘˜ } such that {π‘₯1 (π‘Ž1 +πœπ‘˜(1) )} and {π‘₯2 (π‘Ž1 +πœπ‘˜(1) )} uniformly converge on Z+ for π‘˜ large enough.
Analogously, for π‘Ž2 ∈ 𝐢, we can also choose a subsequence {πœπ‘˜(2) } of {πœπ‘˜(1) } such that {π‘₯1 (π‘Ž2 + πœπ‘˜(2) )} and {π‘₯2 (π‘Ž2 +
πœπ‘˜(2) )} uniformly converge on Z+ for π‘˜ large enough.
6
Journal of Applied Mathematics
Repeating the above process, for π‘Žπ‘š ∈ 𝐢, we get a subsequence {πœπ‘˜(π‘š) } of {πœπ‘˜(π‘š−1) } such that {π‘₯1 (π‘Žπ‘š +πœπ‘˜(π‘š) )} and {π‘₯2 (π‘Žπ‘š +
πœπ‘˜(π‘š) )} uniformly converge on Z+ for π‘˜ large enough.
Now, we choose the sequence {πœπ‘˜(π‘š) } which is a subsequence of {πœπ‘˜ } denoted by {πœπ‘˜ }; then, for all 𝑛 ∈ 𝐢, we obtain
that π‘₯𝑖 (𝑛 + πœπ‘˜ ) → π‘₯𝑖∗ (𝑛), 𝑖 = 1, 2 uniformly in 𝑛 ∈ 𝐢 as
π‘˜ → +∞. Hence, the conclusion is valid by the arbitrary of
𝐢.
Recall the almost periodicity of {π‘Ÿπ‘– (𝑛)}, {π‘Žπ‘– (𝑛)} and {𝑐𝑖 (𝑛)},
𝑖 = 1, 2, for the above sequence {πœπ‘˜ }, πœπ‘˜ → +∞ as π‘˜ → +∞,
there exists a subsequence denoted by {πœπ‘˜ } such that
π‘Ÿπ‘– (𝑛 + πœπ‘˜ ) 󳨀→ π‘Ÿπ‘– (𝑛) ,
π‘Žπ‘– (𝑛 + πœπ‘˜ ) 󳨀→ π‘Žπ‘– (𝑛) ,
(A.3)
Let π‘˜ → +∞, we have
π‘₯1∗ (𝑛 + 𝛼)
𝑛+𝛼−1
= π‘₯1∗ (𝛼) exp ∑ [π‘Ÿ1 (𝑙) − π‘Ž1 (𝑙) π‘₯1∗ (𝑙) −
𝑙=𝛼
𝑐2 (𝑙) π‘₯2∗ (𝑙)
],
1 + π‘₯2∗ (𝑙)
π‘₯2∗ (𝑛 + 𝛼)
𝑛+𝛼−1
= π‘₯2∗ (𝛼) exp ∑ [π‘Ÿ2 (𝑙) − π‘Ž2 (𝑙) π‘₯2∗ (𝑙) −
𝑙=𝛼
𝑐1 (𝑙) π‘₯1∗ (𝑙)
].
1 + π‘₯1∗ (𝑙)
(A.6)
Since 𝛼 is arbitrary, we know that (π‘₯1∗ (𝑛), π‘₯2∗ (𝑛)) is a solution
of system (1) on Z+ , and
𝑐𝑖 (𝑛 + πœπ‘˜ ) 󳨀→ 𝑐𝑖 (𝑛) ,
0 < π‘š1 − πœ€ ≤ π‘₯1∗ (𝑛) ≤ 𝑀1 + πœ€,
as π‘˜ → +∞ uniformly on Z+ .
For any 𝛼 ∈ Z+ , we can assume that πœπ‘˜ + 𝛼 ≥ 𝑛0 for π‘˜ large
enough. Let 𝑛 ∈ Z+ , by an inductive argument of system (1)
from πœπ‘˜ + 𝛼 to 𝑛 + πœπ‘˜ + 𝛼, we obtain
π‘₯1 (𝑛 + πœπ‘˜ + 𝛼)
0 < π‘š2 − πœ€ ≤ π‘₯2∗ (𝑛) ≤ 𝑀2 + πœ€,
𝑛+πœπ‘˜ +𝛼−1
𝑙=πœπ‘˜ +𝛼
0 < π‘š2 ≤ π‘₯2∗ (𝑛) ≤ 𝑀2 ,
𝑐2 (𝑙) π‘₯2 (𝑙)
],
1 + π‘₯2 (𝑙)
π‘₯2 (𝑛 + πœπ‘˜ + 𝛼)
(A.7)
Notice that πœ€ is an arbitrarily small positive constant; it follows
that
0 < π‘š1 ≤ π‘₯1∗ (𝑛) ≤ 𝑀1 ,
= π‘₯1 (πœπ‘˜ +𝛼) exp ∑ [π‘Ÿ1 (𝑙)−π‘Ž1 (𝑙) π‘₯1 (𝑙)−
∀𝑛 ∈ Z+ .
∀𝑛 ∈ Z+ .
(A.8)
Thus, Ω =ΜΈ Φ. This completes the proof.
𝑛+πœπ‘˜ +𝛼−1
𝑐 (𝑙) π‘₯1 (𝑙)
= π‘₯2 (πœπ‘˜ +𝛼) exp ∑ [π‘Ÿ2 (𝑙)−π‘Ž2 (𝑙) π‘₯2 (𝑙)− 1
].
1 + π‘₯1 (𝑙)
𝑙=𝜏 +𝛼
π‘˜
(A.4)
B. Proof of Theorem 8
Denote 𝑝1 (𝑛) = ln π‘₯1 (𝑛), 𝑝2 (𝑛) = ln π‘₯2 (𝑛). It follows from
system (1) that
Thus, it derives that
π‘₯1 (𝑛 + πœπ‘˜ + 𝛼)
𝑛+𝛼−1
= π‘₯1 (πœπ‘˜ + 𝛼) exp ∑ [π‘Ÿ1 (𝑙 + πœπ‘˜ ) − π‘Ž1 (𝑙 + πœπ‘˜ ) π‘₯1 (𝑙 + πœπ‘˜ )
𝑝1 (𝑛 + 1) = 𝑝1 (𝑛) + π‘Ÿ1 (𝑛) − π‘Ž1 (𝑛) 𝑒𝑝1 (𝑛) −
𝑐2 (𝑛) 𝑒𝑝2 (𝑛)
,
1 + 𝑒𝑝2 (𝑛)
𝑝2 (𝑛 + 1) = 𝑝2 (𝑛) + π‘Ÿ2 (𝑛) − π‘Ž2 (𝑛) 𝑒𝑝2 (𝑛) −
𝑐1 (𝑛) 𝑒𝑝1 (𝑛)
.
1 + 𝑒𝑝1 (𝑛)
(B.1)
𝑙=𝛼
−
𝑐2 (𝑙 + πœπ‘˜ ) π‘₯2 (𝑙 + πœπ‘˜ )
],
1 + π‘₯2 (𝑙 + πœπ‘˜ )
According to Theorem 7, we can see that the system (B.1) has
a bounded solution (𝑝1 (𝑛), 𝑝2 (𝑛)) satisfying
π‘₯2 (𝑛 + πœπ‘˜ + 𝛼)
𝑛+𝛼−1
= π‘₯2 (πœπ‘˜ + 𝛼) exp ∑ [π‘Ÿ2 (𝑙 + πœπ‘˜ ) − π‘Ž2 (𝑙 + πœπ‘˜ ) π‘₯2 (𝑙 + πœπ‘˜ )
𝑙=𝛼
−
𝑐1 (𝑙 + πœπ‘˜ ) π‘₯1 (𝑙 + πœπ‘˜ )
].
1 + π‘₯1 (𝑙 + πœπ‘˜ )
ln π‘š1 ≤ 𝑝1 (𝑛) ≤ ln 𝑀1 ,
ln π‘š2 ≤ 𝑝2 (𝑛) ≤ ln 𝑀2 ,
(A.5)
𝑛 ∈ Z+ .
(B.2)
Thus, |𝑝1 (𝑛)| ≤ 𝐴, |𝑝2 (𝑛)| ≤ 𝐡, where 𝐴 = max{| ln π‘š1 |,
| ln 𝑀1 |}, 𝐡 = max{| ln π‘š2 |, | ln 𝑀2 |}. Define the norm
Journal of Applied Mathematics
7
󡄨 󡄨
󡄨 󡄨
󡄨 󡄨
󡄨
󡄨
≤ (󡄨󡄨󡄨𝑝1 (𝑛)󡄨󡄨󡄨 + σ΅„¨σ΅„¨σ΅„¨π‘ž1 (𝑛)󡄨󡄨󡄨 + 󡄨󡄨󡄨𝑝̃1 (𝑛)󡄨󡄨󡄨 + σ΅„¨σ΅„¨σ΅„¨π‘žΜƒ1 (𝑛)󡄨󡄨󡄨)
β€–(𝑝1 (𝑛), 𝑝2 (𝑛))β€– = |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) + π‘Ÿ1 (𝑛) − π‘Ž1 (𝑛) 𝑒𝑝1 (𝑛) −
󡄨
󡄨 󡄨
󡄨
⋅ (󡄨󡄨󡄨𝑝1 (𝑛) − 𝑝̃1 (𝑛)󡄨󡄨󡄨 + σ΅„¨σ΅„¨σ΅„¨π‘ž1 (𝑛) − π‘žΜƒ1 (𝑛)󡄨󡄨󡄨)
𝑐2 (𝑛) 𝑒𝑝2 (𝑛)
,
1 + 𝑒𝑝2 (𝑛)
󡄨
󡄨 󡄨
󡄨 󡄨
󡄨 󡄨
󡄨
+ (󡄨󡄨󡄨𝑝2 (𝑛)󡄨󡄨󡄨 + σ΅„¨σ΅„¨σ΅„¨π‘ž2 (𝑛)󡄨󡄨󡄨 + 󡄨󡄨󡄨𝑝̃2 (𝑛)󡄨󡄨󡄨 + σ΅„¨σ΅„¨σ΅„¨π‘žΜƒ2 (𝑛)󡄨󡄨󡄨)
󡄨
󡄨 󡄨
󡄨
⋅ (󡄨󡄨󡄨𝑝2 (𝑛) − 𝑝̃2 (𝑛)󡄨󡄨󡄨 + σ΅„¨σ΅„¨σ΅„¨π‘ž2 (𝑛) − π‘žΜƒ2 (𝑛)󡄨󡄨󡄨)
𝑐 (𝑛) 𝑒𝑝1 (𝑛)
𝑝2 (𝑛 + 1) = 𝑝2 (𝑛) + π‘Ÿ2 (𝑛) − π‘Ž2 (𝑛) 𝑒𝑝2 (𝑛) − 1
,
1 + 𝑒𝑝1 (𝑛)
π‘ž1 (𝑛 + 1) = π‘ž1 (𝑛) + π‘Ÿ1 (𝑛) − π‘Ž1 (𝑛) π‘’π‘ž1 (𝑛) −
𝑐2 (𝑛) π‘’π‘ž2 (𝑛)
,
1 + π‘’π‘ž2 (𝑛)
π‘ž2 (𝑛 + 1) = π‘ž2 (𝑛) + π‘Ÿ2 (𝑛) − π‘Ž2 (𝑛) π‘’π‘ž2 (𝑛) −
𝑐1 (𝑛) π‘’π‘ž1 (𝑛)
.
1 + π‘’π‘ž1 (𝑛)
(B.3)
We assume that π‘Œ = (𝑝1 (𝑛), 𝑝2 (𝑛)), π‘Š = (π‘ž1 (𝑛), π‘ž2 (𝑛)) are
any two solutions of system (B.1) defined on S; then, β€–π‘Œβ€– ≤ 𝐷,
β€–π‘Šβ€– ≤ 𝐷, where 𝐷 = 𝐴 + 𝐡, and S = {(𝑝1 (𝑛), 𝑝2 (𝑛)) | ln π‘šπ‘– ≤
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, π‘Œ, π‘Š) = (𝑝1 (𝑛) − π‘ž1 (𝑛)) + (𝑝2 (𝑛) − π‘ž2 (𝑛)) .
(B.4)
It is obvious that the norm β€–π‘Œ−π‘Šβ€– = |𝑝1 (𝑛)−π‘ž1 (𝑛)|+|𝑝2 (𝑛)−
π‘ž2 (𝑛)| is equivalent to β€–π‘Œ − π‘Šβ€–∗ = [(𝑝1 (𝑛)−π‘ž1 (𝑛))2 +(𝑝2 (𝑛)−
π‘ž2 (𝑛))2 ]1/2 ; that is, there are two constants 𝐢1 > 0, 𝐢2 > 0,
such that
𝐢1 β€–π‘Œ − π‘Šβ€– ≤ β€–π‘Œ − π‘Šβ€–∗ ≤ 𝐢2 β€–π‘Œ − π‘Šβ€– ,
(B.5)
then,
2
2
(𝐢1 β€–π‘Œ − π‘Šβ€–) ≤ 𝑉 (𝑛, π‘Œ, π‘Š) ≤ (𝐢2 β€–π‘Œ − π‘Šβ€–) .
(B.6)
Let π‘Ž, 𝑏 ∈ 𝐢(R+ , R+ ), π‘Ž(π‘₯) = 𝐢12 π‘₯2 , 𝑏(π‘₯) = 𝐢22 π‘₯2 ; then,
condition (i) of Lemma 4 is satisfied.
Μƒ π‘Š)
Μƒ ∈ Z+ × S × S, we
Moreover, for any (𝑛, π‘Œ, π‘Š), (𝑛, π‘Œ,
have
󡄨󡄨
Μƒ π‘Š)
Μƒ 󡄨󡄨󡄨󡄨
󡄨󡄨𝑉 (𝑛, π‘Œ, π‘Š) − 𝑉 (𝑛, π‘Œ,
󡄨
󡄨
󡄨󡄨
2
2
= 󡄨󡄨󡄨(𝑝1 (𝑛) − π‘ž1 (𝑛)) + (𝑝2 (𝑛) − π‘ž2 (𝑛))
2
2󡄨
−(𝑝̃1 (𝑛) − π‘žΜƒ1 (𝑛)) − (𝑝̃2 (𝑛) − π‘žΜƒ2 (𝑛)) 󡄨󡄨󡄨󡄨
󡄨
2
2󡄨
≤ 󡄨󡄨󡄨󡄨(𝑝1 (𝑛) − π‘ž1 (𝑛)) − (𝑝̃1 (𝑛) − π‘žΜƒ1 (𝑛)) 󡄨󡄨󡄨󡄨
󡄨
2
2󡄨
+ 󡄨󡄨󡄨󡄨(𝑝2 (𝑛) − π‘ž2 (𝑛)) − (𝑝̃2 (𝑛) − π‘žΜƒ2 (𝑛)) 󡄨󡄨󡄨󡄨
󡄨
󡄨
= 󡄨󡄨󡄨(𝑝1 (𝑛) − π‘ž1 (𝑛)) + (𝑝̃1 (𝑛) − π‘žΜƒ1 (𝑛))󡄨󡄨󡄨
󡄨
󡄨
⋅ 󡄨󡄨󡄨(𝑝1 (𝑛) − π‘ž1 (𝑛)) − (𝑝̃1 (𝑛) − π‘žΜƒ1 (𝑛))󡄨󡄨󡄨
󡄨
󡄨
+ 󡄨󡄨󡄨(𝑝2 (𝑛) − π‘ž2 (𝑛)) + (𝑝̃2 (𝑛) − π‘žΜƒ2 (𝑛))󡄨󡄨󡄨
󡄨
󡄨
⋅ 󡄨󡄨󡄨(𝑝2 (𝑛) − π‘ž2 (𝑛)) − (𝑝̃2 (𝑛) − π‘žΜƒ2 (𝑛))󡄨󡄨󡄨
󡄨
󡄨 󡄨
󡄨
≤ 𝐿 {󡄨󡄨󡄨𝑝1 (𝑛) − 𝑝̃1 (𝑛)󡄨󡄨󡄨 + 󡄨󡄨󡄨𝑝2 (𝑛) − 𝑝̃2 (𝑛)󡄨󡄨󡄨
󡄨
󡄨 󡄨
󡄨
+ σ΅„¨σ΅„¨σ΅„¨π‘ž1 (𝑛) − π‘žΜƒ1 (𝑛)󡄨󡄨󡄨 + σ΅„¨σ΅„¨σ΅„¨π‘ž2 (𝑛) − π‘žΜƒ2 (𝑛)󡄨󡄨󡄨}
σ΅„©
Μƒσ΅„©σ΅„©σ΅„©σ΅„©} ,
Μƒ σ΅„©σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©σ΅„©π‘Š − π‘Š
= 𝐿 {σ΅„©σ΅„©σ΅„©σ΅„©π‘Œ − π‘Œ
σ΅„©
σ΅„© σ΅„©
(B.7)
Μƒ = (𝑝̃1 (𝑛), 𝑝̃2 (𝑛)), π‘Š
Μƒ = (Μƒ
where π‘Œ
π‘ž1 (𝑛), π‘žΜƒ2 (𝑛)), and 𝐿 =
4 max{𝐴, 𝐡}. 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(𝐡.3) (𝑛)
= 𝑉 (𝑛 + 1) − 𝑉 (𝑛)
2
= (𝑝1 (𝑛 + 1) − π‘ž1 (𝑛 + 1))
2
+ (𝑝2 (𝑛 + 1) − π‘ž2 (𝑛 + 1))
2
2
− (𝑝1 (𝑛) − π‘ž1 (𝑛)) − (𝑝2 (𝑛) − π‘ž2 (𝑛))
2
2
= [(𝑝1 (𝑛 + 1) − π‘ž1 (𝑛 + 1)) − (𝑝1 (𝑛) − π‘ž1 (𝑛)) ]
2
2
+ [(𝑝2 (𝑛 + 1) − π‘ž2 (𝑛 + 1)) − (𝑝2 (𝑛) − π‘ž2 (𝑛)) ]
= [ (𝑝1 (𝑛) − π‘ž1 (𝑛)) − π‘Ž1 (𝑛) (𝑒𝑝1 (𝑛) − π‘’π‘ž1 (𝑛) )
2
−𝑐2 (𝑛) (
π‘’π‘ž2 (𝑛)
𝑒𝑝2 (𝑛)
2
−
)] −(𝑝1 (𝑛)−π‘ž1 (𝑛))
𝑝
(𝑛)
1+𝑒 2
1 + π‘’π‘ž2 (𝑛)
+ [ (𝑝2 (𝑛) − π‘ž2 (𝑛)) − π‘Ž2 (𝑛) (𝑒𝑝2 (𝑛) − π‘’π‘ž2 (𝑛) )
2
−𝑐1 (𝑛) (
π‘’π‘ž1 (𝑛)
𝑒𝑝1 (𝑛)
2
−
)] −(𝑝2 (𝑛)−π‘ž2 (𝑛))
𝑝
(𝑛)
1+𝑒 1
1 + π‘’π‘ž1 (𝑛)
= −2π‘Ž1 (𝑛) (𝑝1 (𝑛) − π‘ž1 (𝑛)) (𝑒𝑝1 (𝑛) − π‘’π‘ž1 (𝑛) )
− 2𝑐2 (𝑛) (𝑝1 (𝑛) − π‘ž1 (𝑛)) (
+ 2π‘Ž1 (𝑛) 𝑐2 (𝑛) (
π‘’π‘ž2 (𝑛)
𝑒𝑝2 (𝑛)
−
) (𝑒𝑝1 (𝑛) − π‘’π‘ž1 (𝑛) )
1 + 𝑒𝑝2 (𝑛) 1 + π‘’π‘ž2 (𝑛)
+ π‘Ž12 (𝑛) (𝑒𝑝1 (𝑛) − π‘’π‘ž1 (𝑛) )
+ 𝑐22 (𝑛) (
𝑒𝑝2 (𝑛)
π‘’π‘ž2 (𝑛)
−
)
1 + 𝑒𝑝2 (𝑛) 1 + π‘’π‘ž2 (𝑛)
2
2
𝑒𝑝2 (𝑛)
π‘’π‘ž2 (𝑛)
−
)
1 + 𝑒𝑝2 (𝑛) 1 + π‘’π‘ž2 (𝑛)
8
Journal of Applied Mathematics
− 2π‘Ž2 (𝑛) (𝑝2 (𝑛) − π‘ž2 (𝑛)) (𝑒𝑝2 (𝑛) − π‘’π‘ž2 (𝑛) )
− 2𝑐1 (𝑛) (𝑝2 (𝑛) − π‘ž2 (𝑛)) (
+ 2π‘Ž2 (𝑛) 𝑐1 (𝑛) (
𝑝1 (𝑛)
π‘ž1 (𝑛)
𝑒
𝑒
−
)
𝑝
(𝑛)
1
1+𝑒
1 + π‘’π‘ž1 (𝑛)
2
π‘’π‘ž1 (𝑛)
𝑒𝑝1 (𝑛)
−
) (𝑒𝑝2 (𝑛) − π‘’π‘ž2 (𝑛) )
1 + 𝑒𝑝1 (𝑛) 1 + π‘’π‘ž1 (𝑛)
2
2
+ 2[
π‘Ž2 (𝑛) 𝑐1 (𝑛) πœ‰2 (𝑛) πœ‚1 (𝑛)
2
(1 + πœ‚1 (𝑛))
+
By the mean value theorem, it derives that
𝑒𝑝𝑖 (𝑛) − π‘’π‘žπ‘– (𝑛) = πœ‰π‘– (𝑛) (𝑝𝑖 (𝑛) − π‘žπ‘– (𝑛)) ,
πœ‚π‘– (𝑛)
𝑒𝑝𝑖 (𝑛)
π‘’π‘žπ‘– (𝑛)
−
=
(𝑝𝑖 (𝑛) − π‘žπ‘– (𝑛)) ,
2
𝑝
(𝑛)
π‘ž
(𝑛)
𝑖
𝑖
1+𝑒
1+𝑒
(1 + πœ‚π‘– (𝑛))
(B.9)
𝑖 = 1, 2, where πœ‰π‘– (𝑛) and πœ‚π‘– (𝑛) lie between 𝑒𝑝𝑖 (𝑛) and π‘’π‘žπ‘– (𝑛) ,
respectively. Substituting (B.9) into (B.8), we get
2
= −2π‘Ž1 (𝑛) πœ‰1 (𝑛) (𝑝1 (𝑛) − π‘ž1 (𝑛))
−
4
2𝑐2 (𝑛) πœ‚2 (𝑛)
(1 + πœ‚2 (𝑛))
2
2
(𝑝2 (𝑛) − π‘ž2 (𝑛))
(𝑝1 (𝑛) − π‘ž1 (𝑛)) (𝑝2 (𝑛) − π‘ž2 (𝑛))
2π‘Ž1 (𝑛) 𝑐2 (𝑛) πœ‰1 (𝑛) πœ‚2 (𝑛)
2
(1 + πœ‚2 (𝑛))
(𝑝1 (𝑛) − π‘ž1 (𝑛))
− 2π‘Ž2 (𝑛) πœ‰2 (𝑛) (𝑝2 (𝑛) − π‘ž2 (𝑛))
+
−
4
(1 + πœ‚1 (𝑛))
2𝑐1 (𝑛) πœ‚1 (𝑛)
(1 + πœ‚1 (𝑛))
2
2
π‘Ž1 (𝑛) 𝑐2 (𝑛) πœ‰1 (𝑛) πœ‚2 (𝑛)
(1 + πœ‚2 (𝑛))
−
2
2
𝑐2 (𝑛) πœ‚2 (𝑛)
2
(1 + πœ‚2 (𝑛))
]
≤ [−2π‘Ž1𝐿 π‘š1 + π‘Ž1π‘ˆ2 𝑀12 +
𝑐1π‘ˆ2 𝑀12
(1 + π‘š1 )
+ [−2π‘Ž2𝐿 π‘š2 + π‘Ž2π‘ˆ2 𝑀22 +
+[
π‘Ž2π‘ˆπ‘1π‘ˆπ‘€1 𝑀2
2
(1 + π‘š1 )
𝑐2π‘ˆ2 𝑀22
2
4
(1 + π‘š2 )
𝑐1π‘ˆπ‘€1
+
2
] (𝑝1 (𝑛) − π‘ž1 (𝑛))
4
+
2
(1 + π‘š1 )
] (𝑝2 (𝑛) − π‘ž2 (𝑛))
π‘Ž1π‘ˆπ‘2π‘ˆπ‘€1 𝑀2
+
2
(1 + π‘š2 )
2
𝑐2π‘ˆπ‘€2
2
(1 + π‘š2 )
]
2
× {(𝑝1 (𝑛) − π‘ž1 (𝑛)) + (𝑝2 (𝑛) − π‘ž2 (𝑛)) }
= − [2π‘Ž1𝐿 π‘š1 − π‘Ž1π‘ˆ2 𝑀12 −
−
𝑐1π‘ˆπ‘€1
(1 + π‘š1 )
2
𝑐1π‘ˆ2 𝑀12
π‘Ž1π‘ˆπ‘2π‘ˆπ‘€1 𝑀2
−
4
(1 + π‘š1 )
−
2
(1 + π‘š2 )
−
π‘Ž2π‘ˆπ‘1π‘ˆπ‘€1 𝑀2
2
(1 + π‘š1 )
𝑐2π‘ˆπ‘€2
]
2
(1 + π‘š2 )
− [2π‘Ž2𝐿 π‘š2 − π‘Ž2π‘ˆ2 𝑀22 −
−
𝑐1π‘ˆπ‘€1
2
(1 + π‘š1 )
−
𝑐2π‘ˆ2 𝑀22
4
(1 + π‘š2 )
π‘Ž1π‘ˆπ‘2π‘ˆπ‘€1 𝑀2
2
(1 + π‘š2 )
−
−
π‘Ž2π‘ˆπ‘1π‘ˆπ‘€1 𝑀2
2
(1 + π‘š1 )
𝑐2π‘ˆπ‘€2
2
(1 + π‘š2 )
]
2
2
(𝑝1 (𝑛) − π‘ž1 (𝑛))
(𝑝2 (𝑛) − π‘ž2 (𝑛)) (𝑝1 (𝑛) − π‘ž1 (𝑛))
2
+ π‘Ž22 (𝑛) πœ‰22 (𝑛) (𝑝2 (𝑛) − π‘ž2 (𝑛))
+
(1 + πœ‚1 (𝑛))
× (𝑝1 (𝑛) − π‘ž1 (𝑛))
× (𝑝2 (𝑛) − π‘ž2 (𝑛))
𝑐12 (𝑛) πœ‚12 (𝑛)
𝑐1 (𝑛) πœ‚1 (𝑛)
−
2
2
+ π‘Ž12 (𝑛) πœ‰12 (𝑛) (𝑝1 (𝑛) − π‘ž1 (𝑛))
+
]
× (𝑝1 (𝑛) − π‘ž1 (𝑛)) (𝑝2 (𝑛) − π‘ž2 (𝑛))
Δ𝑉(𝐡.3) (𝑛)
(1 + πœ‚2 (𝑛))
4
(1 + πœ‚2 (𝑛))
2
(B.8)
+
𝑐22 (𝑛) πœ‚22 (𝑛)
+ [−2π‘Ž2 (𝑛) πœ‰2 (𝑛) + π‘Ž22 (𝑛) πœ‰22 (𝑛) +
× (𝑝2 (𝑛) − π‘ž2 (𝑛))
𝑒𝑝1 (𝑛)
π‘’π‘ž1 (𝑛)
−
).
𝑝
(𝑛)
1+𝑒 1
1 + π‘’π‘ž1 (𝑛)
𝑐22 (𝑛) πœ‚22 (𝑛)
]
4
(1 + πœ‚1 (𝑛))
× (𝑝1 (𝑛) − π‘ž1 (𝑛))
+ π‘Ž22 (𝑛) (𝑒𝑝2 (𝑛) − π‘’π‘ž2 (𝑛) )
+ 𝑐12 (𝑛) (
𝑐12 (𝑛) πœ‚12 (𝑛)
= [−2π‘Ž1 (𝑛) πœ‰1 (𝑛) + π‘Ž12 (𝑛) πœ‰12 (𝑛) +
2π‘Ž2 (𝑛) 𝑐1 (𝑛) πœ‰2 (𝑛) πœ‚1 (𝑛)
2
(1 + πœ‚1 (𝑛))
× (𝑝1 (𝑛) − π‘ž1 (𝑛))
(𝑝2 (𝑛) − π‘ž2 (𝑛))
× (𝑝2 (𝑛) − π‘ž2 (𝑛))
2
2
= − {𝑠1 (𝑝1 (𝑛) − π‘ž1 (𝑛)) + 𝑠2 (𝑝2 (𝑛) − π‘ž2 (𝑛)) }
2
2
≤ −𝛽 {(𝑝1 (𝑛) − π‘ž1 (𝑛)) + (𝑝2 (𝑛) − π‘ž2 (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 (π‘₯1∗ (𝑛), π‘₯2∗ (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).
References
[1] W. Qin, Z. Liu, and Y. Chen, “Permanence and global stability
of positive periodic solutions of a discrete competitive system,”
Discrete Dynamics in Nature and Society, vol. 2009, Article ID
830537, 13 pages, 2009.
[2] J. Xu, Z. Teng, and H. Jiang, “Permanence and global attractivity
for discrete nonautonomous two-species Lotka-Volterra competitive system with delays and feedback controls,” Periodica
Mathematica Hungarica, vol. 63, no. 1, pp. 19–45, 2011.
[3] W. Qin and Z. Liu, “Permanence and positive periodic solutions
of a discrete delay competitive system,” Discrete Dynamics in
Nature and Society, vol. 2010, Article ID 381750, 22 pages, 2010.
[4] X. Liao, Z. Ouyang, and S. Zhou, “Permanence of species
in nonautonomous discrete Lotka-Volterra competitive system
with delays and feedback controls,” Journal of Computational
and Applied Mathematics, vol. 211, no. 1, pp. 1–10, 2008.
[5] H. Zhou, D. Huang, W. Wang, and J. Xu, “Some new difference
inequalities and an application to discrete-time control systems,” Journal of Applied Mathematics, vol. 2012, Article ID
214609, 14 pages, 2012.
[6] K. Mukdasai, “Robust exponential stability for LPD discretetime system with interval time-varying delay,” Journal of Applied
Mathematics, vol. 2012, Article ID 237430, 13 pages, 2012.
[7] R. P. Agarwal, Difference Equations and Inequalities: Theory,
Methods, and Application, Marcel Dekker, New York, NY, USA,
2000.
[8] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, NY, USA, 1980.
[9] J. D. Murray, Mathematical Biology, vol. 19, Springer, Berlin,
Germany, 1989.
[10] C. Niu and X. Chen, “Almost periodic sequence solutions of a
discrete Lotka-Volterra competitive system with feedback control,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5,
pp. 3152–3161, 2009.
[11] K. Gopalsamy and S. Mohamad, “Canonical solutions and
almost periodicity in a discrete logistic equation,” Applied Mathematics and Computation, vol. 113, no. 2-3, pp. 305–323, 2000.
[12] Z. Huang, X. Wang, and F. Gao, “The existence and global
attractivity of almost periodic sequence solution of discretetime neural networks,” Physics Letters A, vol. 350, no. 3-4, pp.
182–191, 2006.
[13] D. Cheban and C. Mammana, “Invariant manifolds, global
attractors and almost periodic solutions of nonautonomous
difference equations,” Nonlinear Analysis: Theory, Methods and
Applications, vol. 56, no. 4, pp. 465–484, 2004.
9
[14] S. Zhang, “Existence of almost periodic solutions for difference
systems,” Annals of Differential Equations, vol. 16, no. 2, pp. 184–
206, 2000.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Download