Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2013, Article ID 179401, 5 pages
http://dx.doi.org/10.1155/2013/179401
Research Article
Dynamics of a System of Rational Higher-Order
Difference Equation
Banyat Sroysang
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12121, Thailand
Correspondence should be addressed to Banyat Sroysang; banyat@mathstat.sci.tu.ac.th
Received 5 February 2013; Accepted 22 April 2013
Academic Editor: Cengiz Çinar
Copyright © 2013 Banyat Sroysang. 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.
We focus on a system of a rational 𝑚-order difference equation 𝑥𝑛+1 = (𝑥𝑛−𝑚+1 )/(𝐴 + 𝑦𝑛 𝑦𝑛−1 ⋅ ⋅ ⋅ 𝑦𝑛−𝑚+1 ), 𝑦𝑛+1 = (𝑦𝑛−𝑚+1 )/(𝐵 +
𝑥𝑛 𝑥𝑛−1 ⋅ ⋅ ⋅ 𝑥𝑛−𝑚+1 ), 𝑛 = 0, 1, . . ., where 𝐴, 𝐵, 𝑥0 , 𝑥−1 , . . . , 𝑥−𝑚+1 , 𝑦0 , 𝑦−1 , . . . , 𝑦−𝑚+1 ∈ (0, ∞). We investigate the dynamical behavior of
positive solution for the system.
1. Introduction
In 2011, Kurbanli et al. [1] studied the behavior of positive
solutions of the system of rational difference equations
𝑥𝑛+1 =
𝑦𝑛+1 =
where the initial conditions are arbitrary real numbers.
Moreover, Kurbanli [3] studied the behavior of the solutions
of the difference equation system
𝑥𝑛−1
,
𝑦𝑛 𝑥𝑛−1 + 1
𝑦𝑛−1
,
𝑥𝑛 𝑦𝑛−1 + 1
𝑥𝑛−1
,
𝑦𝑛 𝑥𝑛−1 − 1
𝑦𝑛+1 =
𝑦𝑛−1
,
𝑥𝑛 𝑦𝑛−1 − 1
𝑧𝑛+1 =
𝑧𝑛−1
,
𝑦𝑛 𝑧𝑛−1 − 1
𝑥𝑛−1
,
𝑦𝑛 𝑥𝑛−1 − 1
𝑦𝑛+1 =
𝑦𝑛−1
,
𝑥𝑛 𝑦𝑛−1 − 1
(1)
𝑧𝑛+1 =
where the initial conditions are arbitrary nonnegative real
numbers.
In the same year, Kurbanli [2] studied the behavior of
solutions of the system of rational difference equations
𝑥𝑛+1 =
𝑥𝑛+1 =
(3)
1
,
𝑦𝑛 𝑧𝑛
where 𝑥0 , 𝑥−1 , 𝑦0 , 𝑦−1 , 𝑧0 , 𝑧−1 ∈ R such that 𝑦0 𝑥−1 ≠ 1,
𝑥0 𝑦−1 ≠ 1 and 𝑦0 𝑧0 ≠ 1.
In [4], Liu et al. gave more results of the solution of the
system (2) including a new and simple expression of 𝑧𝑛 and
the asymptotical behavior of the solution.
In [5], Stević showed that the system of difference equations
𝑥𝑛+1 =
𝑎𝑥𝑛−1
,
𝑏𝑦𝑛 𝑥𝑛−1 + 𝑐
𝑦𝑛+1 =
𝛼𝑦𝑛−1
,
𝛽𝑥𝑛 𝑦𝑛−1 + 𝛾
(4)
𝑛 = 0, 1, . . . ,
(2)
can be solved.
In 2012, Gu and Ding [6] derived two canonical state
space forms from multiple-input multiple-output systems
described by difference equations.
2
Discrete Dynamics in Nature and Society
The system of two nonlinear difference equations
𝑦
𝑥
𝑥𝑛+1 = 𝐴 + 𝑛 , 𝑦𝑛+1 = 𝐴 + 𝑛 ,
𝑥𝑛−𝑝
𝑦𝑛−𝑞
Definition 1. A point (𝑥, 𝑦) ∈ 𝐼𝑥 × 𝐼𝑦 is called an equilibrium
point of the system (10) if 𝑥 = 𝑓(𝑥, 𝑥, . . . , 𝑥, 𝑦, 𝑦, . . . , 𝑦) and
𝑦 = 𝑔(𝑥, 𝑥, 𝑥, . . . , 𝑦, 𝑦, . . . , 𝑦).
(5)
𝑛 = 0, 1, . . . ,
was studied by Papaschinopoulos and Schinas [7], where
𝑝, 𝑞 ∈ N.
Moreover, the system of rational difference equations
𝑦𝑛
𝑥𝑛
, 𝑦𝑛+1 =
,
𝑥𝑛+1 =
𝑎 + 𝑐𝑦𝑛
𝑏 + 𝑑𝑥𝑛
(6)
Definition 2. The linearized system of the system (10) about
the equilibrium (𝑥, 𝑦) is the system of linear difference
equations
𝑚−1
𝑥𝑛+1 = ∑ (
𝑖=0
𝑛 = 0, 1, . . . ,
+
was studied by Clark et al. [8, 9], where 𝑎, 𝑏, 𝑐, 𝑑 ∈ (0, ∞) and
𝑥0 , 𝑦0 ∈ [0, ∞).
Liu et al. [10] studied the behavior of a system of rational
difference equations
𝑦𝑛−1
𝑥𝑛−1
, 𝑦𝑛+1 =
,
𝑥𝑛+1 =
𝑦𝑛 𝑥𝑛−1 − 1
𝑥𝑛 𝑦𝑛−1 − 1
(7)
1
, 𝑛 = 0, 1, . . . ,
𝑧𝑛+1 =
𝑥𝑛 𝑧𝑛−1
where the initial conditions are nonzero real numbers.
In 2012, Zhang et al. [11] studied the solutions, stability
character, and asymptotic behavior of the system of a rational
third-order difference equation
𝑦𝑛−2
𝑥𝑛−2
,
𝑦𝑛+1 =
,
𝑥𝑛+1 =
𝐴 + 𝑦𝑛 𝑦𝑛−1 𝑦𝑛−2
𝐵 + 𝑥𝑛 𝑥𝑛−1 𝑥𝑛−2 (8)
𝑛 = 0, 1, . . . ,
where 𝐴, 𝐵, 𝑥0 , 𝑥−1 , 𝑥−2 , 𝑦0 , 𝑦−1 , 𝑦−2 ∈ (0, ∞).
In this paper, we studied the solutions, stability character,
and asymptotic behavior of the system of a rational 𝑚-order
difference equation
𝑥𝑛−𝑚+1
,
𝑥𝑛+1 =
𝐴 + 𝑦𝑛 𝑦𝑛−1 ⋅ ⋅ ⋅ 𝑦𝑛−𝑚+1
(9)
𝑦𝑛−𝑚+1
𝑦𝑛+1 =
, 𝑛 = 0, 1, . . . ,
𝐵 + 𝑥𝑛 𝑥𝑛−1 ⋅ ⋅ ⋅ 𝑥𝑛−𝑚+1
where 𝐴, 𝐵, 𝑥0 , 𝑥−1 , . . . , 𝑥−𝑚+1 , 𝑦0 , 𝑦−1 , . . . , 𝑦−𝑚+1 ∈ (0, ∞).
2. Preliminaries
Let 𝑚 ∈ N and let 𝑓 : 𝐼𝑥𝑚 × 𝐼𝑦𝑚 → 𝐼𝑥 and 𝑔 : 𝐼𝑥𝑚 × 𝐼𝑦𝑚 → 𝐼𝑦
be continuously differentiable functions, where 𝐼𝑥 and 𝐼𝑦 are
intervals in R.
For any (𝑥0 , 𝑦0 ), (𝑥−1 , 𝑦−1 ), . . . , (𝑥−𝑚+1 , 𝑦−𝑚+1 ) ∈ 𝐼𝑥 × 𝐼𝑦 ,
the system of difference equations
𝑥𝑛+1 = 𝑓 (𝑥𝑛 , 𝑥𝑛−1 , . . . , 𝑥𝑛−𝑚+1 , 𝑦𝑛 , 𝑦𝑛−1 , . . . , 𝑦𝑛−𝑚+1 ) ,
𝑦𝑛+1 = 𝑔 (𝑥𝑛 , 𝑥𝑛−1 , . . . , 𝑥𝑛−𝑚+1 , 𝑦𝑛 , 𝑦𝑛−1 , . . . , 𝑦𝑛−𝑚+1 ) ,
𝑛 = 0, 1, . . . ,
(10)
has a unique solution
{(𝑥𝑛 , 𝑦𝑛 )}∞
𝑛=−𝑚+1 .
𝜕𝑓 (𝑥, 𝑥, . . . , 𝑥, 𝑦, 𝑦, . . . , 𝑦)
𝑥𝑛−𝑖
𝜕𝑥𝑛−𝑖
𝑦𝑛+1
𝜕𝑓 (𝑥, 𝑥, . . . , 𝑥, 𝑦, 𝑦, . . . , 𝑦)
𝑦𝑛−𝑖 ) ,
𝜕𝑦𝑛−𝑖
𝑚−1
𝜕𝑔 (𝑥, 𝑥, . . . , 𝑥, 𝑦, 𝑦, . . . , 𝑦)
= ∑(
𝑥𝑛−𝑖
𝜕𝑥𝑛−𝑖
𝑖=0
+
(11)
𝜕𝑔 (𝑥, 𝑥, . . . , 𝑥, 𝑦, 𝑦, . . . , 𝑦)
𝑦𝑛−𝑖 ) .
𝜕𝑦𝑛−𝑖
Definition 3. An equilibrium point (𝑥, 𝑦) of the system (10)
is said to be stable relative to 𝐼𝑥 × 𝐼𝑦 if for every 𝜖 > 0,
there exists 𝛿 > 0 such that for any (𝑥0 , 𝑦0 ), (𝑥−1 , 𝑦−1 ), . . . ,
(𝑥−𝑚+1 , 𝑦−𝑚+1 ) ∈ 𝐼𝑥 × 𝐼𝑦 , with
0
0
𝑖=−𝑚+1
𝑖=−𝑚+1
󵄨
󵄨
󵄨
󵄨
max { ∑ 󵄨󵄨󵄨𝑥𝑖 − 𝑥󵄨󵄨󵄨 , ∑ 󵄨󵄨󵄨𝑦𝑖 − 𝑦󵄨󵄨󵄨} < 𝛿.
(12)
One has max{|𝑥𝑛 − 𝑥|, |𝑦𝑛 − 𝑦|} < 𝜖 for all 𝑛 ≥ −𝑚 + 1.
Definition 4. An equilibrium point (𝑥, 𝑦) of the system
(10) is called an attractor relative to 𝐼𝑥 × 𝐼𝑦 if for all
(𝑥0 , 𝑦0 ), (𝑥−1 , 𝑦−1 ), . . . , (𝑥−𝑚+1 , 𝑦−𝑚+1 ) ∈ 𝐼𝑥 × 𝐼𝑦 , one has
lim𝑛 → ∞ 𝑥𝑛 = 𝑥 and lim𝑛 → ∞ 𝑦𝑛 = 𝑦.
Definition 5. An equilibrium point (𝑥, 𝑦) of the system (10) is
said to be asymptotically stable relative to 𝐼𝑥 × 𝐼𝑦 if it is stable,
and it is also an attractor.
Definition 6. An equilibrium point (𝑥, 𝑦) of the system (10) is
said to be unstable if it is not stable.
Theorem 7 (see [12]). Let 𝑋(𝑛 + 1) = 𝐹(𝑋(𝑛)), 𝑛 = 0, 1, . . .,
be a system of difference equations and let 𝑋 be the equilibrium
point of the system. If all eigenvalues of the Jacobian matrix
evaluated at 𝑋 lie inside the open unit disk, then 𝑋 is asymptotically stable. If one of them has a modulus greater than one,
then 𝑋 is unstable.
Theorem 8 (see [13]). Let 𝑋(𝑛 + 1) = 𝐹(𝑋(𝑛)), 𝑛 = 0, 1, . . .,
be a system of difference equations and let 𝑋 be the equilibrium
point of the system. Assume that the characteristic polynomial
of the system about 𝑋 is 𝑎0 𝜆𝑛 + 𝑎1 𝜆𝑛−1 + ⋅ ⋅ ⋅ + 𝑎𝑛−1 𝜆 + 𝑎𝑛 where
𝑎𝑖 ∈ R for all 𝑖 and 𝑎0 > 0. Then all roots of the characteristic
equation lie inside the open unit disk if and only if Δ 𝑘 > 0 for
Discrete Dynamics in Nature and Society
3
all positive integer 𝑘 ≤ 𝑛, where Δ 𝑘 is the principal minor of
order 𝑘 of the 𝑛 × 𝑛 matrix
⋅⋅⋅ 0
⋅⋅⋅ 0
⋅⋅⋅ 0).
.
d ..
0 0 0 ⋅ ⋅ ⋅ 𝑎𝑛
𝑎1
𝑎0
Δ𝑛 = ( 0
..
.
𝑎3
𝑎2
𝑎1
..
.
𝑎5
𝑎4
𝑎3
..
.
Theorem 9. Let (𝑥𝑛 , 𝑦𝑛 ) be positive solution of the system (9).
For all nonnegative integer 𝑘, one has
𝑥−𝑚+1
,
{
{
{
𝐴𝑘+1
{
{
{
{ 𝑥−𝑚+2
{
{
{
{ 𝐴𝑘+1 ,
0 ≤ 𝑥𝑛 ≤ {
{
{ ..
{
{
.
{
{
{
𝑥0
{
{
,
{ 𝑘+1
𝐴
{
𝑦−𝑚+1
,
{
{
{
𝐵𝑘+1
{
{
{
𝑦
{
{
{ −𝑚+2 ,
{
𝑘+1
0 ≤ 𝑦𝑛 ≤ { 𝐵
{
{
..
{
{
{
.
{
{
{
{ 𝑦0
,
{ 𝐵𝑘+1
(13)
3. Results
We note that
(i) if 𝐴 < 1 and 𝐵 < 1 then the system (9) has equilibrium
𝑚
𝑚
1 − 𝐵, √
1 − 𝐴);
(0, 0) and ( √
(ii) if 𝐴 = 1 and 𝐵 < 1 then the system (9) has equilibrium
𝑚
1 − 𝐵, 0);
(0, 0) and ( √
(iii) if 𝐴 < 1 and 𝐵 = 1 then the system (9) has equilibrium
𝑚
1 − 𝐴);
(0, 0) and (0, √
(iv) if 𝐴 > 1 and 𝐵 > 1 then (0, 0) is the unique equilibrium point of the system (9).
𝑛 = 𝑚𝑘 + 1;
𝑛 = 𝑚𝑘 + 2;
𝑛 = 𝑚𝑘 + 𝑚,
(14)
𝑛 = 𝑚𝑘 + 1;
𝑛 = 𝑚𝑘 + 2;
𝑛 = 𝑚𝑘 + 𝑚.
Proof. Obviously, they are true for 𝑘 = 0. Suppose that they
are true for 𝑘 = 𝑙. Then
𝑥𝑚(𝑙+1)−𝑚+1
1 𝑥
1
{
) , 𝑛 = 𝑚 (𝑙 + 1) + 1;
= 𝑥𝑚𝑙+1 ≤ ( −𝑚+1
𝑥𝑚(𝑙+1)+1 ≤
{
{
𝐴
𝐴
𝐴
{
𝐴𝑙+1
{
{
{
{
𝑥𝑚(𝑙+1)−𝑚+2
1 𝑥
1
{
{
) , 𝑛 = 𝑚 (𝑙 + 1) + 2;
𝑥𝑚(𝑚+1)+2 ≤
= 𝑥𝑚𝑙+2 ≤ ( −𝑚+2
{
{
{
𝐴
𝐴
𝐴 𝐴𝑙+1
𝑥𝑛 = {
{
..
{
{
.
{
{
{
{
𝑥𝑚(𝑙+1)
1 𝑥0
1
{
{
𝑛 = 𝑚 (𝑙 + 1) + 𝑚,
{
{𝑥𝑚(𝑙+1)+𝑚 ≤ 𝐴 = 𝐴 𝑥𝑚𝑙+𝑚 ≤ 𝐴 ( 𝐴𝑙+1 ) ,
{
{
𝑦𝑚(𝑙+1)−𝑚+1 1
1 𝑦
{
) , 𝑛 = 𝑚 (𝑙 + 1) + 1;
= 𝑦𝑚𝑙+1 ≤ ( −𝑚+1
{
{ 𝑦𝑚(𝑙+1)+1 ≤
𝐵
𝐵
𝐵
𝐵𝑙+1
{
{
{
{
{
𝑦
{
{ 𝑦4(𝑙+1)+2 ≤ 𝑚(𝑙+1)−𝑚+2 = 1 𝑦𝑚𝑙+2 ≤ 1 ( 𝑦−𝑚+2 ) , 𝑛 = 4 (𝑚 + 1) + 2;
𝑦𝑛 = {
𝐵
𝐵
𝐵 𝐵𝑙+1
{
.
{
{
..
{
{
{
{
𝑦𝑚(𝑙+1) 1
{
1 𝑦0
{𝑦
),
𝑛 = 𝑚 (𝑙 + 1) + 𝑚.
= 𝑦𝑚𝑙+𝑚 ≤ ( 𝑙+1
𝑚(𝑙+1)+𝑚 ≤
𝐵
𝐵
𝐵 𝐵
{
Thus, they are true for 𝑘 = 𝑙 + 1.
By the mathematical induction, this proof is completed.
Corollary 10. Let (𝑥𝑛 , 𝑦𝑛 ) be positive solution of the system
(9). If 𝐴 > 1 and 𝐵 > 1, then the sequence {(𝑥𝑛 , 𝑦𝑛 )} converges
exponentially to the equilibrium point (0, 0).
Theorem 11. Let 𝐴 > 1 and 𝐵 > 1. Then the equilibrium point
(0, 0) of the system (9) is asymptotically stable.
Proof. The linearized system of the system (9) about the
equilibrium (0, 0) is
Φ𝑛+1 = 𝐷Φ𝑛 ,
(16)
where
𝑥𝑛
𝑥𝑛−1
( 𝑥𝑛−2 )
( .. )
( . )
)
(
(𝑥𝑛−𝑚+1 )
)
Φ𝑛 = (
( 𝑦𝑛 ) ,
)
(
( 𝑦𝑛−1 )
)
(
( 𝑦𝑛−2 )
..
.
(𝑦𝑛−𝑚+1 )
(15)
4
Discrete Dynamics in Nature and Society
1
0
𝐴
1 0 ⋅⋅⋅ 0 0 0
0 0 ⋅⋅⋅ 0
(
(
(0
(
( ..
(.
(
(0
𝐷=(
(
(0
(
(
(0
(
(
(0
..
.
1 ⋅⋅⋅ 0
0
0 ⋅⋅⋅ 0
0
0 ⋅⋅⋅ 0
0
0
0 ⋅⋅⋅
.
d d d d d d ..
0 ⋅⋅⋅ 1 0 0 0 0
0
..
.
0 ⋅⋅⋅ 0
0
0
0
0
0
0 ⋅⋅⋅ 0
0
1
0
0
0
0 ⋅⋅⋅ 0
0
0
1
0
0
0
d d d d d d d d
0
( 0 ⋅⋅⋅ 0 0 0 0 0 1
0
1
(0
(
( ..
(.
(
(0
(
𝐺=(
(𝛽
(
(0
(
(0
(
..
.
(0
)
)
0)
)
.. )
.)
)
0)
)
1).
)
𝐵)
)
0)
)
)
0)
..
.
0)
0 ⋅⋅⋅ 0
0 ⋅⋅⋅ 0
1 ⋅⋅⋅ 0
1
0
0
𝛼
0
0
d
0
𝛽
0
0
d
0
𝛽
0
0
d
0
0
1
0
d
⋅⋅⋅
⋅⋅⋅
⋅⋅⋅
⋅⋅⋅
d
1
𝛽
0
0
d d d d d
0 ⋅⋅⋅ 0 0 0
𝛼 ⋅⋅⋅
0 ⋅⋅⋅
0 ⋅⋅⋅
.
d ..
0 0
0 0
0 0
1 0
𝛼
0
0
..
.
0
0
0
0
𝛼
0
0)
)
.. )
.)
)
0)
)
),
1)
)
0)
)
0)
)
..
d d d .
0 0 1 0)
(20)
(17)
in which
𝛼 = − √(1 − 𝐴)𝑚−1 (1 − 𝐵),
𝑚
The characteristic equation of the system (16) is
(21)
1
1
(𝜆 − ) (𝜆𝑚 − ) = 0.
𝐴
𝐵
𝑚
𝛽 = − √(1 − 𝐴) (1 − 𝐵)𝑚−1 .
𝑚
(18)
The characteristic polynomial of the system (19) is
𝑚
Thus, |𝜆| < 1. By Theorem 7, the equilibrium point (0, 0) is
asymptotically stable.
1 − 𝛼𝛽 − ∑ (𝑖𝛼𝛽𝜆𝑖−1 ) − 2𝜆𝑚
𝑖=1
(22)
Theorem 12. Let 𝐴 < 1 and 𝐵 < 1. Then both the equilibrium
𝑚
𝑚
1 − 𝐵, √
1 − 𝐴) of the system (9) are
points (0, 0) and ( √
unstable.
Proof. We note by the characteristic equation (18) that |𝜆| >
1 and then, by Theorem 7, the equilibrium point (0, 0) is
unstable.
𝑚
1 − 𝐵,
Next, we consider the equilibrium point ( √
𝑚
√1 − 𝐴). The linearized system of the system (9) about the
𝑚
𝑚
equilibrium ( √
1 − 𝐵, √
1 − 𝐴) is
Φ𝑛+1 = 𝐺Φ𝑛 ,
(19)
𝑚−1
− ∑ (𝑖𝛼𝛽𝜆2𝑚−(𝑖+1) ) + 𝜆2𝑚 .
𝑖=1
We note the characteristic polynomial 𝑎0 𝜆2𝑚 + 𝑎1 𝜆2𝑚−1 +
⋅ ⋅ ⋅ + 𝑎2𝑚−1 𝜆 + 𝑎2𝑚 that 𝑎1 = 0. Thus, we obtain that not
all of Δ 𝑘 > 0, 𝑘 = 1, 2, . . . , 2𝑚. By Theorems 7 and 8, the
𝑚
𝑚
1 − 𝐵, √
1 − 𝐴) is unstable.
equilibrium point ( √
𝑚
𝑚
Theorem 13. Let 𝐴, 𝐵 < 1 and Ω1 = (0, √
1 − 𝐵) × ( √
1 − 𝐴,
𝑚
𝑚
∞), Ω2 = ( √1 − 𝐵, ∞) × (0, √1 − 𝐴). Assume that {(𝑥𝑛 ,
𝑦𝑛 )}∞
𝑛=−𝑚+1 satisfies the system (9). Then
(i) if {(𝑥𝑛 , 𝑦𝑛 )}0𝑛=−𝑚+1 ⊆ Ω1 , then {(𝑥𝑛 , 𝑦𝑛 )}∞
𝑛=−𝑚+1 ⊆ Ω1 ;
(ii) if {(𝑥𝑛 , 𝑦𝑛 )}0𝑛=−𝑚+1 ⊆ Ω2 , then {(𝑥𝑛 , 𝑦𝑛 )}∞
𝑛=−𝑚+1 ⊆ Ω2 .
where
𝑥𝑛
𝑥𝑛−1
( 𝑥𝑛−2 )
( .. )
( . )
)
(
(𝑥𝑛−𝑚+1 )
)
Φ𝑛 = (
( 𝑦𝑛 ) ,
)
(
( 𝑦𝑛−1 )
)
(
( 𝑦𝑛−2 )
..
.
(𝑦𝑛−𝑚+1 )
Proof. (i) Assume that {(𝑥𝑛 , 𝑦𝑛 )}0𝑛=−𝑚+1 ⊆ Ω1 . Then, for any
𝑖 ∈ {0, 1, . . . , 𝑚 − 1},
𝑥𝑖+1 =
𝑦𝑖+1 =
𝑥𝑖−𝑚+1
𝑥𝑖−𝑚+1
<
𝑚 = 𝑥𝑖−𝑚+1 ,
𝑚
𝐴 + 𝑦𝑖 𝑦𝑖−1 ⋅ ⋅ ⋅ 𝑦𝑖−𝑚+1 𝐴 + ( √
1 − 𝐴)
𝑦𝑖−𝑚+1
𝑦𝑖−𝑚+1
>
𝑚 = 𝑦𝑖−𝑚+1 .
𝑚
𝐵 + 𝑥𝑖 𝑥𝑖−1 ⋅ ⋅ ⋅ 𝑥𝑖−𝑚+1 𝐵 + ( √
1 − 𝐵)
(23)
Then (𝑥1 , 𝑦1 ), (𝑥2 , 𝑦2 ), . . . , (𝑥𝑚 , 𝑦𝑚 ) ∈ Ω1 .
Discrete Dynamics in Nature and Society
Next, we suppose that (𝑥𝑘 , 𝑦𝑘 ), (𝑥𝑘−1 , 𝑦𝑘−1 ), . . . , (𝑥𝑘−𝑚+1 ,
𝑦𝑘−𝑚+1 ) ∈ Ω1 where 𝑘 is a positive integer. Then
𝑥𝑘+1 =
𝑦𝑘+1 =
𝑥𝑘−𝑚+1
𝑥𝑘−𝑚+1
<
𝑚 = 𝑥𝑘−𝑚+1 ,
𝑚
𝐴 + 𝑦𝑘 𝑦𝑘−1 ⋅ ⋅ ⋅ 𝑦𝑘−𝑚+1 𝐴 + ( √
1 − 𝐴)
𝑦𝑘−𝑚+1
𝑦𝑘−𝑚+1
>
𝑚 = 𝑦𝑘−𝑚+1 .
𝑚
𝐵 + 𝑥𝑘 𝑥𝑘−1 ⋅ ⋅ ⋅ 𝑥𝑘−𝑚+1 𝐵 + ( √
1 − 𝐵)
(24)
Then (𝑥𝑘+1 , 𝑦𝑘+1 ) ∈ Ω1 .
By the mathematical induction, {(𝑥𝑛 , 𝑦𝑛 )}∞
𝑛=−𝑚+1 ⊆ Ω1 .
(ii) This is similar to the proof of (i).
Acknowledgment
The author would like to thank the referees for their useful
comments and suggestions.
References
[1] A. S. Kurbanli, C. Çinar, and I. Yalçinkaya, “On the behavior of
positive solutions of the system of rational difference equations
𝑥𝑛+1 = 𝑥𝑛−1 /(𝑦𝑛 𝑥𝑛−1 + 1), 𝑦𝑛+1 = 𝑦𝑛−1 /(𝑥𝑛 𝑦𝑛−1 + 1),” Mathematical and Computer Modelling, vol. 53, no. 5-6, pp. 1261–1267, 2011.
[2] A. S. Kurbanli, “On the behavior of solutions of the system of
rational difference equations: 𝑥𝑛+1 = 𝑥𝑛−1 /(𝑦𝑛 𝑥𝑛−1 − 1), 𝑦𝑛+1 =
𝑦𝑛−1 /(𝑥𝑛 𝑦𝑛−1 −1), and 𝑧𝑛+1 = 𝑧𝑛−1 /(𝑦𝑛 𝑧𝑛−1 −1),” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 932362, 12 pages,
2011.
[3] A. S. Kurbanli, “On the behavior of solutions of the system of
rational difference equations: 𝑥𝑛+1 = 𝑥𝑛−1 /(𝑦𝑛 𝑥𝑛−1 − 1), 𝑦𝑛+1 =
𝑦𝑛−1 /(𝑥𝑛 𝑦𝑛−1 − 1), and 𝑧𝑛+1 = 𝑧𝑛−1 /(𝑦𝑛 𝑧𝑛−1 − 1),” Discrete
Dynamics in Nature and Society, vol. 2011, Article ID 932362,
12 pages, 2011.
[4] K. Liu, Z. Zhao, X. Li, and P. Li, “More on three-dimensional
systems of rational difference equations,” Discrete Dynamics in
Nature and Society, vol. 2011, Article ID 178483, 9 pages, 2011.
[5] S. Stević, “On a system of difference equations,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3372–3378, 2011.
[6] Y. Gu and R. Ding, “Observable state space realizations for multivariable systems,” Computers & Mathematics with Applications, vol. 63, no. 9, pp. 1389–1399, 2012.
[7] G. Papaschinopoulos and C. J. Schinas, “On a system of two
nonlinear difference equations,” Journal of Mathematical Analysis and Applications, vol. 219, no. 2, pp. 415–426, 1998.
[8] D. Clark and M. R. S. Kulenović, “A coupled system of rational
difference equations,” Computers & Mathematics with Applications, vol. 43, no. 6-7, pp. 849–867, 2002.
[9] D. Clark, M. R. S. Kulenović, and J. F. Selgrade, “Global asymptotic behavior of a two-dimensional difference equation modelling competition,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 52, no. 7, pp. 1765–1776, 2003.
[10] K. Liu, Z. Wei, P. Li, and W. Zhong, “On the behavior of a system
of rational difference equations 𝑥𝑛+1 = 𝑥𝑛−1 /(𝑦𝑛 𝑥𝑛−1 − 1), 𝑦𝑛+1 =
𝑦𝑛−1 /(𝑥𝑛 𝑦𝑛−1 −1), 𝑧𝑛+1 = 1/𝑥𝑛 𝑧𝑛−1 ,” Discrete Dynamics in Nature
and Society, vol. 2012, Article ID 105496, 9 pages, 2012.
[11] Q. Zhang, L. Yang, and J. Liu, “Dynamics of a system of rational
third-order difference equation,” Advances in Difference Equations, vol. 2012, article 136, p. 6, 2012.
5
[12] H. Sedaghat, Nonlinear Difference Equations, vol. 15 of Mathematical Modelling: Theory and Applications, Kluwer Academic
Publishers, Dordrecht, The Netherlands, 2003.
[13] V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, vol. 256 of
Mathematics and Its Applications, Kluwer Academic Publishers,
Dordrecht, The Netherlands, 1993.
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