Báo cáo hóa học: "ON THE SYSTEM OF RATIONAL DIFFERENCE EQUATIONS xn+1 = f (xn , yn−k ), yn+1 = f (yn ,xn−k )" potx

ON THE SYSTEM OF RATIONAL DIFFERENCE
EQUATIONS x
n+1
= f (x
n
, y
n−k
), y
n+1
= f (y
n
,x
n−k
)
TAIXIANG SUN, HONGJIAN XI, AND LIANG HONG
Received 15 September 2005; Revised 27 October 2005; Accepted 13 November 2005
We study the global asymptotic behavior of the positive solutions of the system of rational
difference equations x
n+1
= f (x
n
, y
n−k
), y
n+1
= f (y
n
,x
n−k
), n = 0,1,2, , under appro-
priate assumptions, where k

∈{1,2, } and the initial values x
−k
,x
−k+1
, ,x
0
, y
−k
, y
−k+1
,
, y
0
∈ (0,+∞). We give sufficient conditions under which every positive solution of this
equation converges to a positive equilibrium. The main theorem in [1]isincludedinour
result.
Copyright © 2006 Taixiang Sun et al. 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 cite d.
1. Introduction
Recently there has been published quite a lot of works concerning the behavior of posi-
tive solutions of systems of rational difference equations [2–7]. These results are not only
valuable in their own right, but they can provide insight into their differential counter-
parts.
In [1], Camouzis and Papaschinopoulos studied the global asymptotic behavior of the
positive solutions of the system of rational difference equations
x
n+1
= 1+
x

n
y
n−k
,
y
n+1
= 1+
y
n
x
n−k
,
n
= 0,1,2, , (1.1)
where k
∈{1,2, } and the initial values x
−k
,x
−k+1
, ,x
0
, y
−k
, y
−k+1
, , y
0
∈ (0,+∞).
To be motivated by the above studies, in this paper, we consider the more general
equation

x
n+1
= f (x
n
, y
n−k
),
y
n+1
= f (y
n
,x
n−k
),
n
= 0,1,2, , (1.2)
Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2006, Article ID 16949, Pages 1–7
DOI 10.1155/ADE/2006/16949
2 Systemofrationaldifference equations
where k
∈{1,2, }, the initial values x
−k
,x
−k+1
, ,x
0
, y
−k

, y
−k+1
, , y
0
∈ (0,+∞)and f
satisfies the following hypotheses.
(H
1
) f ∈ C(E × E,(0,+∞)) with a = inf
(u,v)∈E×E
f (u,v) ∈ E,whereE ∈{(0,+∞),
[0,+
∞)}.
(H
2
) f (u,v) is increasing in u and decreasing in v.
(H
3
) There exists a decreasing function g ∈ C((a,+∞),(a,+∞)) such that
(i) For any x>a, g(g(x))
= x and x = f (x,g(x));
(ii) lim
x→a
+
g(x) = +∞ and lim
x→+∞
g(x) = a.
A pair of sequences of positive real numbers
{(x
n

, y
n
)}
∞
n=−k
that satisfies (1.2)isapos-
itive solution of (1.2). If a p ositive solution of (1.2) is a pair of positive constants (x, y),
then (x, y) is called a positive equilibrium of (1.2). In this paper, our main result is the
following theorem.
Theorem 1.1. Assume that (H
1
)–(H
3
) hold. Then the following statements are true.
(i) Every pair of positive constant (x, y)
∈ (a,+∞) × (a,+∞) satisfying the equation
y
= g(x) (1.3)
is a positive equilibrium of (1.2).
(ii) Every positive solution of (1.2) converges to a positive equilibrium (x, y) of (1.2)sat-
isfying (1.3)asn
→∞.
2. Proof of Theorem 1.1
In this section we will prove Theorem 1.1. To do this we need the following lemma.
Lemma 2.1. Let
{(x
n
, y
n
)}

∞
n=−k
be a positive solution of (1.2). Then there exists a real num-
ber L
∈ (a,+∞) with L<g(L) such that x
n
, y
n
∈ [L,g(L)] for all n ≥ 1.Furthermore,if
limsupx
n
= M, liminf x
n
= m, limsup y
n
= P, liminf y
n
= p, then M = g(p) and P =
g(m).
Proof. From (H
1
)and(H
2
), we have
x
i
= f

x
i−1

, y
i−1−k

>f

x
i−1
, y
i−1−k
+1

≥
a,
y
i
= f

y
i−1
,x
i−1−k

>f

y
i−1
,x
i−1−k
+1


≥
a,
for ev ery 1
≤ i ≤ k +1. (2.1)
Since lim
x→a
+
g(x) = +∞, there exists L ∈ (a,+∞)withL<g(L)suchthat
x
i
, y
i
∈

L,g(L)

for every 1 ≤ i ≤ k +1. (2.2)
It follows from (2.2)and(H
3
)that
g(L)
= f

g(L),L

≥
x
k+2
= f


x
k+1
, y
1

≥
f

L,g(L)

=
L,
g(L)
= f

g(L),L

≥
y
k+2
= f

y
k+1
,x
1

≥
f


L,g(L)

=
L.
(2.3)
Inductively, we ha v e t hat x
n
, y
n
∈ [L,g(L)] for all n ≥ 1.
Taixiang Sun et al. 3
Let l imsupx
n
= M, liminf x
n
= m,limsupy
n
= P, liminf y
n
= p, then there exist se-
quences l
n
≥ 1ands
n
≥ 1suchthat
lim
n→∞
x
l
n

= M,lim
n→∞
y
s
n
= p. (2.4)
Without loss of generality, we may assume (by taking a subsequence) that there exist
A,D ∈ [m,M]andB,C ∈ [p,P]suchthat
lim
n→∞
x
l
n
−1
= A,
lim
n→∞
y
l
n
−k−1
= B,
lim
n→∞
y
s
n
−1
= C,
lim

n→∞
x
s
n
−k−1
= D.
(2.5)
Thus, from (1.2), (H
2
)and(H
3
), we have
f

M, g(M)

=
M = f (A, B) ≤ f (M, p),
f

p,g(p)

=
p = f (C, D) ≥ f (p,M),
(2.6)
from which it follows that
g(M)
≥ p, g(p) ≤ M. (2.7)
By (H
3

), we obtain
p
= g

g(p)

≥
g(M). (2.8)
Therefore, M
= g(p). By the symmetr y, we have also P = g(m). Lemma 2.1 is proven. 
Proof of Theorem 1.1.
(i) Is obv ious.
(ii) Let
{(x
n
, y
n
)}
∞
n=−k
be a positive solution of (1.2) with the initial conditions x
0
,
x
−1
, ,x
−k
, y
0
, y

−1
, , y
−k
∈ (0,+∞). By Lemma 2.1,wehavethat
a<liminf x
n
= g(P) ≤ limsupx
n
= M<+∞,
a<liminf y
n
= g(M) ≤ limsup y
n
= P<+∞.
(2.9)
Without loss of generality, we may assume (by taking a subsequence) that there exists a
sequence l
n
≥ 4k such that
lim
n→∞
x
l
n
= M,
lim
n→∞
x
l
n

− j
= M
j
∈

g(P), M

,forj ∈{1,2, ,3k +1},
lim
n→∞
y
l
n
− j
= P
j
∈

g(M),P

,forj ∈{1,2,···,3k +1}.
(2.10)
4 Systemofrationaldifference equations
From (1.2), (2.10)and(H
3
), we have
f

M, g(M)


=
M = f

M
1
,P
k+1

≤
f

M
1
,g(M)

≤
f

M, g(M)

, (2.11)
from which it follows that
M
1
= M, P
k+1
= g(M). (2.12)
In a similar fashion, we may obtain that
f


M, g(M)

=
M = M
1
= f

M
2
,P
k+2

≤
f

M
2
,g(M)

≤
f

M, g(M)

, (2.13)
from which it follows that
M
2
= M, P
k+2

= g(M). (2.14)
Inductively, we ha v e t hat
M
j
= M,
P
k+ j
= g(M),
for j
∈{1,2, ,2k +1}, (2.15)
from which it follows that
lim
n→∞
x
l
n
− j
= M,forj ∈{0,1, ,2k +1},
lim
n→∞
y
l
n
− j
= g(M), for j ∈{k +1, ,3k +1}.
(2.16)
In view (2.16), for any 0 <ε<M
− a, there exists some l
s
≥ 4k such that

M
− ε<x
l
s
− j
<M+ ε,ifj ∈{0, 1, ,2k +1},
g(M + ε) <y
l
s
− j
<g(M − ε), if j ∈{k +1, ,2k +1}.
(2.17)
From (1.2)and(2.17), we have
y
l
s
−k
= f

y
l
s
−k−1
,x
l
s
−2k−1

<f


g(M − ε),M − ε

=
g(M − ε). (2.18)
Also (1.2), (2.17)and(2.18) implies
x
l
s
+1
= f

x
l
s
, y
l
s
−k

>f

M − ε,g(M − ε)

=
M − ε. (2.19)
Inductively, it follows that
y
l
s
+n−k

<g(M − ε) ∀n ≥ 0,
x
l
s
+n
>M− ε ∀n ≥ 0.
(2.20)
Taixiang Sun et al. 5
Since limsupx
n
= M and liminf y
n
= g(M), we have
lim
n→∞
x
n
= M,lim
n→∞
y
n
= g(M). (2.21)
Thus lim
n→∞
(x
n
, y
n
) = (M,P)withP = g(M). Theorem 1.1 is proven. 
3. Examples

To illustrate the applicability of Theorem 1.1, we present the following examples.
Example 3.1. Consider equation
x
n+1
=
p + x
n
1+y
n−k
,
y
n+1
=
p + y
n
1+x
n−k
,
n
= 0,1, , (3.1)
where k
∈{1,2,···}, the initial conditions x
−k
,x
−k+1
, ,x
0
, y
−k
, y

−k+1
, , y
0
∈ (0,+∞)
and p
∈ (0,+∞). Let E = [0,+∞)and
f (x, y)
=
p + x
1+y
(x
≥ 0, y ≥ 0), g(x) =
p
x
(x>0). (3.2)
It is easy to verify that (H
1
)–(H
3
)holdfor(3.1). It follows from Theorem 1.1 that
(i) every pair of positive constant (x, y)
∈ (0,+∞) × (0,+∞) satisfy ing the equation
xy
= p (3.3)
is a positive equilibrium of (3.1).
(ii) every positive solution of (3.1) converges to a positive equilibrium (x, y)of(3.1)
satisfying (3.3)asn
→∞.
Example 3.2. Consider equation
x

n+1
= 1+
x
n
y
n−k
,
y
n+1
= 1+
y
n
x
n−k
,
n
= 0,1, , (3.4)
where k
∈{1,2, } and the initial conditions x
−k
,x
−k+1
, ,x
0
, y
−k
, y
−k+1
, , y
0

∈(0,+∞).
Let E
= (0,+∞)and
f (x, y)
= 1+
x
y
(x>0, y>0), g(x)
=
x
x − 1
(x>1). (3.5)
It is easy to verify that (H
1
)–(H
3
)holdfor(3.4). It follows from Theorem 1.1 that
(i) every pair of positive constant (x, y)
∈ (1,+∞) × (1,+∞) satisfy ing the equation
xy
= x + y (3.6)
is a positive equilibrium of (3.4);
6 Systemofrationaldifference equations
(ii) every positive solution of (3.4) converges to a positive equilibrium (x, y)of(3.4)
satisfying (3.6)asn
→∞.
Example 3.3. Consider equation
x
n+1
= p +

A + x
n
q + y
n−k
,
y
n+1
= p +
A + y
n
q + x
n−k
,
n
= 0,1, , (3.7)
where k
∈{1,2, }, the initial conditions x
−k
,x
−k+1
, ,x
0
, y
−k
, y
−k+1
, , y
0
∈ (0,+∞),
A

∈ (0,+∞)andp,q ∈ [0,1] with p + q = 1. Let E = (0,+∞)ifp>0andE = [0,+∞)if
p
= 0and
f (x, y)
= p +
A + x
q + y
, (3.8)
then a
= inf
(x,y)∈E×E
f (x, y) = p.Letg(x) = (pq + px + A)/(x − p)(x>p). It is easy to
verify that (H
1
)–(H
3
)holdfor(3.7). It follows from Theorem 1.1 that
(i) every pair of positive constant (x, y)
∈ (p,+∞) × (p,+∞) satisfying the equation
xy
= pq + px+ py+ A (3.9)
is a positive equilibrium of (3.7);
(ii) every positive solution of (3.7) converges to a positive equilibrium (x, y)of(3.7)
satisfying (3.9)asn
→∞
Acknowledgments
I would like to thank the rev iewers for their constructive comments and suggestions.
Project Supported by NNSF of China (10361001,10461001) and NSF of Guangxi
(0447004).
References

[1] E. Camouzis and G. Papaschinopoulos, Global asymptotic behavior of positive solutions on the sys-
tem of rational difference equations x
n+1
= 1+x
n
/y
n−m
, y
n+1
= 1+y
n
/x
n−m
, Applied Mathematics
Letters 17 (2004), no. 6, 733–737.
[2] C. C¸ inar, On the positive solutions of the difference equation system x
n+1
= 1/y
n
, y
n+1
=
y
n
/x
n−1
y
n−1
, Applied Mathematics and Computation 158 (2004), no. 2, 303–305.
[3] D.ClarkandM.R.S.Kulenovi

´
c, A coupled system of rational diffe rence equations, Computers &
Mathematics with Applications 43 (2002), no. 6-7, 849–867.
[4] D.Clark,M.R.S.Kulenovi
´
c,andJ.F.Selgrade,Global asymptotic behavior of a two-dimensional
difference equation modelling competition, Nonlinear Analysis 52 (2003), no. 7, 1765–1776.
[5] E.A.Grove,G.Ladas,L.C.McGrath,andC.T.Teixeira,Ex istence and behavior of solutions of a
rational system, Communications on Applied Nonlinear Analysis 8 (2001), no. 1, 1–25.
[6] G. Papaschinopoulos and C. J. Schinas, On a system of two nonlinear difference equations,Journal
of Mathematical Analysis and Applications 219 (1998), no. 2, 415–426.
Taixiang Sun et al. 7
[7] X. Yang, On the system of rational diffe rence equations x
n
= A + y
n−1
/x
n−p
y
n−q
, y
n
= A +
x
n−1
/x
n−r
y
n−s
, Journal of Mathematical Analysis and Applications 307 (2005), no. 1, 305–311.

Taixiang Sun: Department of Mathematics, Guangxi University, Nanning, Guangxi 530004, China
E-mail address:
Hongjian Xi: Department of Mathematics, Guangxi College of Finance and Economics, Nanning,
Guangxi 530004, China
E-mail address:
Liang Hong: Department of Mathematics, Guangxi University, Nanning, Guangxi 530004, China
E-mail address: