Global behavior of a higher order rational difference equation

DIFFERENCE EQUATION

Received 17 January 2006; Revised 6 April 2006; Accepted 12 April 2006

We investigate in this paper the global behavior of the following difference equation: xn+1=(Pk(xn i0,xn i1,...,xn i2k) +b)/(Qk(xn i0,xn i1,...,xn i2k) +b), n=0, 1,..., under appropriate assumptions, whereb [0,), k1,i0,i1,...,i2k 0, 1,...withi0< i1< < i2k, the initial conditionsxi

2k,xi 2k+1,...,x0 (0,). We prove that unique equilib-riumx=1 of that equation is globally asymptotically stable.

Copyright © 2006 H. Xi and T. Sun. 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.

1. Introduction

For some difference equations, although their forms (or expressions) look very simple, it is extremely difficult to understand thoroughly the global behaviors of their solutions. Accordingly, one is often motivated to investigate the qualitative behaviors of difference equations (e.g., see [2,3,6,9,10]).

In [6], Ladas investigated the global asymptotic stability of the following rational dif-ference equation:

(E1)

xn+1=xn+xn 1xn 2

xnxn 1+xn 2, n=0, 1,..., (1.1)

where the initial valuesx 2,x 1,x0 R+(0, +).

In [9], Nesemann utilized the strong negative feedback property of [1] to study the following difference equation:

(E2)

xn+1=xn 1+xnxn 2

xnxn 1+xn 2, n=0, 1,..., (1.2)

where the initial valuesx 2,x 1,x0 R+.

Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 27637, Pages1–7

In [10], Papaschinopoulos and Schinas investigated the global asymptotic stability of the following nonlinear difference equation:

(E3)

xn+1=

iZk j 1,jxn i+xn jxn j+1+ 1

iZkxn i

, n=0, 1,..., (1.3)

wherek 1, 2, 3,...,j,j1 Zk0, 1,...,k, and the initial valuesx k,x k+1,..., x0 R+.

Recently, Li [7,8] studied the global asymptotic stability of the following two nonlinear difference equations:

(E4)

xn+1= xn 1xn 2xn 3+xn 1+xn 2+xn 3+a

xn 1xn 2+xn 1xn 3+xn 2xn 3+ 1 +a, n=0, 1,... (1.4)

(E5)

xn+1= xnxn 1xn 3+xn+xn 1+xn 3+a

xnxn 1+xnxn 3+xn 1xn 3+ 1 +a, n=0, 1,..., (1.5)

wherea [0, +) and the initial valuesx 3,x 2,x 1,x0 R+.

Letk1 andi0,i1,...,i2k 0, 1,...withi0< i1<< i2k. LetP0(xn i0)=xn i0and Q0(xn i0)=1, for any 1jk, let

Pjxn i0,...,xn i2j

=xn i2jxn i2j 1+ 1

Pj 1

xn i0,...,xn i2j 2

+xn i2j+xn i2j 1

Qj 1

xn i0,...,xn i2j 2

, Qjxn i0,...,xn i2j

=xn i2jxn i2j 1+ 1

Qj 1

xn i0,...,xn i2j 2

+xn i2j+xn i2j 1

Pj 1

xn i0,...,xn i2j 2

.

(1.6)

In this paper, we consider the following difference equation:

xn+1= Pk

xn i0,xn i1,...,xn i2k

+b Qkxn i0,xn i1,...,xn i2k

+b, n=0, 1,..., (1.7)

whereb [0,) and the initial conditionsx i2

k,x i2k+1,...,x0 (0,). It is easy to see that the positive equilibriumxof (1.7) satisfies

x= Pk(x,x,...,x) +b

Qk(x,x,...,x) +b

=

x2_{+ 1P}

k 1(x,x,...,x) + 2xQk 1(x,x,...,x) +b

x2_{+ 1Q}

k 1(x,x,...,x) + 2xPk 1(x,x,...,x) +b.

Thus, we have

from which one can see that (1.7) has the unique positive equilibriumx=1.

Remark 1.1. Let k=1, then (1.7) is (1.4) when (i0,i1,i2)=(1, 2, 3) and is (1.5) when

(i0,i1,i2)=(0, 1, 3).

2. Properties of positive solutions of (1.7)

In this section, we will study properties of positive solutions of (1.7). Since

Pkxn i0,xn i1,...,xn i2k

n= i2k, then it follows fromProposition 2.2that xn=1 for anyn1an=0 for anyn1 Proof. “If ” part is obvious.

“Only if ” part. Ifxp=1 for somepi2k, thenap=0, wherean

n= i2kis itinerary ofxn

andmt 0, 1,...,is+ 1such that

jtis+ 1=mti2k+ 1+t. (2.3)

Together withProposition 2.2, it follows that

a(is+1)(i2k+1)+t+p=0. (2.4)

Again byProposition 2.2, we havean=0 for anyn(is+ 1)(i2k+ 1) +p, which implies

xn=1 for anyn(is+ 1)(i2k+ 1) +p.

Example 2.5. Consider the equation

xn+1= xn i0xn i1xn 3

+xn i0+xn i1+xn 3+b

xn i0xn i1+xn i0xn 3+xn i1xn 3+ 1 +b, n=0, 1,..., (2.5)

whereb [0,+), 0i0< i1<3, and the initial valuesx 3,x 2,x 1,x0 R+. Letxn n= 3

be a solution of (2.5) whose itinerary isan

n= 3, then the following hold.

(1) If (i0,i1) (0, 1), (1, 2)andxn

n= 3is not eventually equal to 1, thenan n= 3

is a periodic sequence of period 7. (2) If (i0,i1)=(0, 2) andxn

n= 3is not eventually equal to 1, thenan

n= 3is a

peri-odic sequence of period 6. (3)xn=1 for anyn1

0

j= 3(xj1)=0. (4)xn

n= 3is eventually equal to 1xp=1 for somep3.

Proof. (1) If (i0,i1)=(0, 1), then fromProposition 2.2, it follows that for anyn0, an+4=an+3an+2an=an+2an+1an 1an+2an

=an+1anan 1=anan 1an 3anan 1

=an 3.

(2.6)

If (i0,i1)=(1, 2), then in a similar fashion, it is true thatan+4=an 3for anyn0. (2) If (i0,i1)=(0, 2), then fromProposition 2.2, it follows that for anyn0,

an+3=an+2anan 1=an+1an 1an 2anan 1

=an+1anan 2=anan 2an 3anan 2

=an 3.

(2.7)

(3) It follows fromProposition 2.3.

(4) It follows fromProposition 2.4since either gcd(i0+ 1, 4)=1 or gcd(i1+ 1, 4)=1.

3. Global asymptotic stability of (1.7)

Lemma3.1. Let(y0,y1,...,yi2k) Ri2k +1

+ (1, 1,..., 1)andM=maxyj, 1/yj0j i2k, then

1 M < Pk

yi0,yi1,...,yi2k

Qkyi0,yi1,...,yi2k

< M. (3.1)

Proof. Since (y0,y1,...,yi2k) Ri2k +1

+ (1, 1,..., 1)andM=maxyj, 1/yj0ji2k, we haveM >1 and eitherMa >1/M orM > a1/Mfor anya yj, 1/yj0 j i2k.

It is easy to verify that

P1yi0,yi1,yi2=yi1yi2+ 1yi0+yi1+yi2 <yi1yi2+ 1M+yi1+yi2yi0M

=Q1

yi0,yi1,yi2M, P1

yi0,yi1,yi2M=yi1yi2+ 1yi0+yi1+yi2M >yi1yi2+ 1+yi1+yi2yi0

=Q1

yi0,yi1,yi2.

(3.2)

From that we have P2

yi0,yi1,yi2,yi3,yi4=yi3yi4+ 1P1

yi0,yi1,yi2+yi3+yi4Q1

yi0,yi1,yi2 <yi3yi4+ 1Q1

yi0,yi1,yi2M+yi3+yi4P1

yi0,yi1,yi2M

=Q2

yi0,yi1,yi2,yi3,yi4M, P2

yi0,yi1,yi2,yi3,yi4M=yi3yi4+ 1P1

yi0,yi1,yi2+yi3+yi4Q1

yi0,yi1,yi2M >yi3yi4+ 1Q1yi0,yi1,yi2+yi3+yi4P1yi0,yi1,yi2

=Q2

yi0,yi1,yi2,yi3,yi4.

(3.3)

By induction, we have that for any 1jk,

Pjyi0,yi1,...,yi2j

< Qjyi0,yi1,...,yi2j

M,

Pjyi0,yi1,...,yi2j

M > Qjyi0,yi1,...,yi2j

. (3.4)

Thus

1 M < Pk

yi0,yi1,...,yi2k

Qkyi0,yi1,...,yi2k

< M. (3.5)

Letnbe a positive integer and letρdenote the part-metric onRn

and Nussbaum proved that the distances indicated by the part-metric and by the Eu-clidean norm are equivalent onRn

+.

Lemma3.2 [5]. LetT:Rn

+R n

+be a continuous mapping with unique fixed pointx

Rn

+.

Suppose that there exists somel1such that for the part-metricρ,

ρTl_x,x

Theorem3.3. The unique equilibriumx=1of (1.7) is globally asymptotically stable. Proof. Letxn

n= i2kis not eventually equal to 1 since otherwise there is nothing to show. Denoted byT:Ri2k+1

n= i2k of (1.7) is represented by the first component of the solution

yn

n=0of the systemyn+1=T ynwith initial condition y0=(x i2k,x i2k+1,...,x0). It

fol-lows fromLemma 3.1that for alln0, the following inequalities hold:

min

from which it follows that

Thus, forx

=(1, 1,..., 1) and the part-metricρofRi2k+1

+ , we have

ρTi2k+1_y

n,x

=log min

xn+i, _x1 n+i

1ii2k+ 1

<log min

xn i, 1 xn i

0ii2k

=ρyn,x

(3.12)

for alln0. It follows fromLemma 3.2that the positive equilibriumx=1 of (1.7) is

globally asymptotically stable.

Acknowledgments

Project supported by NNSF of China (10461001, 10361001) and NSF of Guangxi (0640205).

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Hongjian Xi: Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530004, China

E-mail address:[email protected]

Taixiang Sun: Department of Mathematics, College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China