ON THE SYSTEM OF RATIONAL DIFFERENCE EQUATIONS xn

x

f ( y

,x

), y

g (x

, y

)

TAIXIANG SUN AND HONGJIAN XI

Received 20 March 2006; Revised 19 May 2006; Accepted 28 May 2006

We study the global behavior of positive solutions of the system of rational difference equations xn+1= f(yn−q,xn−s), yn+1 =g(xn−t,yn−p), n=0, 1, 2,. . ., where p,q,s,t∈ {0, 1, 2,. . .} withs≥t and p≥q, the initial values x_−s,x_−s+1,. . .,x0,y_−p,y_−p+1,. . .,y0∈ (0, +∞). We give sufficient conditions under which every positive solution of this system converges to the unique positive equilibrium.

Copyright © 2006 T. Sun and H. Xi. 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

In this paper, we study the convergence of positive solutions of a system of rational dif- ference equations. Recently there has been published quite a lot of works concerning the behavior of positive solutions of systems of rational difference equations [1–7,9,11]. Not only these results are valuable in their own right, but also they can provide insight into their differential counterparts.

Papaschinopoulos and Schinas [10] studied the oscillatory behavior, the periodicity, and the asymptotic behavior of the positive solutions of systems of rational difference equations

xn+1=A+x_n−1

yn , yn+1=A+ yn−1

xn , n=0, 1,. . ., (1.1) whereA∈(0, +∞) and the initial valuesx₋1,x0,y₋1,y0∈(0, +∞).

Recently, Kulenovi´c and Nurkanovi´c [8] investigated the global asymptotic behavior of solutions of systems of rational difference equations

xn+1=a+x_n

b+yn, yn+1=d+yn

e+xn, n=0, 1,. . ., (1.2) wherea,b,d,e∈(0, +∞) and the initial valuesx0,y0∈(0, +∞).

Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 51520, Pages1–8 DOI 10.1155/ADE/2006/51520

In this paper, we consider the more general equation xn+1=fyn−q,xn−s

, yn+1=gxn−t,yn−p

, (1.3)

where p,q,s,t∈ {0, 1, 2,. . .}withs≥t and p≥q, the initial valuesx₋s,x₋s+1,. . .,x0,y₋p, y₋p+1,. . .,y0∈(0, +∞) and f satisfies the following hypotheses.

(H1) f(u,v),g(u,v)∈C(E×E, (0, +∞)) with a=inf(u,v)∈E×Ef(u,v)∈E and b= inf_(u,v)∈E×Eg(u,v)∈E, whereE∈ {(0, +∞), [0, +∞)}.

(H2) f(u,v) andg(u,v) are decreasing inuand increasing inv.

(H3) Equation

x= f(y,x), y=g(x,y) (1.4)

has a unique positive solutionx=x,y=y.

(H4) f(b,x) has only one fixed point in the interval (a, +∞), denoted byA, andg(a,y) has only one fixed point in the interval (b, +∞), denoted byB.

(H5) For everyw∈E,f(w,x)/xandg(w,x)/xare nonincreasing inxin (0, +∞).

2. Main results

Theorem 2.1. Assume that (H1)–(H5) hold and{(xn,yn)}is a positive solution of (1.3), then there exists a positive integerNsuch that

f(B,a)≤xn≤A, g(A,b)≤yn≤B, forn≥N. (2.1) Proof. Sincea=inf(u,v)∈E×Ef(u,v)∈Eandb=inf(u,v)∈E×Eg(u,v)∈E, we have

x= f(y,x)> f(y+ 1,x)≥a,

y=g(x,y)> g(x+ 1,y)≥b. (2.2) Claim 1. g(A,b)< y < Band f(B,a)< x < A.

Proof ofClaim 1. IfB≤y, then it follows from (H2), (H4), and (H5) that B=g(a,B)> g(x,B)=Bg(x,B)

B ^≥Bg(x,y)

y ⁼B, (2.3)

which is a contradiction. Thereforey < B. In a similar fashion it is true thatx < A.

Sincey < Bandx < A, we have that

f(B,a)< f(y,x)=x, g(A,b)< g(x,y)=y, (2.4)

Claim 1is proven.

Claim 2. (i) For alln≥q+ 1,xn+1≤xn−sifxn−s> Aandxn+1≤Aifxn−s≤A.

(ii) For alln≥t+ 1,yn+1≤yn−pifyn−p> Bandyn+1≤Bifyn−p≤B.

Proof ofClaim 2. We only prove (i) (the proof of (ii) is similar). Obviously xn+1= fyn−q,xn−s

≤fb,xn−s

. (2.5)

Ifxn−s≤A, thenxn+1≤f(b,xn−s)≤f(b,A)=A.

Ifxn−s> A, then

fb,xn−s x_n−s ^≤

f(b,A)

A ⁼1, (2.6)

which impliesxn+1≤ f(b,xn−s)≤xn−s.Claim 2is proven.

Claim 3. (i) There exists a positive integerN1such thatxn≤Afor alln≥N1. (ii) There exists a positive integerN2such thatyn≤Bfor alln≥N2.

Proof ofClaim 3. We only prove (i) (the proof of (ii) is similar). Assume on the contrary thatClaim 3does not hold. Then it follows fromClaim 2that there exists a positive in- tegerRsuch thatx_n(s+1)+R≥x(n+1)(s+1)+R> Afor everyn≥1. Let lim_n→∞x_n(s+1)+R=A1, thenA1≥A.

We know fromClaim 2that{xn}and{yn}are bounded. Letc=limn→∞supyn(s+1)+R−q−1, thenc≥band there exists a sequencen_k→ ∞such that

klim→∞y_{nk(s+1)+R−q−}1=c. (2.7)

By (1.3) we have that

x_nk(s+1)+R=fy_{nk(s+1)+R−q−}1,x_{(nk−1)(s+1)+R}, (2.8) from which it follows that

A1=fc,A1

≤ fb,A1

=A1fb,A1

A1 ≤A1f(b,A)

A ⁼A1. (2.9)

This with (H2) and (H4) impliesc=bandA1=A. Therefore limn→∞yn(s+1)+R−q−1=b.

Since{x_n}and{y_n}are bounded, we may assume (by taking a subsequence) that there exist a sequenceln→ ∞andα,β∈Esuch that

klim→∞xlk(s+1)+R−q−t−2=α, lim

k→∞ylk(s+1)+R−q−p−2=β. (2.10) By (1.3) we have that

ylk(s+1)+R−q−1=gxlk(s+1)+R−q−t−2,ylk(s+1)+R−q−p−2

, (2.11)

from which it follows that

b=g(α,β)> g(α+ 1,β)≥b. (2.12)

This is a contradiction.Claim 3is proven.

LetN=max{N1,N2}+ 2s+ 2p, then for alln > Nwe have that x_n≤A, y_n≤B,

xn= fyn−q−1,xn−s−1

≥ f(B,a), yn=gxn−t−1,yn−p−1

≥g(A,b).

(2.13)

Theorem 2.1is proven.

Theorem 2.2. LetI=[c,d] andJ=[α,β] be intervals of real numbers. Assume that f ∈ C(J×I,I) andg∈C(I×J,J) satisfy the following properties:

(i) f(u,v) andg(u,v) are decreasing inuand increasing inv;

(ii) ifM1,m1∈I withm1≤M1 and M2,m2∈J withm2≤M2 are a solution of the system

M1= fm2,M1

, m1=fM2,m1 , M2=gm1,M2

, m2=gM1,m2

, (2.14)

thenM1=m1andM2=m2. Then the system

xn+1= fyn−q,xn−s

, yn+1=gxn−t,yn−p

, n=0, 1,. . ., (2.15) has a unique equilibrium (S,T) and every solution of (2.15) with the initial valuesx₋s,x₋s+1, . . .,x0∈Iandy_−p,y_−p+1,. . .,y0∈Jconverges to (S,T).

Proof. Let

m⁰₁=c, m⁰₂=α, M₁⁰=d, M⁰₂=β, (2.16) and fori=1, 2,. . ., we define

Mⁱ₁= fmⁱ₂⁻¹,M₁ⁱ⁻¹, mⁱ₁=fM₂ⁱ⁻¹,mⁱ₁⁻¹,

Mⁱ₂=gmⁱ₁⁻¹,M₂ⁱ⁻¹, mⁱ₂=gM₁ⁱ⁻¹,mⁱ₂⁻¹. (2.17) It is easy to verify that

m⁰1≤m¹1= fM2⁰,m⁰1

≤ fm⁰2,M1⁰

=M1¹≤M1⁰,

m⁰₂≤m¹₂=gM₁⁰,m⁰₂≤gm⁰₁,M₂⁰=M₂¹≤M₂⁰. (2.18) From (i) and (2.18) we obtain

m¹₁=fM₂⁰,m⁰₁≤fM₂¹,m¹₁=m²₁, m²₁=fM₂¹,m¹₁≤fm¹₂,M¹₁=M₁², M₁²=fm¹₂,M₁¹≤fm⁰₂,M⁰₁=M₁¹, m¹₂=gM₁⁰,m⁰₂≤gM₁¹,m¹₂=m²₂, m²₂=gM₁¹,m¹₂≤gm¹₁,M₂¹=M₂², M₂²=gm¹₁,M₂¹≤gm⁰₁,M₂⁰=M₂¹.

(2.19)

By induction it follows that fori=0, 1,. . .,

mⁱ₁≤mⁱ⁺¹₁ ≤ ··· ≤M₁ⁱ⁺¹≤M₁ⁱ, mⁱ₂≤mⁱ⁺¹₂ ≤ ··· ≤M₂ⁱ⁺¹≤M₂ⁱ.

(2.20)

On the other hand, we have xn∈[m⁰1,M1⁰] for anyn≥ −s and yn∈[m⁰2,M2⁰] for any n≥ −psincex_−s,x_−s+1,. . .,x0∈[m⁰₁,M₁⁰] andy_−p,y_−p+1,. . .,y0∈[m⁰₂,M₂⁰]. For anyn≥0, we obtain

m¹₁= fM₂⁰,m⁰₁≤x_n+1=fy_n−q,x_n−s≤ fm⁰₂,M₁⁰=M₁¹,

m¹₂=gM₁⁰,m⁰₂≤y_n+1=gx_n−t,y_n−p≤gm⁰₁,M₂⁰=M¹₂. (2.21) Letk=max{s+ 1,p+ 1}. It follows that for anyn≥k,

m²₁= fM₂¹,m¹₁≤xn+1=fyn−q,xn−s

≤ fm¹₂,M₁¹=M₁², m²₂=gM₁¹,m¹₂≤yn+1=gxn−t,yn−p

≤gm¹₁,M₂¹=M²₂.

(2.22)

By induction, forl=0, 1,. . ., we obtain that for anyn≥lk,

m^l+1₁ ≤x_n+1≤M^l+1₁ , m^l+1₂ ≤y_n+1≤M^l+1₂ . (2.23) Let

nlim→∞mⁿ₁=m1, lim

n→∞mⁿ₂=m2,

nlim→∞Mⁿ₁=M1, lim

n→∞Mⁿ₂=M2. (2.24)

By the continuity of f andg, we have from (2.17) that M1=fm2,M1

, M2=gm1,M2

, m2=gM1,m2

, m1=fM2,m1

. (2.25)

Using assumption (ii), it follows from (2.23) that

nlim→∞xn=m1=M1=S, lim

n→∞yn=m2=M2=T. (2.26)

Theorem 2.2is proven.

Theorem 2.3. If (H1)–(H5) hold and the system M1=fm2,M1

, M2=gm1,M2

, m2=gM1,m2

, m1=fM2,m1

, (2.27)

with f(B,a)≤m1≤M1≤A andg(A,b)≤m2≤M2≤B has the unique solutionm1= M1=xandm2=M2=y, then every solution of (1.3) converges to the unique positive equi- librium (x,y).

Proof. Let{(x_n,y_n)}is a positive solution of (1.3). ByTheorem 2.1, there exists a positive integerN such that f(B,a)≤xn= f(yn−q,xn−s)≤Aandg(A,b)≤yn=g(xn−t,yn−p)≤ B for all n≥N. Since f, g satisfy the conditions (i) and (ii) of Theorem 2.2 in I= [f(B,a),A] andJ=[(A,b),B], it follows that{(x_n,y_n)}converges to the unique positive

equilibrium (x,y).

3. Examples

In this section, we will give two applications of the above results.

Example 3.1. Consider equation x_n+1= c+xn−s

a+y_n−q, y_n+1=d+y_n−p

b+x_n−t, (3.1)

where p,q,s,t∈ {0, 1, 2,. . .} with s≥t and p≥q, the initial values x_−s,x_−s+1,. . .,x0, y₋p,y₋p+1,. . .,y0∈(0, +∞) anda,b,c,d∈(0, +∞). Ifa >1 andb >1, then every positive solution of (3.1) converges to the unique positive equilibrium.

Proof. LetE=[0, +∞), it is easy to verify that (H1)–(H5) hold for (3.1). In addition, if M1=c+M1

a+m2, M2=d+M2

b+m1, m2=d+m2

b+M1, m1= c+m1

a+M2,

(3.2)

with 0≤m1≤M1and 0≤m2≤M2, then we have M1−m1

(a−1)=m1M2−M1m2, M2−m2

(b−1)=M1m2−m1M2,

(3.3) from which it follows thatM1=m1andM2=m2. Moreover, it is easy to verify that (3.2) have the unique solution

M1=m1=x=−(a−1)(b−1) +c−d+(a−1)(b−1) +d−c²+ 4c(a−1)(b−1)

2(a−1) ,

M2=m2=y=−(a−1)(b−1) +d−c+(a−1)(b−1) +c−d²+ 4d(a−1)(b−1)

2(b−1) .

(3.4) It follows from Theorems2.1and2.3that every positive solution of (3.1) converges to the

unique positive equilibrium (x,y).

Example 3.2. Consider equation x_n+1=a+ xn−s

y_n−q, y_n+1=b+ y_n−p

x_n−t, (3.5)

where p,q,s,t∈ {0, 1, 2,. . .} with s≥t and p≥q, the initial values x₋s,x₋s+1,. . .,x0, y_−p,y_−p+1,. . .,y0∈(0, +∞) anda,b∈(0, +∞). Ifa >1 andb >1, then every positive so- lution of (3.5) converges to the unique positive equilibrium.

Proof. LetE=(0, +∞), it is easy to verify that (H1)–(H5) hold for (3.5). In addition, if M1=a+M1

m2

, M2=b+M2

m1

, m2=b+ m2

M1, m1=a+m1

M2,

(3.6)

with 0≤m1≤M1and 0≤m2≤M2, then (3.6) have the unique solution M1=m1=x=ab−1

b−1 , M2=m2=y=ab−1

a−1 .

(3.7)

It follows from Theorems2.1and2.3that every positive solution of (3.5) converges to the unique positive equilibrium (x,y)=((ab−1)/(b−1), (ab−1)/(a−1)).

Acknowledgment

The project was supported by NNSF of China (10461001,10361001) and NSF of Guangxi (0447004).

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

E-mail address:[email protected]

Hongjian Xi: Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China

E-mail address:[email protected]