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: stx1963@163.com Hongjian Xi: Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530004, China E-mail address: xhongjian@263.net Liang Hong: Department of Mathematics, Guangxi University, Nanning, Guangxi 530004, China E-mail address: stxhql@gxu.edu.cn . 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. j = M j ∈  g(P ), M  ,forj ∈{ 1,2 , ,3 k +1 }, lim n→∞ y l n − j = P j ∈  g(M),P  ,forj ∈{ 1,2 , · ,3 k +1}. (2.10) 4 Systemofrationaldifference equations From (1.2 ), (2.10)and(H 3 ), we have f  M,. 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