GLOBAL BEHAVIOR OF A HIGHER-ORDER RATIONAL DIFFERENCE EQUATION HONGJIAN XI AND TAIXIANG SUN 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: x n+1 = (P k (x n i 0 ,x n i 1 , ,x n i 2k )+b)/(Q k (x n i 0 ,x n i 1 , ,x n i 2k )+b), n = 0,1, ,under appropriate assumptions, where b [0, ), k 1, i 0 ,i 1 , ,i 2k 0,1, with i 0 <i 1 < <i 2k , the initial conditions x i 2k ,x i 2k +1 , ,x 0 (0, ). We prove that unique equilib- rium x = 1 of that equation is globally asy mptotically 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) x n+1 = x n + x n 1 x n 2 x n x n 1 + x n 2 , n = 0,1, , (1.1) where the initial values x 2 ,x 1 ,x 0 R + (0,+ ). In [9], Nesemann utilized the strong negative feedback property of [1]tostudythe following difference equation: (E2) x n+1 = x n 1 + x n x n 2 x n x n 1 + x n 2 , n = 0,1, , (1.2) where the initial values x 2 ,x 1 ,x 0 R + . Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 27637, Pages 1–7 DOI 10.1155/ADE/2006/27637 2Globalbehaviorofadifference equation In [10], Papaschinopoulos and Schinas investigated the global asymptotic stability of the following nonlinear difference equation: (E3) x n+1 =  i Z k j 1, j x n i + x n j x n j+1 +1  i Z k x n i , n = 0,1, , (1.3) where k 1,2,3, , j, j 1 Z k 0,1, ,k , and the initial values x k ,x k+1 , , x 0 R + . Recently, Li [7, 8] studied the global asymptotic stability of the following two nonlinear difference equations: (E4) x n+1 = x n 1 x n 2 x n 3 + x n 1 + x n 2 + x n 3 + a x n 1 x n 2 + x n 1 x n 3 + x n 2 x n 3 +1+a , n = 0,1, (1.4) (E5) x n+1 = x n x n 1 x n 3 + x n + x n 1 + x n 3 + a x n x n 1 + x n x n 3 + x n 1 x n 3 +1+a , n = 0,1, , (1.5) where a [0,+ ) and the initial values x 3 ,x 2 ,x 1 ,x 0 R + . Let k 1andi 0 ,i 1 , ,i 2k 0,1, with i 0 <i 1 < <i 2k .LetP 0 (x n i 0 ) = x n i 0 and Q 0 (x n i 0 ) = 1, for any 1 j k,let P j  x n i 0 , ,x n i 2j  =  x n i 2j x n i 2j 1 +1  P j 1  x n i 0 , ,x n i 2j 2  +  x n i 2j + x n i 2j 1  Q j 1  x n i 0 , ,x n i 2j 2  , Q j  x n i 0 , ,x n i 2j  =  x n i 2j x n i 2j 1 +1  Q j 1  x n i 0 , ,x n i 2j 2  +  x n i 2j + x n i 2j 1  P j 1  x n i 0 , ,x n i 2j 2  . (1.6) In this paper, we consider the following difference equation: x n+1 = P k  x n i 0 ,x n i 1 , ,x n i 2k  + b Q k  x n i 0 ,x n i 1 , ,x n i 2k  + b , n = 0,1, , (1.7) where b [0, ) and the initial conditions x i 2k ,x i 2k +1 , ,x 0 (0, ). It is easy to see that the positive equilibrium x of (1.7) satisfies x = P k (x, x, ,x)+b Q k (x, x, ,x)+b =  x 2 +1  P k 1 (x, x, ,x)+2xQ k 1 (x, x, ,x)+b  x 2 +1  Q k 1 (x, x, ,x)+2xP k 1 (x, x, ,x)+b . (1.8) H. Xi and T. Sun 3 Thus, we have ( x 1)  x 2 + x  Q k 1 (x, x, ,x)+(x +1)P k 1 (x, x, ,x)+b  = 0, (1.9) from which one can see that (1.7) has the unique positive equilibrium x = 1. Remark 1.1. Let k = 1, then (1.7)is(1.4)when(i 0 ,i 1 ,i 2 ) = (1,2,3) and is (1.5)when (i 0 ,i 1 ,i 2 ) = (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 P k  x n i 0 ,x n i 1 , ,x n i 2k  Q k  x n i 0 ,x n i 1 , ,x n i 2k  =  x n i 2k 1  x n i 2k 1 1  P k 1  x n i 0 , ,x n i 2k 2  Q k 1  x n i 0 , ,x n i 2k 2  = =  x n i 2k 1  x n i 2k 1 1   x n i 2 1  x n i 1 1  P 0  x n i 0  Q 0  x n i 0  =  x n i 0 1  x n i 1 1   x n i 2k 1  , (2.1) it follows from (1.7)thatforanyn 0, x n+1 1 =  x n i 0 1  x n i 1 1   x n i 2k 1  Q k  x n i 0 ,x n i 1 , ,x n i 2k  + b . (2.2) Definit ion 2.1. Let x n n= i 2k be a solution of (1.7)and a n n= i 2k asequencewitha n 1,0,1 for every n i 2k . a n n= i 2k is called itinerary of x n n= i 2k if a n = 1when x n < 1, a n = 0whenx n = 1, and a n = 1whenx n > 1. From (2.2), we get the following. Proposition 2.2. Let x n n= i 2k beasolutionof(1.7) whose itinerary is a n n= i 2k , then a n+1 = a n i 0 a n i 1 a n i 2k for any n 0. Proposition 2.3. Let x n n= i 2k be a solution of (1.7), then it follows that x n = 1 for any n 1  i 2k j=0 (x j 1) = 0. Proof. Let itinerary of x n n= i 2k be a n n= i 2k ,thenitfollowsfromProposition 2.2 that x n = 1foranyn 1 a n = 0foranyn 1  i 2k j=0 a j = 0  i 2k j=0 (x j 1) = 0.  Proposition 2.4. If gcd(i s +1,i 2k +1)= 1 for some s 0,1, ,2k 1 ,thenapositive solution x n n= i 2k of (1.7)iseventuallyequalto1 x p = 1 for some p i 2k . Proof. “If” part is obvious. “Only if” part. If x p = 1forsomep i 2k ,thena p = 0, where a n n= i 2k is itinerary of x n n= i 2k .ByProposition 2.2,wehavea j(i 2k +1)+p = a j(i s +1)+p = 0forany j 0. Since gcd(i s +1,i 2k +1)= 1, we see that for any t 0,1, ,i 2k , there exist j t 1,2, ,i 2k +1 4Globalbehaviorofadifference equation and m t 0,1, ,i s +1 such that j t  i s +1  = m t  i 2k +1  + t. (2.3) Together with Proposition 2.2, it follows that a (i s +1)(i 2k +1)+t+p = 0. (2.4) Again by Proposition 2.2,wehavea n = 0foranyn (i s +1)(i 2k +1)+p, w hich implies x n = 1foranyn (i s +1)(i 2k +1)+p.  Example 2.5. Consider the equation x n+1 = x n i 0 x n i 1 x n 3 + x n i 0 + x n i 1 + x n 3 + b x n i 0 x n i 1 + x n i 0 x n 3 + x n i 1 x n 3 +1+b , n = 0,1, , (2.5) where b [0,+ ), 0 i 0 <i 1 < 3, and the initial values x 3 ,x 2 ,x 1 ,x 0 R + .Let x n n= 3 be a solution of (2.5) whose itinerary is a n n= 3 , then the following hold. (1) If (i 0 ,i 1 ) (0,1),(1,2) and x n n= 3 is not eventually equal to 1, then a n n= 3 is a periodic sequence of period 7. (2) If (i 0 ,i 1 ) = (0,2) and x n n= 3 is not eventually equal to 1, then a n n= 3 isaperi- odic sequence of period 6. (3) x n = 1foranyn 1  0 j = 3 (x j 1) = 0. (4) x n n= 3 is eventually equal to 1 x p = 1forsomep 3. Proof. (1) If (i 0 ,i 1 ) = (0,1), then from Proposition 2.2, it follows that for any n 0, a n+4 = a n+3 a n+2 a n = a n+2 a n+1 a n 1 a n+2 a n = a n+1 a n a n 1 = a n a n 1 a n 3 a n a n 1 = a n 3 . (2.6) If (i 0 ,i 1 ) = (1,2), then in a similar fashion, it is true that a n+4 = a n 3 for any n 0. (2) If (i 0 ,i 1 ) = (0,2), then from Proposition 2.2, it follows that for any n 0, a n+3 = a n+2 a n a n 1 = a n+1 a n 1 a n 2 a n a n 1 = a n+1 a n a n 2 = a n a n 2 a n 3 a n a n 2 = a n 3 . (2.7) (3) It follows from Proposition 2.3. (4) It follows from Proposition 2.4 since either gcd(i 0 +1,4)= 1orgcd(i 1 +1,4)= 1.  3. Global asymptotic stability of (1.7) In this section, we will study global asymptotic stability of (1.7). To do this, we need the following lemmas. H. Xi and T. Sun 5 Lemma 3.1. Let (y 0 , y 1 , , y i 2k ) R i 2k +1 + (1,1, ,1) and M = max y j ,1/y j 0 j i 2k , then 1 M < P k  y i 0 , y i 1 , , y i 2k  Q k  y i 0 , y i 1 , , y i 2k  <M. (3.1) Proof. Sinc e (y 0 , y 1 , , y i 2k ) R i 2k +1 + (1,1, ,1) and M = max y j ,1/y j 0 j i 2k , we have M>1 and either M a>1/M or M>a 1/M for any a y j ,1/y j 0 j i 2k . It is easy to verify that P 1  y i 0 , y i 1 , y i 2  =  y i 1 y i 2 +1  y i 0 +  y i 1 + y i 2  <  y i 1 y i 2 +1  M +  y i 1 + y i 2  y i 0 M = Q 1  y i 0 , y i 1 , y i 2  M, P 1  y i 0 , y i 1 , y i 2  M =  y i 1 y i 2 +1  y i 0 +  y i 1 + y i 2  M >  y i 1 y i 2 +1  +  y i 1 + y i 2  y i 0 = Q 1  y i 0 , y i 1 , y i 2  . (3.2) From that we have P 2  y i 0 , y i 1 , y i 2 , y i 3 , y i 4  =  y i 3 y i 4 +1  P 1  y i 0 , y i 1 , y i 2  +  y i 3 + y i 4  Q 1  y i 0 , y i 1 , y i 2  <  y i 3 y i 4 +1  Q 1  y i 0 , y i 1 , y i 2  M +  y i 3 + y i 4  P 1  y i 0 , y i 1 , y i 2  M = Q 2  y i 0 , y i 1 , y i 2 , y i 3 , y i 4  M, P 2  y i 0 , y i 1 , y i 2 , y i 3 , y i 4  M =  y i 3 y i 4 +1  P 1  y i 0 , y i 1 , y i 2  +  y i 3 + y i 4  Q 1  y i 0 , y i 1 , y i 2  M >  y i 3 y i 4 +1  Q 1  y i 0 , y i 1 , y i 2  +  y i 3 + y i 4  P 1  y i 0 , y i 1 , y i 2  = Q 2  y i 0 , y i 1 , y i 2 , y i 3 , y i 4  . (3.3) By induction, we have that for any 1 j k, P j  y i 0 , y i 1 , , y i 2j  <Q j  y i 0 , y i 1 , , y i 2j  M, P j  y i 0 , y i 1 , , y i 2j  M>Q j  y i 0 , y i 1 , , y i 2j  . (3.4) Thus 1 M < P k  y i 0 , y i 1 , , y i 2k  Q k  y i 0 , y i 1 , , y i 2k  <M. (3.5)  6Globalbehaviorofadifference equation Let n be a positive integer and let ρ denote the par t-metric on R n + (see [11]) which is defined by ρ(x, y) = logmin  x i y i , y i x i 1 i n  for x =  x 1 , ,x n  , y =  y 1 , , y n  R n + . (3.6) It was shown by Thompson [11]that( R n + ,ρ) is a complete metric space. In [4], Krause and Nussbaum proved that the distances indicated by the part-met ric and by the Eu- clidean norm are equivalent on R n + . Lemma 3.2 [5]. Let T : R n + R n + be a continuous mapping with unique fixed point x R n + . Suppose that there exists some l 1 such that for the part-metric ρ, ρ  T l x, x  <ρ  x, x  x = x . (3.7) Then x is globally asymptotically stable. Theorem 3.3. The unique equilibrium x = 1 of (1.7) is globally asymptotically stable. Proof. Let x n n= i 2k be a solution of (1.7) with initial conditions x i 2k ,x i 2k +1 , ,x 0 R i 2k +1 + such that x n n= i 2k is not eventually equal to 1 since otherwise there is nothing to show. Denoted by T : R i 2k +1 + R i 2k +1 + the mapping T  x n i 2k ,x n i 2k +1 , ,x n  =  x n i 2k +1 ,x n i 2k +2 , ,x n , P k  x n i 0 ,x n i 1 , ,x n i 2k  + b Q k  x n i 0 ,x n i 1 , ,x n i 2k  + b  . (3.8) Then solution x n n= i 2k of (1.7) is represented by the first component of the solution y n n=0 of the system y n+1 = Ty n with initial condition y 0 = (x i 2k ,x i 2k +1 , ,x 0 ). It fol- lows from Lemma 3.1 that for all n 0, the following inequalities hold: min  x n i , 1 x n i 0 i i 2k  <x n+1 < max  x n i , 1 x n i 0 i i 2k  . (3.9) Inductively, we obtain that for all n 0andall1 j i 2k +1, min  x n i , 1 x n i 0 i i 2k  <x n+j < max  x n i , 1 x n i 0 i i 2k  , (3.10) from which it follows that min  x n i , 1 x n i 0 i i 2k  < min  x n+i , 1 x n+i 1 i i 2k +1  . (3.11) H. Xi and T. Sun 7 Thus, for x = (1, 1, ,1) and the part-metric ρ of R i 2k +1 + ,wehave ρ  T i 2k +1  y n  ,x  = logmin  x n+i , 1 x n+i 1 i i 2k +1  < logmin  x n i , 1 x n i 0 i i 2k  = ρ  y n ,x  (3.12) for all n 0. It follows from Lemma 3.2 that the positive equilibrium x = 1of(1.7)is globally asymptotically stable.  Acknowledgments Project suppor ted by NNSF of China (10461001, 10361001) and NSF of Guangxi (0640205). References [1] A.M.Amleh,N.Kruse,andG.Ladas,On a class of difference equations with strong negative feedback,JournalofDifference Equations and Applications 5 (1999), no. 6, 497–515. [2] C. C¸ inar, On the positive solutions of the difference equation x n+1 = αx n 1  (1 + bx n x n 1 ), Applied Mathematics and Computation 156 (2004), no. 2, 587–590. [3]H.M.El-Owaidy,A.M.Ahmed,andM.S.Mousa,On the recursive sequences x n+1 = αx n 1  (β x n ), Applied Mathematics and Computation 145 (2003), no. 2-3, 747–753. [4] U. Krause and R. D. Nussbaum, A limit set trichotomy for self-mappings of normal cones in Banach spaces, Nonlinear Analysis 20 (1993), no. 7, 855–870. [5] N. Kruse and T. Nesemann, Global asymptotic stability in some discrete dynamical systems,Jour- nal of Mathematical Analysis and Applications 235 (1999), no. 1, 151–158. [6] G. Ladas, Open problems and c onjectures,JournalofDifference Equations and Applications 4 (1998), no. 1, 497–499. [7] X. Li, Global behavior for a fourth-order rational difference equation, Journal of Mathematical Analysis and Applications 312 (2005), no. 2, 555–563. [8] , Qualitative properties for a fourth-order rational difference equation, Journal of Mathe- matical Analysis and Applications 311 (2005), no. 1, 103–111. [9] T. Nesemann, Positive nonlinear difference equations: some results and applications, Nonlinear Analysis 47 (2001), no. 7, 4707–4717. [10] G. Papaschinopoulos and C. J. Schinas, Global asymptotic stability and oscillation of a family of difference equations, Journal of Mathematical Analysis and Applications 294 (2004), no. 2, 614– 620. [11] A. C. Thompson, On certain contraction mappings in a partially ordered vector space, Proceedings of the American Mathematical Society 14 (1963), no. 3, 438–443. Hongjian Xi: Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530004, China E-mail address: xhongjian@263.com Taixiang Sun: Department of Mathematics, College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China E-mail address: stx1963@163.com . onjectures,JournalofDifference Equations and Applications 4 (1998), no. 1, 497–499. [7] X. Li, Global behavior for a fourth-order rational difference equation, Journal of Mathematical Analysis and Applications. a n+3 a n+2 a n = a n+2 a n+1 a n 1 a n+2 a n = a n+1 a n a n 1 = a n a n 1 a n 3 a n a n 1 = a n 3 . (2.6) If (i 0 ,i 1 ) = (1,2), then in a similar fashion, it is true that a n+4 = a n 3 for any. GLOBAL BEHAVIOR OF A HIGHER-ORDER RATIONAL DIFFERENCE EQUATION HONGJIAN XI AND TAIXIANG SUN Received 17 January 2006; Revised 6 April 2006; Accepted 12 April 2006 We investigate in this paper