Open Access

On the system of rational difference equations xn+1 = f(x n ,yn-k), yn+1 = f(y n , xn-k)

Advances in Difference Equations20062006:016949

Received: 15 September 2005

Accepted: 13 November 2005

Published: 23 February 2006


We study the global asymptotic behavior of the positive solutions of the system of rational difference equations xn+1 = f(x n ,yn-k), yn+1 = f(y n , xn-k), n = 0,1,2,..., under appropriate assumptions, where k {1,2,...} and the initial values x-k, x-k+1,...,x0, y-k, y-k+1, ..., y0 (0,+∞). We give sufficient conditions under which every positive solution of this equation converges to a positive equilibrium. The main theorem in [1] is included in our result.


Authors’ Affiliations

Department of Mathematics, Guangxi University
Department of Mathematics, Guangxi College of Finance and Economics


  1. Camouzis E, Papaschinopoulos G: Global asymptotic behavior of positive solutions on the system of rational difference equations xn+1 = 1+ x n / yn-m, yn+1 = 1+ y n / xn-m. Applied Mathematics Letters 2004,17(6):733–737. 10.1016/S0893-9659(04)90113-9MathSciNetView ArticleMATHGoogle Scholar
  2. Çinar C: On the positive solutions of the difference equation system xn+1 = 1/ y n , yn+1 = y n / xn-1yn-1. Applied Mathematics and Computation 2004,158(2):303–305. 10.1016/j.amc.2003.08.073MathSciNetView ArticleGoogle Scholar
  3. Clark D, Kulenović MRS: A coupled system of rational difference equations. Computers & Mathematics with Applications 2002,43(6–7):849–867. 10.1016/S0898-1221(01)00326-1MathSciNetView ArticleMATHGoogle Scholar
  4. Clark D, Kulenović MRS, Selgrade JF: Global asymptotic behavior of a two-dimensional difference equation modelling competition. Nonlinear Analysis 2003,52(7):1765–1776. 10.1016/S0362-546X(02)00294-8MathSciNetView ArticleMATHGoogle Scholar
  5. Grove EA, Ladas G, McGrath LC, Teixeira CT: Existence and behavior of solutions of a rational system. Communications on Applied Nonlinear Analysis 2001,8(1):1–25.MathSciNetMATHGoogle Scholar
  6. Papaschinopoulos G, Schinas CJ: On a system of two nonlinear difference equations. Journal of Mathematical Analysis and Applications 1998,219(2):415–426. 10.1006/jmaa.1997.5829MathSciNetView ArticleMATHGoogle Scholar
  7. Yang X: On the system of rational difference equations x n = A + yn-1/ xn-pyn-q, y n = A + xn-1/ xn-ryn-s. Journal of Mathematical Analysis and Applications 2005,307(1):305–311. 10.1016/j.jmaa.2004.10.045MathSciNetView ArticleGoogle Scholar


© Taixiang Sun et al. 2006

This article is published under license to BioMed Central Ltd. 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.