Skip to main content
Advances in Continuous and Discrete Models

Theory and Modern Applications

Download PDF
  • Research Article
  • Open access
  • Published:

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

Advances in Difference Equations volume 2006, Article number: 016949 (2006) Cite this article

Abstract

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.

[1234567]

References

  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)90113-9

    Article  MathSciNet  MATH  Google 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.2003.08.073

    Article  MathSciNet  Google 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)00326-1

    Article  MathSciNet  MATH  Google 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)00294-8

    Article  MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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.1997.5829

    Article  MathSciNet  MATH  Google 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.2004.10.045

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Guangxi University, Nanning, Guangxi, 530004, China

    Taixiang Sun & Liang Hong

  2. Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi, 530004, China

    Hongjian Xi

Authors
  1. Taixiang Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongjian Xi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liang Hong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Taixiang Sun.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Sun, T., Xi, H. & Hong, L. On the system of rational difference equations xn+1 = f(x n ,yn-k), yn+1 = f(y n , xn-k). Adv Differ Equ 2006, 016949 (2006). https://doi.org/10.1155/ADE/2006/16949

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/ADE/2006/16949

Keywords