Skip to main content

On the system of rational difference equations xn+1 = f(yn-q, xn-s), yn+1 = g(xn-t, yn-p)

Abstract

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,...} with st and pq, 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.

[1234567891011]

References

  1. 1.

    Agarwal RP, O'Regan D, Wong PJY: Eigenvalue characterization of a system of difference equations. Nonlinear Oscillations 2004,7(1):3–47.

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Agarwal RP, Wong PJY: Advanced Topics in Difference Equations, Mathematics and Its Applications. Volume 404. Kluwer Academic, Dordrecht; 1997:viii+507.

    Book  Google Scholar 

  3. 3.

    Camouzis E, Papaschinopoulos G: Global asymptotic behavior of positive solutions on the system 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 2004,17(6):733–737. 10.1016/S0893-9659(04)90113-9

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Çinar C: 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 2004,158(2):303–305. 10.1016/j.amc.2003.08.073

    MathSciNet  Article  Google Scholar 

  5. 5.

    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

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    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

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    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 

  8. 8.

    Kulenović MRS, Nurkanović M: Asymptotic behavior of a system of linear fractional difference equations. Journal of Inequalities and Applications 2005,2005(2):127–143. 10.1155/JIA.2005.127

    MathSciNet  MATH  Google Scholar 

  9. 9.

    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

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Papaschinopoulos G, Schinas CJ: On the system of two nonlinear difference equations x n+1 = A + x n-1 / y n , y n+1 = A + y n-1 x n . International Journal of Mathematics and Mathematical Sciences 2000,23(12):839–848. 10.1155/S0161171200003227

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Yang X: On the system of rational difference equations x n = A + y n-1 / x n- p y n- q , y n = A + x n-1 / xn-ryn-s. Journal of Mathematical Analysis and Applications 2005,307(1):305–311. 10.1016/j.jmaa.2004.10.045

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

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. On the system of rational difference equations xn+1 = f(yn-q, xn-s), yn+1 = g(xn-t, yn-p). Adv Differ Equ 2006, 051520 (2006). https://doi.org/10.1155/ADE/2006/51520

Download citation

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation