Skip to main content

Global behavior of a higher-order rational difference equation

Abstract

We investigate in this paper the global behavior of the following difference equation: , n = 0,1,..., under appropriate assumptions, where b [0, ∞), k ≥ 1, i0, i1,...,i2k {0,1,...} with i0 <i1 < ... <i2k, the initial conditions . We prove that unique equilibrium of that equation is globally asymptotically stable.

[1234567891011]

References

  1. 1.

    Amleh AM, Kruse N, Ladas G: On a class of difference equations with strong negative feedback. Journal of Difference Equations and Applications 1999,5(6):497–515. 10.1080/10236199908808204

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Çinar C: On the positive solutions of the difference equation xn+1 = αxn-1/1+ bx n xn-1. Applied Mathematics and Computation 2004,156(2):587–590. 10.1016/j.amc.2003.08.010

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    El-Owaidy HM, Ahmed AM, Mousa MS: On the recursive sequences xn+1 = - αxn-1/ β ± x n . Applied Mathematics and Computation 2003,145(2–3):747–753. 10.1016/S0096-3003(03)00271-6

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Krause U, Nussbaum RD: A limit set trichotomy for self-mappings of normal cones in Banach spaces. Nonlinear Analysis 1993,20(7):855–870. 10.1016/0362-546X(93)90074-3

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Kruse N, Nesemann T: Global asymptotic stability in some discrete dynamical systems. Journal of Mathematical Analysis and Applications 1999,235(1):151–158. 10.1006/jmaa.1999.6384

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Ladas G: Open problems and conjectures. Journal of Difference Equations and Applications 1998,4(1):497–499.

    MathSciNet  Article  Google Scholar 

  7. 7.

    Li X: Global behavior for a fourth-order rational difference equation. Journal of Mathematical Analysis and Applications 2005,312(2):555–563. 10.1016/j.jmaa.2005.03.097

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Li X: Qualitative properties for a fourth-order rational difference equation. Journal of Mathematical Analysis and Applications 2005,311(1):103–111. 10.1016/j.jmaa.2005.02.063

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Nesemann T: Positive nonlinear difference equations: some results and applications. Nonlinear Analysis 2001,47(7):4707–4717. 10.1016/S0362-546X(01)00583-1

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Papaschinopoulos G, Schinas CJ: Global asymptotic stability and oscillation of a family of difference equations. Journal of Mathematical Analysis and Applications 2004,294(2):614–620. 10.1016/j.jmaa.2004.02.039

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Thompson AC: On certain contraction mappings in a partially ordered vector space. Proceedings of the American Mathematical Society 1963,14(3):438–443.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding authors

Correspondence to Hongjian Xi or 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

Xi, H., Sun, T. Global behavior of a higher-order rational difference equation. Adv Differ Equ 2006, 027637 (2006). https://doi.org/10.1155/ADE/2006/27637

Download citation

Keywords

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