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Global behavior of a higher-order rational difference equation

Advances in Difference Equations volume 2006, Article number: 027637 (2006) Cite this article

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.

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References

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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 

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Authors and Affiliations

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

    Hongjian Xi & Taixiang Sun

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  1. Hongjian Xi
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  2. Taixiang Sun
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Correspondence to Hongjian Xi or Taixiang Sun.

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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.

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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

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  • DOI: https://doi.org/10.1155/ADE/2006/27637

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