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On the rational difference equation \(x_{n}=1+\frac{(1-x_{n-k})(1-x_{n-l})(1-x_{n-m})}{x_{n-k}+x_{n-l}+x_{n-m}}\)

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Abstract

In this note we consider the following higher order rational difference equations

$$x_{n}=1+\frac{(1-x_{n-k})(1-x_{n-l})(1-x_{n-m})}{x_{n-k}+x_{n-l}+x_{n-m}},\quad n=0,1,\ldots,$$

where 1≤k<l<m, and the initial values x −m ,x −m+1,…,x −1 are positive numbers. We give some sufficient conditions for the persistence of positive solutions for the above equation, and prove that the positive equilibrium point of this equation is globally asymptotically stable.

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References

  1. Agarwal, R.P.: Difference Equations and Inequalities, 1st edn. Dekker, New York (1992); 2nd edn. (2000)

    MATH  Google Scholar 

  2. Amleh, A.M., Kruse, N., Ladas, G.: On a class of difference equation with strong negative feedback. J. Diff. Equ. Appl. 5, 497–515 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berenhaut, K.S., Foley, J.D., Stević, S.: The global attractivity of the rational difference equation \(y_{n}=\frac{y_{n-k}+y_{n-m}}{1+y_{n-k}y_{n-m}}\) . Appl. Math. Lett. 20, 54–58 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berenhaut, K.S., Foley, J.D., Stević, S.: The global attractivity of the rational difference equation y n =1+y n−k/y n−m. Proc. Am. Math. Soc. 135(1), 1133–1140 (2007)

    Article  MATH  Google Scholar 

  5. Berenhaut, K.S., Foley, J.D., Stević, S.: The global attractivity of the rational difference equation y n =A+(y n−k/y n−m)p. Proc. Am. Math. Soc. 136, 103–110 (2008)

    Article  MATH  Google Scholar 

  6. Berenhaut, K.S., Stević, S.: The global attractivity of a high order rational difference equation. J. Math. Anal. Appl. 326(2), 940–944 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Berg, L., Stević, S.: Periodicity of some classes of holomophic difference equations. J. Diff. Equ. Appl. 12(8), 827–835 (2006)

    Article  MATH  Google Scholar 

  8. Berg, L., Stević, S.: Linear difference equation mod 2 with applications to nonlinear equations. J. Diff. Equ. Appl. 14(7), 693–704 (2008)

    Article  MATH  Google Scholar 

  9. Cinar, C., Stević, S., Yalcinkaya, I.: A note on global asymptotic stability of a family of rational difference equation. Rostock. Math. Kolloqu. 59, 41–49 (2004)

    Google Scholar 

  10. Koci, V.L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic, Dordrecht (1993)

    Google Scholar 

  11. Kruse, N., Nesemann, T.: Global asymptotic stability in some discrete dynamical systems. J. Math. Anal. Appl. 326(2), 151–158 (2007)

    MathSciNet  Google Scholar 

  12. Kulenovic, M.R.S., Ladas, G.: Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures. Chapman & Hall, London (2002)

    MATH  Google Scholar 

  13. Li, X., Zhu, D.: Global asymptotic stability of a nonlinear recursive sequence. Appl. Math. Lett. 20, 833–838 (2004)

    Google Scholar 

  14. Li, X.: Global behaviour for a fourth-order rational difference equation. J. Math. Anal. Appl. 312, 555–563 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Li, X., Zhu, D.: Global asymptotic stability in a rational difference equation. J. Diff. Equ. Appl. 9(9), 833–839 (2003)

    Article  MATH  Google Scholar 

  16. Liao, M., Li, X., Tang, X.: On a conjecture for a higher-order rational difference equation. Comput. Math. Appl. 56(2), 305–310 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Putnam Exam, Am. Math. Monthly, 734–736 (1965)

  18. Stević, S.: On the recursive sequence x n+1=g(x n ,x n−1)/(A+x n ). Appl. Math. Lett. 15, 305–308 (2002)

    Article  MathSciNet  Google Scholar 

  19. Stević, S.: On the recursive sequence \(x_{n+1}=\alpha+\frac{x_{n-1}^{p}}{x_{n}^{p}}\) . J. Appl. Math. Comput. 18(1–2), 229–234 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Stević, S.: Global stability and asymptotics of some classes of rational difference equations. J. Math. Anal. Appl. 316(2), 60–68 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Stević, S.: Asymptotics of some classes of higher order difference equations. Discrete Dyn. Nat. Soc. 2007, Article ID 56813 (2007), 20 pp.

  22. Stević, S.: Existence of nontrivial solutions of a rational difference equation. Appl. Math. Lett. 20, 28–31 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Stević, S.: On the recursive sequence \(x_{n+1}=A+\frac{x_{n}^{p}}{x_{n-1}^{r}}\) . Discrete Dyn. Nat. Soc. 2007, Article ID 40963 (2007), 9 pp.

  24. Stević, S.: Nontrivial solutions of a higher-order rational difference equation. Mat. Zametki 84(5), 772–780 (2008)

    MathSciNet  Google Scholar 

  25. Yang, X., Sun, F., Tang, Y.Y.: A new part-metric related inequality chain and application. Discrete Dyn. Nat. Soc. 2007, Article ID 93872 (2007), 10 pp.

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Authors and Affiliations

  1. School of Mathematics and Physics, University of South China, Hengyang, Hunan, 421001, P.R. China

    Maoxin Liao

  2. School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan, 410083, P.R. China

    Maoxin Liao, Xianhua Tang & Changjin Xu

Authors
  1. Maoxin Liao
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  2. Xianhua Tang
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  3. Changjin Xu
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Correspondence to Maoxin Liao.

Additional information

Supported partly by NNSF of China (Grants: 10771215, 10771094) and the Foundation for “Youth Key Teachers of Hunan Province” (2008).

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Liao, M., Tang, X. & Xu, C. On the rational difference equation \(x_{n}=1+\frac{(1-x_{n-k})(1-x_{n-l})(1-x_{n-m})}{x_{n-k}+x_{n-l}+x_{n-m}}\) . J. Appl. Math. Comput. 35, 63–71 (2011). https://doi.org/10.1007/s12190-009-0340-8

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  • DOI: https://doi.org/10.1007/s12190-009-0340-8

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