Skip to main content
SpringerLink
Log in

On the recursive sequencex n +1=α-(x n /x n −1)

  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

We study the global asymptotic stability, global attractivity, boundedness character, and periodic nature of all positive solutions and all negative solutions of the difference equation

$$x_{n + 1} = \alpha - \frac{{x_n }}{{x_{n - 1} }}, n = 0,1,...,$$

where α∈R is a real number, and the initial conditionsx−1,x 0 are arbitrary real numbers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Devault, G. Ladas and S. W. Schultz,On the recursive sequence x n+1=A/x n +1/x n −2, Proc. Amer. Math. Soc.126(11) (1998), 3257–3261.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. M. Amleh, E. A. Grove, D. A. Georgiou and G. Ladas,On the recursive sequence x n +1= α+(x n −1/x n ), J. Math. Anal. Appl.233 (1999), 790–798.

    Article  MATH  MathSciNet  Google Scholar 

  3. H. M. El-Owaidy, A. M. Ahmed and M. S. Mousa,On asymptotic behavior of the difference equation x n +1=α+(x n−k /x n ), Appl. Math. Comput.147 (1) (2004), 163–167.

    Article  MATH  MathSciNet  Google Scholar 

  4. X. X. Yan and W. T. Li,Global attractivity in the recursive sequence x n +1=(α−βx n )/(ψ-x n −1), Appl. Math. Comput.138 (2003), 415–423.

    Article  MATH  MathSciNet  Google Scholar 

  5. X. X. Yan, W. T. Li and H. R. Sun,Global attractivity in a higher order nonlinear difference equation, Appl. Math. E-Notes2 (2002), 51–58.

    MATH  MathSciNet  Google Scholar 

  6. W. A. Kosmala, M. R. S. Kulenovic, G. Ladas and C. T. Teixeira,On the recursive sequence y n +1=(p+y n −1)/(qy n +y n −1) J. Math. Anal. Appl.251 (2000), 571–586.

    Article  MATH  MathSciNet  Google Scholar 

  7. W. T. Li and H. R. Sun,Global attractivity of a rational recursive sequence. Dynamic Systems and Applications11 (3) (2002), 339–345.

    MATH  MathSciNet  Google Scholar 

  8. M. T. Aboutaleb, M. A. El-Sayed and A. E. Hamza,Stability of the recursive sequence x n +1=(α−βx n ) /(ψ+x n −1) J. Math. Anal. Appl.261 (2001), 126–133.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. DeVault, W. Kosmala, G. Ladas and S. W. Schultz,Global behavior of y n +1=(p+y n−k )/(qy n +y n−k ), Nonlinear Analysis TMA,47 (2001), 4743–4751.

    Article  MATH  MathSciNet  Google Scholar 

  10. X. X. Liao,Asymptotic behavior for some class of discrete dynamical systems, Science in China (Series A)32(3) (2002), 205–215.

    Google Scholar 

  11. H. D. Voulov,Existence of monotone solutions of some difference equations with unstable equilibrium, J. Math. Anal. Appl.272 (2002), 555–564.

    Article  MATH  MathSciNet  Google Scholar 

  12. V. L. Kocic, G. Ladas and I. W. Rodrigues,On rational recursive sequences, J. Math. Anal. Appl.173 (1993), 127–157.

    Article  MATH  MathSciNet  Google Scholar 

  13. X. X. Yan and W. T. Li,Global attractivity in a rational recursive sequence, Appl. Math. Comput.145 (2003), 1–12.

    Article  MATH  MathSciNet  Google Scholar 

  14. X. X. Yan and W. T. Li,Global attractivity for a class of higher order nonlinear difference equations, Appl. Math. Comput.149(2) (2004), 533–546.

    Article  MATH  MathSciNet  Google Scholar 

  15. X. X. Yan and W. T. Li,Global attractivity for a class of higher order nonlinear difference equations, Appl. Math. Comput.149(2) (2004), 533–546.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. L. Li, M. S. Wang and J. R. Yan,On oscillation of nonlinear second order differential equation with damping term, J. Appl. Math. & Computing13 (2003), 223–232.

    Article  MATH  MathSciNet  Google Scholar 

  17. Y. R. Duan, P. Tian and S. N. Zhang,Oscillation and stability of nonlinear neutral impulsive delay differential equations, J. Appl. Math. & Computing12 (2003), 243–254.

    Article  MathSciNet  Google Scholar 

  18. H. M. El-Owaidy, A. M. Ahmed and M. S. Mouse,On asymptotic behavior of the difference equation x n +1=α(x p n−1 / p n ), J. Appl. Math. & Computing12 (2003), 31–38.

    Article  MATH  Google Scholar 

  19. V. L. Kocic and G. Ladas,Global Behavior of Nonlinear Difference Equations of Higher Order with Application, Kluwer Academic Publishers, Dordrecht, 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hexi University, 734000, Zhangye, Gansu, P. R. China

    Xing-Xue Yan & Zhu Zhao

  2. Department of Mathematics, Lanzhou University, 730000, Lanzhou, Gansu, P. R. China

    Xing-Xue Yan, Wan-Tong Li & Zhu Zhao

Authors
  1. Xing-Xue Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wan-Tong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhu Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wan-Tong Li.

Additional information

This work was supported by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011-A25-007-Z), the Foundation for University Key Teacher by the Minitry of Education of China, and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China.

Xing-Xue Yan 1,2 received his M. Sc. from Lanzhou University in 2002. He is now a Ph. D. Candidate in Lanzhou University and an Associate Professor at Hexi University. His research interests cover global behavior of difference equations and functional differential equations.

Wan-Tong Li received M. Sc. from Beijing Institution of Technology and Ph. D from Lanzhou University. He is a professor at department of mathematics, Lanzhou University since 1999. His research interests are mainly on functional differential equations with applications, dynamic systems, partial differential equations with delays and dynamic equations on time scale.

Zhu Zhao 1,2 received his BS from Northwest Normal University in 1989. He is now a M. Sc. Candidate in Lanzhou University and an Associate Professor at Hexi University. His research interests are focussed on the global behavior of difference equations.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yan, XX., Li, WT. & Zhao, Z. On the recursive sequencex n +1=α-(x n /x n −1). JAMC 17, 269–282 (2005). https://doi.org/10.1007/BF02936054

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02936054

AMS Mathematics Subject Classification

Key words and phrases

Search

Navigation