Skip to main content
SpringerLink
Log in

On boundedness of solutions of the difference equation \(x_{n+1}=p+\frac{x_{n-1}}{x_{n}}\) for p<1

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

Abstract

In this paper, we study the difference equation

$$x_{n+1}=p+\frac{x_{n-1}}{x_n}, \quad n=0,1,\ldots, $$

where initial values x −1,x 0∈(0,+∞) and 0<p<1, and obtain the set of all initial values (x −1,x 0)∈(0,+∞)×(0,+∞) such that the positive solutions \(\{x_{n}\}_{n=-1}^{\infty}\) are bounded. This answers the Open problem 4.8.11 proposed by Kulenovic and Ladas (Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, 2002).

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Camouzis, E., Ladas, G.: When does local asymptotic stability imply global attractivity in rational equations? J. Differ. Equ. Appl. 12, 863–885 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Camouzis, E., Ladas, G.: When does periodicity destroy boundedness in rational equations? J. Differ. Equ. Appl. 12, 961–979 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. DeVault, R., Kocic, V., Stutson, D.: Global behavior of solutions of the nonlinear difference equation x n+1=p n +x n−1/x n . J. Differ. Equ. Appl. 11, 707–719 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Feuer, J.: On the behavior of solutions of x n+1=p+x n−1/x n . Appl. Anal. 83, 599–606 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  7. Li, X.: Qualitative properties for a fourth-order rational difference equation. J. Math. Anal. Appl. 311, 103–111 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Li, Z., Zhu, D.: Global asymptotic stability of a higher order nonlinear difference equation. Appl. Math. Lett. 19, 926–930 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Papaschinopoulos, G., Schinas, C.J.: Global asymptotic stability and oscillation of a family of difference equations. J. Math. Anal. Appl. 294, 614–620 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Stevic, S.: On the recursive sequence \(x_{n+1}=(\alpha +\sum_{i=1}^{k}\alpha_{i}x_{n-p_{i}})/(1+\sum_{j=1}^{m}\beta_{j}x_{n-q_{j}})\). J. Differ. Equ. Appl. 13, 41–46 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Sun, T., Xi, H.: Global asymptotic stability of a family of difference equations. J. Math. Anal. Appl. 309, 724–728 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Sun, T., Xi, H.: On the system of rational difference equations x n+1=f(y n−q ,x n−s ),y n+1=g(x n−t ,y n−p ). Adv. Differ. Equ. 2006, 51520 (2006)

    MathSciNet  Google Scholar 

  14. Sun, T., Xi, H., Chen, Z.: Global asymptotic stability of a family of nonlinear recursive sequences. J. Differ. Equ. Appl. 11, 1165–1168 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Sun, T., Xi, H., Hong, L.: On the system of rational difference equations x n+1=f(y n ,x n−k ), y n+1=f(x n ,y n−k ). Adv. Differ. Equ. 2006, 16949 (2006)

    MathSciNet  Google Scholar 

  16. Xi, H., Sun, T.: Global behavior of a higher-order rational difference equation. Adv. Differ. Equ. 2006, 27637 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, P.R. China

    Taixiang Sun & Xin Wu

  2. College of Electrical Engineering, Guangxi University, Nanning, Guangxi, 530004, P.R. China

    Qiuli He

  3. Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi, 530003, P.R. China

    Hongjian Xi

Authors
  1. Taixiang Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xin Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qiuli He
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hongjian Xi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Taixiang Sun.

Additional information

Project supported by NNSF of China (11261005, 51267001) and NSF of Guangxi (2011GXNSFA018135, 2012GXNSFDA276040).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sun, T., Wu, X., He, Q. et al. On boundedness of solutions of the difference equation \(x_{n+1}=p+\frac{x_{n-1}}{x_{n}}\) for p<1. J. Appl. Math. Comput. 44, 61–68 (2014). https://doi.org/10.1007/s12190-013-0680-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-013-0680-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation