Skip to main content
SpringerLink
Log in

On the recursive sequence \( {x}_{n+1}=\frac{x_{n-\left(4k+3\right)}}{1+\prod_{t=0}^2{x}_{n-\left(k+1\right)t-k}} \)

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The solution of the difference equation

$$ {x}_{n+1}=\frac{x_{n-\left(4k+3\right)}}{1+\prod_{t=0}^2{x}_{n-\left(k+1\right)t-k}},\kern0.5em n=0,1,2,\dots, $$

where x −(4k+3), x −(4k+2) , . . . , x −1, x 0 ∈ (0, ∞) and k = 0, 1, . . . , is studied.

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. A. M. Amleh, E. A. Grove, G. Ladas, and D. A. Georgiou, “On the recursive sequence \( {y}_{n+1}=\alpha +\frac{y_{n-1}}{y_n} \),” J. Math. Anal. Appl., 233, 790–798 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Cinar, “On the positive solutions of the difference equation \( {x}_{n+1}=\frac{x_{n-1}}{1+{ax}_n{x}_{n-1}} \),” Appl. Math. Comp., 158, 809–812 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Cinar, “On the positive solutions of the difference equation \( {x}_{n+1}=\frac{x_{n-1}}{-1+{ax}_n{x}_{n-1}} \),” Appl. Math. Comp., 158, 793–797 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Cinar, “On the positive solutions of the difference equation \( {x}_{n+1}=\frac{ax_{n-1}}{1+{bx}_n{x}_{n-1}} \),” Appl. Math. Comp., 156, 587–590 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  5. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the difference equation \( {x}_{n+1}={ax}_n-\frac{bx_n}{cx_n-{dx}_{n-1}} \),” Adv. in Differ. Equa., 2006, Article ID 82579, (2006).

  6. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Qualitative behavior of higher order difference equation,” Soochow J. of Math., 33, 861–873 (2007).

    MathSciNet  MATH  Google Scholar 

  7. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Global attractivity and periodic character of a fractional difference equation of order three,” Yokohama Math. J., 53, 89–100(2007).

    MathSciNet  MATH  Google Scholar 

  8. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the difference equation \( {x}_{n+1}=\frac{ax_{n-1}}{\beta +\upgamma \prod_{i=0}^k{x}_{n-1}} \),” J. Conc. Appl. Math., 5, 101–113 (2007).

    MATH  Google Scholar 

  9. E. M. Elabbasy and E. M. Elsayed, “On the global attractivity of difference equation of higher order,” Carpath. J. of Math., 24, 45–53 (2008).

    MathSciNet  MATH  Google Scholar 

  10. E. M. Elsayed, “On the solution of recursive sequence of order two,” Fasciculi Math., 40, 5–13 (2008).

    MathSciNet  MATH  Google Scholar 

  11. E. M. Elsayed, “Dynamics of a recursive sequence of higher order,” Commun. on Appl. Nonlin. Anal., 16, 37–50 (2009).

    MathSciNet  MATH  Google Scholar 

  12. E. M. Elsayed, “Solution and attractivity for a rational recursive sequence,” Discrete Dyn. in Nature and Soc., 2011, Article ID 982309, (2011).

  13. E. M. Elsayed, “On the solution of some difference equation,” Europ. J. of Pure and Appl. Math., 4, 287–303 (2011).

    MathSciNet  MATH  Google Scholar 

  14. E. M. Elsayed, “On the dynamics of a higher order rational recursive sequence,” Commun. Math. Analys., 12, 117–133 (2012).

    MathSciNet  MATH  Google Scholar 

  15. E. M. Elsayed, “Solution of rational difference system of order two,” Math. Comput. Model., 55, 378–384 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  16. C. H. Gibbons, M. R. S. Kulenović, and G. Ladas, “On the recursive sequence \( {x}_{n+1}=\frac{\alpha +\beta {x}_{n-1}}{\chi +{x}_n} \),” Math. Sci. Res. Hot-Line, 4, No. 2, 1–11 (2000).

    MathSciNet  MATH  Google Scholar 

  17. M.R.S. Kulenović, G. Ladas, and W. S. Sizer, “On the recursive sequence \( {x}_{n+1}=\frac{\alpha {x}_n+\beta {x}_{n-1}}{\chi {x}_n+\delta {x}_{n-1}} \),” Math. Sci. Res. Hot-Line, 2, No. 5, 1–16 (1998).

    MathSciNet  MATH  Google Scholar 

  18. S. Stevic, “On the recursive sequence \( {x}_{n+1}=\frac{x_{n-1}}{g\left({x}_n\right)} \),” Taiwan. J. Math., 6, No. 3, 405–414 (2002).

    Google Scholar 

  19. D. Şimşek, C. Çınar, and I. Yalçınkaya, “On the recursive sequence \( {x}_{n+1}=\frac{x_{n-3}}{1+{x}_{n-1}} \),” Int. J. Contemp. Math. Sci., 1, Nos. 9-12, 475–480 (2006).

    MathSciNet  MATH  Google Scholar 

  20. D. Şimşek, C. Çınar, R. Karataş, and I. Yalçınkaya, “On the recursive sequence \( {x}_{n+1}=\frac{x_{n-5}}{1+{x}_{n-2}} \),” Int. J. Pure Appl. Math., 27, No. 4, 501–507 (2006).

    MathSciNet  MATH  Google Scholar 

  21. D. Şimşek, C. Çınar, R. Karataş, and I. Yalçınkaya, “On the recursive sequence \( {x}_{n+1}=\frac{x_{n-5}}{1+{x}_{n-1}{x}_{n-3}} \),” Int. J. Pure Appl. Math., 28, No. 1, 117–124 (2006).

    MathSciNet  MATH  Google Scholar 

  22. D. Şimşek, C. Çınar, and I. Yalçınkaya, “On The Recursive Sequence x(n+1) = x[n-(5k+9)] / 1+x(n-4)x(n-9) ... x[n-(5k+4)],” Taiwan. J. of Math., 12, No. 5, 1087–1098 (2008).

    MATH  Google Scholar 

  23. D. Şimşek and A. Doğan, “On a class of recursive sequence,” Manas J. of Engin., 2, No. 1, 16–22 (2014).

    Google Scholar 

  24. I. Yalcinkaya, B. D. Iricanin, and C. Cinar, “On a max-type difference equation,” Discrete Dyn. in Nature and Soc., 2007, Article ID 47264, doi: 10.1155/2007/47264, (2007).

  25. H. D. Voulov, “Periodic solutions to a difference equation with maximum,” Proc. Am. Math. Soc., 131, 2155–2160 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  26. X. Yang, B. Chen, G. M. Megson, and D. J. Evans, “Global attractivity in a recursive sequence,” Appl. Math. and Comput., 158, 667–682 (2004).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kyrgyz–Turkish Manas University, Bishkek, Kyrgyzstan

    Dağıstan Simsek & Fahreddin Abdullayev

  2. Selcuk University, Konya, Turkey

    Dağıstan Simsek

  3. Mersin University, Mersin, Turkey

    Fahreddin Abdullayev

Authors
  1. Dağıstan Simsek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fahreddin Abdullayev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dağıstan Simsek.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 13, No. 3, pp. 376–387 July–September, 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Simsek, D., Abdullayev, F. On the recursive sequence \( {x}_{n+1}=\frac{x_{n-\left(4k+3\right)}}{1+\prod_{t=0}^2{x}_{n-\left(k+1\right)t-k}} \) . J Math Sci 222, 762–771 (2017). https://doi.org/10.1007/s10958-017-3330-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-017-3330-7

Keywords

Search

Navigation