Skip to main content
SpringerLink
Log in

Global stability of a third-order nonlinear system of difference equations with period-two coefficients

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

The main objective of this paper is to study the global behavior of the following third-order rational system of difference equations

$$\begin{aligned} x_{n+1}=\frac{p_n+y_n}{p_n+y_{n-2}}, \ \ y_{n+1}=\frac{q_n+x_n}{q_n+x_{n-2}}, \qquad n=0,1,2,\ldots , \end{aligned}$$

where \(\{p_n\}\) and \(\{q_n\}\) are 2-periodic sequences of positive numbers, and the initial values \(x_{-i},\, y_{-i} \in [0, \infty )\), for \(i=0,1,2\).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Abo-Zeid, R.: Global attractivity of a higher-order difference equation. Discrete Dyn. Nat. Soc. 2012, 11 (2012) (article ID 930410)

  2. Abo-Zeid, R.: Global behavior of a higher order difference equation. Math. Slovaca 64, 931–940 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abu-Saris, R.M., Devault, R.: Global stability of \(y_{n+1}=\alpha +\frac{y_n}{y_{n-k}}\). Appl. Math. Lett. 16, 173–178 (2003)

  4. Camouzis, E., Ladas, G.: Dynamics of third-order rational difference equations with open problems and conjectures. Chapman Hall/CRC, UK (2007)

  5. Din, Q.: Dynamics of a discrete Lotka-Volterra model. Adv. Differ. Equations 2013, 13 (2013) (article 95)

  6. Din, Q.: Global stability of a population model. Chaos Solitons Fractals 59, 119–128 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Din, Q.: Global behavior of a plant-herbivore model. Adv. Differ. Equations 2015, 12 (2015) (article 119)

  8. Elabbasy, E.M., Elsayed, E.M.: On the global attractivity of difference equation of higher order. Carpathian J. Math. 24(2), 45–53 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Elabbasy, E.M., Eleissawy, S.M.: Global behavior of a higher order rational difference equation. Fasc. Math. 53, 39–52 (2014)

    MathSciNet  MATH  Google Scholar 

  10. El-Metwally, H.: Global behavior of an economic model. Chaos Solitons Fractals 33, 994–1005 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Elsayed, E.M.: On the global attractivity and the periodic character of a recursive sequence. Opusc. Math. 30, 431–446 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Elsayed, E.M.: On the global attractivity and the solution of recursive sequence. Studia Sci. Math. Hung. 47, 401–418 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Elaydi, S.: An introduction to difference equations. Undergraduate texts in mathematics, 3rd edn. Springer, New York (1999)

    Book  Google Scholar 

  14. Grove, E.A., Ladas, G.: Periodicities in nonlinear difference equations. Chapman Hall/CRC, UK (2005)

  15. Hamza, A.E., Ahmed, A.M., Youssef, A.M.: On the recursive sequence \(x_{n+1}=\frac{a+b x_{n}}{A +B x_{n-1}^{k}}\). Arab J. Math. Sci. 17, 31–44 (2011)

    Article  MathSciNet  Google Scholar 

  16. Ibrahim, T.F.: Boundedness and stability of a rational difference equation with delay. Revue Roum. de Math. Pures et Appliquées 57, 215–224 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Hu, L.X., He, W.S., Xia, H.M.: Global asymptotic behavior of a rational difference equation. Appl. Math. Comput. 218, 7818–7828 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hu, L.X., Xia, H.M.: Global asymptotic stability of a second order rational difference equation. Appl. Math. Comput. 233, 377–382 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kocic, V.L., Ladas, G.: Global behavior of nonlinear difference equations of higher order with applications. Kluwer Academic Publishers, Dordrecht (1993)

    Book  MATH  Google Scholar 

  20. Kulenovic, M.R.S., Ladas, G.: Dynamics of second order rational difference equations with open problems and conjectures. Chapman and Hall, CRC Press, UK (2001)

    Book  MATH  Google Scholar 

  21. Ozturk, I., Bozkurt, F., Ozen, S.: On the difference equation \(y_{n+1}= \frac{\alpha +\beta e^{-y_n}}{\gamma + y_{n-1}}\). Appl. Math. Comput. 181, 1387–1393 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Papashinopoulos, G., Radin, M., Shinas, C.J.: Study of the asymptotic behaviour of the solutions of three systems of difference equations of exponential form. Appl. Math. Comput. 218, 5310–5318 (2012)

    Article  MathSciNet  Google Scholar 

  23. Pituk, M.: More on Poincaré’s and Peron’s theorems for difference equations. J. Differ. Equations Appl. 8, 201–216 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Stević, S.: On a discrete epidemic model. Discrete Dyn. Nat. Soc. 2007, 10 (2007) (article ID 87519)

  25. Thai, T.H., Khuong, V.V.: Global asymptotic stability of a second-order system of difference equations. Indian J. Pure Appl. Math. 45(2), 185–198 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Touafek, N.: On a second order rational difference equation. Hacettepe J. Math. Stat. 41, 867–874 (2012)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LMAM Laboratory, Department of Mathematics, Mohamed Seddik Ben Yahia University, Jijel, Algeria

    Imane Dekkar & Nouressadat Touafek

  2. Department of Mathematics, Faculty of Science and Art, Nevsehir Haci Bektas Veli University, Nevsehir, Turkey

    Yasin Yazlik

Authors
  1. Imane Dekkar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nouressadat Touafek
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yasin Yazlik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nouressadat Touafek.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dekkar, I., Touafek, N. & Yazlik, Y. Global stability of a third-order nonlinear system of difference equations with period-two coefficients. RACSAM 111, 325–347 (2017). https://doi.org/10.1007/s13398-016-0297-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-016-0297-z

Keywords

Mathematics Subject Classification

Search

Navigation