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Global behavior of two third order rational difference equations with quadratic terms

  • R. Abo-Zeid EMAIL logo
Veröffentlicht/Copyright: 22. Januar 2019
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Abstract

In this paper, we determine the forbidden sets, introduce an explicit formula for the solutions and discuss the global behaviors of solutions of the difference equations

xn+1=axnxn1bxn1+cxn2,n=0,1,

where a,b,c are positive real numbers and the initial conditions x−2,x−1,x0 are real numbers.

MSC 2010: 39A10
  1. (Communicated by Michal Fečkan)

Acknowledgement

The author is grateful to the anonymous referee for his constructive suggestions.

References

[1] Abo-Zeid, R.: On the solutions of two third order recursive sequences, Arm. J. Math. 6(2) (2014), 64–66.Suche in Google Scholar

[2] Abo-Zeid, R.: Global behavior of a higher order difference equation, Math. Slovaca 64(4) (2014), 931–940.10.2478/s12175-014-0249-zSuche in Google Scholar

[3] Abo-Zeid, R.: Global behavior of a third order rational difference equation, Math. Bohem. 139(1) (2014), 25–37.10.21136/MB.2014.143635Suche in Google Scholar

[4] Abo-Zeid, R.: Attractivity of two nonlinear third order difference equations, J. Egyptian Math. Soc. 21 (2013), 241–247.10.1016/j.joems.2013.03.009Suche in Google Scholar

[5] Abo-Zeid, R.—Cinar, C.: Global behavior of the difference equationxn+1=Axn1BCxnxn2, Bol. Soc. Paran. Mat. 31(1) (2013), 43–49.10.5269/bspm.v31i1.14432Suche in Google Scholar

[6] Abo-Zeid, R.—Al-Shabi, M. A.: Global behavior of a third order difference equation, Tamkang J. Math. 43(3) (2012), 375–383.10.5556/j.tkjm.43.2012.801Suche in Google Scholar

[7] Abo-Zeid, R.: Global asymptotic stability of a higher order difference equation, Bull. Allahabad Math. Soc. 25(2) (2010), 341–351.Suche in Google Scholar

[8] Abo-Zeid, R.: Global asymptotic stability of a second order rational difference equation, J. Appl. Math. Inform. 28(3) (2010), 797–804.Suche in Google Scholar

[9] Agarwal, R. P.: Difference Equations and Inequalities, First Edition, Marcel Dekker, 1992.Suche in Google Scholar

[10] Camouzis, E.—Ladas, G.: Dynamics of Third-Order Rational Difference Equations; With Open Problems and Conjectures, Chapman and Hall/HRC Boca Raton, 2008.10.1201/9781584887669Suche in Google Scholar

[11] Grove, E. A.—Ladas, G.: Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC, 2005.10.1201/9781420037722Suche in Google Scholar

[12] Karakostas, G.: Convergence of a difference equation via the full limiting sequences method, Diff. Equ. Dyn. Sys. 1(4) (1993), 289–294.Suche in Google Scholar

[13] Kocic, V. L.—Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic, Dordrecht, 1993.10.1007/978-94-017-1703-8Suche in Google Scholar

[14] Kruse, N.—Nesemann, T.: Global asymptotic stability in some discrete dynamical systems, J. Math. Anal. Appl. 253(1) (1999), 151–158.10.1006/jmaa.1999.6384Suche in Google Scholar

[15] Kulenović, M. R. S.—Ladas, G.: Dynamics of Second Order Rational Difference Equations; With Open Problems and Conjectures, Chapman and Hall/HRC Boca Raton, 2002.10.1201/9781420035384Suche in Google Scholar

[16] Levy, H.—Lessman, F.: Finite Difference Equations, Dover, New York, NY, USA, 1992.Suche in Google Scholar

[17] Sedaghat, H.: Form Symmetries and Reduction of Order in Difference Equations, CRC Press, Boca Raton, 2011.10.1201/b10907Suche in Google Scholar

[18] Sedaghat, H.: Every homogeneous equation of degree one admits a reduction in order, J. Diff. Equ. Appl. 15 (2009), 621–624.10.1080/10236190802201453Suche in Google Scholar

[19] Sedaghat, H.: Global behaviours of rational difference equations of orders two and three with quadratic terms, J. Diff. Equ. Appl. 15(3) (2009), 215–224.10.1080/10236190802054126Suche in Google Scholar

[20] Sedaghat, H.: On third order rational equations with quadratic terms, J. Diff. Equ. Appl. 14(8) (2008), 889–897.10.1080/10236190802054118Suche in Google Scholar

[21] Stević, S.: On positive solutions of a (k +1)th order difference equation, Appl. Math. Lett. 19(5) (2006), 427–431.10.1016/j.aml.2005.05.014Suche in Google Scholar

[22] Stević, S.: More on a rational recurrence relation, Appl. Math. E-Notes 4 (2004), 80–84.Suche in Google Scholar

Received: 2017-01-27
Accepted: 2018-01-12
Published Online: 2019-01-22
Published in Print: 2019-02-25

© 2019 Mathematical Institute Slovak Academy of Sciences

Heruntergeladen am 15.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0210/pdf
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