Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter February 20, 2018

On the solutions of a higher order difference equation

  • Raafat Abo-Zeid EMAIL logo

Abstract

In this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equation

xn+1=axnxn-kbxn-cxn-k-1,n=0,1,,

where a,b,c are positive real numbers and the initial conditions x-k-1,x-k,,x-1,x0 are real numbers. We show that when a=b=c, the behavior of the solutions depends on whether k is even or odd.

MSC 2010: 39A10; 39A20

References

[1] R. Abo-Zeid, Global asymptotic stability of a second order rational difference equation, J. Appl. Math. Inform. 28 (2010), no. 3–4, 797–804. Search in Google Scholar

[2] R. Abo-Zeid, Global behavior of a rational difference equation with quadratic term, Math. Morav. 18 (2014), no. 1, 81–88. 10.5937/MatMor1401081ASearch in Google Scholar

[3] R. Abo-Zeid, Global behavior of a third order rational difference equation, Math. Bohem. 139 (2014), no. 1, 25–37. 10.21136/MB.2014.143635Search in Google Scholar

[4] R. Abo-Zeid and M. A. Al-Shabi, Global behavior of a third order difference equation, Tamkang J. Math. 43 (2012), no. 3, 375–383. 10.5556/j.tkjm.43.2012.801Search in Google Scholar

[5] R. Abo-Zeid and C. Cinar, Global behavior of the difference equation xn+1=Axn-1B-Cxnxn-2, Bol. Soc. Parana. Mat. (3) 31 (2013), no. 1, 43–49. 10.5269/bspm.v31i1.14432Search in Google Scholar

[6] M. A. Al-Shabi and R. Abo-Zeid, Global asymptotic stability of a higher order difference equation, Appl. Math. Sci. (Ruse) 4 (2010), no. 17–20, 839–847. Search in Google Scholar

[7] K. S. Berenhaut, J. D. Foley and S. Stević, The global attractivity of the rational difference equation yn=yn-k+yn-m1+yn-kyn-m, Appl. Math. Lett. 20 (2007), no. 1, 54–58. 10.1016/j.aml.2006.02.022Search in Google Scholar

[8] E. Camouzis, G. Ladas, I. W. Rodrigues and S. Northshield, The rational recursive sequence xn+1=(βxn2)/(1+xn-12), Comput. Math. Appl. 28 (1994), no. 1–3, 37–43. 10.1016/0898-1221(94)00091-3Search in Google Scholar

[9] M. Dehghan, C. M. Kent, R. Mazrooei-Sebdani, N. L. Ortiz and H. Sedaghat, Dynamics of rational difference equations containing quadratic terms, J. Difference Equ. Appl. 14 (2008), no. 2, 191–208. 10.1080/10236190701565636Search in Google Scholar

[10] E. A. Grove, E. J. Janowski, C. M. Kent and G. Ladas, On the rational recursive sequence xn+1=αxn+β(γxn+δ)xn-1, Comm. Appl. Nonlinear Anal. 1 (1994), no. 3, 61–72. Search in Google Scholar

[11] H. Sedaghat, Global behaviours of rational difference equations of orders two and three with quadratic terms, J. Difference Equ. Appl. 15 (2009), no. 3, 215–224. 10.1080/10236190802054126Search in Google Scholar

[12] S. Stević, Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl. 316 (2006), no. 1, 60–68. 10.1016/j.jmaa.2005.04.077Search in Google Scholar

[13] X. Yang, On the global asymptotic stability of the difference equation xn=xn-1xn-2+xn-3+axn-1+xn-2xn-3+a, Appl. Math. Comput. 171 (2005), no. 2, 857–861. Search in Google Scholar

[14] X. Yang, D. J. Evans and G. M. Megson, On two rational difference equations, Appl. Math. Comput. 176 (2006), no. 2, 422–430. 10.1016/j.amc.2005.09.031Search in Google Scholar

[15] X. Yang, W. Su, B. Chen, G. M. Megson and D. J. Evans, On the recursive sequence xn=axn-1+bxn-2c+dxn-1xn-2, Appl. Math. Comput. 162 (2005), no. 3, 1485–1497. 10.1016/j.amc.2004.03.023Search in Google Scholar

Received: 2015-08-31
Accepted: 2016-05-25
Published Online: 2018-02-20
Published in Print: 2020-06-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 19.4.2024 from https://www.degruyter.com/document/doi/10.1515/gmj-2018-0008/html
Scroll to top button