ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Dynamics of $K^{th}$ Order Rational Difference Equation

Journal of Applied Nonlinear Dynamics 10(1) (2021) 125--149 | DOI:10.5890/JAND.2021.03.008

Department of Mathematics, Birzeit University, West Bank

Abstract

In this paper we will investigate the dynamical behavior of the following rational difference equation $$x_{n+1}= \frac{\alpha + \beta x_{n} + \gamma x_{n-k}} {A +B x_{n} + C x_{n-k}},\quad n=0,1,...$$ where the parameters $\alpha, \beta, \gamma$ and A, B, C and the initial conditions $x_{-k},\dots,x_{-1},x_{0}$ are non-negative real numbers, and the denominator is nonzero. Our concentration here, is on the global stability, the periodic character, the analysis of semi-cycles and the invariant intervals of the positive solution of the above equation. It is worth mentioning that our difference equation is the general case of the rational equation which is studied by Kulenovic and Ladas in their monograph ( Dynamics of Second Order Rational Difference Equation with Open Problems and Conjectures, 2002 ).

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