@article {
author = {Muna, Abu Alhalawa and Saleh, Mohammad},
title = {Dynamics of higher order rational difference equation $x_{n+1}=(\alpha+\beta x_{n})/(A + Bx_{n}+ Cx_{n-k})$},
journal = {International Journal of Nonlinear Analysis and Applications},
volume = {8},
number = {2},
pages = {363-379},
year = {2017},
publisher = {Semnan University},
issn = {2008-6822},
eissn = {},
doi = {10.22075/ijnaa.2017.10822.1526},
abstract = {The main goal of this paper is to investigate the periodic character, invariant intervals, oscillation and global stability and other new results of all positive solutions of the equation$$x_{n+1}=\frac{\alpha+\beta x_{n}}{A + Bx_{n}+ Cx_{n-k}},~~ n=0,1,2,\ldots,$$where the parameters $\alpha$, $\beta$, $A$, $B$ and $C$ are positive, and the initial conditions $x_{-k},x_{-k+1},\ldots,x_{-1},x_{0}$ are positive real numbers and $k\in\{1,2,3,\ldots\}$. We give a detailed description of the semi-cycles of solutions and determine conditions under which the equilibrium points are globally asymptotically stable. In particular, our paper is a generalization of the rational difference equation that was investigated by Kulenovic et al. [The Dynamics of $x_{n+1}=\frac{\alpha +\beta x_{n}}{A+Bx_{n}+ C x_{n-1}}$, Facts and Conjectures, Comput. Math. Appl. 45 (2003) 1087--1099].},
keywords = {stability theory,semi-cycle analysis,invariant intervals,nonlinear difference equations,discrete dynamical systems},
url = {https://ijnaa.semnan.ac.ir/article_2794.html},
eprint = {https://ijnaa.semnan.ac.ir/article_2794_5faa22d45bfb19c931f7a566b1d51774.pdf}
}