Muna, A., Saleh, M. (2017). Dynamics of higher order rational difference equation $x_{n+1}=(\alpha+\beta x_{n})/(A + Bx_{n}+ Cx_{n-k})$. International Journal of Nonlinear Analysis and Applications, 8(2), 363-379. doi: 10.22075/ijnaa.2017.10822.1526

Abu Alhalawa Muna; Mohammad Saleh. "Dynamics of higher order rational difference equation $x_{n+1}=(\alpha+\beta x_{n})/(A + Bx_{n}+ Cx_{n-k})$". International Journal of Nonlinear Analysis and Applications, 8, 2, 2017, 363-379. doi: 10.22075/ijnaa.2017.10822.1526

Muna, A., Saleh, M. (2017). 'Dynamics of higher order rational difference equation $x_{n+1}=(\alpha+\beta x_{n})/(A + Bx_{n}+ Cx_{n-k})$', International Journal of Nonlinear Analysis and Applications, 8(2), pp. 363-379. doi: 10.22075/ijnaa.2017.10822.1526

Muna, A., Saleh, M. Dynamics of higher order rational difference equation $x_{n+1}=(\alpha+\beta x_{n})/(A + Bx_{n}+ Cx_{n-k})$. International Journal of Nonlinear Analysis and Applications, 2017; 8(2): 363-379. doi: 10.22075/ijnaa.2017.10822.1526

Dynamics of higher order rational difference equation $x_{n+1}=(\alpha+\beta x_{n})/(A + Bx_{n}+ Cx_{n-k})$

^{}Department of Mathematics, Faculty of Science, Birzeit University, Palestine

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].