Document Type: Research Paper

**Authors**

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].

$$x_{n+1}=\frac{\alpha+\beta x_{n}}{A + Bx_{n}+ Cx_{n-k}},~~ n=0,1,2,\ldots,$$

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**

Volume 8, Issue 2

Summer and Autumn 2017

Pages 363-379