Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-68228220171201Dynamics of higher order rational difference equation $x_{n+1}=(alpha+beta x_{n})/(A + Bx_{n}+ Cx_{n-k})$363379279410.22075/ijnaa.2017.10822.1526ENAbu Alhalawa MunaDepartment of Mathematics, Faculty of Science, Birzeit University, PalestineMohammad SalehDepartment of Mathematics, Faculty of Science, Birzeit University, PalestineJournal Article20170311The 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<br />$$x_{n+1}=frac{alpha+beta x_{n}}{A + Bx_{n}+ Cx_{n-k}},~~ n=0,1,2,ldots,$$<br />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 $kin{1,2,3,ldots}$. 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].http://ijnaa.semnan.ac.ir/article_2794_5faa22d45bfb19c931f7a566b1d51774.pdf