- - - Department of Mathematics, Faculty of Science, Birzeit University, Palestine
- - - Department of Mathematics, Faculty of Science, Birzeit University, Palestine

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 equationx_{n+۱}=\frac{\alpha+\beta x_{n}}{A + Bx_{n}+ Cx_{n-k}},~~ n=۰,۱,۲,\ldots,where the parameters \alpha, \beta, A, B and C are positive, and the initial conditions x_{-k},x_{-k+۱},\ldots,x_{-۱},x_{۰} are positive real numbers and k\in\{۱,۲,۳,\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+۱}=\frac{\alpha +\beta x_{n}}{A+Bx_{n}+ C x_{n-۱}}, Facts and Conjectures, Comput. Math. Appl. ۴۵ (۲۰۰۳) ۱۰۸۷--۱۰۹۹].

stability theory, semi-cycle analysis, invariant intervals, nonlinear difference equations, discrete dynamical systems