Dynamics of higher order rational difference equation x_{n+۱}=(\alpha+\beta x_{n})/(A + Bx_{n}+ Cx_{n-k})

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].

Keywords:

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