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 | ||
| مقاله 54، دوره 8، شماره 2، 2017، صفحه 363-379 اصل مقاله (457.18 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2017.10822.1526 | ||
| نویسندگان | ||
| Abu Alhalawa Muna؛ Mohammad Saleh* | ||
| Department of Mathematics, Faculty of Science, Birzeit University, Palestine | ||
| چکیده | ||
| 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]. | ||
| کلیدواژهها | ||
| stability theory؛ semi-cycle analysis؛ invariant intervals؛ nonlinear difference equations؛ discrete dynamical systems | ||
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