TY - JOUR
ID - 2794
TI - Dynamics of higher order rational difference equation $x_{n+1}=(\alpha+\beta x_{n})/(A + Bx_{n}+ Cx_{n-k})$
JO - International Journal of Nonlinear Analysis and Applications
JA - IJNAA
LA - en
SN - 2008-6822
AU - Muna, Abu Alhalawa
AU - Saleh, Mohammad
AD - Department of Mathematics, Faculty of Science, Birzeit University, Palestine
Y1 - 2017
PY - 2017
VL - 8
IS - 2
SP - 363
EP - 379
KW - stability theory
KW - semi-cycle analysis
KW - invariant intervals
KW - nonlinear difference equations
KW - discrete dynamical systems
DO - 10.22075/ijnaa.2017.10822.1526
N2 - 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].
UR - https://ijnaa.semnan.ac.ir/article_2794.html
L1 - https://ijnaa.semnan.ac.ir/article_2794_5faa22d45bfb19c931f7a566b1d51774.pdf
ER -