Global Behavior of Two Rational Third Order Difference Equations

Authors : R. Abo-zeid, H. Kamal

Pages : 212-217

Doi:10.32323/ujma.626465

View : 16 | Download : 10

Publication Date : 2019-12-26

Article Type : Research

Abstract :In this paper, we solve and study the global behavior of all admissible solutions of  the two difference equations $$x_{n+1}=\frac{x_{n}x_{n-2}}{x_{n-1}-x_{n-2}}, \quad n=0,1,...,$$ and $$x_{n+1}=\frac{x_{n}x_{n-2}}{-x_{n-1}+x_{n-2}}, \quad n=0,1,...,$$ where  the initial values $x_{-2}$, $x_{-1}$, $x_{0}$ are real numbers.\\ We show that every admissible solution for the first equation converges to zero. For the other equation, we show that every admissible solution is periodic with prime period  six.  Finally we give some illustrative examples.
Keywords : difference equation, forbidden set, periodic solution, convergence