Global Behavior of Two Rational Third Order Difference Equations

Year 2019, Volume: 2 Issue: 4, 212 - 217, 26.12.2019

R. Abo-zeid , H. Kamal

https://doi.org/10.32323/ujma.626465

Cited By: 12

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

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