Global Behavior of Two Rational Third Order Difference Equations
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
References
- [1] R. Abo-Zeid, Behavior of solutions of a second order rational difference equation, Math. Morav., 23 (1) (2019) , 11-25 .
- [2] R. Abo-Zeid, Global behavior of two third order rational difference equations with quadratic terms, Math. Slovaca, 69 (1) (2019) , 147-158 .
- [3] R. Abo-Zeid, Global Behavior of a fourth order difference equation with quadratic term, Bol. Soc. Mat. Mexicana, 25 (1) (2019) , 187-194 .
- [4] R. Abo-Zeid, Behavior of solutions of a higher order difference equation, Alabama J. Math., 42 (2018) , 1-10 .
- [5] R. Abo-Zeid, On the solutions of a higher order difference equation, Georgian Math. J., doi:10.1515/gmj-2018-0008.
- [6] R. Abo-Zeid, On a third order difference equation, Acta Univ. Apulensis, 55 (2018) , 89-103 .
- [7] R. Abo-Zeid Forbidden sets and stability in some rational difference equations, J. Difference Equ. Appl., 24 (2) (2018) , 220-239 .
- [8] R. Abo-Zeid, On the solutions of a second order difference equation, Math. Morav., 21 (2) (2017), 61-75 .
- [9] R. Abo-Zeid, Global behavior of a higher order rational difference equation, Filomat 30 (12) (2016), 3265-3276 .
- [10] R. Abo-Zeid, Global behavior of a third order rational difference equation, Math. Bohem., 139 (1) (2014) , 25-37 .
