Geodesic
Global behavior of a third order rational difference equation
Mathematica Bohemica, Tome 139 (2014) no. 1, pp. 25-37

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In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation $$x_{n+1}=\frac {ax_{n}x_{n-1}}{-bx_{n}+ cx_{n-2}},\quad n\in \mathbb {N}_0 $$ where $a$, $b$, $c$ are positive real numbers and the initial conditions $x_{-2}$, $x_{-1}$, $x_0$ are real numbers. We show that every admissible solution of that equation converges to zero if either $ac$ with ${(a-c)}/{b}1$. \endgraf When $a>c$ with ${(a-c)}/{b}>1$, we prove that every admissible solution is unbounded. Finally, when $a=c$, we prove that every admissible solution converges to zero.
In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation $$x_{n+1}=\frac {ax_{n}x_{n-1}}{-bx_{n}+ cx_{n-2}},\quad n\in \mathbb {N}_0 $$ where $a$, $b$, $c$ are positive real numbers and the initial conditions $x_{-2}$, $x_{-1}$, $x_0$ are real numbers. We show that every admissible solution of that equation converges to zero if either $a$ or $a>c$ with ${(a-c)}/{b}1$. \endgraf When $a>c$ with ${(a-c)}/{b}>1$, we prove that every admissible solution is unbounded. Finally, when $a=c$, we prove that every admissible solution converges to zero.
DOI : 10.21136/MB.2014.143635
Classification : 39A20, 39A21, 39A23, 39A30
Keywords: difference equation; forbidden set; periodic solution; unbounded solution 
Abo-Zeid, Raafat. Global behavior of a third order rational difference equation. Mathematica Bohemica, Tome 139 (2014) no. 1, pp. 25-37. doi: 10.21136/MB.2014.143635
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