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Keywords:
difference equation; forbidden set; periodic solution; unbounded solution
difference equation; forbidden set; periodic solution; unbounded solution
Summary:
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 $a<c$ 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.
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