Raafat Abo-Zeid, Department of Basic Science, The High Institute for Engineering & Modern Technology, Cairo, Egypt, e-mail: abuzead73@yahoo.com
Abstract: 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_nx_{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 ${(a-c)}/b<1$. 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.
Keywords: difference equation; forbidden set; periodic solution; unbounded solution
Classification (MSC 2010): 39A20, 39A21, 39A23, 39A30
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