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Global behavior of a third order rational difference equation. (English) Zbl 1340.39014
A third-order nonlinear difference equation of the form \[ x_{n+1}=\frac {ax_nx_{n-1}}{-bx_n+cx_{n-2}},\quad n\in \mathbb N_0,\tag{1} \] is considered, where \(a, b, c\) are positive constants. The forbidden set \(F\) for equation (1) is meant as a set of initial values \((x_{-2}, x_{-1}, x_0)\in \mathbb R^3\) such that its subsequent iteration under equation (1) does not exist for some \(n\in \mathbb N_0\). In the case when \((x_{-2}, x_{-1}, x_0)\notin F\), the corresponding solution to equation (1) can be defined for all forward iterations \(n\in \mathbb N_0\).
The author provides a description of the forbidden set \(F\). The explicit form of the solutions is then given for initial values \((x_{-2}, x_{-1}, x_0)\notin F\). Based on this representation sufficient conditions in terms of the coefficients \(a,b,c\) are given for all solutions to converge to zero. Another complementary set of conditions is given when the solutions are unbounded. Some partial cases of periodicity and asymptotic periodicity are considered.

MSC:
39A20 Multiplicative and other generalized difference equations
39A21 Oscillation theory for difference equations
39A23 Periodic solutions of difference equations
39A30 Stability theory for difference equations
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