*(English)*Zbl 1103.39004

The authors consider the difference equation

where ${\left\{{p}_{n}\right\}}_{n}$ is positive and periodic with period $k\in \{2,3\}$. The initial conditions are positive. In the case $k=2$ it is proved that there are no solutions of odd period; then stability by the first approximation of the equilibrium is considered. Further global results are given for an associated system of three difference equations. This will lead to a global stability result for the basic equation.

Next, sufficient conditions for the existence of unbounded solutions are given. In the case $k=3$ the following results are obtained: existence of a unique positive equilibrium using such classical results as Theorems of Descartes and Rolle; this equilibrium is stable by the first approximation. Existence of unbounded solutions is obtained also in this case.

##### MSC:

39A11 | Stability of difference equations (MSC2000) |

39A20 | Generalized difference equations |