On some difference equations with eventually periodic solutions (Q1269600)

The authors study the boundedness and the periodic character of the solutions of the equation \[ \begin{aligned} x_n & ={\alpha x_n +\beta x_{n-1} \over 2}, \quad \text{if }x_n+x_{n-1} \text{ is even}; \\ x_n & =\gamma x_n+ \partial x_{n-1}, \quad \text{if } x_n+x_{n-1} \text{ is odd, }n=0,1, \dots \end{aligned} \] where \(x_{-1}\), \(x_0\in Z\) and \(\alpha, \beta,\gamma,\partial \in\{-1,1\}\).