On the periodicity of a difference equation with maximum (Q937019)

Summary: We investigate the periodic nature of solutions of the max difference equation \[ x_{n+1}= \max\{x_n,A\}/(x_nx_{n-1}), \quad n=0,1,\dots, \] where \(A\) is a positive real parameter, and the initial conditions \(x_{-1}=A^{r_{-1}}\) and \(x_0=A^{r_0}\) such that \(r_{-1}\) and \(r_0\) are positive rational numbers. The results in this paper answer the open problem 6.2 posed by \textit{E. A. Grove} and \textit{G. Ladas} in their book ``Periodicities in nonlinear difference equations''. Boca Raton (FL) (2005; Zbl 1078.39009).