On the asymptotic behavior and periodic nature of a difference equation with maximum

In this paper, we investigate the asymptotic behavior and periodic nature of positive solutions of the difference equation xn=max{Axn−1,1xn−2α},n≥0, where A≥0A≥0 and 0≤α≤10≤α≤1. We prove that every positive solution of this difference equation approaches x¯=1 or is eventually periodic with a period 2, 3 or 4.