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 D max A xn􀀀1 ; 1 x n􀀀2 ; n 0; where A 0 and 0 1. We prove that every positive solution of this difference equation approaches x D 1 or is eventually periodic with a period 2, 3 or 4.