On a difference equation with maximum

This paper studies the asymptotic behavior of positive solutions to the difference equationxn=max1xn-1α,Axn-2,n=0,1,…,where 0 < α < 1, A > 0. We prove that every positive solution of this equation approaches x∗ = 1 or is eventually periodic with period 4 according as A ⩽ 1 or A > 1.