Global behavior of the max-type difference equation xn=max1xn-m,Anxn-r

In this paper, we study the global behavior of the following max-type difference equationxn=max1xn-m,Anxn-r,n=0,1,2,…,where {An}n⩾0{An}n⩾0 is a sequence of positive numbers with An∈(0,1)An∈(0,1) for every n⩾0n⩾0 and supAn<1supAn<1, and m,r∈{1,2,3,…}m,r∈{1,2,3,…}, and the initial values x-d,x-d+1,…,x-1∈(0,+∞)x-d,x-d+1,…,x-1∈(0,+∞) with d=max{m,r}d=max{m,r}. We show that: (1) If {xn}n⩾-d{xn}n⩾-d is a positive solution of this equation, then for every 0⩽k⩽2m-1,{x2mn+k}n⩾00⩽k⩽2m-1,{x2mn+k}n⩾0 is eventually monotone. (2) If {An}n⩾0{An}n⩾0 is a periodic sequence, then every positive solution of this equation is eventually periodic with period 2m2m.