Global behavior of the max-type difference equation xn = max { 1 / xn-m, An / xn-r }

In this paper, we study the global behavior of the following max-type difference equation x n = max 1 x n - m , A n x n - r , n = 0 , 1 , 2 , ? , where { A n } n ? 0 is a sequence of positive numbers with A n ? ( 0 , 1 ) for every n ? 0 and sup A n < 1 , and m , r ? { 1 , 2 , 3 , ? } , and the initial values x - d , x - d + 1 , ? , x - 1 ? ( 0 , + ∞ ) with d = max { m , r } . We show that: (1) If { x n } n ? - d is a positive solution of this equation, then for every 0 ≤ k ≤ 2 m - 1 , { x 2 mn + k } n ? 0 is eventually monotone. (2) If { A n } n ? 0 is a periodic sequence, then every positive solution of this equation is eventually periodic with period 2 m .

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