We study the following max-type difference equation x n = max{A n/x n−r, x n−k}, n = 1,2,…, where {A n}n=1 +∞ is a periodic sequence with period p and k, r ∈ {1,2,…} with gcd(k, r) = 1 and k ≠ r, and the initial conditions x 1−d, x 2−d,…, x 0 are real numbers with d = max{r, k}. We show that if p = 1 (or p ≥ 2 and k is odd), then every well-defined solution of this equation is eventually periodic with period k, which generalizes the results of (Elsayed and Stevi (2009), Iričanin and Elsayed (2010), Qin et al. (2012), and Xiao and Shi (2013)) to the general case. Besides, we construct an example with p ≥ 2 and k being even which has a well-defined solution that is not eventually periodic.
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机译：我们研究以下最大类型差分方程xn =max{A n / xn-r，xn-k}，n = 1,2，…，其中{A n} n = 1 +∞是一个周期周期，周期为p和 k ， r ∈{1,2，…}，其中gcd（ k ， r ）= 1和 k ≠ r ，初始条件 x 1− d ，< em> x 2- d ，…， x 0是具有 d =max{ r < / em>， k }。我们证明，如果 p = 1（或 p ≥2并且 k 是奇数），则该方程的每个明确定义的解决方案最终都是周期为 k 的周期，它概括了（Elsayed和Stevi <数学xmlns：mml =“ http://www.w3.org/1998/Math/MathML” id =“ M1”溢出的结果=“ scroll”> c ´ （2009），Iričanin和Elsayed（2010），Qin等人（2012）以及Xiao和Shi（2013））。此外，我们构造一个示例，其中 p ≥2并且 k 甚至是偶数，它具有定义明确的解决方案，该解决方案最终不是周期性的。
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