Eventually Periodic Solutions of a Max-Type Difference Equation

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 [Formula: see text] (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.