Eventually periodic solutions of a max-type difference equation.

We study the following max-type difference equation xn = 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 Stevic (2009), Iricanin 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.