title: Eventually periodic solutions of a max-type difference equation.
abstract: 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 Stević (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.
authors.affiliation: College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China.
authors.forename: Taixiang
authors.identifier: 0000-0001-9819-1615
authors.initials: T
authors.lastname: Sun
authors.affiliation: College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China.
authors.forename: Jing
authors.initials: J
authors.lastname: Liu
authors.affiliation: College of Electrical Engineering, Guangxi University, Nanning, Guangxi 530004, China.
authors.forename: Qiuli
authors.initials: Q
authors.lastname: He
authors.affiliation: College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China.
authors.forename: Xin-He
authors.initials: XH
authors.lastname: Liu
authors.affiliation: College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China.
authors.forename: Chunyan
authors.initials: C
authors.lastname: Tao
mesh_terms: D008962:Models, Theoretical