Eventually Periodic Solutions of a Max-Type Difference Equation

• Main Authors: Taixiang Sun, Jing Liu, Qiuli He, Xin-He Liu, Chunyan Tao
• Format: Article
• Language: English
• Published: Hindawi Limited 2014-01-01
• Series: The Scientific World Journal
• Online Access: http://dx.doi.org/10.1155/2014/219437

We study the following max-type difference equation xn=max⁡{An/xn-r,xn-k}, n=1,2,…, where {An}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 x1-d,x2-d,…,x0 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), 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.