Eventually Periodic Solutions of a Max-Type Difference Equation

Abstract

We study the following max-type difference equationxn=max⁡{An/xn-r,xn-k},n=1,2,…, where{An}n=1+∞is a periodic sequence with periodpandk,r∈{1,2,…}withgcd(k,r)=1andk≠r, and the initial conditionsx1-d,x2-d,…,x0are real numbers withd=max⁡{r,k}. We show that ifp=1(orp≥2andkis odd), then every well-defined solution of this equation is eventually periodic with periodk, 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 withp≥2andkbeing even which has a well-defined solution that is not eventually periodic.