The Scientific World Journal (Jan 2014)

Eventually Periodic Solutions of a Max-Type Difference Equation

  • Taixiang Sun,
  • Jing Liu,
  • Qiuli He,
  • Xin-He Liu,
  • Chunyan Tao

DOI
https://doi.org/10.1155/2014/219437
Journal volume & issue
Vol. 2014

Abstract

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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.