On the Recursive Sequence 𝑥𝑛+𝟏=max{𝑥𝑛,𝐴}/𝑥𝟐𝑛𝑥𝑛−𝟏

Abstract

We investigate the periodic nature of the solution of the max-type difference equation 𝑥𝑛+1=max{𝑥𝑛,𝐴}/𝑥2𝑛𝑥𝑛−1, 𝑛=0,1,2,…, where the initial conditions are 𝑥−1=𝐴𝑟1 and 𝑥0=𝐴𝑟2 for 𝐴∈(0,∞), and that 𝑟1 and 𝑟2 are positive rational numbers. The results in this paper solve the Open Problem proposed by Grove and Ladas (2005).