A Note on the Periodicity of the Lyness Max Equation

Abstract

We investigate the periodic nature of solutions of a “max-type” difference equation sometimes referred to as the “Lyness max” equation. The equation we consider is  xn+1=max{xn,A}/xn−1⁢ . where A is a positive real parameter, x−1=Ar−1, and x0=Ar0 such that r−1 and r0 are positive rational numbers. The results in this paper answer the Open Problem of Grove and Ladas (2005).