On the periodicity of a max-type rational difference equation

Abstract

This paper shows that every well-defined solution of the following max-type difference equation \[{x_{n + 1}} = \max \{ \frac{A}{{{x_n}}},\,\frac{A}{{{x_{n - 1}}}},\,{x_{n - 2}}\} ,\quad n \in {N_0},\] where \(A \in R\) and the initial conditions \({x_{ - 2}},\,{x_{ - 1}},\,{x_0}\) are arbitrary non-zero real numbers is eventually periodic with period three by using new iteration method for the more general nonlinear difference equations and inequality skills as well as the mathematical induction. Our main results considerably improve results appearing in the literature.