Dynamical behavior of one rational fifth-order difference equation

Abstract

In this paper, we study the qualitative behavior of the rational recursive equation \begin{equation*} x_{n+1}=\frac{x_{n-4}}{\pm1\pm x_{n}x_{n-1}x_{n-2}x_{n-3}x_{n-4}}, \quad n \in \mathbb{N}_{0}:=\{0\}\cup\mathbb N, \end{equation*} where the initial conditions are arbitrary nonzero real numbers. The main goal of this paper, is to obtain the forms of the solutions of the nonlinear fifth-order difference equations, where the initial conditions are arbitrary positive real numbers. Moreover, we investigate stability, boundedness, oscillation and the periodic character of these solutions. The results presented in this paper improve and extend some corresponding results in the literature.