Qualitative properties of difference equation of order six (Q515425)

Summary: In this paper we study the qualitative properties and the periodic nature of the solutions of the difference equation \[ x_{n+1}=\alpha x_{n-2}+\frac{\beta x_{n-2}^2}{\gamma x_{n-2}+\delta x_{n-5}},\quad n=0,1,\dots, \] where the initial conditions \(x_{-5}\), \(x_{-4}\), \(x_{-3}\), \(x_{-2}\), \(x_{-1}\), \(x_0\) are arbitrary positive real numbers and \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) are positive constants. In addition, we derive the form of the solutions of some special cases of this equation.