Qualitative behavior of rational difference equations of higher order

Abstract

In this paper we study the behavior of the solution of the following rational difference equation$$x_{n+1}=\frac{a x_{n-r}^2+b x_{n-l} x_{n-k}^2}{c x_{n-r}^2+d x_{n-l} x_{n-k}^2} \quad n=0,1, \ldots,$$where the parameters $a, b, c$ and $d$ are positive real numbers and the initial conditions $x_{-t}, x_{-t+1}, \ldots, x_{-1}$ and $x_0$ are posistive real numbers where $t=\max \{r, k, l\}$.