On the asymptotic stability of $x_{n+1}=(a+x_n{x}_{n-k})/(x_n+x_{n-k})$

Abstract

We prove that the equilibrium solution of the rational difference equation

$$x_{n+1}=\frac{a+x_n{x}_{n-k}}{x_n+x_{n-k}}\text{,}n=0,1,2,\dots$$

where $k$ is a nonnegative integer, $a\ge 0$, and $x_{-k},\dots ,x_0>0$, is globally asymptotically stable.