In this paper, we investigate the local and global stability and the period two solutions of all nonnegative solutions of the difference equation,
$$\begin{aligned} x_{n+1} = \frac{ ax_{n}+bx_{n-k}}{A+Bx_{n-k}} \end{aligned}$$
where
a,
b,
A,
B are all positive real numbers,
\(k \ge 1\) is a positive integer, and the initial conditions
\(x_{-k},x_{-k+1},...,x_{0}\) are nonnegative real numbers. It is shown that the zero equilibrium point is globally asymptotically stable under the condition
\(a+b \le A\), and the unique positive solution is also globally asymptotically stable under the condition
\(a-b \le A \le a+b\). By the end, we study the global stability of such an equation through numerically solved examples.