**MSC:**- 39A10 Difference equations, See also {33Dxx}
- 39A11 Stability of difference equations
- 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.), See also {30E15}

difference equations

\[

x_{n+1}=\frac{1+x_n\sum_{i=1}^k

x_{n-i}}{x_{n}+x_{n-1}+x_{n}\sum_{i=2}^k x_{n-i}},\quad n=0,1,...,

\]

where $k\in {\bf N},$ converge to the positive equilibrium $\bar

x=1.$ The result generalizes the main theorem in the paper: Li

Xianyi and Zhu Deming, Global asymptotic stability in a rational

equation, {\it J.\ Differ.\ Equations Appl.} {\bf 9} (9), (2003),

833-839. We present a very short proof of the theorem. Also, we

find the asymptotics of some of the positive solutions.