## A note on global asymptotic stability of a family of rational equations.(English)Zbl 1083.39003

The authors prove that all positive solutions of the difference equations $x(n+1)= \frac{1+x_n\sum_{i=1}^kx_{n-i}}{x_n+x_{n-1}+x_n\sum_{i=2}^kx_{n-i}}, \;\;n=0,1,\dots,$ where $$k\in N$$, converge to the positive equilibrium $$\overline{x}=1$$. The result generalizes the main theorem in the paper by X. Li and D. Zhu [J. Difference Equ. Appl. 9, No. 9, 833–839 (2003; Zbl 1055.39014)]. The authors present a very short proof of the theorem. At the same time, the authors find the asymptotics of some of the positive solutions.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations

Zbl 1055.39014