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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

Citations:

Zbl 1055.39014
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