论文信息 - A Note on Global Asymptotic Stability of a Family of Rational Equations

A Note on Global Asymptotic Stability of a Family of Rational Equations

In this note we prove that all positive solutions of the difference equations xn+1 = 1 + xn ∑k i=1 xn−i xn + xn−1 + xn ∑k i=2 xn−i , n = 0, 1, ..., where k ∈ N, converge to the positive equilibrium x = 1. The result generalizes the main theorem in the paper: Li Xianyi and Zhu Deming, Global asymptotic stability in a rational equation, J. Differ. Equations Appl. 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.

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