The power form of asymptotic stability in the analytic case (Q1901875)

The author proves that, if the trivial solution of an analytic differential equation \(\dot x= F(x)\), \(F(0)= 0\), \(x\in \mathbb{R}^n\), is asymptotically stable, then the other solutions tend to zero at least as \(t\) to a negative power, i.e., \[ |x(t, x_0, t_0)|\leq a|x_0|^\alpha(1+ b(t- t_0)|x_0|^\beta)^{- 1/\beta},\quad t\geq t_0, \] with some \(\alpha\in (0, 1]\) and \(a, b, \beta> 0\).