A New Example on Lyapunov Stability

Abstract

The purpose of this paper is to present an example of an Ordinary Differential Equation $x'=F(x)$ in the infinite-dimensional Hilbert space $\ell ^2$ with F being of class $\mathcal {C}^1$ in the Fréchet sense, such that the origin is an asymptotically stable equilibrium point but the spectrum of the linearized operator DF(0) intersects the half-plane $\mathfrak {R}(z)>0$. The possible existence or not of an example of this kind has been an open question until now, to our knowledge. An analogous example, but of a non-invertible map instead of a flow defined by an ODE was recently constructed by the authors in Rodrigues and Solà-Morales (J. Differ. Equ. 269:1349–1359, 2020). The two examples use different techniques, but both are based on a classical example in Operator Theory due to S. Kakutani.