A new example on Lyapunov stability

Abstract

The purpose of this paper is to present an example of an Ordinary Differential Equation $x^{\prime }=F(x)$ 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 $\Re (z)>0$. 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 a recent paper. The two examples use different techniques, but both are based on a classical example in Operator Theory due to S. Kakutani.