Example of a first-order differential equation in Hilbert space without continuous dependence of the solution on the initial condition (Q1069014)

Let \(b_ 1>a_ 1>b_ 2>a_ 2..\). be a sequence of real numbers such that \(a_ n=1/(2n+1),\) \(b_ n=1/2n\), and for each n define the continuous function \(q_ n(\eta): {\mathbb{R}}^+\mapsto [0,1]\) given by 0 for \(0\leq \eta \leq a_ n,1\) for \(\eta \geq b_ n\) and linear on the interval \([a_ n,b_ n]\). Consider \(Q: \ell_ 2\mapsto \ell_ 2\), \(Q(x)=\sum^{\infty}_{n=1}q_ n(\| x\|)\cdot <x,e_ n>e_ n,\) where \(\{e_ n\}_ 1^{\infty}\) is the canonical basis in \(\ell_ 2\). The author considers the equation \(x'(t)=F(x(t))\), \(x(t)\in \ell_ 2\), \(t\in {\mathbb{R}}\), where \(F(x)=Q(x)/\sqrt{\| x\|}\) for \(x\neq 0\), and \(F(0)=0\), and he shows that for this equation there exists a unique solution through \((t_ 0,x_ 0)\) without continuous dependence on \(x_ 0\).