Uniqueness without continuous dependence in infinite dimension
Consider an ordinary differential equation with continuous right hand side. It is well known that the Peano's existence theorem for the corresponding initial-value problem (IVP) cannot be generalized to equations in infinite dimensional Banach spaces. And what about the assertion ''Uniqueness implies continuous dependence on the initial state'' valid for finite dimension? It is this question that the author answers in a negative way through an example in \(\ell^{\infty}\). Considering on [0,\(\infty)\) the IVP (1) \(\dot v=g\circ v\), \(v(0)=y\) for each \(y\in \ell^{\infty}\) with an adequate mapping \(g: \ell^{\infty}\to \ell^{\infty}\) explicitly defined, the author: i) proves that (1) has exactly one noncontinuable solution \(\psi_ y\) with domain [0,\(\infty)\), giving the form of the solution; ii) exhibits a sequence of elements \(z_ m\in \ell^{\infty}\) convergent to zero but such that, for every \(t\in (0,\infty)\), \(\psi_{z_ m}(t)\) does not converge to \(\psi_ 0(t)[=0]\). A question is raised: Will the fact that \(\ell^{\infty}\) is neither separable nor reflexive be significant for the failure of the property?
scientific article; zbMATH DE number 3901172