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Uniqueness without continuous dependence. (English) Zbl 0604.34003
Differential equations and their applications, Equadiff 6, Proc. 6th Int. Conf., Brno/Czech. 1985, Lect. Notes Math. 1192, 115-121 (1986).
[For the entire collection see Zbl 0595.00009.]
For x and h in \(R^ n\), continuity of solutions of \(\{\) (1) \(x'=h(t,x)\), \(x(t_ 0)=x_ 0\}\) is equivalent to uniqueness. This is partly because both x and \(x_ 0\) are in \(R^ n\) and have the same topologies. J. J. Schäffer [J. Differ. Equations 56, 426-428 (1985)] constructed an abstract example, using \(\ell^{\infty}\) as the initial condition space, of an equation whose solutions are unique but not continuous in initial conditions. He conjectures that this results from \(\ell^{\infty}\) being neither separable nor reflexive.
In the present paper we consider a scalar problem (2) \(x'=x+\int^{t}_{-\infty}[x(s)/(t-s+1)^ 3]ds,\phi\) which requires initial functions \(\phi:(-\infty,t_ 0]\to R.\) The question of continuity of the solutions in \(\phi\) rests on the topology chosen. We select the initial function space as a subset (X,\(\rho)\) of a topological vector space (Y,\(\rho)\) and show that solutions are unique but not continuous in \(\phi\). Moreover, the initial functions lie in a separable (compact) subset of (Y,\(\rho)\). We also argue (Proposition 2) that the subset can be embedded in a reflexive subspace of (Y,\(\rho)\), but the argument is incorrect since it uses a statement which is generally true only for finite dimensional spaces. We now believe that the conclusion of Proposition 2 is itself false.
The remainder of the paper discusses a fading memory condition pertaining to general systems of the form of (2). We note that B. M. Garay and J. J. Schäffer [ibid. 64, 48-50 (1986)] show that there are equations in any arbitrary Banach space of infinite dimension where uniqueness does not imply continuity.

MSC:
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
45J05 Integro-ordinary differential equations