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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.

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