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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