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Zbl 0604.34003
Burton, T.A.; Dwiggins, D.P.
Uniqueness without continuous dependence. (English)
[A] 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.] \par For x and h in $R\sp n$, continuity of solutions of $\{$ (1) $x'=h(t,x)$, $x(t\sb 0)=x\sb 0\}$ is equivalent to uniqueness. This is partly because both x and $x\sb 0$ are in $R\sp n$ and have the same topologies. {\it J. J. Schäffer} [J. Differ. Equations 56, 426-428 (1985)] constructed an abstract example, using $\ell\sp{\infty}$ as the initial condition space, of an equation whose solutions are unique but not continuous in initial conditions. He conjectures that this results from $\ell\sp{\infty}$ being neither separable nor reflexive. \par In the present paper we consider a scalar problem (2) $x'=x+\int\sp{t}\sb{-\infty}[x(s)/(t-s+1)\sp 3]ds,\phi$ which requires initial functions $\phi:(-\infty,t\sb 0]\to R.$ 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. \par The remainder of the paper discusses a fading memory condition pertaining to general systems of the form of (2). We note that {\it B. M. Garay} and {\it 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 2000:: *34A12 Initial value problems for ODE
45J05 Integro-ordinary differential equations

Keywords: first order differential equation; fading memory condition

Citations: Zbl 0595.00009

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