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Abstract

Herzog and Lemmert have proven that if $E$ is a Fréchet space and $T : E \to E$ is a continuous linear operator, then solvability (in $[0,1]$) of the Cauchy problem $\dot x=Tx$, $x(0)=x_0$ for any $x_0\in E$ implies solvability of the problem $\dot x(t)=Tx(t)+f(t,x(t))$, $x(0)=x_0$ for any $x_0\in E$ and any continuous map $f:[0,1]\times E\to E$ with relatively compact image. We prove the same theorem for a large class of locally convex spaces including:

$\bullet$ DFS-spaces, i.e., strong duals of Fréchet–Schwartz spaces, in particular the spaces of Schwartz distributions $\S'(\mathbb R^n)$, the spaces of distributions with compact support ${\cal E}'({\mit\Omega} )$ and the spaces of germs of holomorphic functions $H(K)$ over an arbitrary compact set $K\subset\mathbb C^n$;

$\bullet$ complete LFS-spaces, i.e., complete inductive limits of sequences of Fréchet–Schwartz spaces, in particular the spaces ${\cal D}({\mit\Omega} )$ of test functions;

$\bullet$ PLS-spaces, i.e., projective limits of sequences of DFS-spaces, in particular, the spaces ${\cal D}'({\mit\Omega} )$ of distibutions and ${\cal A}({\mit\Omega} )$ of real-analytic functions.

Here ${\mit\Omega} $ is an arbitrary open domain in $\mathbb R^n$. We construct an example showing that the analogous statement for (smoothly) time-dependent linear operators is invalid already for Fréchet spaces.