Geodesic
Counterexample to Peano's theorem in infinite-dimensional $F'$-spaces
Matematičeskie zametki, Tome 62 (1997) no. 1, pp. 128-137.

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Let $E$ be a nonnormable Fréchet space, and let $E'$ be the space of all continuous linear functionals on $E$ in the strong topology. A continuous mapping $f\colon E'\to E'$ such that for any $t_0\in\mathbb R,x_0\in E'$, the Cauchy problem $\dot x=f(x)$, $x(t_0)=x_0$ has no solutions is constructed. 
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S. A. Shkarin. Counterexample to Peano's theorem in infinite-dimensional $F'$-spaces. Matematičeskie zametki, Tome 62 (1997) no. 1, pp. 128-137. http://geodesic.mathdoc.fr/item/MZM_1997_62_1_a14/

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