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. 
@article{MZM_1997_62_1_a14,
     author = {S. A. Shkarin},
     title = {Counterexample to {Peano's} theorem in infinite-dimensional $F'$-spaces},
     journal = {Matemati\v{c}eskie zametki},
     pages = {128--137},
     publisher = {mathdoc},
     volume = {62},
     number = {1},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_62_1_a14/}
}
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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/