Geodesic
Peano's theorem is false for any infinite-dimensional Fr\'echet space
Sbornik. Mathematics, Tome 78 (1994) no. 1, pp. 211-214

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that for any nonnormable Fréchet space $E$ a continuous map $f\colon E\to E$ and a closed infinite-dimensional subspace $L$ can be found such that the Cauchy problem $\dot x=f(x)$, $x(0)=u$ has no solution for any $u\in L$. Previous counterexamples to Peano's theorem cover Banach spaces and nonsemireflexive spaces. 
@article{SM_1994_78_1_a12,
     author = {S. G. Lobanov},
     title = {Peano's theorem is false for any infinite-dimensional {Fr\'echet} space},
     journal = {Sbornik. Mathematics},
     pages = {211--214},
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     volume = {78},
     number = {1},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1994_78_1_a12/}
}
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S. G. Lobanov. Peano's theorem is false for any infinite-dimensional Fr\'echet space. Sbornik. Mathematics, Tome 78 (1994) no. 1, pp. 211-214. http://geodesic.mathdoc.fr/item/SM_1994_78_1_a12/