Geodesic
Peano's theorem is false for any infinite-dimensional Fréchet space
Sbornik. Mathematics, Tome 78 (1994) no. 1, pp. 211-214 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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},
     year = {1994},
     volume = {78},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1994_78_1_a12/}
}
TY  - JOUR
AU  - S. G. Lobanov
TI  - Peano's theorem is false for any infinite-dimensional Fréchet space
JO  - Sbornik. Mathematics
PY  - 1994
SP  - 211
EP  - 214
VL  - 78
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1994_78_1_a12/
LA  - en
ID  - SM_1994_78_1_a12
ER  - 
%0 Journal Article
%A S. G. Lobanov
%T Peano's theorem is false for any infinite-dimensional Fréchet space
%J Sbornik. Mathematics
%D 1994
%P 211-214
%V 78
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1994_78_1_a12/
%G en
%F SM_1994_78_1_a12
S. G. Lobanov. Peano's theorem is false for any infinite-dimensional Fréchet space. Sbornik. Mathematics, Tome 78 (1994) no. 1, pp. 211-214. http://geodesic.mathdoc.fr/item/SM_1994_78_1_a12/

[1] Godunov A. N., “O teoreme Peano v banakhovykh prostranstvakh”, Funk. analiz i ego pril., 9:1 (1975), 59–60 | MR | Zbl

[2] Cellina A., “On nonexistence of solution of differential equations in nonreflexive spaces”, Bull. Amer. Math. Soc., 78 (1972), 1069–1072 | DOI | MR | Zbl

[3] Astala K., “On Peano`s theorem in locally convex spaces”, Stud. Math., 73:3 (1982), 213–223 | MR | Zbl

[4] Pich A., Yadernye lokalno vypuklye prostranstva, Mir, M., 1967 | MR

[5] Bessaga C., Pełczynski A., Rolewicz S., “On diametral approximative dimention and linear homogeneity of $F$-spaces”, Bull. Acad. Pol. Sci., 9 (1961), 677–683 | MR | Zbl

[6] Bonet J., Perez Carreras P., Barelled locally convex spaces, Math. studies, 131, North-Holland, 1987 | MR | Zbl

[7] Dubinsky Ed., The structure of nuclear Frechet spaces, Lect. Notes Math., 720, 1979 | MR | Zbl

[8] Dugunji J., “An extension of Titze`s theorem”, Pacific. J. Math., 1 (1951), 353–367 | MR