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Peano's theorem is false for any infinite-dimensional Fréchet space

S. G. Lobanov


Abstract: 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.

UDC: 517.911+517.982.23

MSC: Primary 34G20; Secondary 46A45

Received: 22.08.1991

 English version: Russian Academy of Sciences. Sbornik. Mathematics, 1994, 78:1, 211–214

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