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From the answers to Mathoverflow Question "Continuity in Banach space for non-linear maps", it is possible to infer that the assumption of the Cauchy-Lipschitz theorem for the autonomous equation $$ \dot X=F(X), \tag{$\ast$} $$ could fail to be satisfied even if $F$ is $C^1$: in fact, using the mean-value inequality, the verification of the Cauchy-Lipschitz hypothesis seems to require local boundedness of the derivative, a property which could fail in infinite dimension even when this derivative is continuous.

My question: it is well-known that Peano's theorem (local existence with $F$ continuous, of course without uniqueness) fails in infinite dimensions. Is there a counterexample to existence or uniqueness for ($\ast$) when $F$ is $C^1$ (and of course $F'$ not locally bounded, otherwise CL theorem provides local existence and uniqueness)?

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marked as duplicate by András Bátkai, Alex Degtyarev, Joonas Ilmavirta, Daniel Moskovich, Johannes Hahn Mar 4 '15 at 15:39

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  • $\begingroup$ For a counterexample to Peano's theorem in infinite dimension, check for instance Exercise 18, page IV.41 of the Bourbaki’s volume Fonctions d'une variable réelle. $\endgroup$ – Bazin Mar 3 '15 at 11:08
  • $\begingroup$ I suggest to have a look at the paper I mention in the other question: dx.doi.org/10.1007/BF01078180 $\endgroup$ – András Bátkai Mar 3 '15 at 11:27
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This is not possible. Indeed, since $F$ is $\mathcal{C}^1$, for every $x_0$ there exists $\delta > 0$ such that for every $x \in B_{x_0} (\delta)$, $\Vert F' (x) - F' (x_0)\Vert \le 1$. Hence $\Vert F' (x) \Vert \le \Vert F' (x_0) \Vert + 1$ and thus the derivative $F'$ is bounded on the ball $B_{x_0} (\delta)$. You can then show the existence of a solution on a small time interval.

Compared to the question "Continuity in Banach space for non-linear maps", the radius $\delta > 0$ is not fixed in advance and could be very small.

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  • $\begingroup$ I have edited the answer to explain this explicitly. $\endgroup$ – Jean Van Schaftingen Mar 3 '15 at 12:29

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