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I was wondering if anyone knew of a legitimate citable reference where Picard's Theorem on the existence of solutions to ODEs in Banach space is proven? For some reason I can only find proofs for the case of $\mathbb{R}^n$ in any of the books I have looked at. I realize that the proof is the same, but I would like to have a reference where I do not have to mention this fact.

Thanks in advance

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Try Theorem 6, pp.117-118 of the survey paper

Lobanov, S. G. & Smolyanov, O. G. "Ordinary differential equations in locally convex spaces". Uspekhi Mat. Nauk, 1994, 49, 93-168

(link to paper). It being partially a survey paper, the theorem is stated without proof; if you want the proofs, you'd have to follow the references therein.

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Thank you very much. –  user12014 Jun 6 '12 at 10:26

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