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Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension.

The first one I know is the Peano existence theorem. I give a counterexample here for infinite dimension.

The second one states that is the maximal solution of a differential equation is defined in an interval smaller than the one of definition of the map of the unique value problem, then the solution is "exploding". I give a counterexample here for infinite dimension.

Both are from the mathematician Jean Dieudonné.

Do you know other ODE properties valid in finite dimension spaces but not in Banach spaces of infinite dimension?

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    $\begingroup$ I'd say, those properties that use compactness and not only completeness, like the two you mentioned. A third one should be the statement :" Uniqueness implies continuous dependence", which should have a counter-example in the same space $c_0$. $\endgroup$ – Pietro Majer Jan 11 '15 at 12:05
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    $\begingroup$ there is a nice statement by de giorgi about long-time behaviour of gradient-like systems. i can give you a precise statement if you are interested in this kind of properties. $\endgroup$ – Delio Mugnolo Jan 11 '15 at 15:50
  • $\begingroup$ Yes Delio, I'm interested in the statement you mention. Thanks. $\endgroup$ – mathcounterexamples.net Jan 11 '15 at 18:03
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You can find some relevant information in Godunov, A. N. The Peano theorem in Banach spaces. Functional Anal. Appl. 9 (1975), no. 1, 53–55 and Pasika, E. E. An example of a first-order differential equation in a Hilbert space without continuous dependence of the solution on the initial condition. (Russian) Ukrain. Mat. Zh. 35 (1983), no. 6, 786–788. Also you can check papers citing Godunov's paper in MathSciNet or Google Scholar.

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A nice survey is the paper of Lobanov and Smolyanov:

Sergey Grigorievich Lobanov and Oleg Georgievich Smolyanov, Ordinary differential equations in locally convex spaces. Russian Mathematical Surveys 49.3 (1994): 97–175.

The paper lists several counterexamples in the infinite dimensional setting for classical properties and theorems of ODEs in finite-dimensional spaces, such as Peano's theorem, Kneser's theorem, continuous dependence on initial data, continuation of solutions and Picard's theorem.

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