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](http://en.wikipedia.org/wiki/Peano_existence_theorem). I give a counterexample [here](http://www.mathcounterexamples.net/continuous-differential-equation-with-no-solution/) 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](http://www.mathcounterexamples.net/solution-of-a-differential-equation-not-exploding-in-finite-time/) 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?



ODE properties true in finite dimension but not in Banach spaces of infinite dimension
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. 


