Does Peano's theorem apply to spaces with infinite dimension? Or is there a counterexample?
Here, Peano's theorem is:
Let $E$ be a space with finite dimension. Consider a point $(t_0,x_0) \in \Re \times E$, constants $ a, b > $ 0 and a continuous function $$F: [t_0 – a, t_0 + a] \times B_b[x_0] \longrightarrow E$$
Then for every $ M> $ 0 satisfying
$$\sup \{||F(t,x)||:(t,x) \in [t_0 – a, t_0 + a] \times B_b[x_0]\} < M$$
the Cauchy problem
$$x'(t)=F(t,x(t));\ \ x(t_0)=x_0$$
admits at least one solution in the interval:
$$\big[t_0 – \min(a,\frac{b}{M}),t_0 + \min(a,\frac{b}{M})\big] $$
An infinite-dimensional counterexample would be of great help. Thank you very much.
Best Answer
No, Peano's existence theorem fails completely in infinite-dimensional spaces: there are counterexamples in every infinite-dimensional Banach space. This is a theorem of Godunov (A. N. Godunov, Peano's theorem in Banach spaces, Functional Analysis and its Applications 9 (1975), 53-55, http://dx.doi.org/10.1007/BF01078180), while the first counterexample in some Banach space was due to Dieudonné (J. Dieudonné, Deux exemples singuliers d'équations différentielles, Acta Sci. Math. Szeged. 12:B (1950), 38-40; see http://acta.fyx.hu/).
It's not hard to see that the finite-dimensional proof fails (the weak point is generally when Arzelà–Ascoli is applied), but I still find it surprising that the theorem fails as well, since it naively sounds like it should obviously be true.