Several theorems regarding differential equations are true not only in finite dimension but also in infinite dimension Banach spaces. This is in particular the case for the **Cauchy–Lipschitz theorem**.


On the other end the Peano existence theorem is false for Banach space with infinite dimensions. See [here](http://www.mathcounterexamples.net/continuous-differential-equation-with-no-solution/) for a counterexample.


Do you know other counterexamples in "classical" Banach spaces that are different from $c\_0$ (the space of sequences of reals converging to $0$)? In particular, is there "an easy example" in the space $C([0,1],\mathbb{R})$ with $\sup$ norm?



Examples of continuous differential equations with no solution
Here is an example in $C([-1,1],R)$, which is a continuous analogue to the discrete example you pointed to:
$$
{du(t,x)\over dt} = \operatorname{sign}(u(t,x))\sqrt{|u(t,x)|} + x\;,\qquad u(0,x) = 0\;.
$$
For any $t > 0$, the solution (in $L^\infty$) develops a discontinuity at the origin, so that it doesn't belong to $C([-1,1],R)$.


