I'm Taking a Course in Differential Equations and this is one of the exercises those I have to do at home, I can't come up with these short questions:
Let X be an infinite-dimensional Banach space. It can be shown that there is an (artificially constructed) continuous function $f : \mathbb R \times X \to X$ such that there is no $\varepsilon > 0$ such that the initial value problem $$x′ = f(t,x),\ \ \ \ \ x(0) = 0$$ has a solution on $(−\varepsilon, \varepsilon)$. (In many Banach spaces, one can even construct an autonomous such $f$ i.e. independent of $t$.)
Why does the above assertion not contradict Peano’s theorem? Which step of the proof using Euler polygones breaks down?
Might the above constructed f be locally Lipschitz with respect to x?
Any hint or answer will be really appreciated! Thank you in advance!