Is there a reflexive Banach space $B$ and a continuous map $f:B\to B$ such that the differential equation
$$ \frac{d x (t)}{dt} = f(x(t)) $$
with some initial condition $x(0)=x\_0$ has no solution?



A differential equation with continuous coefficient and no solution in a reflexive Banach space?
You can find a very general example of the desired type in the paper: P. Hajek, M. Johanis, On Peano's theorem in Banach spaces. J. Differential Equations 249 (2010), no. 12, 3342–3351. 


