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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?

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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.

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