  When addressing ordinary differential equations in infinite dimensional
Banach spaces, an interesting question that arises concerns the existence (or
non existence) of blowing up solutions in finite time. In this manuscript we
discuss this question for the fractional differential equation $cD_t^\alpha u =
f(t,u)$ proving that when $f$ is locally Lipschitz, however does not maps
bounded sets into bounded sets, we can construct a maximal local solution that
not "blows up" in finite time.
