Remark

When $H = E$ and the condition (16) of IV, p. 174 is satisfied on $I \times E$, prop. 7 of IV, p. 174 applies to every solution $u$ of $x' = g(t, x)$ on any interval where this solution is defined; one may even assume that $g(t, x)$ is Lipschitz with respect to a regulated function $k(t)$ though not necessarily bounded on this interval.