Proposition 2

If f is a continuous map from $I \times H$ into E, then every solution of (1) on I is a strict solution.

Indeed, such a solution $u$ is a primitive of the continuous function $f(t, u(t))$ (II, p. 66, prop. 3).

Furthermore, we note that a continuous function $f$ on $I \times H$ satisfies the conditions of lemma 1 (Gen. Top., X, p. 286, cor. 3).

In the sequel we shall choose $t_0 \in I$ and $x_0 \in H$ arbitrarily and investigate whether there exist solutions of (1) on $I$ (or on a neighbourhood of $t_0$ in $I$) taking the value $x_0$ at the point $t_0$ (or, what comes to the same, solutions of (6)).