Exercise 3, § 1

Let $S \subset H$ be an open ball with centre $x_0$ and radius $r$, and $f$ a Lipschitz function for the constant $k > 0$ on $I \times S$; suppose further that $t \mapsto f(t, x_0)$ is bounded on $I$, and denote by $M_0$ the supremum of $\|f(t, x_0)\|$ over $t \in I$. Show that for all $t_0 \in I$ there is an integral $u$ of $x' = f(t, x)$, with values in $S$, equal to $x_0$ at the point $t_0$ and defined on the intersection of $I$ and the interval $]t_0 - \lambda, t_0 + \lambda[$, where

(note that one has $\|u(t) - x_0\| \leq M(t - t_0) + k \int_{t_0}^t \|u(s) - x_0\| \, ds$ for $t > t_0$).