Exercise 4, § 1

a) Let $I = ]t_0 - a, t_0 + a[$ be an open interval in $\mathbf{R}$, let $S$ be an open ball with centre $x_0$ and radius $r$ in $E$, and $f$ a locally Lipschitz function on $I \times S$. Let $(s, z) \mapsto h(s, z)$ be a function $\geq 0$ defined and continuous for $0 \leq s \leq a$ and $0 \leq z \leq r$, such that for every $s \in [0, a]$ the map $z \mapsto h(s, z)$ is increasing; suppose that $\|f(t, x)\| \leq h(|t - t_0|, \|x - x_0\|)$ on $I \times S$. Let $\varphi$ be a primitive of a regulated function on an interval $[0, \alpha]$ (with $\alpha < a$), with values in $[0, r[$, such that $\varphi(0) = 0$ and $\varphi'(s) > h(s, \varphi(s))$ on $[0, \alpha]$ except on a countable set. Show that the integral $u$ of $x' = f(t, x)$, equal to $x_0$ at the point $t_0$, is defined on $]t_0 - \alpha, t_0 + \alpha[$, and that $\|u(t) - x_0\| \leq \varphi(|t - t_0|)$ on this interval.

b) Deduce from a) that if $f$ is defined on $I \times E$, and there is a function $h$ defined, continuous, increasing and $> 0$ on $[0, +\infty[$, such that $\int_0^{+\infty} dz/h(z) = +\infty$, and that $\|f(t, x)\| \leq h(\|x\|)$, then every integral of $x' = f(t, x)$ is defined on all of $I$.