Proposition 7

Let $f$ and $g$ be two functions defined on $I \times H$, satisfying the hypotheses of lemma 1 of IV. p. 165, and such that, on $I \times H$

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Suppose further that $g$ is Lipschitz with respect to a constant $k > 0$ on $I \times H$ and that $f$ is locally Lipschitz on $I \times H$, or that $E$ is finite dimensional. Let $(t_0, x_0)$ be a point of $I \times H$, $\mu$ a number $> 0$, and

Let $u$ be an integral of the equation $x' = g(t, x)$ defined on an interval $K = [t_0, b[$ contained in $I$, equal to $x_0$ at the point $t_0$, and such that for all $t \in K$ the closed ball with centre $u(t)$ and radius $\varphi(t)$ is contained in $H$. Under these conditions, for every $y \in H$ such that $\|y - x_0\| \leq \mu$ there exists an integral $v$ of $x' = f(t, x)$, defined on $K$, with values in $H$, and equal to $y$ at the point $t_0$; moreover, $\|u(t) - v(t)\| \leq \varphi(t)$ on $K$.

Let $\mathscr{M}$ be the family of integrals of $x' = f(t, x)$ each of which takes its values in $H$, is equal to $y$ at $t_0$, and is defined on a half-open interval $[t_0, \tau[$ contained in $I$ (depending on the interval considered). By th. 1 of IV, p. 171 (when $f$ is locally Lipschitz) or IV, p. 167 corollary (when $E$ is finite dimensional), $\mathscr{M}$ is not empty, and the same reasoning as in prop. 3 of IV, p. 166 shows that $\mathscr{M}$ is inductive for the order "v is a restriction of w". Let $v_0$ be a maximal element of $\mathscr{M}$ and $[t_0, t_1[$ the interval of definition of $v_0$; by prop. 6 of IV, p. 171, it all comes down to proving that $t_1 \geq b$. If not, one would have

on the interval $[t_0, t_1[$ by prop. 6; now on the compact interval $[t_0, t_1]$ the regulated function $g(t, u(t))$ is bounded, so the function $g(t, v_0(t))$ is bounded on the interval $[t_0, t_1[$, for $\| g(t, v_0(t)) \| \leq \| g(t, u(t)) \| + k \varphi(t)$ on this interval. Since $v_0$ is an approximate solution of $x' = g(t, x)$ to within $\alpha$ on $[t_0, t_1[$ there exists a number $M > 0$ such that $\| v_0'(t) \| \leq M$ on this interval, except at the points of a countable set; the mean value theorem now shows that $\| v_0(s) - v_0(t) \| \leq M |s - t|$ for every pair of points $s, t$ of $[t_0, t_1[$, so (by Cauchy's criterion) $v_0(t)$ has a left limit $c$ at the point $t_1$, and, by continuity, one has $\| c - u(t_1) \| \leq \varphi(t_1)$, and so $c \in H$. Now one sees, from IV, p. 171, th. 1 or IV, p. 167, corollary, that there exists an integral of $x' = f(t, x)$ defined on an interval $[t_1, t_2[$ and equal to $c$ at $t_1$, which contradicts the definition of $v_0$.