Exercise 10, § 1

Let $f$ be a continuous bounded real function defined on the box $P : |t - t_0| < a, |x - x_0| < b$ of $\mathbf{R}^2$. Let M be the supremum of $|f(t, x)|$ over P, and let $I = ]t_0 - \alpha, t_0 + \alpha[$, where $\alpha = \inf(a, b/M)$. Show that the upper and lower envelopes of the set $\Phi$ of integrals of $x' = f(t, x)$ defined on I and taking the value $x_0$ at the point $t_0$, are again integrals of $x' = f(t, x)$, which one calls the maximal and minimal integral of this equation corresponding to the point $(t_0, x_0)$ (remark that the set $\Phi$ is equicontinuous and closed for the topology of uniform convergence on I).

For every $\tau \in I$ let $\xi$ be the value of the minimal integral (corresponding to $(t_0, x_0)$) at the point $\tau$. Show that the minimal integral corresponding to the point $(\tau, \xi)$ is identical to the minimal integral corresponding to the point $(t_0, x_0)$ on an interval of the form $[\tau, \tau + h[$ if $\tau > t_0$, and of the form $]\tau - h, \tau]$ if $\tau < t_0$.

Deduce from this that there exists a largest open interval $]t_1, t_2[$ containing $t_0$ and contained in $]t_0 - a, t_0 + a[$, such that the minimal integral $u$ corresponding to $(t_0, x_0)$ can be extended by continuity to $]t_1, t_2[$, so that at every point $t$ of $]t_1, t_2[$, the value $u(t)$ belongs to $]x_0 - b, x_0 + b[$, and that $u$ is identical with the minimal integral corresponding to the point $(t, u(t))$ on an interval of the form $[t, t + h[$ if $t > t_0$ and of the form $]t - h, t]$ if $t < t_0$; show further that either $t_1 = t_0 - a$ (resp. $t_2 = t_0 + a$) or that $\lim_{t \to t_1^-} u(t) = x_0 \pm b$ (resp. $\lim_{t \to t_2^+} u(t) = x_0 \pm b$).