Exercise 8, § 1

Let $f$ be a function defined on $I \times H$ satisfying the conditions of lemma 1 of IV, p. 165, and such that for a constant $k$ with $0 < k < 1$, and a point $t_0 \in I$, one has, for all $t \in I$ and $x_1, x_2 \in H$,

Show that if $u$ and $v$ are two approximate solutions of the equation $x' = f(t, x)$, to within $\varepsilon_1$ and $\varepsilon_2$ respectively, defined on $I$, with values in $H$, and having the same value at the point $t_0$, then, for every $t \in I$,

Deduce from this that th. 1 (IV, p. 171) and 2 (IV, p. 172) remain valid for the equation $x' = f(t, x)$ under the indicated conditions.