Corollary

Tag 08BQ, permanent. Cite it as [Bourbaki, Tag 08BQ].

Functions of a Real Variable, chapter IV, § 1. EXISTENCE THEOREMS

Let $\mathbf{f}$ be a Lipschitz function for the constant $k > 0$ , defined on $\mathbf{I} \times \mathbf{H}$ . If $\mathbf{u}$ and $\mathbf{v}$ are two approximate solutions of (1) to within $\varepsilon_1$ and $\varepsilon_2$ respectively, defined on $\mathbf{I}$ and taking their values in $\mathbf{H}$ , then, for all $t \in \mathbf{I}$ ,

\| \mathbf{u}(t) - \mathbf{v}(t) \| \leq \| \mathbf{u}(t_0) - \mathbf{v}(t_0) \| e^{k |t - t_0|} + (\varepsilon_1 + \varepsilon_2) \frac{e^{k |t - t_0|} - 1}{k}.

(15)

This inequality is in fact an immediate consequence of (9) when $t \geq t_0$ ; to prove it for $t \leq t_0$ it suffices to apply it to the equation

\frac{d \mathbf{x}}{ds} = -\mathbf{f}(-s, \mathbf{x})

obtained from (1) by the change of variable $t = -s$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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