Proposition 5

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

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

*Let $k(t)$ be a real regulated function and $> 0$ on $I$ , and let $f(t, x)$ be a function that is defined and Lipschitz with respect to $k(t)$ on $I \times H$ . If $u$ and $v$ are two approximate solutions of (1), to within $\varepsilon_1$ and $\varepsilon_2$ respectively, defined on $I$ with values in $H$ , then, for all $t \in I$ such that $t \geq t_0$ ,

\|u(t) - v(t)\| \leq \|u(t_0) - v(t_0)\| \Phi(t) + (\varepsilon_1 + \varepsilon_2)\Psi(t)

where

\begin{cases} \Phi(t) = 1 + \int_{t_0}^t k(s) \exp \left( \int_s^t k(\tau) d\tau \right) ds \\ \Psi(t) = t - t_0 + \int_{t_0}^t (s - t_0)k(s) \exp \left( \int_s^t k(\tau) d\tau \right) ds. \end{cases}

From the relation $\|u'(t) - f(t, u(t))\| \leq \varepsilon_1$ , valid on the complement of a countable set, one deduces, from the mean value theorem, that

\left\| u(t) - u(t_0) - \int_{t_0}^t f(s, u(s))\, ds \right\| \leq \varepsilon_1 (t - t_0)

and similarly

\left\| v(t) - v(t_0) - \int_{t_0}^t f(s, v(s))\, ds \right\| \leq \varepsilon_2 (t - t_0)

whence

\|u(t) - v(t)\| \leq \|u(t_0) - v(t_0)\|

+ \left\| \int_{t_0}^t (f(s, u(s)) - f(s, v(s)))\, ds \right\| + (\varepsilon_1 + \varepsilon_2)(t - t_0).

By the Lipschitz condition (8) one has

\left\| \int_{t_0}^t (f(s, u(s)) - f(s, v(s)))\, ds \right\| \leq \int_{t_0}^t \|f(s, u(s)) - f(s, v(s))\|\, ds

\leq \int_{t_0}^t k(s) \|u(s) - v(s)\|\, ds

whence, putting $w(t) = \|u(t) - v(t)\|$ ,

w(t) \leq w(t_0) + (\varepsilon_1 + \varepsilon_2)(t - t_0) + \int_{t_0}^t k(s)w(s)\, ds.

The proposition is thus a consequence of the following lemma:

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

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