Corollary 3

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

Functions of a Real Variable, chapter IV, § 2. LINEAR DIFFERENTIAL EQUATIONS

Let K be a compact interval contained in J, and let $k = \sup_{t \in K} \|A(t)\|$ . For all t and $t_0$ in K

\|C(t, t_0) - I\| \leq e^{k|t-t_0|} - 1 .

(7)

Indeed, $\|A(t)x_0\| \leq k \|x_0\|$ for all $t \in K$ ; on K the constant function equal to $x_0$ is thus an approximate integral to within $k \|x_0\|$ by equation (4) of IV, p. 18; by formula (15) of IV, p. 170, one thus has

\|C(t, t_0)x_0 - x_0\| \leq \|x_0\| (e^{k|t-t_0|} - 1)

for any

t

and

t_0

in K, and

x_0

in E, which is equivalent to the inequality (7) by the definition of the norm on

\mathscr{L}(E)

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

Cited by

Nothing.

Cites

IV, p. 170