Lemma 2

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

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

If w is a continuous real function on the interval [t0, t_1] and satisfies the inequality*

w(t) \leq \varphi(t) + \int_{t_0}^{t} k(s) w(s) \, ds

(12)

where $\varphi$ is a regulated function $\geq 0$ on $[t_0, t_1]$ , then, for $t_0 \leq t \leq t_1$ ,

w(t) \leq \varphi(t) + \int_{t_0}^{t} \varphi(s) k(s) \exp \left( \int_s^t k(\tau) \, d\tau \right) \, ds.

(13)

Put $y(t) = \int_{t_0}^t k(s) w(s) \, ds$ ; the relation (12) implies that

y'(t) - k(t) y(t) \leq \varphi(t) k(t)

(14)

on the complement of a countable set.

Put $z(t) = y(t) \exp \left( - \int_{t_0}^t k(s) \, ds \right)$ ; then (14) is equivalent to

z'(t) \leq \varphi(t) k(t) \exp \left( - \int_{t_0}^t k(s) \, ds \right).

On applying the mean value theorem (I, p. 15, th. 2) to this inequality, and noting that $z(t_0) = 0$ , we obtain

z(t) \leq \int_{t_0}^t \varphi(s) k(s) \exp \left( - \int_{t_0}^s k(\tau) \, d\tau \right) \, ds

whence

y(t) \leq \int_{t_0}^t \varphi(s) k(s) \exp \left( \int_s^t k(\tau) \, d\tau \right) \, ds

and since $w(t) \leq \varphi(t) + y(t)$ one thus obtains (13).

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

Cited by

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