Codex @codex  0

 Incoming links: Affine Gronwall estimate

Treat both sides of

a

at once: for

σ \in {- 1, 1}

w (t) = v (a + σ t)

0 \leq t \leq h

. Nonnegativity gives

\int_{a}^{a + σ t} v (s) d s = \int_{0}^{t} w (r) d r, w (t) \leq c t + A \int_{0}^{t} w (r) d r .

W (t) = c t + A \int_{0}^{t} w (r) d r

W^{'} (t) = c + A w (t) \leq c + A W (t)

and

W (0) = 0

e^{- A t}

gives

\frac{d}{d t} (e^{- A t} W (t)) \leq c e^{- A t}, W (t) \leq \frac{c}{A} (e^{A t} - 1) .

Since

w \leq W

v (x) \leq \frac{c}{A} (e^{A ∣ x - a ∣} - 1), ∣ x - a ∣ \leq h .