1376 CHAPTER 42. MAXIMAL MONOTONE OPERATORS, HILBERT SPACE
From Lemma 42.2.1 it follows that
|yλ (t +h)− yλ (t)|2−|yλ (h)− y0|2
≤∫ t
0| f (s+h)− f (s)|2 ds+
∫ t
0|yλ (s+h)− yλ (s)|2 ds
By Gronwall’s inequality,
|yλ (t +h)− yλ (t)|2 ≤ et(|yλ (h)− y0|2 +
∫ t
0| f (s+h)− f (s)|2 ds
). (42.2.11)
The last integral equals
∫ t
0
∣∣∣∣∫ s+h
sf ′ (r)dr
∣∣∣∣2 ds≤∫ t
0h∫ s+h
s
∣∣ f ′ (r)∣∣2 drds
= h[∫ h
0
∫ r
0
∣∣ f ′ (r)∣∣2 dsdr+∫ t
h
∫ r
r−h
∣∣ f ′ (r)∣∣2 dsdr+∫ t+h
t
∫ t
r−h
∣∣ f ′ (r)∣∣2]dsdr
≤ h2∫ t+h
0
∣∣ f ′ (r)∣∣2 dr
and now it follows that for all t +h < T,∣∣∣∣yλ (t +h)− yλ (t)h
∣∣∣∣2 ≤ eT
(∣∣∣∣yλ (h)− y0
h
∣∣∣∣2 +∥∥ f ′∥∥2
L2(0,T ;H)
). (42.2.12)
Now return to 42.2.8. From Lemma 42.1.3, |Aλ y0| ≤min{|a| : a ∈ Ay0} .∣∣∣∣yλ (h)− y0
h
∣∣∣∣≤ ∣∣∣∣1h∫ h
0Aλ yλ (s)ds
∣∣∣∣+ ∣∣∣∣1h∫ h
0f (s)ds
∣∣∣∣Then taking limsuph→0 of both sides
lim suph→0
∣∣∣∣yλ (h)− y0
h
∣∣∣∣≤ |Aλ y0|+ | f (0)| ≤min{|a| : a ∈ Ay0}+ | f (0)|
This is where y0 ∈ D(A) is used. Thus from 42.2.12, there exists a constant C independentof t, h and λ such that for h small enough,∣∣∣∣yλ (t +h)− yλ (t)
h
∣∣∣∣2 ≤C
≡ eT(
min{|a| : a ∈ Ay0}+ | f (0)|+1+∥∥ f ′∥∥2
L2(0,T ;H)
)Thus, letting h→ 0,∣∣y′
λ(t)∣∣2 ≤C ≡ eT
(min{|a| : a ∈ Ay0}+ | f (0)|+1+
∥∥ f ′∥∥2
L2(0,T ;H)
)(42.2.13)