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:{
}→
Lemma B.3.19
The map T
u
0
,f
u(t) which is a solution of PVI
(
B.6
)
is
Lipschitz from
H
×
S(
0
,T)
→
I(
0
,T)
.
{
}→
Proof
Observe that
u
0
,f
U
k
is Lipschitz continuous uniformly in
k
:let
{
u
0
,f
∗
}
be a second set of initial data. Then pick
v
=
u
k,m
+
1
in (
B.12b
) and also
u
k,m
we may apply Proposi-
tion
B.3.10
which gives for the extensions of
u
k,m
,
u
k,m
as in (
B.55
)
u
k,m
+
1
v
=
in (
B.12b
). To the difference
w
=
u
k,m
−
S(
0
,
∞
)
.
By (
B.68
), an analogous estimate uniform in
k
holds also true for the linear exten-
sions
U
k
(t)
,
U
k
(t)
.
C
f
k
(t)
u
k
(t
u
k,
1
H
+
u
k
(t
+
k)
−
+
k)
I(
0
,T
+
k)
≤
u
k,
1
−
f
k
(t)
−
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