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Applications
47
So,
"ulw"∞<DE(V(J))+8
.Consequently,
diamAI<DE(V(J))+8
,
which,togetherwiththefactthat
8
isarbitrary,showsthat
DC(V)<
DE(V(J)).
Finally,weshallshowthat
DE(V(J))=max
t∈J
DE(V(t))
.Oneofthe
twoinequalitiesiseasytoobtain.Since
V(t)⊆V(J)
forevery
t∈J
,
weimmediatelyseethat
max
t∈J
DE(V(t))<DE(V(J))
.Letusalsonote
thatwecanuseflmax”insteadofflsup”,becausethefunction
v
is
continuousonJ.
Theproofoftheoppositeinequalityrequiresmorework.Let
8>
0
beanarbitrarynumber.Theset
J
iscompact,andthereforeitcan
becoveredbyafinitenumberofsets
U1
,
...
,
Un⊆J
withdiame-
tersnotgreaterthan
δ
,where
δ:=δ(1
28)
isanumberchosenasin
Definition1.2.4.Ineachset
Ui
letusfixapoint
ti
.Then,forany
t∈Ui
andany
u∈V
wehave
"u(t)lu(ti)"<
1
28
,whichshows
that
u(t)=u(ti)+u(t)lu(ti)∈u(ti)+BE(
0,
1
28)
.Thus,
V(Ui)⊆
V(ti)+BE(
0,
1
28)
,andconsequently
DE(V(Ui))<DE(V(ti))+8
for
any
i∈{
1,
...
,
n}
(seeTheorem1.1.21(g)andRemark1.1.31).From
thisweconcludethat
DE(V(J))=DE(
i=1
U
m
V(Ui))=max
1<i<m
DE(V(Ui))
<max
1<i<m
DE(V(ti))+8<max
t∈J
DE(V(t))+8
(cf.Theorem1.1.20(d)).Finally,notingthat
8
isarbitrary,wehave
DE(V(J))<max
t∈J
DE(V(t)),whichendstheproof.
Asaby-productofAmbrosetti’slemmaweobtaintheclassical
Arzelá–Ascolicompactnesscriterion.
102090Corollary
(Arzelá–Ascolicompactnesscriterion)
0
Let
E
beaBa-
nachspaceandlet
V
beanon-emptysubsetof
C([a
,
b]
;
E)
.If
V
isequicon-
tinuousand
V(t)
isrelativelycompactin
E
forevery
t∈[a
,
b]
,then
V
is
relativelycompactinC([a,b];E).
Proof.
First,letusobservethat,becauseoftherelativecompactness
ofeach
V(t)
in
E
,theset
V
ispointwisebounded,thatis,forevery
1
