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46
Measuresofnon-compactness
1
over,let
U1
,
...
,
Um⊆J
beafinitecoveringofthecompactinter-
val
J
suchthat
diamUj<δ
for
j=
1,
...
,
m
,where
δ:=δ(1
28)
is
apositivenumberchosenasinDefinition1.2.4.Then,itiseasyto
n
m
seethat
V(J)⊆
U
U
Wij
,where
Wij:=Vi(Uj)
for
i=
1,
...
,
n
and
i=1
j=1
j=1,...,m.Furthermore,foru,w∈ViandS,t∈Uj,wehave
"u(t)lw(S)"<"u(t)lw(t)"+"w(t)lw(S)"
<diamVi+
1
2
8<DC(V)+8.
Therefore,
diamWij<DC(V)+8
for
i=
1,
...
,
n
and
j=
1,
...
,
m
.
This,inviewofthearbitrarinessof
8
,impliesthat
DE(V(J))<DC(V)
.
Now,wearegoingtoshowtheoppositeinequality.Let
8>
0and
letusconsiderthecollection
{Ut}t∈J
ofintervals
Ut:=(tlδ
,
t+δ)∩J
openin
J
,where,asbefore,
δ:=δ(1
38)
isanumberchosenasinDefi-
nition1.2.4.Since
J
iscompact,itispossibletofindafinitenumberof
n
points
t1
,
...
,
tn∈J
suchthat
J=
U
Ut
i
(seeTheorem1.1.1).Further,
i=1
bythedefinitionoftheKuratowskimeasureofnon-compactness,
V(J)
canbecoveredbysets
V1
,
...
,
Vm⊆V(J)⊆E
withdiametersnot
exceedingDE(V(J))+
1
38.
By
Φ
letusdenotethefinitesetofallthemappings
ϕ:{
1,
...
,
n}→
{1,...,m},andforeachϕ∈Φletusput
AI:={u∈V|u(ti)∈VI(i)foreveryi=1,...,n}.
Then,
V=U
AI
.Indeed,if
u∈V
,thenforeach
i∈{
1,
...
,
n}
the
I∈Φ
value
u(ti)
belongstosome
Vj
i
,andso
u∈AI
,wherethefunction
ϕ
isgivenbyϕ(i)=ji.
Itremainsnowtoestimatethediametersofthesets
AI
.Let
u
,
w∈
AI
and
t∈J
.Since
t∈Ut
i
forsome
i∈{
1,
...
,
n}
,inviewofthefact
thatu(ti),w(ti)∈VI(i),weget
"u(t)lw(t)"<"u(t)lu(ti)"+"u(ti)lw(ti)"+"w(ti)lw(t)"
<1
38+diamVI(i)+
1
38<DE(V(J))+8.
