Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
42
Measuresofnon-compactness
isitslocalsolutionforsome
b∈(
0,1
]
.Then,thecomponentsof
x
,that
is,thefunctions
Ęi:[
0,
b]→R
,arelocalsolutionstothefollowing
infinitesystemofCauchyproblems
1
Ę!
i(t)=d|Ęi(t)|,Ęi(0)=i−2,
where
t∈[
0,1
]
and
i∈N
.Therefore,
Ęi(t)=(1
2t+1
i)
2
forevery
i∈N
and
t∈[
0,
b]
.Itisclearthatthesequence
(Ęi(b))i∈N
doesnotconverge
tozero,thatis,
x(b)/
∈C0
.Hence,onthecontrarytowhatweassumed,
xcannotbeasolutiontotheCauchyproblem(1.8).
Itisperhapssurprising–terrifying,even–thatsuchcounterexam-
plesexistwheneverwedealwithinfinitelymanydimensions.Func-
tionalanalysisisindeedquiteascarything.Inaninfinite-dimensional
Banachspace,noonehearsyouscream.
102030Theorem
(Godunov)
0
Foranyinfinite-dimensionalBanachspace
E
thereexistapoint
x0∈E
,positivenumbers
r
and
a
andacontinuous
function
f:[
0,
a]XBE(x0
,
r)→E
suchthattheCauchyproblem
(1.8)
admitsnosolutionsonanyinterval[0,8].
Wewilldevotetherestofthissectiontoproveafewtheorems
aboutabstractordinarydifferentialequations.Sincewecannothope
foranexactanalogueofthePeanotheorem,wewillneedsomething
strongerthanthecontinuityoftheright-handsideoftheequation.It
willturnout(seeTheorem1.2.11)thatwemayemployacondition
relatedtothemeasureofnon-compactness(wewillalsostudysimilar
conditionsinSection2.2whentalkingaboutfixedpoints).
Inordertobeabletocomputemeasuresofnon-compactnessof
subsetsofthespaceofcontinuousfunctions,wearefirstgoingto
introduceanimportantdefinition.
102040Definition0
Let
E
beaBanachspace.Anon-emptyset
V⊆
C([a
,
b]
;
E)
issaidtobeequicontinuous,ifforevery
8>
0thereexists
a
δ=δ(8)>
0suchthat
"x(t)lx(S)"<8
forany
x∈V
andallpoints
S,t∈[a,b]satisfyingtheinequality|tlS|<δ.
