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Applications
41
classicalmechanicsisdeterministic:theinitialstateofthesystemfully
determinesitsfuture.)Thefirstquestioninthecase
E=R
wasan-
sweredalongtimeagoinacountryfar,faraway–inthelate19th
century,Peano,anItalianmathematician,provedasuitableexistence
resultforrealfunctions,whichturnedouttobealsovalidforvector
functionstakingvaluesinafinite-dimensionalspaceendowedwith
anynormwhatsoever.(ThefirstversionofPeano’sresultinpaper[212]
dealswithreal-valuedfunctions.Inanotherpaper[213]Peanoappar-
entlyprovedaversionforfunctionswithvaluesin
Rn
,althoughthe
paperisnotexactlytheeasiestthingtoread...)
102010Theorem
(Peano)
0
Letthefunction
f:[
0,
a]XBRn(x0
,
r)→Rn
for
some
r∈(
0,
+∞]
becontinuous.Then,theCauchyproblem
(1.8)
admits
alocalsolution
x:[
0,
b]→BRn(x0
,
r)
,wherethenumber
b
isgivenbythe
formula
b:=min{a
,
rM−1}
with
M
apositivenumbersuchthat
M>
"f(t,u)"forallt∈[0,a]andu∈BRn(x0,r).
Anaturalquestioniswhetheritispossibletosubstituteanyinfinite-
dimensionalBanachspaceinplaceof
Rn
intheabovetheorem,andif
not,whatconditionmustaBanachspacesatisfyinorderforthePeano
theoreminthatspacetobetrue.Theanswertothefirstquestionis
negative,asthefollowingexampleshows.
102020Example0
LetusconsidertheBanachspace
C0
andletthefunc-
tion
f:[
0,1
]XC0→C0
begivenby
f(t
,
u)=(√|u1|
,
√|u2|
,
...)
forevery
t∈[
0,1
]
and
u=(ui)i∈N∈C0
.(Theauthorswouldliketostressthat
thisisnotatypo–wedounderstandthat
f
doesnotflreally”depend
on
t
,butwewantedtobeconsistentwithequation
(1.8)
.)Notethat
f
iswell-defined,i.e.,forany
u∈C0
and
t∈[
0,1
]
thesequence
f(t
,
u)
isalsoanelementof
C0
.Moreover,
f
isalsocontinuous,sinceforany
t,S∈[0,1]andu=(ui)i∈N,w=(Wi)i∈N∈C0wehave
"f(t,u)lf(S,w)"∞=sup
|
||ui|
1/2l|W
i|
1/2|
|
i∈N
<sup
|uilWi|
1/2="ulw"1/2
∞.
i∈N
Now,letusconsidertheCauchyproblem
(1.8)
with
a=
1andthe
initialcondition
x0:=(i−2)i∈N
,andletussupposethat
x:[
0,
b]→C0
1
