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40
Measuresofnon-compactness
Beforeweproceedtothemainpartofthissection,letusbriefly
recallsomebasicnotionsfrominfinite-dimensionalcalculus.Let
E
be
aBanachspace.Afunction
x:[a
,
b]→E
issaidtobedifferentiableata
pointt∈(a,b),ifthelimit
1
h→0
lim
x(t+h)lx(t)
h
(1.7)
exists;ifthisisthecase,thenthelimit
(1.7)
isdenotedby
x!(t)
andis
calledthederivativeof
x
at
t
.(Thisway,wealsodefinethederivative
function
x!:(a
,
b)→E
.)Wecanalsospeakaboutdifferentiability
ofthefunction
x
attheendpointsoftheinterval
[a
,
b]
.Itsufficesto
considerlimitssimilarto
(1.7)
butwith
h
tendingtoeither0
+
or0
−
.
(Insuchacase,wecanspeakofthefunction
x!:[a
,
b]→E
.)Sincethe
onlydifferenceinthedefinitionofadifferentiablemappingbetween
theabstractandclassicalsettingisthatintheabstractsettingthelimits
aretakenwithrespecttothenormin
E
ratherthantheEuclidean
metricin
R
,weleaveallthedetailstothereader.Letusalsorecallthat
wesaythatafunction
x:[a
,
b]→E
iscontinuouslydifferentiable,ifthe
mappingx!:[a,b]→Eiscontinuous.
Now,letusreturntothemaintopicofthissection.Considerthe
followingCauchyproblem(alsocalledtheinitialproblem)
x!(t)=f(t,x(t)),x(0)=x0,
(1.8)
where
t∈[
0,
a]
with
a>
0,
x0
isanelementof
E
and
f:[
0,
a]X
BE(x0
,
r)→E
,where
r∈(
0,
+∞]
,isacontinuousmapping.(Here
weadopttheobviousconventionthat
BE(x0
,
+∞):=E
.)Afunction
x:[
0,
b]→BE(x0
,
r)
,where
b∈(
0,
a]
,issaidtobea(local)solution
to
(1.8)
if
x
iscontinuouslydifferentiableon
[
0,
b]
,satisfiesthediffer-
entialequationx!(t)=f(t,x(t))foreveryt∈[0,b],andx(0)=x0.
ProbablythefirstquestionregardingtheCauchyproblemis:does
ithaveasolutionatall?Andprobablythesecondquestionis:howcan
webesurethatithasexactlyonesolution?(Thisisveryimportantin
certainapplications.Forinstance,ifadifferentialequationdescribes
aphysicalphenomenonlikethemotionofasetofbodiesunderthe
gravityforce,uniquenessofthesolutioncorrespondstothefactthat
