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12
JeanMawhin
ofaconvexcompactsetofthefunctionspaceCn([ajb])orL2([ajb]).Their
motivationwastheobtentionofexistenceofsolutionsoftheboundaryvalue
problem
(2.10)
x(n)=f(tjxjx!j...jx(n11))j
/
ł
b
n11
Σ
j=o
pij(t)x
(j)(t)dt+
n11
Σ
j=o
Σ
k=1
m
qijkx(j)(tk)=ai
(i=1j2j...jn;a≤t1<t2<...tm≤b).Theyprovedinparticularthat
(2.10)hasatleastonesolutionwhenb−aissufficientlysmall.
Inaseriesofbeautifulpaperswrittenbetween1927and1930[88]–[90],
JuliuszSchauderextendedBirkhoff–Kellogg’sfixedpointtheoremtocontin-
uousselfmapsofaconvexcompactsetofanarbitraryBanachspace,aresult
referredasSchauderfixedpointtheorem.Hismotivationandapplicationswere
essentiallypartialdifferentialequationsandhisuniqueexampleforordinary
differentialequationswasanalternativeproofofGiuseppePeano’sexistence
resultforCauchyproblem(forwhichindeednofixedpointtechniqueisreally
needed).
Apaperof1930ofRenatoCaccioppolireproducedBirkhoff–Kellogg’sfixed
pointtheoremforCn([ajb]withoutanyreferencetoBirkhoff,Kelloggor
Schauder.Oneyearlater,anotherone[10]acknowledgedtheworkofthose
authorsandgaveapplicationstoproblem(2.10)withb−aisarbitraryand
fisboundedeverywhereorsublinearatinfinityinx1j...jxn.Thoseresults
stimulatedalotofactivityinItaly.TheywereimprovedbyGiuseppeScorza-
DragonianditsschoolatRomaandPadova,throughCaccioppoli’sfunctional
analyticapproach,andbySilvioCinquiniatPisa,whousedtheshootingap-
proachbaseduponthesolutionoftheassociatedCauchyproblem.Aside
aspectwasastrongfightbetweenScorza-DragoniandCinquinilastingfor
morethantenyears(includingtheSecondWorldWar),aboutthe“topologi-
calornot”or“elementaryornot”characteroftheirrespectiveapproaches.
2.3.Existenceresultsformulti-pointboundaryvalueproblems
(1961–62)
ThefirstpaperofLasotadevotedtoamultipointboundaryvalueproblem,
writtenwithOpialandpublishedin1961[53],wasdirectlymotivatedbyde
LaValléePoussin’sone:
