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JeanMawhin
TheproofofthisresultmotivatedLasotatoconsiderthefixedpointprob-
leminaBanachspaceBoftheform
x=A(x)x+b(x)
when,foreachx∈B,A(x)belongstoasuitableclassoflinearoperators
onBandbiscompletelycontinuousandsublinearatinfinity.Hisresults
canbeseenasextensionsofIvarFredholm’sfirsttheoremforlinearintegral
equations.
In1964,LasotaandOpialconsideredtheexistenceofω-periodicsolutions
forsystemsoftheform
x
!=A(tjx)x+b(tjx)
whereA=(aij),andaijjbi:Rm+1→RsatisfyCarathéodoryconditionsand
areω-periodicwithrespecttot.
Lasotaandcoworkershavealsoconsideredsecondorderdifferentialequa-
tionsorsystems
x
!!=f(tjxjx!)
withvariouslineartwo-pointboundaryconditions,andfirstordersystems
x
!=f(tjx)
withfairlygenerallinearboundaryconditions.
About40papersonordinarydifferentialequationshavebeenwrittenby
Lasota,between1961and1980,thelastonedealingwiththeso-called‘unique-
nessimpliesexistence’methodology.Mostofthosepapersareabeautiful
blendoflinearfunctionalanalysis,fixedpointtheory(essentiallySchauder’s
theorem)andinequalities,namelyingredientswhicharestillbasicinthe
presentdaystudiesofnonlinearboundaryvalueproblemsforordinarydif-
ferentialequations.Manypapersarejointwork,firstwithOpial,andlater
withseveralyoungcollaborators.Theyarelistedinthebibliography,but
onlytheonesdealingwithboundaryvalueproblemsandperiodicsolutions
aredescribedhere.Furthermore,forthesakeofbrevity,wehavenotconsid-
ered,whenanalyzingLasota’slegacy,theextensionsofhisresultstodifference
equations,functionaldifferentialequationsanddifferentialrelations.
