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Differentialinclusions–thetheoryinitiatedbyCracowMathematicalSchool
11
Notethat,forO=f:X→Y,thenotionofuppersemicontinuitycoincides
withthelowersemicontinuitywhichmeansnothingelsethanthecontinuity
off.
Inwhatfollows,wealsosaythatamultivaluedmapO:XOYiscontin-
uousmapifitisbothu.s.c.andl.s.c.
ThemostfamousselectiontheoremisthefollowingresultprovedbyE.A.
Michael.
THEOREM2.12(E.A.Michael).LetXbeaparacompactspace,EaBa-
nachspaceandO:XOEal.s.c.mapwithclosedconvexvalues.Thenthere
existsf:X→E,acontinuousselectionofO(f⊂O).
Wewouldliketopointoutthatinthefollowingpart,wewillpresentthe
Kuratowski–Ryll–Nardzewskiselectiontheoremfrequentlyusedinthetheory
ofdifferentialinclusions.
Apartfromsemicontinuousmultivaluedmappings,multivaluedmeasur-
ablemappingswillbeofthegreatimportanceinthesequel.Throughoutthis
section,weassumethatYisaseparablemetricspace,and(ΩjUjµ)isamea-
surablespace,i.e.,asetΩequippedwithσ-algebraUofsubsetsandacount-
ablyadditivemeasureµonU.AtypicalexampleiswhenΩisabounded
domainintheEuclideanspaceRk,equippedwiththeLebesguemeasure.
DEFINITION2.13.AmultivaluedmapO:ΩOYwithclosedvaluesis
calledmeasurablemapifO11(V)∈U,foreachopenV⊂Y.
Inwhatfollows,weshallusethefollowingKuratowski–Ryll–Nardzewski
selectiontheorem.
THEOREM2.14(Kuratowki–Ryll–Nardzewski).LetYbeaseparablecom-
pletespace.TheneverymeasurableO:ΩOYhasa(single-valued)measur-
ableselection.
LetΩ=[0ja]beequippedwiththeLebesguemeasureandY=Rn.
DEFINITION2.15.AmapO:[0ja]×RnORnwithnonemptycom-
pactvaluesiscalledu-Carathéodorymap(resp.,l-Carathéodorymap;resp.,
Carathéodorymap)ifitsatisfies:
(1)tOO(tjx)ismeasurable,foreveryx∈Rn,
(2)xOO(tjx)isu.s.c.(resp.,l.s.c.;resp.,continuous),foralmostallt∈
[0ja],
(3)|y|≤µ(t)(1+|x|),forevery(tjx)∈[0ja]×Rn,y∈O(tjx),where
µ:[0ja]→[0j+fl)isanintegrablefunction.