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8
LechGórniewicz
AndrzejLasota,ZdzisławOpial,CzesławOlech,JózefMyjak,AndrzejPelczar
andAndrzejPliś.
ItisworthaddingthatWażewskihimselfusedanotherterm,thatisori-
entorequation[Hrównanieorientorowe”].Thefundamentalroleforthetheory
wasplayedbyWażewski)sworktitled:
1.Onanoptimalcontrolproblem,Prague,1964,692–704.
Inthatpaper,Ważewskidemonstratedthateachproblemofcontrolling
ordinarydifferentialequationsofthefirstordercanbearticulatedwithori-
entorequationterms.Thatobservationservedasanessentialstimulusto
studyorientordifferentialequationsandconsequently,itcontributedtothe
introductionofthenewterm,stillvalid,andthatisHdifferentialinclusions”.
Weshallreturntothesubjectoftherelationstothecontroltheoryinthe
forthcomingpassagesofthepresentlecture.
Thetheoryofdifferentialinclusionsislocatedwithinthemainstreamof
non-linearanalysis–ortoputitmoreprecisely–multi-valuedanalysis.This
theoryisintensivelydevelopedespeciallyinthecountriessuchasFrance,
Germany,Russia,Italy,CanadaandUSA.InPoland,thereisalargegroup
ofmathematiciansworkingontheseissues.Thatgroupismainlylocatedin
thefollowingcentres:Gdańsk,Toruń,WarszawaandZielonaGóra.
Moreover,thereferenceslistedbelow–becauseofthenatureofthelecture
–arelimitedtotheworksbyProfessorAndrzejLasotaexclusivelyrelating
directlyorindirectlytothetheoryofdifferentialinclusions.Richliteratureon
thesubjectcanbefoundinthereferencesintheparticularmonographs.
2.Multivaluedmappings
Inthissection,weshallsurveythemostimportantpropertiesofmulti-
valuedmappingswhichweuseinthesequel.Thereareseveralmonographs
devotedtomultivaluedmappings;seee.g.[1]–[3],[7],[8].
Inwhatfollows,weassumethatalltopologicalspacesaretheTikhonov
T31
2
-spaces.
LetXandYbetwospacesandassumethat,foreverypointx∈X,a
nonemptyclosed(sometimeswewillassumeonlythatO(x)/=∅)subsetO(x)
ofYisgiven;inthiscase,wesaythatOisamultivaluedmappingfromX
toYandwewriteO:XOY.Inwhatfollows,thesymbolO:X→Yis
reservedforsingle-valuedmappings,i.e.,O(x)isapointofY.
LetO:XOYbeamultivaluedmap.WeassociatewithOthegraphΓϕ
ofObyputting:
Γϕ={(xjy)∈X×Y|y∈O(x)}
