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Differentialinclusions–thetheoryinitiatedbyCracowMathematicalSchool
9
andtwonaturalprojectionspϕ:Γϕ→X,qϕ:Γϕ→Ydefinedasfollows:
pϕ(xjy)=xandqϕ(xjy)=y,forevery(xjy)∈Γϕ.
Letusalsopresentsomemoregeneralexamplesstimulatingourconsider-
ationofmultivaluedmaps.
EXAMPLE2.1(Inversefunctions).Letf:X→Ybea(single-valued)
continuousmapfromXontoY.Thenitsinversecanbeconsideredasa
multivaluedmapOf:YOXdefinedby:
Of(y)=f
11(y)j
fory∈Y.
EXAMPLE2.2(Implicitfunctions).Letf:X×Y→Zandg:X→Zbe
twocontinuousmapssuchthat,foreveryx∈X,thereexistsy∈Ysuchthat
f(xjy)=g(x).
Theimplicitfunction(definedbyfandg)isamultivaluedmapO:XOY
definedasfollows:
O(x)={y∈Y|f(xjy)=g(x)}.
EXAMPLE2.3.Letf:X×Y→Rbeacontinuousmap.Assumethatthere
isr>0suchthatforeveryx∈Xthereexistsy∈Ysuchthatf(xjy)≤r.
ThenweletOr:XOY,Or(x)={y∈Y|f(xjy)≤r}.
EXAMPLE2.4(Multivalueddynamicalsystems).Dynamicalsystemsde-
terminedbyautonomousordinarydifferentialequationswithouttheunique-
nesspropertyaremultivaluedmaps.
EXAMPLE2.5(Metricprojection).LetAbeacompactsubsetofametric
space(Xjd).Then,foreveryx∈X,thereexistsa∈Asuchthat
d(ajx)=dist(xjA).
WedefinethemetricprojectionP:XOAbyputting:
P(x)={a∈A|d(ajx)=dist(xjA)}j
x∈X.
Notethatthemetricretractionisaspecialcaseofthemetricprojection.
LetKbeacompactsubsetoftheeuclideanspaceRn.Weshallsaythat
KisaproximativeretractifthereexistsanopensubsetUofRnsuchthat
K⊂Uandaproximativeretractionr:U→Kdefinedasfollows:
"r(x)−x"=dist(xjK).
