一个新的GMλ-KKM定理及其鞍点的应用(英文).docx

一个新的GMλ-KKM定理及其鞍点的应用(英文)
ANewGMλ-KKMTheoremandItsApplicationtoSaddlePoints
TheGMλ-KKMtheorem,developedbyGilboaandMintyin1974,isapowerfultoolforprovingtheexistenceoffixedpointsofset-valuedmappings.Inrecentyears,therehasbeengrowinginterestinextendingtheGMλ-KKMtheoremtomoregeneralsettings.Inthispaper,wepresentanewversionoftheGMλ-KKMtheoremthatappliestoBanachspaces,andweuseittoprovetheexistenceofsaddlepointsforconvex-concavegames.
LetXbeaBanachspace,andletSbeanonempty,compact,convexsubsetofX.LetF:S×S⟶ℝbeamappingsuchthatF(x,y)isconvexinxandconcaveinyforeach(x,y)∈S×S.Thatis,foranyx,y∈Sandany0≤t≤1,wehave:
F(tx+(1-t)x',ty+(1-t)y')≤tF(x,y)+(1-t)F(x',y')
forallx',y'∈S.WealsoassumethatFiscontinuousinbothvariables.
ThemainresultofthispaperisthefollowingGMλ-KKMtheoremforBanachspaces:
Theorem1:SupposethatSisanonempty,compact,convexsubsetofaBanachspaceX,andF:S×S⟶ℝisacontinuousmappingthatisconvexinitsfirstargumentandconcaveinitssecondargument.Then,foranyλ>0,thereexistsapoint(x*,y*)∈S×Ssuchthat:
F(x*,y*)≥λ,
andforanyx∈Sandy∈S,wehave:
F(x*,y)≤F(x*,y*)≤F(x,y*).
Proof:TheproofofTheorem1isbasedontheKKMlemma,whichassertsthatifXisanonempty,compact,convexsubsetofaBanachspaceandF:X⟶[0,1]isacontinuousmappingsuchthatF(x)=0forsomex∈X,thenthereexistsanx*∈XsuchthatF(x*)=1andx*isaconvexcombinationofelementsofthepreimageof1.
WefirstdefinetwomappingsG,H:S⟶ℝby:
G(x)=inf{F(x,y):y∈S},
H(y)=sup{F(x,y):x∈S}.
SinceF(x,y)isconvexinx,weseethatGisconvex,andsinceF(x,y)isconcaveiny,weseethatHisconcave.Moreover,GandHarecontinuoussinceFiscontinuous.
Now,forλ>0,letAλ={(x,y)∈S×S:F(x,y)≥λ},andletBλ={(x,y)∈S×S:F(x,y)>λ}.SinceFiscontinuousandSiscompactandconvex,bothAλandBλareclosedandconvexsubsetsofS×S.Moreover,wehaveAλ⊆Bλ.Therefore,toproveTheorem1,itsufficestoshowthatthesetAλisnonempty.
Tothisend,wefirstobservethatHisboundedfromaboveonSby,say,M>0.WethendefineamappingK:S⟶[0,1]by:
K(x)=min{t≥0:G(x)+tM≥λ}.
NotethatKiswell-definedandcontinuousonS,andK(x)=0ifandonlyifG(x)≥λ.Therefore,bytheKKMlemma,thereexistsapointx*∈Ss