Contributions to the degree theory for perturbation of maximal monotone maps

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Contributions to the degree theory for perturbation of maximal monotone maps

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Contributions to the degree theory for perturbation of maximal monotone maps
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Quarcoo, Joseph
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Compact
Densely defined
Eigenvalues
Homotopy
Quasimonotone
Dissertations, Academic -- Mathematics -- Doctoral -- USF
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ABSTRACT: Let x be a real reflexive separable locally uniformly convex banach space with locally uniformly convex dual spacex^*. Let t:x\supset d(t)\rightarrow 2^{x^*} be maximal monotone with 0\in t(0), 0\in intd(t) and c:x\supset d(c)\rightarrow x^*. Assume that $l\subset d(c)$ is a dense linear subspace of x, c is of class (s_+)_l and \langle cx,x\rangle\geq-\psi(\lx\l), x\in d(c), where \psi:\mathbb{r}^+\rightarrow\mathbb{r}^+ is nondecreasing. a new topological degree is developed for the sum t+c in chapter one. This theory extends the recent degree theory for the operators c of type (s_+)_{0,l} in [15]. unlike such a recent extension to multivalued (s_+)_{0,l}-type operators, the current approach utilizes the approximate degree d(t_t+c,g,0), t\downarrow 0, where t_t = (T^{-1}+tJ^{-1})^{-1}and G is an open bounded subset of X and is such that $0\in G$, for the single-valued mapping $T_t+C$. The subdifferential\partial\varphi, for \varphi belonging to a large class of proper c onvex lower semicontinuous functions, gives rise to operators T to which this degree theory applies. Theoretical applications to problems of Nonlinear Analysis are included, as well as applications from the field of partial differential equations. Let T:X\supset D(T)\rightarrow 2^{X^*} be maximal monotone with compact resolvents, i.e, the operator $(T+\epsilonJ)^{-1}:X^*\rightarrow X is compact for every \in 0. We present a relevant result in chapter 2 that says there exists an open ball around zero in the image of a relatively open set by a continuous and bounded perturbation of a maximal monotone operator with compact resolvents. The generalized degree function for compact perturbations of m-accretive operators established by Y. -Z Chen in [7] isextended to the case of a multivalued compact perturbations of maximal monotone maps by appealing to the topological degree forset-valued compact fields in locally convex spaces introduced by Tsoy Wo-Ma in [25]. Such is the content of the thi rd chapter. A unified eigen value theory is developed for the pair(T,S), where T:X\supset D(T)\rightarrow 2^{X^*} is aquasimonotone-type operator which belong to the so-called A_G(QM) class introduced by Arto Kittila in [23] and S is abounded demicontinuous mapping of class (S)_+. Conditions are given for the existence of a pair (x,\lambda)\in (0,\infty)\times(D(T+S)\cap\partial G)$ such that Tx+\lambda Sx\ni 0$. This is the content of Chapter 4.
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Dissertation (Ph.D.)--University of South Florida, 2006.
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by Joseph Quarcoo.

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ABSTRACT: Let x be a real reflexive separable locally uniformly convex banach space with locally uniformly convex dual spacex^*. Let t:x\supset d(t)\rightarrow 2^{x^*} be maximal monotone with 0\in t(0), 0\in intd(t) and c:x\supset d(c)\rightarrow x^*. Assume that $l\subset d(c)$ is a dense linear subspace of x, c is of class (s_+)_l and \langle cx,x\rangle\geq-\psi(\lx\l), x\in d(c), where \psi:\mathbb{r}^+\rightarrow\mathbb{r}^+ is nondecreasing. a new topological degree is developed for the sum t+c in chapter one. This theory extends the recent degree theory for the operators c of type (s_+)_{0,l} in [15]. unlike such a recent extension to multivalued (s_+)_{0,l}-type operators, the current approach utilizes the approximate degree d(t_t+c,g,0), t\downarrow 0, where t_t = (T^{-1}+tJ^{-1})^{-1}and G is an open bounded subset of X and is such that $0\in G$, for the single-valued mapping $T_t+C$. The subdifferential\partial\varphi, for \varphi belonging to a large class of proper c onvex lower semicontinuous functions, gives rise to operators T to which this degree theory applies. Theoretical applications to problems of Nonlinear Analysis are included, as well as applications from the field of partial differential equations. Let T:X\supset D(T)\rightarrow 2^{X^*} be maximal monotone with compact resolvents, i.e, the operator $(T+\epsilonJ)^{-1}:X^*\rightarrow X is compact for every \in 0. We present a relevant result in chapter 2 that says there exists an open ball around zero in the image of a relatively open set by a continuous and bounded perturbation of a maximal monotone operator with compact resolvents. The generalized degree function for compact perturbations of m-accretive operators established by Y. -Z Chen in [7] isextended to the case of a multivalued compact perturbations of maximal monotone maps by appealing to the topological degree forset-valued compact fields in locally convex spaces introduced by Tsoy Wo-Ma in [25]. Such is the content of the thi rd chapter. A unified eigen value theory is developed for the pair(T,S), where T:X\supset D(T)\rightarrow 2^{X^*} is aquasimonotone-type operator which belong to the so-called A_G(QM) class introduced by Arto Kittila in [23] and S is abounded demicontinuous mapping of class (S)_+. Conditions are given for the existence of a pair (x,\lambda)\in (0,\infty)\times(D(T+S)\cap\partial G)$ such that Tx+\lambda Sx\ni 0$. This is the content of Chapter 4.
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ContributionstotheDegreeTheoryforPerturbationsofMaximalMonotoneMapsbyJosephQuarcooAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophyDepartmentofMathematicsCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:AthanassiosG.Kartsatos,Ph.D.YunchengYou,Ph.D.Wen-XiuMa,Ph.D.ArunavaMukherjea,Ph.D.EvgueniiRakhmanov,Ph.D.DateofApproval:July6,2006Keywords:compact,eigenvalues,denselydened,homotopy,quasimonotonecCopyright2006,JosephQuarcoo

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DedicationToMyBelovedFamilyandProfessorF.K.A.Allotey,forwithoutthem,Iwouldn'thavecomethisfar

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AcknowledgementsIwouldlikerstofalltoexpressmyprofoundgratitudetomyadvisorProfessorAthanassiosG.Kartsatosforhiscontinuousencouragementandsupportovertheyears.Specialthankstomymentors,ProfessorF.K.A.AlloteyandProfessorC.E.Chidume.IwouldliketothankDr.RaymondAtta-Fynn,whoencouragedmetotakethequan-tumleaptopursueamajorinmathematicswhenIdoubtedmyowncondence.Ialsothankmycommitteemembers,Dr.YunchengYou,Dr.Wen-XiuMa,Dr.ArunavaMukherjeaandDr.EvgueniiRakhmanovfortheimportantroletheyplayed.Finally,Ithankmyparentsfortheirunconditionalloveandsupportandalltheob-viousreasons.

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TableofContentsAbstractiii1Introduction12DegreeTheoryforMultivaluedS+L-PerturbationofMaximalMonotoneOperators42.1Preliminaries................................42.2ConstructionoftheDegree........................72.3Propertiesofthedegreemapping.....................212.4ApplicationsinNonlinearAnalysis....................312.5FurtherApplications...........................422.6FurtherMappingTheoremsfortheNewDegree.............563BallsintheRangeofaContinuousBoundedPerturbationofaMaximalMonotoneOperatorwithCompactResolvents644TheGeneralizedTopologicalDegreeforMultivaluedCompactPerturbationofMaximalMonotoneOperators744.1TheTsoy-WoMaDegree.........................744.2TheGeneralizeddegreedT)]TJ/F26 11.955 Tf 11.955 0 Td[(G;DT;y.............754.3PropertiesoftheGeneralizedDegree...................795AnEigenvalueProblemforS+PerturbationofNonlinearOperatorsApproximatedbyQuasimonotoneMappings815.1ClassesofMultivaluedMappingsofMonotoneType..........82i

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5.2TheDegreeforOperatorsofClassAGS+...............845.3StatementoftheEigenvalueProblem..................88References94AbouttheAuthorEndPageii

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ContributionstotheDegreeTheoryforPerturbationsofMaximalMonotoneMapsJosephQuarcooABSTRACTLetXbearealreexiveseparablelocallyuniformlyconvexBanachspacewithlocallyuniformlyconvexdualspaceX.LetT:XDT!2Xbemaximalmonotonewith02T,02intDTandC:XDC!X.AssumethatLDCisadenselinearsubspaceofX,CisofclassS+LandhCx;xi)]TJ/F26 11.955 Tf 31.434 0 Td[(kxk,x2DC,where:R+!R+isnondecreasing.AnewtopologicaldegreeisdevelopedforthesumT+Cinchapterone.ThistheoryextendstherecentdegreetheoryfortheoperatorsCoftypeS+0;Lin[15].UnlikesucharecentextensiontomultivaluedS+0;L-typeoperators,thecurrentapproachutilizestheapproximatedegreedTt+C;G;0,t#0,whereTt=T)]TJ/F24 7.97 Tf 6.586 0 Td[(1+tJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1)]TJ/F24 7.97 Tf 6.587 0 Td[(1andGisanopenboundedsubsetofXandissuchthat02G,forthesingle-valuedmappingTt+C.Thesubdierential@',for'belongingtoalargeclassofproperconvexlowersemicontinuousfunctions,givesrisetooperatorsTtowhichthisdegreetheoryapplies.TheoreticalapplicationstoproblemsofNonlinearAnalysisareincluded,aswellasapplicationsfromtheeldofpartialdierentialequations.LetT:XDT!2Xbemaximalmonotonewithcompactresolvents,i.e,theoperatorT+J)]TJ/F24 7.97 Tf 6.586 0 Td[(1:X!Xiscompactforevery>0.Wepresentarelevantresultinchapter2thatsaysthereexistsanopenballaroundzerointheimageofarelativelyopensetbyacontinuousandboundedperturbationofamaximalmonotoneoperatorwithcompactresolvents.Thegeneralizeddegreefunctionforcompactperturbationsofm-accretiveopera-iii

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torsestablishedbyY.-ZChenin[7]isextendedtothecaseofamultivaluedcompactperturbationsofmaximalmonotonemapsbyappealingtothetopologicaldegreeforset-valuedcompacteldsinlocallyconvexspacesintroducedbyTsoyWo-Main[25].Suchisthecontentofthethirdchapter.AuniedeigenvaluetheoryisdevelopedforthepairT;S,whereT:XDT!2Xisaquasimonotone-typeoperatorwhichbelongtotheso-calledAGQMclassintroducedbyArtoKittilain[23]andSisaboundeddemicontinuousmappingofclassS+.Conditionsaregivenfortheexistenceofapairx;2;1DT+S@GsuchthatTx+Sx30.ThisisthecontentofChapter4.iv

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1IntroductionFunctionalanalysisprovidesanabstractframeworktoinvestigatedierentialorin-tegralequations.Byanadequatechoiceofunderlyingfunctionspaces,theproblemsfortheseequationscanberewrittenasanoperatorequationTx=y,whereTisamappingfromaBanachspaceXintoanotherBanachspaceYandy2Y.Ausefulwaytoobtaininformationaboutthesolutionsetistopologicaldegreetheory.SuchatheoryconsistsofanalgebraiccountofthenumberofsolutionsoftheequationTx=y.Thevalueofthedegreefunctionforanysuchcountisanordinaryinteger.Thisintegermaybepositive,negativeorzero.Fornitedimensionalsituations,posi-tivecountscorrespondstosolutionsatwhichthemappingTisorientation-preserving,whilenegativecountcorrespondstosolutionsatwhichTisnotorientation-preserving.Althoughtopologicaldegreewasoneoftheearliesttoolsindealingwithsuchop-eratorequations,ithasstoodinthecoreofallnonlinearanalysis.Foralongtime,theconceptofdegreewasstudiedespeciallywithinalgebraictopology.Toenlargeitsaudience,signicantcontributionsforananalyticapproachofthetopologicaldegreehavebeenrevealedinthelastdecades.TheclassicaltopologicaldegreeinthespaceRnforcontinuousmappingT:Rn!RnwasintroducedbyL.E.J.Brouwerin1912,andwasuniquelydeterminedbyfouraxioms:thenormalizationproperty,existencecondition,additivitywithrespecttothedomainandinvarianceunderhomotopies.In1934LerayandSchaudergeneral-izedtheBrouwerdegreetoinnitedimensionalBanachspacesformappingsoftheformI+T,whereIistheidentitymapandTiscompact.TheirconstructionwasbasedonthenitedimensionalBrouwerdegree.Since1934,variousgeneralizationsofthedegreetheoryhavebeendened.I.V.1

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Skrypnikin1973andF.E.Browderintheearlyeightiesinaseriesofarticles[27]and[3]-[5]respectively,whichextendedtheconceptofatopologicaldegreetononlinearmappingsofmonotonetype.TheirmethodswerebasedonGalerkin-typeapproxima-tionsforwhichthenitedimensionalBrouwerdegreeiswell-dened.Theinterestinthedenitionsofnonlinearmappingsofmonotonetypearisesfromthefactthatthesepropertiescanbeveriedunderconcretehypothesesforthemapsbe-tweenSobolevspacesobtainedfromellipticoperatorsingeneralizeddivergenceform.In1999,A.G.KartsatosandI.V.Skrypnikin[15]introducedforthersttimethetopologicaldegreetheoriesfordenselydenedmappingsinvolvingoperatorsoftypeS+0;L.By"denselydened",wemeanthedomainofsuchamappingcontainsalineardensesubspace.Here,again,theconstructionofthedegreewasbasedontheclassicalBrouwerdegree.In2005[16],thesameauthorsintroducedanewtopologicaldegreefordenselyde-nedquasibounded~S+-perturbationsofmultivaluedmaximalmonotoneoperatorsinreexiveBanachspaces.ThisnewdegreetheorywasasubstantialextensionofBrowder'sdegreetheoryin[4].Thisworkisorganizedinfourchapters.InChapteronewedeneanewtopolog-icaldegreefordenselydenedS+Lperturbationofmultivaluedmaximalmonotoneoperators.Thisnewtopologicaldegreetheoryisinthespiritof[15]and[16]butwedonotusethenitedimensionalBrouwerdegreeasin[15],andtheconditionthattheperturbationisquasiboundedwithrespecttoamaximalmonotoneoperatorisnolongerassumed.Theconstructionofthisnewdegreetheoryfollowsthatof[16].Wehaveappliedthisnewdegreeinstudyofexistence,surjectivityandmappingthe-orems.InChapter2,wepresentarelevantresultsthatsaysthereexistsanopenballaroundzerointheimageofarelativelyopensetbyacontinuousandboundedper-turbationofamaximalmonotoneoperatorwithcompactresolventsundervariousboundaryconditions.RecentlyY.-Z.Chenin[7]establishedageneralizeddegreefunctionforcompactperturbationsofm-accretiveoperatorsandshowedthatthisdegreefunctionhasthe2

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crucialpropertiesofthetopologicaldegree.X.FuandS.Songin[9]extended[7]tothecasewherethecompactperturbationismultivalued.Z.GuanandA.G.Kartsatosin[11]extendedittocompactpertur-bationsofmaximalmonotonemaps.Weextendtheresultsin[11]tomultivaluedcompactperturbationsofmaximalmonotonemapsbyappealingtothetopologicaldegreeforset-valuedcompacteldsinlocallyconvexspacesintroducedbyTsoyWo-Main[25].Theeigenvalueproblem,Tx+Sx30for>0,hadbeenconsideredbymanyauthorsincluding[8],[12],[14]and[17]forvariousclassesofTandS.InChapterfour,weconsiderthisinclusionforT2AGQM,aclassofquasimonotone-typemap-pingintroducedbyArtoKittilain[23]andSisaboundeddemicontinuousmappingofclassS+.3

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2DegreeTheoryforMultivaluedS+L-PerturbationofMaximalMonotoneOperators2.1PreliminariesInordertodiscussthedegreeforthemapsfromXtoX,whereXisareexiveBanachspace,weneedtointroducersttheappropriatemappings.Denition2.1.1AmapT:XDT!2Xissaidtobe"monotone"ifhx)]TJ/F26 11.955 Tf 11.956 0 Td[(y;x)]TJ/F26 11.955 Tf 11.956 0 Td[(yi0forallx;x;y;y2GrT.HereGrTdenotesthegraphofTandh:;:ithedual-itybracketforthepairhX;Xi,i.e,hx;xiisthevalueofthefunctionalxatx.WesaythatTis"maximalmonotone"ifitismonotoneandforanyu;u2XXforwhichhu)]TJ/F26 11.955 Tf 11.955 0 Td[(x;u)]TJ/F26 11.955 Tf 11.955 0 Td[(xi0,forallx;x2GrT,wehaveu;u2GrT.Denition2.1.2LetBXandf:B!X.WesaythatfisofclassS+ififisdemicontinuousi.e.xn!xinBimpliesfxn*fxinXandiiiffxn:n1gBandxn*xforsomex2Xandlimsupn!1hfxn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(xi0;impliesxn!xinX.4

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Denition2.1.3AsequenceofoperatorsTn:X!X,n1,issaidtosatisfyconditionSqonasetAXifforeverysequencefxngA,suchthatxn*x,andTnxn!y,wehavexn!x.Denition2.1.4TheoperatorT:XDT!2Xissaidtobe"pseudomonotone"ifxn*xasn!1andlimsupn!1hvn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi0;vn2Txn;impliesx2DTandhv;x)]TJ/F26 11.955 Tf 11.955 0 Td[(yiliminfn!1hvn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(yiforally2X,v2Tx.Denition2.1.5TheoperatorT:XDT!2Xissaidtobe"generalizedpseu-domonotone"ifxn*xandvn*v,withvn2Txnasn!1andlimsupn!1hvn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi0;thenhvn;xni!hv;xi,x2DTandv2Tx.Denition2.1.6TheoperatorT:XDT!2XissaidtobequasimonotoneifforanysequencexninDTwithxn*xinXwehavelimsupn!1hvn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi0withvn2Txn.Byawell-knownrenormingtheoremduetoTroyanski[29],givenareexiveBanachspace,wecanalwaysrenormitequivalentlysothatbothXandXarelocallyuni-formlyconvex.ThuswithoutlossofgeneralityweassumewithoutfurthermentionthatbothXandXarelocallyuniformlyconvex.5

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WedeneJ:X!X,"theduality"mapofX,byJx=fx2X:hx;xi=kxk2=kxk2gJisawell-dened,singled-valuedmapfromXtoXwhichisahomeomorphism,maximalmonotoneandofclassS+.AnoperatorT:XDT!Y;withYanotherrealBanachspace,isbounded"ifitmapsboundedsubsetsofDTontoboundedsets.Itiscompact"ifitiscontinuousandmapsboundedsubsetsofDTontorelativelycompactsubsetsofY:Itisdemicontinuous"completelycontinuous"ifitisstrong-weakweak-strongcontinuousonDT:Denition2.1.7T:XDT!2Xisstronglyquasibounded"ifforeachS>0thereexistsKS>0suchthatkxkS;hu;xiS;forsomeu2Tx;implykukKS:BrowderandHesshaveshownin[6,Proposition14]that,amonotoneoperatorTisstronglyquasiboundedif02intDT:ThefollowinglemmacanbefoundinZeidler[31,p.915].Lemma2.1.8LetT:XDT!2Xbemaximalmonotone.ThenthefollowingaretrueifxngDT;xn!x0andTxn3yn*y0implyx02DTandy02Tx0:iifxngDT;xn*x0andTxn3yn!y0implyx02DTandy02Tx0:Fromlemma1.1.7weseethateitheroneofi,iiimpliesthatthegraphGToftheoperatorTisclosed,i.e.GTfx;u;x2DT;u2Txgisaclosedsubset6

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ofXX:2.2ConstructionoftheDegreeInwhatfollowsXisaseparablereexiveBanachspace.LetA:XDA!XwithDAdenseinX:WeassumethatthereexistsasubspaceLofthespaceXsuchthatLDA; L=X::1DenotebyFLthesetofallnitedimensionalsubspacesofL:WecanchooseasequencefFng;n2N;suchthat,foreachn2N;Fn2FL;FnFn+1;dimFn=n;and [nFn=X::2WeletLFn=1[n=1Fn::3Denition2.2.1WesaythattheoperatorC:XDC!XsatisesConditionS+0;L,ifforeverysequencefFngsatisfying.2andeverysequencefxngLwithxn*x0;limsupn!1hCxn;xni0;limn!1hCxn;yi=0;forsomex02Xandanyy2LFn;itfollowsthatxn!x0;x02DCandCx0=0:WesaythattheoperatorC:XDC!XsatisesconditionS+LiftheoperatorCh:DC!X;denedbyChu=Cu)]TJ/F26 11.955 Tf 10.153 0 Td[(h;satisesconditionS+0;Lforanyh2X.WeneedthefollowingthreeconditionsontheoperatorC:c1thereexistsasubspaceLofXwhichsatises.1andissuchthattheoperatorCsatisesConditionS+L;7

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c2foreveryF2FL;y2LthemappingaF;y:F!R+;denedbyaF;yx=hCx;yi;iscontinuous.c3Thereexistsafunction:R+!R+whichisnondecreasingandsuchthathCx;xi)]TJ/F26 11.955 Tf 29.888 0 Td[(kxk;x2DC:Lemma2.2.2LetT:XDT!2Xbemaximalmonotoneandsuchthat02DTand02T:Thenthemappingt;x!Ttxiscontinuousontheset;1X,whereTt=T)]TJ/F24 7.97 Tf 6.587 0 Td[(1+tJ)]TJ/F24 7.97 Tf 6.587 0 Td[(1)]TJ/F24 7.97 Tf 6.587 0 Td[(1.Proof.Fixa>0.LetfxngX,ftng[;1besuchthatxn!x0andtn!t0.Letyn=Ttnxn.Thenforsomezn2DTwithyn2Tzn,T)]TJ/F24 7.97 Tf 6.587 0 Td[(1+tnJ)]TJ/F24 7.97 Tf 6.587 0 Td[(1yn3xn=zn+tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1yn::4UsingthemonotonicityoftheoperatorTandthecondition02Twegethyn;xni=hyn;zni+tnhyn;J)]TJ/F24 7.97 Tf 6.587 0 Td[(1ynikynk2whichgivestheboundedofthesequencefyng,andhencetheboundednessoffzngby1:4.SinceXandXarereexive,wemayassumethatyn*y0,zn*z0andJ)]TJ/F24 7.97 Tf 6.587 0 Td[(1yn*j0.UsingthisandthemonotonicityofthedualitymappingJ)]TJ/F24 7.97 Tf 6.587 0 Td[(1:X!X=X,weobtainlimn!1hyn)]TJ/F26 11.955 Tf 11.956 0 Td[(y0;xni=0;liminfn!1hyn)]TJ/F26 11.955 Tf 11.955 0 Td[(y0;tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1yni0::5Thesecondof:5followsfromhyn)]TJ/F26 11.955 Tf 11.955 0 Td[(y0;tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1yn)]TJ/F26 11.955 Tf 11.955 0 Td[(tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1y0itnkynk)-222(ky0k2;8

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whichimplieshyn)]TJ/F26 11.955 Tf 11.955 0 Td[(y0;tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1ynihyn)]TJ/F26 11.955 Tf 11.955 0 Td[(y0;tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1y0i:From:4and:5wehavelimsupn!1hyn)]TJ/F26 11.955 Tf 11.955 0 Td[(y0;znilimn!1hyn)]TJ/F26 11.955 Tf 11.955 0 Td[(y0;xni+limsupn!1)]TJ/F19 11.955 Tf 9.299 0 Td[([hyn)]TJ/F26 11.955 Tf 11.955 0 Td[(y0;tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1yni];whichsayslimsupn!1hyn)]TJ/F26 11.955 Tf 11.955 0 Td[(y0;zni0:Buthyn;zni=hyn)]TJ/F26 11.955 Tf 11.955 0 Td[(y0;zni+hy0;zni:Thisyieldslimsupn!1hyn;znihy0;z0i::6Letv2DT,v02Tv.UsingthemonotonicityoftheoperatorT,wegethyn)]TJ/F26 11.955 Tf 11.955 0 Td[(v0;zn)]TJ/F26 11.955 Tf 11.955 0 Td[(vi0;orhyn;znihyn;vi+hv0;zni)-222(hv0;vi:Weconcludethatliminfn!1hyn;znihy0;vi+hv0;z0i)-222(hv0;vi::7:6and:7implyhy0)]TJ/F26 11.955 Tf 11.955 0 Td[(v0;z0)]TJ/F26 11.955 Tf 11.955 0 Td[(vi0:Sincethepointv;v02GrTisarbitrary,wehavez02DTandy02Tz0bythemaximalmonotonicityoftheoperatorT.Thus,wemaytakein:7v=z0toarriveatliminfn!1hyn;znihy0;z0i::89

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From:4,therstequalityin:5and1:8weobtainlimsupn!1hyn)]TJ/F26 11.955 Tf 11.955 0 Td[(y0;tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1yni0:UsingtheS+-propertyoftheoperatorJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1weobtainyn!y0.Passingtothelimitin:4andtakingintoconsiderationthaty02Tz0,wegetx0=z0+t0J)]TJ/F24 7.97 Tf 6.586 0 Td[(1y02T)]TJ/F24 7.97 Tf 6.587 0 Td[(1+t0J)]TJ/F24 7.97 Tf 6.586 0 Td[(1y0:Thus,y0=T)]TJ/F24 7.97 Tf 6.586 0 Td[(1+t0J)]TJ/F24 7.97 Tf 6.587 0 Td[(1)]TJ/F24 7.97 Tf 6.587 0 Td[(1x0=Tt0x0,andwearedone.Lemma2.2.3AssumethattheoperatorsT:XDT!2X;T0:XDT0!Xaremaximalmonotonewith02DTDT0and02TT0:Assumefurtherthat,T+T0ismaximalmonotone.Assumethatthereisapositivesequenceftngsuchthattn#0;asequencefxngDT0andasequencewn2T0xnsuchthatxn*x02XandTtnxn+wn*y02X:Thenthefollowingaretrue:itheinequalitylimn!1hTtnxn+wn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i<0isimpossible;iiiflimn!1hTtnxn+wn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=0;thenx02DT+T0andy02T+T0x0:ThefollowinglemmafollowsfromtheproofofalemmaofBrowderandHess[6,Proposition12].Lemma2.2.4AssumethatT:XDT!2Xismaximalmonotoneandbounded10

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i.e.ifMisaboundedsetinX;thentheset[x2DTMTxisbounded.ThenifMisaboundedsetinXthesetfTtx:t;x2;1DTMgisalsobounded.Proof.Fromtheproofofalemmaof[27],weseethatift;x2;1DTM;thenkTtxkkzkforanyz2Tx:Thus,ifK>0isanupperboundforthesetTDTM;wehavekTtxkK;x2DTM;aswell.ThenextlemmaispracticallycontainedintheproofofTheorem7ofBrowderandHessin[6].Wegivethefullproofhereforcompletenessandfuturereference.Lemma2.2.5LetT:XDT!2Xbemaximalmonotoneandsuchthat02intDTand02T:Letftng;1andfungXbesuchthatkunkS;hTtnun;uniS1;whereS;S1arepositiveconstants.ThenthereexistsanumberK>0suchthatkTtnunkKforalln=1;2;::::Proof.Lettheassumptionsofthelemmabesatised.Wesetwn=Ttnun=T)]TJ/F24 7.97 Tf 6.587 0 Td[(1+tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1)]TJ/F24 7.97 Tf 6.587 0 Td[(1un:11

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ThisimpliesthatT)]TJ/F24 7.97 Tf 6.587 0 Td[(1wn+tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1wn3unT)]TJ/F24 7.97 Tf 6.586 0 Td[(1wn3un)]TJ/F26 11.955 Tf 11.955 0 Td[(tnJ)]TJ/F24 7.97 Tf 6.587 0 Td[(1wn=un)]TJ/F26 11.955 Tf 11.955 0 Td[(tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1Ttnun=JtnunThenwehavewn2TJtnun:Ifweletxn=Jtnun;thenT)]TJ/F24 7.97 Tf 6.586 0 Td[(1wn3Jtnun=xn=un)]TJ/F26 11.955 Tf 11.955 0 Td[(tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1wn:ThisimpliestnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1wn=un)]TJ/F26 11.955 Tf 11.955 0 Td[(xnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1wn=1 tnun)]TJ/F26 11.955 Tf 11.955 0 Td[(xnwn=J1 tnun)]TJ/F26 11.955 Tf 11.955 0 Td[(xntnwn=Jun)]TJ/F26 11.955 Tf 11.955 0 Td[(xn:Thus,hwn;xni=hwn;un)]TJ/F26 11.955 Tf 11.955 0 Td[(tnJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1wni=hwn;uni)]TJ/F26 11.955 Tf 19.261 0 Td[(tnhwn;J)]TJ/F24 7.97 Tf 6.586 0 Td[(1wni=hwn;uni)]TJ/F26 11.955 Tf 19.261 0 Td[(tnkwnk2hTtnun;uniS1::9From.9,wealsoobtaintnkwnk2=hwn;uni)-222(hwn;xni:Since02Tandwn2Txn;wehavehwn;xni0;whichimpliestnkwnk2S1:Now,iffwngisunbounded,wemayassumethatkwnk!1andkwnkkwnk2for12

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alln:Thus,tnkwnkS1andtnkwnk=kJun)]TJ/F26 11.955 Tf 11.955 0 Td[(xnk=kun)]TJ/F26 11.955 Tf 11.955 0 Td[(xnkimpliesthatfxngisbounded.Wenotethatsince02intDT;theoperatorTisstronglyquasibounded.Thus,theboundednessoffxngandfhwn;xnigwn2Txnimply,bythestrongquasiboundednessofT;theboundednessoffwng;i.e.,acontra-diction.ItfollowsthatfTtnungisbounded.Lemma2.2.6AssumethattheoperatorCsatisesconditionsc1andc2.AssumethattheoperatorT:XDT!2Xismaximalmonotonesuchthat02DTand02T:Then,foreacht2;1;theoperatorTt+CsatisestheconditionS+0;L:Proof.Weknowthat,foreacht2;1;theoperatorTtisbounded,continuousandmaximalmonotone.ToshowthattheoperatorTt+CsatisestheconditionS+0;L;weassumethatfxngLissuchthatxn*x02X;limsupn!1hTtxn+Cxn;xni0;limn!1hTtxn+Cxn;yi=0:10foreveryy2LFn:SinceTtisbounded,wemayassumethatTtxn*v2X:Thus,fromthelastequalityof.10,alongwithhCxn;yi=hTtxn+Cxn;yi)-222(hTtxn;yi;y2LFn;wegetlimn!1hCxn+v;yi=0;y2LFn::11Now,weobservethathCxn+v;xni=hTtxn+Cxn;xni)-222(hTtxn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0i)-222(hTtxn;x0i+hv;xni;:1213

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which,inviewof1.10andthemonotonicityofTt;implieslimsupn!1hCxn+v;xnilimsupn!1hTtxn+Cxn;xni)]TJ/F19 11.955 Tf 19.261 0 Td[(liminfn!1hTtxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i)]TJ/F19 11.955 Tf 30.552 0 Td[(liminfn!1hTtxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0::13Wehaveusedthefactthatliminfn!1hTtxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0;:14whichisanimmediateconsequenceofhTtxn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0i=hTtxn)]TJ/F26 11.955 Tf 11.955 0 Td[(Ttx0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i+hTtx0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0ihTtx0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i:SinceCisoftypeS+L;.11and.13implyxn!x0;x02DC=DTt+CandCx0+v=0;orTtx0+Cx0=0:ItfollowsthattheoperatorTt+CsatisesconditionS+0;L:Wenowgiveatheoremthatwillallowustodeneourdegreefrom[16]ontheoper-atorTt+C:Theorem2.2.7AssumethattheoperatorsT;CareasinLemma1.2.6with02intDTandCsatisfyingconditionc3.LetGXbeopenandboundedandassumethatTx+Cx6=0;x2DT+C@G:Thenthereexistst0>0suchthatTtx+Cx6=0foranyt;x2;t0]L@G:14

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Proof.Assumethatourassertionisfalse.Thenthereexistsequencesftng;1;fxngL@Gsuchthattn#0;xn*x02XandTtnxn+Cxn=0:Sincefxngisbounded,weapplyconditionc3togethTtnxn;xni=hCxn;xnikxnkS;whereSisboundforthesequencefxng:Thus,Lemma1.2.5impliesfTtnxngisbounded,andwemayassumethatTtnxn*v0:Now,ifweassumethatliminfn!1hTtnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i<0;:15thenthereexistsasubsequenceoffxng;denotedagainbyfxng;suchthatlimn!1hTtnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=)]TJ/F26 11.955 Tf 9.299 0 Td[(q<0:ByiofLemma1.2.3,wehaveacontradiction.Itfollowsthatliminfn!1hTtnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0::16ThisandhCxn+v;xni=hTtnxn+Cxn;xni)-30(hTtnxn;xn)]TJ/F26 11.955 Tf 9.665 0 Td[(x0i)-30(hTtnxn;x0i+hv;xni:17implylimsupn!1hCxn+v;xni)]TJ/F19 11.955 Tf 31.88 0 Td[(liminfn!1hTtnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0::18Now,lety2LFn:Then,sinceCxn*)]TJ/F26 11.955 Tf 9.299 0 Td[(v;limn!1hCxn+v;yi=0::1915

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SincetheoperatorCisoftypeS+L;wehavexn!x02DCandCx0+v=0.Thus,sinceTtnxn*v;limn!1hTtnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=0:InvokingLemma1.2.3,weseenowthatx02DTandv=)]TJ/F26 11.955 Tf 9.298 0 Td[(Cx02Tx0;orTx0+Cx030:This,however,isacontradictiontoourboundaryconditionbecausex02DT+C@G:Inordertobeabletodeneournewdegreemapping,weneedtoshowthatthedegreedTt+C;G;0isconstantforallsucientlysmallvaluesoft:WeneedthefollowingthreedenitionsandTheoremAfrom[15].Denition2.2.8LetA:XDA!Xsatisfyconditionsc1;c2andassumethatAx6=0;x2DA@G;whereGXisopenandbounded.ThenthedegreedA;G;0isdenedbydA;G;0=limn!1degAn;Gn;0:20wheredegAn;Gn;0istheBrouwerdegreeofthenite-dimensionalmappingAn;denedbyAnx=nXi=1hAx;viivi;x2Fn;Gn=GFnandFn=spanfv1;v2;:::;vng:Remark2.2.9Aswecaneasilyseefromtheabovedenition,theonlypointsinDA@GthatmatterherearethepointsinFn@Gn:Thus,anyboundaryconditionofthetypex62DA@Gcanactuallybereplacedbytheconditionx62L@G:Inwhatfollows,thesymboldT;G;0denotesthedegreefunctiondenedin[15]foroperatorsofthetypeS+0;LandopenboundedsetsGX:LetGXbeopenandbounded.LetAt:XDAt!X;t2[0;1];beaone-parameterfamilyof16

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nonlinearoperators.WeassumethatthereexistsasubspaceLofXandasequencefFngasin1.2and1.3suchthat L=X;LfFngDAt;t2[0;1]::21Denition2.2.10WesaythatthefamilyfAtgsatisesConditionS+t0;Lifwhen-everfxngLfFng;ftng[0;1]aresuchthatxn*x0;tn!t0andlimsupn!1hAtnxn;xni0;limn!1hAtnxn;yi=0;:22forsomex02Xandanyy2LfFng;itfollowsthatxn!x0;x02DAt0andAt0x0=0:Denition2.2.11LetAi:XDAi!X;i=0;1;satisfytheconditionsc1;c2withacommonspaceL:TheoperatorsA;Aarecalledhomotopic"withrespecttotheboundedopensetGXifthereexistsaone-parameterfamilyofoperatorsAt:XDAt!X;t2[0;1];suchthat1.Ai=Ai;i=0;1;Atx6=0;t;x2[0;1]DAt@G;2.thefamilyfAtgsatisesConditionS+t0;LwithrespecttothespaceL;3.foreveryspaceFLfFngandeveryy2LfFngthemapping~aF;y:F[0;1]!Rdenedby~aF;yx;t=hAtx;yiiscontinuous.IfAtisreplacedbyAt)]TJ/F26 11.955 Tf 12.699 0 Td[(st;wheres:[0;1]!Xisacontinuouscurve,anditsatisesconditionS+t0;L;thenwesaythatfAtg;t2[0;1];isahomotopyofclassS+tL:Theorem2.2.12LetAi:XDAi!X;i=0;1;satisfytheconditionsc1,c2withacommonspaceL;andassumethattheoperatorsA;Aarehomotopic17

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withrespecttotheopenandboundedsetGX:ThendA;G;0=dA;G;0::23Theorem2.2.13AssumethatT:XDT!2Xismaximalmonotonewith02intDTand02T:AssumethattheoperatorC:XDC!Xsatisesconditionsc1)]TJ/F19 11.955 Tf 13.287 0 Td[(c3.LetGXbeopenandboundedandsuchthatTx+Cx6=0;x2DT+C@G:Lett0>0betheconstantofTheorem1.2.7.ThenthedegreedTt+C;G;0isconstantfort2;t0]:Proof.FollowingTheorem1.2.12,itsucestoshowthatforanytwonumberst1;t22;t0]wehavedTt1+C;G;0=dTt2+C;G;0::24Obviously,thedegreedTt;G;0iswelldenedbecause,accordingtoTheorem1.2.7seealsoRemark1.2.9,Ttx+Cx6=0foreveryx2L@G:Now,lett1;t22;t0]begivenwitht1
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whereS1isboundforthesequencefxng:UsingLemma1.2.5,weobtainthebound-ednessofthesequencefTstnxng:WemaythusassumethatTstnxn*v:Then,sincehTstnxn;yi!hv;yi;weobtainfromthelastof.25limn!1hCxn+v;yi=0::26WenowobservethathTstnxn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0i=hTstnxn)]TJ/F26 11.955 Tf 11.956 0 Td[(Tstnx0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i+hTstnx0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i:BythemonotonicityofTstnandLemma1.2.2,weobtainliminfn!1hTstnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0iliminfn!1hTstnx0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=limn!1hTstnx0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=hTs~t0x0;0i=0:WenextshowthatlimsuphCxn+v;xni0:WehavehCxn+v;xni=hCxn+Tstnxn;xni)-222(hTstnxn;xni+hv;xni=hCxn+Tstnxn;xni)-222(hTstnxn;xni+hTstnxn;x0ihTstnxn;x0i+hv;xni=hCxn+Tstnxn;xni)-222(hTstnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i)-222(hTstnxn;x0i+hv;xni:HencelimsuphCxn+v;xnilimsuphCxn+Tstnxn;xni)]TJ/F19 11.955 Tf 19.261 0 Td[(liminfhTstnxn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0i19

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hv;x0i+hv;x0i0:SinceCisoftypeS+L,weseethatxn!x02DC=DA~t0andCx0=)]TJ/F26 11.955 Tf 9.299 0 Td[(v:Thus,byLemma1.2.2,Tstnxn!Ts~t0x0andA~t0x0=Ts~t0x0+Cx0=0:Condition3ofDenition1.2.11istriviallysatisedbecausetheoperatort;x!TtxiscontinuousandCsatisesconditionc2,hence.24istrueandtheproofoftheS+t0;L-propertyoftheoperatorfamilyAtiscomplete.ThusbyTheorem1.2.12,wehavedA1;G;0=dA0;G;0;whichmeansthatdTs+C;G;0=dTs+C;G;0dTt1+C;G;0=dTt2+C;G;0andthiscompletestheproof.Wearenowreadyforthedenitionofthenewdegree.Denition2.2.14LettheassumptionsofTheorem1.2.13besatised.Thenthenewdegreemapping,degT+C;G;0;isdenedbydegT+C;G;0=dTt+C;G;0;t2;t0]:wheredisthedegreefordenselydenedS+0;Loperators.Ifp2Xissuchthatp62T+CDT+C@G;thendegT+C;G;p:=dTt+C)]TJ/F26 11.955 Tf 11.956 0 Td[(p;G;0:ItiseasytoseethatiftheoperatorCsatisestheconditionsc1)]TJ/F19 11.955 Tf 12.672 0 Td[(c3,thensodoestheoperatorCx)]TJ/F26 11.955 Tf 13.04 0 Td[(p:Infact,c2istriviallysatised.Thepropertyc120

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followsfromthefactthatsinceC)]TJ/F26 11.955 Tf 11.963 0 Td[(hisoftypeS+0;Lforallh2X;wehavethatC)]TJ/F26 11.955 Tf 12.109 0 Td[(p)]TJ/F26 11.955 Tf 12.109 0 Td[(h=C)]TJ/F19 11.955 Tf 12.109 0 Td[(p+hisalsooftypeS+0;Lforallh2X:Thepropertyc3followsfromhCx)]TJ/F26 11.955 Tf 11.955 0 Td[(p;xi)]TJ/F26 11.955 Tf 29.888 0 Td[(kxk)-222(kpkkxk)]TJ/F19 11.955 Tf 31.216 0 Td[(+kpkekxk;whereet=maxft;tg;t2R+:Inwhatfollows,wewillusethesymboldT+C;G;0insteadofdegT+C;G;0:WeshouldnoteherethatifDT=XandTisbounded,thenwedonotneedtheconditioninvolvingthe-functionontheoperatorC:Thisisbecause,accordingtoLemma1.2.4,iffxngisboundedandftng;1;thenfTtnxngisalsobounded.2.3PropertiesofthedegreemappingWeneedthefollowingcondition.ConditionT1T:XDT!2Xismaximalmonotonewith02intDTand02T:Theorem2.3.1AssumethattheoperatorTsatisesT1:LetGXbeopenandbounded.LetddenotethedegreemappingdenedinDenition1.2.14.Thenthefollowingstatementsaretrue.iIf02G;then,forevery>0;dT+J;G;0=1:If062J G;dJ;G;0=0:iiifp62T+CDT+C@GanddT+C;G;p6=0;thenthereexistsx2DT+CGsuchthatT+Cx3p;21

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iiiif02G;thenthedegreedHt;;G;0iswell-denedandinvariant,wherethehomotopyHisofthetypeHt;xtT+C1x+)]TJ/F26 11.955 Tf 11.955 0 Td[(tC2x;t2[0;1];:27providedthat062Ht;@G;t2[0;1]:Here,C1satisesc1)]TJ/F19 11.955 Tf 12.312 0 Td[(c3withthefunction1andC2:X!Xisbounded,demicontinuous,strictlymonotone,oftypeS+;andsatisesC2=0andhC2x;xi2kxk;x2DC2;where2:!R+!R+isstrictlyincreasing,continuousandsuchthat2=0:Inparticular,dT+C1;G;0=dC2;G;0=1:ivUndertheassumptionsonT;C1;C2andGiniii,thedegreedHt;;G;0iswell-denedandinvariant,wherethehomotopyHisofthetypeHt;xtT+C1x+C2x;t2[0;1];:28providedthat062Ht;@G;t2[0;1]:Inparticular,dT+C1+C2;G;0=dC2;G;0=1:vThedegreedHt;;G;0isinvariantunderhomotopiesofthetypeHt;xT+Cx)]TJ/F26 11.955 Tf 11.955 0 Td[(yt;t2[0;1];wherey:[0;1]!Xisacontinuouscurve.Here,062Ht;@G;t2[0;1]:viIfG1;G2areopenandsubsetsofGsuchthatG1G2=;and0=2T+CDT+C GnG1[G2;thendT+C;G;0=dT+C;G1;0+dT+C;G2;0:Proof.PropertyifollowsfromthefactthatdTt+J;G;0=1forallsucientlysmallt>0;becauseourdegreeforTt+JistheSkrypnikdegreefrom[28].Thisfollowsalsofromthemoregeneralhomotopyargumentaboutiiibelow.22

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Propertyiifollowsalsofromthefactthatifp62T+CDT+C@Gthenp62Tt+CL@Gforallsucientlysmallt>0;whichimplies,byouroriginaldegreetheory,thatp2RTt+Cforthesamevaluesoft:Toseethis,letxnbesuchasolution,xn2G.Thenwecanndtn2;1suchthatTtnxn+Cxn=p.Wemayassumethatxn*x0andtn#0.ThenhTtnxn;xni=hp;xni)-222(hCxn;xnikpkkxnk+kxnkkpk+S:=S1Bylemma1.2.5,Ttnxnisbounded.HenceweassumethatTtnxn*v,whichimpliesCxn=p)]TJ/F26 11.955 Tf 11.955 0 Td[(Ttnxn*p)]TJ/F26 11.955 Tf 11.955 0 Td[(v.Wearegoingtoshowthatlimn!1hTtnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=0:Lety2LFn.ThenhCxn+v;yi=hCxn+Ttnxn;yi)-222(hTtnxn;yi+hv;yi:Thereforelimn!1hCxn+v;yi=hp;yiorlimn!1hCxn+v)]TJ/F26 11.955 Tf 11.955 0 Td[(p;yi=0:NowhCxn+v)]TJ/F26 11.955 Tf 9.298 0 Td[(p;xni=hCxn+Ttnxn)]TJ/F26 11.955 Tf 9.298 0 Td[(p;xni+hv;xnihTtnxn;xn)]TJ/F26 11.955 Tf 9.299 0 Td[(x0ihTtnxn;x0i:SinceTtnismonotoneliminfn!1hTtnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0:23

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Hencelimsupn!1hCxn+v)]TJ/F26 11.955 Tf 11.955 0 Td[(p;xni0:SinceCisoftypeS+L,wehavethatxn!x0,x02 GandCx0=p)]TJ/F26 11.955 Tf 12.078 0 Td[(v.Bythestrongconvergenceofxnwehavethatlimn!1hTtnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=0:ByLemma1.2.3,x02DTandv2Tx0.Butp=2T+CDT+C@G,hencep=v+Cx02T+Cx0andthereforex02DT+CG.Wearegoingtoelaborateonpropertiesiiiandiv.Tothisend,weletrsttheassumptionsonT;C1;C2andGiniiibesatised.WeconsiderthehomotopymappingH1t;s;x=sTt+C1x+)]TJ/F26 11.955 Tf 11.955 0 Td[(sC2x;s2[0;1];andshowthatthereexistst0>0suchthattheequationH1t;s;x=0hasnosolutionx2L@Gforallsucientlysmallt2;t0]andalls2[0;1]:Letusassumethatthisisnottrue.Thenthereexistsequencesfsng[0;1];ftng;1andfxngL@Gsuchthatsn!s02[0;1];tn#0;xn*x02X;C2xn*h2andsnTtn+C1xn+)]TJ/F26 11.955 Tf 11.955 0 Td[(snC2xn=0::29Wenoticerstthatsn6=0becauseC2xn=0impliesxn=0=2@GbythestrictmonotonicityoftheoperatorC2:Weconsidertwocases:js0=0;jjs02;1]:Forthecasei,wedivide.29bysntogethTtnxn;xni=hC1xn;xni)]TJ/F31 11.955 Tf 19.261 16.856 Td[(1 sn)]TJ/F19 11.955 Tf 11.955 0 Td[(1hC2xn;xnihC1xn;xni1kxnk1S;:3024

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where1isthefunctionfromconditionc3fortheoperatorC1;andSisanupperboundforthesequencefxng:UsingLemma1.2.5andthestrongquasiboundednessoftheoperatorT;weseethatthesequencefTtnxngisbounded.WemaythusassumeTtnxn*v2X:Inaddition,from.30wealsoseethat)]TJ/F26 11.955 Tf 9.299 0 Td[(1S)]TJ/F26 11.955 Tf 21.918 0 Td[(1kxnkhC1xn;xni)]TJ/F31 11.955 Tf 30.552 16.857 Td[(1 sn)]TJ/F19 11.955 Tf 11.955 0 Td[(1hC2xn;xni)]TJ/F31 11.955 Tf 31.88 16.857 Td[(1 sn)]TJ/F19 11.955 Tf 11.956 0 Td[(12kxnk::31Sincethesequencefkxnkgisbounded,ithasasubsequence,denotedagainbyfkxnkg;whichconvergestoanumberq0:Ifq>0;then2kxnk!2q>0and)]TJ/F19 11.955 Tf 14.199 0 Td[(limn!11 sn)]TJ/F19 11.955 Tf 11.956 0 Td[(12kxnk=:However,thiscontradictswith.31becausetheright-handsideofitisboundedbelow.Consequently,kxnk!0:Thisisacontradictionagainbecauseitimpliesxn!x0=02G;whilexn2@G:Forthecasejj,lets02;1]:Again,wemaytakeTtnxn*v;which,alongwith.29,givesC1xn*)]TJ/F26 11.955 Tf 9.299 0 Td[(v)]TJ/F31 11.955 Tf 11.955 16.856 Td[(1 s0)]TJ/F19 11.955 Tf 11.956 0 Td[(1h2::32aFixingy2LFn,weseethat.32implieslimn!1C1xn+v+1 s0)]TJ/F19 11.955 Tf 11.955 0 Td[(1h2;y=0::32bWearegoingtoshowthatliminfn!1hsnTtnxn+)]TJ/F26 11.955 Tf 11.955 0 Td[(snC2xn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0::33Tothisend,werewrite.33asfollowsliminfn!1an+bn0;:3425

PAGE 33

wherebothsequencesan;bnareboundedandan=snhTtnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i;bn=)]TJ/F26 11.955 Tf 11.956 0 Td[(snhC2xn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0i::35Assumethat.34isnottrue.Thenthereexistsasubsequenceoffng;denotedbyfngagain,suchthatlimn!1an+bn<0::36Obviously,thisimpliesthatthereexistsafurthersubsequenceoffng;denotedagainbyfng;suchthatoneofthefollowingistrue:limn!1an=)]TJ/F26 11.955 Tf 9.299 0 Td[(q;limn!1bn=)]TJ/F26 11.955 Tf 9.299 0 Td[(q;:37whereqisapositiveconstant.Letusassumethattherstof.37istrue.Then,sincesn!s0;wehavelimn!1hTtnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=)]TJ/F26 11.955 Tf 12.808 8.088 Td[(q s0<0:However,thisisimpossiblebyiofLemma1.2.3.Letusassumethatthesecondof.37istrue.Then,again,limn!1f)]TJ/F26 11.955 Tf 11.955 0 Td[(snhC2xn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0ig<0::38Ifs0=1;then.38isimpossible,becausethefactortotherightof)]TJ/F26 11.955 Tf 11.876 0 Td[(sninitisbounded.Thus,s02;1:This,alongwith.38,sayslimn!1hC2xn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=)]TJ/F26 11.955 Tf 23.04 8.088 Td[(q 1)]TJ/F26 11.955 Tf 11.955 0 Td[(s0<0;:39whereqisapositiveconstant.BytheS+-propertyofC2;wehavexn!x0;whichcontradicts.39.Wehaveshownthat.33istrue.Naturally,sincesn!s0226

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;1];wealsohaveliminfn!1hTtnxn+1 sn)]TJ/F19 11.955 Tf 11.955 0 Td[(1C2xn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0::40Now,letu=v+1 s0)]TJ/F19 11.955 Tf 11.955 0 Td[(1h2:WehavehC1xn+u;xni=hTtnxn+C1xn+[)]TJ/F26 11.955 Tf 11.955 0 Td[(sn=sn]C2xn;xnihTtnxn+[)]TJ/F26 11.955 Tf 11.955 0 Td[(sn=sn]C2xn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0ihTtnxn+[)]TJ/F26 11.955 Tf 11.955 0 Td[(sn=sn]C2xn;x0i+hu;xni::41Werecallthatfxngsatisestheequation.29.Havingthisinmind,aswellas.40,wearriveatlimsupn!1hC1xn+u;xni)]TJ/F19 11.955 Tf 45.164 0 Td[(liminfn!1hTtnxn+1 sn)]TJ/F19 11.955 Tf 11.955 0 Td[(1C2xn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0ihu;x0i+hu;x0i=liminfn!1hTtnxn+1 sn)]TJ/F19 11.955 Tf 11.955 0 Td[(1C2xn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0::42Using.32b,.42andtheS+L-propertyoftheoperatorC1;weobtainxn!x02DC1andC1x0=)]TJ/F26 11.955 Tf 9.298 0 Td[(v)]TJ/F19 11.955 Tf 11.955 0 Td[([)]TJ/F26 11.955 Tf 11.955 0 Td[(s0=s0]h2:Sincelimn!1hTtnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=0;Lemma1.2.3saysthatx02DTandv2Tx0:SinceC2xn!C2x0=h2;wehaves0Tx0+Cx0+)]TJ/F26 11.955 Tf 12.051 0 Td[(s0C2x030;i.e.,acontradictionwithourboundarycondition0=2Ht;@G;t2[0;1]:Itfollowsthatthereexistst0>0suchthattheequationH1t;s;x=0hasnosolutionx2@Gforanyt;s2;t0][0;1].WemustnowshowthatthedegreedH1t;s;;G;0isindependentofs2[0;1]27

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foraxedt2;t0]:Tothisend,wext2;t0]andshowthatthemappingAsx:=H1t;s;xsatisestheconditionsS+s0;Landc3.Thelatterisobviouslytruebecausetheoperatorx!Ttxiscontinuous,theoperatorC2isdemicontinuous,andtheoperatorC1satisesc2.ToshowthatH1t;s;xisoftypeS+s0;L;letusassumethatforasequencefsng2[0;1],asequencexn2L;andanyfunctionaly2LFn,wehavesn!s02[0;1];xn*x0,Ttxn*v,C2xn*h2,limsupn!1hAsnxn;xni=limsupn!1hsnTtxn+C1xn+)]TJ/F26 11.955 Tf 11.955 0 Td[(snC2xn;xni0andlimn!1hAsnxn;yi=limn!1hsnTtxn+C1xn+)]TJ/F26 11.955 Tf 11.955 0 Td[(snC2xn;yi=0::43Letasassumethats0=0:Ifthereexistsasubsequencefsnkgsuchthatsnk=0;k=1;2;:::;then0limsupk!12kxnkklimsupk!1hC2xnk;xnki01:44implieskxnkk!0;i.e.xnk!0:Thissaysx0=02X=DAs0andAs0x0=0:Wemaythusassumethatsn>0;n=1;2;::::Usingtherstof.43wenowseethatsnhC1xn;xni=hAsnxn;xni)]TJ/F26 11.955 Tf 19.261 0 Td[(snhTtxn;xni)]TJ/F19 11.955 Tf 19.261 0 Td[()]TJ/F26 11.955 Tf 11.955 0 Td[(snhC2xn;xnihAsnxn;xni)]TJ/F19 11.955 Tf 19.261 0 Td[()]TJ/F26 11.955 Tf 11.955 0 Td[(sn2kxnk::45Sincethesequencefkxnkgisbounded,ithasasubsequencethatconvergestoanumberq0:Ifq>0;thendenotingthissubsequencebyfkxnkgweobtainfrom.450=)]TJ/F19 11.955 Tf 14.2 0 Td[(limn!1sn1S)]TJ/F19 11.955 Tf 26.818 0 Td[(limn!1sn1kxnklimsupn!1snhC1xn;xni)]TJ/F19 11.955 Tf 33.461 0 Td[(limn!12kxnk=)]TJ/F26 11.955 Tf 9.298 0 Td[(2q<0;wherekxnkSforalln:Consequently,wemusthavekxnk!0;whichimpliesxn!x0=02DAs0=XandAs0x0=0:28

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Wenowhavetoconsiderthecases02;1]:Werstnotethat1.43impliesthefollowingtworelationslimsupn!1hTtxn+C1xn+1 sn)]TJ/F19 11.955 Tf 11.955 0 Td[(1C2xn;xni0andlimn!1hTtxn+C1xn+1 sn)]TJ/F19 11.955 Tf 11.955 0 Td[(1C2xn;yi=0::46Fromthesecondof.46weget,forally2LFn;limn!1hC1xn+v+~sh2;yi=0;:47where~s=)]TJ/F26 11.955 Tf 11.955 0 Td[(s0=s0:Weneedtoshowthatlimsupn!1hC1xn+v+~sh2;xni0::48Weset~sn=)]TJ/F26 11.955 Tf 11.955 0 Td[(sn=snandobservethatliminfn!1hTtxn+~snC2xn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0::49Infact,thisfollowsfromhTt+~snC2xn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=hTt+~snC2xn)]TJ/F19 11.955 Tf 11.955 0 Td[(Tt+~snC2x0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i+hTt+~snC2x0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0ihTt+~snC2x0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i:Consequently,hC1xn+v+~sh2;xni=hTtxn+C1xn+~snC2xn;xni)-222(hTtxn+~snC2xn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0ihTtxn+~snC2xn;x0i+hv+~sh2;x0i::5029

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Again,limsupn!1hC1xn+v+~sh2;xni)]TJ/F19 11.955 Tf 31.88 0 Td[(liminfn!1hTtxn+~snC2xn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0;thus,.48istrue.From1.47,.48andtheS+L-propertyofC1;weobtainxn!x02DC1;Ttxn!Ttx0andC2xn*C2x0:SinceC1x0+v+~sh2=0;wehaves0Ttx0+C1x0+)]TJ/F26 11.955 Tf 11.955 0 Td[(s0C2x0=0;whichconcludestheproofofthefactthatthefamilyofoperatorsAs;s2[0;1];satisestheconditionS+s0;L:Since,inourcase,AandAmaybereplacedbyanytwooperatorsAs0;As1fromthefamilyAs;s2[0;1];.23saysdH1t;s;;G;0=const:;s2[0;1]:Now,lets1;s22[0;1]withs16=s2andt12;t0]begiven.Then,fromwhatwehaveshownabove,dHs1;;G;0=dH1t1;s1;;G;0=dH1t1;s2;;G;0=dHs2;;G;0:Inparticular,dT+C1;G;0=dH;;G;0=dH;;G;0=dC2;G;0=1:ThelastequalityfollowsfromthefactthatthedegreedC2;G;0istheclassicalSkrypnikdegree[28]whichequals1cf.Browder[4,Theorem3,iv].Propertyvfollowseasilyfromthedenitionofthedegreemapping.Toprovepropertyviwenotethatif0=2T+C)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(DT+C GnG1[G2then0=2Tt+C)]TJ/F19 11.955 Tf 5.48 -9.684 Td[(DT+C GnG1[G2:Hencefort2[0;t0andbydenition30

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ofourdegreewehavethatdT+C;G;0=dTt+C;G;0=dTt+C;G1;0+dTt+C;G2;0=dT+C;G1;0+dT+C;G2;0:2.4ApplicationsinNonlinearAnalysisExampleAsanexampleweconsiderthefollowingoperators.LetbeaboundedopensubsetofRnwithboundary@belongingtoC2;,forsome>0.Fori=0;1;:::;n,letai:R!R,besuchthat,aix;uismeasurablew.r.tx2forallu2R,andcontinuousw.r.tu2Rforalmostallx2.Wealsoassumethatfori=0;1;:::;n,thefollowinginequalitiesaresatised.jaix;uj1;i=1;2;:::;nja0x;uj1juj+axwhere,1isapositiveconstantanda2L2.WithX=L2,wedenetheoperatorsSandCasfollows.Su=u;DS=W2;2W1;20;andCu=nXi=1aix;u@u @xi+a0x;u;DC=W1;2:NowconsiderrstaclosedandconvexsetKX:Let'K:X!R+[f1gbedenedby'Kx=8<:0;ifx2K;1;otherwise.31

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Thefunction'Kisproper,convexandlowersemicontinuousonX:Foranyx2K,thesubdierentialof'withrespecttotheconvexsetKisdenedas@'Kx=fx:hx;y)]TJ/F26 11.955 Tf 11.956 0 Td[(xi0;y2Kg:Also,8<:D@'K=Kand02@'Kx;x2K@'Kx=f0g;x2intK:Theoperator@'K:X!2Xismaximalmonotoneand02intD@'Kand02@'KwithintK6=;.IfweletK= Brthen@' Bru=8>>><>>>:0;kukr.ThenTu=)]TJ/F26 11.955 Tf 9.298 0 Td[(Su+@' Bruismaximalmonotonewith02intDT=intD@'whichisanontrivialexampleofanoperatorTthatcanbecoveredbyourpresenttheory.Also,bytheconditionsonaix;u,i=0;1;:::;n,denedbytheaboveinequalities,itiseasilyveriedthattheoperatorCsatisesconditionc1)]TJ/F26 11.955 Tf 12.51 0 Td[(c3.Wecheckc2andc3below.BythedenitionoftheoperatorC,wehavehCu;ui=ZnXi=1aix;uu@u @xidx+Za0x;uudx:HencejhCu;uij1nXi=1kukL2k@u @xikL2+1kuk2L2+kakL2kukL2:32

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SincekukW1;2=Xjj1kDuk2L21 2=kuk2L2+nXi=1k@u @xik2L21 2;itfollowsthatkukL2kukW1;2andnXi=1k@u @xikL2kukW1;2:HencewehavejhCu;uij21kuk2W1;2+kakL2kukW1;2:Let:R+!R+bedenedbyr=21r2+r,where2R+.Thenisnondecreasing.ItfollowsfromthathCu;ui)]TJ/F26 11.955 Tf 29.888 0 Td[(kukW1;2:Nextweshowthatc2isalsosatised.WechooseL=f2C1:kkW1;2<1g.Then,LisadenselinearsubspaceofW1;2.Lety2LandF2FL.WeshowthatthemappingaF;y:F!RdenedbyaF;yu=hCu;yiiscontinuous.Letuk!uinW1;2ask!1.Thenkuk)]TJ/F26 11.955 Tf 11.955 0 Td[(ukW1;2=Xjj1kDuk)]TJ/F26 11.955 Tf 11.955 0 Td[(uk2L21 2!0=kuk)]TJ/F26 11.955 Tf 11.956 0 Td[(uk2L2+nXi=1k@uk)]TJ/F26 11.955 Tf 11.955 0 Td[(u @xik2L21 2!0:Hence@uk @xi!@u @xiinL2forall1inandkuk)]TJ/F26 11.955 Tf 12.172 0 Td[(ukL2!0:Thereforethereexistsasubsequenceoffukg,stilldenotedbyfukgandafunctionv2L2such33

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thatukx!uxa.e.inandjukxjjvxja.e.inforallk1.Sincetheaix;uarecontinuousinuforalli=0;1;:::;n;wehaveaix;ukx!ax;uxa.e.infori=0;1;:::;n:Hence,forasubsequence,wehavejhCuk)]TJ/F26 11.955 Tf 11.955 0 Td[(Cu;yijnXi=1Zjaix;ukxjj@uk @xi)]TJ/F26 11.955 Tf 14.836 8.088 Td[(@u @xijjyjdx+nXi=1Zjaix;ukx)]TJ/F26 11.955 Tf 11.955 0 Td[(aix;uxjjyjj@u @xijdx+Zja0x;ukx)]TJ/F26 11.955 Tf 11.955 0 Td[(a0x;uxjjyjdx:ThisimpliesjhCuk)]TJ/F26 11.955 Tf 11.955 0 Td[(Cu;yij1nXi=1Zk@uk @xi)]TJ/F26 11.955 Tf 14.836 8.088 Td[(@u @xikL2kykL2dx+nXi=1Zjaix;ukx)]TJ/F26 11.955 Tf 11.955 0 Td[(aix;uxjjyjj@u @xijdx+Zja0x;ukx)]TJ/F26 11.955 Tf 11.955 0 Td[(a0x;uxjjyjdx:Fori=0;1;:::;n;wealsohavejaix;uk)]TJ/F26 11.955 Tf 11.955 0 Td[(aix;uj21andja0x;uk)]TJ/F26 11.955 Tf 11.955 0 Td[(a0x;uj21jvj+ax:Therefore,bythedominatedconvergencetheorem,wehaveZjaix;ukx)]TJ/F26 11.955 Tf 11.955 0 Td[(aix;uxj2dx!0foralli=0;1;::;n.Since,also,k@uk @xi)]TJ/F27 7.97 Tf 14.951 4.707 Td[(@u @xikL2!0andjhCuk)]TJ/F26 11.955 Tf 12.243 0 Td[(Cu;yij!0,theresultsfollows.Inwhatfollows,wegiverstgiveanabstractresultinvolvingtheexistenceofzeros.34

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Theorem2.4.1ExistenceofZerosAssumethatGXisopenandboundedwith02G:AssumethatT:XDT!2XsatisesconditionT1.AssumethatC:XDC!Xsatisesconditionsc1)]TJ/F26 11.955 Tf 11.956 0 Td[(c3.AssumethatTx+Cx+Jx630;;x2;DT+C@G;:51foraxedconstantsuchthat>Q2 Q21:=Q;:52whereQ1=infx2@Gfkxkg;Q2=supx2@Gfkxkg::53ThentheinclusionTx+Cx30:54hasasolutionx2DT+C G.If.51holdsalsowith=0;thenx2DT+CG:Proof.Supposetheconclusionofthetheoremisfalse.Thatis,Tx+Cx63055aforx2DT+C G.Weshowrstthatthereexists"0>0suchthatTx+Cx+Jx630;;x2;0]DT+C@G::55bInfact,ifthisisnottrue,thenthereexistsequencesf"ng;1;fxng@G;vn2Txnsuchthatn#0;xn*x02Xandvn+Cxn+nJxn=0;:5635

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whichimplieslimsupn!1hvn+Cxn+nJxn;xni=0:Thenhvn;xni=hCxn;xni)]TJ/F26 11.955 Tf 19.261 0 Td[(nhJxn;xnikxnk)]TJ/F26 11.955 Tf 11.955 0 Td[(nkxnk2S;whereSistheboundedforthesequencefxng.FromthequasiboundednessofTandconditionc3wegettheboundednessoffvng:Assumingthatvn*v;wealsogetCxn*)]TJ/F26 11.955 Tf 9.299 0 Td[(vandhCxn+v;yi!0;wherey2LFn.Ontheotherhand,letv02Tx0.Thenhvn+nJxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=hvn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i+nhJxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=hvn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i+hv0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i+nhJxn)]TJ/F26 11.955 Tf 11.955 0 Td[(Jx0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i+nhJx0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i:Hence,bythemonotonicityofTandJ,wehaveliminfn!1hvn+nJxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0:Therefore,wehavehCxn+v;xni=hvn+Cxn+nJxn;xni)-136(hvn+nJxn;xn)]TJ/F26 11.955 Tf 10.927 0 Td[(x0i)-136(hvn)]TJ/F26 11.955 Tf 10.928 0 Td[(v+nJxn;x0i;andthefactthatnkJxnk!0asn!1,implieslimsupn!1hCxn+v;xni0:ConditionS+LonCimpliesthatxn!x0andCx0=)]TJ/F26 11.955 Tf 9.298 0 Td[(v:FromLemma1.1.7,36

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wegetx02DTandTx0+Cx030;i.e.,acontradictionto.55abecausex02DT+C@G:Therefore1.55bistrue.Wepick0furthersothatQ+"0<;x2;0];andconsiderthehomotopyinclusiontTx+Cx+Jx+)]TJ/F26 11.955 Tf 11.955 0 Td[(tJx30::57Wearegoingtoshowthat.57hasnosolutionx2DT+C@Gforanyt2[0;1]:Tothisend,assumethatthecontraryistrue.Thenthereexistsequencesftng[0;1];fxngDT+C@G;vn2Txnsuchthattn!t02[0;1];xn*x02X;Jxn*j2Xandtnvn+Cxn+Jxn+)]TJ/F26 11.955 Tf 11.955 0 Td[(tnJxn=0::58Bywhatwehavejustshownabove,tn6=1:Also,tn6=0because02G;J=0andJisinjective.Itfollowsthattn>0:Now,foralln1,kxnksupkxnkkxnksupkxnk=Q2:Also,fromc3,themonotonicityofTandthefactthat02T,wehave)]TJ/F26 11.955 Tf 9.299 0 Td[(kxnkhCxn;xnihCxn;xni+hvn;xni=hvn+Cxn;xni:From.58andabovewehave)]TJ/F26 11.955 Tf 9.299 0 Td[(Q2)]TJ/F26 11.955 Tf 28.56 0 Td[(kxnkhvn+Cxn;xni=)]TJ/F26 11.955 Tf 9.299 0 Td[("kxk2)]TJ/F19 11.955 Tf 11.955 0 Td[([)]TJ/F26 11.955 Tf 11.955 0 Td[(tn=tn]kxnk237

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<)]TJ/F19 11.955 Tf 9.299 0 Td[([)]TJ/F26 11.955 Tf 11.955 0 Td[(tn=tn]kxnk2)]TJ/F19 11.955 Tf 21.918 0 Td[([)]TJ/F26 11.955 Tf 11.955 0 Td[(tn=tn]Q21;:59and1)]TJ/F26 11.955 Tf 11.956 0 Td[(tn tn1 Q+1:Itfollowsthatt02;1]:From.58,wehavehvn;xni=hCxn;xni)]TJ/F26 11.955 Tf 19.261 0 Td[(kxnk2)]TJ/F19 11.955 Tf 13.151 8.088 Td[(1)]TJ/F26 11.955 Tf 11.955 0 Td[(tn tnkxnk2kxnk)]TJ/F31 11.955 Tf 11.955 16.857 Td[(+1)]TJ/F26 11.955 Tf 11.955 0 Td[(tn tnkxnk2S;whereSistheboundforfxng,whichsaysthatfvngisbounded.Letusassumethatvn*v:Then1.58saysthatCxn*)]TJ/F26 11.955 Tf 9.299 0 Td[(v)]TJ/F19 11.955 Tf 11.955 0 Td[(+~t0j;where~t0=[)]TJ/F26 11.955 Tf 11.955 0 Td[(t0=t0]:Consequently,limn!1hCxn+v++~t0j;yi=0foreveryy2LFn:Itisnoweasytoobtain,aswedidseveraltimesbefore,thatlimsupn!1hCxn+v++~t0j;xni0:WeonlynoteherethatwehaveusedLemma1.2.3i,toconcludethatliminfn!1hvn++~t0j;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0i0:BytheS+L-propertyofC;weobtainxn!x02DC@GandCx0=)]TJ/F26 11.955 Tf 9.299 0 Td[(v)]TJ/F19 11.955 Tf 11.675 0 Td[(+38

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~t0Jx0:Lemma1.1.7saysthatx02DTandTx0+Cx0++~t0Jx030,i.e,acontradictiontoourassumedhypothesis.51because0<+~t00+~t00+Q<:ItiseasytoseethattheoperatorT+"JsatisesconditionT1.ItisalsoeasytoseethattheoperatorJsatisestheconditionsonC2ofTheorem1.3.1,iii.Thus,accordingtoiiiofTheorem1.3.1,themappingHt;x:=tTx+Cx+"Jx+)]TJ/F26 11.955 Tf 11.955 0 Td[(tJxisanadmissiblehomotopyforourdegree.ItfollowsthatdH;;G;0=dT+C+J;G;0=dH;;G;0=dJ;G;0=1:ThissaysthattheinclusionTx+Cx+Jx30hasasolutionxforevery>0:Welet=n=1=nandxn=xn:Thenwehave,forsomevn2Txn;vn+Cxn+=nJxn=0:Wemayassumethatxn*x02X:LetSbetheboundforfxng,thenhvn;xni=hCxn;xni)]TJ/F19 11.955 Tf 21.024 8.088 Td[(1 nkxnk2kxnkS:BythequasiboundednessofT,weobtainthatfvngisbounded.Letvn*v:Then,Cxn*)]TJ/F26 11.955 Tf 9.299 0 Td[(vandhCxn+v;yi!0:But1 nJxn!0asn!1.Thereforelimn!1vn+Cxn=0:39

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Wearegoingtoshowthatlimsupn!1hCxn+v;xni0:ObviouslyhCxn+v;xni=hvn+Cxn;xni)-222(hvn;xni+hv;xni)-222(hvn;x0i+hvn;x0i=hvn+Cxn;xni)-222(hvn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i+hv;xni)-222(hvn;x0i:Letu02Tx0.ThensinceTismaximalmonotonewehavehvn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0ihu0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i:Thereforeliminfn!1hvn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0iliminfn!1hu0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=0:Hencelimsupn!1hCxn+v;xnilimsupn!1hvn+Cxn;xni)]TJ/F19 11.955 Tf 13.948 0 Td[(liminfn!1hvn;xn)]TJ/F26 11.955 Tf 9.298 0 Td[(x0i+hv;x0ihv;x0i0and,consequently,xn!x02DCandCx0=)]TJ/F26 11.955 Tf 9.299 0 Td[(v:Onceagain,byLemma1.1.7,x02DT,Tx0+Cx030andx02 G,andtheproofiscomplete.Ifweassumethat.51isalsotruewith=0;then,obviously,x02G.Interestinglyenough,itturnsoutthattheboundarycondition1.51,for=+1;isthesameasTx+Cx+@' Brxnf0g630;x2@Br::6040

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Infact,asKenmochinotesin[22],@' Brx=8>>><>>>:0;kxkr.ThefollowingcorollarytoTheorem1.5.1containsasurjectivityresult.Corollary2.4.2AssumethattheoperatorT:XDT!2XsatisesT1.AssumethatC:XDC!Xsatisesconditionsc1)]TJ/F19 11.955 Tf 12.206 0 Td[(c3.AssumethatforanopenboundedsetGXcontainingzeroandsomef2Xwehavehw+Cx)]TJ/F26 11.955 Tf 11.956 0 Td[(f;xi0;x;w2DT+C@GTx::61Thenthereexistsasolutionx2DT+CGoftheinclusionTx+Cx3f::62If.61isreplacedbylimkxk!1x2DT+C@Gw2Txhw+Cx;xi kxk=+1;:63thentheoperatorT+Cissurjective.Proof.Werstnotethatunderthecondition.61wehavehw+Cx)]TJ/F26 11.955 Tf 11.955 0 Td[(f+Jx;xihJx;xi=kxk2>0forany>0;anyx2DT+C@Gandanyw2Tx.Consequently,.51istruefor=+1andC)]TJ/F26 11.955 Tf 12.318 0 Td[(finplaceoftheoperatorC:Aswehavenotedearlier,theoperatorx!Cx)]TJ/F26 11.955 Tf 11.027 0 Td[(fsatisestheconditionsc1)]TJ/F26 11.955 Tf 11.027 0 Td[(c3foranyf2X:OurconclusionfollowsinthiscasefromTheorem1.5.1.41

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ToshowthatT+Cissurjectiveunder.63,wexf2X:Then,by.63,thereexistsaballG=BrXsuchthathw+Cx;xi kxk>kfk;x;w2DT+C@GTx::64Consequently,forthesamex;wasaboveandall0hw+Cx)]TJ/F26 11.955 Tf 11.955 0 Td[(f+Jx;xihw+Cx;xi)-222(kfkkxk=kxkhw+Cx;xi kxk)-222(kfk>0:Thus,again,1.51istrueforall0andTheorem1.5.1appliestoobtain.62forsomex2DT+CBr.2.5FurtherApplicationsInwhatfollowsdenoteJthedualitymappingwiththegaugefunction.Thefunc-tion:R+!R+iscontinuous,strictlyincreasingandsuchthat=0andr!1asr!1.ThemappingJiscontinuous,bounded,surjective,strictlyandmaximalmonotoneandsatisesconditionS+.AlsohJx;xi=kxkkxkandkJxk=kxk.IfGisanopenboundedsubsetinXcontainingzero,thendJ;G;0=1.ThefollowingpropositionshowshowwecansolveimportantapproximateproblemfortheoperatorT+C.Thisapproximateproblem,inclusion1.65belowcanbeusedinavarietyofproblemsinnonlinearanalysiswhichincludesproblemsofsolvability,existenceofeigenvalues,rangesofsums,invarianceofdomainandbifurcation.Inwhatfollows,weassumethatTsatisestheconditionT1andCsatisescondi-tionsc1)]TJ/F26 11.955 Tf 11.955 0 Td[(c3.Proposition2.5.1AssumetheoperatorTsatisestheconditionT1andCsatises42

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conditionsc1)]TJ/F26 11.955 Tf 12.215 0 Td[(c3.LetGbeanopenandboundedsubsetofXwith02G.AssumealsothatHt;:@G63p,t2[0;1],whereHt;x=tT+C)]TJ/F26 11.955 Tf 11.955 0 Td[(p+Jx+)]TJ/F26 11.955 Tf 11.955 0 Td[(tJx;withp2Xxedandapositiveconstant.ThenthedegreedHt;:;G;0iswelldenedandconstantforallt2[0;1].Inparticular,theinclusionTx+Cx+Jx3p:65issolvableinG.Proof.Theconclusionofthispropositionfollowsfromi-iiiofTheorem1.3.1.Infact,onemaytakehereC1=C)]TJ/F26 11.955 Tf 12.229 0 Td[(p+JandC2=J.WewillshowinthenexttheoremthatC1satisesc1)]TJ/F26 11.955 Tf 10.244 0 Td[(c3andJsatisestheconditionsofC2inTheorem1.3.1.ThenthehomotopyinvarianceiniiiinTheorem1.3.1saysthatdHt;:;G;0isconstantforallt2[0;1].HenceitfollowsthatdT+C+J;G;0=dJ;G;0=1;andbyiiofTheorem1.3.1implies:65.Theorem2.5.2SurjectivityLetTsatisfyT1.LetC:XDC!Xsatisfyc1)]TJ/F26 11.955 Tf 12.139 0 Td[(c3.AssumethatthereisaconstantQ>0and:[Q;1!R+,withr!0asr!1,suchthatforeveryx2DTDCwithkxkQandeveryu2Txwehavehu+Cx;xi)]TJ/F26 11.955 Tf 29.888 0 Td[(kxkkxkkxk;:66whereisagaugefunction.Then,forevery>0,RT+C+J=X.43

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If,inaddition,liminfkxk!1jTx+Cxj kxk:x2DTDC>0;:67thenRT+C=X.Proof.Wexp2X,>0,andconsidertheproblemHt;x=tT+C)]TJ/F26 11.955 Tf 11.955 0 Td[(p+Jx+)]TJ/F26 11.955 Tf 11.955 0 Td[(tJx30;t2[0;1];andapplyiiiTheorem1.3.1.Tothisend,weneedtoshowthattheoperatorU=C+J)]TJ/F26 11.955 Tf 12.013 0 Td[(psatisesc1)]TJ/F26 11.955 Tf 12.013 0 Td[(c3andtheoperatorJsatisestheconditionsonC2asinTheorem1.3.1.Jiscontinuous,bounded,surjective,strictlyandmaximalmonotoneandsatisesconditionS+.WealsohaveJ=0.Now,hJx;xi=kxkkxk:Ifwetake2r=rrasiniiiTheorem1.3.1,thenhJx;xi2kxk,2:R+!R+isstrictlyincreasing,continuousandsuchthat2=0.ToshowthatUsatisesc1,letLDC, L=XandsuchthatCsatisesc1)]TJ/F26 11.955 Tf 12.153 0 Td[(c3.LetfFngbeasequencesatisfying:2,fxngLsuchthatxn*x0forsomex02X.FixhinXandletlimsupn!1hUxn)]TJ/F26 11.955 Tf 11.955 0 Td[(h;xni0;limn!1hUxn)]TJ/F26 11.955 Tf 11.955 0 Td[(h;yi=0foranyy2LFn.Wearegoingtoshowthatxn!x0,x02DU=DCandUx0=h.BydenitionofU,wehaveCxn)]TJ/F26 11.955 Tf 11.956 0 Td[(p)]TJ/F26 11.955 Tf 11.955 0 Td[(h=Uxn)]TJ/F26 11.955 Tf 11.955 0 Td[(h)]TJ/F26 11.955 Tf 11.955 0 Td[(Jxn:HencehCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p)]TJ/F26 11.955 Tf 11.955 0 Td[(h;yi=hUxn)]TJ/F26 11.955 Tf 11.955 0 Td[(h;yi)-222(hJxn;yi:44

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SinceJisbounded,weassumethatJxn*j.Thenlimn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p)]TJ/F26 11.955 Tf 11.955 0 Td[(h;yi=hj;yi:Henceweconcludethatlimn!1hCxn)]TJ/F26 11.955 Tf 11.956 0 Td[(p)]TJ/F26 11.955 Tf 11.955 0 Td[(h+j;yi=0:Now,hCxn)]TJ/F26 11.955 Tf 11.956 0 Td[(p)]TJ/F26 11.955 Tf 11.955 0 Td[(h+j;xni=hUxn)]TJ/F26 11.955 Tf 11.955 0 Td[(h;xni)]TJ/F26 11.955 Tf 19.261 0 Td[(hJxn;xni+hj;xni=hUxn)]TJ/F26 11.955 Tf 11.955 0 Td[(h;xni)]TJ/F26 11.955 Tf 19.261 0 Td[(hJxn;xni+hj;xni)]TJ/F26 11.955 Tf 19.261 0 Td[(hJxn;x0i+hJxn;x0i=hUxn)]TJ/F26 11.955 Tf 11.956 0 Td[(h;xni)]TJ/F26 11.955 Tf 19.261 0 Td[(hJxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i)]TJ/F26 11.955 Tf 19.261 0 Td[(hJxn;x0i+hj;xni:ButhJxn)]TJ/F26 11.955 Tf 11.955 0 Td[(Jx0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0sinceJismonotone.Henceliminfn!1hJxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0iliminfn!1hJx0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=0:Thereforelimsupn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p)]TJ/F26 11.955 Tf 11.955 0 Td[(h+j;xnilimsupn!1hUxn)]TJ/F26 11.955 Tf 11.955 0 Td[(h;xni)]TJ/F26 11.955 Tf 19.261 0 Td[(liminfn!1hJxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0:SinceCisoftypeS+L,wehavexn!x0,x02DC=DC)]TJ/F26 11.955 Tf 10.474 0 Td[(p+J=DU,andCx0)]TJ/F26 11.955 Tf 11.955 0 Td[(p)]TJ/F26 11.955 Tf 11.955 0 Td[(h)]TJ/F26 11.955 Tf 11.955 0 Td[(j=0:45

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SinceJiscontinuous,wehaveJxn!Jx0=jbytheuniquenessoflimits.ThereforeUx0=Cx0)]TJ/F26 11.955 Tf 11.955 0 Td[(p)]TJ/F26 11.955 Tf 11.955 0 Td[(Jx0=h:Weshowthatc2issatised.Tothisend,letF2FL,y2L.WeshowthatthemappingbF;y:F!RdenedbybF;yx=hCx)]TJ/F26 11.955 Tf 11.956 0 Td[(p+Jx;yi;iscontinuous.LetfxngFsuchthatxn!x0.ThensinceJiscontinuouswehaveJxn!Jx0.SinceaF;yx=hCx;yiiscontinuouswehavethathCxn)]TJ/F26 11.955 Tf 12.174 0 Td[(p;yi!hCx)]TJ/F26 11.955 Tf 12.173 0 Td[(p;yi.ThereforebF;yxn=hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p+Jxn;yi!hCx)]TJ/F26 11.955 Tf 11.955 0 Td[(p+Jx;yi=bF;yx:FinallyhCx)]TJ/F26 11.955 Tf 11.956 0 Td[(p+Jx;xi=hCx;xi+hJx;xi)-222(hp;xi)]TJ/F26 11.955 Tf 28.559 0 Td[(1kxk+kxkkxk)-222(kpkkxk)]TJ/F26 11.955 Tf 28.559 0 Td[(1kxk)-222(kpkkxk=)]TJ/F19 11.955 Tf 9.298 0 Td[(1kxk+kpkkxk:Letr=1r+r0;r2R+:Sinceisnondecreasing,isalsonondecreasing.HenceUsatisesc1)]TJ/F26 11.955 Tf 11.955 0 Td[(c3.Wenextshowthatallthesolutionsofareboundedbyaconstantwhichisinde-pendentoft2[0;1].Tothisend,assumethatthereisasequenceftmg[0;1],andasequencefxmgDHsuchthatkxmk!1asm!1.46

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Ifthereexistasubsequenceftmkgofftmgsuchthattmk=0,k=1,2,...,thenxmk=0forallk,whichcontradictskxmkk!1ask!1.Wemayassumethattm>0,m=1,2,...,.ThenDH=DTDCandtmTxm+Cxm)]TJ/F26 11.955 Tf 11.956 0 Td[(p+Jxm+)]TJ/F26 11.955 Tf 11.955 0 Td[(tJxm30;or,forsomeum2Txm,tmum+Cxm)]TJ/F26 11.955 Tf 11.955 0 Td[(p+[1)]TJ/F26 11.955 Tf 11.955 0 Td[(tm)]TJ/F26 11.955 Tf 11.955 0 Td[(]Jxm=0:Bythehypothesis,andassumingthatkxmkQforallm,wendhum+Cxm)]TJ/F26 11.955 Tf 11.955 0 Td[(p;xmihp;xmi)]TJ/F26 11.955 Tf 19.261 0 Td[(kxmkkxmkkxmkkpkkxmk)]TJ/F26 11.955 Tf 20.589 0 Td[(kxmkkxmkkxmk=)]TJ/F31 11.955 Tf 9.298 13.27 Td[(kpk kxmk+kxmkkxmkkxmk=)]TJ/F19 11.955 Tf 10.983 3.155 Td[(~kxmkkxmkkxmk;where~kxmk!0asm!1.Usingthisalongwithwehavekxmkkxmk[1)]TJ/F26 11.955 Tf 11.955 0 Td[(tm)]TJ/F26 11.955 Tf 11.955 0 Td[(]kxmkkxmk)]TJ/F26 11.955 Tf 28.559 0 Td[(tmhum+cxm)]TJ/F26 11.955 Tf 11.955 0 Td[(p;xmitm~kxmkkxmkkxmk:Thisimpliestm~kxmk!0asm!1,i.e.,acontradiction.Thus,thereexistr>0suchthatallpossiblesolutionsoflieintheballBr.Con-sequently,nosolutionofliesin@Br,andthedegreemappingdHt;:;Br;0iswelldened.Therefore,byiiiofTheorem1.3.1,wehavedHt;:;Br;0isxedforallt2[0;1].ThismeansdH;:;Br;0=dH;:;Br;047

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ordT+C)]TJ/F26 11.955 Tf 11.955 0 Td[(p+J;Br;0=dJ;Br;0=1:ByiiofTheorem1.3.1,wehavethattheinclusionTx+Cx+Jx3pissolvableforevery>0.LetxnbeasolutionofTx+Cx+1 nJx3p:nWeassumethat:67holdsandshowthatthesequencefxngisbounded.Tothisend,assumethereexistasubsequenceoffxng,denotedagainbyfxng,suchthatkxnk!1.Thenthereexists>0suchthatliminfn!1jTxn+Cxnj kxnkliminfkxk!1jTx+Cxj kxk:x2DTDC=:However,forsomeun2Txn,wehavefromnkun+Cxnk=kp)]TJ/F19 11.955 Tf 13.718 8.088 Td[(1 nJxnk1 nkxnk+kpkand=liminfn!1jTxn+Cxnj kxnkliminfn!1kTxn+Cxnk kxnkliminfn!11 n+kpk kxnk=0;i.e.,acontradiction.Sincefxngisbounded,thenk1 nJxnk=1 nkxnk1 nS;48

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wherekxnkS.Hencelimn!1k1 nJxnk=0:Thenfromnandforsomesequenceun2Txn,wehavelimn!1un+Cxn=p:Againfromnhun;xni=hp;xni)-222(hCxn;xni)]TJ/F19 11.955 Tf 21.024 8.088 Td[(1 nhJxn;xnikpkS+1kxnk)]TJ/F19 11.955 Tf 13.718 8.088 Td[(1 nkxnkkxnkkpkS+1S:BythestrongquasiboundednessofT,wehavethatthesequencefungisbounded,sowemayassumethatun*u0.HenceCxn*p)]TJ/F26 11.955 Tf 11.955 0 Td[(u0.Lety2LFn.ThenhCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;yi=hun+Cxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;yi)-222(hun;yi:Itfollowsthatlimn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;yi=hu0;yi;orlimn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p+u0;yi=0:Wenextshowthatlimsupn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p+u0;xni0:Letv2Tx0.ThenbythemonotonicityofT,wehavehun)]TJ/F26 11.955 Tf 11.955 0 Td[(v;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0i0:49

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Henceliminfn!1hun;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0iliminfn!1hv;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=0:NowhCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p+u0;xni=hun+Cxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;xni)-222(hun;xni+hu0;xni)-222(hun;x0i+hun;x0i=hun+Cxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;xni)-222(hun;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i+hu0;xni)-222(hu0;xni:Hencelimsupn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p+u0;xnilimsupn!1hun+Cxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;xni)]TJ/F19 11.955 Tf 19.261 0 Td[(liminfn!1hun;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0i:Thereforelimsupn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p+u0;xni0:SinceCisoftypeS+L,wehavexn!x0,x02DCandCx0=p)]TJ/F26 11.955 Tf 11.956 0 Td[(u0.ByLemma1.1.7,wehavex02DTandu02Tx0.Thereforex02DTDC=DT+CandTx0+Cx03p.ThissaysthatRT+C=X.AnothercorollaryofTheorem1.6.2anditsproofisthefollowing.Corollary2.5.3AssumeTsatisesT1,andC:XDC!Xsatisesc1)]TJ/F26 11.955 Tf 10.994 0 Td[(c3.AssumethataThereexistsaconstantk>0,Q>0suchthathu+Cx;xi)]TJ/F26 11.955 Tf 29.888 0 Td[(kkxk;x2DTDC;u2Tx;andkxkQ;:68bT+C)]TJ/F24 7.97 Tf 6.587 0 Td[(1isbounded.ThenRT+C=X50

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Proof.Werstobservefromathathu+Cx;xi)]TJ/F26 11.955 Tf 9.299 0 Td[(k kxkkxk2=)]TJ/F26 11.955 Tf 9.299 0 Td[(kxkkxk2;x2DTDC;kxkQwherer=k r.Thus:66istrueforr=r.Consequently,Theorem1.6.2impliesRT+C+J=X,i.e,givenp2X,theinclusionTx+Cx+Jx3pissolvablefor>0.HereJ=Jisthenormalizeddualitymapping.Letusxp2XandconsiderasolutionxnoftheinclusionTxn+Cxn+1 nJxn3p::69Toshowthatfxngisbounded.Letassumethatthecontraryistrue.Then,withoutlossofgenerality,wemayassumethatkxnkQ,n=1;2;:::;:Thenby:69forsomeyn2Txn,yn+Cxn+1 nJxn=p::70Thenby.68hyn+Cxn;xni=hp;xni)]TJ/F19 11.955 Tf 21.024 8.088 Td[(1 nkxnk2)]TJ/F26 11.955 Tf 21.918 0 Td[(kkxnk;or)]TJ/F26 11.955 Tf 9.298 0 Td[(kkxnk)]TJ/F19 11.955 Tf 32.979 8.088 Td[(1 nkxnk2+hp;xni)]TJ/F19 11.955 Tf 31.65 8.088 Td[(1 nkxnk2+kpkkxnk:Thisimplies1 nkxnkkpk+k:From:70wehavekyn+Cxnkkpk+1 nkxnk2kpk+k::71Sinceyn+Cxn2T+Cxn,xn2T+C)]TJ/F24 7.97 Tf 6.587 0 Td[(1yn+Cxn:From:71andtheboundednessofT+C)]TJ/F24 7.97 Tf 6.587 0 Td[(1,wehavefxngisbounded,i.e.,a51

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contradiction.Since1 nkJxnk=1 nkxnk!0:asn!1,weconcludefrom:70thatlimn!1yn+Cxn!p:By1:70again,hyn;xni=hp;xni)-222(hCxn;xni)]TJ/F19 11.955 Tf 21.024 8.088 Td[(1 nhJxn;xnikpkS+1kxnk)]TJ/F19 11.955 Tf 13.718 8.088 Td[(1 nkxnk2kpkS+1S;whereSistheboundforfxng.BythestrongquasiboundednessofT,wehavethatthesequencefyngisbounded,sowemayassumethatyn*y0.HenceCxn*p)]TJ/F26 11.955 Tf 10.578 0 Td[(y0.Lety2LFn.ThenhCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;yi=hyn+Cxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;yi)-222(hyn;yi:Itfollowsthatlimn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;yi=hy0;yi;orlimn!1hCxn)]TJ/F26 11.955 Tf 11.956 0 Td[(p+y0;yi=0:BythesameargumentasinTheorem1.6.2wehave,limsupn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p+y0;xni0:Weconcludethatx02DT+Candp2T+Cx0.52

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Theorem2.5.4Leray-SchauderConditionLetTsatisfyT1,andC:XDC!Xsatisesc1)]TJ/F26 11.955 Tf 12.283 0 Td[(c3.AssumefurtherthatthereexistsanopenboundedconvexsetGXcontainingzeroandsuchthattheinclusionTx+Cx3Jx:72hasnosolutionx2DT+C@Gforany0.ThentheinclusionTx+Cx30hasasolutionx2DT+CG.Proof.WeconsiderthehomotopyequationHt;x=tTx+Cx+Jx+)]TJ/F26 11.955 Tf 11.955 0 Td[(tJx30::73Fort=1wehaveTx+Cx+Jx30,whichsaysthatTx+Cx3)]TJ/F26 11.955 Tf 20.591 0 Td[(Jx.By:72,thisinclusionhasnosolutionx2@G.Also,fort=0,Jx=0.SinceJisinjectiveandJ=0,weconcludethatx=0=2@G.Letusnowassumethatforsomet2;1theinclusion:73hasasolutionx2@G.ThenTx+Cx+[1 t)]TJ/F19 11.955 Tf 11.955 0 Td[(1+]Jx30;whichcontradictstheassumptionofthetheorem.Thusbyproposition1.6.1,theinclusionTx+Cx+Jx30issolvableinGforevery>0.Letxn2GbeasolutionofTxn+Cxn+1 nJxn30;:74Thenforsomeyn2Txn,wehavelimn!1kyn+Cxnk=limn!11 nkJxnk=limn!11 nkxnk=0:53

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Weconcludethatlimn!1yn+Cxn=0:From:74,wehaveforsomeyn2Txnhyn;xni=hCxn;xni)]TJ/F19 11.955 Tf 21.024 8.087 Td[(1 nhJxn;xni1kxnk)]TJ/F19 11.955 Tf 13.718 8.088 Td[(1 nkxnk21S;whereSistheboundforfxng.BythestrongquasiboundednessofT,wehavethatthesequencefyngisbounded,sowemayassumethatyn*y0.HenceCxn*)]TJ/F26 11.955 Tf 9.298 0 Td[(y0.Lety2LFn.ThenhCxn;yi=hyn+Cxn;yi)-222(hyn;yi:Itfollowsthatlimn!1hCxn;yi=hy0;yi;orlimn!1hCxn+y0;yi=0:Wearegoingtoshowthatlimsupn!1hCxn+y0;xni0:WeobservethathCxn+y0;xni=hyn+Cxn;xni)-222(hyn;xni+hy0;xni)-222(hyn;x0i+hyn;x0i=hyn+Cxn;xni)-222(hyn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i+hy0;xni)-222(hyn;x0i:Letu02Tx0.ThensinceTismaximalmonotonewehavehyn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0ihu0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i:54

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Thereforeliminfn!1hyn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0iliminfn!1hu0;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=0:Hencelimsupn!1hCxn+y0;xnilimsupn!1hyn+Cxn;xni)]TJ/F19 11.955 Tf 19.261 0 Td[(liminfn!1hyn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0i0:SinceCisoftypeS+L,wehavexn!x0,x02DCandCx0+y0=0.ByLemma1.1.7,wehavex02DTandy02Tx0.Thereforex02DTDC=DT+CandTx0+Cx030.Obviouslyx02 coG= G.Butx0=2@Gbytheassumptionofthetheoremandhencewehaveourconclusion.Corollary2.5.5LetTsatisfyT1,andC:XDC!Xsatisesc1)]TJ/F26 11.955 Tf 12.928 0 Td[(c3.AssumefurtherthatthereexistsanopenboundedconvexsetGXcontainingzeroandsuchthatforeveryx2DT+C@Gandeveryu2Txwehavehu+Cx;xi0:ThentheinclusionTx+Cx30hasasolutionx2DT+CG.Proof.SupposethattheinclusionTx+Cx3Jx:75hasasolutionx2DT+C@Gfor0.Thenforsomeu2Tx,wehavefrom:75thatu+Cx=Jx:Thisimplieshu+Cx;xi=hJx;xi=kxk20;55

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i.e.,acontradictiontothehypothesisofthecorollary.HenceTx+Cx3Jxhasnosolutionx2DT+C@G.ByTheorem1.6.5,theresultfollows.2.6FurtherMappingTheoremsfortheNewDegreeDenition2.6.1LetGXbeopen.AnoperatorT:XDT!2Xiscalled"locallymonotone"onGifforeveryx02DTG,thereexistsaball Brx0GsuchthatTismonotoneonDT Brx0.Theorem2.6.2EquivalentConditionsfortheExistenceofzerosLetTsatisfyT1,andC:XDC!Xsatisesc1)]TJ/F26 11.955 Tf 10.785 0 Td[(c3.AssumefurtherthatforsomeopenboundedsetGX,theoperatorT+CislocallymonotoneonG.Thenthefollowingareequivalent:a02T+CDTG;bthereexistsr>0andx02DTGsuchthat Brx0Gandhu+Cx;x)]TJ/F26 11.955 Tf 10.078 0 Td[(x0i0,foreveryx;u2DT+C@Brx0Tx;cthereexistsr>0andx02DTGsuchthat Brx0GandT+Cx63Jx)]TJ/F26 11.955 Tf 11.955 0 Td[(x0,forevery;x2;0DT+C@Brx0:Proof.Assumethat02T+CDTG.Thenthereexistsx02DTGsuchthat02T+Cx0.SinceTislocallymonotoneonG,thereexistsaball Brx0GsuchthatTismonotoneonDT+C Brx0.Consequentlywehavehu+Cx;x)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0;foreveryx;u2DT+C@Brx0Tx.Itfollowsthatab.Toshowthatbc,assumethatbholdsandletT+Cx3Jx)]TJ/F26 11.955 Tf 12.405 0 Td[(x0,for56

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some;x2;0DT+C@Brx0:Thenforsomeu2Tx0hu+Cx;x)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=hJx)]TJ/F26 11.955 Tf 11.955 0 Td[(x0;x)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=kx)]TJ/F26 11.955 Tf 11.955 0 Td[(x0k2<0:Thisisacontradiction.Hencebc.Letchold.WeconsidertheapproximateproblemTx+Cx+1 nJx)]TJ/F26 11.955 Tf 11.955 0 Td[(x030::76ItisactuallypossibletoreplaceJxbyJx)]TJ/F26 11.955 Tf 11.884 0 Td[(x0invarioushomotopiesprovidedx02G.HencebyTheorem1.6.5,:76issolvableinBrx0foranyn=1;2;:::;whereinthiscase=)]TJ/F24 7.97 Tf 6.587 0 Td[(1 n:Letxnbeasolutionof:76lyinginBrx0.Wemayassumethatxn*~x2 Brx0.Thenforsomeyn2Txn,wehaveyn+Cxn+1 nJxn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0=0::77Hencelimn!1kyn+Cxnk=limn!11 nkJxn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0k=limn!11 nkxn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0k=0:Thereforelimn!1yn+Cxn=0:Lety2LFn.Thenlimn!1hyn+Cxn;yi=0:From:77,wehavehyn;xni=hCxn;xni)]TJ/F19 11.955 Tf 21.024 8.088 Td[(1 nhJxn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0;xnikxnk+1 nkJxn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0kkxnkr0+1 nrr0r0+rr057

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wherer0istheboundforthesequencefxng.SinceTisstronglyquasibounded,wehavethatfyngisbounded.Hencewemayassumethatthatyn*y0.ThereforeCxn=)]TJ/F19 11.955 Tf 11.062 8.088 Td[(1 nJxn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0)]TJ/F26 11.955 Tf 11.955 0 Td[(yn*)]TJ/F26 11.955 Tf 9.299 0 Td[(y0:Lety2LFn.ThenhCxn;yi=hyn+Cxn;yi)-222(hyn;yi:Itfollowsthatlimn!1hCxn;yi=hy0;yi;orlimn!1hCxn+y0;yi=0:Wenextshowthatlimsupn!1hCxn+y0;xni0:NowhCxn+y0;xni=hyn+Cxn;xni)-222(hyn;xni+hy0;xni)-222(hyn;~xi+hyn;~xi=hyn+Cxn;xni)-222(hyn;xn)]TJ/F19 11.955 Tf 12.68 0 Td[(~xi+hy0;xni)-222(hyn;~xi:Letu02T~x.ThensinceTismaximalmonotonewehavehyn;xn)]TJ/F19 11.955 Tf 12.68 0 Td[(~xihu0;xn)]TJ/F19 11.955 Tf 12.68 0 Td[(~xi:Thereforeliminfn!1hyn;xn)]TJ/F19 11.955 Tf 12.68 0 Td[(~xiliminfn!1hu0;xn)]TJ/F19 11.955 Tf 12.68 0 Td[(~xi=0:58

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Hencelimsupn!1hCxn+y0;xnilimsupn!1hyn+Cxn;xni)]TJ/F19 11.955 Tf 15.019 0 Td[(liminfn!1hyn;xn)]TJ/F19 11.955 Tf 10.559 0 Td[(~xi+hy0;~xi)-44(hy0;~xi0:SinceCisoftypeS+L,wehavexn!~x,~x2DCandC~x+y0=0.ByLemma1.1.7,wehave~x2DTandy02T~x.Therefore~x2DTDC=DT+CandT~x+C~x30.However~x=2@Gbecause Brx0G:Thus~x2DT+CG.Therefore02T+CDT+CG.Theorem2.6.3BallsintheRangeofT+CLetTsatisfyT1,andC:XDC!Xsatisesc1)]TJ/F26 11.955 Tf 12.283 0 Td[(c3.AssumefurtherthatthereisaboundedopensubsetGofXwith02G,andthereexistsaconstantr>0andz02Xsuchthatkz0k
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andH2t;x=Tx+Cx+1 nJx)]TJ/F26 11.955 Tf 11.955 0 Td[(tz0)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F26 11.955 Tf 11.955 0 Td[(tp::82Weshowthat:81and:82areadmissiblehomotopies.ThatisHit;:@G630,fori=1;2:SupposerstthatHit;:@G30andthatfori=1,H1t;xhasasolutionxt2@G.Thenforu2Txt,wehavetu+Cxt)]TJ/F26 11.955 Tf 11.955 0 Td[(z0+1 nJxt=0:Ift=0,thenJxt=0.SinceJisinjectivewehavext=0,i.e.,acontradictionsince02intG.Considert2;1].Then0=hu+Cxt)]TJ/F26 11.955 Tf 11.955 0 Td[(z0;xti+1 nthJxt;xti1 nthJxt;xti=1 ntkxtk2>0;i.e.,acontradiction.HencedH1t;:;G;0iswell-denedforallt2[0;1].WenextshowthatthedegreedH2t;:;G;0iswell-dened.Tothisend,letxt2@Gbeasolutionof:82.Thenforu2Txtwehaveu+Cxt+1 nJxt)]TJ/F26 11.955 Tf 11.955 0 Td[(tz0)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F26 11.955 Tf 11.955 0 Td[(tp=0::83LetQ=supfkxk:x2@Ggandx2;randanintegern0>0sothatr)]TJ/F26 11.955 Tf 11.955 0 Td[(>1 n0Q+maxfkpk;kz0kg::84ThenjTx+Cxjr>r)]TJ/F26 11.955 Tf 11.955 0 Td[(;x2@GDT:60

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Since:84holdsforn>n0,insteadofn0,weconsideronlyvaluesofsuchthatnn0.Thenfrom:83,wehave0=ku+Cxt+1 nJxt)]TJ/F26 11.955 Tf 11.955 0 Td[(tz0)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F26 11.955 Tf 11.955 0 Td[(tpkjTxt+Cxtj)]TJ/F19 11.955 Tf 19.696 8.088 Td[(1 nkxtk)]TJ/F19 11.955 Tf 20.59 0 Td[(tkz0k+)]TJ/F26 11.955 Tf 11.956 0 Td[(tkpkjTxt+Cxtj)]TJ/F19 11.955 Tf 17.933 0 Td[([1 nkxtk+maxfkpk;kz0kg]jTxt+Cxtj)]TJ/F19 11.955 Tf 17.933 0 Td[([1 nQ+maxfkpk;kz0kg]>r)]TJ/F26 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 11.955 0 Td[([1 nQ+maxfkpk;kz0kg]>0:Thisisacontradiction.HencedH2t;:;G;0iswell-denedforallt2[0;1]andnn0.Thusweseethatwhennissucientlylarge,saynn0,bothhomotopiesareadmissibleandthedegreesdH1t;:;G;0anddH2t;:;G;0arewell-denedandconstantfort2[0;1].HoweverdH1;:;G;0=d1 nJ;G;0=1:ItfollowsthatdH2;:;G;0=dH2;:;G;0=dH1;:;G;0=dH1;:;G;0=1;ordT+C+1 nJ)]TJ/F26 11.955 Tf 11.955 0 Td[(p;G;0=dT+C+1 nJ)]TJ/F26 11.955 Tf 11.955 0 Td[(z0;G;0=dT+C+1 nJ;G;0=d1 nJ;G;0=1:Thisimpliesthattheinclusion:80issolvableinGforalllargen.Letusassumethatthisistrueforalln=1;2;:::;andconsiderasolutionxn2Gof1:80.Wemay61

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assumethatxn*x02 coG.Forsomeyn2Txnand:80,wehavelimn!1kyn+Cxn)]TJ/F26 11.955 Tf 11.955 0 Td[(pk=limn!11 nkJxnk=limn!11 nkxnk=0:By1:80,wehavehyn;xni=hp;xni)-222(hCxn;xni)]TJ/F19 11.955 Tf 21.024 8.088 Td[(1 nhJxn;xnikpkS+1kxnk)]TJ/F19 11.955 Tf 13.718 8.087 Td[(1 nkxnk2kpkS+1S;whereSisanupperboundforfxng.SinceTisstronglyquasiboundedness,wehavethatthesequencefyngisbounded,sowemayassumethatyn*y0.HenceCxn*p)]TJ/F26 11.955 Tf 11.955 0 Td[(y0.Lety2LFn.ThenhCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;yi=hyn+Cxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;yi)-222(hyn;yi:Itfollowsthatlimn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;yi=hy0;yi;orlimn!1hCxn)]TJ/F26 11.955 Tf 11.956 0 Td[(p+y0;yi=0:Wenextshowthatlimsupn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p+y0;xni0:Letv2Tx0.ThenbythemonotonicityofT,wehavehyn)]TJ/F26 11.955 Tf 11.955 0 Td[(v;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i0:62

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Henceliminfn!1hyn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0iliminfn!1hv;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=0:NowhCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p+y0;xni=hyn+Cxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;xni)-222(hyn;xni+hy0;xni)-222(hyn;x0i+hyn;x0i=hyn+Cxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;xni)-222(hyn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(x0i+hy0;xni)-222(hy0;xni:Hencelimsupn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p+y0;xnilimsupn!1hyn+Cxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p;xni)]TJ/F19 11.955 Tf 19.261 0 Td[(liminfn!1hyn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i:Thereforelimsupn!1hCxn)]TJ/F26 11.955 Tf 11.955 0 Td[(p+y0;xni0:SinceCisoftypeS+L,wehavexn!x0,x02DCandCx0=p)]TJ/F26 11.955 Tf 11.955 0 Td[(y0.ByLemma1.1.7,wehavex02DTandy02Tx0.Thereforex02DTDC=DT+CandTx0+Cx03p.ConsequentlyBrT+CDT+C coGwhichnishestheproofoftherstinclusion.If,inadditionGisconvex,then coG= G.HenceBrT+CDT+C G:ButtheboundaryofGisexcludedfromthisinclusionbecausep2Brimpliesku+CxkjTx+Cxj>kpk;x2DT+C@G;u2Txorku+Cx)]TJ/F26 11.955 Tf 11.955 0 Td[(pkku+Cxk)-222(kpk>0;x2DT+C@G;u2Tx:Thus,BrT+CDT+CG:63

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3BallsintheRangeofaContinuousBoundedPerturbationofaMaximalMonotoneOperatorwithCompactResolventsInwhatfollows,whenwesaythatamaximalmonotoneoperatorT:XDT!2Xhascompactresolvents"completelycontinuousresolvents"wemeanthatforevery">0theoperatorT+"J)]TJ/F24 7.97 Tf 6.586 0 Td[(1:X!Xiscompactcompletelycontinuous.Actually,ifT+"J)]TJ/F24 7.97 Tf 6.586 0 Td[(1iscompactforsome">0;thenitiscompactforall">0cf.[19,Lemma3andtheresolventidentityonpage1690].Thefollowingrelevantresultsaysthatthereexistsanopenballaroundzerointheimageofarelativelyopensetbyacontinuousandboundedperturbationofamaximalmonotoneoperatorwithcompactresolvents.ForasetAX;wesetjAj=inffkyk:y2Ag:Westatethefollowingimportantlemmawithoutproof.Lemma3.0.4LetXandXbelocallyuniformlyconvexBanachspacesandletfxngbeasequenceinXforwhichhJxn)]TJ/F26 11.955 Tf 12.446 0 Td[(Jx;xn)]TJ/F26 11.955 Tf 12.445 0 Td[(xi!0foragivenelementxofX.Thenxn!xinX.Theorem3.0.5LetGXbeopenandboundedandcontainingzero.Letthefol-lowingassumptionsbesatised:iT:XDT!2X;with02DT;ismaximalmonotoneandhascompactresolvents;iiC: G!Xiscontinuousandbounded64

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iiithereexistaconstantr>0andz02Xsuchthatkz0k0;>0;arecompactcf.,forexample,Kartsatos[19,p.1684].Weareplanningtosolvetheapproximateequation~Tx+~Cx+=nJx3pTx)]TJ/F26 11.955 Tf 11.955 0 Td[(v0+Cx+v0+=nJx3pTx+Cx+=nJx3pn65

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forx2D~T;wherep2Brisaxedvector.Tothisend,weconsiderthehomotopiesH1t;xx)]TJ/F19 11.955 Tf 11.956 0 Td[(t~T+=nJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1)]TJ/F26 11.955 Tf 9.298 0 Td[(t~Cx)]TJ/F26 11.955 Tf 11.955 0 Td[(z0;:2andH2t;xx)]TJ/F19 11.955 Tf 11.955 0 Td[(~T+=nJ)]TJ/F24 7.97 Tf 6.587 0 Td[(1)]TJ/F19 11.955 Tf 9.299 0 Td[(~Cx)]TJ/F26 11.955 Tf 11.956 0 Td[(tz0+)]TJ/F26 11.955 Tf 11.955 0 Td[(tp;:3foraxedpositiveintegernandt2[0;1]:WeletQ=supfkxk:x2@Ggandx2;randtheintegern0>0sothatr)]TJ/F26 11.955 Tf 11.955 0 Td[(>=n0Q+maxfkpk;kz0kg::4Thisispossiblebecausewealsohavekpkr)]TJ/F26 11.955 Tf 11.955 0 Td[(;x2@GDT:Since.4holdsforanyn>n0insteadofn0;weconsideronlyvaluesofnsuchthatnn0:WenowshowthatthemappingsHit;xx)]TJ/F26 11.955 Tf 9.368 0 Td[(Fit;xareactuallyhomotopiesofcompacttransformations.ThismeansthateachmappingFi:[0;1] G!X;i=1;2;iscompact.ToshowthatF1t;x=t~T+=nJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1)]TJ/F26 11.955 Tf 9.299 0 Td[(t~Cx)]TJ/F26 11.955 Tf 11.956 0 Td[(z0iscontinuous,lettm;xm2;1] Gbesuchthattm!t00andxm!x02X:Wemayassumethattm>0:Wesetym=tm~T+=nJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1)]TJ/F26 11.955 Tf 9.298 0 Td[(tm~Cxm)]TJ/F26 11.955 Tf 11.956 0 Td[(z0andy0=t0~T+=nJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1)]TJ/F26 11.955 Tf 9.298 0 Td[(t0~Cx0)]TJ/F26 11.955 Tf 11.955 0 Td[(z0:Thisimpliesthattm~Tym=)]TJ/F19 11.955 Tf 9.298 0 Td[(=nJym)]TJ/F26 11.955 Tf 11.956 0 Td[(tmCxm)]TJ/F26 11.955 Tf 11.955 0 Td[(z066

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andt0~Ty0=)]TJ/F19 11.955 Tf 9.299 0 Td[(=nJy0)]TJ/F26 11.955 Tf 11.955 0 Td[(t0Cx0)]TJ/F26 11.955 Tf 11.955 0 Td[(z0:Henceitfollowsthathtm~Tym)]TJ/F26 11.955 Tf 11.955 0 Td[(t0~Ty0;ym)]TJ/F26 11.955 Tf 11.955 0 Td[(y0i+=nhJym)]TJ/F26 11.955 Tf 11.955 0 Td[(Jy0;ym)]TJ/F26 11.955 Tf 11.955 0 Td[(y0i=hDm;ym)]TJ/F26 11.955 Tf 11.955 0 Td[(y0i;whereDm=t0~Cx0)]TJ/F26 11.955 Tf 11.956 0 Td[(tm~Cxm+tm)]TJ/F26 11.955 Tf 11.955 0 Td[(t0z0!0asm!1:Here,andinwhatfollows,wesometimesusethesymbol~Tx;x2D~T;todenoteasingleappropriateelementoftheset~Tx:Buttm~Tym)]TJ/F26 11.955 Tf 11.956 0 Td[(t0~Ty0=tm~Tym)]TJ/F19 11.955 Tf 14.248 3.022 Td[(~Ty0+tm)]TJ/F26 11.955 Tf 11.955 0 Td[(t0~Ty0:Henceusingthemonotonicitypropertyof~T;weobtainhtm~Tym)]TJ/F26 11.955 Tf 11.955 0 Td[(t0~Ty0;ym)]TJ/F26 11.955 Tf 11.955 0 Td[(y0i=htm~Tym)]TJ/F19 11.955 Tf 14.248 3.022 Td[(~Ty0+tm)]TJ/F26 11.955 Tf 11.955 0 Td[(t0~Ty0;ym)]TJ/F26 11.955 Tf 11.955 0 Td[(y0itm)]TJ/F26 11.955 Tf 11.955 0 Td[(t0h~Ty0;ym)]TJ/F26 11.955 Tf 11.955 0 Td[(y0i:Thereforeitfollowsthat=nhJym)]TJ/F26 11.955 Tf 11.955 0 Td[(Jy0;ym)]TJ/F26 11.955 Tf 11.956 0 Td[(y0i)]TJ/F19 11.955 Tf 43.171 0 Td[(tm)]TJ/F26 11.955 Tf 11.955 0 Td[(t0h~Ty0;ym)]TJ/F26 11.955 Tf 11.955 0 Td[(y0i+hDm;ym)]TJ/F26 11.955 Tf 11.955 0 Td[(y0ijtm)]TJ/F26 11.955 Tf 11.955 0 Td[(t0jk~Ty0k+kDmkkym)]TJ/F26 11.955 Tf 11.955 0 Td[(y0k::5Tondaboundforthesequencefkymkg;weevaluatetm~Tym+=nJym=)]TJ/F26 11.955 Tf 9.299 0 Td[(tm~Cxm)]TJ/F26 11.955 Tf 11.955 0 Td[(z0:6atymandusethefactthath~Tym;ymi0toobtain=nkymk2==nhJym;ymi)]TJ/F26 11.955 Tf 29.888 0 Td[(tmh~Cxm)]TJ/F26 11.955 Tf 11.955 0 Td[(z0;ymitmk~Cxm)]TJ/F26 11.955 Tf 11.956 0 Td[(z0kkymk;:767

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fromwhichfollowstheboundednessofthesequencefymg:Usingthisin.5,weseethatlimm!1hJym)]TJ/F26 11.955 Tf 11.955 0 Td[(Jy0;ym)]TJ/F26 11.955 Tf 11.955 0 Td[(y0i=0:Lemma2.0.4,impliesfymgconvergesstronglytoy0asm!1,i.e.thecontinuityofF1t;xontheset[0;1]X:WenowhavetoshowthatthemappingF1t;xmaps[0;1]GintoarelativelycompactsubsetofX:Lettm;xm2[0;1]G:Sinceftmgisbounded,wemayassumethattm!t02[0;1]:Ift0=0;thentheboundednessofthesequencesfymgandf~Cxmgand2.7implyym!0.Therefore,itsucestoassumethatt0>0.Weobserverstthat.7impliesagaintheboundednessofthesequencefymg:Wealsonotethat.6issatised.Thus,theboundednessoffymg;f~Cymgandt0>0implytheboundednessof~Tym:Werewrite.6asfollows:~Tym+=nJym=)]TJ/F26 11.955 Tf 9.299 0 Td[(tm~Cxm)]TJ/F26 11.955 Tf 11.955 0 Td[(z0+)]TJ/F26 11.955 Tf 11.955 0 Td[(tm~Tym::8ThensinceJymand~Tymarebounded,)]TJ/F26 11.955 Tf 9.299 0 Td[(tm~Cxm)]TJ/F26 11.955 Tf 11.955 0 Td[(z0+)]TJ/F26 11.955 Tf 11.956 0 Td[(tm~Tymisbounded.Thisimpliesym=~T+=nJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1h)]TJ/F26 11.955 Tf 9.298 0 Td[(tm~Cxm)]TJ/F26 11.955 Tf 11.955 0 Td[(z0+)]TJ/F26 11.955 Tf 11.955 0 Td[(tm~Tymi;i.e.,fymgliesinacompactset.Itthereforecontainsaconvergentsubsequence.ThiscompletestheproofofthecompactnessoftheoperatorF1t;xon[0;1] G:ToshowthecompactnessoftheoperatorF2t;x;werstobservethatitscontinuityfollowseasilyfromthecontinuityoftheoperator~Candthecontinuityoftheresolvent~T+=nJ)]TJ/F24 7.97 Tf 6.586 0 Td[(1:ThefactthatF2t;xmaps[0;1] Gintoarelativecompactsetfollowseasilyfromthecontinuityandboundednessof~Candthecompactnessoftheaboveresolvent.WearenowgoingtoshowthatdH1;;G;0=1:968

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andthendH2;;G;0=dH1;;G;0=1;:10whered=d;;denotestheLeray-Schauderdegree.Toshow.9,weshowrstthatthedegreethereiswell-dened.Tothisend,letusassumethatthehomotopyequationH1t;xx)]TJ/F26 11.955 Tf 11.955 0 Td[(F1t;x=0hasasolutionxt2@G;forsomet2[0;1]:Thent~Txt+~Cxt)]TJ/F26 11.955 Tf 11.955 0 Td[(z0+=nJxt=0::11Obviously,t=0impliesxt=0;i.e.acontradictionbecause0isaninteriorpointofG:Thus,t2;1]and,afterdividing2.11bytandusing;weobtain,0=h~Txt+~Cxt)]TJ/F26 11.955 Tf 11.956 0 Td[(z0;xti+h[1=nt]Jxt;xti[1=nt]hJxt;xti=[1=nt]kxtk2>0:Consequently,theLeray-SchauderdegreedH1t;;G;0iswell-denedforallt2[0;1]andequals1becausewehave02GanddH1;;G;0=dI;G;0=1:WenowshowthatthedegreedH2t;;G;0iswelldened.Tothisend,let2.3haveasolutionxt2@G:Thenwehave~Txt+~Cxt+=nJxt)]TJ/F26 11.955 Tf 11.955 0 Td[(tz0)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F26 11.955 Tf 11.955 0 Td[(tp=0:Thisimplies0=k~Txt+~Cxt+=nJxt)]TJ/F26 11.955 Tf 11.955 0 Td[(tz0)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F26 11.955 Tf 11.955 0 Td[(tpkj~Txt+~Cxtj)]TJ/F19 11.955 Tf 17.933 0 Td[(=nkxtk)]TJ/F19 11.955 Tf 20.589 0 Td[(tkz0k+)]TJ/F26 11.955 Tf 11.955 0 Td[(tkpkj~Txt+~Cxtj)]TJ/F19 11.955 Tf 17.933 0 Td[([=nkxtk+maxfkpk;kz0kg]69

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j~Txt+~Cxtj)]TJ/F19 11.955 Tf 17.933 0 Td[([=nQ+maxfkpk;kz0kg]>r)]TJ/F26 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 11.955 0 Td[([=nQ+maxfkpk;kz0kg]>0:ThiscontradictionsaysthatdH2t;;G;0iswell-denedforallt2[0;1]andequalsthedegreesdH1;;G;0anddH2;;G;0:WeconcludethatdH2;;G;0=1;whichimpliesx)]TJ/F19 11.955 Tf 11.955 0 Td[(~T+=nJ)]TJ/F24 7.97 Tf 6.587 0 Td[(1~Cx+p=0;or~Tx+~Cx+=nJx3p::12forsomex2G:Thus,wehavethesolvabilityofnforeachnn0;i.e.thesolvabilityoftheinclusionTx+Cx+=nJx3p:13withsolutionxn2G;nn0:SinceGisbounded,fxngisbounded.From2.13wehavekTxn+Cxn)]TJ/F26 11.955 Tf 11.955 0 Td[(pk==nkJxnk!0orTxn+Cxn!pasn!1.Thisimpliesp2 T+CDTG:ThissaysthatBr T+CDTGandimplies Br T+CDTG:Toshowthesecondconclusionofthetheorem,assumethatT+CisoftypeS:Letxn2DTGsolvetheinclusion.13.Sincefxngisbounded,wemayassumethatxn*x2 coG= G:Here,coGdenotestheconvexhullofthesetG:Wehavefrom.13hTxn+Cxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=)]TJ/F19 11.955 Tf 9.299 0 Td[(=nhJxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i+hp;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i70

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andlimn!1hTxn+Cxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(x0i=0:SinceT+CisoftypeS;wehavexn!x0:ButCiscontinuousimpliesCxn!Cx0.From.13andforsomeyn2Txn,wehaveyn=)]TJ/F26 11.955 Tf 9.298 0 Td[(Cxn)]TJ/F19 11.955 Tf 11.955 0 Td[(=nJxn+p!)]TJ/F26 11.955 Tf 24.575 0 Td[(Cx0+p:SinceTisclosedandCxn!Cx0;wehavex02DTandTx03)]TJ/F26 11.955 Tf 23.451 0 Td[(Cx0+p:Thus,p2T+CDT G:However,x062T+CDT@GbecausejTx0+Cx0jr>kpk:Theproofiscomplete.Corollary3.0.6LetT:XDT!2X;withDTunboundedandcontainingzero,bemaximalmonotonewithcompactresolventsandC: DT!Xcontinuousandbounded.Assumethatthereexistz02Xandaconstantr>0suchthatkz0k0suchthatforeveryx2DTwithkxkr1andeveryu2Txwehavehu+Cx)]TJ/F26 11.955 Tf 11.955 0 Td[(z0;xi0:Then Br RT+C:If,moreover,T+CisoftypeS;wehaveBrRT+C:Proof.TheprooffollowsthestepsoftheproofofTheorem2.0.5.TheonlydierenceisthatanumberQr1isnowpickedsothat2.4istrueandthedegreesarecom-putedonBQinsteadofG:If,inaddition,02TinTheorem2.0.5,thenwechoosev0=0:AsimilarremarkholdsforCorollaries2.0.6and2.0.7.Theorem2.0.5isofcoursetrueifthecompact-71

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nessoftheresolventsofTisreplacedbythecompactnessoftheoperatorC:However,thiscaseiscoveredbytheresultsofYang[30]whousedBrowder'sdegreein[4].If,inthiscase,wetakeC=0;weobtainthefollowingcorollary.WewouldalsoliketomentionthatifthedomainofamaximalmonotoneoperatorTisbounded,thenT)]TJ/F24 7.97 Tf 6.587 0 Td[(1islocallybounded,whichimpliesthatTissurjectivecf.,forexample,PascaliandSburlan[27,p.147].Corollary3.0.7LetT:XDT!2X;withDTunboundedandcontaining0;bemaximalmonotone.Assumethatforsomepositiveconstantsr;r1andsomez02Xwehavekz0k
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holdsforeveryfunctionalz02X;thenwedon'tneedthesecondhomotopyfunctionH2t;xintheproofofTheorem2.0.5.Infact,thersthomotopywillensurethateveryz02Brliesintheappropriaterangeset.Thissituationisguaranteedifwereplacebythefollowing:hu+Cx;xi)]TJ/F26 11.955 Tf 29.888 0 Td[(rkxk;foreveryx2DT@G;u2Tx::16Thisisbecausewewouldthenhavehu+Cx)]TJ/F26 11.955 Tf 11.955 0 Td[(z0;xi=hu+Cx;xi)-222(hz0;xi>hu+Cx;xi)]TJ/F26 11.955 Tf 19.261 0 Td[(rkxk0::1773

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4TheGeneralizedTopologicalDegreeforMultivaluedCompactPerturbationofMaximalMonotoneOperatorsLetXbearealreexiveBanachspacewithdualX.LetGisanopenboundedsub-setonX.LetT:XDT!2Xbem-accretiveandC: G!Xbeacompactmapping.RecentlyY.-ZChenin[7]developedageneralizeddegreetheoryforT)]TJ/F26 11.955 Tf 11.131 0 Td[(C.HisresultswereextendedbyX.FuandS.Songin[9]tothecasewherethecompactperturbationCwasmultivalued.Theauthors,Z.GuanandA.G.Kartsatosin[11]extendedthisgeneralizeddegreetothecasewhereTismaximalmonotoneandC,asingle-valuedcompactmapping.Weextendtheresultsin[11]tothecasewhereCisamultivaluedcompactmapping.Unlike[7],[9]and[11],wheretheyappealedtotheLeray-Schauderdegreetheory,inourcaseweappealtothedegreeofTsoy-WoMain[25].Denition4.0.8AmultifunctionG:BX!2XissaidtobelongtoclassPifitmapsboundedsetstorelativelycompactsets,foreveryx2B,GxisaclosedandconvexsubsetofX,andG:isu.s.cinthesensethatforeveryclosedsetCX,G)]TJ/F24 7.97 Tf 6.587 0 Td[(1C=fx2B:GxC6=;gisclosedinX.4.1TheTsoy-WoMaDegreeLetAbeanopensubsetofalocallyconvexHausdorspaceE,p2Eand\050EthefamilyofallnonemptycompactconvexsubsetofE.Letf: A!\050Ebeaset-valuedcompacteldsuchthatp=2f@A.Byacompacteldwemeanf=I)]TJ/F26 11.955 Tf 10.44 0 Td[(F,whereF74

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isacompactmapping.Letg: A!\050Ebeanitedimensionalset-valuedcompacteld,i.e,g=I)]TJ/F26 11.955 Tf 12.304 0 Td[(G,whereGiscompactandnitedimensional.Aset-valuedmaph:[0;1] A!\050Eiscalledaset-valuedhomotopyifthemapH:[0;1] A!\050EdenedbyHt;x=x)]TJ/F26 11.955 Tf 12.569 0 Td[(ht;xiscompactandp=2h[0;1] A.Twoset-valuedcompacteldsfandf0aresaidtobehomotopicifthereexistsaset-valuedhomotopyhsuchthath0=fandh1=f0,wherehtx=ht;xforallt;x2[0;1] A.NowletE1beanynitedimensionalvectorspacecontainingG Aandp.Letgbehomotopicfinthesenseoftheabovedenitionandp=2g@A.ThentheTsoy-WoMadegree,wedenotebydMA,oftheset-valuedcompacteldf,isdenedasdMAf;A;p=d1gj AE1;AE1;p;wherethedegreed1isevaluatedinthenitedimensionalvectorspaceE1.InwhatfollowsisanopenboundedsubsetofthereexiveBanachspaceX.4.2TheGeneralizeddegreedT)]TJ/F26 11.955 Tf 11.955 0 Td[(G;DT;yDenition4.2.1LetT:XDT!2XbemaximalmonotoneandG: !2XamultifunctionofclassP.Lety2Xandapositivenumberwithy=2J+T)]TJ/F26 11.955 Tf -424.077 -20.922 Td[(GDT@.WedenethedegreedJ+T)]TJ/F26 11.955 Tf 11.956 0 Td[(G;DT;y=dMAI)]TJ/F19 11.955 Tf 11.955 0 Td[(T+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1G+y;;0;whenDT6=;,dJ+T)]TJ/F26 11.955 Tf 11.955 0 Td[(G;DT;y=0whenDT=;:Itiseasytoseethaty=2J+T)]TJ/F26 11.955 Tf 12.847 0 Td[(GDT@impliesthat0=2I)]TJ/F19 11.955 Tf 12.847 0 Td[(T+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1G+y@.Indeed,if02I)]TJ/F19 11.955 Tf 12.4 0 Td[(T+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1G+y@,thenthereexistsx2@suchthatx=T+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1g+yforsomeg2Gx.Thisimpliesthatx2DTandTx+Jx3g+y.Henceitfollowsthaty2J+T)]TJ/F26 11.955 Tf 11.015 0 Td[(GDT@,75

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i.e,acontradiction.SinceT+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1iscontinuousandGisofclassP,henceT+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1G+yisamultivaluedset-valuedcompactmapping.Thereforethede-greedMAI)]TJ/F19 11.955 Tf 11.955 0 Td[(T+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1G+y;;0iswell-dened.Wewillshowinthesequelthatthedegreeisindependentof>0.Thefollowingthe-oremisahomotopypropertyfromtheTsoy-WoMadegree.Westateitwithoutproof.Theorem4.2.2LetH:[0;1] !2XbedenedbyHt;x=x)]TJ/F26 11.955 Tf 12.278 0 Td[(ht;x,whereh:[0;1] !2Xiscompact.If0=2Ht;xforallt2[0;1]andx2@,thendMAHt;:;;0isindependentoft2[0;1].Proposition4.2.3LetT:XDT!2XbemaximalmonotoneandG: !2XamultivaluedofclassP,y2Xwithy=2 T)]TJ/F26 11.955 Tf 11.955 0 Td[(GDT@.Thenthereexist0>0suchthaty=2J+T)]TJ/F26 11.955 Tf 12.265 0 Td[(GDT@for2;0]andthedegreedMAI)]TJ/F19 11.955 Tf 11.955 0 Td[(T+J)]TJ/F24 7.97 Tf 6.586 0 Td[(1G+y;;0isindependentof2;0].Proof.Supposesuch0doesnotexist,thenthereexistfxngDT@,n!0suchthaty=nJxn+vn)]TJ/F26 11.955 Tf 13.214 0 Td[(gnforsomevn2Txnandgn2Gxn.Thenvn)]TJ/F26 11.955 Tf 12.233 0 Td[(gn=y)]TJ/F26 11.955 Tf 12.233 0 Td[(nJxn:Passingtosubsequenceifnecessary,weassumethatxn*x0andgn!g.SinceJisboundedwehavenJxn!0.Hencevn)]TJ/F26 11.955 Tf 12.3 0 Td[(gn!y.Hencey2 T)]TJ/F26 11.955 Tf 11.956 0 Td[(GDT@,i.e.,acontradiction.LetHt;x:[0;1] !2XbedenedbyHt;x=x)]TJ/F19 11.955 Tf 11.956 0 Td[([t1+)]TJ/F26 11.955 Tf 11.956 0 Td[(t2J+T])]TJ/F24 7.97 Tf 6.586 0 Td[(1Gx+y;for1;22;0]and1>2.Wenotethat0=t1+)]TJ/F26 11.955 Tf 11.544 0 Td[(t22;0].Weshowthat0=2Ht;xforallt2[0;1]andx2@:Supposethisisfalse.Thenthereexistst;x2[0;1]@suchthat02Ht;x=x)]TJ/F19 11.955 Tf 11.955 0 Td[([t1+)]TJ/F26 11.955 Tf 11.955 0 Td[(t2J+T])]TJ/F24 7.97 Tf 6.587 0 Td[(1Gx+y;or0=x)]TJ/F19 11.955 Tf 11.955 0 Td[([t1+)]TJ/F26 11.955 Tf 11.955 0 Td[(t2J+T])]TJ/F24 7.97 Tf 6.587 0 Td[(1g+y:76

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forsomeg2Gx.Thus,x2DTandg+y=t1+)]TJ/F26 11.955 Tf 11.955 0 Td[(t2Jx+v;forsomev2Tx.Thereforey20J+T)]TJ/F26 11.955 Tf 11.955 0 Td[(GDT@;where0=t1+1)]TJ/F26 11.955 Tf 12.514 0 Td[(t22;0],i.e.,acontradiction.HencebythehomotopyinvariancepropertyoftheMa-degree,wehavedMAHt;:;;0=dMAI)]TJ/F19 11.955 Tf 11.955 0 Td[([t1+)]TJ/F26 11.955 Tf 11.955 0 Td[(t2J+T])]TJ/F24 7.97 Tf 6.587 0 Td[(1G+y;;0isxedforallt2[0;1].HencedMAH;:;;0=dMAH;:;;0;meaningthatdMAI)]TJ/F19 11.955 Tf 11.955 0 Td[(T+1J)]TJ/F24 7.97 Tf 6.587 0 Td[(1G+y;;0=dMAI)]TJ/F19 11.955 Tf 11.955 0 Td[(T+2J)]TJ/F24 7.97 Tf 6.587 0 Td[(1G+y;;0:Hencetheproposition.Denition4.2.4LetT:XDT!2XbemaximalmonotoneandG: !2XamultifunctionofclassP,y2Xandy=2 T)]TJ/F26 11.955 Tf 11.955 0 Td[(GDT@;Wedenethegeneralizedtopologicaldegreeasfollows.dT)]TJ/F26 11.955 Tf 11.956 0 Td[(G;DT;y=lim!0dJ+T)]TJ/F26 11.955 Tf 11.955 0 Td[(G;DT;y=lim!0dMAI)]TJ/F19 11.955 Tf 11.955 0 Td[(T+J)]TJ/F24 7.97 Tf 6.586 0 Td[(1G+y;;0:77

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Itisthenclearthat,theabovepropositionjustiesthisdenition.Theorem4.2.5LetT:XDT!2XbemaximalmonotoneandGi: !2X,fori=1;2aremappingsofclassP.LetH:[0;1] !2XbedenedbyHt;x=x)]TJ/F19 11.955 Tf 11.002 0 Td[(T+J)]TJ/F24 7.97 Tf 6.586 0 Td[(1tG1x+)]TJ/F26 11.955 Tf 11.003 0 Td[(tG2x+ytforallt2[0;1]andx2 .Supposethatfyt;0t1gisacontinuouscurveinXwithyt=2J+T)]TJ/F26 11.955 Tf 11.484 0 Td[(GtDT@,whereGt=tG1+)]TJ/F26 11.955 Tf 11.955 0 Td[(tG2.ThendMAHt;:;;ytisindependentoft2[0;1].Proof.Werstshowthatht;x=T+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1tG1x+1)]TJ/F26 11.955 Tf 12.612 0 Td[(tG2x+ytisacompactmap.Tothisend,lettn;xn2[0;1] andletwn=T+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1tnG1xn+)]TJ/F26 11.955 Tf 11.955 0 Td[(tnG2xn+ytn:Letgn2G1xnandhn2G2xn.Bypassingtoasubsequenceweassumethattn!t02[0;1],gn!gandhn!h.Hencetngn+)]TJ/F26 11.955 Tf 11.955 0 Td[(tnhn+ytn!t0g+)]TJ/F26 11.955 Tf 11.955 0 Td[(t0h+yt0:SinceT+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1iscontinuous,wehavewnhasaconvergentsubsequence.Wenextshowthat0=2Ht;xforallt2[0;1]andx2@.Supposethisisfalse.Thenthereexistst;x2[0;1]@suchthat02Ht;x=x)]TJ/F19 11.955 Tf 11.955 0 Td[(J+T)]TJ/F24 7.97 Tf 6.586 0 Td[(1tG1x+)]TJ/F26 11.955 Tf 11.955 0 Td[(tG2x+yt;or0=x)]TJ/F19 11.955 Tf 11.955 0 Td[(J+T)]TJ/F24 7.97 Tf 6.587 0 Td[(1tg1x+)]TJ/F26 11.955 Tf 11.955 0 Td[(tg2x+yt:forsomeg12G1xandg22G2x.Thus,x2DTandtg1x+)]TJ/F26 11.955 Tf 11.955 0 Td[(tg2x+yt=Jx+v;78

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forsomev2Tx.Thereforeyt2J+T)]TJ/F26 11.955 Tf 11.094 0 Td[(Gtxandx2DT@,i.e.,acon-tradiction.ByTheorem3.2.2,dMAHt;:;;ytisindependentoft2[0;1].ThismeansbytheDenition3.2.4,wehavedT)]TJ/F26 11.955 Tf 9.62 0 Td[(Gt;DT;ytisxedforallt2[0;1].4.3PropertiesoftheGeneralizedDegreeTheorem4.3.1LetXbeareexiveBanachspaceandsuchthatXanditsdualXarelocallyuniformlyconvex.LetT:XDT!2XbemaximalmonotoneandG: !2XamultifunctionofclassP.Lety2Xwithy=2 T)]TJ/F26 11.955 Tf 11.955 0 Td[(GDT@.WehaveiIfy2T+JDT,thendJ+T;DT;y=1iiIfdT)]TJ/F26 11.955 Tf 11.955 0 Td[(G;DT;y6=0,theny2 T)]TJ/F26 11.955 Tf 11.955 0 Td[(GDT.iiiIf1and2aretwodisjointopensubsetofsuchthaty=2 T)]TJ/F26 11.955 Tf 11.955 0 Td[(GDT n1[2;thendT)]TJ/F26 11.955 Tf 11.955 0 Td[(G;DT;y=dT)]TJ/F26 11.955 Tf 11.956 0 Td[(G;DT1;y+dT)]TJ/F26 11.955 Tf 11.955 0 Td[(G;DT2;y:Proof.iSincey2T+JDT,thenT+J)]TJ/F24 7.97 Tf 6.586 0 Td[(1y2DT.ThereforewehavedJ+T;DT;y=dMAI)]TJ/F19 11.955 Tf 11.955 0 Td[(T+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1y;;0=1:iiByDenition3.2.4,ifdT)]TJ/F26 11.955 Tf 9.983 0 Td[(G;DT;y6=0,thendJ+T)]TJ/F26 11.955 Tf 9.982 0 Td[(G;DT;y6=0for>0small.ThisimpliesalsobyDenition3.2.1wehavedMAI)]TJ/F19 11.955 Tf 11.307 0 Td[(T+J)]TJ/F24 7.97 Tf 6.586 0 Td[(1G+y;;06=0.Thenthereexistx2andg2Gxsuchthatx)]TJ/F19 11.955 Tf 11.955 0 Td[(T+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1g+y=0:SinceT+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1:X!DT,wehavex2DTandg+y2Tx+Jx,soy2Tx+Jx)]TJ/F26 11.955 Tf 12.441 0 Td[(g:Hencey=v+Jx)]TJ/F26 11.955 Tf 12.44 0 Td[(g,v2Tx.As!0,79

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Jx!0andv)]TJ/F26 11.955 Tf 11.955 0 Td[(g!y.Thereforey2 T)]TJ/F26 11.955 Tf 11.955 0 Td[(GDT.iiiSincey=2 T)]TJ/F26 11.955 Tf 11.955 0 Td[(GDT n1[2;wehavebytherstpartofproposition3.2.3y=2J+T)]TJ/F26 11.955 Tf 11.955 0 Td[(GDT n1[2for>0small.Then0=2I)]TJ/F19 11.955 Tf 11.955 0 Td[(T+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1G+y n1[2:HencedJ+T)]TJ/F26 11.955 Tf 11.955 0 Td[(G;DT;y=dMAI)]TJ/F19 11.955 Tf 11.955 0 Td[(T+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1G+y;;0=dMAI)]TJ/F19 11.955 Tf 11.955 0 Td[(T+J)]TJ/F24 7.97 Tf 6.587 0 Td[(1G+y;1;0+dMAI)]TJ/F19 11.955 Tf 11.955 0 Td[(T+J)]TJ/F24 7.97 Tf 6.586 0 Td[(1G+y;2;0=dJ+T)]TJ/F26 11.955 Tf 11.955 0 Td[(G;DT1;y+dJ+T)]TJ/F26 11.955 Tf 11.955 0 Td[(G;DT1;y:Allow!0,thentheresultfollowimmediately.80

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5AnEigenvalueProblemforS+PerturbationofNonlinearOperatorsApproximatedbyQuasimonotoneMappingsLetXbeareexiveBanachspaceandGanopenboundedsubsetofX.LetT:XDT!2X,S: G!2Xand,bepositiverealnumbers.TheeigenvalueproblemTu+Su30Ehadbeenconsideredbymanyauthorsrecently.Z.GuanandA.G.Kartsatosin[12]consideredthecaseswheretheoperatorTismaximalmonotoneandSiseitherbounded,demicontinuousandofclassS+orSisdenselydenedsatisfyingcertainconditionsorquasiboundedw.r.ttoT.AlsoinanotherpaperbyA.G.KartsatosandI.V.Skrypnikin[17],theyconsideredthecaseswheretheoperatorTisS+,maximalmonotone,m-accretive,maximalmonotonewithcompactresolventsandm-accretivewithcompactresolventwithSeithercompactorcontinuousandbounded.Theyshowedthatthereexists;x2;]DT@GsuchthattheinclusionEissatised.P.M.FitzpatrickandW.V.Petryshynin[8]consideredthecasewhereTisA-properandScompact.WegiveanotherresultoftheeigenvalueproblemE,whereT2AGQM,aclassofquasimonotone-typemappingsintroducedbyArtoKittilain[23]whenSisboundeddemicontinuousmappingofclassS+.81

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5.1ClassesofMultivaluedMappingsofMonotoneTypeWeintroduceinthissection,theclassesofmappingsweshallbedealingwithinthesequel.DenotebyFGS+=fT: G!XjTisbounded,demicontinuousandoftypeS+g;FGPM=fT: G!XjTisboundedandpseudomonotoneg;FGQM=fT: G!XjTisbounded,demicontinuousandquasimonotoneg:NextwedenenewclassesofmapsAGS+,AGPMandAGQMofmultival-uedmappingswhichareapproximatedbysingle-valuedmappingsofclassFGS+,FGPMandFGQM,respectively.TheS+,pseudomonotonicityandquasimono-tonicityconditionsareappliedtotheapproximatingsequencesratherthemappingitself.Denition5.1.1LetGbeanopensubsetofXandletT:XDT!2X.TisofclassAGS+ifthereisasequenceTnanapproximatingsequenceofTinFGS+withthefollowingproperties:A1ForeachC>0thereexistsaK0suchthathTnu;ui)]TJ/F26 11.955 Tf 31.799 0 Td[(Kforallu2 G,kukCandforalln2N:A2Lettn[0;1],un2 GandTmnbeasubsequenceofTn.Iftn!0,un!uinXandtnTmnun*zinX,thenz=0.A3Letun2 GandTmnbeasubsequenceofTn.Ifun*uinXandTmnun*winXandlimsupn!1hTmnun;unihw;ui,thenun!uinX,u2DTandw2Tu.Denition5.1.2TisofclassAGPMifthereisanapproximatingsequenceTninFGPMsatisfyingA1,A2andthefollowingcondition:A4Letun2 GandletTmnbeanysubsequenceofTn.Ifun*uinXand82

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Tmnun*winXandlimsupn!1hTmnun;unihw;ui,thenhTmnun;uni!hw;ui,andifu2 G,thenalsou2DTandw2Tu.Denition5.1.3TisofclassAGQMifthereisanapproximatingsequenceTninFGQMsatisfyingA1,A2andthefollowingcondition:A5Letun2 GandletTmnbeanysubsequenceofTn.Ifun*uinXandTmnun*winX,thenliminfn!1hTmnun;unihw;ui.Ifun!uinXandTmnun*winX,thenu2DTandw2Tu.IfT2FGS+,wechooseTn=Tforalln2N.ThenTnsatisesA1,A2andA3;henceT2AGS+andthusFGS+AGS+.ByasimilarargumentFGPMAGPMandFGQMAGQM.Inwhatfollowsallsubsequencesoffungorfxngwillstillbedenotedbyfung;fxng.Lemma5.1.4LetGbeanopensubsetofX.IfS2FGS+andT2AGQM,thenS+T2AGS+.Proof.LetTnbeanapproximatingsequenceofTandletRn=S+Tn.WeshowthatRn2FGS+.SinceSandTnareboundedanddemicontinuoussoisRn.Nextsupposethatun2 Gsuchthatun*uinXandlimsupn!1hRnun;un)]TJ/F26 11.955 Tf 12.065 0 Td[(ui0.Thenlimsupn!1hSun;un)]TJ/F26 11.955 Tf 11.955 0 Td[(uilimsupn!1hRnun;un)]TJ/F26 11.955 Tf 11.956 0 Td[(ui)]TJ/F19 11.955 Tf 19.261 0 Td[(liminfn!1hTnun;un)]TJ/F26 11.955 Tf 11.955 0 Td[(ui0:Wehaveusedthefactthat,sinceTnisdemicontinuousandquasimonotoneitimpliesliminfn!1hTnun;un)]TJ/F26 11.955 Tf 11.955 0 Td[(ui0:SinceSisoftypeS+,wehaveun!u.HenceRn2FGS+.SinceSisboundedandTnsatisesA1soRnsatisesA1.Lettn[0;1],83

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un2 GandRmnbeasubsequenceofRn.ThenbydenitionofRntnTmnun=tnRmnun)]TJ/F26 11.955 Tf 11.955 0 Td[(tnSun:Iftn!0,un!uinXtnRmnun*zinX,thenbytheboundednessofSwehavetnTmnun*zinX.SinceTnsatisesA2,z=0.ToshowthatRnsatisesA3,letun2 GandletRmnbeanysubsequenceofRn.Assumethatun*uinXandRmnun*winX.SinceSisbounded,wemayassumethatforasubsequenceofun,alsodenotedbyun,Sun*yinXtosomey2Xandlimsupn!1hRmnun;unihw;ui.ThenTmnun=Rmnun)]TJ/F26 11.955 Tf 11.293 0 Td[(Sun*w)]TJ/F26 11.955 Tf 11.708 0 Td[(yinX.SinceT2AGQM,wehaveliminfn!1hTmnun;unihw)]TJ/F26 11.955 Tf 11.707 0 Td[(y;uiandhencelimsupn!1hSun;un)]TJ/F26 11.955 Tf 11.955 0 Td[(uilimsupn!1hRmnun;un)]TJ/F26 11.955 Tf 11.955 0 Td[(ui)]TJ/F19 11.955 Tf 19.261 0 Td[(liminfn!1hTmnun;un)]TJ/F26 11.955 Tf 11.955 0 Td[(ui0:SinceSisoftypeS+wehaveun!uinXandSun*Su=yinX.Henceu2DTandw)]TJ/F26 11.955 Tf 11.955 0 Td[(Su2Tu,i.e,u2DS+Tandw2S+Tu.Remark5.1.5ItiseasilyseenthatconditionA3impliesA4andthatA4impliesA5.ThustheinclusionsAGS+AGPMAGQMarevalid.5.2TheDegreeforOperatorsofClassAGS+Denition5.2.1LetGbeaboundedopensubsetofX.AfamilyfTtj0t1gofmappingsof GintoXisanS+-homotopyifforanysequenceun2 Gand84

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tn[0;1]withtn!t,un*uinXandlimsupn!1hTtnun;un)]TJ/F26 11.955 Tf 11.955 0 Td[(ui0;wehaveun!uinXandTtnun*Ttu.Denition5.2.2AfamilyfHtj0t1gofmappingsHt:DHt!2XiscalledaahomotopyofclassHAGS+ifthereexistsasequenceHn;tofboundedS+-homotopiesHn;t:[0;1] G!XsuchthatHn;tsatisesconditionsA1andA2foreachxedt2[0;1]andthefollowingcondition:A6Iftn[0;1],tn!t,un2 G,un*uinX,Hmn:tnun*winXandlimsupn!1hHmn:tnun;unihw;ui,thenun!uinX,u2DHtandw2Htu.Thefollowinglemmacanbefoundin[23]whichwestateitwithoutproof.Lemma5.2.3LetS: G!XbeaboundeddemicontinuousS+-mappingandletT2AGS+.ThentheanehomotopyHt=)]TJ/F26 11.955 Tf 11.955 0 Td[(tS+tTisofclassHAGS+.Denition5.2.4LetGbeanopenboundedsubsetofXandletT2AGS+.Ify=2T@GDT,wedeneDT;G;y,thedegreeofToverGwithrespecttoy,byDT;G;y=fk2 Z=Z[f;+1gjkisaclusterpointofthesequencedS+Tn;G;yforsomeapproximatingsequenceTnofTg.HerethedegreedS+isthedegreeforboundeddemicontinuousS+mappingdenedbySkypnikin[28].ItisobviousthatDT;G;yisalwaysanonemptysubseton Z.IfDT;G;ycontainsonlyoneelementk,weshortlywriteDT;G;y=kinsteadoftheprecisenotationDT;G;y=fkg.ThisdegreeisthegeneralizationofthedS+sinceifT2FGS+,thenDT;G;y=dS+T;G;yWestatethefollowingtheoremwithoutproof.Forthedetailsoftheproofwerefer85

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thereaderto[23].Theorem5.2.5LetGbeanopenboundedsubsetofXandT2AGS+.Then:aIfDT;G;y6=0,thenthereexistsu2GDTsuchthaty2Tu.bLetG1andG2bedisjointopensubsetofG.Ify=2T GnG1[G2,thenDT;G;yDT;G1;y+DT;G2;y:cLetHt2HAGS+andletfyt:0t1gbeacontinuouscurveinX.Ifyt=2Ht@G,thenDHt;G;ytisconstantinforallt2[0;1].dLetJ:X!XbethedualitymappingofX.ThenDJ;G;y=8<:1ify2JG;0ify=2JG.Lemma5.2.6LetGbeanopenboundedsubsetofXsuchthat02G.IfT2AGS+andhw;ui>kwkkuk:1forallu2@GDTandw2Tu,thenDT;G;0=1andthereexistsu02GDTsuchthat02Tu0.Proof.WeconsiderthehomotopyHt=)]TJ/F26 11.955 Tf 12.159 0 Td[(tJ+tT2HAGS+,byLemma4.2.3forallu2 G.ThenH0u=Ju6=0forallu2@Gand0=2Tu=H1uforallu2@GDTby:1.If)]TJ/F26 11.955 Tf 11.956 0 Td[(tJu+tw=0:2forsomet2;1,u2@GDTandw2Tu,thenhw;ui=)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F26 11.955 Tf 11.955 0 Td[(t thJu;ui=)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F26 11.955 Tf 11.955 0 Td[(t tkuk2:86

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By4:2tkwk=)]TJ/F26 11.955 Tf 11.955 0 Td[(tkJuk=)]TJ/F26 11.955 Tf 11.955 0 Td[(tkuk:Thus,hw;ui=)]TJ/F26 11.955 Tf 10.494 8.087 Td[(tkwk tkuk=kwkuk:Thiscontradicts:1.Hence0=2Htuforallt2[0;1]andu2@GDT.ThereforeDT;G;0=DJ;G;0=dS+J;G;0=1andhencetheconclusionofthelemmafollows.Corollary5.2.7LetGbeanopenboundedsubsetofXwith02G.LetT2AGQMsatisfying.1forallu2@GDTandw2Tu.Letbeapositivenumber.ThenthedegreeDT+J;G;0=1Proof.SinceT2AGQMandJ2FGS+thenbyLemma4.1.4andfor>0,T+J2AGS+.Letw2Tuforu2@GDT.Thenhw+Ju;ui=hw;ui+hJu;ui>kwkkuk+kJukkuk=)]TJ/F19 11.955 Tf 9.299 0 Td[(kwk)]TJ/F26 11.955 Tf 20.59 0 Td[(kJukkuk>)]TJ/F19 11.955 Tf 9.299 0 Td[(kw+Jukkuk:Thisshowsthatw+JusatisesthehypothesisofLemma4.2.6.Henceifwecon-siderthehomotopyHt=)]TJ/F26 11.955 Tf 11.589 0 Td[(tJ+tT+J,thenHt2HAGS+byLemma4.2.3.Itfollowsthatforallt2[0;1]andu2@GDT,wehave0=2Htubythesameargumentinlemma4.2.6.ThereforeDT+J;G;0=dS+J;G;0=1.Wealsonoteherethat,theconclusionofthiscorollaryimplies,theinclusionTx+87

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Jx3hasnosolutioninDT@Gforevery>0.5.3StatementoftheEigenvalueProblemTheorem5.3.1LetGbeopenandboundedsubsetofXwith02G.LetS: G!XbeamappingofclassFGS+.LetT:XDT!2XbeofclassAGQMsatisfyingthecondition.1.Let0,bepositivenumbers.AssumethatPthereexists2;]suchthattheinclusionTx+Sx+Jx30:3hasnosolutionx2DTG:Thenithereexists0;x02;]DT@GsuchthatTx0+0Sx0+Jx030;4:4iiIf0=2TDT@G,TnsatisfyconditionSqonthe@GandpropertyPissatisedforevery2;0],thenthereexists0;x02;]DT@GsuchthatTx0+0Sx030.Proof.Assume.4isnottrue.Thenforevery2;],theinclusionTx+Sx+Jx30:5hasnosolutionx2DT@G:ConsiderthehomotopyinclusionHt;xTx+tSx+Jx30:688

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forallt2[0;1].WearegoingtoshowrstthatHtx=Ht;xisahomotopyofclassHAGS+.DeneHn;tx=Tnx+tSx+JxwhereTnisanapproximatingsequenceofT.ClearlyHn;tisboundedforallt2[0;1]andn2N.WenextshowthatitisanS+-homotopy.Foranysequencetj2[0;1]andxj2 G,considerthesequenceHn;tjxj=Tnxj+tjSxj+Jxj:7foreachn2N.Passingtosubsequenceifnecessary,wemayassumethattj!t,xj*xinXandlimsupj!1hHn;tjxj;xj)]TJ/F26 11.955 Tf 11.956 0 Td[(xi0:Then,itfollowsthatlimsupj!1hJxj;xj)]TJ/F26 11.955 Tf 11.955 0 Td[(xi=limsupj!1hHn;tjxj;xj)]TJ/F26 11.955 Tf 11.955 0 Td[(xi)]TJ/F19 11.955 Tf 19.261 0 Td[(liminfj!1hTnxj;xj)]TJ/F26 11.955 Tf 11.955 0 Td[(xi)]TJ/F26 11.955 Tf 19.261 0 Td[(tliminfj!1hSxj;xj)]TJ/F26 11.955 Tf 11.956 0 Td[(xi0:Wehaveusedthefactthatforeachn2N,Tn2FGQMandthatS2FGS+.SinceJisofclassS+wehavexj!x.HenceJxj!Jx,Tnxj*TnxandSxj*Sx.HenceHn;tjxj*Hn;tx.NextweshowthatHn;tsatisesconditionsA1,A2foreachxedt2[0;1]aswellasA6.BydenitionofHn;t,hHn;tx;xi=hTnx;xi+thSx;xi+hJx;xi)]TJ/F26 11.955 Tf 28.56 0 Td[(K+thSx;xi+kxk2)]TJ/F26 11.955 Tf 28.56 0 Td[(K+thSx;xi)]TJ/F26 11.955 Tf 28.56 0 Td[(M89

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forsomeM>0.HerewehaveusedthefactthatSisboundedandTsatisesA1.ToshowA2,letsn[0;1],xn2 GandHmn;tbeasubsequenceofHn;t.Supposesn!0,xn!xinXandsnHmn;t*zinX.Weshowthatz=0.BythedenitionofHn;t,snTmnxn=snHmn;txn)]TJ/F26 11.955 Tf 11.955 0 Td[(tsnSxn)]TJ/F26 11.955 Tf 11.955 0 Td[(snJxn:SinceJandSarebounded,snTmn*z.ButTsatisesA2.Hencez=0.NextweshowA6isalsosatised.Tothisend,letftng[0;1],xn2 Gbesuchthattn!t,xn*xinXandHmn;tnxn*winXwithlimsupn!1hHmn;tnxn;xnihw;xi;whichimplieslimsupn!1hHmn;tnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi0:ForasubsequenceweassumethatJxn*jandSxn*y.NowTmnxn=Hmn;tnxn)]TJ/F26 11.955 Tf 11.955 0 Td[(tnSxn)]TJ/F26 11.955 Tf 11.955 0 Td[(Jxn*w)]TJ/F26 11.955 Tf 11.955 0 Td[(j)]TJ/F26 11.955 Tf 11.955 0 Td[(ty:SinceT2AGQMwehaveliminfn!1hTmnxn;xnihw)]TJ/F26 11.955 Tf 11.955 0 Td[(j)]TJ/F26 11.955 Tf 11.955 0 Td[(ty;xi:Henceliminfn!1hTmnxn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(xi0:From.7,wehavehJxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi=hHmn;tnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi)]TJ/F26 11.955 Tf 19.261 0 Td[(tnhSxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi)-222(hTmnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi:Thereforelimsupn!1hJxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi0:90

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SinceJisoftypeS+,wehavethatxn!xandx2 G.HenceitfollowsthatJxn!Jx=jandSxn*Sx=y.SinceT2AGQM,wealsohavex2DTandw)]TJ/F26 11.955 Tf 11.955 0 Td[(Jx)]TJ/F26 11.955 Tf 11.956 0 Td[(tSx2Tx.Thus,wehavew2Tx+tSx+Jx=Htxandx2DT G.Nextweshowthat8t2[0;1],theinclusionHtx30isnotsolvableforallx2DT@G.Supposethisisfalse.ThenforanapproximatingsequenceTnofTwithtn2[0;1]andxn2@G,wehaveTnxn+tnSxn+Jxn=0:Iftn=0,wehaveTnxn+Jxn=0.Passingtosubsequenceifnecessary,weassumethatxn*xinX.SinceJisbounded,Jxn*j.HenceTmnxn*)]TJ/F26 11.955 Tf 9.299 0 Td[(j.SinceT2AGQMweconcludethatliminfn!1hTmnxn;xnih)]TJ/F26 11.955 Tf 34.537 0 Td[(j;xi;fromwhichfollowsliminfn!1hTmnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi0:Hencelimsupn!1hJxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi)]TJ/F19 11.955 Tf 31.88 0 Td[(liminfn!1hTmnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi0:SinceJisoftypeS+,wehavethatxn!xandx2@G.ThereforeJxn!Jx=j.AgainsinceT2AGQM,wehavex2DTand)]TJ/F26 11.955 Tf 9.299 0 Td[(Jx2TxorTx+Jx30forx2DT@G,i.e.,acontradictiontoourassumptionofthetheorem.Thereforetn6=0.Hencetn>0.Assumethattn!t=0.ThenTmnxn=)]TJ/F26 11.955 Tf 9.298 0 Td[(tnSxn)]TJ/F26 11.955 Tf 13.171 0 Td[(Jxn*)]TJ/F26 11.955 Tf 9.299 0 Td[(j,sinceSisbounded.Bythesameargumentabove,weget02T+JDT@G,i.e.,acontradiction.Itfollowsthatt>0.SinceSisbounded,wemayassumeSxn*y.ThenwehaveTmnxn=)]TJ/F26 11.955 Tf 9.299 0 Td[(tnSxn)]TJ/F26 11.955 Tf 11.955 0 Td[(Jxn*)]TJ/F26 11.955 Tf 9.298 0 Td[(ty)]TJ/F26 11.955 Tf 11.955 0 Td[(j:91

PAGE 99

Again,sinceT2AGQM,liminfn!1hTmnxn;xnih)]TJ/F26 11.955 Tf 34.537 0 Td[(ty)]TJ/F26 11.955 Tf 11.956 0 Td[(j;xi;whichimpliesliminfn!1hTmnxn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(xi0:SinceS2FGS+,limsupn!1hJxn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(xi)]TJ/F19 11.955 Tf 31.88 0 Td[(liminfn!1hTmnxn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(xi)]TJ/F26 11.955 Tf 19.261 0 Td[(tliminfn!1hSxn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(xi0:SinceJisoftypeS+,xn!xandx2@G.ThereforeJxn!Jx=jandSxn*Sx=y.AlsoTmnxn*)]TJ/F26 11.955 Tf 9.299 0 Td[(tSx)]TJ/F26 11.955 Tf 11.955 0 Td[(Jx;andhencex2DTand)]TJ/F26 11.955 Tf 9.299 0 Td[(tSx)]TJ/F26 11.955 Tf 12.393 0 Td[(Jx2TxorTx+tSx+Jx30forx2DT@G.ThisisacontradictiontoourassumptionthattheinclusionTx+tSx+Jx30hasnosolutionforx2DT@G.ThereforewehaveshownthatHt;xisanadmissiblehomotopyforwhich0=2HtDT@Gforallt2[0;1].HencethedegreeDHt;G;0isxedforallt2[0;1].ThereforeDT+S+J;G;0=DH;:;G;0=DH;:;G;0=DT+J;G;0=1:ThissaysthattheinclusionTx+Sx+Jx30hasasolutionx2DTGfor2;].ThisisacontradictiontoPandtheproofofiiscomplete.iiLetxn2DT@G,TnanapproximatingsequenceofT,n2;]besuch92

PAGE 100

thatTnxn+nSxn+1 nJxn=0:Weassumethatn!,xn*x,Sxn*yandJxn*j.Weconsiderthefollowtwocases.a=0andb>0.Forcasea,wehaveforasubsequenceTmnofTn;Tmnxn=)]TJ/F26 11.955 Tf 9.299 0 Td[(nSxn)]TJ/F19 11.955 Tf 13.718 8.088 Td[(1 nJxn!0::8SinceTnsatisesconditionSqonthe@G,wehavexn!xandhencex2@G.Alsoby.8,wehaveliminfn!1hTmnxn;xni=0:Itfollowsthatx2DTand02Tx.Thisimpliesthat02TDT@G,i.e,acontradictionwhichindicatesthatcaseaisimpossible.For>0,wehaveTmnxn=)]TJ/F26 11.955 Tf 9.298 0 Td[(nSxn)]TJ/F19 11.955 Tf 13.718 8.088 Td[(1 nJxn*)]TJ/F26 11.955 Tf 9.298 0 Td[(y:SinceT2AGQM,wehaveliminfn!1hTmnxn;xnih)]TJ/F26 11.955 Tf 34.537 0 Td[(y;xi;fromwhichfollowsliminfn!1hTmnxn;xn)]TJ/F26 11.955 Tf 11.956 0 Td[(xi0:Nowby.8hnSxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi=h1 nJxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi)-222(hTmnxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xiThereforelimsupn!1hSxn;xn)]TJ/F26 11.955 Tf 11.955 0 Td[(xi0:93

PAGE 101

SinceSisofclassS+,xn!x2@GandSxn*Sx=y:HenceTmnxn*)]TJ/F26 11.955 Tf 9.299 0 Td[(Sx.Itfollowsthatx2DTand)]TJ/F26 11.955 Tf 9.299 0 Td[(Sx2Tx.WeconcludethatTx+Sx30forx2DT@Gand2;].94

PAGE 102

References[1]V.Barbu,NonlinearsemigroupsanddierentialequationsinBanachspaces,Noordho,Leyden,1975.MR52:11166[2]J.BerkovitsandV.Mustonen,Onthetopologicaldegreeformappingsofmono-tonetype,NonlinearAnal.10,12986,1373-1389.[3]F.EBrowder,Fixedpointtheoryandnonlinearproblems,Bull.Amer.Math.Soc.9983,No.1,1-39.[4]F.EBrowder,Degreeofmappingfornonlinearmappingsofmonotonetype,Proc.Nat.Acad.Sci,USA80983,1771-1773.[5]F.EBrowder,Thedegreeofmappinganditsgeneralization,Contemp.Math.21983,15-40.[6]F.EBrowderandP.Hess,NonlinearmappingsofmonotonetypeinBanachspaces,J.Funct.Anal.11972,251-294.[7]Y.ZChen,Thegeneralizeddegreeforcompactperturbationsofm-accretiveop-eratorsandapplications,Nonlinear.Anal.TMA13989,393-403.[8]P.M.FitzpatrickandW.VPetryshyn,OnthenonlineareigenvalueproblemTu=Cu,involvingnoncompactabstractanddierentialoperators,Boll.Un.Math.Ital.,1597880-107.[9]X.FuandS.Song,Thegeneralizeddegreeformultivaluedcompactperturbationsofm-accretiveoperatorsandapplications,Nonlinear.Anal.,43001,767-776.[10]Z.Guan,Solvabilityofsemilinearequationswithcompactperturbationsofoper-atorsofmonotonetype,Proc.Amer.Math.Soc.121994,93-102.95

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[11]Z.GuanandA.G.Kartsatos,Adegreeformaximalmonotoneoperators,LecturenotesinPureandAppliedMathematics,178,115-130[12]Z.GuanandA.GKartsatos,OntheeigenvalueproblemforperturbationsofnonlinearaccretiveandmonotoneoperatorsinBanachspaces,Nonlinear.Anal.,27,No.2996,125-141[13]Z.Guan,A.G.KartsatosandI.V.SkypnikRangesfordenselydenedgener-alizedpseudomontoneperturbationsofmaximalmonotonemaps,J.DierentialEquations,188003,332-351.[14]F.-LHuangandH.-ZLi,Onthenonlineareigenvaluesforperturbationsofmono-toneandaccretiveoperatorsinBanachspaces,SichuanDaxueXuebaoJ.SichuanUniv.[15]A.G.KartsatosandI.V.Skypnik,TopologicaldegreetheoriesfordenselydenedmappingsinvolvingoperatorsoftypeS+,Adv.DierentialEquations,4999,413-456.[16]A.G.KartsatosandI.V.Skypnik,Atopologicaldegreefordenselydenedqua-sibounded~S+-perturbationsofmultivaluedmaximalmonotoneoperatorsinre-exiveBanachspaces,Abstr.Appl.Anal.,2005121-158.[17]A.G.KartsatosandI.V.Skypnik,Normalizedeigenvectorsfornonlinearabstractellipticoperators,J.DierentialEquations,155999,443-475.[18]A.G.KartsatosandI.V.Skypnik,InvarianceofdomainforperturbationsofmaximalmonotoneoperatorsinBanachspaces,Toappear.[19]A.G.Kartsatos,Newresultsintheperturbationstheoryofmaximalmonotonemapsandm-accretiveoperatorsinBanachspaces,Trans.Amer.Math.Soc.,348996,1663-1707.[20]A.G.Kartsatos,Ontheconnectionbetweentheexistenceofzerosandtheasymp-toticbehaviorofresolventsofmaximalmonotoneoperatorsinreexiveBanachspaces,Trans.Amer.Math.Soc.,350998,3967-3987.96

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[21]A.G.Kartsatos,ZerosofademicontinuousaccretiveoperatorsinBanachSpaces,J.IntegralEquations,8985,175-184.[22]N.Kenmochi,NonlinearoperatorsofmonotonetypeinreexiveBanachspacesandNonlinearPerturbations,HiroshimaMathJ.,4974229-263.[23]A.Kittila,Onthetopologicaldegreeofaclassofmappingsofmonotonetypeandapplicationstostronglynonlinearellipticproblems,Ann.Acad.Sci.Fenn.Ser.,AIMath.Dissertationes,91994,1-48.[24]N.G.Lloyd,DegreeTheory,CambridgeUniv.Press,Cambridge,1978.[25]T.W.Ma,Topologicaldegreeforset-valuedcompactvectoreldsinlocallyconvexspaces,DissertationesMath.,92972,1-43.[26]N.S.PapageorgiouandHuShouchan,GeneralizationsoftheBrowderdegreetheory,Trans.Amer.Math.Soc.,347,No.1995,233-259.[27]D.PascaliandS.Sburlan,Nonlinearmappingsofmonotonetype,SijthoandNoordhoIntern.Publ.,Bucuresti,Romania,andSijtho&Noordho,AlphenaandenRijn,1978.[28]I.V.Skrypnik,NonlinearHigherOrderEllipticEquations,NaukovaDumka,Kiev,1973.[29]S.L.Troyanski,Onlocallyuniformlyconvexanddierentialnormsincertainnon-separableBanachspaces,StudiaMath,37970/1971,173-180.[30]G.H.Yang,Therangesofnonlinearmappingsofmonotonetype,J.Math.Anal.Appl,173,165-172.[31]E.Zeidler,NonlinearFunctionalAnalysisanditsApplications,II/B,Springer-Verlag,NewYork,1990.97

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AbouttheAuthorTheauthor,JosephNiiOfoliQuarcoo,isanativeofGhanainWestAfrica.HehadiselementaryeducationuptotheBachelorofsciencedegreelevelinmath-ematicsinGhana.InAugust2000,hewasawardedtheUNESCOscholarshiptoparticipateintheMathematicsDiplomaprogramatthefamousAbdusSalamInter-nationalCenterforTheoreticalPhysicsinTrieste,Italy.HenishedtheDiplomasuccessfullyandagainawardedagraduateteachingassistantshiptotheUniversityofSouthFloridatopursueaPh.DinMathematicswhichhehassuccessfullycom-pleted.Hesincerelywishtothankallwhoinanywayhadcontributedtothisgreatachievement.


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