USF Libraries
USF Digital Collections

The Leray-Schauder approach for the degree of perturbed maximal monotone operators

MISSING IMAGE

Material Information

Title:
The Leray-Schauder approach for the degree of perturbed maximal monotone operators
Physical Description:
Book
Language:
English
Creator:
Boubakari, Ibrahimou
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla.
Publication Date:

Subjects

Subjects / Keywords:
Demicontinuous
Eigenvalue
Invariance of domain
Quasimonotone
Strongly quasibounded
Yosida Approximant
Dissertations, Academic -- Mathematics and Statistics -- Doctoral -- USF   ( lcsh )
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Dissertation (Ph.D.)--University of South Florida, 2007.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by Ibrahimou Boubakari.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 92 pages.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001969426
oclc - 271690468
usfldc doi - E14-SFE0002198
usfldc handle - e14.2198
System ID:
SFS0026516:00001


This item is only available as the following downloads:


Full Text
xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam Ka
controlfield tag 001 001969426
003 fts
005 20081114150106.0
006 m||||e|||d||||||||
007 cr mnu|||uuuuu
008 081114s2007 flu sbm 000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0002198
035
(OCoLC)271690468
040
FHM
c FHM
d FHM
049
FHMM
090
QA36 (ONLINE)
1 100
Boubakari, Ibrahimou.
4 245
The Leray-Schauder approach for the degree of perturbed maximal monotone operators
h [electronic resource] /
by Ibrahimou Boubakari.
260
[Tampa, Fla.] :
b University of South Florida,
2007.
502
Dissertation (Ph.D.)--University of South Florida, 2007.
504
Includes bibliographical references.
516
Text (Electronic dissertation) in PDF format.
538
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
500
Title from PDF of title page.
Document formatted into pages; contains 92 pages.
590
Adviser: Athanassios Kartsatos, Ph.D.
599
001969426
653
Demicontinuous.
Eigenvalue.
Invariance of domain.
Quasimonotone.
Strongly quasibounded.
Yosida Approximant.
0 690
Dissertations, Academic
z USF
x Mathematics and Statistics
Doctoral.
773
t USF Electronic Theses and Dissertations.
856
u http://digital.lib.usf.edu/?e14.2198



PAGE 1

Operators by IbrahimouBoubakari Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:AthanassiosKartsatos,Ph.D. MarcusMcWaters,Ph.D. ArunavaMukherjea,Ph.D. BorisShekhtman,Ph.D. YunchengYou,Ph.D. DateofApproval: June8,2007 Keywords:Demicontinuous,Eigenvalue,Invarianceofdomain,Quasimonotone,Stronglyquasibounded,YosidaApproximant cCopyright2007,IbrahimouBoubakari

PAGE 2

TomyfatherYerimaBoubakariAhmadou

PAGE 3

IwishtoexpressmysincerethankstoalltheothermembersofmyPh.D.committee,ProfessorsMcWaters,Mukherjea,Shekhtman,YouandRabson,fortheirinterestandsupportduringthepreparationofthiswork. IwishtothanktheentirestaoftheUSFmathematicsdepartmentfortheirkindhospitalityduringmystayatUSF. Manythanksgotomyfriendsandfamilyfortheirpatience,interestandsupportthroughoutmystudentcareer. Aboveall,IthankALLAHwhomadeeverythingpossibleforme.

PAGE 4

Introduction1 1Preliminaries4 1.1MappingsofMonotoneType......................4 1.2DenitionoftheTopologicalDegree...................9 2ConstructionoftheDegree11 2.1DegreeforBoundedPerturbationsofType(S+)............11 2.2BasicPropertiesoftheDegree......................24 2.3DegreeforUnboundedPerturbationofType(S+)...........39 2.4DegreeforQuasimonotonePerturbation................42 3Applications46 3.1TheSubdierentialOperator.......................46 3.2ApplicationinPartialDierentialEquations..............48 3.3InvarianceofDomain...........................57 3.4EigenvalueProblem............................68 4ExistenceandSurjectivityResult76 4.1NoncoerciveMappings..........................76 4.2OddMappings..............................84 References88i

PAGE 6

IbrahimouBoubakari Abstract Berkovitshasdevelopedatopologicaldegreefordemicontinuousmappingsoftype(S+);andhasshownthatthedegreemappingisuniqueundertheassumptionthatitsatisescertaingeneralproperties.Heprovedthatiffisaboundeddemicontinousmappingoftype(S+),GisanopenboundedsubsetofX,and0=2f(@G);thenthereexists"0>0suchthatforevery"2(0;"0)wehave0=2(I+1

PAGE 8

Thesolvabilityofsuchproblemsisoftenachievedbyestimatingvaluesofthe\degree"functiond(F;G;y),whichisalwaysanintegerorzero.Here,GisanopenboundedsubsetofXandy=2F(@G).Theinteger-valuedfunctiondissaidtobeatopologicaldegreeifitsatisescertainnormalizing,additivityandhomotopyproperties.Itturnsoutthatifthisdegreeisnotzero,thentheequationF(x)=yhasasolutionx2G.Themostimportantpropertyofadegreefunctionisthehomotopyinvariance,whichwasoriginallydevelopedbyHenriPoincareandwhichconsistsofembeddingtheprobleminaparametrizedfamilyofproblemsandthenshowingthatsuchparametricfamiliesofoperatorsare,inasense,homotopictooperatorswithknowndegrees. Thenotionof\topologicaldegree"wasrstintroducedbyBrouwerin1912[7]forcontinuousmappingsbetweennitedimentionalspaces.Thistheorywasgeneralizedin1934byLerayandSchauderforoperatorsofthetypeITininnitedimentional1

PAGE 9

TherehavebeenvariousextensionsandgeneralizationsoftheLeray-Schauderde-greetheoryindierentdirections.Forexample,Skrypnikdevelopedin[49]atopolog-icaldegreeforboundeddemicontinuousmappingsofclass(S+),whichmapanopenboundedsubsetofaBanachspaceXintoitsdualX.Ontheotherhand,Browder[8]developedadegreetheoryfordemicontinuous(S+)-perturbationsofmaximalmono-toneoperators.ThisBrowderdegreeisobtainedasthelimitoftheSkypnikdegreesforassociatedboundeddemicontinuous(S+)-mappings.KartsatosandSkrypnikde-velopeddegreetheoriesin[26]fordenselydenedmappingsofmonotonetype. InthispaperweareconcentratingonthedemonstrationofthefactthattheLeray-Schaudertopologicaldegreetheorycanbeusedforthedevelopmentofatopologicaldegreetheoryformaximalmonotoneperturbationsofdemicontinuousoperatorsoftype(S+)inseparablereexiveBanachspaces.Thisisanextensionofthecorre-spondingBerkovits[5]degreedevelopmentforjustdemicontinuousoperatorsoftype(S+):ItisalsoavariantapproachtotheassociatedBrowderdegreedescribedabove.WealsodemonstratethepossibleapplicabilityofourresultsintheeldofPartialDierentialEquationsandinvarianceofdomainandeigenvalueproblems.Sincethe2

PAGE 10

Ourstudyisstructuredasfollows.Inchapteronewegivesomepreliminariesanddenitionsneededforourstudy.Chaptertwoisdevotedforthedevelopmentofourdegreetheoryforvariousperturbationsofmaximalmonotonemappingandthestudyofsomeadmissiblehomotopies.ChapterthreedealswithapplicationsintheeldofPartialDierentialEquations,thediscussionofproblemsofinvarianceofdomainandeigenvalues.Inchapterfourwediscussfurtherapplicationsbylookingatnoncoerciveaswellasoddmappings.3

PAGE 12

Itissaidtobe\cyclicallymonotone"ifhx0x1;u0i+:::+hxn1xn;un1i+hxnx0;uni0

PAGE 14

ThefollowingresultcanbefoundinKartsatosandSkrypnik[28].Proposition1.1.12

PAGE 16

LetXandYbetopologicalspacesandletObeaclassofopensubsetsGofX.ForeachGinO,weassociateaclassFGofmapsof DG2Oandf2FG,thentherestrictionfj

PAGE 17

ApplicationsofdegreetheoriesinvariousproblemsofNonlinearAnalysismaybefound,e.g.,inAdhikariandKartsatos[3],Berkovits[5],Browder[9],[14],Kartsatos[21],[22],Kartsatosetal.[23]-[30],Kittila[32],KobayashiandOtani[33],PascaliandSburlan[40],Petryshyn[41],[42],Skrypnik[48],[49],andZeidler[52]andthereferencestherein.Wenowstatethefollowingtheorem.Itisassumedthatallthehomotopiesinitareadmissibleandthedegreemappingdiswelldened.Theorem1.2.2 Proof.Itiseasytoseethatfy2FG:ConsiderthehomotopyT+(1s)f+s(fy),0s1,andthecontinuouscurvey(s)=(1s)y;0s1.Wecaneasilyseethaty(s)=2(T+(1s)f+s(fy))(@G),foranys2[0;1].Hencetheconclusionfollowsfromthehomotopyinvarianceproperty(iii)ofDenition1.2.1. 10

PAGE 18

Itshouldbenotedthatinthecaseofaboundeddemicontinuous(S+)-perturbationf;thisdegreecoincideswiththeBrowderdegree.Weshouldalsomentionthatif,inaddition,theoperatorTisboundeddemicontinuousmaximalmonotoneanddenedonthespaceX;thenthisdegreecoincideswiththeSkrypnik-Browder-Berkovitsde-greebecausethemappingT+fisthenaboundeddemicontinuous(S+)-mappimg.2.1DegreeforBoundedPerturbationsofType(S+) LetXbeaninnitedimensionalrealreexiveseparableBanachspace.WefurtherassumethatXandXarelocallyuniformlyconvex.LetGbeaboundedopensetinX.LetT:XD(T)!2Xbeamaximalmonotoneoperatorandf: WerstrecallthefollowingembeddingPropositionduetoBrowderandTon.Proposition2.1.1

PAGE 19

TheoperatorQisalsolinearandcompact,andsinceQ(H)isdenseinX,itfollowsthatQisinjective. Fortheconstructionofthedegreewewillneedthefollowinglemmas.TheproofofthenextonecanbefoundinZeidler[52,p.915].Lemma2.1.2

PAGE 20

Thefollowinglemmawas,essentially,rstprovedbyBrezis,CrandallandPazyin13

PAGE 22

Proof.Supposethattheconclusionisfalse.Thenthereexistsequencesf"ngR+;ftngR+andfxngAwithn#0,tn#0andI+1 foralln.Thissaysxn=1 andh(Ttn+f)(xn);xni=h(Ttn+f)(xn);1

PAGE 23

By(2.1.5)h(Ttn+f)(xn);xi!0 wendlimsupn!1h(Ttn+f)(xn);xnxi=limsupn!1h(Ttn+f)(xn);xni0: TheproofofthisinequalityfollowsexactlyasinTheorem1ofKartsatosandSkrypnik[28].Itisincludedhereinforcompleteness.Assumethatthisinequalityisnottrue.16

PAGE 24

Thus,hu;yi+hy;xyi
PAGE 25

Proof.Letx2X,s>0,andt>0.Theny=Tstx=(sT)1+tJ11x()x2(sT)1y+tJ1y()x2T1y s+tsJ1y s()y=sT1+stJ11x=sTstx:

PAGE 26

istruefors2(0;1);x2X.Letsn2(0;1);xn2Xbesuchthatsn!s02(0;1);xn!x0.From(2.1.9)weobtaintheexistenceofyn2TJsnxn;y02TJs0x0suchthatsnyn=J(xnJsnxn);s0y0=J(x0Js0x0):(2.1.10) Usingthis,themonotonicityoftheoperatorTandtheassumptions02D(T);02T(0);wehavekxnJsnxnk2=hJ(xnJsnxn);xnJsnxni=snhyn;xnJsnxnihJ(xnJsnxn);xni;

PAGE 28

Manytimes,theLeray-SchauderdegreedLSbelowisactuallytheNagumode-greefrom[39](cf.alsoRothe[46]).IthasthefourbasicpropertiesoftheoriginalLeray-Schauderdegree,butitenjoysthefollowingadvantage.IntheLeray-SchauderhomotopyxH(t;x);t2[0;1];x2 themappingx!H(t;x)iscompacton

PAGE 29

Weknowthat(i)and(ii)togetherimply(iii),but(iii)doesnotimply(i). FortherelevantdiscussionandexamplesthereaderisreferredtoRothe[46,pp.56-57].Severaltimesbelow,thehomotopyH(t;x)doessatisfy(i)and(ii)above,andisaLeray-Schauderhomotopy.TheNagumodegreeisofcoursetheLeray-Schauderdegree,bytheuniquenessoftheLeray-Schauderdegree,onhomotopiessatisfying(i)and(ii)above. ThenextlemmacontainsabasicinvariancepropertyoftheassociatedLeray-Schauderdegree.Theorem2.1.13 (ii)Itsucestoshowthatforanytwonumberst1;t22(0;t0]wehavedLS(U0(t1);G;0)=dLS(U0(t2);G;0):(2.1.11)22

PAGE 30

iscontinuouson[0;1] iscompactinX.Infact,thecontinuityoftheoperatorSfollowsimmediatelyfromthecontinuityofthemapping(t;x)!Ttx(seeProposition1.1.12),thedemicontinuityoffandthecompletecontinuityofthelinearcompactoperatorQQ:TherelativecompactnessofthesetS([0;1] Wearenowreadyforthedenitionofourdegreemapping.Denition2.1.14

PAGE 31

Wegivethefollowinglemmasforcompletenessandfuturereference.Lemma2.2.3

PAGE 32

Proof.Werstshowthatliminfn!1hhsn(xn);xnxi0(2.2.16)25

PAGE 33

thenlimsupn!1hfsn(xn);xnxilimsupn!1hhsn(xn)+fsn(xn);xnxiliminfn!1hhsn(xn);xnxilimsupn!1hhsn(xn)+fsn(xn);xnxi0:

PAGE 35

Proof.(i)WenotethatH(t;x):=tx+(1t)I+1 "QQ(J)i=jjxjj2+1t "jjQJ(x)jj2W>0 forallx6=0.Thus,forsmall">0;dLS(H(1;);G;0)=dLS(I;G;0)=dLS(H(0;);G;0)=d(J;G;0); (ii)Weshowonlyhecasey=0.Thecasey6=0issimilar.Assumethatd(T+f;G;0)6=0.Ifwealsohave0=2(T+f)(G),then,byLemma2.1.8,0=2(U0(t))( forallsmallt>0;whichimpliesdLS(U0(t);G;0)=0:28

PAGE 36

(iii)Westshow(a)thatthereexistst0>0suchthat0=2(I+1 (a)WehavegivenanevenmoreelaborateproofofsuchasituationinPart(vii)below,wherefs=sf1+(1s)f2:TheproofofPart(a)of(iii)isthereforeomitted. (b)WeobservethatthefunctionH1(s;x)=1 G) iscompact.Using(2.2.17),wenddLS(H1(s;);G;0)=dLS(H1(0;);G;0)=d(T+f+y(0);G;0);s2[0;1]: (iv)Weconsideronlythecasey=0:LetG1;G2beasin(iv).ByLemma2.1.8,whereA=

PAGE 37

(v)Wepicks02(0;1)andconsiderthefunctionH0(s;t;x):=[s0(1s)+s](Tt+f1)(x)+(1s)(1s0)f2(x): foreveryt2(0;t0];s2[0;1]:Assumethatthisnottrue.Thenthereexistsequencestn#0;sn2[0;1]withsn!~s;andxn2@Gsuchthatxn+1

PAGE 38

Weclaimthatlimsupn!1hfq(sn)(xn);xnx0i0:(2.2.21) Assumethatitisnottrue.Thenthereexistsubsequencesoffsngfxng;respectively,denotedagainbyfsng;fxng;respectively,suchthatlimn!1hfq(sn)(xn);xnx0i>0:

PAGE 39

InvokingLemma2.1.6,(i),withSequaltothezerooperator,weseethat(2.2.23)isimpossible.Consequently,(2.2.21)istrueandthefactthatfq(sn)isa(S+)-homotopyimpliesthatxn!x02@G:Startingnowwithlimsupn!1hfq(sn)(xn);xnx0i=0;(2.2.24) weobtain,possiblyforsuitablesubsequences,limn!1hTq(sn)tnxn;xnx0i0:(2.2.25) This,however,andLemma2.1.6implyx02D(Tq(~s))=D(T)andh2Tq(~s)x0:Thus,w=0=h+v2(Tq(~s)+fq(~s))(x0)(Tq(~s)+fq(~s))(D(T)\@G): Now,weseethatforeacht2(0;t0]themapping(s;x)!Tq(s)tx=q(s)Tq(s)txiscontinuousbyProposition1.1.12.ThisimpliesthecontinuityofthemappingH1:(s;x)!1 on[0;1] G):32

PAGE 40

iswelldenedandconstantforalls2[0;1]:Wealsopickt0sucientlysmallsothatd(T+f1;G;0)=dLS(I+1 forallt2(0;t0]:Weseenowthattheconstantdegreein(2.2.26)equals,fors=1;thedegreein(2.2.27),whichisindependentoftheparametert:Itfollowsthatforeverys2[0;1]andeveryt2(0;t0]thedegreein(2.2.26)isconstant.Takingthelimitinitast!0;weobtainthedesiredconclusiond(q(s)(T+f1)+(1q(s))f2;G;0)=d(T+f1;G;0): (vi)Weknowfrom(v)thatd(H(s;);G;0)=d(T+f1;G;0);s2(0;1]:

PAGE 41

Weclaimthat,eventually,sn>0:Otherwise,fromxn+1 Ifs0=0,thenwegetfrom(2.2.30)(1sn)hf2(xn);xnisnhTtnxn;xnisnhf1(xn);xnisnhf1(xn);xni;

PAGE 43

Wenowpickt0sucientlysmallsothatbesidesthevalidityof(2.2.28)wealsohavethevalidityofd(T+f1;G;0)=dLSI+1 Itiseasytoseenow,asbefore,thatthemappingH1(s;x)=x+1 isconstantfors2[0;1]:Thus,foraxedt2(0;t0];dLS(H1(1;);G;0)=dLS(I+1 (vii)Werstshowthatthereexistst0>0suchthat0=2I+1

PAGE 44

iswelldenedandconstantfors2[0;1]:ThisfollowsfromthefactthatthemappingH1(s;x)=1 iscontinuouson[0;1] G)37

PAGE 45

foralls2[0;1]:Takingthelimitaboveast!0;weobtaind(T+sf1+(1s)f2;G;0)=d(T+f1;G;0); ThehomotopystatementofTheorem2.2.6,(vi),isapplicableinmanyexistenceproblemsonNonlinearAnalysis.Infact,suchhomotopiesHcanbedenedasH(t;x)=t(T+f+"J)+(1t)"J;whereTismaximalmonotoneandfisdemicon-tinuous,boundedandoftype(S+):Inmanycases,homotopiesliketheonein(vii)ofTheorem1mayalsobeveryusefulinobtainingthesolvabilityofvariousrelevantproblemsinNonlinearAnalysis.

PAGE 46

InthissectionwedealwiththeextensionofthedenitionofthetopologicaldegreetheorytomapsoftheformT+f;whereTisastronglyquasiboundedmaximalmonotoneoperatorwith02T(0);andfisapossiblyunboundeddemicontinuousmapoftype(S+).Wedoassumethatfisstronglyquasibounded.TheideahereistosuitablyreduceT+ftodeneauniquetopologicaldegreefortheresultingboundedfunction.ThiswasdonebyBerkovitsin[5]forT=0.Weremindthereaderthat\demicontinuous,stronglyquasiboundedand(S+)"doesnotnecessarilyimply\bounded".Forexample,f(x)=ln(x+1);x2(1;1);isdemicontinuous,stronglyquasiboundedandoftype(S+)withconstantS=S(M)=M+1;wherejxjMandhf(x);xi=xf(x)M:However,itisnotbounded.TheextensionofthedegreeisaconsequenceofthefollowingfourLemmas.Lemma2.3.1 Proof.IfKisemptyornite,wearedone.Otherwise,letfxngKbeaninnitesequence.BythedenitionofK:wehaveTxn+fxn30.Sincexn2

PAGE 47

Assumethatthisisnottrue.Thenthereexistasubsequenceoffxng,denotedagainbyfxng,suchthatlimn!1hf(xn);xnxi>0: toobtainlimsupn!1hyn;xni
PAGE 48

TheproofofthefollowingLemmacanbefoundin[5,p.25].Lemma2.3.2 (T+f)1(0)G0G,(b) Proof.LetG1andG2betwoopensetssatisfying(a)and(b)ofLemma2.3.2.ItiseasytoseethatG1\G2alsosatisesthesameconditions,andbytheadditivity41

PAGE 49

Ify=2(T+f)(@G),wedene^d(T+f;G;y)=^d(T+fy;G;0):(2.3.38) Inparticular,iffisbounded,thenwecanchooseG0tobeG;andthus^d(T+f;G;0)=d(T+f;G;0).Thismeansthat^danddcoincideonboundeddemicontinuoustype(S+)-perturbationsofmaximalmonotoneoperators. IftheoperatorTisdemicontinuouswithintD(T)

PAGE 50

Proof.Supposethatthisisnottrue.Thenthereexistsequencesfng(0;1)withn#0andfxngAsuchthat(T+f+nJ)(xn)3y: 43

PAGE 51

Proof.Let1;2besuchthat0<1<2<0andconsiderthehomotopyT+(1s)(f+1J)+s(f+2J) orT+f+sJ; Nowwearereadytogivethefollowingdenition.Denition2.4.4 Proof.Letusnowverifythecondition(i0)-(iv0)givenatthebeginningofthissection.44

PAGE 52

(ii0)LetG1andG2betwodisjointopensubsetsofGandassumethaty=2 (iii0)LetH:[0;1]X!2Xbeaquasimonotoneperturbationofmaximalmonotonehomotopy,andletfy(s);0s1gbeacontinuouscurveinX.DenoteH(s;:)byTs(:)+fs(:)andsupposethatthereexistsr>0suchthat(Br(y(s)))\((Ts+fs)(@G))=;forall0s1.ProceedingasinLemma2.4.2wend"0>0suchthaty(s)=2((Ts+fs)+"J)(@G)for0s1and0<"<"0.Hence,byDenition2.4.4andthehomotopyinvarianceof^d,wegetdq(H(s;:);G;y(s))=dq(H(s;:)+"J;G;y(s))=constant (iv0)Foreverysucientlysmall">0,wehavedq(J;G;y)=^d(J+"J;G;y)=+1ify2J(G).Theproofisnowcomplete. 45

PAGE 53

LetCbeaconvexsetofX,afunctionisproperonCif(x)>forallx2Cand(x)<1atleastinonepointx2C. Let:X! @(x)iscalledthesubdierentialofatx.Remark3.1.1

PAGE 54

LetCXaclosedconvexsetandconsiderC:X!R+[f1gdenedbyC(x)=8<:0;ifx2C;1;otherwise.(3.1.1) ThefunctionCisproper,convexandlowersemicontinuousonX,andx2@C(x),forx2C,ifandonlyifhx;yxi0;y2C:

PAGE 55

whereH(t;x):=t(T+f+"J)x+(1t)Jx;(t;x)2[0;1] Weassumerstthattn>0;t0=0:Wehavehf(xn);xnihxn+f(xn)+"nJxn;xni=1

PAGE 56

whichimplieslimsupn!1hxn+f(xn);xnx0i0:(3.2.6) Ifweassumethatlimsupn!1hf(xn);xnx0i>0;(3.2.7) weobtainacontradiction,foranappropriatesubsequenceoffxng,ifnecessary,fromLemma2.1.6forT=0:Thus,wemusthavelimsupn!1hf(xn);xnx0i0:(3.2.8) Sincefisoftype(S+);wemusthavexn!x02@Gandf(x)*f(x0):Usingthisin3.2.4,wegetxn*f(x0)1 Alltheotherpossibilitiesforftngcanbehandledeithertrivially,orasabove.Theyarethereforeomitted.Itfollowsthat(3.2.3)istrueforallsucientlysmall">0;say,forall"2(0;"0]:49

PAGE 57

Workingwithsubsequences,ifnecessary,weseethat(3.2.9)impliesthat(3.2.7)isimpossible,andthat(3.2.8)impliesthatxn!x02 WeconsiderthespaceX=Wm;p0()withtheintegerm1,thenumberp2(1;1),andthedomainRN.WeletN0denotethenumberofallmulti-indices=(1;:::;N)suchthatjj=1+:::+Nm.Forevery=()jjm2RN0wehavetherepresentation=(;),where=()jjm2RN1,=()jj=m2RN2andN0=N1+N2.Welet(u)=(Du)jjm;(u)=(Du)jjm1;(u)=(Du)jj=m50

PAGE 58

@xii:Wealsosetq=p=(p1). Weconsiderthepartialdierentialoperatorindivergenceform(Au)(x)=Xjjm(1)jjDA(x;u(x);:::;Dmu(x));x2: Thereexistp2(1;1);c1>0and12Lq()suchthatjA(x;)jc1jjp1+1(x);x2;2RN0;jjm:(A2) TheLeray-LionsconditionXjj=m(A(x;;1)A(x;;2))(12)>0 issatisedforeveryx2,2RN1,1;22RN2with16=2.(A3) issatisedforeveryx2;1;22RN0.(A4) Thereexistc2>0,22L1()suchthatXjjmA(x;)c2jjp2(x);x2;2RN051

PAGE 59

Similarly,condition(A1),withAreplacedbyB;impliesthattheoperatorf:Wm;p0()!Wm;q();denedbyhf(u);vi=ZXjjmB(x;(u))Dv;u;v2Wm;p0(); WeconsideraproperclosedconvexsubsetKofXsuchthat02intK:Let'K:X!R+[f1gbedenedby Thefunction'Kisproper,convexandlowersemicontinuousonX;andx2@'K(x);forx2K;ifandonlyifhx;yxi0;y2K:

PAGE 61

50A(1;0)(x;y;)=x+1A(0;1)(x;y;)=0A(2;0)(x;y;)=xy3+3A(1;1)(x;y;)=34A(0;2)(x;y;)=0 Thepartialdierentialoperatorindivergenceform(Au)(x)=Xjjm(1)jjDA(x;u(x);:::;Dmu(x));x2: 5(x;y)@2u @x2(x;y)+@4u @x4(x;y)+3@4u @x2@y2(x;y)(x;y)2 CoecientsAareclearlyCaratheodoryfunctions.54

PAGE 62

50jx2+y2+j3 50jx2+y2+1+jjjA(1;0)(x;y;)j=jx+1jjxj+j1jjA(0;1)(x;y;)j=0jA(2;0)(x;y;)j=jxy3+3jjxy3j+jjjA(1;1)(x;y;)j=j34j3jjjA(0;2)(x;y;)j=0 Ifwechoosec1=3and1(x;y)=x2+y2+1+jxy3j,conditionA1isveried.(A2) Since(33)2+3(44)2>0forevery,2R3,with6= From(3 503 50)(00)+(11)2+(33)2+3(44)20 weobtaincondition(A3)(A4) 50)0+(x+1)1+(xy3+3)3+324=(x2+y2)0+x1+xy33+8 50+21+23+324=jj22 5022+224+(x2+y2)0+x1+xy3355

PAGE 63

50+0+1+22+3+224) Nowifwechoose2(x;y)=cu2 5(x;y)+u(x;y)+@u(x;y) NowwedeneanoperatoreT:W2;20()!W2;2by=Z(x2+y2+u3 5)v+(x+@u @x)@v @x+(xy3+@2u @x2)@2v @x2+3@2u @x@y@2v @x@ydxdyu;v2W2;20: 5)v+(x+@u @x)@v @x+(xy3+@2u @x2)@2v @x2+3@2u @x@y@2v @x@ydxdyjXjj2Zj(+1(x;y))jjv+@v @x+:::+@2v @y2dxdyjXjj2Z(jj2)1 2+Z(j1j2)1 2jjvjjW2;20C(jjujjW2;20)jjvjjW2;20 Nowforu;v2D(eT)wehaveheTueTv;uvi=Z(u3 5v3 5)(uv)+(@u @x@v @x)2+(@2u @x2@2v @x2)2+3(@2u @x@y@2v @x@y)2056

PAGE 64

Similarly,condition(A1);(A2)and(A4),withAreplacedbyB,impliesthattheoperatorf:W2;20()!W2;2,denedby=Z(B(0;0)((x;y);(u)))v+(B(1;0)((x;y);(u)))@v @x+:::+(B(0;2)((x;y);(u)))@2v @y2dxdy Ifwechoose,B(0;0)(x;y;)=y2B(1;0)(x;y;)=x+0B(0;1)(x;y;)=1B(2;0)(x;y;)=0B(1;1)(x;y;)=0B(0;2)(x;y;)=x3y+5 @y)@2u @x@y+@4u @y43.3InvarianceofDomain

PAGE 65

Wemayassumethatxn*x0andf(xn)*f.Wearegoingtoshowthatlimsupn!1hf(xn);xnx0i0:(3.3.15) Weassumeinsteadthatlimsupn!1hf(xn);xnx0i>0;

PAGE 66

Also,(3.3.14)impliesun*fandlimsupn!1hun;xnihf;x0i:

PAGE 67

ProofofTheorem3.3.1.Letp2(T+f)(D(T)\G).Weshowthatthereisaneighborhoodofplyingin(T+f)(D(T)\G).Withoutlossofgenerality,wemayassumethatp=0,02D(T)\G,02T(0),andf(0)=0.SinceT+fislocallyinjectiveonG,thereexistsanopenballBq(0)suchthat ByLemma3.3.2weknowthatthereexistaballBr(0)suchthat(T+f)(D(T)\@Bq(0))\Br(0)=;: hasnosolutionx2@Bq(0)forany0<<0,0
PAGE 68

andlimn!1hTsnxn+f(xn)h(tn);x0i=0 implylimsupn!1hTsnxn+f(xn)h(tn);xnx0i0: Assumethatitisnottrue.Thenthereexistasubsequenceoffxng,denotedagainbyfxng,suchthatlimn!1hf(xn);xnxi>0:61

PAGE 69

Then,by(3.3.20)hu;yi+hy;x0yi"isreplacedby"".BythemaximalmonotonicityofTwehavethatx2D(T)andu2Tx.HencewegetTtnxn+f(n;xn)+Jxn*0=u+j2Tx+Jx; Now,wexs2(0;s0],2(0;],2(0;]andconsiderthehomotopyfunctionx+1 whereH2(s;x)Ttx+f(s;x)+Jx(3.4.39) (3.4.39)isahomotopyoftype(S+)seeKartsatosandSkrypnik[29].Andusingthefactthat(Tt+J)(0)=0,wenotethat0=2H2(s;@G)foranys2[0;1]andthereforedB(H2(s;:);G;0)=dB(H2(1;:);G;0)=dB(H2(0;:);G;0)=dB(Tt+J;G;0)=1:

PAGE 80

becauseH1(s;;x)=H2(1;x).Thus02(T+f(;:)+J)(D(T)\G); (ii)TheProofgoestrueexactlyasinKartsatosandSkrypnikin[29]itisrepeatedhereforcompleteness. LetthesequencesfxngD(T)\@G,un2Txn,n2(0;1]besuchthatun+f(n;xn)+(1=n)Jxn=0(3.4.40) Wemayassumethatn!02[0;],xn*x0,f(n;xn)*fandJxn*j.Weconsidertwocases:(j) (jj).Wearegoingtoshowrstthatlimsupn!1hf(n;xn);xnx0i0:(3.4.41) Assumethecontrary.Thenwemayalsochoosefxng,orasubsequenceofitdenotedagainbyfxng,sothatlimn!1hf(n;xn);xnx0i>0:(3.4.42)73

PAGE 81

Since,by(3:4:40),un*c,wealsohavehun;xni=hun;xnx0i+hun;x0i; Nowwex(x;x)2G(T)andexaminehunx;xnxi0: SinceTismaximalmonotoneand(x;x)2G(T)isarbitrary,wegetx02D(T)andc2Tx0.However,lettingx=x0in(3:4:45)wegetacontradiction.Thus,(3:4:44)istrue.Weobservethathf(n;xn);xnx0i=hf(n;xn)f(0;xn);xnx0i+hf(0;xn);xnx0i:74

PAGE 83

SinceJ( ConsiderrstananehomotopyH(s;x)betweenT+fand Jx=(1s)Tx+(1s)f(x)+s Jx76

PAGE 84

thisimplythatforsomey12Tx1wehave(1s1)y1+(1s1)f(x1)+s1 Ifs1=0,then(4.1.3)isequivalenttoy1+f(x1)=0,whichcontradict(4.1.1).Ifs1=1,(4.1.3)gives

PAGE 85

J(x)+(1s)(J(x)J( Ifs1=0ors1=1wegetbytheinjectivityofJthatx1= xihJ( butontheotherhandwehavehs1

PAGE 86

Finallytheassertion(4.1.2)followsfrom(4.1.4)and(4.1.6). Proof.Assumerstthat02 Assumenextthat0=2 Itiseasytoseethatfisdemicontinuousandoftype(S+),andforanyy2Txwehavehy+f(x);x

PAGE 87

Weclaimthatthereexist1>0suchthat0=2Tx+(1s)(f(x)+J(x))+sf(x) forall0s1,0<<1andx2@G.Infact,ifthisisnottruethenthereexistsequencesfsng[0;1],fngwithn#0andfxng@Gsuchthat02Txn+(1sn)f(xn)+nJ(xn)+snfn(xn) whichfurtherimpliesthatthereexistyn2Txnsuchthatyn+(1sn)f(xn)+nJ(xn)+snfn(xn)=0 oryn+f(xn)=nJxnnJ(xn andsinceJisboundedandn#0,wehavethatyn+f(xn)!02 whichisacontradiction.Hence,byDenition2.4.4d(T+f;G;0)=d(T+f+J;G;0)80

PAGE 88

forany>0,with0<<1.Thus02 AsapplicationoftheaboveTheorem,welookatthefollowingCorollary.Corollary4.1.4 Weclaimthat limsupn!1hf(xn);xnxi0:(4.1.9) Infactifitisnottruethenthereexistasubsequenceoffxng,denotedagainbyfxng,suchthatlimn!1hf(xn);xnxi>0:

PAGE 89

Then,by(4.1.10)hy;zi+hz;xzi
PAGE 90

WecanderivethefollowingTheorem.Theorem4.1.5 jjxjj+jjTx+f(x)jj>0foralljjxjjR;y2Tx(4.1.12) Proof.Letp2Xbexed,wecanchooseR0Randk>0suchthatjjy+f(x)tpjjkforallt2[0;1]andjjxjjR0(4.1.13) Indeed,Ifitisnottrue,thenthereexistssequencesfxngXwithjjxnjj!1,yn2Txnandftng[0;1]suchthatjjyn+f(xn)tnpjj!0asn!1.Wecanassumethattn!t0,whichimpliesthatyn+f(xn)!t0p.Bytheproperty(B),fxngisbounded,whichisacontradictionwithourassumption.Thusbytheinvarianceunderhomotopywecanconcludethatd(T+f;BR0;p)=d(T+f;BR0;0).By(4.1.12)wehavehy+f(x);xi>jjTx+f(x)jjjjxjjforalljjxjj=R0:

PAGE 92

Proof.Obvious (T+f)1(0)eGG(ii) WecanstatenowthefollowingtheoremTheorem4.2.6

PAGE 93

2(T(x)T(x)) andef(x)=1 2(f(x)f(x)) clearlyeTismaximalmonotonewith02intD(eT),efisdemicontinuousoftype(S+),eTandefareoddon NowbyLemma(4.2.5)thereexistsanopensymmetricsubseteGofGsuchthat(eT+ef)1(0)eGandtherestrictionef: forall0
PAGE 94

87

PAGE 95

H.Amann:FixedPointEquationsandNonlinearEigenvalueProblemsinOr-deredBanachSpaces,SIAMRev.18(1976),no.4,620{709.[2] H.AmannandS.Weiss:Ontheuniquenessofthetopologicaldegree,Math.Z.130,(1973)39-54[3] D.R.AdhikariandA.G.Kartsatos:TopologicaldegreetheoriesandnonlinearoperatorequationsinBanachspaces,NonlinearAnal.(toappear).[4] V.Barbu:NonlinearSemigroupsandDierentialEquationsinBanachSpaces,NoordhoInt.Publ.,Leyden(TheNetherlands),1975.[5] J.Berkovits:Onthedegreetheoryfornonlinearmappingsofmonotonetype,Ann.Acad.Sci.Fenn.Ser.AI,Math.Dissertationes58(1986).[6] H.Brezis,M.G.CrandallandA.Pazy:Perturbationsofnonlinearmaximalmono-tonesetsinBanachspaces,Comm.PureAppl.Math.23(1970)123{144.[7] L.E.J.Brouwer:UberAbbildungvonMannigfaltigkeiten.Math.Ann.71(1972),97-115.[8] F.E.Browder:Multi-ValuedMonotoneNonlinearMappingsandDualityMap-pingsinBanachSpaces,Trans.Amer.Math.Soc.118(1965),338-351.[9] F.E.Browder:NonlinearoperatorsandnonlinearequationsofevolutioninBa-nachspaces,NonlinearFunctionalAnalysis,Proc.Sympos.PureAppl.Math.18(1976),1-308.88

PAGE 96

F.E.Browder:DegreeofmappingfornonlinearmappingsofmonotonetypeProc.Nat.Acad.Sci.80(1983),1771{1773.[11] F.E.Browder:Degreeofmappingfornonlinearmappingsofmonotonetype:denselydenedmapping,Proc.Nat.Acad.Sci.80(1983),2405{2407.[12] F.E.Browder:Degreeofmappingsfornonlinearmappingsofmonotonetype:stronglynonlinearmapping,Proc.Nat.Acad.Sci.80,(1983),2408{2409.[13] F.E.Browder:Fixedpointtheoryandnonlinearproblems,Bull.Amer.Math.Soc.9(1983),1{39.[14] F.E.Browder:ThedegreeofmappinganditsgeneralizationsContemp.Math.21(1983),15{40.[15] F.E.BrowderandB.A.Ton:NonlinearfunctionalequationsinBanachspacesandellipticsuper-regularization,Math.Z.105(1968),177{195.[16] F.E.BrowderandP.Hess:NonlinearmappingsofmonotonetypeinBanachspaces:J.Funct.Anal.11(1972),251{294.[17] R.F.Brown:ATopologicalIntroductiontoNonlinearAnalysis,SecondEditionBirhauser,Boston.Basel.Berlin.[18] I.Cioranescu:GeometryofBanachSpaces,DualityMappingsandNonlinearProblems,KluwerAcad.Publ.,Dordrecht,1990.[19] L.Fuhrer:EinelementareranalytischerBeweiszurEindeutigkeitdesAbbil-dungsgradesimRn.(German)Math.Nachr.54(1972),259-267.[20] Z.Guan,andA.G.Kartsatos:OntheeigenvalueProblemforPerturbationsofNonlinearAccretiveandMonotoneOperatorsinBanachSpaces.NonlinearAnal.,27,No.2(1996),125-141.[21] A.G.Kartsatos:Newresultsintheperturbationtheoryofmaximalmonotoneandm-accretiveoperatorsinBanachspaces,Trans.Amer.Math.Soc.,348(1996),1663{1707.89

PAGE 97

A.G.Kartsatos:Ontheconnectionbetweentheexistenceofzerosandtheasymp-toticbehaviorofresolventsofmaximalmonotoneoperatorsinreexiveBanachspaces,Trans.Amer.Math.Soc.350(1998),3967{3987.[23] A.G.KartsatosandJ.Lin:Homotopyinvarianceofparameter-dependentdo-mainsandperturbationtheoryformaximalmonotoneandm-accretiveoperatorsinBanachspaces,Adv.DierentialEquations8(2003),129{160.[24] A.G.KartsatosandJ.Quarcoo:AnewtopologicaldegreetheoryfordenselydenedperturbationsofmultivaluedmaximalmonotoneoperatorsinreexiveBanachspaces,nonlinearanal.(toappear).[25] A.G.KartsatosandI.V.Skrypnik:Normalizedeigenvectorsfornonlinearabstractandellipticoperators,J.DierentialEquations155(1999),443{475.[26] A.G.KartsatosandI.V.Skrypnik:Topologicaldegreetheoriesfordenselydenedmappingsinvolvingoperatorsoftype(S+),Adv.ierentialEquations4(1999),413{456.[27] A.G.Kartsatos,andI.V.Skrypnik:Theindexofacriticalpointfordenselydenedoperatorsoftype(S+)LinBanachspaces.Trans.Amer.Math.Soc.354(2002),no.4,1601-1630.[28] A.G.KartsatosandI.V.Skrypnik:Anewtopologicaldegreetheoryfordenselydenedquasibounded(~S+)-perturbationsofmultivaluedmaximalmonotoneop-eratorsinreexiveBanachspaces,Abstr.Appl.Anal.(2005),no.2,121{158.[29] A.G.KartsatosandI.V.Skrypnik:OntheeigenvalueproblemforperturbednonlinearmaximalmonotoneoperatorsinreexiveBanachspaces,Trans.Amer.Math.Soc.358(2006),3851{3881.[30] A.G.KartsatosandI.V.Skrypnik:DegreetheoriesandinvarianceofdomainforperturbedmaximalmonotoneoperatorsinBanachspaces,Adv.DierentialEquations(toappear).90

PAGE 98

N.Kenmochi:NonlinearoperatorsofmonotonetypeinreexiveBanachspacesandnonlinearperturbations,HiroshimaMath.J.4(1974),229{263.[32]A.Kittila:Onthetopologicaldegreeforaclassofmappingsofmonotonetypeandapplicationstostronglynonlinearellipticproblems,Ann.Acad.Sci.Fenn.Ser.AI,Math.Dissertationes91(1994).[33] J.KobayashiandM.Otani:Topologicaldegreefor(S)+-mappingswithmaximalmonotoneperturbationsanditsapplicationstovariationalinequalitiesNonlinearAnal.59(2004),147{172.[34] M.A.Kranosel'skii:TopologicalMethodsintheTheoryofNonlinearintegralEquations:PergamonPress,Oxford,1964.[35] I.Lakshmikantham,andS.Leela:NonlinearDierentialEquationsinAbstractSpaces.PergamonPress,Oxford(1981).[36] J.LerayandJ.Schauder:Topologieetequationsfonctionnelles,Ann.Sci.EcoleNorm.Sup.51(1934),45{78.[37] N.G.Lloyd,DegreeTheory,CambridgeUniversityPress,Cambridge,1978.[38] C.MorticiandS.Sburlan:BifurcationsforMonotoneTypeOperators:An.St.Univ.OvidiusConstanta,6(2),1998,87-94.[39] M.Nagumo:Degreeofmappingsinconvexlineartopologicalspaces,Amer.J.Math.73(1951),497{511.[40] D.PascaliandS.Sburlan:NonlinearMappingsofMonotoneType,SijthoandNoordhoof,Bucharest,1978.[41] W.V.Petryshyn:Approximation-solvabilityofNonlinearFunctionalandDier-entialEquations,MarcelDekker,NewYork,1993.[42] W.V.Petryshyn:GeneralizedTopologicalDegreeandSemilinearEquations.Cam-bridgeTractsinMathematics,117,CambridgeUniv.Press,Cambridge,1995.91

PAGE 99

R.T.Rockafellar:Onthemaximalityofsumsofnonlinearmonotoneoperators,Trans.Amer.Math.Soc.,149(1970),75-88.[44]R.T.Rockafellar:Onthemaximalityofthesubdierentialmappings,PacicJournalofMathematics,33,No.1,1970[45] R.T.Rockafellar:Convexanalysis,PrincetonUniversityPress,1969.[46] E.H.Rothe:IntroductiontoVariousAspectsofDegreeTheoryinBanachSpaces,MathematicalSurveysandMonographs,23,Providence,RhodeIsland,1986.[47] S.Simons:MinimaxandMonotonicity,Lect.NotesinMath.#1693,Springer-Verlag,Berlin,1998.[48] I.V.Skrypnik:NonlinearEllipticEquationsofHigherOrder,NaukovaDumka,Kiev,1973.[49] I.V.Skrypnik:MethodsforAnalysisofNonlinearEllipticBoundaryValueProb-lems,TranslationsofMathematicalMonographs,139,Providence,RhodeIsland,1994.[50] S.L.Troyanski:Onlocallyuniformlyconvexanddierentialnormsincertainnon-separableBanachspaces.StudiaMath,37(1970/1971),173-180[51] K.Yosida:FunctionalAnalysis,Springer-Verlag,Berlin,1965.[52] E.Zeidler,FunctionalAnalysisandItsApplications,II/B,Springer-Verlag,NewYork,1990.92