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

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

Material Information

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

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.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 92 pages.
Statement of Responsibility:
by Ibrahimou Boubakari.

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:
001969426 ( ALEPH )
271690468 ( OCLC )
E14-SFE0002198 ( USFLDC DOI )
e14.2198 ( USFLDC Handle )

Postcard Information

Format:
Book

Downloads

This item has 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


printinsert_linkshareget_appmore_horiz

Download Options

close
Choose Size
Choose file type
Cite this item close

APA

Cras ut cursus ante, a fringilla nunc. Mauris lorem nunc, cursus sit amet enim ac, vehicula vestibulum mi. Mauris viverra nisl vel enim faucibus porta. Praesent sit amet ornare diam, non finibus nulla.

MLA

Cras efficitur magna et sapien varius, luctus ullamcorper dolor convallis. Orci varius natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Fusce sit amet justo ut erat laoreet congue sed a ante.

CHICAGO

Phasellus ornare in augue eu imperdiet. Donec malesuada sapien ante, at vehicula orci tempor molestie. Proin vitae urna elit. Pellentesque vitae nisi et diam euismod malesuada aliquet non erat.

WIKIPEDIA

Nunc fringilla dolor ut dictum placerat. Proin ac neque rutrum, consectetur ligula id, laoreet ligula. Nulla lorem massa, consectetur vitae consequat in, lobortis at dolor. Nunc sed leo odio.