Applied Nonlinear Functional Analysis: an Introduction

621 Pages·2018·2.968 MB·English

by  Papageorgiou, Nikolaos S., Winkert, Patrick

NikolaosS.PapageorgiouandPatrickWinkert
AppliedNonlinearFunctionalAnalysis
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Nikolaos S.Papageorgiou and Patrick Winkert
Applied Nonlinear
Functional
Analysis
|
An Introduction
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MathematicsSubjectClassification2010
26-XX,28-XX,46-XX,47-XX,49-XX
Authors
Prof.Dr.NikolaosS.Papageorgiou
NationalTechnicalUniversityofAthens
DepartmentofMathematics
ZografouCampus
15780Athens
Greece
[e-mail]
Dr.PatrickWinkert
TechnischeUniversitätBerlin
InstitutfürMathematik
Straßedes17.Juni136
10623Berlin
Germany
[e-mail]
ISBN978-3-11-051622-7
e-ISBN(PDF)978-3-11-053298-2
e-ISBN(EPUB)978-3-11-053183-1
LibraryofCongressControlNumber:2018939852
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2018WalterdeGruyterGmbH,Berlin/Boston
Coverimage:NikolaosS.Papageorgiou,PatrickWinkert
Typesetting:le-texpublishingservicesGmbH,Leipzig
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
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|
Thisbookisdedicatedinmemoryofthefirstauthor’smother
M.S.Papageorgiou
andinmemoryofthesecondauthor’sfather
WolfgangWinkert
whobothpassedawayduringitspreparation.
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Preface
TheaimofthisbookistopresentthefoundationsofmodernNonlinearFunctional
Analysisandequipthereaderwithallthenecessarytoolstocontinuewiththeoretical
and/orappliedresearchinthefield.NonlinearFunctionalAnalysisisaverybroad
subjectandhasapplicationsinmanydifferentareasofphysics,mechanics,engineering,
andeconomics.Infact,itemergedasadistinctdisciplinewithinmathematicalanalysis
specificallyasawaytoaddresstheseneedsinamathematicallyrigorousway.Thisway
NonlinearFunctionalAnalysisdistinguisheditselffromtheclassicalLinearFunctional
Analysisandacquiredaninterdisciplinarycharacter.
Thepresentbookprovidesastartingpointtofollowsomeofthemainpathsof
NonlinearFunctionalAnalysis,especiallythoseleadingtoapplications.Thegoalis
topresentthetheoriesandtechniquestothenewcomer,whichwillallowhim/herto
proceedtomorespecializedtopics.Thefirstthreechapterspresentthemainelements
oftopology,measuretheory,andBanachspacetheory,whichareneededtoproceed
further.Inthelastthreechapterswepresentmoreadvancedandspecializedtopicsthat
aremotivatedbytheapplications.InChapter4weexaminecertainspacesoffunctions
andmeasuresthatprovidethefunctionalframeworkintheappliedproblems.Wedeal
withLebesgue,Lebesgue-Bochner,andSobolevspaces,whicharethebasictoolsinthe
studyofboundaryvaluedproblems.Wealsostudyspacesofabsolutelycontinuous
functions,offunctionsofboundedvariation,andofmeasuresthateventuallyleadto
Youngmeasures.Alltheseconstitutethemoderntoolsindealingwithproblemsof
thecalculusofvariations,controltheoryandoptimization,aswellasmathematical
economics.InChapter5wedealwithnonsmoothandmultivaluedanalysis,twofieldsof
mathematicalanalysisthatemergedsimultaneouslyintheearly1960’sanddeveloped
inparallel,feedingeachotherwithnewnotionsandmethods.Asaresult,wedealwith
convexfunctionsandtheirdualityandsubdifferentialtheory.Wealsoexaminethe
approximationpropertiesofsetsandextendthesubdifferentialtheorytothenonconvex
oneintermsoflocallyLipschitzfunctionsinthesenseofClarke.Furthermore,we
presentthemaintopologicalandmeasuretheoreticaspectsofset-valuedmapswith
applications to integral functionals. In Chapter 6 we finally study topics that are
traditionallyassociatedwithwhatiscalled“NonlinearAnalysis.”Theseareoperators
ofmonotonetype,degreetheory,fixedpointtheory,variationalprinciplessuchas
Ekeland’sVariationalPrinciple,andvariationalconvergencesuchasΓ-orepigraphical
convergence.Withthischoiceofmaterial,webelievethatthereaderwillbeproperly
equippedattheendtodoresearchinthisexcitingfieldofmathematicalanalysis.Each
chapterisfollowedbyatleast50problems.Weencouragethereadertotrythemin
ordertotesthis/herunderstandingofthematerial.Thesolutionstotheproblemswill
bepostedonthepersonalsiteofthesecondauthor.Ourhopeisthatthereader,with
thehelpofthematerialinthisbook,canproceedwithconfidenceinthemanydifferent
partsofthisfield.
https://doi.org/10.1515/9783110532982-201
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VIII | Preface
FinallytheauthorswishtothankDr.ApostolosDamialis,MariaDassing,andNadja
SchedensackofDeGruyterfortheirkindsupportandhelpduringthepreparationof
thisbook.
NikolaosS.Papageorgiou,Athens,Greece
PatrickWinkert,Berlin,Germany
January2018
Brought to you by | University of Groningen
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