Implicit Functions and Solution Mappings: A View from Variational Analysis

388 Pages·2009·3.115 MB·English

by  Asen L. Dontchev, R. Tyrrell Rockafellar (auth.)

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Asen L. Dontchev (cid:129) R. Tyrrell Rockafellar
Implicit Functions
and Solution Mappings
A View from Variational Analysis
With 12 Illustrations
ABC
AsenL.Dontchev R.TyrrellRockafellar
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ISSN1439-7382 e-ISSN
ISBN978-0-387-87820-1 e-ISBN978-0-387-87821-8
DOI10.1007/978-0-387-87821-8
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Preface
Settingupequationsandsolvingthemhaslongbeenso importantthat,inpopular
imagination, it has virtually come to describe what mathematical analysis and its
applicationsare all about. A centralissue in the subjectis whetherthe solution to
anequationinvolvingparametersmaybeviewedasafunctionofthoseparameters,
andifso,whatpropertiesthatfunctionmighthave.Thisisaddressedbytheclassical
theoryofimplicitfunctions,whichbeganwithsinglerealvariablesandprogressed
through multiple variables to equations in infinite dimensions, such as equations
associatedwithintegralanddifferentialoperators.
A major aim of the book is to lay out that celebrated theory in a broader way
thanusual, bringingto lightmanyof its lesser knownvariants,forinstance where
standardassumptionsof differentiabilityare relaxed.However,anothermajor aim
is to explain how the same constellation of ideas, when articulated in a suitably
expanded framework, can deal successfully with many other problems than just
solvingequations.
Thesedays,formsofmodelinghaveevolvedbeyondequations,interms,forex-
ample, of problemsof minimizing or maximizing functions subject to constraints
whichmayincludesystemsofinequalities.Thequestioncomesupofwhetherthe
solution to such a problem may be expressed as a function of the problem’s pa-
rameters, but differentiability no longer reigns. A function implicitly obtainable
this manner may only have one-sided derivatives of some sort, or merely exhibit
Lipschitz continuity or something weaker. Mathematical models resting on equa-
tionsare replacedby “variationalinequality”models,whichare furthersubsumed
by“generalizedequation”models.
Thekeyconceptforworkingatthislevelofgenerality,butwithadvantageseven
in the context of equations, is that of the set-valued solution mapping which as-
signstoeachinstanceoftheparameterelementinthemodelallthecorresponding
solutions,if any.Thecentralquestioniswhethera solutionmappingcanbe local-
ized graphicallyin order to achievesingle-valuednessand in that sense producea
function,thedesiredimplicitfunction.
Inmodernvariationalanalysis,set-valuedmappingsareanacceptedworkhorse
in problem formulation and analysis, and many tools have been developed for
v
vi Preface
handlingthem.Therearehelpfulextensionsofcontinuity,differentiability,andre-
gularityofseveraltypes,togetherwithpowerfulresultsabouthowtheycanbeap-
plied. A corresponding further aim of this book is to bring such ideas to wider
attentionbydemonstratingtheiraptnessforthefundamentaltopicathand.
Inlinewithclassicalthemes,weconcentrateprimarilyonlocalpropertiesofso-
lutionmappingsthatcanbecapturedmetrically,ratherthanonresultsderivedfrom
topologicalconsiderationsorinvolvingexoticspaces.Inparticular,weonlybriefly
discusstheNash–Moserinversefunctiontheorem.Wekeeptofinitedimensionsin
Chapters 1 to 4, but in Chapters 5 and 6 provide bridges to infinite dimensions.
Global implicit functiontheorems, including the classical Hadamard theorem,are
notdiscussedinthebook.
InChapter1weconsidertheimplicitfunctionparadigmintheclassicalcaseof
thesolutionmappingassociatedwithaparameterizedequation.Wegivetwoproofs
oftheclassicalinversefunctiontheoremandthenderivetwoequivalentformsofit:
the implicitfunctiontheoremand the correctionfunctiontheorem.Then we grad-
ually relax the differentiability assumption in various ways and even completely
exitfromit,relyinginsteadontheLipschitzcontinuity.We also discusssituations
in which an implicit function fails to exist as a graphical localization of the so-
lution mapping, but there nevertheless exists a function with desirable properties
servinglocallyasaselectionoftheset-valuedsolutionmapping.Thischapterdoes
notdemandofthereadermorethancalculusandsomelinearalgebra,anditcould
thereforebeusedbybothteachersandstudentsinanalysiscourses.
Motivated by optimization problems and models of competitive equilibrium,
Chapter 2 movesinto wider territory.The questionsare essentially the same as in
the first chapter, namely, when a solution mapping can be localized to a function
withsomecontinuityproperties.Butitisnolongeranequationthatisbeingsolved.
Insteaditisaconditioncalledageneralizedequationwhichcapturesamorecom-
plicateddependenceandcovers,asaspecialcase,variationalinequalityconditions
formulatedintermsoftheset-valuednormalconemappingassociatedwithacon-
vex set. Although our prime focus here is variational models, the presentation is
self-contained and again could be handled by students and others without special
background.Itprovidesanintroductiontoasubjectofgreatapplicabilitywhichis
hardlyknowntothemathematicalcommunityfamiliarwithclassicalimplicitfunc-
tions,perhapsbecauseofinadequateaccessibility.
In Chapter 3 we departfrom insisting on localizationsthat yield implicitfunc-
tionsandapproachsolutionmappingsfromtheangleofa“varyingset.”Weidentify
continuitypropertieswhichsupporttheparadigmoftheimplicitfunctiontheoremin
aset-valuedsense.Thischaptermaybereadindependentlyfromthefirsttwo.Chap-
ter4continuestoviewsolutionmappingsfromthisanglebutinvestigatessubstitutes
forclassicaldifferentiability.Byutilizingconceptsofgeneralizedderivatives,weare
abletogetimplicitmappingtheoremsthatreachfarbeyondtheclassicalscope.
Chapter5 takesa differentdirection.ItpresentsextensionsoftheBanachopen
mappingtheoremwhichareshowntofitinfinite-dimensionallyintotheparadigmof
thetheorydevelopedfinite-dimensionallyinChapter3.Somebackgroundinbasic
functionalanalysisisrequired.Chapter6goesfurtherdownthatroadandillustrates
Preface vii
howsome ofthe implicitfunction/mappingtheoremsfromearlierin the bookcan
beusedinthestudyofproblemsinnumericalanalysis.
This bookis targetedata broadaudienceof researchers,teachersand graduate
students,alongwithpractitionersinmathematicalsciences,engineering,economics
andbeyond.Insummary,itconcernsoneofthechieftopicsinallofanalysis,his-
toricallyandnow,anaidnotonlyintheoreticaldevelopmentsbutalsoinmethods
forsolvingspecificproblems.Itcrossesthroughseveraldisciplinessuchasrealand
functionalanalysis,variationalanalysis,optimization,andnumericalanalysis,and
canbeusedinpartasagraduatetextaswellasareference.Itstartswithelemen-
tary results and with each chapter, step by step, opens wider horizonsby increas-
ing the complexity of the problems and concepts that generate implicit function
phenomena.
Manyexercisesareincluded,mostofthemsuppliedwithdetailedguides.These
exercises complement and enrich the main results. The facts they encompass are
sometimesinvokedinthesubsequentsections.
Eachchapterendswithashortcommentarywhichindicatessourcesintheliter-
aturefortheresultspresented(butisnotasurveyofalltherelatedliterature).The
commentariesto someofthe chaptersadditionallyprovidehistoricaloverviewsof
pastdevelopments.
WhidbeyIsland,Washington AsenL.Dontchev
August,2008 R.TyrrellRockafellar

Acknowledgements
SpecialthanksareowedtoourreadersMariusDurea,ShuLu,YoshiyukiSekiguchi
and Hristo Sendov, who gave us valuable feedback on the entire manuscript,
and to Francisco J. Arago´n Artacho, who besides reviewing most of the book
helpedusmasterfullywithallthe figures.Duringvariousstagesofthewritingwe
also benefited from discussions with Aris Daniilidis, Darinka Dentcheva, He´le`ne
Frankowska,MichelGeoffroy,AlexanderIoffe,StephenRobinson,VladimirVeliov,
andConstantinZa˘linescu.WearealsogratefultoMaryAnglinforherhelpwiththe
finalcopy-editingofthebook.
Theauthors
ix