Topological Analysis

495 Pages·2012·4.406 MB·English

by  Martin Väth

De Gruyter Series in Nonlinear Analysis
and Applications 16
Editor in Chief
Jürgen Appell, Würzburg, Germany
Editors
Catherine Bandle, Basel, Switzerland
Alain Bensoussan, Richardson, Texas, USA
Avner Friedman, Columbus, Ohio, USA
Karl-Heinz Hoffmann, Munich, Germany
Mikio Kato, Kitakyushu, Japan
Umberto Mosco, Worcester, Massachusetts, USA
Louis Nirenberg, New York, USA
Boris N. Sadovsky, Voronezh, Russia
Alfonso Vignoli, Rome, Italy
Katrin Wendland, Freiburg, Germany

Martin Väth
Topological Analysis
From the Basics to the Triple Degree
for Nonlinear Fredholm Inclusions
De Gruyter
Mathematics Subject Classification 2010: Primary: 47H11, 47H05, 47H08; Secondary:
54C60, 54C55, 54C20, 47H10, 55M20, 54E35, 46T05, 47A53, 54F45, 54H25, 46T20,
54C15, 54C25, 54C10, 54E50, 55M15, 55M10, 55M25, 55P05.
ISBN 978-3-11-027722-7
e-ISBN 978-3-11- 027733-3
ISSN 0941-813X
Library of Congress Cataloging-in-Publication Data
A CIP catalog record for this book has been applied for at the Library of Congress.
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available in the internet at http://dnb.dnb.de.
© 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston
Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de
Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen
 Printed on acid-free paper
Printed in Germany
www.degruyter.com
Preface
Topological methods are among the most important theoretical tools in analysis,
usefulformanyquestionsarisinginanalysis,concerningfinite-dimensional prob-
lems or also infinite-dimensional problems like differential or integral equations.
One particularly important tool is degree theory which was originally developed
byLerayandSchauderforequationsofaratherspecificformandwhichismean-
while available for a much larger class of equations, even those which involve
noncompact situations ormultivaluedmaps.
Thismonographaimstogiveaself-contained introductionintothewholefield:
Requiring essentially only basic knowledge of elementary calculus and linear al-
gebra, it provides all required background from topology, analysis, linear and
nonlinearfunctionalanalysis,andmultivaluedmaps,containing evenbasictopics
likeseparation axioms, inverse andimplicit function theorems, theHahn-Banach
theorem,Banachmanifolds, orthemostimportantconceptsofcontinuity ofmul-
tivaluedmaps. Thus,itcanbeusedasadditionalmaterialinbasiccoursesonsuch
topics. Themainintention, however, is toprovide also additional information on
somefinepointswhichareusuallynotdiscussed insuchintroductory courses.
The selection of the topics is mainly motivated by the requirements for de-
gree theory which is presented in various variants, starting from the elementary
Brouwer degree (in Euclidean spaces and on manifolds) with several of its fa-
mous classical consequences, up to a general degree theory for function triples
which applies for a large class of problems in a natural manner. Although it has
been known to specialists that, in principle, such a general degree theory must
exist, this is probably the first monograph in which the corresponding theory is
developedindetail.
Berlin,March2012 MartinVäth

Contents
Preface v
1 Introduction 1
I Topologyand MultivaluedMaps
2 MultivaluedMaps 9
2.1 NotationsforMultivalued MapsandAxioms ................ 9
2.1.1 Notations ...................................... 9
2.1.2 Axioms ....................................... 11
2.2 Topological NotationsandBasicResults ................... 17
2.3 Separation Axioms .................................... 24
2.4 UpperSemicontinuous MultivaluedMaps .................. 43
2.5 ClosedandProperMaps ................................ 52
2.6 Coincidence PointSetsandClosedGraphs .................. 55
3 MetricSpaces 59
3.1 NotationsandBasicResultsforMetricSpaces ............... 59
3.2 ThreeMeasuresofNoncompactness ....................... 67
3.3 Condensing Maps ..................................... 75
3.4 Convexity ............................................ 84
3.5 TwoEmbeddingTheoremsforMetricSpaces ............... 89
3.6 SomeOldandNewExtensionTheoremsforMetricSpaces ..... 96
4 SpacesDefinedbyExtensions,Retractions,orHomotopies 105
4.1 AEandANESpaces ................................... 105
4.2 ANRandARSpaces ................................... 107
4.3 ExtensionofCompactMapsandofHomotopies ............. 114
4.4 UV1 andR SpacesandHomotopicCharacterizations ....... 122
ı
5 AdvancedTopologicalTools 129
5.1 SomeCoveringSpaceTheory ............................ 129
viii Preface
5.2 AGlimpseonDimensionTheory ......................... 133
5.3 VietorisMaps......................................... 140
II Coincidence Degree forFredholm Maps
6 SomeFunctionalAnalysis 147
6.1 BoundedLinearOperatorsandProjections .................. 147
6.2 LinearFredholmOperators .............................. 160
7 OrientationofFamiliesofLinearFredholmOperators 169
7.1 OrientationofaLinearFredholmOperator .................. 169
7.2 OrientationofaContinuous Family ....................... 178
7.3 OrientationofaFamilyinBanachBundles ................. 182
8 SomeNonlinearAnalysis 197
8.1 ThePointwiseInverseandImplicitFunctionTheorems ........ 197
8.2 OrientedNonlinear FredholmMaps ....................... 203
8.3 OrientedFredholm MapsinBanachManifolds .............. 204
8.4 APartialImplicitFunctionTheoreminBanachManifolds ..... 214
8.5 TransversalSubmanifolds ............................... 220
8.6 Parameter-Dependent Transversality andPartialSubmanifolds .. 226
8.7 OrientationonSubmanifolds andonPartialSubmanifolds ...... 229
8.8 ExistenceofTransversal Submanifolds ..................... 231
8.9 PropernessofFredholmMaps ............................ 234
9 TheBrouwerDegree 237
9.1 Finite-Dimensional Manifolds ............................ 237
9.2 OrientationofContinuous MapsandofManifolds ............ 248
9.3 TheCr BrouwerDegree ................................ 255
9.4 Uniqueness oftheBrouwerDegree ........................ 261
9.5 ExistenceoftheBrouwerDegree ......................... 279
9.6 SomeClassicalApplications oftheBrouwerDegree .......... 293
10 TheBenevieri–FuriDegrees 309
10.1 FurtherPropertiesoftheBrouwerDegree ................... 310
10.2 TheBenevieri–Furi C1 Degree ........................... 318
Preface ix
10.3 TheBenevieri–Furi Coincidence Degree ................... 324
III Degree Theory forFunctionTriples
11 FunctionTriples 339
11.1 FunctionTriplesandTheirEquivalences ................... 341
11.2 TheSimplifierProperty ................................. 355
11.3 HomotopiesofTriples .................................. 361
11.4 LocallyNormalTriples ................................. 365
12 TheDegreeforFinite-DimensionalFredholmTriples 367
12.1 TheTripleVariantoftheBrouwerDegree .................. 367
12.2 TheTripleVariantoftheBenevieri–Furi Degree ............. 380
13 TheDegreeforCompactFredholmTriples 391
13.1 TheLeray–Schauder TripleDegree ........................ 391
13.2 TheLeray–Schauder Coincidence Degree .................. 404
13.3 ClassicalApplications oftheLeray–Schauder Degree ......... 407
14 TheDegreeforNoncompactFredholmTriples 413
14.1 TheDegreeforFredholmTripleswithFundamental Sets ....... 414
14.2 HomotopicTestsforFundamentalSets ..................... 429
14.3 TheDegreeforFredholmTripleswithConvex-fundamental Sets 437
14.4 Countably Condensing Triples ........................... 448
14.5 ClassicalApplications intheGeneralFramework ............ 456
14.6 ASampleApplication forBoundaryValueProblems .......... 462
Bibliography 465
IndexofSymbols 475
Index 477