Topological Analysis (eBook)

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Martin Väth, Freie Universität Berlin, Germany.

Preface 5
1 Introduction 11
I Topology and Multivalued Maps 17
2 Multivalued Maps 19
2.1 Notations for Multivalued Maps and Axioms 19
2.1.1 Notations 19
2.1.2 Axioms 21
2.2 Topological Notations and Basic Results 27
2.3 Separation Axioms 34
2.4 Upper Semicontinuous Multivalued Maps 53
2.5 Closed and Proper Maps 62
2.6 Coincidence Point Sets and Closed Graphs 65
3 Metric Spaces 69
3.1 Notations and Basic Results for Metric Spaces 69
3.2 Three Measures of Noncompactness 77
3.3 Condensing Maps 85
3.4 Convexity 94
3.5 Two Embedding Theorems for Metric Spaces 99
3.6 Some Old and New Extension Theorems for Metric Spaces 106
4 Spaces Defined by Extensions, Retractions, or Homotopies 115
4.1 AE and ANE Spaces 115
4.2 ANR and AR Spaces 117
4.3 Extension of Compact Maps and of Homotopies 124
4.4 UV8 and Rd Spaces and Homotopic Characterizations 132
5 Advanced Topological Tools 139
5.1 Some Covering Space Theory 139
5.2 A Glimpse on Dimension Theory 143
5.3 Vietoris Maps 150
II Coincidence Degree for Fredholm Maps 155
6 Some Functional Analysis 157
6.1 Bounded Linear Operators and Projections 157
6.2 Linear Fredholm Operators 170
7 Orientation of Families of Linear Fredholm Operators 179
7.1 Orientation of a Linear Fredholm Operator 179
7.2 Orientation of a Continuous Family 188
7.3 Orientation of a Family in Banach Bundles 192
8 Some Nonlinear Analysis 207
8.1 The Pointwise Inverse and Implicit Function Theorems 207
8.2 Oriented Nonlinear Fredholm Maps 213
8.3 Oriented Fredholm Maps in Banach Manifolds 214
8.4 A Partial Implicit Function Theorem in Banach Manifolds 224
8.5 Transversal Submanifolds 230
8.6 Parameter-Dependent Transversality and Partial Submanifolds 236
8.7 Orientation on Submanifolds and on Partial Submanifolds 239
8.8 Existence of Transversal Submanifolds 241
8.9 Properness of Fredholm Maps 244
9 The Brouwer Degree 247
9.1 Finite-Dimensional Manifolds 247
9.2 Orientation of Continuous Maps and of Manifolds 258
9.3 The Cr Brouwer Degree 265
9.4 Uniqueness of the Brouwer Degree 271
9.5 Existence of the Brouwer Degree 289
9.6 Some Classical Applications of the Brouwer Degree 303
10 The Benevieri-Furi Degrees 319
10.1 Further Properties of the Brouwer Degree 320
10.2 The Benevieri-Furi C1 Degree 328
10.3 The Benevieri-Furi Coincidence Degree 334
III Degree Theory for Function Triples 347
11 Function Triples 349
11.1 Function Triples and Their Equivalences 351
11.2 The Simplifier Property 365
11.3 Homotopies of Triples 371
11.4 Locally Normal Triples 375
12 The Degree for Finite-Dimensional Fredholm Triples 377
12.1 The Triple Variant of the Brouwer Degree 377
12.2 The Triple Variant of the Benevieri-Furi Degree 390
13 The Degree for Compact Fredholm Triples 401
13.1 The Leray-Schauder Triple Degree 401
13.2 The Leray-Schauder Coincidence Degree 414
13.3 Classical Applications of the Leray-Schauder Degree 417
14 The Degree for Noncompact Fredholm Triples 423
14.1 The Degree for Fredholm Triples with Fundamental Sets 424
14.2 Homotopic Tests for Fundamental Sets 439
14.3 The Degree for Fredholm Triples with Convex-fundamental Sets 447
14.4 Countably Condensing Triples 458
14.5 Classical Applications in the General Framework 466
14.6 A Sample Application for Boundary Value Problems 472
Bibliography 475
Index of Symbols 485
Index 487

lt;P>"The book invites the reader to go on a journey from the basics of topology, through the fields of multivalued maps and Fredholm operators, up to constructions of topological degrees (both those which have been known for decades as well as more recent ones) which are presented in great detail. […] The book gives a very detailed study of different topological concepts that play important roles in a modern approach to nonlinear analysis problems, with special emphasis put on the study of problems involving set-valued maps and operators. It can surely be recommended to researchers and students interested in homotopy invariants and their applications in the theory of differential equations/inclusions." Zentralblatt für Mathematik