Väth Topological Analysis

From the Basics to the Triple Degree for Nonlinear Fredholm Inclusions
1. Auflage 2012
ISBN: 978-3-11-027733-3
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 499 Seiten

Reihe: ISSN

Zielgruppe

Researchers, Lecturers, and Graduate Students in Mathematics; Academic Libraries

Autoren/Hrsg.

Väth, Martin

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis
• Mathematik | Informatik Mathematik Topologie Analytische Topologie

Inhaltsverzeichnis

1;Preface;5
2;1 Introduction;11
3;I Topology and Multivalued Maps;17
3.1;2 Multivalued Maps;19
3.1.1;2.1 Notations for Multivalued Maps and Axioms;19
3.1.1.1;2.1.1 Notations;19
3.1.1.2;2.1.2 Axioms;21
3.1.2;2.2 Topological Notations and Basic Results;27
3.1.3;2.3 Separation Axioms;34
3.1.4;2.4 Upper Semicontinuous Multivalued Maps;53
3.1.5;2.5 Closed and Proper Maps;62
3.1.6;2.6 Coincidence Point Sets and Closed Graphs;65
3.2;3 Metric Spaces;69
3.2.1;3.1 Notations and Basic Results for Metric Spaces;69
3.2.2;3.2 Three Measures of Noncompactness;77
3.2.3;3.3 Condensing Maps;85
3.2.4;3.4 Convexity;94
3.2.5;3.5 Two Embedding Theorems for Metric Spaces;99
3.2.6;3.6 Some Old and New Extension Theorems for Metric Spaces;106
3.3;4 Spaces Defined by Extensions, Retractions, or Homotopies;115
3.3.1;4.1 AE and ANE Spaces;115
3.3.2;4.2 ANR and AR Spaces;117
3.3.3;4.3 Extension of Compact Maps and of Homotopies;124
3.3.4;4.4 UV8 and Rd Spaces and Homotopic Characterizations;132
3.4;5 Advanced Topological Tools;139
3.4.1;5.1 Some Covering Space Theory;139
3.4.2;5.2 A Glimpse on Dimension Theory;143
3.4.3;5.3 Vietoris Maps;150
4;II Coincidence Degree for Fredholm Maps;155
4.1;6 Some Functional Analysis;157
4.1.1;6.1 Bounded Linear Operators and Projections;157
4.1.2;6.2 Linear Fredholm Operators;170
4.2;7 Orientation of Families of Linear Fredholm Operators;179
4.2.1;7.1 Orientation of a Linear Fredholm Operator;179
4.2.2;7.2 Orientation of a Continuous Family;188
4.2.3;7.3 Orientation of a Family in Banach Bundles;192
4.3;8 Some Nonlinear Analysis;207
4.3.1;8.1 The Pointwise Inverse and Implicit Function Theorems;207
4.3.2;8.2 Oriented Nonlinear Fredholm Maps;213
4.3.3;8.3 Oriented Fredholm Maps in Banach Manifolds;214
4.3.4;8.4 A Partial Implicit Function Theorem in Banach Manifolds;224
4.3.5;8.5 Transversal Submanifolds;230
4.3.6;8.6 Parameter-Dependent Transversality and Partial Submanifolds;236
4.3.7;8.7 Orientation on Submanifolds and on Partial Submanifolds;239
4.3.8;8.8 Existence of Transversal Submanifolds;241
4.3.9;8.9 Properness of Fredholm Maps;244
4.4;9 The Brouwer Degree;247
4.4.1;9.1 Finite-Dimensional Manifolds;247
4.4.2;9.2 Orientation of Continuous Maps and of Manifolds;258
4.4.3;9.3 The Cr Brouwer Degree;265
4.4.4;9.4 Uniqueness of the Brouwer Degree;271
4.4.5;9.5 Existence of the Brouwer Degree;289
4.4.6;9.6 Some Classical Applications of the Brouwer Degree;303
4.5;10 The Benevieri-Furi Degrees;319
4.5.1;10.1 Further Properties of the Brouwer Degree;320
4.5.2;10.2 The Benevieri-Furi C1 Degree;328
4.5.3;10.3 The Benevieri-Furi Coincidence Degree;334
5;III Degree Theory for Function Triples;347
5.1;11 Function Triples;349
5.1.1;11.1 Function Triples and Their Equivalences;351
5.1.2;11.2 The Simplifier Property;365
5.1.3;11.3 Homotopies of Triples;371
5.1.4;11.4 Locally Normal Triples;375
5.2;12 The Degree for Finite-Dimensional Fredholm Triples;377
5.2.1;12.1 The Triple Variant of the Brouwer Degree;377
5.2.2;12.2 The Triple Variant of the Benevieri-Furi Degree;390
5.3;13 The Degree for Compact Fredholm Triples;401
5.3.1;13.1 The Leray-Schauder Triple Degree;401
5.3.2;13.2 The Leray-Schauder Coincidence Degree;414
5.3.3;13.3 Classical Applications of the Leray-Schauder Degree;417
5.4;14 The Degree for Noncompact Fredholm Triples;423
5.4.1;14.1 The Degree for Fredholm Triples with Fundamental Sets;424
5.4.2;14.2 Homotopic Tests for Fundamental Sets;439
5.4.3;14.3 The Degree for Fredholm Triples with Convex-fundamental Sets;447
5.4.4;14.4 Countably Condensing Triples;458
5.4.5;14.5 Classical Applications in the General Framework;466
5.4.6;14.6 A Sample Application for Boundary Value Problems;472
6;Bibliography;475
7;Index of Symbols;485
8;Index;487

Über Autor(innen)

Väth, Martin
Martin Väth, Freie Universität Berlin, Germany.