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: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, Band 16, 499 Seiten

Reihe: De Gruyter Series in Nonlinear Analysis and ApplicationsISSN

This monograph aims to give a self-contained introduction into the whole field of topological analysis: Requiring essentially only basic knowledge of elementary calculus and linear algebra, it provides all required background from topology, analysis, linear and nonlinear functional analysis, and multivalued maps, containing even basic topics like separation axioms, inverse and implicit function theorems, the Hahn-Banach theorem, Banach manifolds, or the most important concepts of continuity of multivalued maps. Thus, it can be used as additional material in basic courses on such topics. The main intention, however, is to provide also additional information on some fine points which are usually not discussed in such introductory courses.The selection of the topics is mainly motivated by the requirements for degree theory which is presented in various variants, starting from the elementary Brouwer degree (in Euclidean spaces and on manifolds) with several of its famous 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 the first monograph in which the corresponding theory is developed in detail.

Zielgruppe

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

Autoren/Hrsg.

Väth, Martin

Fachgebiete

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

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.

Martin Väth, Freie Universität Berlin, Germany.