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