Topological analysis. From the basics to the triple degree for nonlinear Fredholm inclusions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1248.47001
DOI: 10.1515/9783110277333
author: Martin Väth
publication date: 2012-07-13T00:00:00Z
Mathematics Subject Classification ID: 47-01
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 46T05
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 55M25
zbMATH DE Number: 6055190
zbMATH Keywords: topological degree
zbMATH Keywords: manifold
zbMATH Keywords: function triple
zbMATH Keywords: Fredholm operators
zbMATH Keywords: multivalued maps
zbMATH Keywords: topological methods in nonlinear analysis
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110277333
OpenAlex ID: W4233643834
title: Topological analysis. From the basics to the triple degree for nonlinear Fredholm inclusions
published in: De Gruyter Series in Nonlinear Analysis and Applications
review text: This book focuses on one of the main topological tool of nonlinear analysis, the topological degree. One of the aims of the author is to provide a modern treatment of the topic, with the use of function triples, but he does not restrict himself to this aim. 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.NEWLINENEWLINE The material is divided into three main parts. The first part gives a self-contained presentation of multivalued maps, measures of noncompactness, extension properties, and several other crucial topological tools. The material of this part, and consequently the rest of the book, is presented in a way well suited even for undergraduate students. It is worth pointing out that a comprehensive description of the set-theoretical axiomatic treatment is a very characteristic feature of the book. In the whole text, usage of the axiom of choice (AC) is always indicated. Special attention should be paid to Chapter~4 where, among properties of AE, ANE, AR and ANR spaces, a~detailed characterization of \(UV^\infty\) and \(R_\delta\) spaces is provided. These sets are crucial for further investigations. The so-called Vietoris* maps are defined in Chapter~5 and are used in the sequel to obtain a homotopy reduction in a construction of a degree. This notion is strictly motivated by \textit{W. Kryszewski} [Homotopy Properties of Set-Valued Mappings. Univ. N. Copernicus Publishing, Toruń (1997)], where some homotopic versions of the Vietoris theorem are obtained.NEWLINENEWLINE The second part of the book (Chapters 6--10) is headed for the Furi-Pera coincidence degree for Fredholm maps on Banach manifolds, but along the way familiarizes the reader with suitable information on Banach manifolds and elements of the analysis of Fred…
reviewed by: Grzegorz Gabor
description: scientific article; zbMATH DE number 6055190