Overview
- Discusses topics on basic fixed-point theorems due to Banach, Brouwer, Schauder and Tarski, their variants and applications
- Introduces finite-dimensional degree theory based on Heinz's approach and some geometric coefficients for Banach spaces
- Explains Sharkovsky’s theorem on periodic points and Thron’s results on the convergence of iterates of certain real functions
- Presents two classic counter-examples in fixed-point theory: one due to Huneke and other due to Kinoshita
- Elaborates Manka’s proof on the fixed-point property of arcwise connected hereditarily unicoherent continua
- Offers a detailed treatment of Ward’s theory of partially ordered topological spaces culminating in Sherrer theorem
Part of the book series: Forum for Interdisciplinary Mathematics (FFIM)
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Table of contents (12 chapters)
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Front Matter
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Back Matter
Keywords
About this book
This book provides a primary resource in basic fixed-point theorems due to Banach, Brouwer, Schauder and Tarski and their applications. Key topics covered include Sharkovsky’s theorem on periodic points, Thron’s results on the convergence of certain real iterates, Shield’s common fixed theorem for a commuting family of analytic functions and Bergweiler’s existence theorem on fixed points of the composition of certain meromorphic functions with transcendental entire functions. Generalizations of Tarski’s theorem by Merrifield and Stein and Abian’s proof of the equivalence of Bourbaki–Zermelo fixed-point theorem and the Axiom of Choice are described in the setting of posets. A detailed treatment of Ward’s theory of partially ordered topological spaces culminates in Sherrer fixed-point theorem. It elaborates Manka’s proof of the fixed-point property of arcwise connected hereditarily unicoherent continua, based on the connection he observed between set theory and fixed-point theory viaa certain partial order. Contraction principle is provided with two proofs: one due to Palais and the other due to Barranga. Applications of the contraction principle include the proofs of algebraic Weierstrass preparation theorem, a Cauchy–Kowalevsky theorem for partial differential equations and the central limit theorem. It also provides a proof of the converse of the contraction principle due to Jachymski, a proof of fixed point theorem for continuous generalized contractions, a proof of Browder–Gohde–Kirk fixed point theorem, a proof of Stalling's generalization of Brouwer's theorem, examine Caristi's fixed point theorem, and highlights Kakutani's theorems on common fixed points and their applications.
Reviews
“This monograph is written by a well-known expert in fixed point theory and presents his choice of results from this wide area of research. ... The monograph can serve as a very useful introduction into the fixed point topic, which is one of the most applicable parts, both of Topology and Nonlinear Analysis.” (Zoran Kadelburg, zbMath 1412.54001, 2019)
Authors and Affiliations
About the author
Bibliographic Information
Book Title: Elementary Fixed Point Theorems
Authors: P.V. Subrahmanyam
Series Title: Forum for Interdisciplinary Mathematics
DOI: https://doi.org/10.1007/978-981-13-3158-9
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Singapore Pte Ltd. 2018
Hardcover ISBN: 978-981-13-3157-2Published: 21 January 2019
eBook ISBN: 978-981-13-3158-9Published: 10 January 2019
Series ISSN: 2364-6748
Series E-ISSN: 2364-6756
Edition Number: 1
Number of Pages: XIII, 302
Number of Illustrations: 5 b/w illustrations
Topics: Analysis