Elementary Fixed Point Theorems

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.

List of contents

Chapter 1. Prerequisites.- Chapter 2. Fixed Points of Some Real and Complex Functions.- Chapter 3. Fixed Points and Order.- Chapter 4. Partially Ordered Topological Spaces and Fixed Points.- Chapter 5. Contraction Principle.- Chapter 6. Applications of the Contraction Principle.- Chapter 7. Caristi's fixed point theorem.- Chapter 8. Contractive and Nonexpansive Mappings.- Chapter 9. Geometric Aspects of Banach Spaces and Nonexpansive Mappings.- Chapter 10. Brouwer's Fixed Point Theorem.- Chapter 11. Schauder's Fixed Point Theorem and Allied Theorems.- Chapter 12. Basic Analytic Degree Theory af a Mapping.

About the author

P.V. SUBRAHMANYAM is a Professor Emeritus at the Indian Institute of Technology Madras (IIT Madras), India. He received his PhD in Mathematics from IIT Madras, for his dissertation on “Topics in Fixed- Point Theory” under the supervision of(late)Dr. V. Subba Rao. He received his MSc degree in Mathematics from IIT Madras, and BSc degree in Mathematics from Madras University. He has held several important administrative positions, such as senior professor and head of the Department of Mathematics at IIT Madras; founder and head of the Department of Mathematics at the Indian Institute of Technology Hyderabad (IIT Hyderabad); Executive Chairman of the Association of Mathematics Teachers of India(AMTI); president of the Forum for Interdisciplinary Mathematics (FIM).Before joinining IIT Madras he served as a faculty member  at Loyola College, Madras University, and Hyderabad Central University. His areas of interest include classical analysis, nonlinear analysis and fixed-point theory,fuzzy- set theory, functional equations and mathematics education. He has published over 70 papers and served on the editorial board of the Journal of Analysis and the Journal of Differential Equations and Dynamical Systems. He received an award for his outstanding contributions to mathematical sciences in 2004 and the Lifetime Achievement Award from the FIM in 2016. He has given various invited talks at international conferences and completed brief visiting assignments in many countries such as Canada, Czech Republic, Germany, Greece, Japan, Slovak Republic and the USA. He is also a life member of the Association of Mathematics Teachers of India, FIM, Indian Mathematical Society and the  Society for Industrial and Applied Mathematics(SIAM).

Summary

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.

Additional text

“The chapters are relatively independent. For this reason, this would be a nice book to hand to an advanced undergraduate or a beginning graduate student to supplement coursework or to find a topic for a project. … Overall, the author succeeds in providing an intriguing collection of theorems beyond those widely known from basic study.” (Michele Intermont, MAA Reviews, September 15, 2019)

“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)

Report

"The chapters are relatively independent. For this reason, this would be a nice book to hand to an advanced undergraduate or a beginning graduate student to supplement coursework or to find a topic for a project. ... Overall, the author succeeds in providing an intriguing collection of theorems beyond those widely known from basic study." (Michele Intermont, MAA Reviews, September 15, 2019)
"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)

Product details

• Authors: P V Subrahmanyam, P. V. Subrahmanyam, P.V. Subrahmanyam
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 01.01.2019
• EAN: 9789811331572
• ISBN: 978-981-1331-57-2
• No. of pages: 302
• Dimensions: 156 mm x 243 mm x 24 mm
• Weight: 612 g
• Illustrations: XIII, 302 p. 5 illus.
• Series: Forum for Interdisciplinary Mathematics; Forum for Interdisciplinary Ma; Forum for Interdisciplinary Mathematics
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Analysis