Elementary Fixed Point Theorems

Beschreibung

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.

Discusses topics on basic fixed-point theorems due to Banach, Brouwer, Schauder and Tarski, their variants and their 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

Autor*in

P.V. Subrahmanyam

Themen in »Elementary Fixed Point Theorems«

Partial order Fixed Points quasi-order Contraction Principle Cauchy-Kowalevsky Theorem Brouwer’s Fixed Point Theorem Schauder’s Fixed Point Theorem

Stimmen zu »Elementary Fixed Point Theorems«

“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)
()

Details

Link teilen