Fixed Point Theory in Distance Spaces

This is a monograph on fixed point theory, covering the purely metric aspects of the theory-particularly results that do not depend on any algebraic structure of the underlying space. Traditionally, a large body of metric fixed point theory has been couched in a functional analytic framework. This aspect of the theory has been written about extensively. There are four classical fixed point theorems against which metric extensions are usually checked. These are, respectively, the Banach contraction mapping principal, Nadler's well known set-valued extension of that theorem, the extension of Banach's theorem to nonexpansive mappings, and Caristi's theorem. These comparisons form a significant component of this book. This book is divided into three parts. Part I contains some aspects of the purely metric theory, especially Caristi's theorem and a few of its many extensions. There is also a discussion of nonexpansive mappings, viewed in the context of logical foundations. Part I also contains certain results in hyperconvex metric spaces and ultrametric spaces. Part II treats fixed point theory in classes of spaces which, in addition to having a metric structure, also have geometric structure. These specifically include the geodesic spaces, length spaces and CAT(0) spaces. Part III focuses on distance spaces that are not necessarily metric. These include certain distance spaces which lie strictly between the class of semimetric spaces and the class of metric spaces, in that they satisfy relaxed versions of the triangle inequality, as well as other spaces whose distance properties do not fully satisfy the metric axioms.

List of contents

Preface.- Part 1. Metric Spaces.- Introduction.- Caristi's Theorem and Extensions.- Nonexpansive Mappings and Zermelo's Theorem.- Hyperconvex metric spaces.- Ultrametric spaces.- Part 2. Length Spaces and Geodesic Spaces.- Busemann spaces and hyperbolic spaces.- Length spaces and local contractions.- The G-spaces of Busemann.- CAT(0) Spaces.- Ptolemaic Spaces.- R-trees (metric trees).- Part 3. Beyond Metric Spaces.- b-Metric Spaces.- Generalized Metric Spaces.- Partial Metric Spaces.- Diversities .- Bibliography.- Index.

About the author

W. A. Kirk(1) Naseer Shahzad(2) (1) Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, USA E-mail address: [e-mail] (2) Department of Mathematics, King Abdulaziz University, PO.Box 80203, Jeddah 21589, Saudi Arabia E-mail address: [e-mail]

Summary

This is a monograph on fixed point theory, covering the purely metric aspects of the theory–particularly results that do not depend on any algebraic structure of the underlying space. Traditionally, a large body of metric fixed point theory has been couched in a functional analytic framework. This aspect of the theory has been written about extensively. There are four classical fixed point theorems against which metric extensions are usually checked. These are, respectively, the Banach contraction mapping principal, Nadler’s well known set-valued extension of that theorem, the extension of Banach’s theorem to nonexpansive mappings, and Caristi’s theorem. These comparisons form a significant component of this book. This book is divided into three parts. Part I contains some aspects of the purely metric theory, especially Caristi’s theorem and a few of its many extensions. There is also a discussion of nonexpansive mappings, viewed in the context of logical foundations. Part I also contains certain results in hyperconvex metric spaces and ultrametric spaces. Part II treats fixed point theory in classes of spaces which, in addition to having a metric structure, also have geometric structure. These specifically include the geodesic spaces, length spaces and CAT(0) spaces. Part III focuses on distance spaces that are not necessarily metric. These include certain distance spaces which lie strictly between the class of semimetric spaces and the class of metric spaces, in that they satisfy relaxed versions of the triangle inequality, as well as other spaces whose distance properties do not fully satisfy the metric axioms.

Additional text

“The authors of this interesting monograph are
concerned with purely metric aspects of fixed point theory. … this book can
serve not only as a timely introduction to metric fixed point theory, but also
as a catalyst for further research in this fertile area.” (Simeon Reich,
Mathematical Reviews, October, 2015)
“The book is clearly written and contains a very good selection of results in this rapidly growing area of research – fixed points in metric spaces and their generalizations. The sources of the presented results are carefully mentioned as well as references to related results and further investigation … . an essential reference tool for researchers working in fixed point theory as well as for those interested in applications of metric spaces and their generalizations to other areas … .” (S. Cobzaş, Studia Universitatis Babes-Bolyia, Mathematica, Vol. 60 (1), 2015)
“This monograph treats the purely metric aspects of fixed point theory. … This book provides a concise accessible document as an introduction to the metric fixed point theory for readers interested in this area.” (In-Sook Kim, zbMATH 1308.58001, 2015)

Report

"The authors of this interesting monograph are concerned with purely metric aspects of fixed point theory. ... this book can serve not only as a timely introduction to metric fixed point theory, but also as a catalyst for further research in this fertile area." (Simeon Reich, Mathematical Reviews, October, 2015)

"The book is clearly written and contains a very good selection of results in this rapidly growing area of research - fixed points in metric spaces and their generalizations. The sources of the presented results are carefully mentioned as well as references to related results and further investigation ... . an essential reference tool for researchers working in fixed point theory as well as for those interested in applications of metric spaces and their generalizations to other areas ... ." (S. Cobzas, Studia Universitatis Babes-Bolyia, Mathematica, Vol. 60 (1), 2015)

"This monograph treats the purely metric aspects of fixed point theory. ... This book provides a concise accessible document as an introduction to the metric fixed point theory for readers interested in this area." (In-Sook Kim, zbMATH 1308.58001, 2015)

Product details

• Authors: Willia Kirk, William Kirk, Naseer Shahzad
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 01.01.2014
• EAN: 9783319109268
• ISBN: 978-3-31-910926-8
• No. of pages: 173
• Dimensions: 161 mm x 244 mm x 16 mm
• Weight: 397 g
• Illustrations: XI, 173 p.
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Geometry; B, Mathematik für Ingenieure, Mathematische Modellierung, Topologie, Mathematics and Statistics, Topology, Differential Geometry, Maths for engineers, Mathematical modelling, Mathematical Modeling and Industrial Mathematics, Mathematical models