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Fixed Point Theory in Distance Spaces

  • Book
  • © 2014

Overview

Authors:
  1. William Kirk

    1. Department of Mathematics, University of Iowa, Iowa City, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Naseer Shahzad

    1. Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

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    You can also search for this author in PubMed   Google Scholar

  • Covers four classical fixed point theorems against which metric extensions are usually checked and several new modern theorems on fixed point theory and distances.

  • Presents a concise accessible document which can be used as an introduction to the subject and its central themes, featuring material collected in a single volume for the first time.

  • Introduces concepts addressing both mathematical and applied problems, including more formal concepts which have yet to find formal application.

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Table of contents (15 chapters)

  1. Front Matter

    Pages I-XI
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  2. Metric Spaces

    1. Front Matter

      Pages 1-1
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    2. Introduction

      • William Kirk, Naseer Shahzad
      Pages 3-5

    3. Caristi’s Theorem and Extensions

      • William Kirk, Naseer Shahzad
      Pages 7-18

    4. Nonexpansive Mappings and Zermelo’s Theorem

      • William Kirk, Naseer Shahzad
      Pages 19-22

    5. Hyperconvex Metric Spaces

      • William Kirk, Naseer Shahzad
      Pages 23-24

    6. Ultrametric Spaces

      • William Kirk, Naseer Shahzad
      Pages 25-36

  3. Length Spaces and Geodesic Spaces

    1. Front Matter

      Pages 37-37
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    2. Busemann Spaces and Hyperbolic Spaces

      • William Kirk, Naseer Shahzad
      Pages 39-45

    3. Length Spaces and Local Contractions

      • William Kirk, Naseer Shahzad
      Pages 47-59

    4. The G-Spaces of Busemann

      • William Kirk, Naseer Shahzad
      Pages 61-64

    5. CAT(0) Spaces

      • William Kirk, Naseer Shahzad
      Pages 65-94

    6. Ptolemaic Spaces

      • William Kirk, Naseer Shahzad
      Pages 95-98

    7. \(\mathbb{R}\)-Trees (Metric Trees)

      • William Kirk, Naseer Shahzad
      Pages 99-110

  4. Beyond Metric Spaces

    1. Front Matter

      Pages 111-111
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    2. b-Metric Spaces

      • William Kirk, Naseer Shahzad
      Pages 113-131

    3. Generalized Metric Spaces

      • William Kirk, Naseer Shahzad
      Pages 133-139

    4. Partial Metric Spaces

      • William Kirk, Naseer Shahzad
      Pages 141-152

    5. Diversities

      • William Kirk, Naseer Shahzad
      Pages 153-158

  5. Back Matter

    Pages 159-173
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Keywords

About this book

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.

Reviews

“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)

Authors and Affiliations

  • Department of Mathematics, University of Iowa, Iowa City, USA

    William Kirk

  • Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

    Naseer Shahzad

About the authors

W. A. Kirk(1) Naseer Shahzad(2) (1) Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, USA E-mail address: william-kirk@uiowa.edu (2) Department of Mathematics, King Abdulaziz University, PO.Box 80203, Jeddah 21589, Saudi Arabia E-mail address: nshahzad@kau.edu.sa

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