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Fixed Point Theory in Distance Spaces

  • William Kirk
  • Naseer Shahzad

Table of contents

  1. Front Matter
    Pages I-XI
  2. Metric Spaces

    1. Front Matter
      Pages 1-1
    2. William Kirk, Naseer Shahzad
      Pages 3-5
    3. William Kirk, Naseer Shahzad
      Pages 7-18
    4. William Kirk, Naseer Shahzad
      Pages 19-22
    5. William Kirk, Naseer Shahzad
      Pages 23-24
    6. William Kirk, Naseer Shahzad
      Pages 25-36
  3. Length Spaces and Geodesic Spaces

    1. Front Matter
      Pages 37-37
    2. William Kirk, Naseer Shahzad
      Pages 39-45
    3. William Kirk, Naseer Shahzad
      Pages 47-59
    4. William Kirk, Naseer Shahzad
      Pages 61-64
    5. William Kirk, Naseer Shahzad
      Pages 65-94
    6. William Kirk, Naseer Shahzad
      Pages 95-98
    7. William Kirk, Naseer Shahzad
      Pages 99-110
  4. Beyond Metric Spaces

    1. Front Matter
      Pages 111-111
    2. William Kirk, Naseer Shahzad
      Pages 113-131
    3. William Kirk, Naseer Shahzad
      Pages 133-139
    4. William Kirk, Naseer Shahzad
      Pages 141-152
    5. William Kirk, Naseer Shahzad
      Pages 153-158
  5. Back Matter
    Pages 159-173

About this book

Introduction

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.

Keywords

fixed point theory metric spaces and geodesic shapes metric spaces and geometric conditions semimetric spaces triangle inequalities

Authors and affiliations

  • William Kirk
    • 1
  • Naseer Shahzad
    • 2
  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA
  2. 2.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia

Bibliographic information