Fixed Point Theory in Distance Spaces

This is a monograph on fixed point theory, covering the purely metric aspects of the theoryparticularly 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.

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.

Autorentext
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

Inhalt
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.