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2001 | Buch

Contraction Mappings and Extensions Kapitel lesen Erstes Kapitel lesen

Handbook of Metric Fixed Point Theory

herausgegeben von: William A. Kirk, Brailey Sims

Verlag: Springer Netherlands

Enthalten in: Professional Book Archive

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Über dieses Buch

Metric fixed point theory encompasses the branch of fixed point theory which metric conditions on the underlying space and/or on the mappings play a fundamental role. In some sense the theory is a far-reaching outgrowth of Banach's contraction mapping principle. A natural extension of the study of contractions is the limiting case when the Lipschitz constant is allowed to equal one. Such mappings are called nonexpansive. Nonexpansive mappings arise in a variety of natural ways, for example in the study of holomorphic mappings and hyperconvex metric spaces.
Because most of the spaces studied in analysis share many algebraic and topological properties as well as metric properties, there is no clear line separating metric fixed point theory from the topological or set-theoretic branch of the theory. Also, because of its metric underpinnings, metric fixed point theory has provided the motivation for the study of many geometric properties of Banach spaces. The contents of this Handbook reflect all of these facts.
The purpose of the Handbook is to provide a primary resource for anyone interested in fixed point theory with a metric flavor. The goal is to provide information for those wishing to find results that might apply to their own work and for those wishing to obtain a deeper understanding of the theory. The book should be of interest to a wide range of researchers in mathematical analysis as well as to those whose primary interest is the study of fixed point theory and the underlying spaces. The level of exposition is directed to a wide audience, including students and established researchers.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Contraction Mappings and Extensions
Abstract
A complete survey of all that has been written about contraction mappings would appear to be nearly impossible, and perhaps not really useful. In particular the wealth of applications of Banach’s contraction mapping principle is astonishingly diverse. We only attempt to touch on some of the high points of this profound and seminal development in metric fixed point theory.
W. A. Kirk
Chapter 2. Examples of Fixed Point Free Mappings
Abstract
In this short chapter we collect together examples of fixed point free nonexpansive mappings in a variety of Banach spaces. These examples help delineate the class of spaces enjoying the fpp, the w-fpp, or the w*-fpp. We begin by recalling the relevant definitions.
Brailey Sims
Chapter 3. Classical Theory of Nonexpansive Mappings
Abstract
Mappings which are defined on metric spaces and which do not increase distances between pairs of points and their images are called nonexpansive. Thus an abstract metric space is all that is needed to define the concept. At the same time, the more interesting results seem to require some notion of topology; more specifically a topology which assures that closed metric balls are compact. This is not a serious limitation, however, because many spaces which arise naturally in functional analysis possess such topologies; most notably the weak and weak* topologies in Banach spaces.
Kazimierz Goebel, W. A. Kirk
Chapter 4. Geometrical Background of Metric Fixed Point Theory
Abstract
The interplay between the geometry of Banach spaces and fixed point theory has been very strong and fruitful. In particular, geometrical properties play key roles in metric fixed point problems. In this text we discuss the most basic of these geometrical properties. Since many fixed point results have a quantitative character, we place special emphasis on the scaling coefficients and functions corresponding to the properties considered. The material we cover is far from exhaustive, in particular we do not consider applications. These are treated elsewhere in the Handbook. The interested reader may also consult [5], [44] and [1].
Stanisław Prus
Chapter 5. Some Moduli and Constants Related to Metric Fixed Point Theory
Abstract
Indeed, there are a lot of quantitative descriptions of geometrical properties of Banach spaces. The most common way for creating these descriptions, is to define a real function (a “modulus” depending on the Banach space under consideration, and from this define a suitable constant or coefficient closely related to this function. The moduli and/or the constants are attempts to get a better understanding about two things:
  • The shape of the unit ball of a space, and
  • The hidden relations between weak and strong convergence of sequences.
Enrique Llorens Fuster
Chapter 6. Ultra-Methods in Metric Fixed Point Theory
Abstract
Over the last two decades ultrapower techniques have become major tools for the development and understanding of metric fixed point theory. In this short chapter we develop the Banach space ultrapower and initiate its use in studying the weak fixed point property for nonexpansive mappings. For a more extensive and detailed treatment than is given here the reader is referred to [1] and [21].
M. A. Khamsi, B. Sims
Chapter 7. Stability of the Fixed Point Property for Nonexpansive Mappings
Abstract
In 1971 Zidler [Zi 71] showed that every separable Banach space (X, ‖·‖) admits an equivalent renorming, (X, ‖·‖0), which is uniformly convex in every direction (UCED), and consequently it has weak normal structure and so the weak fixed point property (WFPP) [D-J-S 71].
Jesús Garcia-Falset, Antonio Jiménez-Melado, Enrique Llorens-Fuster
Chapter 8. Metric Fixed Point Results Concerning Measures of Noncompactness
Abstract
Fixed Point Theory has two main branches: on the one hand we can consider the results that are deduced from topological properties and on the other hand those which can be obtained using metric properties.
T. Domínguez, M. A. Japón, G. López
Chapter 9. Renormings of ℓ1 and C 0 and Fixed Point Properties
Abstract
As has been noted in previous chapters, there are many geometric conditions on a Banach space strong enough to imply that the Banach space has the fixed point property. Geometric conditions such as uniform rotundity, uniform smoothness, or normal structure together with reflexivity are sufficient to imply the fixed point property. Each of these conditions also implies (or assumes in the last case) that the Banach space is reflexive.
P. N. Dowling, C. J. Lennard, B. Turett
Chapter 10. Nonexpansive Mappings: Boundary/Inwardness Conditions and Local Theory
Abstract
Boundary and inwardness conditions have been particularly useful in extending fixed point theory for nonexpansive mappings to broader classes of mappings, particularly to mappings satisfying local contractive and pseudocontractive assumptions. At the same time these conditions often enable one to relax the assumption that the mapping takes values in its own domain.
W. A. Kirk, C. H. Morales
Chapter 11. Rotative Mappings and Mappings with Constant Displacement
Abstract
This part is a continuation of the chapter by K. Goebel, so we adopt all his notations and definitions.
Wieslawa Kaczor, Małgorzata Koter-Mórgowska
Chapter 12. Geometric Properties Related to Fixed Point Theory in Some Banach Function Lattices
Abstract
The aim of this chapter is to present criteria for the most important geometric properties related to the metric fixed point theory in some classes of Banach function lattices, mainly in Orlicz spaces and Cesaro sequence spaces. We also give some informations about respective results for Musielak-Orlicz spaces, Orlicz-Lorentz spaces and Calderón-Lozanovsky spaces.
S. Chen, Y. Cui, H. Hudzik, B. Sims
Chapter 13. Introduction to Hyperconvex Spaces
Abstract
The notion of hyperconvexity is due to Aronszajn and Panitchpakdi [1] (1956) who proved that a hyperconvex space is a nonexpansive absolute retract, i.e. it is a non-expansive retract of any metric space in which it is isometrically embedded. The corresponding linear theory is well developed and associated with the names of Gleason, Goodner, Kelley and Nachbin (see for instance [19, 29, 42, 46]). The nonlinear theory is still developing. The recent interest into these spaces goes back to the results of Sine [54] and Soardi [57] who proved independently that fixed point property for nonexpansive mappings holds in bounded hyperconvex spaces. Since then many interesting results have been shown to hold in hyperconvex spaces.
R. Espínola, M. A. Khamsi
Chapter 14. Fixed Points of Holomorphic Mappings: A Metric Approach
Abstract
Let X 1 and X 2 be two complex normed linear spaces and let D 1 be a domain (that is, a nonempty open connected subset) in X 1. A mapping f : D 1 → X 2 is said to be holomorphic in D 1 if it is Fréchet differentiable at each point of D 1. If D 1 and D 2 are domains in X 1 and X 2, respectively, then H(D 1, D 2) will denote the family of all holomorphic mappings from D 1 into D 2.
Tadeusz Kuczumow, Simeon Reich, David Shoikhet
Chapter 15. Fixed Point and Non-Linear Ergodic Theorems for Semigroups of Non-Linear Mappings
Abstract
Let S be a semigroup, ℓ∞(S) be the Banach space of bounded real valued functions on S with the supremum norm. There is a strong relation between the existence of an invariant mean (or submean) on an invariant subspace of ℓ∞(S) and fixed point or ergodic properties of S when S is represented as a semigroup of nonexpansive mappings on a closed convex subset of a Banach space. It is the purpose of this chapter of the Handbook to exhibit on some recent results on such relations. Since this handbook is intended for researchers and graduate students, detailed proofs for central results, historical remarks, open problems and many references will be included. It is our hope that our effort will generate further research in this direction of non-linear analysis which depends on the ideal theory of S,and existence of an invariant mean (or submean) on a subspace of ℓ∞(S) of a semigroup S.
Anthony To-Ming Lau, Wataru Takahashi
Chapter 16. Generic Aspects of Metric Fixed Point Theory
Abstract
Let X be a complete metric space. According to Baire’s theorem, the intersection of every countable collection of open dense subsets of X is dense in X. This rather simple, yet powerful result has found many applications. In particular, given a property which elements of X may have, it is of interest to determine whether this property is generic, that is, whether the set of elements which do enjoy this property contains a countable intersection of open dense sets. Such an approach, when a certain property is investigated for the whole space X and not just for a single point in X, has already been successfully applied in many areas of Analysis. We mention, for instance, the theory of dynamical systems [12, 18, 24, 35, 33, 52], optimization [22, 44], variational analysis [2, 9], [20, 211, the calculus of variations [4, 14, 55] and optimal control [56, 57].
Simeon Reich, Alexander J. Zaslavski
Chapter 17. Metric Environment of the Topological Fixed Point Theorems
Abstract
In metric fixed point theory the term the fixed point property is usually related to a certain class of mappings described by some metric conditions. In topological part of the theory however, we use this term with respect to the wide class of spaces and families of continuous transformations. Let us begin with recalling the classical definition and facts.
Kazimierz Goebel
Chapter 18. Order-Theoretic Aspects of Metric Fixed Point Theory
Abstract
This chapter is intended to present connections between two branches of fixed point theory: The first, using metric methods which is the main subject of this Handbook, and the second, involving partial ordering techniques. We shall concentrate here on the following problem: Given a space with a metric structure (e.g., uniform space, metric space or Banach space) and a mapping satisfying some geometric conditions, define a partial ordering (depending on a structure of a space and/or a mapping) so that one of fundamental ordering principles — the Knaster-Tarski Theorem, Zermelo’s Theorem or the Tarski-Kantorovitch Theorem — can be applied to deduce the existence of a fixed point. We emphasize that all the above principles are independent of the Axiom of Choice (abbr., AC) so the above approach to metric fixed point theory is wholly constructive. It seems that such studies were initiated by H. Amann [5] and B. Fuchssteiner [33] in 1977. Subsequently, they were continued among others by S. Hayashi [37], R. Mańka ([59], [60]), R. Lemmert and P. Volkmann [58], A. Baranga [8], T. Büber and W. A. Kirk [20] and J. Jachymski ([41], [42], [43], [44], [46], [47]). On the other hand, some authors have also studied a reciprocal of the above problem: Given a partially ordered set and a mapping on it, define a metric depending on this order so that some theorems of metric fixed point theory could be applied. That was done recently by Y.-Z. Chen [22], who used Thompson’s [81] metric generated by an order. However, in this chapter, we shall not discuss these problems.
Jacek Jachymski
Chapter 19. Fixed Point and Related Theorems for Set-Valued Mappings
Abstract
In this chapter, we focus in the discussion of fixed point theory for set, valued mappings by using Knaster-Kuratowski-Mazurkiewicz (KKM) principle in both topological vector spaces and hyperconvex metric spaces. In particular, the fixed point theory of set-valued mappings of Browder-Fan and Fan-Glicksberg type has been extensively studied in the setting of locally convex spaces, H-spaces, G-convex spaces and metric hyperconvex spaces. By using its own feature of hyperconvex metric spaces being a special class of H-spaces, we also establish its general KKM theory and then its various applications. In section 2, we first discuss some recent developments of KKM theory itself and the general Ky Fan minimax principle is given in section 3. In sections 4 and 5, two types of Ky Fan minimax inequalities and their equivalent fixed point forms for set-valued mappings are given. In section 6, the general Fan-Glicksberg type fixed point theorem is discussed in G-Convex spaces. These spaces include locally convex H-spaces, locally convex topological vector spaces and metric hyperconvex metric spaces as special cases. Finally, the general KKM theory and its various applications in metric hyperconvex spaces and the generic stability of fixed points are discussed in section 7.
George Xian-Zhi Yuan
Backmatter
Metadaten
Titel
Handbook of Metric Fixed Point Theory
herausgegeben von
William A. Kirk
Brailey Sims
Copyright-Jahr
2001
Verlag
Springer Netherlands
Electronic ISBN
978-94-017-1748-9
Print ISBN
978-90-481-5733-4
DOI
https://doi.org/10.1007/978-94-017-1748-9