Fixed points of nonexpansive mappings
type: Item
instance of: scholarly article
title: Fixed points of nonexpansive mappings
zbMATH Open document ID: 0865.47041
DOI: 10.1007/BF02362840
author: N. M. Gulevich
published in: Journal of Mathematical Sciences (New York)
publication date: 1997-07-13T00:00:00Z
review text: This is a brief, but very informative survey article of the basic results concerning the question of existence of fixed points of nonexpansive mappings in Banach spaces. The structure of this article is the following: Introduction. Principal notations. Chapter 1: Normal structure in Banach spaces and its generalizations; 1.1. Normal structure; 1.2. Uniformly normal structure; 1.3. Close-to-normal structure; 1.4. Weakly normal structures; 1.5. Fixed points and product spaces. Chapter 2: The method of asymptotic center; 2.1. Asymptotic centers of bounded sequences; 2.2. Multi-valued mappings; 2.3. Uniformly Lipschitzian mappings. Chapter 3: Nonstandard methods in the fixed point theory; 3.1. The example of Alspach; 3.2. Ultrapower and superreflexivity; 3.3. Ultrapower and fixed points. Some unsolved problems. Literature cited (228 articles). In Chapter 3, the author, using the nonstandard method, gives a positive answer to a question of \textit{D. Amir} [Pac. J. Math. 118, 1-15 (1985; Zbl 0557.46011)]: every Banach space with uniformly normal structure is superreflexive. As supplement we refer to: Recent advances on metric fixed point theory (\textit{T. Dominguez Benavides}, ed.), Univ. de Sevilla (1996; Zbl 0859.00024).
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 903623
zbMATH Keywords: existence of fixed points of nonexpansive mappings
zbMATH Keywords: Banach space with uniformly normal structure
zbMATH Keywords: superreflexive
reviewed by: Jarosław Górnicki
MaRDI profile type: Publication
cites work: Measures of noncompactness and condensing operators. Transl. from the Russian by A. Iacob
cites work: Inner functions on compact spaces
cites work: Q3843864
cites work: Q3364185
cites work: An estimate for the distance from a fixed point set
cites work: Q3799218
cites work: Q3254057
cites work: Q4142342
cites work: Q3314428
cites work: Nonstandard methods in fixed point theory
cites work: A Fixed Point Free Nonexpansive Map
cites work: On Jung's constant and related constants in normed linear spaces
cites work: Fixed point theorems for set-valued mappings of contractive type
cites work: Reflexivity of a Banach Space with a Uniformly Normal Structure
cites work: Q3487997
cites work: Q3049390
cites work: Asymptotic Normal Structure and Fixed Points of Nonexpansive Mappings
cites work: On Modulus of Noncompact Convexity and Its Properties
cites work: Banach-Saks properties and spreading models.
cites work: Théorème de point fixe et d'approximation. (Fixed point and approximation theorem)
cites work: Normal structure in Banach spaces
cites work: Q3483890
cites work: Points fixes dans les espaces des operateurs nucleaires
cites work: Q3709373
cites work: EXISTENCE OF PERIODIC SOLUTIONS FOR NONLINEAR EQUATIONS OF EVOLUTION
cites work: FIXED-POINT THEOREMS FOR NONCOMPACT MAPPINGS IN HILBERT SPACE
cites work: NONEXPANSIVE NONLINEAR OPERATORS IN A BANACH SPACE
cites work: Fixed point theorems for nonlinear semicontractive mappings in Banach spaces
cites work: Nonlinear mappings of nonexpansive and accretive type in Banach spaces
cites work: Semicontractive and semiaccretive nonlinear mappings in Banach spaces
cites work: Q4092518
cites work: Asymptotic behavior of nonexpansive mappings
cites work: On J-Convexity and Some Ergodic Super-Properties of Banach Spaces
cites work: Q5655874
cites work: Normal structure of Banach spaces
cites work: Normal structure coefficients for Banach spaces
cites work: Fixed point theorems for multivalued mappings in Banach spaces
cites work: Q3974686
cites work: Q3828578
cites work: Fixed Point Theorems for Mappings Satisfying Inwardness Conditions
cites work: On a Geometric Condition Related to Boundedly Complete Bases and Normal Structure in Banach Spaces
cites work: Q3695811
cites work: Q3990227
cites work: Application des ultraproduits à l'étude des espaces et des algèbres de Banach
cites work: Strict Convexity and Smoothness of Normed Spaces
cites work: Q3213662
cites work: Normed Linear Spaces that are Uniformly Convex in Every Direction
cites work: Common fixed points for commuting contraction mappings
cites work: Normal structure coefficients of Lp(Ω)
cites work: Q4150254
cites work: Some remarks on set-valued mappings
cites work: Some properties of the characteristic of convexity relating to fixed point theory
cites work: Q4200693
cites work: Q3961313
cites work: Some more Banach spaces with normal structure
cites work: Q4751192
cites work: (KK)-Properties, Normal Structure and Fixed Points of Nonexpansive Mappings in Orlicz Sequence Spaces
cites work: A Theorem on Fixed Points Under Isometries
cites work: The construction of an asymptotic center with a fixed-point property
cites work: Fixed Point Theorems in Uniformly Convex Banach Spaces
cites work: Q4134286
cites work: Remarks on the fixed point problem for nonexpansive maps
cites work: Q4039048

Fixed points of nonexpansive mappings (cites work)
cites work: Q3799136

Fixed points of nonexpansive mappings (cites work)
cites work: On two classes of Banach spaces with uniform normal structure

Fixed points of nonexpansive mappings (cites work)
cites work: A geometrically aberrant Banach space with normal structure

Fixed points of nonexpansive mappings (cites work)
cites work: Fixed point theorem for non-expansive mappings on banach spaces with uniformly normal structure

Fixed points of nonexpansive mappings (cites work)
cites work: Q5601137

Fixed points of nonexpansive mappings (cites work)
cites work: Q4141721

Fixed points of nonexpansive mappings (cites work)
cites work: Q4141722

Fixed points of nonexpansive mappings (cites work)
cites work: A fixed point theorem for transformations whose iterates have uniform Lipschitz constant

Fixed points of nonexpansive mappings (cites work)
cites work: Q3998720

Fixed points of nonexpansive mappings (cites work)
cites work: Uniformly Lipschitzian Families of Transformations in Banach Spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Irregular convex sets with fixed-point property for nonexpansive mappings

Fixed points of nonexpansive mappings (cites work)
cites work: Q3746314

Fixed points of nonexpansive mappings (cites work)
cites work: Zum Prinzip der kontraktiven Abbildung

Fixed points of nonexpansive mappings (cites work)
cites work: Q3985563

Fixed points of nonexpansive mappings (cites work)
cites work: Q3831548

Fixed points of nonexpansive mappings (cites work)
cites work: Q5587383

Fixed points of nonexpansive mappings (cites work)
cites work: Some geometric properties related to the fixed point theory for nonexpansive mappings

Fixed points of nonexpansive mappings (cites work)
cites work: A Fixed-Point Theorem for Inward and Outward Maps

Fixed points of nonexpansive mappings (cites work)
cites work: A fixed point theorem for a class of star-shaped sets in \(c_ 0\)

Fixed points of nonexpansive mappings (cites work)
cites work: Ultraproducts in Banach space theory.

Fixed points of nonexpansive mappings (cites work)
cites work: Q4194007

Fixed points of nonexpansive mappings (cites work)
cites work: Banach spaces which are nearly uniformly convex

Fixed points of nonexpansive mappings (cites work)
cites work: On nearly uniformly convex and k-uniformly convex spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Single-valued mappings, multivalued mappings and fixed-point theorems

Fixed points of nonexpansive mappings (cites work)
cites work: A Non-Reflexive Banach Space Isometric With Its Second Conjugate Space

Fixed points of nonexpansive mappings (cites work)
cites work: Uniformly non-square Banach spaces

Fixed points of nonexpansive mappings (cites work)
cites work: On Nonexpansive Mappings

Fixed points of nonexpansive mappings (cites work)
cites work: Existence of fixed points of nonexpansive mappings in a space without normal structure

Fixed points of nonexpansive mappings (cites work)
cites work: La propriété du point fixe dans les espaces de Banach avec base inconditionelle. (The fixed point properties in Banach spaces with unconditional bases)

Fixed points of nonexpansive mappings (cites work)
cites work: Q3485501

Fixed points of nonexpansive mappings (cites work)
cites work: Normal Structure for Banach Spaces with Schauder Decomposition

Fixed points of nonexpansive mappings (cites work)
cites work: James quasi reflexive space has the fixed point property

Fixed points of nonexpansive mappings (cites work)
cites work: On Normal Structure, Fixed Point Property and Contractions of Type (γ)

Fixed points of nonexpansive mappings (cites work)
cites work: Fixed Points of Nonexpansive Mappings in Banach Lattices

Fixed points of nonexpansive mappings (cites work)
cites work: A Fixed Point Theorem for Mappings which do not Increase Distances

Fixed points of nonexpansive mappings (cites work)
cites work: Nonexpansive mappings and the weak closure of sequences of iterates

Fixed points of nonexpansive mappings (cites work)
cites work: Fixed point theorems for nonlinear nonexpansive and generalized contraction mappings

Fixed points of nonexpansive mappings (cites work)
cites work: An Abstract Fixed Point Theorem for Nonexpansive Mappings

Fixed points of nonexpansive mappings (cites work)
cites work: Q3938101

Fixed points of nonexpansive mappings (cites work)
cites work: Fixed point theory for nonexpansive mappings. II

Fixed points of nonexpansive mappings (cites work)
cites work: Q3725043

Fixed points of nonexpansive mappings (cites work)
cites work: An Iteration Process for Nonexpansive Mappings with Applications to Fixed Point Theory in Product Spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Nonexpansive and locally nonexpansive mappings in product spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Q3353650

Fixed points of nonexpansive mappings (cites work)
cites work: Fixed-point theorems for mapping defined on unbounded sets in Banach spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Q3338819

Fixed points of nonexpansive mappings (cites work)
cites work: Q3913946

Fixed points of nonexpansive mappings (cites work)
cites work: Fixed Point Theorems in Product Spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Q3211860

Fixed points of nonexpansive mappings (cites work)
cites work: Q3665748

Fixed points of nonexpansive mappings (cites work)
cites work: Q3472661

Fixed points of nonexpansive mappings (cites work)
cites work: Q4772256

Fixed points of nonexpansive mappings (cites work)
cites work: Q5678074

Fixed points of nonexpansive mappings (cites work)
cites work: Multivalued Nonexpansive Mappings and Opial's Condition

Fixed points of nonexpansive mappings (cites work)
cites work: Q3927742

Fixed points of nonexpansive mappings (cites work)
cites work: Nonexpansive maps in generalized Orlicz spaces

Fixed points of nonexpansive mappings (cites work)
cites work: A characterization of totally normal structure

Fixed points of nonexpansive mappings (cites work)
cites work: Permanence properties of normal structure

Fixed points of nonexpansive mappings (cites work)
cites work: Normal Structure and Weakly Normal Structure of Orlicz Sequence Spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Normal structure and the sum-property

Fixed points of nonexpansive mappings (cites work)
cites work: Sets with Fixed Point Property for Isometric Mappings

Fixed points of nonexpansive mappings (cites work)
cites work: Quasi-normal structures for certain spaces of operators on a Hilbert space

Fixed points of nonexpansive mappings (cites work)
cites work: Normal Structure in Dual Banach Spaces Associated With a Locally Compact Group

Fixed points of nonexpansive mappings (cites work)
cites work: A new convexity property that implies a fixed point property for $L_{1}$

Fixed points of nonexpansive mappings (cites work)
cites work: A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space

Fixed points of nonexpansive mappings (cites work)
cites work: Remarks on Some Fixed Point Theorems

Fixed points of nonexpansive mappings (cites work)
cites work: Fixed Point Theorems for Mappings of Nonexpansive Type

Fixed points of nonexpansive mappings (cites work)
cites work: On Asymptotic Centers and Fixed Points of Nonexpansive Mappings

Fixed points of nonexpansive mappings (cites work)
cites work: Asymptotic centers and nonexpansive mappings in conjugate Banach spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Fixed point theorems for uniformly Lipschitzian mappings in LP spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Asymptotic centers in 𝑐₀, 𝑐 and 𝑚

Fixed points of nonexpansive mappings (cites work)
cites work: On the Normal Structure Coefficient and the Bounded Sequence Coefficient

Fixed points of nonexpansive mappings (cites work)
cites work: Q3739752

Fixed points of nonexpansive mappings (cites work)
cites work: Unconditional bases and fixed points of nonexpansive mappings

Fixed points of nonexpansive mappings (cites work)
cites work: Q3782295

Fixed points of nonexpansive mappings (cites work)
cites work: Q4136031

Fixed points of nonexpansive mappings (cites work)
cites work: Q4188284

Fixed points of nonexpansive mappings (cites work)
cites work: Locally Uniformly Convex Banach Spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Q3915469

Fixed points of nonexpansive mappings (cites work)
cites work: Uniformly normal structure and related coefficients

Fixed points of nonexpansive mappings (cites work)
cites work: Fixed points for multivalued mappings defined on unbounded sets in Banach spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Q4021243

Fixed points of nonexpansive mappings (cites work)
cites work: A fixed point theorem for set valued mappings

Fixed points of nonexpansive mappings (cites work)
cites work: Existence of a Fixed Point for Nonexpansive Mappings with Closed Values

Fixed points of nonexpansive mappings (cites work)
cites work: A remark on weakly inward contractions

Fixed points of nonexpansive mappings (cites work)
cites work: Q3668350

Fixed points of nonexpansive mappings (cites work)
cites work: Q3673529

Fixed points of nonexpansive mappings (cites work)
cites work: Multi-valued contraction mappings

Fixed points of nonexpansive mappings (cites work)
cites work: Q4204696

Fixed points of nonexpansive mappings (cites work)
cites work: A fixed point theorem in \(c_ 0\).

Fixed points of nonexpansive mappings (cites work)
cites work: Weak convergence of the sequence of successive approximations for nonexpansive mappings

Fixed points of nonexpansive mappings (cites work)
cites work: Equivalent norms on spaces of bounded functions

Fixed points of nonexpansive mappings (cites work)
cites work: Q3902045

Fixed points of nonexpansive mappings (cites work)
cites work: Q4014478

Fixed points of nonexpansive mappings (cites work)
cites work: Q3211805

Fixed points of nonexpansive mappings (cites work)
cites work: Some estimates for the normal structure coefficient in Banach spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Banach spaces with the uniform opial property

Fixed points of nonexpansive mappings (cites work)
cites work: The Fixed Point Property and Unbounded Sets in Hilbert Space

Fixed points of nonexpansive mappings (cites work)
cites work: Fixed Points of Non-Expansive Functions

Fixed points of nonexpansive mappings (cites work)
cites work: Q4095442

Fixed points of nonexpansive mappings (cites work)
cites work: Approximate selections, best approximations, fixed points, and invariant sets

Fixed points of nonexpansive mappings (cites work)
cites work: Neue Existenz- und Konvergenzsätze in der Fixpunkttheorie nichtlinearer Operatoren

Fixed points of nonexpansive mappings (cites work)
cites work: Some fixed point iteration procedures

Fixed points of nonexpansive mappings (cites work)
cites work: On Commuting Families of Nonexpansive Operators

Fixed points of nonexpansive mappings (cites work)
cites work: On normal structure, stability of fixed point property and the modulus of noncompact convexity

Fixed points of nonexpansive mappings (cites work)
cites work: Q4205971

Fixed points of nonexpansive mappings (cites work)
cites work: The Weak* Karlovitz lemma for dual lattices

Fixed points of nonexpansive mappings (cites work)
cites work: On nonlinear contraction semigroups in sup norm spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Q3310460

Fixed points of nonexpansive mappings (cites work)
cites work: Q3361450

Fixed points of nonexpansive mappings (cites work)
cites work: On an inequality of Bynum and Drew

Fixed points of nonexpansive mappings (cites work)
cites work: Some examples concerning rotundity in Banach spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Normal structure in Bochner \(L^ p\)-spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Q4713040

Fixed points of nonexpansive mappings (cites work)
cites work: Existence of Fixed Points of Nonexpansive Mappings in Certain Banach Lattices

Fixed points of nonexpansive mappings (cites work)
cites work: Schauder bases and fixed points of nonexpansive mappings

Fixed points of nonexpansive mappings (cites work)
cites work: A Generalization of Uniformly Rotund Banach Spaces

Fixed points of nonexpansive mappings (cites work)
cites work: Normal structure in Banach spaces and its generalisations

Fixed points of nonexpansive mappings (cites work)
cites work: The asymptotic behavior of nonlinear semigroups and invariant means

Fixed points of nonexpansive mappings (cites work)
cites work: On Fixed Point Theorems of Nonexpansive Mappings in Product Spaces

Fixed points of nonexpansive mappings (cites work)
cites work: The normal structure of James quasi reflexive space

Fixed points of nonexpansive mappings (cites work)
cites work: Q3965159

Fixed points of nonexpansive mappings (cites work)
cites work: Q4010106

Fixed points of nonexpansive mappings (cites work)
cites work: On Connections between Set and Ball Measures of Noncompactness

Fixed points of nonexpansive mappings (cites work)
cites work: Close-to-normal structure and its applications

Fixed points of nonexpansive mappings (cites work)
cites work: On Kannan Maps

Fixed points of nonexpansive mappings (cites work)
cites work: A fixed point theorem for multivalued nonexpansive mappings in Banach spaces of normal structure

Fixed points of nonexpansive mappings (cites work)
cites work: Q5753027

Fixed points of nonexpansive mappings (cites work)
cites work: Inequalities in Banach spaces with applications

Fixed points of nonexpansive mappings (cites work)
cites work: Q3359379

Fixed points of nonexpansive mappings (cites work)
cites work: On some fixed point theorems for multivalued mappings

Fixed points of nonexpansive mappings (cites work)
cites work: A geometrically aberrant Banach space with uniformly normal structure

Fixed points of nonexpansive mappings (cites work)
cites work: Weakly Convergent Sequence Coefficient of Product Space

Fixed points of nonexpansive mappings (cites work)
cites work: Q4713309

Fixed points of nonexpansive mappings (cites work)
cites work: Q5638789

Fixed points of nonexpansive mappings (Recommended article)
Recommended article: On fixed points of nonexpansive mappings
Similarity Score: 0.9866483
Recommender Run: Recommender Run 3

Fixed points of nonexpansive mappings (Recommended article)
Recommended article: Q4274030
Similarity Score: 0.9786014
Recommender Run: Recommender Run 3

Fixed points of nonexpansive mappings (Recommended article)
Recommended article: Q5471194
Similarity Score: 0.97829497
Recommender Run: Recommender Run 3

Fixed points of nonexpansive mappings (Recommended article)
Recommended article: Fixed points of nonexpansive and quasi-nonexpansive mappings
Similarity Score: 0.9762019
Recommender Run: Recommender Run 3

Fixed points of nonexpansive mappings (Recommended article)
Recommended article: Fixed point theorems for nonexpansive mappings
Similarity Score: 0.974207
Recommender Run: Recommender Run 3

Fixed points of nonexpansive mappings (Recommended article)
Recommended article: Fixed point theory and nonexpansive mappings
Similarity Score: 0.9738561
Recommender Run: Recommender Run 3

Fixed points of nonexpansive mappings (Recommended article)
Recommended article: Fixed points of nonexpansive mappings in spaces of continuous functions
Similarity Score: 0.96679807
Recommender Run: Recommender Run 3

Fixed points of nonexpansive mappings (Recommended article)
Recommended article: Q4981076
Similarity Score: 0.9665401
Recommender Run: Recommender Run 3

Fixed points of nonexpansive mappings (Recommended article)
Recommended article: Fixed points of nonexpanding maps
Similarity Score: 0.96416104
Recommender Run: Recommender Run 3

Fixed points of nonexpansive mappings (Recommended article)
Recommended article: Q3819582
Similarity Score: 0.9618231
Recommender Run: Recommender Run 3