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Tables of Contents for Handbook of Metric Fixed Point Theory
Chapter/Section Title
Page #
Page Count
Preface
xi
Contraction mappings and extensions
1
34
W. A. Kirk
Introduction
1
2
The contraction mapping principle
3
4
Further extensions of Banach's principle
7
7
Caristi's theorem
14
1
Set-valued contractions
15
3
Generalized contractions
18
2
Probabilistic metrics and fuzzy sets
20
3
Converses to the contraction principle
23
2
Notes and remarks
25
10
Examples of fixed point free mappings
35
14
B. Sims
Introduction
35
1
Examples on closed bounded convex sets
36
4
Examples on weak* compact convex sets
40
3
Examples on weak compact convex sets
43
4
Notes and remarks
47
2
Classical theory of nonexpansive mappings
49
44
K. Goebel
W. A. Kirk
Introduction
49
1
Classical existence results
50
14
Properties of the fixed point set
64
5
Approximation
69
9
Set-valued nonexpansive mappings
78
1
Abstract theory
79
14
Geometrical background of metric fixed point theory
93
40
S. Prus
Introduction
93
1
Strict convexity and smoothness
93
5
Finite dimensional uniform convexity and smoothness
98
10
Infinite dimensional geometrical properties
108
10
Normal structure
118
9
Bibliographic notes
127
6
Some moduli and constants related to metric fixed point theory
133
44
E. L. Fuster
Introduction
133
1
Moduli and related properties
134
23
List of coefficients
157
20
Ultra-methods in metric fixed point theory
177
24
M. A. Khamsi
B. Sims
Introduction
177
1
Ultrapowers of Banach spaces
177
9
Fixed point theory
186
7
Maurey's fundamental theorems
193
2
Lin's results
195
2
Notes and remarks
197
4
Stability of the fixed point property for nonexpansive mappings
201
38
J. Garcia-Falset
A. Jimenez-Melado
E. Llorens-Fuster
Introduction
201
3
Stability of normal structure
204
8
Stability for weakly orthogonal Banach lattices
212
5
Stability of the property M(X) > 1
217
6
Stability for Hilbert spaces. Lin's theorem
223
5
Stability for the τ-FPP
228
3
Further remarks
231
5
Summary
236
3
Metric fixed point results concerning measures of noncompactness
239
30
T. Dominguez
M. A. Japon
G. Lopez
Preface
239
1
Kuratowski and Hausdorff measures of noncompactness
240
4
&phis;-minimal sets and the separation measure of noncompactness
244
4
Moduli of noncompact convexity
248
4
Fixed point theorems derived from normal structure
252
5
Fixed points in NUS spaces
257
3
Asymptotically regular mappings
260
4
Comments and further results in this chapter
264
5
Renormings of l1 and co and fixed point properties
269
30
P. N. Dowling
C. J. Lennard
B. Turett
Preliminaries
269
2
Renormings of l1 and co and fixed point properties
271
23
Notes and remarks
294
5
Nonexpansive mappings: boundary/inwardness conditions and local theory
299
24
W. A. Kirk
C. H. Morales
Inwardness conditions
299
2
Boundary conditions
301
7
Locally nonexpansive mappings
308
2
Locally pseudocontractive mappings
310
10
Remarks
320
3
Rotative mappings and mappings with constant displacement
323
16
W. Kaczor
M. Koter-Morgowska
Introduction
323
1
Rotative mappings
323
7
Firmly lipschitzian mappings
330
3
Mappings with constant displacement
333
3
Notes and remarks
336
3
Geometric properties related to fixed point theory in some Banach function lattices
339
52
S. Chen
Y. Cui
H. Hudzik
B. Sims
Introduction
339
4
Normal structure, weak normal structure, weak sum property, sum property and uniform normal structure
343
13
Uniform rotundity in every direction
356
2
B-convexity and uniform monotonicity
358
4
Nearly uniform convexity and nearly uniform smoothness
362
5
WORTH and uniform nonsquareness
367
1
Opial property and uniform opial property in modular sequence spaces
368
9
Garcia-Falset coefficient
377
1
Cesaro sequence spaces
378
2
WCSC, uniform opial property, k-NUC and UNS for cesp
380
11
Introduction to hyperconvex spaces
391
46
R. Espinola
M. A. Khamsi
Preface
391
2
Introduction and basic definitions
393
1
Some basic properties of hyperconvex spaces
394
5
Hyperconvexity, injectivity and retraction
399
6
More on hyperconvex spaces
405
6
Fixed point property and hyperconvexity
411
4
Topological fixed point theorems and hyperconvexity
415
3
Isbell's hyperconvex hull
418
4
Set-valued mappings in hyperconvex spaces
422
6
The KKM theory in hyperconvex spaces
428
3
Lambda-hyperconvexity
431
6
Fixed points of holomorphic mappings: a metric approach
437
80
T. Kuczumow
S. Reich
D. Shoikhet
Introduction
437
1
Preliminaries
438
2
The Kobayashi distance on bounded convex domains
440
7
The Kobayashi distance on the Hilbert ball
447
3
Fixed points in Banach spaces
450
4
Fixed points in the Hilbert ball
454
6
Fixed points in finite powers of the Hilbert ball
460
5
Isometries on the Hilbert ball and its finite powers
465
4
The extension problem
469
3
Approximating sequences in the Hilbert ball
472
9
Fixed points in infinite powers of the Hilbert ball
481
2
The Denjoy-Wolff theorem in the Hilbert ball and its powers
483
7
The Denjoy-Wolff theorem in Banach spaces
490
6
Retractions onto fixed point sets
496
6
Fixed points of continuous semigroups
502
5
Final notes and remarks
507
10
Fixed point and non-linear ergodic theorems for semigroups of non-linear mappings
517
40
A. Lau
W. Takahashi
Introduction
517
1
Some preliminaries
518
1
Submean and reversibility
519
4
Submean and normal structure
523
4
Fixed point theorem
527
5
Fixed point sets and left ideal orbits
532
6
Ergodic theorems
538
7
Related results
545
12
Generic aspects of metric fixed point theory
557
20
S. Reich
A. J. Zaslavski
Introduction
557
1
Hyperbolic spaces
557
1
Successive approximations
558
3
Contractive mappings
561
3
Infinite products
564
3
(F)-attracting mappings
567
1
Contractive set-valued mappings
568
1
Nonexpansive set-valued mappings
569
1
Porosity
570
7
Metric environment of the topological fixed point theorems
577
36
K. Goebel
Introduction
577
2
Schauder's theorem
579
7
Minimal displacement problem
586
11
Optimal retraction problem
597
7
The case of Hilbert space
604
4
Notes and remarks
608
5
Order-theoretic aspects of metric fixed point theory
613
30
J. Jachymski
Introduction
613
1
The Knaster-Tarski theorem
614
9
Zermelo's fixed point theorem
623
7
The Tarski-Kantorovitch theorem
630
13
Fixed point and related theorems for set-valued mappings
643
48
G. Yuan
Introduction
643
1
Knaster-Kuratowski-Mazurkiewicz principle
644
7
Ky Fan minimax principle
651
2
Ky Fan minimax inequality-I
653
6
Ky Fan minimax inequality-II
659
3
Fan-Glicksberg fixed points in G-convex spaces
662
4
Nonlinear analysis of hyperconvex metric spaces
666
25
Index
691