Table Of ContentFixed Point Theory and Best Approximation:
The KKM-map Principle
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 424
Fixed Point Theory and
Best Approximation:
TheKKM-map
Principle
by
Sankatha Singh
Department ofM athematics and Statistics,
Memorial University ofNewfoundland,
St John's, Newfoundland, Canada
Bruce Watson
Department of Mathematics and Statistics,
Memorial University ofN ewfoundland,
St John's, Newfoundland, Canada
and
Pramila Srivastava
Allahabad Mathematical Society,
Allahabad, India
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4918-6 ISBN 978-94-015-8822-5 (eBook)
DOI 10.1007/978-94-015-8822-5
Printed on acid-free paper
Ali Rights Reserved
© 1997 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1997
Softcover reprint of the hardcover 1s t edition 1997
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
Dedication
To our parents and especially to Shri Mahadeo Singh on the occasion
of his 95th birthday.
Table of Contents
FIXED POINT THEORY AND BEST APPROXIMATION:
THE KKM-MAP PRINCIPLE
Preface ix
1 Introductory Concepts and Fixed Point Theorems 1
1.1 Topological Preliminaries 1
1.1.1 Metric Spaces . . . . . . . 1
1.1.2 Hilbert Spaces .... . . 3
1.1.3 Topological Vector Spaces 7
1.1.4 Locally Convex Spaces . 8
1.2 Normal Structure . . . . . . . . . 9
1.3 Fixed Points ... . .. . . . . . 10
1.4 The Banach Contraction Principle 12
1.5 Fixed Point Theorems for Nonexpansive Mappings 22
1.6 Quasi-nonexpansive Mappings and Fixed Points . 27
1.7 Densifying Maps and Fixed Points . . . 28
1.8 Multivalued Mappings and Fixed Points . . 34
1.9 Integral Equations . . . . . . . . . . . . . . 45
1.10 The Method of Successive Approximations. 53
1.11 The Iteration Process for Continuous Functions 54
1.11.1 The Mann Iterative Process . . . . . . . 56
1.11.2 The Sequence of Iterates of Nonexpansive Mappings 58
1.11.3 Convergence Criteria in Convex Metric Spaces 63
1.11.4 Iterative Methods for Variational Inequalities 66
2 Ky Fan's Best Approximation Theorem 73
2.1 Introduction . .. ... .. . . . . .. .. 73
2.2 Ky Fan Type Theorems in Hilbert Space . 77
2.3 Applications to Fixed Point Theorems . . 85
2.4 Prolla's Theorem and Extensions . . . . . 92
2.5 Ky Fan's Best Approximation Theorem for Multifunctions. 99
2.6 Kakutani Factorizable Maps and Applications . . . . . . . . 113
vii
viii
3 Principle and Applications of KKM-maps 121
3.1 Introduction......................... 121
3.2 The KKM-Map Principle ................. 121
3.3 Extensions of the KKM-Map Principle and Applications 124
3.4 Two function Theorems and Applications . . . 127
3.5 Application to Variational Inequalities . . . . . 131
3.6 Further Extensions of the KKM-Map Principle 135
3.7 Open-Valued KKM-Map and Related Results. 144
3.8 Further Applications . . . . . . . . . . . . . . . 146
3.9 Equivalent Formulation of the KKM-Map Principle. 151
3.10 Theory of the H-KKM-Map Principle ........ 156
4 Partitions of Unity and Applications 159
4.1 Introduction.......... ... .......... 159
4.2 Browder's Theorem and its Applications . . . . . . . 160
4.3 Ky Fan's Theorem, Its Extensions and Applications 167
4.4 Existence Theorems and Consequences . . . . . . . . 170
4.5 Coincidence Theorems and Applications . . . . . . . 174
4.6 Further Results on Variational and Minimax Inequalities. 177
5 Application of Fixed Points to Approximation Theory 191
5.1 Introduction.. ......... .. 191
5.2 Preliminaries and Basic Definitions 191
5.3 Existence of Best Approximations 191
5.4 Invariance of Best Approximation. 193
5.5 Invariance of Best Approximation in Locally Convex Spaces 194
5.6 Some Further Extensions ................... 196
5.7 The Problem of Convexity of Chebyshev Sets . . . . . . .. 197
5.8 Best Simultaneous Approximations and Distance Between
Two Sets ...................... .... 198
5.9 Variational Inequalities and Complementarity Problems 199
Bibliography 205
Preface
Recently a great deal of work has been done in the field of nonlinear
analysis. The topic has grown very rapidly and has many interesting applications
in various fields.
In this book, an attempt is made to pick up only a small section of
the growing field and give an up-to-date development so that any young
researcher can get enough literature to start with the work, and those who
are in the field, can get ready references of most recent work.
The book has five main chapters. The first chapter gives a brief survey
of results on fixed point theory. This includes most recent work in great
detail. In the end, a brief section is devoted to a set of applications.
The second chapter deals with the best approximation and fixed point
theorems. The starting point in this chapter is the well-known result of Ky
Fan. Several extensions and unifications of this important result are given.
This theorem has many applications in fixed theory, and therefore, a good
deal of work is in fixed point theory, especially for non-self maps. Many
interesting results of this nature dealing with random approximation and
random fixed point theorems could not be accommodated because of the
size of the text.
The third chapter is devoted to the study of KKM-map principle where
its extensions and a series of its applications in various fields are also
included. Again, the main result given in infinite dimensional space is due to
Ky Fan. Several extensions, unifications, and equivalence of such results are
given in detail. As results in fixed point theory, variational inequality and
KKM-theory are very closely related, and therefore, this topic has numerous
important applications in a variety of areas of mathematics, mathematical
economics, game theory, and in applied mathematics/engineering.
Chapter 4 deals with results on the partition of unity argument and the
Brouwer fixed point theorems. In this chapter, most of the results are given
where compact sets are considered. The main theorem of Browder is used
to give several interesting applications. The basic tools used simplify the
proof and have been used in various results. A few results are also given
where paracompact set has been considered.
In Chapter 5, applications of the nonlinear analysis are given in the areas
like approximation theory, variational inequalities, and complementarity
problems. Recently, it has been shown by a few researchers that fixed point
theory, optimization problems, approximation theory, complementarity problems,
variational inequalities, and KKM-map principle are equivalent. Thus, a
great deal of interest is generated in the field and researchers in applied
mathematics, engineering, economics, and applied physics are working together to
obtain interesting results.
ix
x
We thank our colleagues for their support, especially P.P.Narayanaswami
and Ruby and Tony Kocurko for going through a part of the manuscript and
making valuable suggestions. The encouragement given by Professor Allasia
(University of Torino) and Professor A. Carbone (University of Calabria)
is greatly appreciated.
It is with pleasure that we express our thanks to Professors Bardaro,
Browder, Ceppitelli, Cheney, Edelstein, Fan, Granas, Isac, Kim, Kirk, Lassonde,
Lin Mehta, Noor, Park, Sehgal, Sessa, Takahashi, Tan, Tarafdar, Vetrivel,
Waters, and Yuan for providing their reprints/preprints and encouragements.
We express our sincere thanks to Ms. Angelique Hempel and the staff
of the Kluwer Academic Publishers for their cooperation, patience, and
understanding.
In the end, we express our thanks and appreciation to Ms. Philomena
French who typed the entire manuscript. She was always ready to make
changes/c orrections/ alterations cheerfully.
Introductory Concepts and Fixed Point Theorems
1.1. Topological Preliminaries
1.1.1. METRIC SPACES
We begin with some basic definitions.
Definition 1.1 Let X be a set and d a function from X X X -t 1R+
(nonnegative reals) such that for all x, y, z E X we have
M1: d(x, y) = 0 if and only if x = y,
M2: d(x, y) = d(y, x), and
+
M3: d(x, y) ~ d(x, z) d(z, y).
A function d satisfying the above conditions is said to be a distance
function or metric and the pair (X, d) a metric space. We write X for a
metric space (X, d).
The real line IR with d(x, y) = Ix - yl is a metric space. The metric dis
called the usual metric for lR.
Definition 1.2 A sequence, {xn}, in metric space X is said to be a Cauchy
sequence if for each c > 0, there exists a positive integer N(c) such that
d(xn,xm ) < c for all m,n 2: N.
Definition 1.3 A sequence, {xn}, of points of X is said to converge to a
point y if, for given c > 0, there exists a natural number N(c) such that
d(xn, y) < € whenever n 2: N.
In this case we write either limn--+oo d(xn, y) = 0 or lim Xn = y or Xn -t y.
Note: A convergent sequence is a Cauchy sequence.
Definition 1.4 A metric space X is said to be complete if every Cauchy
sequence in X converges to a point in X.
With the usual metric, IR is a complete metric space and so is C[a, b],
the space of continuous functions on [a, b], with metric given by
d(J,g) = max IJx - gxl.
xE[a,bj
Note: If f:X t-+ Y and a reference is made to f(x), it is standard practice
in fixed point theory to suppress the brackets and write fx instead. We
S. Singh et al., Fixed Point Theory and Best Approximation: The KKM-map Principle
© Springer Science+Business Media Dordrecht 1997
