Elementary fixed point theorems

306 Pages·2018·1.773 MB·English

by  Subrahmanyam P.V.

Forum for Interdisciplinary Mathematics
P. V. Subrahmanyam
Elementary
Fixed Point
Theorems
Forum for Interdisciplinary Mathematics
Editor-in-chief
P. V. Subrahmanyam, Indian Institute of Technology Madras, Chennai, India
Editorial Board
Yogendra Prasad Chaubey, Concordia University, Montreal (Québec), Canada
Jorge Cuellar, Principal Researcher, Siemens, Germany
Janusz Matkowski, University of Zielona Góra, Poland
ThiruvenkatachariParthasarathy,ChennaiMathematicalInstitute,Kelambakkam,India
Mathieu Dutour Sikirić, Rudjer Boúsković Institute, Zagreb, Croatia
Bhu Dev Sharma, Jaypee Institute of Information Technology, Noida, India
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P. V. Subrahmanyam
Elementary Fixed Point
Theorems
123
P. V.Subrahmanyam (emeritus)
Department ofMathematics
Indian Institute of Technology Madras
Chennai, Tamil Nadu,India
ISSN 2364-6748 ISSN 2364-6756 (electronic)
Forumfor Interdisciplinary Mathematics
ISBN978-981-13-3157-2 ISBN978-981-13-3158-9 (eBook)
https://doi.org/10.1007/978-981-13-3158-9
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Preface
Fixed point theorems constitute an important and interesting aspect of applicable
mathematicsandprovidesolutionstoseverallinearandnonlinearproblemsarising
in biological, engineering and physical sciences. This book, as the title suggests,
deals with some fixed point theorems and applications. The choice of the topics is
largely guided by personal preferences, and the book will serve as a subjective
sampler of topics in fixed point theorems.
This volume neither exhausts all the important fixed point theorems nor elabo-
ratesthemostgeneralformulationsofthetheoremspresented.Noattemptismadeto
provideahistorical/chronologicalbackgroundforthetopicscovered.Nevertheless,
it hopefully supplements the extant publications and promotes interest for further
study among thereaders.
A quick run-through of the highlights of the individual chapters is not out of
place. While Chap. 1 collects the analytic and topological preliminaries needed in
the sequel, Chap. 2 describes the basic properties of iterates of real and complex
functions. It details the theorems of Thron on the rates of convergence of certain
classes of real functions. Cohen’s common fixed point theorem for commuting
continuousfullsurjectionsonacompactinterval,Shield’stheoremontheexistence
ofacommonfixedpointforacommutingfamilyofanalyticfunctionsontheclosed
unit disc, an elementary proof of Sharkovsky’s theorem on periodic points of real
functions and Bergweiler’s theorem on the existence offixed points of meromor-
phic functions are other noteworthy features of Chap. 2. Chapter 3 explores the
existence of fixed points in the setting of partially ordered sets. Knaster–Tarski
principle is formulated leading to Tarski’s theorem, and its applications to set
theory,generaltopology,nonlinearcomplementarityproblem,parabolicdifferential
equations and formal languages are described. Also discussed are some general-
izations due to Merrifield and Stein.
MostofChap.4dealswithWard’stheoryofpartiallyorderedtopologicalspaces
culminating in Schweigert’s fixed point theorem. Manka’s fixed point theorem on
inductively and acyclically ordered posets is proved and used to deduce the fixed
v
vi Preface
point property of continuous functions on dendroids. Also highlighted is Klee’s
counterexampleinfixedpointtheory.Contractionprincipleandsomeofitsvariants
aretakenupinChap.5,includingKupka’stopologicalgeneralizationandNadler’s
extension to multifunctions. Jachymski’s proof of the converse of the contraction
principle is also elaborated. Applications to differential equations, functional
equations, algebraic Weierstrass preparation theorem, Cauchy–Kowalevsky theo-
rem and the central limit theorem are detailed in Chap. 6. Chapter 7 is devoted to
Caristi’sfixedpointtheorem,ageneralizationofthecontractionprinciple.Separate
proofs of Caristi’s theorem due to Siegel, Penot and Kirk are provided. The
equivalence of Caristi’s theorem, Ekeland’s variational principle and Takahashi’s
minimization theorem is proved. That these three equivalent theorems characterize
metrical completeness is also established.
Chapter 8 is on fixed points of contractive and non-expansive maps. Goebel’s
proof of Browder–Gohde–Kirk fixed point theorem for nonexpansive mappings in
thesettingofuniformlyconvexBanachspacesisprovided.Pasicki’stheoryofbead
and discus spaces and his theorem on the fixed points of nonexpansive maps with
an application of Matkowski to certain functional equations are other highlights.
Ishikawa’s theorems on iterates are detailed as also a fixed point theorem due to
Merrifield et al. on generalized contractions using combinatorial ideas.
Chapter 9 using the concepts of the geometry of Banach spaces establishes that
convexweaklycompactsubsetsofnearlyuniformlysmooth(NUS)andnon-square
Banach spaces have fixed point property for nonexpansive mappings. Brouwer’s
theoremistreatedinChap.10.BesidesananalyticproofduetoRogers,proofbased
onSperner’slemmaisprovided.TheexistenceofWalrasianandNashequilibriafor
economies isdeduced. Whyburn’sproof of Stallings'sgeneralization ofBrouwer’s
fixed point theorem for connectivity maps has also been elaborated. Schauder’s
theorem and its extensions and applications constitute the major part of Chap. 11.
Besides Tychonoff’s fixed point theorem and Ky Fan–Browder–Glicksberg theo-
rem for multifunctions, Kakutani–Markov and Ryll–Nardzewski theorems are
described. The existence of Banach limits, Haar integral on compact groups
and Nash equilibria is provided besides an introduction to measures of
noncompactness. Chapter 12 describes the finite-dimensional degree theory as
presentedbyHeinz.AppendicesAandBsummarizetheclassicalcounterexamples
due to Huneke and Kinoshita, respectively. Appendix C deals with fractals and
fixed points.
The text closely follows the notation and organization of the papers quoted
extensively,withinreasonablelimits,toaidthereaderstoperusethesepaperswith
ease.AjudiciouschoiceofsectionsfromChaps.2–12canconstituteabasiccourse
on fixed point theorems.
Noexpressionofgratitudeisadequatefortheawardofasabbaticalleavegranted
bythecompetent authorityof theIndian Institute of TechnologyMadrastomefor
the preparation of an earlier concise version of this tract. I am thankful to my
colleagues for their support during the preparation of this tract and to my wife
Preface vii
Dr. S. Alamelu Bai for proofreading the content and for the moral support. I am
thankful to Dr. N. Sivakumar (Texas A&M University) and Dr. Antony Vijesh
(IIT Indore) for getting several papers for reference. I am grateful to Profs. M. S.
Rangachari, G. Rangan, (Late) K. S. Padmanabhan and (Late) K. N. Venkataraman
of the Madras University, Chennai, for initiating me into different aspects of mathe-
matical analysis. Thanks are due to Mr. E. Boopal for typesetting the manuscript.
Suggestions for improvement and corrections of errors and misprints are earn-
estly solicited.
Chennai, India P. V. Subrahmanyam
Contents
1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Topological Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Fixed Points of Some Real and Complex Functions . . . . . . . . . . . . 23
2.1 Fixed Points of Continuous Maps on Compact
Intervals of R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Iterates of Real Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Periodic Points of Continuous Real Functions . . . . . . . . . . . . . 31
2.4 Common Fixed Points, Commutativity and Iterates. . . . . . . . . . 35
2.5 Common Fixed Points and Full Functions . . . . . . . . . . . . . . . . 41
2.6 Common Fixed Points of Commuting Analytic Functions . . . . . 44
2.7 Fixed Points of Meromorphic Functions. . . . . . . . . . . . . . . . . . 48
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Fixed Points and Order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1 Fixed Points in Linear Continua . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Knaster–Tarski Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Tarski’s Lattice Theoretical Fixed Point Theorem
and Related Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Partially Ordered Topological Spaces and Fixed Points . . . . . . . . . 73
4.1 A Precis of Partially Ordered Topological Spaces. . . . . . . . . . . 73
4.2 Schweigert–Wallace Fixed Point Theorem . . . . . . . . . . . . . . . . 81
4.3 Set Theory, Fixed Point Theory and Order. . . . . . . . . . . . . . . . 86
4.4 Multifunctions and Dendroids . . . . . . . . . . . . . . . . . . . . . . . . . 90
ix
x Contents
4.5 Some Spaces with Fixed Point Property. . . . . . . . . . . . . . . . . . 92
4.6 An Example in Fixed Point Theory . . . . . . . . . . . . . . . . . . . . . 94
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5 Contraction Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1 A Simple Proof of the Contraction Principle. . . . . . . . . . . . . . . 97
5.2 Metrical Generalizations of the Contraction Principle . . . . . . . . 100
5.3 Fixed Points of Multivalued Contractions. . . . . . . . . . . . . . . . . 105
5.4 Contraction Principle in Gauge Spaces. . . . . . . . . . . . . . . . . . . 107
5.5 A Converse to the Contraction Principle . . . . . . . . . . . . . . . . . 109
5.6 A Topological Contraction Principle . . . . . . . . . . . . . . . . . . . . 114
5.7 Another Proof of the Contraction Principle. . . . . . . . . . . . . . . . 116
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Applications of the Contraction Principle . . . . . . . . . . . . . . . . . . . . 121
6.1 Linear Operator Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3 A Functional Differential Equation. . . . . . . . . . . . . . . . . . . . . . 128
6.4 A Classical Solution for a Boundary Value Problem
for a Second Order Ordinary Differential Equation . . . . . . . . . . 129
6.5 An Elementary Proof of the Cauchy–Kowalevsky
Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.6 An Application to a Discrete Boundary Value Problem. . . . . . . 133
6.7 Applications to Functional Equations . . . . . . . . . . . . . . . . . . . . 135
6.8 An Application to Commutative Algebra . . . . . . . . . . . . . . . . . 140
6.9 A Proof of the Central Limit Theorem. . . . . . . . . . . . . . . . . . . 143
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7 Caristi’s Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2 Siegel’s Proof of Caristi’s Fixed Point Theorem . . . . . . . . . . . . 151
7.3 Ekeland’s Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . 156
7.4 A Minimization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.5 An Application of Ekeland’s Principle . . . . . . . . . . . . . . . . . . . 161
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8 Contractive and Non-expansive Mappings . . . . . . . . . . . . . . . . . . . 165
8.1 Contractive Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.2 Non-expansive Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.3 Browder–Gohde–Kirk Fixed Point Theorem. . . . . . . . . . . . . . . 173
8.4 A Generalization to Metric Spaces. . . . . . . . . . . . . . . . . . . . . . 176
8.5 An Application to a Functional Equation . . . . . . . . . . . . . . . . . 181
8.6 Convergence of Iterates in Normed Spaces. . . . . . . . . . . . . . . . 184