Background and Recent Developments of Metric Fixed Point Theory

279 Pages·2018·2.973 MB·English

by  Dhananjay Gopal, Poom Kumam, Mujahid Abbas

Background and
Recent Developments of
METRIC FIXED
POINT THEORY

Background and
Recent Developments of
METRIC FIXED
POINT THEORY
Edited by
Dhananjay Gopal
Poom Kumam
Mujahid Abbas
CRC Press
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Contents
Foreword vii
Preface ix
Editors and Authors xiii
Symbol Descriptions xv
1 Banach Fixed Point Theorem and Its Generalizations 1
Dhananjay Gopal, Deepesh Kumar Patel and Satish Shukla
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Banach fixed point theorem . . . . . . . . . . . . . . . . . . . 1
1.3 Some other generalizations of BCP . . . . . . . . . . . . . . 9
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Metrical Common Fixed Points and Commuting Type
Mappings 29
Dhananjay Gopal and Ravindra K Bisht
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Comparison of weaker forms of commuting mappings . . . . 32
2.3 Motivation and further scope . . . . . . . . . . . . . . . . . . 59
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 α-Admissibility and Fixed Points 69
Deepesh Kumar Patel and Wutiphol Sintunavarat
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 Fixed point of α-ψ-contractive type mappings . . . . . . . . 70
3.3 Fixed and common fixed point of Meir-Keeler α-contractive
type mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4 Fixed point of α-type F-contractive mappings . . . . . . . . 105
3.5 Fixed point of α-ψ-contractive type mappings along with
weakly α-admissible mappings . . . . . . . . . . . . . . . . . 123
3.6 Conclusions and future investigations . . . . . . . . . . . . . 129
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
v
vi Contents
4 Fixed Point Theory in Fuzzy Metric Spaces 135
Dhananjay Gopal
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2 KramosilandMichalekfuzzymetricspacesandGrabiec’sfixed
point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.3 George and Veeramani’s fuzzy metric space and fuzzy
contractive mappings . . . . . . . . . . . . . . . . . . . . . . 140
4.4 Fuzzy metric-like spaces and fixed point results . . . . . . . . 149
4.5 Fuzzy Preˇsi´c-C´iri´coperators and unified fixed point theorems 160
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5 Fixed Point Theory in Soft Metric Spaces: Rise and Fall 179
Mujahid Abbas, Ghulam Murtaza and Salvador Romaguera
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.2 Banach contraction theorem with restriction . . . . . . . . . 183
5.3 Role of restricted condition . . . . . . . . . . . . . . . . . . . 189
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6 Best Proximity Point Theorems for Cyclic Contractions
Mappings 201
Chirasak Mongkolkeha and Poom Kumam
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.2 Best proximity point theorems for cyclic contractions . . . . 202
6.3 Best proximity point with property UC . . . . . . . . . . . . 206
6.4 Best proximity point with proximally complete property . . 216
6.5 Common best proximity points for proximity commuting
mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7 Applications of Metric Fixed Point Theory 229
Satish Shukla
7.1 Root finding problems . . . . . . . . . . . . . . . . . . . . . . 229
7.2 Solution of system of linear algebraic equations . . . . . . . . 230
7.3 Markov process and steady state vector . . . . . . . . . . . . 233
7.4 Solutions of integral equations . . . . . . . . . . . . . . . . . 235
7.5 Solutions of initial value and boundary value problems . . . 239
7.6 Difference equations and cyclic systems . . . . . . . . . . . . 252
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
Index 259
Foreword
Thisbook,“BackgroundandRecentDevelopmentsofMetricFixedPointThe-
ory,”isanattempt to presentanattractiveresearchareaofmetric fixed point
theory in a simple way with focus on clarity of arguments in the proofs of
classical and recent results in this area. A collection of examples illustrates a
varietyofessentialconcepts.Fixedpointtheorymakesthisbookaneasilyac-
cessiblesourceofknowledge.Thisbook touchesonseveralresearchdirections
within the fixed point theory and opens new avenues of investigation to ex-
tend and explore it further. Written by well known fixed point theorists with
vast teaching experience, this book is particularly suitable for young mathe-
maticians who want to study fixed point theory and to pursue their research
careers in that area.
The presentation of Banach fixed point theorem and its generalizations
provides a handy account of development and progress in metric fixed point
theorydealingwithsinglevaluedmapping.Thebookalsoincludesinteresting
common fixed point results for families of commuting mappings. A thorough
discussion on recently introduced concepts of α-admissible mappings and re-
latedresultsinvitesactiveresearchersto contributeandto extendthis theory
further.
A fixed point theory discussed in the framework of fuzzy and soft metric
willnotonlyprovideanadequatebackgroundbutalsowilldescribeadvanced
results in this direction. Best proximity point theory has attracted the atten-
tionofseveralmathematiciansworkinginoptimizationtheory.Thismotivates
us to include a chapter on cyclic mappings satisfying certain contractivecon-
ditions and best proximity point results in the setup of metric spaces. The
last chapter of this book highlights the significance of fixed point theory in
connection with numerical analysis, economics, integral equations, boundary
value problems and difference equations. We believe that this effort will in-
spireyoungmathematicianstoexploremoreapplicationsofmetricfixedpoint
theory and extend the boundaries of this theory further.
Sompong Dhompongsa
Chiang Mai University
Chiang Mai, Thailand
Yeol Je Cho
Gyeongsang National University
Jinju, Korea
vii

Preface
Functional analysis is an important branch of mathematics divided mainly
into linear and nonlinear categories. We live in a nonlinear world in which
infinitesimal inputs may result in macroscopic outputs and vice versa. For
this reason,functional analysis has evolvedinto a separateand practicalarea
of mathematics.
Functional analysis is an important branch of mathematics which can
mainlybedividedintotwocategories:LinearandNonlinear.As,weareliving
inanonlinearworldwhereinfinitesimalinputsmayresultinmacroscopicout-
putsorviceversa,sononlinearfunctionalanalysishasbecomeanindependent
and a useful subject.
The theory of fixed points deals with the conditions which guarantee the
existence of points x of a set X which solve an operator equation x = Tx,
where T is a transformation defined on a set X. The solution set of such a
problem can be empty, a finite set, a countable or uncountable infinite set.
Fixedpointtheoryprovidesessentialtoolsforsolvingproblemsarisinginvar-
ious branchesof mathematicalanalysis.Split feasibility problems,variational
inequality problems, nonlinear optimization problems, equilibrium problems,
complementarityproblems,selectionandmatchingproblems,andprovingso-
lutionsofintegralanddifferentialequationsarecloselyrelatedwithfixedpoint
theory. A long list of problems fall into the category of solving a fixed point
problem. In particular, the solutions have deep roots in nonlinear functional
analysis.
Research in fixed point theory generally includes (a) the investigation of
lessrestrictiveconditionsonmappingsthatguaranteestheexistenceofafixed
point, (b) study of conditions which assure uniqueness of a fixed point, (c)
modification,enrichmentandextensionofthestructuresofdomainsofdefini-
tion to obtain more general spaces, (d) identification of characterization, (e)
constructionsorapproximationsoffixedpoints and(f) the study ofstructure
of the set of fixed points of mapping under consideration.
The Banach contraction principle proved in 1922 lies at the heart of met-
ric fixed point theory and has played a fundamental role in many aspects of
nonlinear functional analysis. The principle arises from the use of successive
approximations (an idea initiated by Picard in 1890) to establish the exis-
tence and uniqueness of the solution of an operator equation T(x)=x. This
principle serves as a powerful tool with a wide range of applications; in par-
ticular itcanbe employedto provethe existence ofsolutions ofdifferentialor
ix