Table Of ContentSynthesis Lectures on
Mathematics & Statistics
Erdal Karapınar · Ravi P. Agarwal
Fixed Point Theory
in Generalized
Metric Spaces
Synthesis Lectures on Mathematics &
Statistics
SeriesEditor
StevenG.Krantz,DepartmentofMathematics,WashingtonUniversity,SaintLouis,
MO,USA
This series includes titles in applied mathematics and statistics for cross-disciplinary
STEMprofessionals,educators,researchers,andstudents.Theseriesfocusesonnewand
traditionaltechniquestodevelopmathematicalknowledgeandskills,anunderstandingof
core mathematical reasoning, and the ability to utilize data in specific applications.
Erdal Karapınar · Ravi P. Agarwal
Fixed Point Theory
in Generalized Metric
Spaces
ErdalKarapınar RaviP.Agarwal
DepartmentofMathematics DepartmentofMathematics
ÇankayaUniversity TexasA&MUniversity-Kingsville
Ankara,Turkey Kingsville,TX,USA
ISSN1938-1743 ISSN1938-1751 (electronic)
SynthesisLecturesonMathematics&Statistics
ISBN978-3-031-14968-9 ISBN978-3-031-14969-6 (eBook)
https://doi.org/10.1007/978-3-031-14969-6
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG
2022
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whetherthewhole
orpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorage
andretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownor
hereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoes
notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective
lawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbookare
believedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsortheeditorsgive
awarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissionsthat
mayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictionalclaimsinpublishedmapsand
institutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
ErdalKarapınar:TomywifeSenemPinar
andourchildrenCanandUlas¸ Ege.
RaviP.Agarwal:TomywifeSadhna.
Preface
Thefixedpointtheoryisoneofthemajorresearchareasinnonlinearfunctionalanalysis,
aswellastopologyandappliedmathematics.Indeed,afixedpointproblemisverysimple
to state, and hence, a vast of real-world problems can be expressed in the framework of
the fixed point. Naturally, the solution of a fixed point problem turns into the solution of
a real-world problem. In other words, fixed point theory has wide application potential
in almost all quantitative sciences due to its nature. Consequently, both the theoretical
advances and possible applications have been studied densely.
In this book, we collect the basic metric fixed point results in the setting of metric
spaces,aswellas,inthecontextofthegeneralizedmetricspaces,suchasb-metricspaces
and partial metric spaces. The book consists of two parts: In the first part, we consider
the fundamental properties of the standard metric space together with the basic fixed
point theorems in the context of metric spaces. In the second part, we consider some
generalization of the notion of the metric and collect some crucial fixed point theorems
in this new structures. In particular, we focus on the fixed point theorems in the context
of “partial metric spaces” and “b-metric spaces”.
Thebookcontainssixchapters.Thefirstthreechapterspresentsomepreliminariesand
historicalnotesonmetricspacesandonmappings.Inthesecondchapter,werecollectthe
basic notions such as metric, norm, and auxiliary functions. The third chapter is devoted
to collect significant metric fixed point theorems. Chapter 4 aims to collect remarkable
generalizationofthemetricspace.Fixedpointtheoremsinthecontextofb-metricspaces
and partial metric spaces are discussed in Chaps. 5–6, respectively. In general, we aim
to bring the historically important fixed point theorems in the metric spaces with giving
the proofs of a few of them. Later, we indicate how these results can be generalized in
thecontextofnewabstractspaces,suchasb-metricspacesandpartialmetricspaces.We
avoid the proof of all theorems and corollaries due to analogy.
Ankara, Turkey Erdal Karapınar
Kingsville, USA Ravi P. Agarwal
July 2022
vii
viii Preface
Acknowledgements Wewouldliketoexpressourthankstoallresearcherswhohavecontributed
significantlytoFixedPointTheoryanditsApplications.
Contents
PartI FixedPointTheoremsintheFrameworkofMetricSpaces
1 Introduction ......................................................... 3
2 MetricSpaces ........................................................ 5
2.1 Basic Topological Properties of the Metric Spaces .................... 6
2.2 Connection Between the Normed Spaces and Metric Spaces ........... 9
2.3 Some Interesting Auxiliary Functions ............................... 12
2.4 Bessage, Janos and Picard Operators ................................ 13
3 MetricFixedPointTheory ............................................ 15
3.1 Banach Contraction Mapping Principle .............................. 15
3.2 Further Consequences of Banach Fixed Point Theorem ............... 22
3.2.1 Linear Extensions of Banach Contraction Principle ............. 22
3.2.2 Nonlinear Extensions of Banach Contraction Principle .......... 25
3.3 Locally Contractive Mappings and Related Fixed Point Results ........ 29
3.4 Discussion on the Mappings Whose Iteration form a Contraction ....... 30
3.5 Extension on Compact Metric Spaces ............................... 36
3.6 Nonunique Fixed Point Theorems .................................. 41
3.7 Fixed Point Results via Admissible Mappings ....................... 42
3.7.1 Standard Fixed Point Theorems .............................. 49
3.7.2 Fixed Point Theorems on Metric Spaces Endowed
with a Partial Order ........................................ 53
3.7.3 Fixed Point Theorems for Cyclic Contractive Mappings ........ 58
3.8 Fixed Point Results via Simulation Functions ........................ 62
3.8.1 Immediate Consequences ................................... 67
3.9 Fixed Points via Two Metrics ...................................... 69
ix
x Contents
PartII FixedPointTheoremsonVariousMetricSpaces
4 GeneralizationofMetricSpaces ....................................... 73
4.1 b-Metric Space .................................................. 73
4.2 Partial Metric Spaces ............................................. 76
4.3 On Further Extension of the Metric Notion: Overview ................ 80
5 FixedPointTheoremsonb-MetricSpaces .............................. 83
5.1 Fixed Point Theorems via Admissible Mappings ..................... 83
5.2 Nonunique Fixed Points Theorems ................................. 87
6 FixedPointTheoremsinPartialMetricSpaces .......................... 97
6.1 Fixed Point Theorems via Admissible Mappings ..................... 98
6.2 Nonunique Fixed Points Theorems ................................. 105
References .............................................................. 123
Index ................................................................... 135
