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Synthesis 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. 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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