Fixed point theory in metric type spaces

395 Pages·2015·1.75 MB·English

by  Agarwal

Ravi P. Agarwal · Erdal Karapınar
Donal O’Regan
Antonio Francisco Roldán-López-de-Hierro
Fixed Point
Theory in
Metric Type
Spaces
Fixed Point Theory in Metric Type Spaces

Ravi P. Agarwal • Erdal Karapınar (cid:129) Donal O’Regan
Antonio Francisco Roldán-López-de-Hierro
Fixed Point Theory in Metric
Type Spaces
123
RaviP.Agarwal ErdalKarapınar
DepartmentofMathematics DepartmentofMathematics
TexasA&MUniversity-Kingsville AtılımUniversity
Kingsville,Texas,USA Incek,Ankara,Turkey
DonalO’Regan AntonioFranciscoRoldán-López-de-Hierro
NationalUniversityofIreland DepartmentofQuantitativeMethods
Galway,Galway,Ireland forEconomicsandBusiness
UniversityofGranada
Granada,Spain
ISBN978-3-319-24080-0 ISBN978-3-319-24082-4 (eBook)
DOI10.1007/978-3-319-24082-4
LibraryofCongressControlNumber:2015953665
SpringerChamHeidelbergNewYorkDordrechtLondon
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RaviP.Agarwal:TomywifeSadhna.
ErdalKarapınar:TomywifeSenemPınar
andourchildUlas¸ Ege.
DonalO’Regan:TomywifeAliceandour
childrenAoife,Lorna,Daniel,andNiamh.
AntonioF.Roldán-López-de-Hierro: Tomy
wifeMaríaJosé,ourdaughtersAnaand
Sofía,andmyparentsMaríaDoloresand
Antonio.

Preface
Fixed-pointtheoryisoneofthemajorresearchareasinnonlinearanalysis.Thisis
partly due to the fact that in many real-world problems, fixed-point theory is the
basic mathematical tool used to establish the existence of solutions to problems
whicharisenaturallyinapplications.Asaresult,fixed-pointtheoryisanimportant
areaofstudyinpureandappliedmathematics,anditisaflourishingareaofresearch.
Asthetitlestates,thisisabookonmetricfixed-pointtheorywherethebasicideas
comefrommetricspacetopology.Wepresentaself-containedaccountofthetheory
(techniquesandresults)inmetric-typespaces(inparticularinG-metricspaces).
Thebookconsistsof12chapters.Thefirstthreechapterspresentsomeprelim-
inaries and historical notes on metric spaces (in particular G-metric spaces) and
onmappings.AvarietyofBanach-typecontractiontheoremsinmetric-typespaces
are established in Chaps.4, 6, 7, and 8. Fixed-point theory in partially ordered
G-metric spaces is discussed in Chaps.5 and 8. Fixed-point theory for expansive
mappings in metric-type spaces is presented in Chap.9. The final three chapters
discussgeneralizationsandpresentresultsandtechniquesinaverygeneralabstract
settingandframework.
Wewouldliketoexpressourthankstoourfamilyandfriends.
RaviP.Agarwal,ErdalKarapınar,DonalO’ReganandAntonioF.Roldán-López-
de-Hierro.
Kingsville,USA RaviP.Agarwal
Incek,Turkey ErdalKarapınar
Galway,Ireland DonalO’Regan
Granada,Spain AntonioFranciscoRoldán-López-de-Hierro
vii

Contents
1 IntroductionwithaBriefHistoricalSurvey............................. 1
1.1 2-MetricSpaces ...................................................... 1
1.2 D-MetricSpaces...................................................... 2
1.3 SomeProblemswithD-MetricSpaces.............................. 3
2 Preliminaries................................................................ 5
2.1 Sets,MappingsandSequences ...................................... 5
2.2 Fixed,CoincidenceandCommonFixedPoints..................... 9
2.3 ControlFunctions .................................................... 10
2.3.1 ComparisonFunctions ..................................... 11
2.3.2 AlteringDistanceFunctionsandAssociatedFunctions .. 13
2.3.3 C´iric´ Functions ............................................. 22
2.3.4 PropertiesofControlFunctions............................ 25
2.4 MetricStructures..................................................... 26
2.5 Quasi-metricSpaces.................................................. 28
2.6 TopologicalStructures ............................................... 30
3 G-MetricSpaces ............................................................ 33
3.1 G-MetricSpaces...................................................... 33
3.1.1 BasicProperties............................................. 35
3.1.2 SomeRelationshipsBetweenMetricsandG-Metrics .... 38
3.1.3 SymmetricG-MetricSpaces............................... 40
3.2 TopologyofaG-MetricSpace....................................... 42
3.2.1 ConvergentandCauchySequences........................ 44
3.2.2 ContinuityofMappingsBetweenG-MetricSpaces ...... 48
3.3 G-MetricsandQuasi-metrics........................................ 49
4 BasicFixedPointResultsintheSettingofG-MetricSpaces .......... 51
4.1 TheBanachProcedure............................................... 51
4.1.1 TheBanachProcedure ..................................... 51
4.1.2 AboutAsymptoticallyRegularSequences
thatarenotCauchy......................................... 54
ix