Fixed Point Theory and Graph Theory. Foundations and Integrative Approaches

429 Pages·2016·3.99 MB·English

by  Monther Alfuraidan

Fixed Point Theory
and Graph Theory
Fixed Point Theory
and Graph Theory
Foundations and Integrative
Approaches
Editedby
Monther Rashed Alfuraidan
KingFahdUniversityofPetroleum&Minerals,Departmentof
MathematicsandStatistics,Dhahran,SaudiArabia
Qamrul Hasan Ansari
AligarhMuslimUniversity,DepartmentofMathematics,Aligarh,
India,and
KingFahdUniversityofPetroleum&Minerals,Department
ofMathematicsandStatistics,Dhahran,SaudiArabia
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FOREWORD
Thepresentbookdiffersinimportantwaysfromprevioustextswhichdeallargelywith
metricfixedpointtheory. Foremost,thetopicscollectedherearenotfoundinanysin-
gleprevioustreatmentsandmostareentirelynew. Theseincludedetaileddiscussions
ofCaristi’stheoreminanumberofsettings,aswellasnumerousapproximationtech-
niques in Banach spaces, hyperbolic spaces, and more general geodesic spaces. In
recent years metric fixed point theory has increasingly moved away from traditional
functionalanalytic settings, thusopeningupnewavenuesofapplication. Thepresent
book reflects this trend by discussing in detail aspects of the theory which involve a
blendofmetricfixedpointtheoryandgraphtheory,aswellashowretractionsareused
toprovefixedpointresultsinorderedsets. Manyexamplesareincludedaswellaspo-
tential applications. Because of the comprehensive background material this book
is self-contained and easily accessible. A notable feature is the thoroughness of the
bibliographiccitations,manyofwhicharenotfoundinanyprevioussources.
Thisbookshouldbeaworthwhileresourceforstudentsandresearchersinterested
inalmostanyaspectofmetricfixedpointtheory,andespeciallyforthoseinterestedin
manyofthemorerecenttrends.
WilliamA.Kirk
IowaCity,IA,USA
ix
ACKNOWLEDGMENTS
WewouldliketothankourillustriousfriendsandcolleaguesintheFixedPointTheory
Research Group at the Department of Mathematics and Statistics at King Fahd Uni-
versity of Petroleum & Minerals who, through their encouragement and help, were
instrumental in the development of this book. In particular, we are grateful to Prof.
Khalid Al-Sultan, the Rector of King Fahd University of Petroleum & Minerals, and
Prof. Mohammad Al-Homoud, the Vice Rector for Academic Affairs of King Fahd
University of Petroleum & Minerals, for their unflinching support in the organiza-
tion of the International Workshop on Fixed Point Theory and Applications during
December 22–24, 2014. This Workshop provided the platform to contact and invite
thedelegatestocontributetothisbookasauthors. Itishearteningtonotethatmostof
themacceptedourofferandgenerouslycontributedtheirresearchforinclusioninthe
book. The book in its presentformis, in large measure, a fruit of their wholehearted
cooperation. Weareprofoundlygratefultothem.
ThecombinationofFixedPointTheoryandGraphTheoryisthebrainchildofour
friend Prof. M. A. Khamsi, University of Texas at El Paso. He is the prime mover
behind the international workshop on this topic and the subsequent plan to publish a
bookwhichcombinesthesetwo subjects. Wearedeeplythankfulandgratefulto him
for his encouragement, support and help. Thanks are also due to Prof. Suliman Al-
Homidan, Dean,CollegeofScience,King FahdUniversityofPetroleum&Minerals,
for his motivating advice and help to publish this book. We would like to convey
our special thanks to Mr. Graham Nisbet, Senior Acquisitions Editor, Mathematical
Sciences,Elsevier,fortakingakeeninterestinpublishingthisbook.
MontherRashedAlfuraidan
QamrulHasanAnsari
December2015
xi
PREFACE
Fixedpointtheoryiswithoutdoubtoneofthemostimportanttoolsofmodernmathe-
matics as attested by Browder, who is consideredasone of the pioneersin the devel-
opmentofthenonlinearfunctionalanalysis. Theflourishingfieldoffixedpointtheory
startedintheearlydaysoftopologythroughtheworkofPoincare´,Lefschetz-Hopfand
Leray-Schauderforexample. Fixedpointtheoryiswidelyusedindifferentareassuch
as ordinaryand partialdifferentialequations, economics,logic programming,convex
optimization, control theory, etc. In metric fixed point theory, successiveapproxima-
tions are rooted in the work of Cauchy, Fredholm, Liouville, Lipschitz, Peano and
Picard. It is well accepted among experts of this subarea that Banach is responsible
forlaying the groundforanabstractframeworkwellbeyondthescopeofelementary
differentialandintegralequations. Thefixedpointtheoryofmultivalued mappingsis
asimportantasthetheoryofsingle-valuedmappings. AfterNadlergavethemultival-
uedversionoftheBanachfixedpointtheoremforcontractionmappings,manyauthors
generalizedhis fixedpointtheoremin differentways. Aconstructiveproofof afixed
pointtheoremmakestheconclusionmorevaluableinviewofthefactthatityieldsan
algorithmforcomputingafixedpoint.
The last decade has seen some excitement about monotone mappings after the
publicationofRanandReuringsoftheanaloguetotheBanachfixedpointtheoremin
metricspacesendowedwithapartialorder. Theirresultwasmotivatedbytheproblem
of finding the solutions to some matrix equations, which often arise in the analysis
of ladder networks, dynamic programming, control theory, stochastic filtering, statis-
tics andmanyotherapplications. Jachymskiis recognizedformakingtheconnection
betweenclassicalmetricfixedpointtheoryandgraphtheory. Thisbookcanbetreated
asafoundationforboththeoriesandpresentsthemostup-to-dateresultsbridgingthe
twotheories.
Each chapter is written by differentauthor(s)who attempt to renderthe major re-
sults understandable to a wide audience, including nonspecialists, and at the same
time to provide a source for examples, references, open questions, and sometimes
newapproachesforthosecurrentlyworkingintheseareasofmathematics. Thisbook
should be of interest to graduate students seeking a field of interest, to mathemati-
cians interested in learning about the subject, and to specialists. All the chapters are
self-contained.
Chapter 1 describes the classical Caristi fixed point theorem and its various ver-
sions,andstudiesCaristi-Browderoperatorsinthesettingofmetricspaces,Rm-metric
+
spaces,s(R+)-metricspaces,andKasaharaspaces. Italsoprovidessomenewresearch
directionsintheCaristi-Browderoperatortheory.
xiii
xiv Preface
Chapter 2 is devoted to the fixed point theorems for self and nonself almost con-
tractionmappings,andcommonfixedpointtheoremsforalmostcontractionmappings.
Iterative approximation of fixed points is also studied for implicit almost contraction
mappings.
The approximate fixed points are useful when a computational approach is in-
volved. ThissubjectisaddressedinChapter3. Someknownresultsabouttheconcept
ofapproximatefixedpointsofamappingarepresented. Mostoftheapproximatefixed
points discussed in this chapter are generated by an algorithm that allows its compu-
tational implementation. An application of these results to the case of a nonlinear
semigroupofmappingsisgiven.
Chapter 4 presents the mathematical idea of viscosity methods for solving some
appliednonlinearanalysisproblems,namely,fixedpointproblems,splitcommonfixed
point problems, split equilibrium problems, hierarchical fixed point problems, etc.
together with examples in linear programming, semi-coercive elliptic problems and
theareaoffinance. Somenumericalexperimentsforaninverseheatequationarealso
given.
Chapter 5 deals with several kinds of extragradient iterative algorithms for some
nonlinear problems, namely, fixed point problems, variational inequality problems,
hierarchicalvariationalinequalityproblemsandsplitfeasibilityproblems. Severalex-
tragradientiterativealgorithmsforfindingacommonsolutionofafixedpointproblem
andavariationalinequalityproblemarealsopresentedandinvestigated.
Chapter6concernswiththefixedpoint(commonfixedpoint)problemsfornonex-
pansive (asymptotically quasi-nonexpansive) mappings in Banach spaces and
certain important classes of metric spaces. Some results about weak convergence,
(cid:2)-convergence and strong convergence of explicit and multistep schemes of
nonexpansive-type mappings to a common fixed point in the context of uniformly
convex Banach spaces, CAT(0) spaces, hyperbolic spaces and convex metric spaces
arepresented.
Chapter 7 discusses a new area that overlaps between metric fixed point theory
andgraphtheory. ThisnewareayieldsinterestinggeneralizationsoftheBanachcon-
traction principle in metric and modular spaces endowed with a graph. The bridge
betweenboththeoriesismainly motivatedbytheapplicationsofmatrix anddifferen-
tialequations.
The last chapter gives an overview how retractions are used to prove fixed point
theoremsforrelation-preservingmapsincontinuousaswellasdiscretesettings. After
firstdiscussingcomparativeretractionsinthecontextofinfiniteorderedsetsandanal-
ysis, the remainderof thechapterfocusesonretractionsthatremovea single pointin
finiteorderedsets,graphsandsimplicialcomplexes.
MontherRashedAlfuraidan
QamrulHasanAnsari
Dhahran,SaudiArabia;andAligarh,India
December2015
ABOUT THE AUTHORS
Monther Rashed Alfuraidan is Associate Professor of Mathematics in the Depart-
ment of Mathematics & Statistics at King Fahd University of Petroleum & Minerals
atDhahran,SaudiArabia. HeobtainedhisPh.D.(Mathematics)fromMichiganState
University. Hehaswrittenmorethantwentyarticlesongraphtheory,algebraicgraph
theoryandmetricfixedpointtheory. Hehaspeer-reviewedmanyarticles(amongoth-
ers)for: AlgebraicJournalofCombinatorics,ArabianJournalofMathematics,Fixed
PointTheoryandApplicationsandJournalofInequalityandApplications.
Qamrul Hasan Ansari is Professor of Mathematics at Aligarh Muslim University,
Aligarh, India. He is also joint professor at King Fahd University of Petroleum &
Minerals,Dhahran,SaudiArabia. HeobtainedhisPh.D.(Mathematics)fromAligarh
Muslim University, India. He is an associate editor of Journal of Optimization The-
ory and Applications, Fixed Point Theory and Applications, Fixed Point Theory and
Carpathian Journal of Mathematics. He has also edited several special issues of
several journals, namely, Journal of Global Optimization, Fixed Point Theory and
Applications,AbstractandAppliedAnalysis,JournalofInequalitiesandApplications,
ApplicableAnalysis,Positivity,Filomat,etc. Hehaswrittenmorethan180articleson
variational inequalities, fixed point theory and applications, vector optimization, etc.
in variousinternationalpeer-reviewedjournals. He hasedited sixbooksforSpringer,
Taylor& FrancisandNarosa,India. He is anauthorof a bookon metric spacespub-
lished by Narosa, India and has co-authoredone book on variationalinequalities and
nonsmoothoptimizationforTaylor&Francis.
Mostafa Bachar received his Ph.D. from the University of Pau (France)in Applied
Mathematics; he is currently working as associate professorat King Saud University
(KSA). He has previously held the position of Research Associate at the Institute of
Mathematics and Scientific Computing at the University of Graz (Austria). Recently
hehaseditedtwo booksinLectureNotesin Mathematics: Mathematical Biosciences
Subseries,Springer. Hiscurrentresearchisinmathematicalmodelinginmathematical
biologyandmathematicalanalysis.
VasileBerinde isProfessorofMathematicsandDirectoroftheDepartmentofMath-
ematicsandComputerScienceintheFacultyofSciences,NorthUniversityCenterat
Baia Mare, Technical University of Cluj-Napoca, Romania. He obtained his B.Sc.,
M.Sc. (inComputerScience)andPh.D.(inMathematics)fromtheBabes-BolyaiUni-
versityofCluj-Napoca,underthesupervisionofProfessorIoanA.Rus. Hismainsci-
entificinterestsareinnonlinearanalysisandfixedpointtheory. Heisthefounderand
xv
xvi AbouttheAuthors
current Editor-in-Chief of the journals Carpathian Journal of Mathematics and Cre-
ativeMathematicsandInformaticsandeditorialboardmemberforthejournalsFixed
Point Theory, Fixed Point Theory and Applications, Journal of Applied Mathemat-
ics, GeneralMathematics, Linearand NonlinearAnalysis, CarpathianMathematical
Publications,DidacticaMathematica,GazetaMatematica˘,RevistadeMatematica˘ din
Timis¸oara,etc. He haspublishedmorethan 150scientific articleson the iterativeap-
proximation of fixed points, fixed point theory and its applications to differentialand
integral equations. He has published several books, textbooks and monographs with
Romanianpublishers. Twoofthemhavebeenundertakenbyinternationalpublishers:
IterativeApproximationofFixedPoints(Springer,2007)andExploring,Investigating
andDiscoveringinMathematics(Birkha¨user,2004).
ButhinahA.BinDehaish isAssociateProfessorofMathematicsatKingAbdullaziz
University. She obtained her Ph.D. (Mathematics) from King Abdullaziz University.
Shehaswrittenmorethan15articlesonharmonicanalysisandfixedpointtheoryand
itsapplicationsinvariousinternationalpeer-reviewedjournals.
HafizFukhar-ud-din isanassociateprofessorattheDepartmentofMathematicsand
Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia.
He obtained his Ph.D. (Mathematics) fromTokyoInstitute of Technology, Japan. He
has supervised three Ph.D. and five M.Phil. students. He has written more than 40
articles on BCH-algebras and fixed point iterative methods in Banach spaces, hyper-
bolicspacesandconvexmetricspaces. HehascontributedachaptertothebookFixed
PointTheory,VariationalAnalysis,andOptimization, Chapman&Hall/CRC, 2014.
Mohamed Amine Khamsi is Professor of Mathematics at the University of Texas
at El Paso. He obtained his Ph.D. (Mathematics) from the University of Paris. He
has written more than 70 articles on fixedpoint theory andits applications in various
internationalpeer-reviewedjournals. Hehasco-authoredfivebooks.
Abdul Rahim Khan is Professor of Mathematics in the Department of Mathemat-
ics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi
Arabia. HeobtainedaPh.D.(Mathematics)fromtheUniversityofWales,UK.Hehas
supervisedfivePh.D.andelevenM.Phil. students. HeisaneditorofCarpathianJour-
nal of Mathematics and Arabian Journal of Mathematics. He has written more than
115 articles on fixed point theory and iterative methods of nonlinear mappings on a
Banachspace,CAT(0)spaceandconvexmetric space,in variouspeer-reviewedjour-
nals. He has contributed a book chapter in Advances in Non-Archimedean Analysis,
Contemporary Mathematics, American Mathematical Society, 2011 and Fixed Point
Theory,VariationalAnalysis,andOptimization, Chapman&Hall/CRC, 2014.
AbouttheAuthors xvii
Paul-Emile Mainge´ is an assistant professorat the Universite´ des Antilles, Campus
de Schoelcher, Martinique, France. He holds a Ph.D. degree from Bordeaux Univer-
sity, France (1996) in the field of Applied Mathematics and an accreditation to su-
perviseresearchinMathematics (Habilitation a` DirigerdesRecherches)fromFrench
WestIndiesUniversity,France(2008).Hehasvastexperienceofteachingandresearch
at university level in various institutions including Bordeaux, Paris, Guadeloupe and
Martinique. His field coversseveralareasof Applied Mathematics such as optimiza-
tion,variationalanalysis,convexanalysis,dynamicalsystems,variationalinequalities,
fixed-point problems and partial differential equations (slow and fast diffusion prob-
lems). He has obtained the Grant of Scientific Excellence for the periods 2006/2010
and2011/2015. Healreadyhasmorethan30researchpaperstohiscreditwhichwere
published in leading world class journals and acts as a refereefor many international
journals.
Abdellatif Moudafi is a Full Professor,first class. He obtained a Ph.D. degreefrom
Montpellier University, France (1991) in the field of Applied Mathematics (Nonlin-
ear Analysis and Optimization), a doctorate (Doctorat d’e´tat) from Casablanca Uni-
versity, Morocco (1995) and an accreditation to supervise research in Mathematics
(Habilitation a` Diriger des Recherches) from Montpellier University, France (1997).
He has vast experience of teaching and research at university level in various insti-
tutions including Montpellier, Clermont-Ferrand, Limoges, Marrakech, Guadeloupe,
Montreal, Martinique, New York and Aix-Marseille. His field covers many areas of
Applied Mathematics such as optimization, applied nonlinear analysis, variational
analysis, convex analysis, asymptotic analysis, discretization of gradient dynamical
systems, proximalalgorithms, variational inequalities, approximation and regulariza-
tion of fixed-point problems and equilibria. He has supervised successfully several
Ph.D. and M.S./M.Phil. students. He has obtained the Grantof Scientific Excellence
fortheperiods2001/2005,2006/2010,and2011/2015. Heiscurrentlymemberofthe
editorialboardofseveralreputedinternationaljournalsofmathematics. Hehasmore
than 100 research papers to his credit which were published in leading world class
journalsandactsasarefereeforover50internationaljournals.
Ma˘da˘lina Pa˘curar is Associate Professor of Mathematics, Department of Statis-
tics, Forecast and Mathematics, Faculty of Economics and Business Administration,
Babes-BolyaiUniversity ofCluj-Napoca, Romania. She obtainedherPh.D. in Math-
ematics from the Babes-Bolyai University of Cluj-Napoca, under the supervision of
Professor Ioan A. Rus. Her scientific interest is in fixed point theory. She has pub-
lished more than 40 scientific articles in the field of fixed point theory on product
spaces. Shehasco-authoredsometextbooks(Mathematicsfor Economics)andhada
monographpublished(Iterativemethodsforfixedpointapproximation,EdituraRiso-
print,Cluj-Napoca,2009)onhermainresearchinterestarea.