Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications

202 Pages·2016·0.97 MB·English

by  Afif Ben Amar, Donal O'Regan

Afi f Ben Amar · Donal O'Regan
Topological Fixed Point
Theory for Singlevalued
and Multivalued
Mappings and
Applications
Topological Fixed Point Theory for Singlevalued
and Multivalued Mappings and Applications

Afif Ben Amar • Donal O’Regan
Topological Fixed Point
Theory for Singlevalued
and Multivalued Mappings
and Applications
123
AfifBenAmar DonalO’Regan
DepartmentofMathematics SchoolofMathematics
UniversityofSfax,FacultyofSciences NationalUniversityofIreland,Galway
Sfax,Tunisia Galway,Ireland
ISBN978-3-319-31947-6 ISBN978-3-319-31948-3 (eBook)
DOI10.1007/978-3-319-31948-3
LibraryofCongressControlNumber:2016936666
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TomyparentsFathiandMounira
TomywifeFatenandourchildrenHadil,
Hiba,andYoussef
and
TomybrothersImed,Aref,andmysister
Alyssa
AfifBenAmar
TomywifeAliceandourchildrenAoife,
Lorna,Daniel,andNiamh
DonalO’Regan

Preface
Fixedpointtheoryisapowerfulandfruitfultoolinmodernmathematicsandmay
beconsideredasacoresubjectinnonlinearanalysis.Inthelast50years,fixedpoint
theoryhasbeenaflourishingareaofresearch.Inthisbook,weintroducetopological
fixedpointtheoryforseveralclassesofsingle-andmultivaluedmaps.Theselected
topicsreflectourparticularinterests.
The text is divided into seven chapters. In Chap.1, we present basic notions
in locally convex topological vector spaces. Special attention is devoted to weak
compactness,inparticulartothetheoremsofEberlein–Šmulian,Grothendieck,and
Dunford–Pettis. Leray–Schauder alternatives and eigenvalue problems for decom-
posablesingle-valuednonlinearweaklycompactoperatorsinDunford–Pettisspaces
areconsideredinChap.2.InChap.3,wepresentsomevariantsofSchauder,Kras-
noselskii, Sadovskii, and Leray–Schauder-type fixed point theorems for different
classesofweaklysequentiallycontinuous(resp.sequentiallycontinuous)operators
ongeneralBanachspaces(resp.locallyconvexspaces).Sadovskii,Furi–Pera,and
KrasnoselskiifixedpointtheoremsandnonlinearLeray–Schauderalternativesinthe
framework of weak topologies and involving multivalued mappings with weakly
sequentially closed graph are considered in Chap.4. The results are formulated
in terms of axiomatic measures of weak noncompactness. In Chap.5, we present
somefixedpointtheoremsinanonemptyclosedconvexofanyBanachalgebrasor
Banachalgebrassatisfyingasequentialcondition.P/forthesumandtheproduct
of nonlinear weakly sequentially continuous operators. We illustrate the theory
by considering functional integral and partial differential equations. The existence
of fixed points and nonlinear Leray–Schauder alternatives for different classes of
nonlinear .ws/-compact operators (weakly condensing, 1-set weakly contractive,
strictlyquasi-bounded)definedonanunboundedclosedconvexsubsetofaBanach
spaceisdiscussedinChap.6.Wealsodiscusstheexistenceofnonlineareigenvalues
andeigenvectorsandsurjectivityofquasi-boundedoperators.InChap.7,wepresent
some approximate fixed point theorems for multivalued mappings defined on
Banach spaces. Weak and strong topologies play a role here and both bounded
and unbounded regions are considered. A method is developed indicating how to
vii
viii Preface
use approximate fixed point theorems to prove the existence of approximate Nash
equilibriafornoncooperativegames.
Wehopethebookwillbeofusetograduatestudentsandtheoreticalandapplied
mathematicians who work in fixed point theory, integral equations, ordinary and
partialdifferentialequations,gametheory,andotherrelatedareas.
Sfax,Tunisia AfifBenAmar
Galway,Ireland DonalO’Regan
Contents
1 BasicConcepts ............................................................... 1
1.1 TopologicalSpaces:SomeFundamentalNotions.................... 1
1.2 NormedSpacesandBanachSpaces .................................. 3
1.3 ConvexSets............................................................ 5
1.3.1 Cones......................................................... 5
1.3.2 OrderedVectorSpaces....................................... 6
1.3.3 VectorLattices ............................................... 7
1.3.4 OrderedNormedSpaces..................................... 7
1.3.5 NormedVectorLatticesandBanachLattices .............. 8
1.4 LocallyConvexVectorSpaces........................................ 9
1.5 WeakandWeak(cid:2) Topologies.......................................... 11
1.6 ConvergenceandCompactnessinWeakTopologies ................ 12
1.7 MetrizabilityofWeakTopologies .................................... 16
1.8 WeakCompactnessinL1.X;(cid:2)/:TheDunford–PettisTheorem..... 19
1.9 TheDunford–PettisProperty.......................................... 23
1.9.1 WeaklyCompactOperators ................................. 23
1.9.2 TheDunford–PettisProperty................................ 24
1.10 AngelicSpaces......................................................... 25
1.11 NormedAlgebras...................................................... 26
1.12 MeasuresofWeakNoncompactness.................................. 27
1.13 TheSuperpositionOperator........................................... 31
1.14 SomeAspectsofContinuityinL1-Spaces............................ 32
1.15 FixedPointTheory .................................................... 35
1.15.1 TheKrasnosel’skii’sFixedPointTheorem................. 35
1.15.2 Leray–SchauderTheory ..................................... 36
1.15.3 MultivaluedMaps............................................ 37
2 NonlinearEigenvalueProblemsinDunford–PettisSpaces.............. 39
2.1 Introduction............................................................ 39
2.2 NonlinearEigenvalueProblems....................................... 40
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