不动点理论

EditorsPreface

Foreword

CHAPTER1.TopologicalSpacesandTopologicalLinearSpaces

1.1.MetricSpaces

1.2.CompactnessinMetricSpaces.MeasuresofNoncompactness

1.3.BaireCategoryTheorem

1.4.TopologicalSpaces

1.5.LinearTopologicalSpaces.LocallyConvexSpaces

CHAPTER2.HilbertspacesandBanachspaces

2.1.NormedSpaces.BanachSpaces

2.2.HilbertSpaces

2.3.ConvergenceinX,X*andL(X)

2.4.TheAdjointofanOperator

2.5.ClassesofBanachSpaces

2.6.MeasuresofNoncompactnessinBanachSpaces

2.7.ClassesofSpecialOperatorsonBanachSpaces

CHAPTER3.TheContractionPrinciple

3.0.Introduction

3.1.ThePrincipleofContractionMappinginCompleteMetricSpaces

3.2.LinearOperatorsandContractionMappings

3.3.SomeGeneralizationsoftheContractionMappings

3.4.HilbertsProjectiveMetricandMappingsofContractiveType

3.5.ApproximateIteration

3.6,AConverseoftheContractionPrinciple

3.7.SomeApplicationsoftheContractionPrinciple

CHAPTER4.BrouwersFixedPointTheorem

4.0.Introduction

4.1.TheFixedPointProperty

4.2.BrouwersFixedPointtheorem.EquivalentFormulations

4.3.RobbinsComplementsofBrouwersTheorem

4.4.TheBorsuk-UlamTheorem

4.5.AnElementaryProofofBrouwersTheorem

4.6.SomeExamples

4.7.SomeApplicationsofBrouwersFixedPointTheorem

4.8.TheComputationofFixedPoints.ScarfsTheorem

CHAPTER5.SchaudersFixedPointTheoremandSomeGeneralizations

5.0.Introduction

5.1.TheSchauderFixedPointTheorem

5.2.DarbosGeneralizationofSchaudersFixedPointTheorem

5.3.Krasnoselskiis,RothesandAltmansTheorems

5.4.BrowdersandFansGeneralizationsofSchaudersandTychonoffsFixedPointTheorem

5.5.SomeApplications

CHAPTER6.FixedPointTheoremsforNonexpansiveMappingsandRelatedClassesofMappings

6.0.Introduction

6.1.NonexpansiveMappings

6.2.TheExtensionofNonexpansiveMappings

6.3.SomeGeneralPropertiesofNonexpansiveMappings

6.4.NonexpansiveMappingsonSomeClassesofBanachSpaces

6.5.ConvergenceofIterationsofNonexpansiveMappings

6.6.ClassesofMappingsRelatedtoNonexpansiveMappings

6.7.ComputationofFixedPointsforClassesofNonexpansiveMappings

6.8.ASimpleExampleofaNonexpansiveMappingonaRotundSpaceWithoutFixedPoints

CHAPTER7.SequencesofMappingsandFixedPoints

7.0.Introduction

7.1.ConvergenceofFixedPointsforContractionsorRelatedMappings

7.2.SequencesofMappingsandMeasuresofNoncompactness

CHAPTER8.DualityMappingsandMonotoneOperators

8.0.Introduction

8.1.DualityMappings

8.2.MonotoneMappingsandClassesofNonexpansiveMappings

8.3.SomeSurjectivityTheoremsonRealBanachSpaces

8.4.SomeSurjectivityTheoremsinComplexBanachSpaces

8.5.SomeSurjectivityTheoremsinLocallyConvexSpaces

8.6.DualityMappingsandMonotonicityforSet-ValuedMappings

8.7.SomeApplications

CHAPTER9.FamiliesofMappingsandFixedPoints

9.0.Introduction

9.1.MarkovsandKakutanisResults

9.2.TheRyll-NardzewskiFixed~PointTheorem

9.3.FixedPointsforFamiliesofNonexpansiveMappings

9.4.lnvariantMeansonSemigroupsandFixedPointforFamiliesofMappings

CHAPTER10.FixedPointsandSet-ValuedMappings

10.0Introduction

10.1ThePompeiu-HausdorffMetric

10.2.ContinuityforSet-ValuedMappings

10.3.FixedPointTheoremsforSomeClassesofSet-valuedMappings

10.4.Set-ValuedContractionMappings

10.5.SequencesofSet-ValuedMappingsandFixedPoints

CHAPTER11.FixedPointTheoremsforMappingsonPM-Spaces

11.0.Introduction

11.1.PM-Spaces

11.2.ContractionMappingsinPM-Spaces

11.3.ProbabilisticMeasuresofNoncompactness

11.4.SequencesofMappingsandFixedPoints

CHAPTER12.TheTopologicalDegree

12.0.Introduction

12.1.TheTopologicalDegreeinFinite-DimensionalSpaces

12.2.TheLeray-SchauderTopo!ogicalDegree

12.3.LeraysExample

12.4.TheTopologicalDegreefork-SetContractions

12.5.TheUniquenessProblemfortheTopologicalDegree

12.6.TheComputationoftheTopologicalDegree

12.7.SomeApplicationsoftheTopologicalDegree

BIBLIOGRAPHY

INDEX

　　Thisbookisintendedasanintroductiontofixedpointtheoryanditsapplications.Thetopicstreatedrangefromfairlystandardresults(suchasthePrincipleofContractionMapping,BrouwersandSchaudersfixedpointtheorems)tothefrontierofwhatisknown,butwehavenottriedtoachievemaximalgeneralityinallpossibledirections.Wehopethatthereferencesquotedmaybeusefulforthispurpose.