Table Of ContentDevelopments in Mathematics
VOLUME34
SeriesEditors:
KrishnaswamiAlladi,UniversityofFlorida,Gainesville,FL,USA
HershelM.Farkas,HebrewUniversityofJerusalem,Jerusalem,Israel
Forfurthervolumes:
www.springer.com/series/5834
Simeon Reich (cid:2) Alexander J. Zaslavski
Genericity in
Nonlinear Analysis
SimeonReich AlexanderJ.Zaslavski
DepartmentofMathematics DepartmentofMathematics
Technion-IsraelInstituteofTechnology Technion-IsraelInstituteofTechnology
Haifa,Israel Haifa,Israel
ISSN1389-2177 ISSN2197-795X(electronic)
DevelopmentsinMathematics
ISBN978-1-4614-9532-1 ISBN978-1-4614-9533-8(eBook)
DOI10.1007/978-1-4614-9533-8
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013955299
MathematicsSubjectClassification: 06-xx,34-xx,46-xx,47-xx,49-xx,54-xx,65-xx,90-xx
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Preface
In recent years it has become more and more evident that Nonlinear Functional
Analysis is of crucial importance in the Mathematical Sciences. This is because
functional analytic ideas and methods have turned out to be essential tools in the
analysisofnonlinearphenomenainmanyareasofMathematicsanditsapplications.
AmongtheseareasonecanmentionOrdinaryDifferentialEquations,PartialDiffer-
entialEquations,theGeometryofBanachSpaces,NonlinearOperatorTheory,the
Calculus of Variations, Optimal Control Theory, Optimization and Mathematical
Economics.
Oneofthemainfeaturesofthefunctionalanalyticapproachistheinvestigation
andsolutionofgeneralclassesofproblemsratherthanofmorespecificindividual
ones.Whenoneusesthisapproach,thefollowingquestionarises:
Weconsideraclassofproblemswhichisidentifiedwithsomefunctionalspace
equipped with a natural complete metric. We know that for some elements of the
functionalspacethecorrespondingproblemspossessasolution(orasolutionwith
some desirable properties) and for some elements such solutions do not exist. We
usually know some sufficient conditions for the existence of solutions, but often
these conditions are difficult to verify or they hold for rather small subsets of the
whole space. In such situations it is natural to ask if a solution (or a solution with
some desirable properties) exists for most elements of the functional space in the
sense of Baire category. This means that the functional space under consideration
contains an everywhere dense G subset such that for all its elements a solution
δ
exists.
It turns out that this generic approach is very useful and many interesting and
importantproblemscanbesolvedusingit.Thegoalofourbookistodemonstrate
this. Althoughit is, of course, impossible to cover the whole spectrum of present-
day trends in Nonlinear Analysis and its applications where the generic approach
isused,wedopresentquiteafewofthemaintopicswhichareofcurrentresearch
interest. They include fixed point theory of both single- and set-valued mappings,
convergence analysis of infinite products, best approximation problems, discrete
and continuous descent methods for minimization in a general Banach space, and
v
vi Preface
thestructureofminimalenergyconfigurationswithrationalnumbersintheAubry-
Mathertheory.
Nowwedescribethestructureofthebook.WebegininChap.1withtheapplica-
tionsoftheBairetheorytofixedpointtheory.Aself-mappingofacompletemetric
space is called nonexpansive if it is Lipschitz with Lipschitz constant one. If the
Lipschitzconstantisless thanone,thenitis calledastrictcontraction.According
to Banach’s celebrated result, a strict contraction has a unique fixed point and all
itsiteratesconvergetoit.Itwasunclearwhathappenswhenamappingactingona
closedandconvexsubsetof a generalBanachspaceis just nonexpansiveuntilthe
classical paper by De Blasi and Myjak of 1976 [49], where they show, using the
Baireapproach,thatmostmappingsintheclassofnonexpansiveself-mappingsof
a bounded, closed and convex subset of a general Banach space possess a unique
fixedpointwhichattractsuniformlyalltheiriterates.Notethattheyalsoshowthat
thesubclassofstrictcontractionsisasmallsetinthewholeclassofnonexpansive
mappings.
Chapter 2 is devoted to further generalizations, extensions and developments
concerningthisresultofDeBlasiandMyjak.UsingtheBaireapproach,weestab-
lish existenceand uniquenessof a fixed point for a generic mapping,convergence
of iterates of a generic nonexpansive mapping, stability of the fixed point under
smallperturbationsofamapping,convergenceofKrasnosel’skii-Manniterationsof
nonexpansivemappings,genericpowerconvergenceoforderpreservingmappings,
and existence and uniqueness of positive eigenvalues and eigenvectors of order-
preservinglinearoperators.Inthischapterwealsostudyconvergenceofiteratesof
nonexpansivemappingsinthepresenceofcomputationalerrors.
Chapter3isdevotedtoanimportantsubclassoftheclassofnonexpansivemap-
pingswhichconsistsoftheso-calledcontractivemappings.Acontractivemapping
is obtained if in the definition of a strict contraction the constant is replaced by a
monotonically decreasing function with nonnegative values which do not exceed
oneandwhichisafunctionofthedistancebetweentwopoints.Thistopichasre-
cently become rather popular. In Chap. 3 we study different types of contractive
mappings,existenceoffixedpointsforsuchmappings,convergenceoftheirpowers
toafixedpoint,stabilityofafixedpointundersmallperturbationsofthemapping,
andusetheBaireapproachtoshowthatmostnonexpansivemappingsarecontrac-
tive.
InChap.4weusethegenericapproachinordertostudytheasymptoticbehavior
oftrajectoriesofacertaindynamicalsystemwhichoriginatesinaconvexminimiza-
tion problem. Usually, an algorithm for the minimization of an objective function
onasetcanbeconsideredaself-mappingofthesetforwhichtheobjectivefunction
is a Lyapunov function. In our case the set is a closed subset of a Banach space.
Theresultspresentedinthischaptershowthatformostalgorithms,thevaluesofthe
objectivefunctionalongallthetrajectoriestendtoitsinfimum.
InChap.5wegeneralizesomeoftheresultsofChap.2formappingswhichare
relativelynonexpansivewithrespecttoBregmandistances.Suchmappingsappear
inoptimizationtheoryandinstudiesoffeasibilityproblems[37,39].
Preface vii
Chapter6isdevotedtothestudyofconvergenceofinfiniteproductsofdifferent
classes of mappings. The convergence of infinite products of nonexpansive map-
pingsisofmajorimportancebecauseoftheirmanyapplicationsinthestudyoffea-
sibility and optimization problems. We study the convergence of typical (generic)
infinite products of mappings to the set of their common fixed points, and estab-
lishweakergodictheorems(atermwhichoriginatesinpopulationbiology),which
roughly mean that all trajectories generated by infinite products converge to each
other. We study convergence and its stability for generic infinite products of non-
expansive mappings, uniformly continuous mappings, order-preserving mappings,
order-preservinglinearmappings,homogeneousorder-preservingmappings,prod-
uctsofaffinemappings,aswellasproductsofresolventsofaccretiveoperators.
In Chap. 7 we study best approximation problems in a general Banach space.
A best approximation problem is determined by a pair consisting of a point and a
closed (convex)subset of a Banach space. We consider the complete metric space
ofsuchpairsequippedwithanaturalcompletemetricandshowthatformost(inthe
senseofBairecategory)pairsthecorrespondingbestapproximationproblemhasa
uniquesolution.Wealsoprovidesomegeneralizationsandextensionsofthisresult.
InChap.8westudydiscreteandcontinuousdescentmethodsforminimizinga
convex(Lipschitz)functiononageneralBanachspace.Weconsideraspaceofvec-
torfieldsV suchthatforanypointx intheBanachspace,thedirectionalderivative
in the direction Vx is nonpositive. This space of vector fields is equipped with a
completemetric.Eachvectorfieldgeneratestwogradienttypealgorithms(discrete
descent methods) and a flow which consists of the solutions of the corresponding
evolutionequation(continuousdescentmethod).Weshowthatmost(inthesenseof
Baire category) vector fields produce algorithms for which values of the objective
functiontendtoitsinfimumasttendstoinfinity.Actually,weintroducethesubclass
ofregularvectorfields,showthattheconvergencepropertystatedaboveholdsfor
themandthatagenericvectorfieldisregular.Wealsoshowthatthisconvergence
propertyisstableundersmallperturbationsofagivenregularvectorfield.
Chapter9isdevotedtoset-valuedmappings.Westudyapproximatefixedpoints
of such mappings, existence of fixed points, and the convergence and stability of
iteratesofset-valuedmappings.
Chapter10isdevotedtotheAubry-MathertheoryappliedtothefamousFrenkel-
Kontorovamodel,aninfinitediscretemodelofsolid-statephysicsrelatedtodislo-
cations in one-dimensional crystals. In this model a configuration of a system is
a sequence of real numbers with indices from −∞ to +∞. We are interested in
(h)-minimal configurations with respect to an energy function h. A configuration
is called (h)-minimal if its total energy cannot be made less by changing its final
states. Classical Aubry-Mather theory is concerned with finding and investigating
h-minimal configurations with a given rotation number, where the function h is
fixed. It implies that the set of all periodic h-minimal configurations of a rational
rotationnumberp/qistotallyordered.Moreover,betweenanytwoneighboringpe-
riodich-minimalconfigurationswithrotationnumberp/q,thereare(non-periodic)
h-minimalheteroclinicconnectionshavingthesamerotationnumberp/q.Wecon-
sider a complete metric space of energy functions h equipped with a certain C2
viii Preface
topology and show that for most energy functions in this space, there exist three
different h-minimal configurations with rotation number p/q such that any other
h-minimalconfigurationwiththesamerotationnumberp/q isatranslationofone
ofthesethree.
Haifa SimeonReich
December31,2012 AlexanderJ.Zaslavski
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 HyperbolicSpaces . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 SuccessiveApproximations . . . . . . . . . . . . . . . . . . . . 2
1.3 ContractiveMappings . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 InfiniteProducts . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 ContractiveSet-ValuedMappings . . . . . . . . . . . . . . . . . 7
1.6 NonexpansiveSet-ValuedMappings . . . . . . . . . . . . . . . 9
1.7 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 FixedPointResultsandConvergenceofPowersofOperators . . . 15
2.1 ConvergenceofIteratesforaClassofNonlinearMappings . . . 15
2.2 ConvergenceofIteratesofTypicalNonexpansiveMappings . . . 23
2.3 AStabilityResultinFixedPointTheory . . . . . . . . . . . . . 29
2.4 Well-PosedNullandFixedPointProblems . . . . . . . . . . . . 34
2.5 MappingsinaFinite-DimensionalEuclideanSpace . . . . . . . 37
2.6 ApproximateFixedPoints . . . . . . . . . . . . . . . . . . . . . 42
2.7 GenericExistenceofSmallInvariantSets . . . . . . . . . . . . . 47
2.8 ManyNonexpansiveMappingsAreStrictContractions . . . . . 51
2.9 Krasnosel’skii-MannIterationsofNonexpansiveOperators . . . 55
2.10 PowerConvergenceofOrder-PreservingMappings. . . . . . . . 63
2.11 PositiveEigenvaluesandEigenvectors . . . . . . . . . . . . . . 72
2.12 ProofofTheorem2.48 . . . . . . . . . . . . . . . . . . . . . . . 75
2.13 AuxiliaryResultsforTheorems2.49–2.51 . . . . . . . . . . . . 78
2.14 ProofsofTheorems2.49and2.50 . . . . . . . . . . . . . . . . . 83
2.15 ProofofTheorem2.51 . . . . . . . . . . . . . . . . . . . . . . . 85
2.16 ConvergenceofInexactOrbitsforaClassofOperators . . . . . 87
2.17 ProofsofTheorem2.65andCorollary2.66 . . . . . . . . . . . . 89
2.18 ProofofTheorem2.67 . . . . . . . . . . . . . . . . . . . . . . . 92
2.19 ProofofTheorem2.68 . . . . . . . . . . . . . . . . . . . . . . . 93
2.20 ProofofTheorem2.69 . . . . . . . . . . . . . . . . . . . . . . . 96
ix
x Contents
2.21 InexactOrbitsofNonexpansiveOperators . . . . . . . . . . . . 97
2.22 ConvergencetoAttractingSets . . . . . . . . . . . . . . . . . . 100
2.23 NonconvergencetoAttractingSets . . . . . . . . . . . . . . . . 103
2.24 ConvergenceandNonconvergencetoFixedPoints . . . . . . . . 106
2.25 ConvergencetoCompactSets . . . . . . . . . . . . . . . . . . . 110
2.26 AnExampleofNonconvergencetoCompactSets . . . . . . . . 113
3 ContractiveMappings . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.1 ManyNonexpansiveMappingsAreContractive . . . . . . . . . 119
3.2 AttractiveSets . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.3 AttractiveSubsetsofUnboundedSpaces . . . . . . . . . . . . . 124
3.4 AContractiveMappingwithnoStrictlyContractivePowers . . . 129
3.5 APowerConvergentMappingwithnoContractivePowers . . . 132
3.6 AMappingwithNonuniformlyConvergentPowers . . . . . . . 134
3.7 TwoResultsinMetricFixedPointTheory . . . . . . . . . . . . 136
3.8 AResultonRakotchContractions. . . . . . . . . . . . . . . . . 144
3.9 AsymptoticContractions . . . . . . . . . . . . . . . . . . . . . 149
3.10 UniformConvergenceofIterates . . . . . . . . . . . . . . . . . 153
3.11 Well-PosednessofFixedPointProblems . . . . . . . . . . . . . 157
3.12 AClassofMappingsofContractiveType . . . . . . . . . . . . . 159
3.13 AFixedPointTheoremforMatkowskiContractions . . . . . . . 166
3.14 Jachymski-Schröder-SteinContractions . . . . . . . . . . . . . . 170
3.15 TwoResultsonJachymski-Schröder-SteinContractions . . . . . 175
4 DynamicalSystemswithConvexLyapunovFunctions . . . . . . . 181
4.1 MinimizationofConvexFunctionals . . . . . . . . . . . . . . . 181
4.2 ProofofProposition4.3 . . . . . . . . . . . . . . . . . . . . . . 183
4.3 ProofsofTheorems4.1and4.2 . . . . . . . . . . . . . . . . . . 185
4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.5 NormalMappings . . . . . . . . . . . . . . . . . . . . . . . . . 190
4.6 ExistenceofaNormalA∈A . . . . . . . . . . . . . . . . . . . 192
c
4.7 AuxiliaryResults . . . . . . . . . . . . . . . . . . . . . . . . . 193
4.8 ProofofTheorem4.12 . . . . . . . . . . . . . . . . . . . . . . . 194
4.9 ProofofTheorem4.13 . . . . . . . . . . . . . . . . . . . . . . . 195
4.10 ProofofTheorem4.14 . . . . . . . . . . . . . . . . . . . . . . . 196
4.11 NormalityandPorosity . . . . . . . . . . . . . . . . . . . . . . 197
4.12 ProofofTheorem4.18 . . . . . . . . . . . . . . . . . . . . . . . 198
4.13 ProofofTheorem4.19 . . . . . . . . . . . . . . . . . . . . . . . 200
4.14 ConvexFunctionsPossessingaSharpMinimum . . . . . . . . . 202
5 Relatively NonexpansiveOperatorswith Respect to Bregman
Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5.1 PowerConvergenceofOperatorsinBanachSpaces . . . . . . . 205
5.2 PowerConvergenceforaClassofContinuousMappings. . . . . 206
5.3 PreliminaryLemmataforTheorems5.1–5.6 . . . . . . . . . . . 209
5.4 ProofsofTheorems5.1–5.6 . . . . . . . . . . . . . . . . . . . . 213
