Table Of ContentNONLINEAR ILL-POSED PROBLEMS OF MONOTONE TYPE
Nonlinear Ill-posed Problems
of Monotone Type
by
YAKOV ALBER
and
IRINA RYAZANTSEVA
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-4395-3 (HB)
ISBN-13 978-1-4020-4395-6 (HB)
ISBN-10 1-4020-4396-1 (e-book)
ISBN-13 978-1-4020-4396-3 (e-book)
Published by Springer,
P.O. Box 17, 3300 AADordrecht, The Netherlands.
www.springer.com
Printed on acid-free paper
All Rights Reserved
© 2006 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming, recording
or otherwise, without written permission from the Publisher, with the exception
of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
Printed in the Netherlands.
Contents
PREFACE vii
ACKNOWLEDGMENTS xiii
1 INTRODUCTION INTO THE THEORY OF MONOTONE AND
ACCRETIVE OPERATORS 1
1.1 ElementsofNonlinearFunctionalAnalysis. . . . . . . . . . . . . . . . . . . 1
1.2 Subdifferentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 MonotoneOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 MaximalMonotoneOperators. . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5 DualityMappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.6 BanachSpacesGeometryandRelatedDualityEstimates. . . . . . . . . . . 40
1.7 EquationswithMaximalMonotoneOperators. . . . . . . . . . . . . . . . . 52
1.8 SummationofMaximalMonotoneOperators . . . . . . . . . . . . . . . . . 59
1.9 EquationswithGeneralMonotoneOperators . . . . . . . . . . . . . . . . . 66
1.10 EquationswithSemimonotoneOperators . . . . . . . . . . . . . . . . . . . 70
1.11 VariationalInequalitieswithMonotoneOperators. . . . . . . . . . . . . . . 72
1.12 VariationalInequalitieswithSemimonotoneOperators . . . . . . . . . . . . 79
1.13 VariationalInequalitieswithPseudomonotoneOperators. . . . . . . . . . . 84
1.14 VariationalInequalitieswithQuasipotentialOperators . . . . . . . . . . . . 90
1.15 EquationswithAccretiveOperators . . . . . . . . . . . . . . . . . . . . . . 95
1.16 Equationswithd-AccretiveOperators . . . . . . . . . . . . . . . . . . . . . 108
2 REGULARIZATION OF OPERATOR EQUATIONS 117
2.1 EquationswithMonotoneOperatorsinHilbertSpaces . . . . . . . . . . . . 117
2.2 EquationswithMonotoneOperatorsinBanachSpaces . . . . . . . . . . . . 123
2.3 EstimatesoftheRegularizedSolutions . . . . . . . . . . . . . . . . . . . . . 129
2.4 EquationswithDomainPerturbations . . . . . . . . . . . . . . . . . . . . . 133
2.5 EquationswithSemimonotoneOperators . . . . . . . . . . . . . . . . . . . 136
2.6 EquationswithNon-MonotonePerturbations . . . . . . . . . . . . . . . . . 138
2.7 EquationswithAccretiveandd-AccretiveOperators . . . . . . . . . . . . . 142
3 PARAMETERIZATION OF REGULARIZATION METHODS 151
3.1 ResidualPrincipleforMonotoneEquations . . . . . . . . . . . . . . . . . . 152
vi Contents
3.2 ResidualPrincipleforAccretiveEquations . . . . . . . . . . . . . . . . . . . 163
3.3 GeneralizedResidualPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . 167
3.4 ModifiedResidualPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
3.5 MinimalResidualPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.6 SmoothingFunctionalPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . 183
4 REGULARIZATION OF VARIATIONAL INEQUALITIES 191
4.1 VariationalInequalitiesonExactlyGivenSets. . . . . . . . . . . . . . . . . 191
4.2 VariationalInequalitiesonApproximatelyGivenSets. . . . . . . . . . . . . 204
4.3 VariationalInequalitieswithDomainPerturbations . . . . . . . . . . . . . . 215
4.4 ExamplesofVariationalInequalities . . . . . . . . . . . . . . . . . . . . . . 219
4.5 VariationalInequalitieswithUnboundedOperators . . . . . . . . . . . . . . 227
4.6 VariationalInequalitieswithNon-MonotonePerturbations . . . . . . . . . . 233
4.7 VariationalInequalitieswithMosco-ApproximationoftheConstraintSets . 236
4.8 VariationalInequalitieswithHypomonotoneApproximations . . . . . . . . 245
4.9 VariationalInequalitieswithPseudomonotoneOperators. . . . . . . . . . . 249
4.10 VariationalInequalitiesofMixedType . . . . . . . . . . . . . . . . . . . . . 254
5 APPLICATIONS OF THE REGULARIZATION METHODS 259
5.1 ComputationofUnboundedMonotoneOperators . . . . . . . . . . . . . . . 259
5.2 ComputationofUnboundedSemimonotoneOperators . . . . . . . . . . . . 268
5.3 ComputationofUnboundedAccretiveOperators . . . . . . . . . . . . . . . 271
5.4 HammersteinTypeOperatorEquations . . . . . . . . . . . . . . . . . . . . 278
5.5 Pseudo-SolutionsofMonotoneEquations . . . . . . . . . . . . . . . . . . . 283
5.6 MinimizationProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
5.7 OptimalControlProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
5.8 FixedPointProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
6 SPECIAL TOPICS ON REGULARIZATION METHODS 311
6.1 Quasi-SolutionMethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
6.2 ResidualMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
6.3 PenaltyMethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
6.4 ProximalPointMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
6.5 IterativeRegularizationMethod . . . . . . . . . . . . . . . . . . . . . . . . 340
6.6 Iterative-ProjectionRegularizationMethod . . . . . . . . . . . . . . . . . . 352
6.7 ContinuousRegularizationMethod . . . . . . . . . . . . . . . . . . . . . . . 363
6.8 Newton−KantorovichRegularizationMethod . . . . . . . . . . . . . . . . . 376
7 APPENDIX 385
7.1 RecurrentNumericalInequalities . . . . . . . . . . . . . . . . . . . . . . . . 385
7.2 DifferentialInequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
BIBLIOGRAPHY 391
INDEX 407
PREFACE
Manyappliedproblemscanbereducedtoanoperatorequationofthefirstkind
Ax=f, x∈X, f ∈Y, (1)
where an operator A maps from a metric space X into a metric space Y. We are inter-
ested in those problems (1) which belong to the class of ill-posed problems. The concept
ofawell-posednesswasintroducedbyJ.Hadamardatthebeginningofthe20thcentury[91].
Definition 1 Let X and Y be metric spaces. The problem (1) is called well-posed if the
following conditions are satisfied:
(i) the problem is solvable in X for all f ∈Y;
(ii) its solution x∈X is unique;
(iii) x∈X continuously depends on perturbations of the element f ∈Y.
Problems that do not satisfy even one of the above requirements (i)-(iii) collectively
form the class of ill-posed problems. Emphasize that solutions of ill-posed problems (if
they exist) are unstable to small changes in the initial data. In connection with this, a
commonbeliefofmanymathematicians inthepastwasthatwell-posedness is anecessary
condition for the problems (1) to bemathematicallyor physicallymeaningful. This raised
adebateaboutwhetherornotthereisanyneedformethodsofsolvingill-posedproblems.
Thetremendousdevelopmentofscienceandtechnologyofthelastdecadesled,moreoften
thannot,topracticalproblemswhichareill-posedbytheirnature. Solvingsuchproblems
becameanecessityand,thus,inventingmethodsforthatpurposebecameafieldofresearch
intheintersectionoftheoreticalmathematicswiththeappliedsciences.
Thefactisthat,inpracticalcomputations,thedateAandfoftheproblem(1),asarule,
arenotpreciselyknown. Therefore,itisimportanttostudythecontinuousdependenceof
theapproximatesolutionsontheintrinsicerrorsinvolvedintheproblemdate. Withoutsuch
knowledge,directnumericalresolutionof(1)isimpossible. Atthesametime,establishing
the continuous dependence mentioned above is not an easy task. Attempts to avoid this
difficulty led investigators to the new theory and conceptually new methods for stable
solution of ill-posed problems. The expansion of computational technology contributed in
largeextenttotheaccelerationofthisprocess.
Building the so-called regularization methods for solving ill-posed problems was initi-
atedbyA.N.Tikhonov[216]. Heprovedtherethefollowingimportantresult:
Theorem 2 A continuous one-to-one operator A which maps acompactsubsetMof
themetricspaceX intometricspaceY hasthecontinuousinverseoperatorA−1 ontheset
N =AM⊂Y.
Onthebasisofthistheorem,M.M.Lavrent’evintroducedtheconceptofconditionally
well-posedproblems[125].
viiiPREFACE
Definition 3 Theproblem(1)issaidt obeconditionallywell-posedifthereexistsnonempty
setX 1⊆Xsuchthat
(i)theequation(1)hasasolutioni nX 1;
(ii)thissolutionisuniqueforeachf∈Y 1=AX1⊂Y;
(iii)theoperatorA −1 iscontinuousonY 1.
ThesubsetX 1 iscalledthewell-posednesssetofproblem(1).AccordingtoTheorem2
andDefinition3,weshouldimposeconditionsontheoperatorAinequation(1)suchthat
theywilldefinethecompactsetX 1.Evenifthoseconditionsaregiven,therearestilldiffi-
cultiesthatarisewhenwetrytoestablishsolvabilityof(1)withf∈Y 1.Thesedifficulties
canbeovercomeifX 1isunderstoodasthequasi-solutionsetofequation(1)withf∈Y 1[97].
Definition 4 Anelementx∈X 1 iscall ed aquasi-solutionofequation(1)ifitmini-
mizestheresidualρ Y(Ax,f)onthesetX 1,whereρ Y isametricinthespaceY.
ProblemsforwhichitisnotpossibletoconstructthecompactsetX 1 ofadmissibleso-
lutionsaresaidtobeessentiallyill-posed.Inordertosolvesuchproblems,A.N.Tikhonov
proposedtheso-calledregularizationmethod[217].Todescribethismethodweestablish
thefollowingdefinitionassuming,forsimplicity,thatonlytheright-handsidefinequation
(1)isgive nwithsomeerrorδ.
Definition 5 AnoperatorR(α,f δ) :Y→Xisca lled aregularizingoperatorforequa-
tion(1)ifitsatisfiestworequirements:
(i)R(α,f δ)isdefinedforallα>0andallf δ ∈Ysuchthatρ (f,fδ)≤δ;
Y
(ii)Thereexistsafunctionα=α(δ)suchthatR(α(δ),f δ)=xδ →xasδ→0,wherexis
α
asolutionof(1).
OperatorsR(α,f)leadtoavarietyofregularizationmethods.Anelementx δ iscalled
α
theregularizedsolutionandαiscalledtheregularizationparameter.Thus,byDefinition
5,anyregularizationmethodhastosolvetwomainproblems:
A)toshowhowt oconstructaregularizingoperatorR(α,f δ),
B)toshowhowtheregularizationparameterα=α(δ)shouldbechoseninordertoensure
convergenceofx δ tosomexasδ→0.
α
Theregularizationmethodproposedin[217]isgive ninvariationalform(seealso[43,99,
130,167,218]).Itproduces asolutionx δ asaminimumpointofthefollowingsmoothing
α
functional:
Φα(x)=ρ2(Ax,fδ)+αφ(x),(2)
δ Y
whereφ(x):X→R 1isastabilizingfunctionalwhichhasthepropertythatφ(z)≥0forall
z∈D(φ)⊂X,whereD(φ)isadomainoffunctionalφ.Moreover,asolutionxofequation
(1)needstobeaninclusionx∈D(φ).
Besidestheregularizationmethoddescribedbythefunctional(2),twomorevariational
methodsforsolvingill-posedproblemsareknown[99]:thequasi-solutionmethodgive nby
PREFACE ix
theminimizationproblem
(cid:7)Axδ−fδ(cid:7)=min{(cid:7)Ax−fδ(cid:7)|x∈X1},
and the residual method in which approximate solutions xδ for equation (1) are found by
solvingtheminimizationproblem:
(cid:7)xδ(cid:7)=min{(cid:7)x(cid:7)|x∈Gδ},
where
Gδ ={x∈D(A)|(cid:7)Ax−fδ(cid:7)≤σ(δ)}.
Here D(A) denotes a domain of operator A and the positive scalar function σ(δ) has the
propertythatσ(δ)→0asδ→0.
The increasing variety of variational regularization methods and growing interest in
them are due to the following facts: (i) a priori information about (1) can be used in the
process of their construction and (ii) there exists quite a wide choice of different means of
solvingoptimizationproblems.
Iftheelementxδ isfoundbysolvingaparametricequationofthetype
α
Ax+αBx=fδ, (3)
where B is some stabilizing operator, then this procedure is an operator regularization
method[54,124,125],andtheequation(3)isaregularizedoperatorequation.
The methods presented above were intensively studied in the case of linear equations.
Thisisconnectedwiththefactthatforlinearoperatorsthereareverypowerfulinvestigation
toolscontainedinthespectraltheoryandoptimizationtheoryforquadraticfunctionals. In
thisdirectionmanyratherdelicatecharacteristicsandpropertiesofthementionedmethods
havebeenalreadyobtained[42,99,125,153,217].
Inspiteofthefactthatsomenonlinearproblemsweresolvedbyregularizationmethods
a long time ago, the progress in this sense was slow. This seems to be due to researchers’
wishtocoveraswideclassesofnonlinearproblemsaspossible. Itisclearthatthespectral
theory can not be applied directly to nonlinear operators. Moreover, if A is a nonlinear
operator,thefunctionalsρ (Ax,fδ)andΦα(x)arenonconvex,ingeneral. Inaddition,the
Y δ
questionofconvexityforthesetGδ remainsstillopen,thatproducesenormousdifficulties
insolvingvariationalproblems.
At the same time, the separate researches into nonlinear problems of the monotone
type, like operator equations and variational inequalities with monotone, maximal mono-
tone,semimonotone,pseudomonotoneandaccretiveoperators,turnedouttobemuchmore
successful. Fromtheappliedpointofviewtheseclassesareextremelywide(itissufficient
torecallthatgradientsandsubgradientsofconvexfunctionalsaremonotoneandmaximal
monotoneoperators,respectively,andnonexpansivemappingsareaccretive.)
Aconsiderablepartofthisbookisdevotedtostudyofnonlinearill-posedproblemswith
operators in these classes. We use the fundamental results in the theory of monotone and
accretiveoperatorsdiscoveredduringthelatterhalfofthe20thcenturybyR.I.Kachurovskii
[102,103],M.M.Vainberg[220,221], F.E. Browder[52]-[58],G.Minty[149], J.-L. Lions
[128],H.Brezis[49],R.Rockafellar[175]-[181],T.Kato[106,109]andmanyothers.
x PREFACE
From the very beginning, the concept of ill-posedness was included in the construction
and theoretical foundation of approximative methods for solving nonlinear problems with
properly monotone and properly accretive operators, and according to this concept, it is
necessary:
1)toassumethatasolutionoftheoriginalproblemexists;
2)toconstructasequenceofregularizedsolutions;
3) to determine whether this sequence converges to a solution of problem (1) and the
convergenceisstablewhentheinitialdataareperturbed.
Let us briefly describe the contents of the book. It consists of seven chapters that
comprise fifty-seven sections. The numeration of sections is in the form X.Y, where X
indicatesthenumberofachapterandYdenotesthecurrentnumberofthesectionwithin
the given chapter. The numeration of definitions, theorems, lemmas, corollaries, remarks
andpropositionsisintheformX.Y.Z,whereXandYareasaboveandZindicatesanumber
ofthestatementinSectionX.Y.Formulasarenumberedanalogously.
Chapter1containsthebasicknowledgefromconvexandfunctionalanalysis,optimiza-
tion,Banachspacegeometryandtheoryofmonotoneandaccretiveoperatorswhichallows
thestudyoftheproposedmaterialwithoutturning,generally,toothersources. Equations
and variational inequalities with maximal monotone operators are main subjects of our
investigations due to their most important applications. Recall that maximal monotone
operators are set-valued, in general. In spite of this, the properties of maximal monotone
operators,solvabilitycriteriaforequationsandvariationalinequalitieswithsuchoperators
andalsostructuresoftheirsolutionsetshavebeenstudiedfullyenough. However,itisnot
always easy to establish the maximal monotonicity of a map. Besides that, the numerical
realizationofmaximalmonotoneoperatormeansitschangebysingle-valuedsectionswhen
the property of maximal monotonicity is lost. Finally, the monotone operator of the orig-
inal problem is not always maximal. Therefore, in this chapter we devote significant time
toequationsandvariationalinequalitieswithsemimonotone,pseudomonotoneandgeneral
monotone mappings. We describe in detail the known properties of the most important
canonical monotone operators, so-called, normalized duality mapping J and duality map-
ping with gauge function Jµ. We also present new properties of these mappings expressed
bymeansofgeometriccharacteristicsofaBanachspace,itsmodulusofconvexityandmod-
ulus of smoothness. Weintroducethenew Lyapunovfunctionals andshow that they have
similar properties. In the theory of monotone operators this approach is not traditional
and it is first stated in monographic literature. The two last sections of the chapter are
dedicatedtoequationswithmultiple-valuedaccretiveandd−accretiveoperators.
Chapter2isdevotedtotheoperatorregularizationmethodformonotoneequations(that
is,forequationswithmonotoneoperators). Inordertopreservethemonotonicityproperty
in a regularized problem, we add to the original operator some one-parameter family of
monotone operators. Thus, the resulting operators are monotone again. As a stabilizing
operator, we use a duality mapping which has many remarkable properties. In the case
ofaHilbertspace,wherethedualitymappingistheidenticaloperator,thisregularization
method for linear equations was first studied by M.M. Lavrent’ev [125]. In this chapter,
weexploretheoperatorregularizationmethodsformonotone,semimonotoneandaccretive
nonlinear equations in Hilbert and Banach spaces. The equations with single-valued and
