Iterative Methods for Fixed Point Problems in Hilbert Spaces

312 Pages·2013·1.85 MB·English

by  Andrzej Cegielski (auth.)

Lecture Notes in Mathematics 2057
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
•
Andrzej Cegielski
Iterative Methods
for Fixed Point Problems
in Hilbert Spaces
123
AndrzejCegielski
UniversityofZielonaGo´ra
FacultyofMathematics,
ComputerScienceandEconometrics
ZielonaGo´ra,Poland
ISBN978-3-642-30900-7 ISBN978-3-642-30901-4(eBook)
DOI10.1007/978-3-642-30901-4
SpringerHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2012945521
MathematicsSubjectClassification(2010):47-02,49-02,65-02,90-02,47H09,47J25,37C25,
65F10,65K15,90C25
(cid:2)c Springer-VerlagBerlinHeidelberg2012
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
To myfamily:
Elz˙bieta,Joanna,Gustaw,
Szymon andKatarzyna
•
Motto:
Ifyou want tofind thesource, you havetogo
up,againstthecurrent
[JohnPaulII]
•
Preface
In this monographwe dealwith iterationmethodsforfindingfixed points(if they
exist)ofnonexpansiveoperatorsdefinedonaHilbertspace,i.e.,operatorsT having
theproperty
d.Tx;Ty/(cid:2)cd.x;y/forallx;y 2X (1)
andforsome constantc 2 Œ0;1(cid:2). Theoriginofthese methodsdatesbackto 1920,
when Stefan Banach (1892–1945) formulated his famous contraction mapping
principle.BanachprovedthatifT WX !X isacontraction(anoperatorsatisfying
(1)withc <1)definedonacompletemetricspace,thenT hasauniquefixedpoint
x(cid:3), i.e., a point for which Tx(cid:3) D x(cid:3). Furthermore, for any x 2 X a sequence
fTkxg1 convergesgeometricallytox(cid:3).Manypracticalproblemscanbereduced
kD0
to finding a fixed point of a nonexpansiveoperator or a common fixed point of a
familyofnonexpansiveoperators.Asimpleexampleisasystemoflinearequations
Ax Db.Anysolutionofthissystemcanbeidentifiedwithacommonfixedpointof
(nonexpansive)operatorsoforthogonalprojectionsontohyperplanescorresponding
toparticularequationsofthesystem.
Undersomeadditionalconditionsanonexpansiveoperatorhasafixedpoint(see,
e.g.,theBrowder–Go¨hde–Kirkfixedpointtheorem),buttheBanachtheoremdoes
notguaranteethe convergenceof thesequencefTkxg1 . Therefore,it isof great
kD0
interest to develop methods for finding fixed points of nonexpansive operators.
The first iterative method for solving a linear system was proposed in 1937 by
a Polish mathematician, Stefan Kaczmarz (1890–1945), in a very short paper (3
pages) “Angena¨herte Auflo¨sung von Systemen linearer Gleichungen” (“Approxi-
mate solution of systems of linear equations”) published in Bulletin International
del’Acade´miePolonaisedesSciencesetdesLettres. Ayearlater,anItalianmath-
ematician, Gianfranco Cimmino (1908–1989),proposed another iterative method
for a linear system. He published his result in a paper “Calcolo approssimato per
le soluzioni dei sistemi di equazioni lineari” (“Approximate computation of the
solutions of linear systems”) in a La Ricerca Scientifica. As opposed to earlier
methodsforsolvingsystemsoflinearequations,bothmethodsaremotivatedrather
bygeometricaloperationsthanalgebraicones.Themainoperationinbothmethods
ix