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1、JournalofMathematicalResearch&ExpositionAug.,2008,Vol.28,No.3,pp.579–588DOI:10.3770/j.issn:1000-341X.2008.03.017Http://jmre.dlut.edu.cnStrongConvergenceTheoremsofViscosityApproximationforAccretiveOperatorsYANLiXia,ZHOUHaiYun(DepartmentofMathematics,NorthChinaElectricPowerUniversity,Hebe
2、i071003,China)(E-mail:yanlixia99@yahoo.com.cn;witman66@yahoo.com.cn)AbstractLetEbearealBanachspaceandletAbeanm-accretiveoperatorwithazero.Defineasequence{xn}asfollows:xn+1=αnf(xn)+(1−αn)Jrnxn,where{αn},{rn}are−1sequencessatisfyingcertainconditions,andJrdenotestheresolvent(I+rA)forr>1.Str
3、ongconvergenceofthealgorithm{xn}isobtainedprovidedthatEeitherhasaweaklycontinuousdualitymaporisuniformlysmooth.Keywordsfixedpoint;nonexpansivemapping;m-accretiveoperator;viscosityapproximation;weaklycontinuousdualitymap;uniformlysmoothBanachspace.DocumentcodeAMR(2000)SubjectClassification
4、47H06;47H10ChineseLibraryClassificationO1771.IntroductionInthesequel,weassumethatEisarealBanachspacewithnormk·k,denotethefixedpointsetbyF(T)={x∈E;Tx=x},theweakconvergenceby⇀,thestrongconvergenceby→.AmappingTwithitsdomainD(T)andrangeR(T)inEiscallednonexpansive(respectivelycontractive)iffor
5、allx,y∈D(T)suchthatkTx−Tyk≤kx−yk(respectivelykTx−Tyk≤αkx−ykforsome0<α<1).LetΠCdenotethesetofallcontractionsonC.Aclassicalwaytostudythenonexpansivemappingsistousethefollowing[1,2]:fort∈(0,1),fdefineamappingTt:Ttx=tu+(1−t)Tx,x∈C,whereu∈Cisafixedpoint.Banach’scontractionmappingPrincipleguara
6、nteesthatTthasafixedpointxtinC.InthecasethatThasafixedpoint,Browder[1]provedthatifEisaHilbertspace,thenxdoesconvergetstronglytoafixedpointofTthatisnearesttou.Reich[2]extendedBrowder’sresulttoauniformlyBanachspaceandthelimitdefinestheuniquesunnynonexpansiveretractionfromContoF(T).Veryrecentl
7、yXu[3]extendedReich’sresulttoaBanachspacewhichhasaweaklycontinuousdualitymap.AndXu[3]provedstrongconvergencetheoremsbythefollowingiterativemethodassumingthateitherEisuniformlysmoothorEhasaweaklycontinuousdualitymap:xn+1=αnu+(1−αn)Jrnxn,n≥0,where{αn}isasequencein(0,1),{rn}isaseq
