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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.De?neasequence{xn}asfollows:xn+1=αnf(xn)+(1?αn)Jrnxn,where{αn},{rn}are?1sequencessatisfyingcertainconditions,andJrdenotestheresolvent(I+rA)forr>1.Str
3、ongconvergenceofthealgorithm{xn}isobtainedprovidedthatEeitherhasaweaklycontinuousdualitymaporisuniformlysmooth.Keywords?xedpoint;nonexpansivemapping;m-accretiveoperator;viscosityapproximation;weaklycontinuousdualitymap;uniformlysmoothBanachspace.DocumentcodeAMR(2000)SubjectClassi?cation
4、47H06;47H10ChineseLibraryClassi?cationO1771.IntroductionInthesequel,weassumethatEisarealBanachspacewithnormk·k,denotethe?xedpointsetbyF(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),fde?neamappingTt:Ttx=tu+(1?t)Tx,x∈C,whereu∈Cisa?xedpoint.Banach’scontractionmappingPrincipleguara
6、nteesthatTthasa?xedpointxtinC.InthecasethatThasa?xedpoint,Browder[1]provedthatifEisaHilbertspace,thenxdoesconvergetstronglytoa?xedpointofTthatisnearesttou.Reich[2]extendedBrowder’sresulttoauniformlyBanachspaceandthelimitde?nestheuniquesunnynonexpansiveretractionfromContoF(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
