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Yaoetal.
JournalofInequalitiesandApplications2014,2014:206http://www.
journalofinequalitiesandapplications.
com/content/2014/1/206RESEARCHOpenAccessConstructionofminimum-normxedpointsofpseudocontractionsinHilbertspacesYonghongYao1,GiuseppeMarino2,Hong-KunXu3andYeong-ChengLiou4,5**Correspondence:simplex_liou@hotmail.
com4DepartmentofInformationManagement,ChengShiuUniversity,Kaohsiung,833,Taiwan5CenterforGeneralEducation,KaohsiungMedicalUniversity,Kaohsiung,807,TaiwanFulllistofauthorinformationisavailableattheendofthearticleAbstractAniterativealgorithmisintroducedfortheconstructionoftheminimum-normxedpointofapseudocontractiononaHilbertspace.
Thealgorithmisprovedtobestronglyconvergent.
MSC:47H05;47H10;47H17Keywords:xedpoint;minimum-norm;pseudocontraction;nonexpansivemapping;projection1IntroductionConstructionofxedpointsofnonlinearmappingsisaclassicalandactiveareaofnonlin-earfunctionalanalysisduetothefactthatmanynonlinearproblemscanbereformulatedasxedpointequationsofnonlinearmappings.
TheresearchofthisareadatesbacktoPi-card'sandBanach'stime.
Asamatteroffact,thewell-knownBanachcontractionprinciplestatesthatthePicarditerates{Tnx}convergetotheuniquexedpointofTwheneverTisacontractionofacompletemetricspace.
However,ifTisnotacontraction(nonexpan-sive,say),thenthePicarditerates{Tnx}fail,ingeneral,toconverge;hence,otheriterativemethodsareneeded.
In,Mann[]introducedthenowcalledMann'siterativemethodwhichgeneratesasequence{xn}viatheaveragedalgorithmxn+=(–αn)xn+αnTxn,n≥,(.
)where{αn}isasequenceintheunitinterval[,],Tisaself-mappingofaclosedconvexsubsetCofaHilbertspaceH,andtheinitialguessxisanarbitrary(butxed)pointofC.
Mann'salgorithm(.
)hasextensivelybeenstudied[–],andinparticular,itisknownthatifTisnonexpansive(i.
e.
,Tx–Ty≤x–yforallx,y∈C)andifThasaxedpoint,thenthesequence{xn}generatedbyMann'salgorithm(.
)convergesweaklytoaxedpointofTprovidedthesequence{αn}satisesthecondition∞n=αn(–αn)=∞.
(.
)Thisalgorithm,however,doesnotconvergeinthestrongtopologyingeneral(see[,Corollary.
]).
2014Yaoetal.
;licenseeSpringer.
ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommonsAttribu-tionLicense(http://creativecommons.
org/licenses/by/2.
0),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
Yaoetal.
JournalofInequalitiesandApplications2014,2014:206Page2of14http://www.
journalofinequalitiesandapplications.
com/content/2014/1/206BrowderandPetryshyn[]studiedweakconvergenceofMann'salgorithm(.
)fortheclassofstrictpseudocontractions(inthecaseofconstantstepsizesαn=αforalln;see[]forthegeneralcaseofvariablestepsizes).
However,Mann'salgorithmfailstoconvergeforLipschitzianpseudocontractions(seethecounterexampleofChidumeandMutan-gadura[]).
ItisthereforeaninterestingquestionofinventingiterativealgorithmswhichgenerateasequenceconverginginthenormtopologytoaxedpointofaLipschitzianpseudocontraction(ifany).
Theinterestofpseudocontractionsliesintheirconnectionwithmonotoneoperators;namely,TisapseudocontractionifandonlyifthecomplementI–Tisamonotoneoperator.
Wealsonoticethatitisquiteusualtoseekaparticularsolutionofagivennonlinearproblem,inparticular,theminimum-normsolution.
Forinstance,givenaclosedconvexsubsetCofaHilbertspaceHandaboundedlinearoperatorA:H→H,whereHisanotherHilbertspace.
TheC-constrainedpseudoinverseofA,ACisthendenedastheminimum-normsolutionoftheconstrainedminimizationproblemAC(b):=argminx∈CAx–b(.
)whichisequivalenttothexedpointproblemx=PCx–λA(Ax–b),(.
)wherePCisthemetricprojectionfromHontoC,AistheadjointofA,λ>isaconstant,andb∈HissuchthatPA(C)(b)∈A(C).
Itisthereforeaninterestingproblemtoinventiterativealgorithmsthatcangeneratesequenceswhichconvergestronglytotheminimum-normsolutionofagivenxedpointproblem.
Thepurposeofthispaperistosolvesuchaproblemforpseudocontractions.
Moreprecisely,weshallintroduceaniterativealgorithmfortheconstructionofxedpointsofLipschitzianpseudocontractionsandprovethatouralgorithm(see(.
)inSec-tion)convergesinthestrongtopologytotheminimum-normxedpointofthemapping.
Fortheexistingliteratureoniterativemethodsforpseudocontractions,thereadercanconsult[,–];forndingminimum-normsolutionsofnonlinearxedpointandvariationalinequalityproblems,see[–];andforrelatediterativemethodsfornonex-pansivemappings,see[,,,]andthereferencestherein.
2PreliminariesLetHbearealHilbertspacewiththeinnerproduct·,·andthenorm·,respectively.
LetCbeanonemptyclosedconvexsubsetofH.
Theclassofnonlinearmappingswhichwewillstudyistheclassofpseudocontractions.
RecallthatamappingT:C→CisapseudocontractionifitsatisesthepropertyTx–Ty,x–y≤x–y,x,y∈C.
(.
)ItisnothardtondthatTisapseudocontractionifandonlyifTsatisesoneofthefollowingtwoequivalentproperties:(a)Tx–Ty≤x–y+(I–T)x–(I–T)yforallx,y∈C;or(b)I–TismonotoneonC:x–y,(I–T)x–(I–T)y≥forallx,y∈C.
Yaoetal.
JournalofInequalitiesandApplications2014,2014:206Page3of14http://www.
journalofinequalitiesandapplications.
com/content/2014/1/206RecallthatamappingT:C→CisnonexpansiveifTx–Ty≤x–y,x,y∈C.
Itisimmediatelyclearthatnonexpansivemappingsarepseudocontractions.
Recallalsothatthenearestpoint(ormetric)projectionfromHontoCisdenedasfollows:Foreachpointx∈H,PCxistheuniquepointinCwiththepropertyx–PCx≤x–y,y∈C.
NotethatPCischaracterizedbytheinequalityPCx∈C,x–PCx,y–PCx≤,y∈C.
(.
)Consequently,PCisnonexpansive.
Inthesequelweshallusethefollowingnotations:Fix(S)standsforthesetofxedpointsofS;xnxstandsfortheweakconvergenceof(xn)tox;xn→xstandsforthestrongconvergenceof(xn)tox.
Belowistheso-calleddemiclosednessprinciplefornonexpansivemappings.
Lemma.
(cf.
[])LetCbeanonemptyclosedconvexsubsetofarealHilbertspaceH,andletS:C→Cbeanonexpansivemappingwithxedpoints.
If{xn}isasequenceinCsuchthatxnxand(I–S)xn→y,then(I–S)x=y.
Wealsoneedthefollowinglemmawhoseproofcanbefoundinliterature(cf.
[]).
Lemma.
LetCbeanonemptyclosedconvexsubsetofarealHilbertspaceH.
As-sumethatamappingF:C→Hismonotoneandweaklycontinuousalongsegments(i.
e.
,F(x+ty)→F(x)weaklyast→,wheneverx+ty∈Cforx,y∈C).
Thenthevariationalinequalityx∈C,Fx,x–x≥,x∈C(.
)isequivalenttothedualvariationalinequalityx∈C,Fx,x–x≥,x∈C.
(.
)Finally,westatethefollowingelementaryresultonconvergenceofrealsequences.
Lemma.
([])Let{an}beasequenceofnonnegativerealnumberssatisfyingan+≤(–γn)an+γnσn,n≥,where{γn}(,)and{σn}satisfy(i)∞n=γn=∞;(ii)eitherlimsupn→∞σn≤or∞n=|γnσn|isaconstantsuchthatM>(–ρ)supf(xt)–xt:t∈(,).
Inparticular,wegetfrom(.
)xn–u≤–ρf(u)–u,xn–u+tnM,u∈Fix(S).
(.
)Yaoetal.
JournalofInequalitiesandApplications2014,2014:206Page6of14http://www.
journalofinequalitiesandapplications.
com/content/2014/1/206Since{xn}isbounded,withoutlossofgenerality,wemayassumethat{xn}convergesweaklytoapointx∈C.
Noticing(.
)wecanuseLemma.
togetx∈Fix(S).
Thereforewecansubstitutexforuin(.
)togetxn–x≤–ρfx–x,xn–x+tnM.
(.
)However,xnx.
Thistogetherwith(.
)guaranteesthatxn→x.
Thenet{xt}isthere-forerelativelycompact,ast→+,inthenormtopology.
Nowwereturnto(.
)andtakethelimitasn→∞togetx–u≤–ρf(u)–u,x–u,u∈Fix(S).
Inparticular,xsolvesthefollowingvariationalinequality:x∈Fix(S),(I–f)u,u–x≥,u∈Fix(S).
ByLemma.
,weseethatxsolvesthevariationalinequalityx∈Fix(S),(I–f)x,u–x≥,u∈Fix(S).
(.
)Therefore,x=(PFix(S)f)x.
Thatis,xistheuniquexedpointinFix(S)ofthecontractionPFix(S)f.
Clearlythisissucienttoconcludethattheentirenet{xt}convergesinnormtoxast→+.
Finally,ifwetakef=,thenvariationalinequality(.
)isreducedto≤x,u–x,u∈Fix(S).
Equivalently,x≤x,u,u∈Fix(S).
Thisclearlyimpliesthatx≤u,u∈Fix(S).
Therefore,xistheminimum-normxedpointofS.
Thiscompletestheproof.
Wearenowinapositiontoprovethestrongconvergenceofalgorithm(.
).
Theorem.
LetCbeanonemptyclosedconvexsubsetofarealHilbertspaceH,andletT:C→CbeL-LipschitzianandpseudocontractivewithFix(T)=.
Supposethatthefollowingconditionsaresatised:(i)limn→∞αn=and∞n=αn=∞;(ii)limn→∞αnβn=limn→∞βnαn=;(iii)limn→∞αnβn––αn–βnαnβn–=.
Yaoetal.
JournalofInequalitiesandApplications2014,2014:206Page7of14http://www.
journalofinequalitiesandapplications.
com/content/2014/1/206Thenthesequence{xn}generatedbyalgorithm(.
)convergesstronglytotheminimum-normxedpointofT.
ProofFirstweprovethatthesequence{xn}isbounded.
Wewillshowthisfactbyinduc-tion.
Accordingtoconditions(i)and(ii),thereexistsasucientlylargepositiveintegermsuchthat–(L+)(L+)αn+βn+βnαn>,n≥m.
(.
)Fixp∈Fix(T)andtakeaconstantM>suchthatmaxx–p,x–p,.
.
.
,xm–p,p≤M.
(.
)Next,weshowthatxm+–p≤M.
Setym=(–αm–βm)xm+βmTxm;thusxm+=PC[ym].
Then,byusingproperty(.
)ofthemetricprojection,wehavexm+–ym,xm+–p≤.
(.
)BythefactthatI–Tismonotone,wehave(I–T)xm+–(I–T)p,xm+–p≥.
(.
)From(.
),(.
)and(.
),weobtainxm+–p=xm+–p,xm+–p=xm+–ym,xm+–p+ym–p,xm+–p≤ym–p,xm+–p=xm–p,xm+–p–αmxm,xm+–p+βmTxm–xm,xm+–p=xm–p,xm+–p+αmxm+–xm,xm+–p–αmp,xm+–p–αmxm+–p,xm+–p+βmTxm–Txm+,xm+–p+βmxm+–xm,xm+–p–βmxm+–Txm+,xm+–p≤xm–pxm+–p+αmxm+–xmxm+–p+αmpxm+–p–αmxm+–p+βmTxm–Txm++xm+–xmxm+–p≤xm–pxm+–p+αmpxm+–p–αmxm+–p+(L+)(αm+βm)xm+–xmxm+–p.
Yaoetal.
JournalofInequalitiesandApplications2014,2014:206Page8of14http://www.
journalofinequalitiesandapplications.
com/content/2014/1/206Itfollowsthat(+αm)xm+–p≤xm–p+αmp+(L+)(αm+βm)xm+–xm.
(.
)By(.
),wehavexm+–xm=PC(–αm–βm)xm+βmTxm–PC[xm]≤(–αm–βm)xm+βmTxm–xm≤αmp+xm–p+βmTxm–p+xm–p≤αmp+xm–p+(L+)βmxm–p≤(L+)(αm+βm)xm–p+αmp≤(L+)(αm+βm)M.
(.
)Substitute(.
)into(.
)toobtain(+αm)xm+–p≤xm–p+αmp+(L+)(L+)(αm+βm)M≤+αmM+(L+)(L+)(αm+βm)M,thatis,xm+–p≤–(αm/)–(L+)(L+)(αm+βm)+αmM=–(αm/)[–(L+)(L+)(αm+βm+(βm/αm))]+αmM≤M.
Byinduction,wegetxn–p≤M,n≥,(.
)whichimpliesthat{xn}isboundedandsois{Txn}.
NowwetakeaconstantM>suchthatM=supnxn∨Txn–xn.
[Herea∨b=max{a,b}fora,b∈R.
]SetS=(I–T)–(i.
e.
,SisaresolventofthemonotoneoperatorI–T).
WethenhavethatSisanonexpansiveself-mappingofCandFix(S)=Fix(T)(cf.
Theoremof[]).
ByLemma.
,weknowthatwhenever{γn}(,)andγn→+,thesequence{zn}denedbyzn=SPC(–γn)zn(.
)Yaoetal.
JournalofInequalitiesandApplications2014,2014:206Page9of14http://www.
journalofinequalitiesandapplications.
com/content/2014/1/206convergesstronglytotheminimum-normxedpointxofS(andofTasFix(S)=Fix(T)).
Withoutlossofgenerality,wemayassumethatzn≤Mforalln.
Itsucestoprovethatxn+–zn→asn→∞(forsomeγn→+).
Tothisend,werewrite(.
)as(I–T)zn=PC(–γn)zn,n≥.
Byusingthepropertyofmetricprojection(.
),wehave(–γn)zn–(zn–Tzn),xn+–(zn–Tzn)≤–γnzn,xn+–zn–(zn–Tzn)+Tzn–zn,xn+–zn–(zn–Tzn)≤–γnzn+Tzn–zn,xn+–zn+zn–Tzn≤γnzn,Tzn–zn–γnzn+Tzn–zn,xn+–zn≤γnznTzn–zn–zn+Tzn–znγn,xn+–zn≤znTzn–zn.
Notethatzn–Tzn=PC(–γn)zn–zn≤(–γn)zn–zn=γnzn.
Hence,weget–zn+Tzn–znγn,xn+–zn≤γnzn.
(.
)From(.
)wehavezn+–zn=SPC(–γn+)zn+–SPC(–γn)zn≤(–γn+)zn+–(–γn)zn=(–γn+)(zn+–zn)+(γn–γn+)zn≤(–γn+)zn+–zn+|γn+–γn|zn.
Itfollowsthatzn+–zn≤|γn+–γn|γn+zn.
(.
)Setγn:=αnβn.
Yaoetal.
JournalofInequalitiesandApplications2014,2014:206Page10of14http://www.
journalofinequalitiesandapplications.
com/content/2014/1/206Bycondition(ii),γn→+andγn∈(,)fornlargeenough.
Hence,by(.
)and(.
)wehave–zn+βn(Tzn–zn)αn,xn+–zn≤αnβnzn≤αnβnM(.
)andzn–zn–≤αnβn––αn–βnαnβn–M.
(.
)By(.
)wehavexn+–xn=PC(–αn–βn)xn+βnTxn–PCxn≤αnxn+βnTxn–xn≤(αn+βn)M.
(.
)Next,weestimatexn+–zn+.
Sincexn+=PC[yn],xn+–yn,xn+–zn≤.
Using(.
)andbythefactthatTisL-Lipschitzianandpseudocontractive,weinferthatxn+–zn=xn+–zn,xn+–zn=xn+–yn,xn+–zn+yn–zn,xn+–zn≤yn–zn,xn+–zn=(–αn–βn)xn+βnTxn–zn,xn+–zn=(–αn–βn)xn–zn,xn+–zn+βnTxn–Txn+,xn+–zn+βnTxn+–Tzn,xn+–zn+–αnzn+βn(Tzn–zn),xn+–zn,whichleadstoxn+–zn≤(–αn–βn)xn–znxn+–zn+βnLxn–xn+xn+–zn+βnxn+–zn+αn–zn+βnαn(Tzn–zn),xn+–zn≤–αn–βnxn–zn+xn+–zn+βnxn+–zn+Lxn–xn++βnxn+–zn+αnβnzn.
Itfollowsthat,using(.
),(.
)and(.
),wegetxn+–zn≤–αn–βn+αn–βnxn–zn+L+αn–βnxn+–xn+αn(+αn–βn)βnzn+βn+αn–βnxn+–zn≤–αn+αn–βnxn–zn+(αn+βn)+αn–βnLMYaoetal.
JournalofInequalitiesandApplications2014,2014:206Page11of14http://www.
journalofinequalitiesandapplications.
com/content/2014/1/206+αn(+αn–βn)βnM+βn+αn–βnM≤–αn+αn–βnxn–zn–+zn–zn–+(αn+βn)+αn–βn+αn(+αn–βn)βn+βn+αn–βnM≤–αn+αn–βnxn–zn–++αn–βnzn–zn–xn–zn–+zn–zn–+(αn+βn)+αn–βn+αn(+αn–βn)βn+βn+αn–βnM≤–αn+αn–βnxn–zn–++αn–βnαnβn––αn–βnαnβn–M+(αn+βn)+αn–βn+αn(+αn–βn)βn+βn+αn–βnM,(.
)wheretheniteconstantM>isgivenbyM:=maxLM,M,Msupnxn–zn–+zn–zn–.
Letδn=αn+αn–βn≈αn(asn→∞)andnotethatby(.
)itfollowsthat{δn}(,).
Moreover,setθn=αnβn––αn–βnαnβn–+αn+βn+βnαn+αnβn+βnαnM.
Thenrelation(.
)isrewrittenasxn+–zn≤(–δn)xn–zn–+δnθn.
(.
)Byconditions(i),(ii)and(iii),itiseasilyfoundthatlimn→∞δn=,∞n=δn=∞,limn→∞θn=.
WecanthereforeapplyLemma.
to(.
)andconcludethatxn+–zn→asn→∞.
Thiscompletestheproof.
Remark.
Choosethesequences(αn)and(βn)suchthatαn=(n+)aandβn=(n+)b,n≥,Yaoetal.
JournalofInequalitiesandApplications2014,2014:206Page12of14http://www.
journalofinequalitiesandapplications.
com/content/2014/1/206whereItisclearthatconditions(i)and(ii)ofTheorem.
aresatised.
Toverifycondition(iii),wecomputeαnβn––αn–βnαnβn–=αn–αn–βnαnβn–=(n+)a–(n+)a–bna–b=(n+)a+na–b–≈a–bn(n+)a→.
Therefore,{αn}and{βn}satisfyallthreeconditions(i)-(iii)inTheorem.
.
4ApplicationToshowanapplicationofourresults,wedealwiththefollowingproblem.
Problem.
Let(.
)Atwhichvaluedoes{xn}approachasngoestoinnityWeclaimthatlimn→∞xn=anditcanbeeasilyderivedbyapplyingTheorem.
.
ProofInordertoapplyourresult,letH=R,C=[,]anddeneT:C→CbyTx:=x+x.
ObservethatTisLipschitzian,pseudocontractiveandthatFix(T)={}.
Moreover,ifwesetαn=n–/andβn=n–/,then(i)limn→∞αn=and∞n=αn=∞;(ii)limn→∞αnβn=limn→∞βnαn=;(iii)limn→∞αnβn––αn–βnαnβn–=.
ThenTheorem.
ensuresthatlimn→∞xn=.
CompetinginterestsTheauthorsdeclarethattheyhavenocompetinginterests.
Authors'contributionsAllauthorscontributedequallyandsignicantlyinwritingthispaper.
Allauthorsreadandapprovedthenalmanuscript.
Authordetails1DepartmentofMathematics,TianjinPolytechnicUniversity,Tianjin,300387,China.
2DipartimentodiMatematica,UniversitádellaCalabria,ArcavacatadiRende(CS),87036,Italy.
3DepartmentofAppliedMathematics,NationalSunYat-SenUniversity,Kaohsiung,80424,Taiwan.
4DepartmentofInformationManagement,ChengShiuUniversity,Kaohsiung,833,Taiwan.
5CenterforGeneralEducation,KaohsiungMedicalUniversity,Kaohsiung,807,Taiwan.
Yaoetal.
JournalofInequalitiesandApplications2014,2014:206Page13of14http://www.
journalofinequalitiesandapplications.
com/content/2014/1/206AcknowledgementsYonghongYaowassupportedinpartbyNSFC71161001-G0105.
Yeong-ChengLiouwassupportedinpartbyNSC101-2628-E-230-001-MY3andNSC101-2622-E-230-005-CC3.
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