Banach空间中强增生算子的非线性方程的解的迭代构造(英文)

1、JournalofMathematicalResearch&ExpositionVol.18,No.3,329-334,August1998IterativeConstructionofSolutionstoNonlinearEquationsofStronglyAccretiveOperatorsinBanachSpaces3ZengLuchuan(Inst.ofMath.,FudanUniversity,Shanhai200433)(Dept.ofMath.,ShanghaiNormalUniversity,Shanghai200234)limn=0orlimn=limn=0nnn

2、intheirtheorems.ThesealsoextendTheorems1and2ofDeng6tothep-uniformlysmoothBanachspacesetting.Keywordsstronglyaccretive,strictlypseudocontractive,p-uniformlysmoothBanachspace.ClassificationAMS(1991)47H15,47H05/CCLO177.21.IntroductionandpreliminariesLetTbeanonlinearoperatorwithdomainD(T)andrangeR(T)ina

3、realBanachspace.Tissaidtobeaccretive2iftheinequalityx-yx-y+r(Tx-Ty)monotone3.IfXisaHilbertspacethentheaccretivecondition(1)reducesto(1)holdsforeachxandyinD(T)andforallr0.If(1)holdsonlyforsomer>0,TissaidtobeTx-Ty,x-y03(2)ReceivedMay28,1994.ProjectsupportedbytheScienceandTechnologyDevelopmentFundat

4、ionofShanghaiHigherLearning.329© 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.forallx,yinX.Tisaccretiveifandonlyifforanyx,yD(T),thereexistsjJ(x-y)suchthatTx-Ty,j0,whereJ(x)=x33232X3:x,x=x=x,xX,(3)isthenormalizeddualitymappingofXand,denotesthedualitypairingbetweenXandX3

5、.TheaccretiveoperatorswereintroducedindependentlybyBrowder2andKato3in1967.LetCbeanonemptysubsetofarealBanachspaceX.AmappingT:CXissaidtobestronglyaccretiveifforeachx,yinCthereisjJ(x-y)suchthat2Tx-Ty,jkx-y(4)forsomerealconstantk>0.Withoutlossofgenerality,weassumethatk(0,1).LetCbeanonemptysubsetofar

6、ealBanachspacex.AmappingT:CXissaidtobesrictlypseudocontractiveifthereexistst>1suchthattheinequalityx-y(1+r)(x-y)-rt(Tx-Ty)(5)holdsforallx,yinCandr>0.If,intheabovedefinition,t=1,thenTissaidtobeapseudocon2tractivemapping.Recently,Deng6answeredpositivelyProblem2inChidume7,byremovingtherestriction

7、n8studiedboththeMannandtheIshikawaitera2nandlimnn=0.Ontheotherhand,TanandXutionprocessinap-uniformlysmoothBanachspaceXandprovedthatthetwoprocessesconvergestronglytotheuniquesolutionoftheequationTx=fincaseTisaLipschitzianandstronglyaccretiveoperatorfromXtoX,ortotheuniquefixedpointofTincaseTisaLipschi

8、tzianandstrictlypseu2docontractivemappingfromaboundedclosedconvexsubsetCofXintoitself.Hence,TanandXu8gaveaffirmativeanswerstoProblems1and2ofChidume7respectively,andalsoextendedallresultsofChidume7tothep-uniformlysmoothBanachspacesetting.ofsmoothnessofXisdefinedby)=sup(x+y+x-y)-1:x,yX,x=1,y=,x(2>0

9、,andthatXissaidtobeuniformlysmoothiflim0x()/=0.Recallalsothatforarealnum2p)dber1<p2,aBanachspaceXissaidtobep-uniformlysmoothiffor>0wheredx(>0isaconstant.Itisknown(cf.1)thatforaHilbertspaceH,21/2H()=(1+)-1(6)andhenceHis2-uniformlysmooth.Itisalsoknownthatif1<p<2,Lp(orlp)isp-uniformlysmo

10、oth;whileif2p<,Lp(orlp)is2-uniformlysmooth.Xu5gavethefollowingcharacteri2zationforap-uniformlysmoothBanachspace.330© 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.Lemma1LetXbeasmoothBanachspaceandpafixednumberin(1,2.Then,Xisp-uni2formlysmoothifandonlyifthereexistsaco

11、nstantdp>0suchthatx+ypxp+py,Jp(x)+dpypforallx,yinX,whereJp(x)isthesubdifferentiableatxofthefunctionalp-1(7)p.ItisknownthatJp(x)=xp-2J(x)forxX,x0,and3p3p-1X3:x,x=x,x=x,xX.WhenxisanLp(orlp)space,theconstantdpin(1.7)hasbeencalculated.Jp(x)=x32.MainresultsTheorem1letxbeap-uniformlysmoothBanachspacewi

12、th1<p2andT:XXbeaLipschitzianandstronglyaccretiveoperatorwithLipschitzconstantL.DefineS:XXbySx=f-Tx+x.Letnn=0andnn=0betwosequencesofrealsin0,1satisfying(i)an=andlimn=0;n=0n(ii)0mintp,np1/)4p2L0(1+L0pforeachn0,whereL0istheLipschitzconstantofSwithL01+L,tpisthe(smaller)solutionoftheequation(8)pk=0(t&

13、gt;0),2andk(0,1),dparetheconstantsappearingin(4)and(7),respectively.Thenforeachx0inXtheIshikawasequencexndefinedbyxn+1=(1-n)xn+nSynandyn=(1-n)xn+nSxn,n0pp-1f(t)=p(p-1)(1-k)t-(1+dpL0)t+convergesstronglytotheuniquesolutionoftheequationTx=f.ProofWefirstobservethattheequationTx=fhasauniquesolutionwhichw

14、edenotebyq.Infact,theexistencefollowsfromMorales4andtheuniquenessfromthestrongaccretivenessofT.Wealsoobservethatforx,yX,Sx-Sy,Jp(x-y)=-Tx-Ty,Jp(x-y)+x-ypp-2p=-x-yTx-Ty,J(x-y)+x-y2-kx-yp-2x-y+x-yp=(1-k)x-yp.Itfollowsthatpxn+1-qp=(1-n)(xn-q)+n(Syn-q)p-1(1-n)pxn-qp+pSyn-q,Jp(x-q)n(1-n)pp(9)+dpnSyn-q.33

15、1© 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.SincepSyn-qpL0yn-qpSxn-q,Jp(xn-q)(1-k)xn-qp,pyn-qp=(1-n)(xn-q)+n(Sxn-q)pp-1(1-xn-qp+pSxn-q,Jp(xn-q)n)n(1-n)pp+dpnSxn-qpp-1ppp(1-+p(1-k)+dpL0xn-qn)n(1-n)n)p=tnxn-q,pp-1ppwheretn=(1-+p(1-k)+dpL0n)n(1-n)n,ppppyn-xnp=nxn-Sxn=

16、n(xn-q)+(q-Sxn)ppp2pn(xn-q+Sxn-q)ppp)2p(1+L0nxn-q,Syn-Sxn,Jp(xn-q)L0yn-xnxn-qp-1p1/p2L0xn-qp,n(1+L0)andSyn-q,Jp(xn-q)=Syn-Sxn,Jp(xn-q)+Sxn-q,Jp(xn-q)p1/p(2L0+(1-k)xn-qp,n(1+L0)weobtainfrom(8)pp-1p1/p(1-k+2L0)xn+1-qp(1-+pn)n(1-n)n(1+L0)+dpL0ntn)xn-q.pppSince1<p2,(1-t)p1-pt+tpand(1-t)weobtainp-11-(

17、p-1)tfor0t1,pp-1pptn=(1-+p(1-k)+dpL0n)n(1-n)n2pp(10)1-pkn-p(p-1)(1-k)n+(1+dpL0)n.2ppSincentpforalln0,wehavefrom(8)p(p-1)(1-k)n-(1+dpL0)n-pkpkn.Henceitfollowsthattn1-nforeachn0.Ontheotherhand,since22limnn=0,thereexistsapositiveintegerNsuchthat0ntpforeachnN.Thisimpliesthatpt+p(1-k)n=(1-n)n(1-n)1-pknfo

18、reachnN.2p-1pp+dpL0n332© 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.Therefore,weobtainthatforeachnN,xn+1-qppp-1p-1p1/p(1-k)+p(1-+p2L0n)n(1-n)n(1-n)n(1+L0)ppp+dpL0pkn(1-n)xn-q22p1/ppppt2L0-pkdpL0n+p(n-(p-1)n)n(1+L0)nnxn-q2p1/pp1/p2n1-pkn+2pL0(1+L0)nn-2p(p-1)L0(1+L0)n2

19、ppp-pkdpL0nnxn-q.2Sincelimnn=0implieslimnnn=0,wehave-11/pp1/p2plim2pL0(1+LppkdpLpnn0)nn-2p(p-1)L0(1+L0)nn-0nnn21/p1/p=2pL0(1+Lp<2p2L0(1+Lp.0)p)Fromthisandthecondition(ii),wederivethatthereisapositiveintegerN0>NsuchthatforeachnN0,xn+1-qp2p1/pp1-pkn+2pL0(1+L0)nnxn-q22p1/pp1-pk2n+2pL0(1+L0)nxn-qp

20、1/p24pL0(1+L0)p(p-1)k1-nxn-q2p(p-1)kexp(-n)xn-q2np(p-1)kexp(-j)xN0-q.2j=N0Thisimmediatelyimpliesthestrongconvergenceofxntoqsincetheseriesnndi2verges.Theproofiscomplete.ReviewingtheproofofTheorem1,wecanseethatthefollowingconsequenceistrue.Theorem2LetCbeanonemptyboundedclosedconvexsubsetofap-uniformly

21、smoothBanachspaceXwith1<p2andT:CCbeaLipschitzianandstrictlypseu2docontractivemappingwithLipschitzconstantL.Letnquencesofrealsin0,1satisfying(i)n=andlimn=0;n=0nn=0andnn=0bese2333© 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.4p2L(1+Lp)1/pwheretpisthe(smaller)solutionoftheequation(ii)0mintp,nforeachn0,pk=0(t>0),2),dparetheconstantsappearingin(5)and(7),respective2k=(t-1)/tandt(1,ly.Then,fo