【doc】有界半变差和有界变差C0-群

【doc】有界半变差和有界变差C0-群 有界半变差和有界变差C0-群 云南大学(自然科学版)2005,27(2):93~96 JournalofYunnanUniversity CN53—1045/N;SNO258—79l71 Onboundedsemivariationandvariationofc0-groups JLYun—feng (CollegeofMathematicsandInformationScience,ShanxiNormalUniversity,Xi'an710062,China) Abstract:Byintroducingtheconceptsofboundedsemivariationandvariationofaco?groups{T(,)}tc-RonaBa? nachspace,somepropertiesof{T(,)}tc-Rarecharacterized.Furthermore,afewrelationsb etweentheinfinitesimal generatorAof{T(t)}tc-RandthesetwoconceptsarealsoestablishedintermsoftheboundednessofA. Keywords:boundedsemivariation;boundedvariation;co?groups;infinitesimalgenerator CLCnumber:0175.3;0177.1Documentcode:AArticleID:0258—7971(2005)02—0o93 —04 ~tSC(2000):47A10;471303 Inreference[1],usingtheboundedsemivariationofco-semigroups,theauthorprovedthewell—posedness ofanabstractinhomogeneousCauchyproblemofoneorder;In[2],theexistenceanduniquenessofthestrong solutionofanabstractinhomogeneousCauchyproblemoftwoorderwassolvedbytheboundedsemivariation ofcosineoperatorfunctions.Inthispaper,byintroducingtheboundedsemivariationandvariationofaco— group{T(t)},?RonaBanachspace,weonlydiscussessomemoreresultsof{T(t)},? R,thecontentswhich relevanttodifferentialequationsarenotconceredatpresent. Inthefollowingdisctkssion,weuseR,X,I,B(X),D(A)and{T(t)},? RtOdenotethesetofallreal numbers,anyBanachspace,theidentityoperatoronX,thesetofallboundedlinearoperatorsonX,thedo— mainofoperatorA,andacO-grouponXrespectively.Meanwhile,usingC([a,b];X)wedenotethesetof allcontinuotksoperatorsform[a,b]RintoX. Definition1Supposethat{T(t)}tERB(X)isaco-grouponX.Fors?[a,b]R,let SVar(T(?),s):sup{ll?(T(ti)一T(,一1))ll:xi?X,llxll?1,{}?D[a,s]),i=l Var(T(?),s):sup{?IT(ti)一T(,i一1)l1.{,i}?D[a,s]), whereD[a,s]denotesallsubdivisionsof[a,s]. IfSVar(T(?),5)<oo(resp.,Var(T(?),s)<oo),thenwe.saythatT(?)isofboundedsemivariation (resp.,boundedvariation).ItiseasytoseethattheconceptofSVar(T(?),s)isanaturalgeneralizationof thatofVar(T(?),s). Byreference[2]and[4],wecaneasilyobtainfollowingconclusions(proposition1andproposition2): PropositionISuplx>sethatT(?)isofboundedsemivariationonanysubinterval[.,b]ofRandthe rb operatorB?C([a,b];X).ThentheRiemann—Stieltjesintegralld(T(,))B(,)existsand r limld(T(t))B(t)=0,s.+n+Jn ?P~cdveddate:2004——09——29 Foundationitem:ThisworkwassupportedbytheNaturalScienceFoundationofChina(10071048) 云南大学(自然科学版)第27卷 whered(T(,))denotesthederivativeofT(,). Proposition2Let{Tn(t)},? [_6](n=1,2,…)beasequenceDfoperatorsofboundedserniation satisfyingSVar(Tn('),S)?M<?forsomeconstantM.IfthereexistsanoperatorT(t)?B(X)such that , l r. imTn(,)T(t)x,Vt?[n,6]andX?XthenT(,)isofboundedsemivariationsatisfyingSVar (T('),s)?M.MoreoverforanyoperatorB?C([n,b];X),thefollowingholds limId(Tn(t)B(t))=Id(T(t))B(t). Reference[3]saysthefollowingfacts:IfthelinearoperatorsAand—Aaretheinfinitesimalgenerators ofC0一semigroups{T十(t)},?0and{T一(t)},?0,respectively,thenAistheinfinitesimalgeneratorofaC0- group{T(,)},?Rgivenby T?={TT一+((一t),forfort~,?O,.. Combiningtheabovefacts,theadditivityofintervalsofintegralandlT(,一 s)xds=lT(s)xdswithJ0J0 thetheorem1.2.4in[3],itiseasytoobtainthefollowinglemma1. Lemma1IfAistheinfinitesimalgeneratorofboundedsemivariationC0-group{T(t)},?Ronany subinter弋,al[n,b]ofR,thenfor?x,wehaverT(,一s)ds?D(A)whichimplieSthatrT(s)ds ?D(A))and r,r AIT(,一s)xds=AIT(s)xds=T(,)X—T(n). Proposition3LetAbetheinfinitesimalgeneratorof{T(t)},?R(ofcourse,Aislinearbutnotneces— sarilybounded),then (1)FDrn<,?rf0rS0merandz?x,wehaye(r—r)T(r)xdr?D(A)and r, AI(r—r)T(r)xdr=((r一,)T(,)一(r—n)T(n))Jn +fT()d.Jn (2)Forn<6,n,6?Rand?XwehavefrT(r—r)dr?D(A)and r6,rr一"AIrT(r—r)xdr:(aT(r—n)一bT(r—b))+IT(r)xdr. Proof(1)Since J(r—r)T(r)dr:J(T(r)ddr=(T(r)dr)d+(T(r)dr)d= (r—t)T(r)dr+(T(r)dr)d. r,/-, ByLamina1,weknowthatIT(r)xdr?D(A),thusI(r—r)T(r)xdr?D(A),andJndn A(r—r)T(r)dr:(r—t)AI— t T( d r)dr+ d(AdT(r)dr)d=Jn— n,n, (r一,)(T(,)一T(n))+I(T()—T(n)x)d/~=r, Jn ((r—t)T(t)一(r—n)T(n))X (2)Takingr=r—t,thenwehave (r-r)=- b (rT?础=r一 +rT()xdJn rr—b ,)T(,)xdt—I(r一,)T(,)xdtJ0 第2期贾云锋:有界半变差和有界变差CO一群 ThuS,by(1),weseethatIrT(r—r)xdr?D(A)and 广brr—nrr—b AJrT(r—r)xdrAJ.(r一,)T(t)xdt—AJ.(r一,)T(,)d: rr—n厂r—b aT(r一口)x—rx+lT(r)xdr一6T(r—b)x+rx—lT(11")xdr= (aT(r一口)一6T(r—b))x+IT(11")xdr.Theproof啪scompleted.rr—n Jr—D Proposition4SupposethatT(?)isofboundedsemivaritaionofanysubinterval[口,b]ofR.The nfor B?C([口,b];X),theRiemann-StieltjesintegralId,(T(11"一,))B(,)iscontinuousin11"on[口,b]. ProofByProposition1weknowthattheintegralId,(T(11"一,))B(,)exists.Let口?r?band At>0suchthatr+?r<b.Then. rr+?r?厂Id(T(11-+Ar—t))B(t)一ld(T(11"一t))B(t)= Id(T(11"+Ar—t))B(t)一ld(T(11",t))B(t)+ld(T(11"+Ar—t))B(t).rrrrrr+? Byproposition1andproposition2,forB?C([n,b];X),taking=11"+?r—t,weobtain Jdt(T(.r+?.r—t))B(t)J.d(T())B(11.+?r—)=0, and limldt(T(r+?r—t))B(t):ldt(T(r—t))B(,),?r+0+JnJ0 respectively.Thus limId(T(r+?r—t))B(t)=ld(T(r—t))B(t),?r0+JnJ0 Similarly,wecanprovethat rr一?rr limId(T(r—Ar—t))B(t)=ld(T(r—t))B(t).?r.0一 JnJ0 Theproofwascompleted. Proposition5LetAbetheinfinitesimalgeneratorof{T(t)},? R.IfAisbounded,thenthefollowing statementshold: (1)lirn .<2】lim<3)lim咎<4) r(T(.)'s)一0.(5)<6】l… imSVar(T(.)'s)_0.(7)SVar(T(.)'r),<oo forsomer>口;(8)lltAI.T(,一s)xdsII<oofor,?[口,r];(9)hIItAlT(,一s)xdsII=0.Jnl一0J0 Inaddition.thefactthat(1)holdsisequivalenttothatofAisbounded. ProofItisobviousthatif(3)holdsthen(4)and(5)hold,(6)holdsif(4)or(5)holds.Moreover, (7)holdsif(6)isvalid,and(8)holdswhenever(9)isvalid.Thusweonlyneedtoshowthat(1),(2) ,(3) and(9)holdsinceAisbounded. (1)By[3]and[6],weknowthatthereexsistconstants??0,M?1suchthatllT(t)ll?Metl rt andAIT(11")xdr=T(t)x—x,x?X.So J0 IIT(,)一III:IIAf2T()d()II?IIAIIIriiT()IIdI:IIAIIMemI,I. 云南大学(自然科学版)第27卷 SinceAisbounded,(1)holds; (2)By[3],[6]andlemma1,thereexsistconstants??0,M?1suchthatIIT(z)II?Me"and AIT(t—s)xds=T(t)X—T(口)X,z?x.So IIT(z)一T(口)II:IIA[IT(t—s)dslI?lIIf,IIT(t—s)lIdsI:IIAIIMemI卜s}I,一aI. SinceAisbounded,(2)holds; (3)Foranydivision{ti}(i=1,2,…,竹)of[口,t],by[3]stillwehave lJT(ti)一T-1)lJ i1lJAj.T(r)drlJ?lJA1ti.lJT(r)lJd?lJAlJMe{=l=0一?i:一 Thus(3)holdssinceAiSbounded. (9)Bytheproofof(2),theconclusionisobviousinlightoftheboundednessofA. Finally,if(1)holds,thenbythedefinitionofAwecaneasilyseethatAisbounded.HenceAisbounded isequivalenttocondition(1).Theproofwascompleted. In[3]and[5],theauthorgivesthefollowingfacts:Let{S(t)},beaCO— semigroupwithgeneratorA onX.Ifforeacht>0,S(t),existsandisbounded,then{S(,)一}彦 0isac0-semigroupwithgenerator — AonX.Nowwedefine G?: ,; then{G(,)},?RisaCO—groupwithgeneratorAonX.Thus,forCO—group{G(,)} ,ERanditsgeneratorA,we obtaintheconclusionwhichiSsimilartoProlx~ition5. [1]TRAVISCC.Differentiabilityofweaksolutionstoanabstractinhomogeneousdifferentialequation[J].ProeAmerMath Soe.1981.(82):425--430. [2]HONIGCS.Volterrastidtjes-integralequations[M].Amsterdan:North-HollandPublComp,1975. [3]PAZYA.Semigroupsoflinearoperatorsandapplicationstopartialdifferetialequations[M].NewYork:Springer-Verlag. 1983. [4]CHYANDK,SHAWSY,PISKAREVS.Onmaximalregularityandsemivariationofcosineoperatorfunctions[J].JLon. donMathSoc.1999.59(2):1023—1032. [5]GOLD6TEINJA.Semigroupsoflinearoperatorsandapplications[M].Oxford:OxfordUniversityPress,1985. [6]郎开禄.杨光俊.a一次积分半群的Trotter-Kato定理与完全二阶Cauchy问 题 快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题 (英)[J].云南大学(自然科学版). 2O00.22(5):321—324. 有界半变差和有界变差C0一群 贾云锋 (陕西师范大学数学与信息科学学院,陕西西安71O062) 摘要:通过引入有界半变差和有界变差co一群的概念对co一群进行了系列刻画;同时在co,群的无 穷小生成元为有界时给出了生成元与有界半变差和有界变差co一群一些性质之间的关系. 关键词:有界半变差;有界变差;c0一群;无穷小生成元 ?作者简介:贾云锋(1972一),男,陕西人,博士生,主要从事泛函微分方程方面的研究.