ON A MULTIPOINT BOUNDARY VALUE PROBLEM FOR LINEAR

ARCHIVUMMATHEMATICUM(BRNO)Tomus30(1994),171{206ONAMULTIPOINTBOUNDARYVALUEPROBLEMFORLINEARORDINARYDIFFERENTIALEQUATIONSWITHSINGULARITIESG.D.TskhovrebadzeAbstract.AcriterionfortheuniquesolvabilityofandsucientconditionsforthecorrectnessofthemodiedVallee-Poussinproblemareestablishedforthelinearordinarydierentialequationswithsingularities.IntroductionThispaperisdevotedtotheinvestigationofacertainmodicationoftheVallee-Poussin’sboundaryvalueproblem,anditseemsnaturaltoexplaininrstplacewhichmodicationismeantandwhichfactorshaveledtoit.Letusconsiderthelinearordinarydierentialequation(1)u(n)=lXk=1pk(t)u(k1)+q(t);wheren2isanaturalnumber,p1;:::;pl;qarecontinuousfunctionsonthesegment[a;b].Letm2f2;:::;ng,ni2f1;:::;n1g(i=1;:::;m),mPi=1ni=n,1a=t1tm=b+1.Asiswell-known,theclassicalVallee-Poussin’sboundaryvalueproblemisfor-mulatedasfollows:Findasolutionofthedierentialequation(1)satisfyingtheconditions(21)u(k1)(ti)=0(k=1;:::;ni;i=1;:::;m):Thesolution,naturally,issoughtforintheclassofn-timescontinuouslydieren-tiablefunctionsonthesegment[a;b].1991MathematicsSubjectClassication:34B15.Keywordsandphrases:linearordinarydierentialequationwithsingularities,modiedVallee-Poussinproblem,uniquesolvability,correctness.ReceivedJanuary21,1993.172G.D.TSKHOVREBADZETherearealotofworksdevotedtotheinvestigationoftheVallee-Poussin’sboundaryvalueprobleminthisclassicalformulation(see,forexample,[1]andReferencesfrom[3]).Thisproblemhasalsobeenstudiedwithsucientthoroughnessinthecasewhenthecoecientsoftheequation(1)havesingularitiesatthepointst1;:::;tm(see,forexample,[2,3,5]).However,inallworksdevotedtothestudyoftheVallee-Poussin’sproblemitisassumedthat(*)Zba(ta)nn11(bt)nnm1mYi=1jttijnikjpk(t)jdt+1(k=1;:::;l);wherenik=nik+1forkni0forkni;andZba(ta)nn11(bt)nnm1jq(t)jdt+1:Thisassumptionisnotcasual.Thematteristhatiffunctionspk(k=1;:::;l)havesingularitiesofordernn1+n1kandnnm+nmkatthepointsaandb,respectively(inparticular,thefunctionp1hassingularitiesofordernatthepointsaandb),thenProblem(1),(21)isnot,generallyspeaking,uniquelysolvableeveninthesimplestcase.Forexample,givenboundaryconditions(21),theequationu(n)=(1)n1(ta)nuhasaninnitenumberofsolutionsforn1=1andsucientlysmall0.Therefore,toprovidethesolutionuniqueness,wehavetointroduceanaddi-tionaland,ofcourse,naturalconditionsuchas,forexample,(22)sup(ta)l11(bt)l12ju(l1)(t)j:atb+1;where12]n11,n1[,22]nm1,nm[.Thisconditionisnaturalbecauseifthecondition(*)isfullled,than(21),yields(22),i.e.Problem(1),(21),(22)coincideswiththeVallee-Poussin’sprob-lem.However,ifthecondition(*)isnotfullled,than,asfollowsfromtheaboveexample,thisisnotso.Problem(1),(21),(22)isthegeneralizationoftheVallee-Poussin’sboundaryvalueproblemandhasnotyetbeenstudiedwithsucientcompleteness.Hereanattemptismadetollupsomehowthisgap.Inparticular,theconditionsareestab-lished,guaranteeingProblem(1),(21),(22)tobeFredholmiananditssolutiontobestablewithrespecttointegrallysmallperturbationsofthecoecientsofequa-tion(1).Itisassumedthatthefunctionspk:Im!R(k=1;:::;l),q:]a;b[!RbelocallyintegrableonImand]a;b[,respectively,whereIm=[a;b]nft1;:::;tmg,andONAMULTIPOINTB.V.P.FORL.O.D.E.WITHSINGULARITIES173thecondition(*)isnotfullled.NotethatthesolutionofProblem(1),(21),(22)issoughtforintheclassoffunctionsu:]a;b[!Rabsolutelycontinuoustogetherwithu(k)(k=1;:::;n1)inside]a;b[.1)Thefollowingnotationwillbeusedthroughoutthispaper:k;1;2(t)=(ta)1k+1(bt)2k+1m1Yi=2jttijnik;1)k;n(t)=(ta)nk(bt)nkm1Yi=2jttijnik;kl()=j(k1):::(l2)j(k=1;:::;l1);ll()=1;L([a;b];R)isasetofLebesgue-integrablefunctionsg:[a;b]!R;Lloc(]a;b[;R)isasetoffunctionsg:]a;b[!RwhichareLebesgue-integrableinside]a;b[;Ln11;n12(]a;b[;R)isasetofmeasurablefunctionsg:]a;b[!Rsuchthatjg()jn11;n12==sup(ta)n11(bt)n12Zta+b2g()d:atb+1:1.LemmasonAPrioriEstimatesInthissectionweconsiderProblemu(n)=lXk=1pk(t)u(k1)+q(t);(1)u(k1)(ti)=0(k=1;:::;ni;i=1;:::;m);(21)sup(ta)l11(bt)l12ju(l1)(t)j:atb+1;(22)where12]n11;n1[,22]nm1,nm[,q2Ln11;n12(]a;b[;R)andpk:]a;b[!R(k=1;:::;l)aremeasurablefunctionssatisfyinginequalities(3)p1k(t)pk(t)p2k(t)foratb(k=1;:::;l):Onimposingcertainrestrictionsonthevectorfunction(p11;:::;p1l;p21;:::;p2l),weobtainanaprioriestimateofthesolutionofProblem(1),(21),(22)whichisuniquefortheconsideredsetofcoecients.Beforeformulatingthemainlemma,somedenitionswillbegiven.1)i.e.,oneachsegmentcontainedin]a;b[.2)inthecasem=2Qm1i=2jttijnikdenotesunity.174G.D.TSKHOVREBADZEDenition1.Letn02f1;:::;n1gand2]n01;n0[.Thevectorfunction(h1;:::;hl)withmeasurablecomponentshk:]a;b[!R(k=1;:::;l)willbesaidtobelongtothesetS+(a;b;n;n0;)8:S(a;b;n;n0;)9;ifthereexists2]a;b[suchthatwehavetheinequalitylimt!asup(ta)l1(nl)!lXk=11kl()Zt(t)nl(a)k+1jhk()jd18:limt!bsup(bt)l1(nl)!lXk=11kl()Zt(t)nl(b)k+1jhk()jd19;inthecasel2fn0+1;:::;ngandtheinequalitylimt!asup(ta)l1(nn01)!(n0l)!lXk=11kl()Zta(ts)n0lZs(s)nn01(a)k+1jhk()jdds1