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ARCHIVUMMATHEMATICUM(BRNO)Tomus30(1994),171{206ONAMULTIPOINTBOUNDARYVALUEPROBLEMFORLINEARORDINARYDIFFERENTIALEQUATIONSWITHSINGULARITIESG.D.TskhovrebadzeAbstract.Acriterionfortheuniquesolvabilityofandsu cientconditionsforthecorrectnessofthemodi edVall ee-Poussinproblemareestablishedforthelinearordinarydi erentialequationswithsingularities.IntroductionThispaperisdevotedtotheinvestigationofacertainmodi cationoftheVall ee-Poussin’sboundaryvalueproblem,anditseemsnaturaltoexplainin rstplacewhichmodi cationismeantandwhichfactorshaveledtoit.Letusconsiderthelinearordinarydi erentialequation(1)u(n)=lXk=1pk(t)u(k 1)+q(t);wheren 2isanaturalnumber,p1;:::;pl;qarecontinuousfunctionsonthesegment[a;b].Letm2f2;:::;ng,ni2f1;:::;n 1g(i=1;:::;m),mPi=1ni=n, 1a=t1 tm=b+1.Asiswell-known,theclassicalVall ee-Poussin’sboundaryvalueproblemisfor-mulatedasfollows:Findasolutionofthedi erentialequation(1)satisfyingtheconditions(21)u(k 1)(ti)=0(k=1;:::;ni;i=1;:::;m):Thesolution,naturally,issoughtforintheclassofn-timescontinuouslydi eren-tiablefunctionsonthesegment[a;b].1991MathematicsSubjectClassi cation:34B15.Keywordsandphrases:linearordinarydi erentialequationwithsingularities,modi edVall ee-Poussinproblem,uniquesolvability,correctness.ReceivedJanuary21,1993.172G.D.TSKHOVREBADZETherearealotofworksdevotedtotheinvestigationoftheVall ee-Poussin’sboundaryvalueprobleminthisclassicalformulation(see,forexample,[1]andReferencesfrom[3]).Thisproblemhasalsobeenstudiedwithsu cientthoroughnessinthecasewhenthecoe cientsoftheequation(1)havesingularitiesatthepointst1;:::;tm(see,forexample,[2,3,5]).However,inallworksdevotedtothestudyoftheVall ee-Poussin’sproblemitisassumedthat(*)Zba(t a)n n1 1(b t)n nm 1mYi=1jt tijnikjpk(t)jdt+1(k=1;:::;l);wherenik= ni k+1fork ni0forkni;andZba(t a)n n1 1(b t)n nm 1jq(t)jdt+1:Thisassumptionisnotcasual.Thematteristhatiffunctionspk(k=1;:::;l)havesingularitiesofordern n1+n1kandn nm+nmkatthepointsaandb,respectively(inparticular,thefunctionp1hassingularitiesofordernatthepointsaandb),thenProblem(1),(21)isnot,generallyspeaking,uniquelysolvableeveninthesimplestcase.Forexample,givenboundaryconditions(21),theequationu(n)=( 1)n 1 (t a)nuhasanin nitenumberofsolutionsforn1=1andsu cientlysmall 0.Therefore,toprovidethesolutionuniqueness,wehavetointroduceanaddi-tionaland,ofcourse,naturalconditionsuchas,forexample,(22)sup (t a)l 1 1(b t)l 1 2ju(l 1)(t)j:atb +1;where 12]n1 1,n1[, 22]nm 1,nm[.Thisconditionisnaturalbecauseifthecondition(*)isful lled,than(21),yields(22),i.e.Problem(1),(21),(22)coincideswiththeVall ee-Poussin’sprob-lem.However,ifthecondition(*)isnotful lled,than,asfollowsfromtheaboveexample,thisisnotso.Problem(1),(21),(22)isthegeneralizationoftheVall ee-Poussin’sboundaryvalueproblemandhasnotyetbeenstudiedwithsu cientcompleteness.Hereanattemptismadeto llupsomehowthisgap.Inparticular,theconditionsareestab-lished,guaranteeingProblem(1),(21),(22)tobeFredholmiananditssolutiontobestablewithrespecttointegrallysmallperturbationsofthecoe cientsofequa-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(*)isnotful lled.NotethatthesolutionofProblem(1),(21),(22)issoughtforintheclassoffunctionsu:]a;b[!Rabsolutelycontinuoustogetherwithu(k)(k=1;:::;n 1)inside]a;b[.1)Thefollowingnotationwillbeusedthroughoutthispaper: k; 1; 2(t)=(t a) 1 k+1(b t) 2 k+1m 1Yi=2jt tijnik;1) k;n(t)=(t a)n k(b t)n km 1Yi=2jt tijnik; kl( )=j(k 1 ):::(l 2 )j(k=1;:::;l 1); ll( )=1;L([a;b];R)isasetofLebesgue-integrablefunctionsg:[a;b]!R;Lloc(]a;b[;R)isasetoffunctionsg:]a;b[!RwhichareLebesgue-integrableinside]a;b[;Ln 1 1;n 1 2(]a;b[;R)isasetofmeasurablefunctionsg:]a;b[!Rsuchthatjg( )jn 1 1;n 1 2==sup (t a)n 1 1(b t)n 1 2 Zta+b2g( )d :atb +1:1.LemmasonAPrioriEstimatesInthissectionweconsiderProblemu(n)=lXk=1pk(t)u(k 1)+q(t);(1)u(k 1)(ti)=0(k=1;:::;ni;i=1;:::;m);(21)sup (t a)l 1 1(b t)l 1 2ju(l 1)(t)j:atb +1;(22)where 12]n1 1;n1[, 22]nm 1,nm[,q2Ln 1 1;n 1 2(]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)whichisuniquefortheconsideredsetofcoe cients.Beforeformulatingthemainlemma,somede nitionswillbegiven.1)i.e.,oneachsegmentcontainedin]a;b[.2)inthecasem=2Qm 1i=2jt tijnikdenotesunity.174G.D.TSKHOVREBADZEDe nition1.Letn02f1;:::;n 1gand 2]n0 1;n0[.Thevectorfunction(h1;:::;hl)withmeasurablecomponentshk:]a;b[!R(k=1;:::;l)willbesaidtobelongtothesetS+(a;b;n;n0; )8:S (a;b;n;n0; )9;ifthereexists 2]a;b[suchthatwehavetheinequalitylimt!asup(t a)l 1 (n l)!lXk=11 kl( )Z t( t)n l( a) k+1jhk( )jd 18:limt!bsup(b t)l 1 (n l)!lXk=11 kl( )Zt (t )n l(b ) k+1jhk( )jd 19;inthecasel2fn0+1;:::;ngandtheinequalitylimt!asup(t a)l 1 (n n0 1)!(n0 l)! lXk=11 kl( )Zta(t s)n0 lZ s( s)n n0 1( a) k+1jhk( )jd ds1