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SINGULAR INTEGRAL EQUATIONS ON THE REAL AXIS WITH SOLUTIONS HAVING SINGULARITIES OF HIGHER ORDER

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SINGULAR INTEGRAL EQUATIONS ON THE REAL AXIS WITH SOLUTIONS HAVING SINGULARITIES OF HIGHER ORDER

SINGULAR INTEGRAL EQUATIONS ON

    THE REAL AXIS WITH SOLUTIONS

    HAVING SINGULARITIES OF HIGHER

    ORDER

    Availableonlineat1^,\^n^f.sciencedirect.com

    

    静秽ScienceDirect

    ActaMathematicaScientia2010,30B(4):10931099

    数学物理

    http://actams.wipm.ac.cn

    SINGULARINTEGRALEQUATIONS0NTHE

    REALAXISWITHSoLUT10NSHAVING

    SINGULARITIESoFHIGHER0RDER

    ZhongShouguo(钟寿国)

    SchoolofMathematicsandStatistics,WuhanUniversity,Wuhan430072,China AbstractWetransformthesingularintegralequationswithsolutionssimultaneously havingsingularitiesofhigherorderatinfinitepointandatseveralfinitepointsonthereal axisintoonesalongaclosedcontourwithsolutionshavingsingularitiesofhigherorder, andfortheformerobtaintheextendedNeothertheoremofcompleteequationaswellas thesolutionsandthesolvableconditionsofcharacteristicequationfromthelatter.The conclusionsdrawnbythisarticlecontainspecialcasesdiscussedbefore. KeywordsSingularintegralequation;solutionwithsingularitiesofhigherorder;real axis;infinitepoint;classHI1,.,An,A

    2000MRSubjectClassification45E05

    1Preliminaries

    SolutionclassofsingularintegralequationsinclassicaltheoryisrestrictedinorH[1,

    andisgeneralizedtoclasswithsingularitiesofhigherorderalongacurvein[26].whenintegral curveisreplacedbytherealaxisX(includingo.),thesingularintegralequationonXinspecial caseswerediscussedin79].Inthisarticle,thegeneralcase,thatis,thesingularintegral equationswithsolutionssimultaneouslyhavingsingularitiesofhigherorderatinfinitepoint andatseveralfinitepointsonX,isdiscussed.Byvirtueoffractional1ineartransformation,we reducetheequationsonXtothesingularintegralequationsalongacircumferencewithsolutions

    havingsingularitiesofhigherorder,andfortheformerobtaintheextendedNoethertheoremof completeequationaswellasthesolutionsandthesolvableconditionsofcharacteristicequation

    fromthelatter.Theconclusionsdrawnbythisarticlecontainallspecialcasesinl791. Theequationsonpossessthefollowingcharacteristics:1.Havetoconsiderthatthe ordersofsingularitiesofseveralfinitepointsinfluencetheorderatinfinitepointinthestructure

    ofsingularitiesofsolution;2.WhentheequationsonXaretransformedintotheequations alongaclosedcontourwewillabandonthoseresultsofthesolutionsandthesolvablecondtions

    forcharacteristicequationin4].Reasonliesinthattheresultsin4lareincomplicatedform,

    ReceivedDecember25,2007,revisedJanuary9,2008.SupportedbytheNNSFofChina(10471107)and

    RFDPofHigherEducationofChina(20060486001)

    1094ACTAMATHEMATICASCIENTIAVb1.30Ser.B

    wllichisn'tconvenienttosimplifyfurther.Forthisreason,weusecertainmethoddifferent fromthatinf41suchthatwemayobtainconciseunifiedresults;3.Whilethekerneldensities simultaneouslyhavethesingularitiesatinfinitepointandatseveralfinitepointsonX,we havetodiscussthepropertiesofcorrespondingCauchytypeintegralsandCauchyprincipal

    valueintegralson:4.Removeobstaclesinoperationalconversionwhenthefractionallinear transformationisused.

    

    Denotep(x):1-I(x-)wheredifferentXk?X(xk?Co),all'sarepositiveintegers, =1

    k=1,,.

    Definition1.1If(+i)+[(+i)m-1,()](?/=I(seesymbolin[1),misa

    nonnegativeinteger,then,denotef(x)?日,specially,=H.

    Definition1.2Ifx?X:X\{.1,,x,Co},f(x)=(+i)A+~oo,()/Jp(),where A=l++n,o.isanonnegativeintegerandf(x)?日一,A:max{A,,,n,o.+1.,

    then,wecall/(x)hassingularitiesoforderA1,,n,..atXl,,n,?respectivelyand denotef(x)E[-I*,Xn,

    ,,

    ?lfr

    ,J)一,,..^

    o.,

    Definition1.3If/(x)?Hand/(co)=0,then,denote/(x)?月_0.If/(x)?Hnear

    =o.and/(co)=0,then,wecal1.()?H0near=...

    Definition1.4Iff(x)?H0near=?andf(x)?HintherestonX,then,denote /(x)?.

    Lemma1.19,11Assumethat/(x)?H(misanonnegativeinteger),undert+i=

    -

    1/(x+i),andletf(x)=F(t),t?F:lt+(i/2)I=1/2.Then,wehavesymmetricderivative

    formulae

    F()()=(x+)+[(+),()](),k=0,1,,m,

    ,())=0+i)k+l[@+)F(t)J(,k=0,1,,m.

    (1.1)

    (1.2)

    So,F()(t)?H(F)f(x)?疗(),.()()hasazerooforderk+1at=?,1km,

    thatis,,()()?百_0,1?m.

    Lemma1.2Assume/(x)?一,,..,then,

    )=1

    exists,thePlemeljformulaisestablished

F():+l/~/(x)+F(x)?X

    (1.3)

    (1.4)

    andwhenz?X,F(x)?andF(x)hasazerooforderlessthan1atx=co. ProofMakingtransformation+i=-1/(~+i),+i=1/(z+i),denotingtk=

    )

    i1/(xk+i),k:1,?--,+1(when+1=oo,tn+l=i)andletting(7_)=,(),wehave F(z)=(1)A+II

    =1

    (u1F(r)dr

    ?()(7-)'F,(1.5)

    n+1

    where?()=1-I(丁一tk)k,tn--kl:i,n+1:o.+1.ByDefinition1.1andLemma1.1, =1

    

    (r)?日(r),then,therightsideof(1.5)existsunderthesenseofsingularintegralof

    No.4Zhong:SINGULARINTEGRALEQUATIONSONTHEREALAXIS1095 higherorderattl,,tn,i[1].So,(1.3)exists.LettingZ+(z)betheupper(1ower)half

    plane,_ing?z+(),__??,inotherwords,?D+:J+(j/2)J<1/2(D=

    co.\D+),__?t?F=r\{tl,?,t,i),applyingthePlemeljformula(3.6)in[10to

    (1.5),weobtain(1.4)throughcalculation.Replacingzin(1.5)byX?X,onaccountof

    similarreasonasinprovingtheexistenceof(1.3),wemightverifytheexistenceofF().By

    2TheExtendedNoetherTheorem

    Considerthecompleteequation

    (?兰+f(x)X?X

    where()=(+i)A+..()/p()?,,,,,()=(+)A+..,()/p()?

    ,,,..,0(z),k(x,?)?膏A(Seethenoteatbelow).

    ByLemma1.2,theintegraltermin(2.1)?琊,SO,theordersofbothsidesof(2.1)at Xl,,n,..areequa1.Thisshowsthatformulationoftheequation(2.1)isrationa1.By

    makingtransformation

(2.1)isreducedto

    t+i=l/(+i),7.+i=l/+i)

    ()(?)A(t)(?)+=F(?),t?r

    (2.2)

    (2.3)

    where(?)=O(t)/II(t),F(t)=F(t)/II(t)((),F(t)havesingularitiesoforder1,,,

    n+1=Aoo+lattl,,tn,tn+1:i?F,respectively).Onaccountofsymbolin[3,weshould denote(),F(t)?,,,..+l,e(t)=(),F(t)=,(),A(t)=n(),K(t,7.)=k(x,?);

    moreover,(A-1)(),()~--1)(?),(A-1)(),andallpartialderivativesoforder1ofK(t,) belongto(r).

    Lemma2?1Equation(2.1)withsolutionsin..,,..isequivalenttoequation(2.3)

    withsolutionsin

    ,,..+1.

    ProofWehavejustverifiednecessity.Sincetheaboveprocessisinvertible,thus,the

    sufficiencyisalsotrue.

    Payattentiontotheorderofsingularityofsolutionofequation(2.3)aRertakingtrans formationat-ibeing..+1,whichisincreasedby1thanthatoftheoriginalequation(2.1)

    at...

    Corollary2.1Assume0(),k(x,?)?H,then,

    0,1

    k(x,?)?疗一meanx+i)k+1+)){+i)k-1+i),x-k-2k(x,?)一一)?疗(,),k= .

    .A1.

    1096ACTAMATHEMATICASCIENTIAVo1.30Ser.B 1.Solving(2.1)inH?Solving(2.3)inH(i)

    2.Solving(2.1)in凰兮Solving(2.3)inH(onlyneari);

    3.Solving(2.1)in?Solving(2.3)inH(r).

    Proof1.isobvious.Now,weprove2..When(2.1)hassolution()=()?Ho,

    then,f(x)=I(x)?H0by(2.1)andLemma2.1.Now,correspondingsolutionof(2.3)may

bewrittenas@)=()/(+i),wheree(t)=(),(i)=(..):(..)=0.So,

    )?Hnear-i.Similarly,F(t)=F(t)/(t+i)?Hneari,thus,2.holds.Intheprocess ofproofofLemma2.1,wedidn'tusethefactthat1,?,areintegers,then,wemaytake 0<1,k:1,?,n,.o=0inLemma2.1andby2.,3.isverified. Below,alwaysassumea2()b2()?0(regulartype,b(x)=k(x,)).Forconvenienceto discuss,alsoassumea(xk)?0,k=1,?,+1.

    Theorem2.1(TheExtendedNoetherTheorem)

    1.Thesolutionclassofhomogenousequations(K)()=0and(K)()=0doesn't

    expand,thatis,

    ,,

    A..H0.Bothsolutionspacesarefinitedimensiona1. 2.lz=,wherel(1)isthedimensionofsolutionspaceof(K)():0((K)()'=O).

    3.Anecessaryandsuncientconditionforthesolvabilityof(3.1)is +

    a(x)pk(x))

    州删一一

    where{())isthecompletesystemof(K)():0in

    1,,n.

    pk(x)=p()/()k,k:

    Proof1.Under(2.2),(K)()=0inHI,,..istransformedinto(K)@)=0in l,,,..+1?By[3]thesolutionclassforthelatterdoesn'texpand,thatis,1,,,..+1

    HH?By3.ofCorollary2.1,thesolutionspaceof(K)()=0is/4o==,h,A.

    Becausethesolutionspaceof(K)(t)=0inHisfinitedimensional,thus,thesolutionspace

    of(K)()=0hasthesamedimension.Theconclusionfor(K)()=0maybeverified

    similarly.

    2.Assumethat(K)()=0isreducedto(K)(?)=0under(2.2).By1.,z(f)alsois thedimensionofsolutionspaceof(K)()0((K)():0).By[3,11:.

    3.By[3)anecessaryandsufficientconditionforthesolvabilityof(KO)(t)=F(t)in

    ?..,,..+1

    is

(1)!B(t)F(?)(?一tk)q2j(t)lA(t)n(t)1)J=1,,Z(2.5)

    where{vAt)Ifisthecompletesystemof()()=0inH.OnaccountofLemma2.1,(2.5) isstilltheconditionofsolvabilityof(2.1).Applying(2.2)to(2.5),weobtain(2.4)through

    complicatedcalculation.

    N0.4Zhong:SINGULARINTEGRALEQUATIONSONTHEREALAXIS1097

    3SolvingCharacteristicEquation

    Considerthecharacteristicequation

    wherethehypothesesarethesameasinSection2.Under(2.2),byLemma2.1,solving(3.1) in

    ,

    A,..isequivalenttosolvingthefollowingequation

    ))+q)(T)dr=

    脚?(3.2)

    in

    ,,+1.AsthementionedreasoninSection1,now,weabandontheresultsin[4

    andusecertainmethoddifferentfromthatin[4]tosolve(3.2)inwhichinterpolationisstill

    essentia1.Supposethat(3.2)hasasolution.ByCorollary2ofLemma3in[3],theintegral termin(3.2)belongtoHnear,=l,?,n+1,thus,

    ()(?%)[()/()],r0,l,?,1,1,?,-t-1.(3.3)

    MaketheHermiteinterpolationpolynomialT(t)ofdegreeA+Aoo,suchthat ()()()(tk),r=0,1,,I,k=l,,+1

    ()=e(t)T(t)/II(t)=[e(t)T(t)]/II(t)(3.4)

    ByCorollary1ofLemma3in[3],()?Hneartk,k=l,,n+1.Substituting(3.4)into

    (3.2),(3.2)isrewrittenasfollows

    A(t)w(t)+B(t)

    7rl

    fW(T)dr

    .

    jF(t)A(t)T(t)?H(t)F(t)(3.5)

    (inwhichbytheExtendedResidueTheorem[1].(7_)/[?(7.)t)]dv=0).By(3.3)and Corollary1ofLemma3in[3],r(t)?Hneartk,=1,-,n+1.So,(3.5)becomesthe characteristicequationinH.Byclassicaltheory[1j

    (1)When0,(3.5)isalwayssolvablewiththegeneralsolution

    ()=KFB()z()l(),(3.6)

    where=Indr{[A(t)B()/(?)+B()])=Indx{[a(x)6()/0()+6()]),1isan arbitrarypolynomialofdegree1,KF=A()F(?)B()()(7ri).F()/[z(7_)(7_ t)]dr,A()=A(t)/[A()B(t)],B(?)=B(t)/[A(t)B(?),z(t)=A(t)+B(t)](t)= [A(t)B(?)](?),()(F)isacanonicalfunctionof(3.5)in[1]. By(3.4)(3.6),thegeneralsolutionof(3.2)is e(t)=KFB()z(t)1(t)

    Byvirtueoftheidentities

    B(t)B(t)T(t).B()z(t)

    II(t1'7ri(3.7)

    X+()]+X(t)]=2A(t)/Z(t),[X+(?)]一一[X()】一=2B(t)/Z(t)(3.8)

    

    

    A一一

    ,

    1098ACTAMATHEMATICASCIENTIA

    andtheExtendedResidueTheorem[1J7wehave 厂二!!:

    7riz(7-)?()(7-t)27ri./r+(7_)?(7_)t)

    )

    

    !(t).11a()]

    2X+()?()2(Ak1)!【丁A(7-)+(7-)(7-t)j

    

    +

三薹[筹丽两

    -

    p.p.

    (,?)}

    )皇22.:laB(T)F(T)1

    Z(t)II(t).()1)!lA(r)YIk(r)Z(r)(r一?ij

    (,..),9,

    whereYIk(t):II(t)/(t)tk),=1,,n+l,P?P?()istheprincipalpartof()/[(t)?(t) at??Itisapolynomialofdegree1andmaybemergedinto

    1(0in(3.7).Theterm

    aftermergenceisstilldenotedby)

    1()or1(@+i))forconveniencetosimplifylatter. Substituting(3.9)into(3.7),weobtain n刚喜[

    

    P.-l(-c,.

    Applyingtheinversetransformationof(2.2)to(3.10),throughcomplicatedcalculatiol1s,we

    obtainthegeneralsolutionof(3.1)?Lsfollows

    ,=K*f-b*?,nak-11

    k=lr=O['L?山,\,\J=七山一山

    +

    (c)?r

    ++...+,cs.

    whereK,(),())b()z()(i)J,()/(?)(?))]d?,.()=o()/[0()b2()], 6()=b(x)/[a.()b2(),(z)=0()+6(z)]y+():f0()6f)1y(),yf):er(z)f foz?Z+),y(2)=[(+i)/(zi)]er()(forz?Z),r(z)=(2i),log{[(+i)/(x

    i)[0())6(z)]/()6())()dx,,cI,,carearbitraryconstants. (2)When0,similart.0,if(3.1)i88.Ivable,then(2.1).nlyhastheunique

    .u0n(3.11)+)appears)and(3?1)iss.Ivableifandonlyif

/):0,:0,1,,一一1,(3-l2)

    No.4Zhong:SINGULARINTEGRALEQUATIONSONTHEREALAXIS1099 or,equivalently

    (i)F()(+i)/z()d=.,_7=.,1,?,一一l(3.13)

    By(3.5),(3.8),andtheExtendedResidueTheorem[1].(3.13)isrewrittenas 1F()(?+i)Jdt

    irz(t)

    =

    :

    r?,J..,_1_(3.14)A(t)Z(t)g(1)!k(t)J'

    (notingRes{T(t)(t+i)J/[X()?()],?)=0when0J一一1).Applying(2.2)to(3.14),

    throughcomplicatedcalculations,weobtaintheconditionofsolvabilityof(3.1)when<0,

    +!

    f(x)dx

    ()(+i)J

    11..+

    (A..)c+o.+[-=l]..):..,J=,?,.c3.5

    Theorem3.1When0thecharacteristicequation(3.1)withsolutionsin,..,,..

    isalwayssolvablewiththegeneralsolution(3.11)containing,carbtraryconstants;when<

    0

    (3.1)hastheuniquesolution(3.11)(?cd(x+i)Jdisappears)ifandonlyif(3.15)holds. J=1

    Remark3.1[7-9]arethespecialcasesinthisarticleandtheirresultsmaybeobtained

    fromthisarticle(thecheckisomitted).

    [8]

    9

    [1O]

    lI

    References

    LuJianke.Boundaryvalueproblemsforanalyticfunctions.Singapore:WorldScientific,199

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