二階線性微分方程英文翻譯

第第頁(yè)二階線性微分方程英文翻譯

SomePropertiesofSolutionsofPeriodicSecondOrder

LinearDifferentialEquations

1.Introductionandmainresults

Inthispaper,weshallassumethatthereaderisfamiliarwiththefundamentalresultsandthestardardnotationsoftheNevanlinna'svaluedistributiontheoryofmeromorphicfunctions[12,14,

(f)and(f)todenoterespectivelytheorder16].Inaddition,wewillusethenotation(f)，

ofgrowth,thelowerorderofgrowthandthee*ponentofconvergenceofthezerosofameromorphicfunctionf，e(f)〔[see8]〕，thee-typeorderoff(z),isdefinedtobe

e(f)limlogT(r,f)rr

Similarly,e(f)，thee-typee*ponentofconvergenceofthezerosofmeromorphicfunctionf,isdefinedtobe

logN(r,1/f)e(f)limrr

Wesaythatf(z)hasregularorderofgrowthifameromorphicfunctionf(z)satisfies

(f)limlogT(r,f)rlogr

Weconsiderthesecondorderlineardifferentialequation

fAf0

WhereA(z)B(ez)isaperiodicentirefunctionwithperiod2i/.Thecomple*oscillationtheoryof(1.1)wasfirstinvestigatedbyBankandLaine[6].Studiesconcerning(1.1)haveeencarriedonandvariousoscillationtheoremshavebeenobtained[2{11,13,17{19].WhenA(z)isrationaline，BankandLaine[6]provedthefollowingtheorem

TheoremALetA(z)B(ez)beaperiodicentirefunctionwithperiod2i/andrationalinezz.IfB()haspolesofoddorderatbothand0,thenforeverysolutionf(z)(0)of(1.1),(f)

Bank[5]generalizedthisresult:Theaboveconclusionstillholdsifwejustsupposethatbothand0arepolesofB(),andatleastoneisofoddorder.Inaddition,thestrongerconclusion

logN(r,1/f)o(r)(1.2)

holds.WhenA(z)istranscendentaline,Gao[10]provedthefollowingtheorem

TheoremBLetB()g(1/)zpjb,whereg(t)isatranscendentalentirefunctionjj1

zwith(g)1,pisanoddpositiveintegerandbp0，LetA(z)B(e).Thenany

non-triviasolutionfof(1.1)musthave(f).Infact,thestrongerconclusion(1.2)holds.

Ane*amplewasgivenin[10]showingthatTheoremBdoesnotholdwhen(g)isanypositiveinteger.Iftheorder(g)1,butisnotapositiveinteger,whatcanwesay?ChiangandGao[8]obtainedthefollowingtheorems

zTheoremCLetA(z)B(e),whereB()g1(1/)g2(),g1andg2areentire

functionsg2transcendentaland(g2)notequaltoapositiveintegerorinfinity,andg1arbitrary.(i)(g2)1.(a)Iffisanon-trivialsolutionof(1.1)withe(f)(g2);

thenf(z)andf(z2i)arelinearlydependent.(b)Iff1andf2areanytwolinearlySuppose

independentsolutionsof(1.1),then

(ii)Supposee(f)(g2).(g2)1(a)Iffisanon-trivialsolutionof(1.1)

withe(f)1,f(z)andf(z2i)arelinearlydependent.Iff1andf2areanytwolinearlyindependentsolutionsof(1.1),thene(f1f2)1.

TheoremDLetg()beatranscendentalentirefunctionanditsorderbenotapositiveintegerorinfinity.LetA(z)B(ez);whereB()g(1/)jbj1jandpisanoddpositivep

integer.Then(f)oreachnon-trivialsolutionfto(1.1).Infact,thestrongerconclusion(1.2)holds.

E*ampleswerealsogivenin[8]showingthatTheoremDisnolongervalidwhen(g)isinfinity.

ThemainpurposeofthispaperistoimproveaboveresultsinthecasewhenB()istranscendental.Specially,wefindaconditionunderwhichTheoremDstillholdsinthecasewhen(g)isapositiveintegerorinfinity.WewillprovethefollowingresultsinSection3.

Theorem1LetA(z)B(e),whereB()g1(1/)g2(),g1andg2areentirefunctionswithg2transcendentalandz(g2)notequaltoapositiveintegerorinfinity,andg1arbitrary.IfSomepropertiesofsolutionsofperiodicsecondorderlineardifferentialequationsf(z)andf(z2i)aretwolinearlyindependentsolutionsof(1.1),then

e(f)

Or

e(f)1(g2)12

WeremarkthattheconclusionofTheorem1remainsvalidifweassume(g1)

isnotequaltoapositiveintegerorinfinity,andg2arbitraryandstillassumeB()g1(1/)g2(),Inthecasewheng1istranscendentalwithitslowerordernotequaltoanintegerorinfinityandg2isarbitrary,weneedonlytoconsiderB*()B(1/)g1()g2(1/)in0,1/.

Corollary1LetA(z)B(ez),whereB()g1(1/)g2(),g1andg2are

entirefunctionswithg2transcendentaland

(a)

(b)(g2)nomorethan1/2,andg1arbitrary.Iffisanon-trivialsolutionof(1.1)withe(f),thenf(z)andf(z2i)arelinearlydependent.Iff1andf2areanytwolinearlyindependentsolutionsof(1.1),

thene(f1f2).

Theorem2Letg()beatranscendentalentirefunctionanditslowerorderbenomorethan1/2.

zLetA(z)B(e),whereB()g(1/)j1bj

ppjandpisanoddpositiveinteger,then(f)foreachnon-trivialsolutionfto(1.1).Infact,thestrongerconclusion(1.2)holds.Weremarkthattheaboveconclusionremainsvalidif

B()g()bjj

j1

WenotethatTheorem2generalizesTheoremDwhen(g)isapositiveintegerorinfinitybut(g)1/2.CombiningTheoremDwithTheorem2,wehave

Corollary2Letg()beatranscendentalentirefunction.LetA(z)B(ez)whereB()g(1/)j1bjjandpisanoddpositiveinteger.Supposethateither(i)or(ii)belowholds:

(i)(g)isnotapositiveintegerorinfinity;

(ii)(g)1/2;

then(f)foreachnon-trivialsolutionfto(1.1).Infact,thestrongerconclusion(1.2)holds.

2.LemmasfortheproofsofTheorems

Lemma1([7])Supposethatk2andthatA0,Ak2areentirefunctionsofperiod2i,andthatfisanon-trivialsolutionofp

y(k)Aj(z)y(j)(z)0

i0k2

SupposefurtherthatfsatisfieslogN(r,1/f)o(r);thatA0isnon-constantandrational

zine，andthatifk3，thenA1,Ak2areconstants.Thentheree*istsanintegerqwith1qksuchthatf(z)andf(zq2i)arelinearlydependent.Thesameconclusion

zholdsifA0istranscendentaline，andfsatisfieslogN(r,1/f)o(r)，andifk3，then

throughasetL1r

k2.haveT(r,Aj)o(T(r,Aj))forj1,as

z

zofinfinitemeasure,1weandbeLemma2([10])LetA(z)B(e)beaperiodicentirefunctionwithperiod2itranscendentaline,B()istranscendentalandanalyticon0.IfB()hasapoleof

oddorderator0(includingthosewhichcanbechangedintothiscasebyvaryingthe

periodofA(z)andEq.(1.1)hasasolutionf(z)0whichsatisfieslogN(r,1/f)o(r),

thenf(z)andf(z)arelinearlyindependent.

3.Proofsofmainresults

Theproofofmainresultsarebasedon[8]and[15].

ProofofTheorem1Letusassumee(f).Sincef(z)andf(z2i)arelinearlyindependent,Lemma1impliesthatf(z)andf(z4i)mustbelinearlydependent.LetE(z)f(z)f(z2i),ThenE(z)satisfiesthedifferentialequation

E(z)2E(z)c2

,(2.1)4A(z)()22E(z)E(z)E(z)

Wherec0istheWronskianoff1andf2(see[12,p.5]or[1,p.354]),andE(z2i)c1E(z)orsomenon-zeroconstantc1.Clearly,E/E

andE/Earebothperiodicfunctionswithperiod2i,whileA(z)isperiodicbydefinition.

2Hence(2.1)showsthatE(z)isalsoperiodicwithperiod2i.Thuswecanfindananalytic

function()in0

yields,sothatE(z)2(ez)Substitutingthise*pressioninto(2.1)c234B()2()22(2.2)4

SincebothB()and()areanalyticinC*:1,theValirontheory[21,p.15]givestheirrepresentationsas

nB()R()b()，()n1R1()()，〔2.3〕

n1aresomeintegers,R()andR1()arefunctionsthatareanalyticandnon-vanishingwheren，

onC*{}，b()and()areentirefunctions.Followingthesameargumentsasusedin[8],wehave

T(,)N(,1/)T(,b)S(,)，〔2.4〕

whereS(,)o(T(,)).Furthermore,thefollowingpropertieshold[8]

e(f)e(E)e(E2)ma*{eR(E2),eL(E2)},

eR(E2)1()(),

WhereeR(E2)(resp,eL(E2))isdefinedtobe

logNR(r,1/E2)logNR(r,1/E2)lim(resp,lim),rrrr

Somepropertiesofsolutionsofperiodicsecondorderlineardifferentialequations

)(resp.NL(r,1/E2)denotesacountingfunctionthatonlycountsthezeros

2ofE(z)intheright-halfplane(resp.intheleft-halfplane),1()isthee*ponentofconvergenceofthezerosofinC*,whichisdefinedtobe

logN(,1/)1()limlog

Recalltheconditione(f),weobtain().whereNR(r,1/E

Nowsubstituting(2.3)into(2.2)yields2

n1R132n1R12c2

4R()b()n1()()R14R1R1()()

R1n1R1n1R1R12n1(n11)222)〔2.5〕(2R1R1R1n

ProofofCorollary1WecaneasilydeduceCorollary1(a)fromTheorem1.

ProofofCorollary1(b).Supposef1andf2arelinearlyindependentande(f1f2),thene(f1),and

Corollary1(a)that

Letfj(z)ande(f2).Wededucefromtheconclusionoffj(z2i)arelinearlydependent,j=1;2.E(z)f1(z)f2(z).Thenwecanfindanon-zeroconstantc2suchthatE(z2i)c2E(z).RepeatingthesameargumentsasusedinTheorem1byusingthefactthatE(z)2isalsoperiodic,weobtain

e(E)1(g2)12,acontradictionsince(g2)1/2.Hencee(f1f2).

ProofofTheorem2Supposetheree*istsanon-trivialsolutionfof(1.1)thatsatisfieslogN(r,1/f)o(r).Wededucee(f)0,sof(z)andf(z2i)arelinearlydependentbyCorollary1(a).However,Lemma2impliesthatf(z)andf(z2i)arelinearly

independent.Thisisacontradiction.HencelogN(r,1/f)o(r)holdsforeachnon-trivial

solutionfof(1.1).ThiscompletestheproofofTheorem2.

AcknowledgmentsTheauthorswouldliketothanktherefereesforhelpfulsuggestionstoimprovethispaper.

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一些周期性的二階線性微分方程解的方法

1．簡(jiǎn)介和主要成果

在本文中，我們假設(shè)讀者熟識(shí)的函數(shù)的數(shù)值分布理論[12，14，16]的基本成果和數(shù)學(xué)符號(hào)。此外，我們將運(yùn)用的符號(hào)(f)，(f)and(f)，表示的順次分別增長(zhǎng)，低增長(zhǎng)的一個(gè)純函數(shù)的零點(diǎn)收斂指數(shù)，f，e(f)〔[8]〕，E型的f(z)，被定義為

e(f)limlogT(r,f)rr

同樣，e(f)，E型的亞純函數(shù)f的零點(diǎn)收斂指數(shù)，被定義為

logN(r,1/f)e(f)limrr

我們說(shuō)，假如一個(gè)亞純函數(shù)f(z)滿意增長(zhǎng)的正常秩序

logT(r,f)(f)limrlogr

我們考慮的二階線性微分方程

fAf0

在A(z)B(ez)是一個(gè)整函數(shù)在2i/。在〔1.1〕的反復(fù)波動(dòng)理論的第一次探討中由銀行和萊恩[6]。已經(jīng)進(jìn)行了討論在〔1.1〕中，并已取得各種波動(dòng)定理在[2{11，13，17{19]。在e函數(shù)中A(z)正確的，銀行和萊恩[6]證明白如下定理

定理A設(shè)A(z)B(ez)這函數(shù)是一個(gè)周期性函數(shù)，周期為2i/在整個(gè)函數(shù)e存zz在。假如B()有奇數(shù)階極點(diǎn)在和0，然后對(duì)于任何一個(gè)結(jié)果答案f(z)(0)在(1.1)中(f)

廣義這樣的結(jié)果：上述結(jié)論仍舊認(rèn)為，假如我們只是假設(shè)，既0和B()的極點(diǎn)，并且至少有一個(gè)是奇數(shù)階。此外，較強(qiáng)的結(jié)論

log

zN(r,1/f)o(r)〔1.2〕認(rèn)為。當(dāng)A(z)是超越在e，高[10]證明白如下定理

定理B

設(shè)B()g(1/)pjb，其中g(shù)(t)是一個(gè)超越整函數(shù)與(g)1，p是奇正整jj1

并且bp0，設(shè)A(z)B(e)，那么任何微分解在〔1.1〕的函數(shù)f需要有(f)。事實(shí)上，在〔1.2〕已經(jīng)有證明的結(jié)論。

是在[10]一個(gè)例子說(shuō)明當(dāng)定理B不成立時(shí)，(g)是任意正整數(shù)。假如在另一方面z(g)1，但假如沒(méi)有一個(gè)正整數(shù)，我們可以說(shuō)些什么呢？蔣和高[8]得到以下定理定理C

設(shè)A(z)B(e)，其中B()g1(1/)g2()，函數(shù)g1和函數(shù)g2是整函數(shù)g2先驗(yàn)和(g2)不等于一個(gè)正整數(shù)或無(wú)窮大，并函數(shù)g1任意。

〔一〕假設(shè)(g2)1〔a〕假如函數(shù)f是一個(gè)非平凡解e(f)(g2)在〔1.1〕，那么f(z)

和f(z2i)是線性相關(guān)。

〔b〕假如函數(shù)f1和函數(shù)f2在〔1.1〕是兩個(gè)線性無(wú)關(guān)函數(shù)，那么存在這樣一個(gè)條件ze(f)(g2)。

〔二〕假設(shè)(g2)1〔a〕假如函數(shù)f有一個(gè)非平凡解在〔1.1〕且e(f)1，f(z)和

f(z2i)是線性相關(guān)的。

假如函數(shù)f1和函數(shù)f2在〔1.1〕在〔1.1〕是兩個(gè)線性無(wú)關(guān)函數(shù)，那么存在這樣一個(gè)條件e(f1f2)1。

定理D

讓g()是一個(gè)超越整函數(shù)和它的秩序是正整數(shù)或無(wú)窮大。設(shè)A(z)B(ez)，B()g(1/)j1bjj和p是一個(gè)奇正整數(shù)。然后(f)或F得到每一個(gè)非平凡解在〔1.1〕。事實(shí)上，在〔1.2〕中已經(jīng)有證明的結(jié)論。

例子說(shuō)明在高[8]定理D不再成立，當(dāng)(g)是無(wú)窮的。

本文的主要目的是改善上述結(jié)果的狀況下，當(dāng)B()是超越。特別地，我們找到的條件下定理D仍舊成立的狀況下，當(dāng)(g)是一個(gè)正整數(shù)或無(wú)窮大。

我們將證明在第3節(jié)的結(jié)果如下：

定理1

設(shè)A(z)B(e)，其中B()g1(1/)g2()，g1和g2g2先驗(yàn)和(g2)不等于一個(gè)正整數(shù)或無(wú)窮，g1任意整函數(shù)。假如定期二階線性微分方程f(z)和f(z2i)的解不是一些屬性是兩個(gè)線性無(wú)關(guān)的解在〔1.1〕，然后zp

e(f)

或者

e(f)1(g2)12

我們的說(shuō)法，定理1的結(jié)論仍舊有效，假如我們假設(shè)函數(shù)(g1)不等于一個(gè)正整數(shù)或無(wú)窮大，任意和承受的狀況下B()g1(1/)g2()，當(dāng)其低階不等于一個(gè)整數(shù)或無(wú)窮超然是任意的，我們只需要考慮B*()B(1/)g1()g2(1/)在0，1/。推論1

設(shè)A(z)B(e)，其中B()g1(1/)g2()，函數(shù)g1和函數(shù)g2是整個(gè)g2先驗(yàn)和z

(g2)不超過(guò)1/2，并且g1任意的。

〔一〕假如函數(shù)f是一個(gè)非平凡解e(f)在〔1.1〕中，那么f(z)和f(z2i)是線

性相關(guān)。

〔二〕假如f1和f2是兩個(gè)線性無(wú)關(guān)解在〔1.1〕中，那么e(f1f2)。

定理2

設(shè)g()是一個(gè)超越整函數(shù)及其低階不超過(guò)1/2。設(shè)A(z)B(e)，其中zB()g(1/)j1bjj和p是一個(gè)奇正整數(shù)，那么(f)為每個(gè)非平凡解F到在〔1.1〕中。事實(shí)上，在〔1.2〕中證明正確的結(jié)論。

我們留意到，上述結(jié)論仍舊有效的假設(shè)p

B()g()bjj

j1p

我們留意到，我們得出定理2推廣定理D，當(dāng)是一個(gè)正整數(shù)或無(wú)窮，但(g)1/2結(jié)合定理2定理的討論。

推論2

z設(shè)g()是一個(gè)超越整函數(shù)。設(shè)A(z)B(e)，其中B()g(1/)pjb和pjj1

是一個(gè)奇正整數(shù)。假設(shè)要么〔一〕或〔二〕中認(rèn)為：

〔一〕(g)不是正整數(shù)或無(wú)窮;

〔二〕(g)1/2

然后為每一個(gè)非平凡解在〔1.1〕中函數(shù)f對(duì)于(f)。事實(shí)上，在〔1.2〕中已經(jīng)有證明的結(jié)論。

2．引理為定理的證明

引理1

〔[7]〕，k2和的假設(shè)A0,Ak2是整個(gè)周期2i，并且函數(shù)f是有一個(gè)非平凡解

y(k)Aj(z)y(j)(z)0

i0k2

進(jìn)一步假設(shè)函數(shù)f滿意logN(r,1/f)o(r);，A0是在e非恒定和理性的，而且，假如z

k3，且A1,Ak2是常數(shù)。那么存在一個(gè)整數(shù)q與1qk，f(z)和f(zq2i)是線

z性相關(guān)。相同的結(jié)論認(rèn)為，假如A0是超越e，和f滿意logN(r,1/f)o(r)，假如k3，

然后通過(guò)一個(gè)無(wú)限措施的集合L1為r，T(r,Aj)o(T(r,Aj))且j1,k2引理2

〔[10]〕設(shè)A(z)B(ez)是一個(gè)周期為2i

極奇數(shù)階設(shè)B()是定期與整函數(shù)周期2i

無(wú)關(guān)的解。

3．主要結(jié)果的證明

主要結(jié)果的證明的基礎(chǔ)上[8]和[15]。11在ez〔包括那些可以轉(zhuǎn)變這種狀況下的先驗(yàn)。在〔1.1〕中由不同的時(shí)期f(z)0，logN(r,1/f)o(r)有一個(gè)滿意，那么f(z)和f(z)是線性在0

定理1的證明

讓我們假設(shè)e(f)。正弦f(z)和f(z2i)是線性無(wú)關(guān)的，引理1意味著f(z)和f(z4i)需要是線性相關(guān)的。設(shè)E(z)f(z)f(z2i)，那么E(z)滿意微分方程

E(z)2E(z)c2

,(2.1)4A(z)()22E(z)E(z)E(z)

其中c0是f1和f2(見(jiàn)[12,p.5]or[1,p.354])，且E(z2i)c1E(z)或某些非零的常數(shù)c1。顯著，E/E和E/E是兩個(gè)周期2i，而A(z)是定義函數(shù)。在〔2.1〕，E(z)2也定期與周期2i。因此，我們可以找到一個(gè)解析函數(shù)()在0，使E(z)2(ez)代入〔2.1〕得這種表達(dá)

c23224B()2()(2.2)4

由于B()和()在C*:1，理論[21，p.15]給出了他們的結(jié)論

B()nR()b()，()n1R1()()，〔2.3〕

其中n，n1是一些整數(shù)，R()和R1()函數(shù)分析和C*{}上非零，b()和()是整函數(shù)。根據(jù)相同的[8]中，我們得出

T(,)N(,1/)T(,b)S(,)，〔2.4〕

其中S(,)o(T(,))，此外，以下結(jié)論由[8]得

e(f)e(E)e(E2)ma*{eR(E2),eL(E2)},

eR(E2)1()(),

其中eR(E2)是定義為

logNR(r,1/E2)logNR(r,1/E2)lim(resp,lim),rrrr

定期二階線性微分方程解的一些性質(zhì)

其中，NR(r,1/E2)(resp.NL(r,1/E2)表示一個(gè)計(jì)數(shù)功能，只計(jì)算在右半平面的E(z)2零點(diǎn)〔在左半平面〕，1()是在的C*零點(diǎn)收斂指數(shù)，它的定義為

logN(,1/)1()limlog

由條件e(f)，我們得到()。

現(xiàn)在〔2.3〕代入〔2.2〕中

2n1R132n1R12cn4R()b()n1()()R14R1R1()()

R1n1R1n1R1R12n1(n11)222)〔2.5〕(2R1R1R1

推論1的證明

我們可以很簡(jiǎn)單地推導(dǎo)出定理1的推論1〔一〕推論1的證明〔B〕。假設(shè)f1和f2與e(f1f2)線性無(wú)關(guān)，那么e(f1)，我們證明推論1的結(jié)論〔一〕，fj(z)與fj(z2i)線性相關(guān)，J=1;2。假設(shè)E(z)f1(z)f2(z)，然后我們可以找到

2E(z2i)c2E(z)的一個(gè)非零的常數(shù)c2，重復(fù)同樣的論點(diǎn)定理1中運(yùn)用的事實(shí)，E(z)

也是能找到，我們得到e(E)1(g2)12與(g2)1/2自沖突，因此e(f1f2)。

定理2的證明

N(r,1/f)o(r)。我們推斷e(f)0，

f(z)和f(z2i)的線性依靠推論1〔a〕。然而，引理2意味著f(z)和f(z2i)是線

性無(wú)關(guān)的。這是一對(duì)沖突。因此logN(r,1/f)o(r)，認(rèn)為都有非平凡解的F在〔1.1〕假設(shè)存在一個(gè)非平凡解的f在〔1.1〕中，滿意log

中，這就完成了定理2的證明。

SomePropertiesofSolutionsofPeriodicSecondOrder

LinearDifferentialEquations

1.Introductionandmainresults

Inthispaper,weshallassumethatthereaderisfamiliarwiththefundamentalresultsandthestardardnotationsoftheNevanlinna'svaluedistributiontheoryofmeromorphicfunctions[12,14,

(f)and(f)todenoterespectivelytheorder16].Inaddition,wewillusethenotation(f)，

ofgrowth,thelowerorderofgrowthandthee*ponentofconvergenceofthezerosofameromorphicfunctionf，e(f)〔[see8]〕，thee-typeorderoff(z),isdefinedtobe

e(f)limlogT(r,f)rr

Similarly,e(f)，thee-typee*ponentofconvergenceofthezerosofmeromorphicfunctionf,isdefinedtobe

logN(r,1/f)e(f)limrr

Wesaythatf(z)hasregularorderofgrowth