数学一些周期性的二阶线性微分方程解的方法大学毕业论文外文文献翻译及原文Word文档格式.docx

数学一些周期性的二阶线性微分方程解的方法大学毕业论文外文文献翻译及原文Word文档格式.docx

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2017.02.14

毕业设计（论文）附录

（翻译）

课题名称一些周期性的二阶线性微分方程解的方法

目录

1．毕业设计（论文）附录（翻译）英文

2．毕业设计（论文）附录（翻译）中文

SomePropertiesofSolutionsofPeriodicSecondOrderLinearDifferentialEquations

1.Introductionandmainresults

Inthispaper,weshallassumethatthereaderisfamiliarwiththefundamentalresultsandthestardardnotationsoftheNevanlinna'

svaluedistributiontheoryofmeromorphicfunctions[12,14,16].Inaddition,wewillusethenotation，andtodenoterespectivelytheorderofgrowth,thelowerorderofgrowthandtheexponentofconvergenceofthezerosofameromorphicfunction，（[see8]），thee-typeorderoff（z）,isdefinedtobe

Similarly,，thee-typeexponentofconvergenceofthezerosofmeromorphicfunction,isdefinedtobe

Wesaythathasregularorderofgrowthifameromorphicfunctionsatisfies

Weconsiderthesecondorderlineardifferentialequation

Whereisaperiodicentirefunctionwithperiod.Thecomplexoscillationtheoryof（1.1）wasfirstinvestigatedbyBankandLaine[6].Studiesconcerning（1.1）haveeencarriedonandvariousoscillationtheoremshavebeenobtained[2{11,13,17{19].Whenisrationalin，BankandLaine[6]provedthefollowingtheorem

TheoremALetbeaperiodicentirefunctionwithperiodandrationalin.Ifhaspolesofoddorderatbothand,thenforeverysolutionof（1.1）,

Bank[5]generalizedthisresult:

Theaboveconclusionstillholdsifwejustsupposethatbothandarepolesof,andatleastoneisofoddorder.Inaddition,thestrongerconclusion

（1.2）

holds.Whenistranscendentalin,Gao[10]provedthefollowingtheorem

TheoremBLet,whereisatranscendentalentirefunctionwith,isanoddpositiveintegerand，Let.Thenanynon-triviasolutionof（1.1）musthave.Infact,thestrongerconclusion（1.2）holds.

Anexamplewasgivenin[10]showingthatTheoremBdoesnotholdwhenisanypositiveinteger.Iftheorder,butisnotapositiveinteger,whatcanwesay?

ChiangandGao[8]obtainedthefollowingtheorems

TheoremCLet,where,andareentirefunctionstranscendentalandnotequaltoapositiveintegerorinfinity,andarbitrary.

（i）Suppose.（a）Iffisanon-trivialsolutionof（1.1）with;

thenandarelinearlydependent.（b）Ifandareanytwolinearlyindependentsolutionsof（1.1）,then.

（ii）Suppose（a）Iffisanon-trivialsolutionof（1.1）with,andarelinearlydependent.Ifandareanytwolinearlyindependentsolutionsof（1.1）,then.

TheoremDLetbeatranscendentalentirefunctionanditsorderbenotapositiveintegerorinfinity.Let;

whereandpisanoddpositiveinteger.Thenoreachnon-trivialsolutionfto（1.1）.Infact,thestrongerconclusion（1.2）holds.

Exampleswerealsogivenin[8]showingthatTheoremDisnolongervalidwhenisinfinity.

Themainpurposeofthispaperistoimproveaboveresultsinthecasewhenistranscendental.Specially,wefindaconditionunderwhichTheoremDstillholdsinthecasewhenisapositiveintegerorinfinity.WewillprovethefollowingresultsinSection3.

Theorem1Let,where,andareentirefunctionswithtranscendentalandnotequaltoapositiveintegerorinfinity,andarbitrary.IfSomepropertiesofsolutionsofperiodicsecondorderlineardifferentialequationsandaretwolinearlyindependentsolutionsof（1.1）,then

Or

WeremarkthattheconclusionofTheorem1remainsvalidifweassume

isnotequaltoapositiveintegerorinfinity,andarbitraryandstillassume,Inthecasewhenistranscendentalwithitslowerordernotequaltoanintegerorinfinityandisarbitrary,weneedonlytoconsiderin,.

Corollary1Let,where,andare

entirefunctionswithtranscendentalandnomorethan1/2,andarbitrary.

（a）Iffisanon-trivialsolutionof（1.1）with,thenandarelinearlydependent.

（b）Ifandareanytwolinearlyindependentsolutionsof（1.1）,then.

Theorem2Letbeatranscendentalentirefunctionanditslowerorderbenomorethan1/2.Let,whereandpisanoddpositiveinteger,thenforeachnon-trivialsolutionfto（1.1）.Infact,thestrongerconclusion（1.2）holds.

Weremarkthattheaboveconclusionremainsvalidif

WenotethatTheorem2generalizesTheoremDwhenisapositiveintegerorinfinitybut.CombiningTheoremDwithTheorem2,wehave

Corollary2Letbeatranscendentalentirefunction.Letwhereandpisanoddpositiveinteger.Supposethateither（i）or（ii）belowholds:

（i）isnotapositiveintegerorinfinity;

（ii）;

thenforeachnon-trivialsolutionfto（1.1）.Infact,thestrongerconclusion（1.2）holds.

2.LemmasfortheproofsofTheorems

Lemma1（[7]）Supposethatandthatareentirefunctionsofperiod,andthatfisanon-trivialsolutionof

Supposefurtherthatfsatisfies;