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.docx
- 文档编号:844840
- 上传时间:2022-10-13
- 格式:DOCX
- 页数:13
- 大小:385.07KB
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.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.docx》由会员分享,可在线阅读,更多相关《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.docx(13页珍藏版)》请在冰豆网上搜索。
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
毕业设计(论文)
外文文献翻译
文献、资料中文题目:
一些周期性的二阶线性微分方程解的方法
文献、资料英文题目:
文献、资料来源:
文献、资料发表(出版)日期:
院(部):
专业:
班级:
姓名:
学号:
指导教师:
翻译日期:
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;
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 数学 一些 周期性 线性 微分方程 方法 大学毕业 论文 外文 文献 翻译 原文