数学专业毕业论文翻译

英文翻译专业数学与应用数学班级学号姓名指导教师2012年5月4日SomePropertiesofSolutionsofPeriodicSecondOrderLinearDifferentialEquations1.IntroductionandmainresultsInthispaper,weshallassumethatthereaderisfamiliarwiththefundamentalresultsandthestardardnotationsoftheNevanlinna'svaluedistributiontheoryofmeromorphicfunctions[12,14,16].Inaddition,wewillusethenotation)(f，)(fand)(ftodenoterespectivelytheorderofgrowth,thelowerorderofgrowthandtheexponentofconvergenceofthezerosofameromorphicfunctionf，)(fe（[see8]），thee-typeorderoff(z),isdefinedtoberfrTfre),(loglim)(Similarly,)(fe，thee-typeexponentofconvergenceofthezerosofmeromorphicfunctionf,isdefinedtoberfrNfre)/1,(loglim)(Wesaythat)(zfhasregularorderofgrowthifameromorphicfunction)(zfsatisfiesrfrTfrlog),(loglim)(Weconsiderthesecondorderlineardifferentialequation0AffWhere)()(zeBzAisaperiodicentirefunctionwithperiod/2i.Thecomplexoscillationtheoryof(1.1)wasfirstinvestigatedbyBankandLaine[6].Studiesconcerning(1.1)haveeencarriedonandvariousoscillationtheoremshavebeenobtained[2{11,13,17{19].When)(zAisrationalinze，BankandLaine[6]provedthefollowingtheoremTheoremALet)()(zeBzAbeaperiodicentirefunctionwithperiod/2iandrationalinze.If)(Bhaspolesofoddorderatbothand0,thenforeverysolution)0)((zfof(1.1),)(fBank[5]generalizedthisresult:Theaboveconclusionstillholdsifwejustsupposethatbothand0arepolesof)(B,andatleastoneisofoddorder.Inaddition,thestrongerconclusion)()/1,(logrofrN(1.2)holds.When)(zAistranscendentalinze,Gao[10]provedthefollowingtheoremTheoremBLetpjjjbgB1)/1()(,where)(tgisatranscendentalentirefunctionwith1)(g,pisanoddpositiveintegerand0pb，Let)()(zeBzA.Thenanynon-triviasolutionfof(1.1)musthave)(f.Infact,thestrongerconclusion(1.2)holds.Anexamplewasgivenin[10]showingthatTheoremBdoesnotholdwhen)(gisanypositiveinteger.Iftheorder1)(g,butisnotapositiveinteger,whatcanwesay?ChiangandGao[8]obtainedthefollowingtheoremsTheoremCLet)()(zeBzA,where)()/1()(21ggB,1gand2gareentirefunctions2gtranscendentaland)(2gnotequaltoapositiveintegerorinfinity,and1garbitrary.(i)Suppose1)(2g.(a)Iffisanon-trivialsolutionof(1.1)with)()(2gfe;then)(zfand)2(izfarelinearlydependent.(b)If1fand2fareanytwolinearlyindependentsolutionsof(1.1),then)()(2gfe.(ii)Suppose1)(2g(a)Iffisanon-trivialsolutionof(1.1)with1)(fe,)(zfand)2(izfarelinearlydependent.If1fand2fareanytwolinearlyindependentsolutionsof(1.1),then1)(21ffe.TheoremDLet)(gbeatranscendentalentirefunctionanditsorderbenotapositiveintegerorinfinity.Let)()(zeBzA;wherepjjjbgB1)/1()(andpisanoddpositiveinteger.Then)(foreachnon-trivialsolutionfto(1.1).Infact,thestrongerconclusion(1.2)holds.Exampleswerealsogivenin[8]showingthatTheoremDisnolongervalidwhen)(gisinfinity.Themainpurposeofthispaperistoimproveaboveresultsinthecasewhen)(Bistranscendental.Specially,wefindaconditionunderwhichTheoremDstillholdsinthecasewhen)(gisapositiveintegerorinfinity.WewillprovethefollowingresultsinSection3.Theorem1Let)()(zeBzA,where)()/1()(21ggB,1gand2gareentirefunctionswith2gtranscendentaland)(2gnotequaltoapositiveintegerorinfinity,and1garbitrary.IfSomepropertiesofsolutionsofperiodicsecondorderlineardifferentialequations)(zfand)2(izfaretwolinearlyindependentsolutionsof(1.1),then)(feOr2)()(121gfeWeremarkthattheconclusionofTheorem1remainsvalidifweassume)(1gisnotequaltoapositiveintegerorinfinity,and2garbitraryandstillassume)()/1()(21ggB,Inthecasewhen1gistranscendentalwithitslowerordernotequaltoanintegerorinfinityand2gisarbitrary,weneedonlytoconsider)/1()()/1()(*21ggBBin0,/1.Corollary1Let)()(zeBzA,where)()/1()(21ggB,1gand2gareentirefunctionswith2gtranscendentaland)(2gnomorethan1/2,and1garbitrary.(a)Iffisanon-trivialsolutionof(1.1)with)(fe,then)(zfand)2(izfarelinearlydependent.(b)If1fand2fareanytwolinearlyindependentsolutionsof(1.1),then)(21ffe.Theorem2Let)(gbeatranscendentalentirefunctionanditslowerorderbenomorethan1/2.Let)()(zeBzA,wherepjjjbgB1)/1()(andpisanoddpositiveinteger,then)(fforeachnon-trivialsolutionfto(1.1).Infact,thestrongerconclusion(1.2)holds.WeremarkthattheaboveconclusionremainsvalidifpjjjbgB1)()(WenotethatTheorem2generalizesTheoremDwhen)(gisapositiveintegerorinfinitybut2/1)(g.CombiningTheoremDwithTheorem2,wehaveCorollary2Let)(gbeatranscendentalentirefunction.Let)()(zeBzAwherepjjjbgB1)/1()(andpisanoddpositiveinteger.Supposethateither(i)or(ii)belowholds:(i))(gisnotapositiveintegerorinfinity;(ii)2/1)(g;then)(fforeachnon-trivialsolutionfto(1.1).Infact,thestrongerconclusion(1.2)holds.2.LemmasfortheproofsofTheoremsLemma1([7])Supposethat2kandthat20,.....kAAareentirefunctionsofperiodi2,andthatfisanon-trivialsolutionof0)()()(20)(kijjzyzAkySupposefurtherthatfsatisfies)()/1,(logrofrN;that0Aisnon-constantandrationalinze，andthatif3k，then21,.....kAAareconstants.Thenthereexistsanintegerqwithkq1suchthat)(zfand)2(iqzfarelinearlydependent.Thesameconclusionholdsif0Aistranscendentalinze，andfsatisfie