IntroductionandmainresultsInthispaper,weshallassumethatthereaderisfamiliarwiththefundamentalresultsandthestardardnotationsoftheNevanlinna'svaluedistributiontheoryofmeromorphicfunctions[12,14,16].Inaddition,wewillusethenotation，andtodenoterespectivelytheorderofgrowth,thelowerorderofgrowthandtheexponentofconvergenceofthezerosofameromorphicfunction，（[see8]），thee-typeorderoff(z),isdefinedtobeSimilarly,，thee-typeexponentofconvergenceofthezerosofmeromorphicfunction,isdefinedtobeWesaythathasregularorderofgrowthifameromorphicfunctionsatisfiesWeconsiderthesecondorderlineardifferentialequationWhereisaperiodicentirefunctionwithperiod.Thecomplexoscillationtheoryof(1.1)wasfirstinvestigatedbyBankandLaine[6].Studiesconcerning(1.1)haveeencarriedonandvariousoscillationtheoremshavebeenobtained[2{11,13,17{19].Whenisrationalin，BankandLaine[6]provedthefollowingtheoremTheoremALetbeaperiodicentirefunctionwithperiodandrationalin.Ifhaspolesofoddorderatbothand,thenforeverysolutionof(1.1),Bank[5]generalizedthisresult:Theaboveconclusionstillholdsifwejustsupposethatbothandarepolesof,andatleastoneisofoddorder.Inaddition,thestrongerconclusion(1.2)holds.Whenistranscendentalin,Gao[10]provedthefollowingtheoremTheoremBLet,whereisatranscendentalentirefunctionwith,isanoddpositiveintegerand，Let.Thenanynon-triviasolutionof(1.1)musthave.Infact,thestrongerconclusion(1.2)holds.Anexamplewasgivenin[10]showingthatTheoremBdoesnotholdwhenisanypositiveinteger.Iftheorder,butisnotapositiveinteger,whatcanwesay?ChiangandGao[8]obtainedthefollowingtheoremsTheorem1Let,where,andareentirefunctionswithtranscendentalandnotequaltoapositiveintegerorinfinity,andarbitrary.IfSomepropertiesofsolutionsofperiodicsecondorderlineardifferentialequationsandaretwolinearlyindependentsolutionsof(1.1),thenOrWeremarkthattheconclusionofTheorem1remainsvalidifweassumeisnotequaltoapositiveintegerorinfinity,andarbitraryandstillassume,Inthecasewhenistranscendentalwithitslowerordernotequaltoanintegerorinfinityandisarbitrary,we