一阶中立型时滞微分方程的振动性英文

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1、OscillatorybehavioroffirstorderneutraldelaydifferentialequationsLIXue2mei(DepartmentofMathematicsHunanNormalUniversity,Changsha,Hunan410081,China)Abstract:Inthispaper,thesuffcientconditionsrespectivelyfortheoscillationandnonoscillationoftheneutraldelaydifferentialequationfx(t)+px(t-frq(t)x(t-0=0

2、,areobtainedwherep>0,q(t)isapositiveperiodicfunction.Keywords»scillation,nonoscillation,neutralequations1991MSC:34C1034KI5DocumentCode:AArticleID:100825513(2000)0120075279Aneutraldelaydifferentialequation(NDDEforshort)isanequationinwhichthehighestorderderivativeoftheunknownfunctionappearsbothwit

3、handwithoutdelays・InthispaperweconsiderthefollowingfirstorderNDDEwithperiodiccoefficientwherepGE(0,°5>,q(t)isapositivecontinuous7^)eriodicfunction.TheoscillatorybehaviorofEq.(1)where-1

4、uraiminthispaperistoobtainsufficientconditionsoftheoscillationwhenp>(whicharcintermofrelationshipofparametersofEq.(1)andcanbeverifiedwhentheparametersaregiven.Asiscustomary,asolutionofEq.(1)iscalledoscillatoryifithasarbitrarilylargezeros・Otherwise,thesolutioniscallednonoscillatory.Inthesequel,fo

5、rconvenience,whenwewriteaninequalitywithoutspecifyingitsdomainofwillassumethatitholdsfor1SomeBasicLemmasS收稿日期:1999204202.作者简介:李雪梅（19642）•女•讲师•研究方向:泛函做分方程.Thefollowinglemmaswillplayanimportantroleintheproofsofthemaintheorems.Leminal4Assumethatq(t)isapositivecontinuous72periodicfunction,andthereex

6、istpositiveintegers,nzsuchthaty=nT9(7=112T.ThenallsolutionsofEq.(1)areoscillatoryifandonlyifthecharacteristicequation"1+pa*)+qQe'n=0(2)hasnorealroots・Lemma2Assumethatp>09y>0q$0,QGC([t(),,R^)andQ(t)Me>0.Supposealsothatx(t)isaneventuallypositivesolutionoftheNDDE・[x(t)+px(i-1^-Q(i)x(t-=0andsetz(t)

7、=x(t)+px(t•冷.Thenz(t)WO,z(〃>0,hmz(t)=0,hmx0(4)weonlyprovethatlimz(t)=0andlimx(t)=0.From(4),wehavelimz(t)=/$0.Jf8Byintegratingbothsidesoftheequalityz(t)=・Q(/丿x(/・Cfromroto°°,vveobtain8