西南大学学报(自然科学版)JournalofSouthwestUniversity(NaturalScienceEdition)文章编号：1673-9868(2009)06-0116-04ExistenceofPeriodicSolutionforSecond-OrderDiscreteHamiltonianSystemsLIChun1,TANGChun-Iei1,SHENXiao-ke27.SchoolofMathematicsandStatistics9SouthwestUniversityfChongqing400715.China；2HuzhouTechnicianInstitute,HuzhouZhejiang313216,ChinaAbstract:Asolvabilityconditionofperiodicsolutionisobtainedforsubquadraticnon-autonomoussecond-orderdiscreteHamiltoniansystembyminimaxmethodsincriticalpointtheory.Keywords：discreteHamiltoniansystem；periodicsolution；criticalpoints；SaddlePointTheoremCLCnumber：0176.3Documentcode：AConsiderthenonlinearsecondorderdiscreteHamiltoniansystemVF(r,u(D)=0Vr€z(1)where△u(t)=“(r+1)—“(t),zA2u(r)=△(△w(/)),F：ZXRNR,F(/♦工)iscontinuouslydifferenti-alinxforeveryf€ZandT-periodicintforallx€R".7、isapositiveintegersZisthesetofallintegers.▽jr)denotesthegradientofF(/・x)inweareinterestedintheexistenceofaperiodicsolutionforsuchasystem(1).Inrecentyearsttheexistenceandmultiplicityofperiodicsolutionforsecond-orderHamiltoniansystemhavebeenstudiedinmanypapersviathecriticalpointtheory,see.c.g.[1—3].However>therearefewknowntechniquesforstudyingtheexistenceofperiodicsolutionsofdiscreteHamiltoniansystem.Recentlyt[4—5]developedanewmethodtostudytheexistenceandmultiplicityofperiodicsolutionsofdifferenceequationsbyusingcriticalpointtheorytwhichseemstobeaverypowerfultooltodealwithsuchproblems.Inalloftheirarticles,theyconstructavariationalframeworkandgetthecorrespondingfunc・tionbythematrixtheory.Now.[_6—8]usedtheoperatortheorywhichisdifferentfromtheirstostudytheexistenceandmultiplicityofperiodicsolutionsofdifferenceequations.Theyalsobuiltasuitablevariationalstructure・sothatthecriticalpointsofthevariationalfunctionalcorrespondtotheperiodicsolutionsofthedifferenceequation.Motivatedbyreferences[4,6—5],weconsideredtheexistenceofperiodicsolutionforthesecond-orderdiscreteHamiltoniansystem.First>weshallstatesomebasicnotations.ForanygivenpositiveintegerT,HrisdefinedbyHr=(u：Z------ARN|“(Z+T)=U(T)9TWZ}HrcanbeequippedwiththeinnerproductT<u9(“(/),)VGHy收稿H期,2008-11・18帛金项Eh国家fl然科学恋金资助项目（10771173）・作者简介：李祚（1980・）・男・浙江湖州人・讲帅・硕1:研究牛.，主要从寧非线性分析研究.通讯作者I席存甫•教授•博士生錚师・bywhichnorm||•||canbeinducedbyII“II=(SII2)1V^eHrr-1第31卷第6期Vol.31No.62009年6月Jun.2009where(•■•)andj•|denotetheusualinnerproductandtheusualnorminR、respectively.Itiseasytosecthat(HT9〈•■•>)isafinitedimensionalHilbertspaceandlinearhomeomorphicto.Now>westateourmainresultbelow.Theorem1SupposethatF⑺x)satisfies(Fl)ThereexistsapositiveintegerT,suchthatF(f+T,h〉=F(/»x)forall(“x)€ZXR,vj(F2)0冬liminf書)£limsup尸,"芒〉V孕>foralltG乙whereAi=2(1—cos军)$Ix\1*1—8\x\2V丄/(F3)(VF(z,x)♦x)—2F(/,才)ooas|x|-*ooforallzEZ[l,T],whereZ\_a,6]=Zf][a,forevery€Zwitha,Thentheproblem(1)hasatleastone7-periodicsolution.ProofLetTT<P(U)=yI△U(a)—另F(“«(/))foreachuEHT.So,onehasTT〈屛(u),v>=—2(△%(£一1)，v(t))—另(▽F(z,u(O),v(O)forallIOP€ItiswellknownthattheproblemoffindingT-periodicsolutionofproblem(1)isequaltotheoneo£seekingthecriticalpointof(p(u).Asweallknown,adeformationlemmacanbeprovedwiththeweaker(C)conditionwhichisdueto[9]replacingtheusual(PS)condition,anditturnsoutthattheSaddlePointTheoremholdsunderthe(C)condition.First,weprovethat...