二阶离散哈密尔顿系统周期解的存在性

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1、第31卷第6期Vol.31No.62009年6月Jun.2009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.SchoolofMathematicsandStatistics9SouthwestU

2、niversityfChongqing400715.China；2HuzhouTechnicianInstitute,HuzhouZhejiang313216,ChinaAbstract:Asolvabilityconditionofperiodicsolutionisobtainedforsubquadraticnon-autonomoussecond-orderdiscreteHamiltoniansystembyminimaxmethodsincriticalpointtheory.Keywords：discreteHamiltoni

3、ansystem；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-periodicin

4、tforallx€R".7、isapositiveintegersZisthesetofallintegers.▽jr)denotesthegradientofF(/・x)inweareinterestedintheexistenceofaperiodicsolutionforsuchasystem(1).Inrecentyearsttheexistenceandmultiplicityofperiodicsolutionforsecond-orderHamiltoniansystemhavebeenstudiedinmanypapers

5、viathecriticalpointtheory,see.c.g.[1—3].However>therearefewknowntechniquesforstudyingtheexistenceofperiodicsolutionsofdiscreteHamiltoniansystem.Recentlyt[4—5]developedanewmethodtostudytheexistenceandmultiplicityofperiodicsolutionsofdifferenceequationsbyusingcriticalpointt

6、heorytwhichseemstobeaverypowerfultooltodealwithsuchproblems.Inalloftheirarticles,theyconstructavariationalframeworkandgetthecorrespondingfunc・tionbythematrixtheory.Now.[_6—8]usedtheoperatortheorywhichisdifferentfromtheirstostudytheexistenceandmultiplicityofperiodicsolution

7、sofdifferenceequations.Theyalsobuiltasuitablevariationalstructure・sothatthecriticalpointsofthevariationalfunctionalcorrespondtotheperiodicsolutionsofthedifferenceequation.Motivatedbyreferences[4,6—5],weconsideredtheexistenceofperiodicsolutionforthesecond-orderdiscreteHam

8、iltoniansystem.First>weshallstatesomebasicnotations.ForanygivenpositiveintegerT,Hrisdefin