含超前与滞后量的2n阶p-Laplace差分方程边值问题周展，王倩广州大学数学与信息科学学院，广州510006广州大学数学与交叉科学广东普通高校重点实验室，广州510006摘要：本文考虑含超前与滞后量的2n阶p-Laplace差分方程边值问题。利用临界点理论，得到了边值问题解的存在性的一些充分条件。结果推广和改进了最近文献的相关结论。关键词：边值问题;p-Laplace差分方程;2n阶;临界点理论.中图分类号：O175.1Boundaryvalueproblemsfor2n-orderp-LaplaciandifferenceequationscontainingbothadvanceandretardationZHOUZhan,WANGQianSchoolofMathematicsandInformationScience,GuangzhouUniversity,Guangzhou510006KeyLaboratoryofMathematicsandInterdisciplinarySciencesofGuangdongHigherEducationInstitutes,GuangzhouUniversity,Guangzhou510006Abstract:Inthispaper,weconsidertheboundaryvalueproblemsfor2n-orderp-Laplaciandifferenceequationcontainingbothadvanceandretardation.Byusingcriticalpointtheory,sufficientconditionsoftheexistenceofsolutionsoftheboundaryvalueproblemsareobtained.Ourresultsgeneralizeandimprovesomerecentones.Keywords:Boundaryvalueproblem;p-Laplaciandifferenceequation;2n-order;Criticalpointtheory.0IntroductionLetN,ZandRdenotethesetsofallnaturalnumbers,integersandrealnumbers,re-spectively.Fora,b∈Z,defineZ(a)={a,a+1,···},andZ(a,b)={a,a+1,···,b}whena≤b.基金项目：TheSpecializedFundfortheDoctoralProgramofHigherEducationofChina(no.20114410110002)andChangjiangScholarsandInnovativeResearchTeaminUniversity(no.IRT1226).作者简介：ZHOUZhan(1965-),male,professor,majorresearchdirection:ordinarydifferentialequations.-1-Considerthefollowing2nth-ordernonlineardifferenceequation∆n(rt−nϕp(∆nxt−n))+(−1)n+1f(t,xt+1,xt,xt−1)=0,withtheboundaryvalueconditionst∈Z(1,T)(1)xt=0,t∈Z(1−n,0)∪Z(T+1,T+n),(2)whereTandnaregivenpositiveintegerswithT>n,r1−n,r2−n,···,rTarepositivenumbersand∆istheforwarddifferenceoperatordefinedby∆xt=xt+1−xt,∆mxt=∆(∆m−1xt)form∈N,ϕpisthep-Laplacianoperator,i.e.,ϕp(s)=|s|p−2s(p>1),f∈C(Z(1,T)×R3,R).Wenoticethat,(1)isakindofdifferenceequationscontainingbothadvanceandretarda-tion.Thiskindofdifferenceequationshasmanyapplicationsbothintheoryandpractice.Forexample,in[1],Agarwalconsideredthefollowingdifferenceequation−ω2Mx(t)+f(−x(t−1)+2x(t)−x(t+1))=0,withtheboundaryvalueconditionsx(0)=x(T+1)=0.t∈Z(1,T)(3)(4)Equation(3)representstheamplitudeofthemotionofeveryparticleinthestring.Andin[2],theauthorsconsideredthefollowingsecondorderfunctionaldifferenceequationLxt=f(t,xt+1,xt,xt−1),withdifferentboundaryvalueconditions∆x0=A,xT+1=B,t∈Z(1,T)(5)wheretheoperatorListheJacobioperatorgivenbyLxt=atxt+1+at−1xt−1+btxt.Inthispaper,wewillconsidertheexistenceofsolutionsoftheboundaryvalueproblemof(1)with(2).First,wewillconstructafunctionalJsuchthatsolutionsoftheboundaryvalueproblem(1)with(2)correspondtocriticalpointsofJ.Then,byusingMountainpasslemma,weobtaintheexistenceofcriticalpointsofJ.Wementionthat,forthecasewheren=1,theboundaryvalueproblemof(1)with(2)wasconsideredin[3]byShi,LiuandWang.Whenn=2,(1)with(2)wasconsideredin[4]byZhang.Ouraiminthispaperistodiscusstheboundaryvalueproblemsofgeneral2n-orderp-Laplaciandifferenceequationscontainingbothadvanceandretardation.Asaspecialcaseofn=2,ourresultsimprovethecorrespondingonesof[4].-2-(b∑(b,c(m)=1forp∈(2,∞).−p−2∗∑∑(∆uj)≤λ(b−a+1)∑∑∑∑Fortheperiodicandsubharmonicsolutionsofp-Laplaciandifferenceequationscontain-ingbothadvanceandretardation,wereferto...