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1、HindawiPublishingCorporationAbstractandAppliedAnalysisVolume2012,ArticleID903281,20pagesdoi:10.1155/2012/903281ResearchArticleSubharmonicSolutionsofNonautonomousSecondOrderDifferentialEquationswithSingularNonlinearitiesN.Daoudi-Merzagui,1F.Derrab,2andA.Boucherif31DepartmentofMathematics,Univers
2、ityofTlemcen,Tlemcen13000,Algeria2DepartmentofMathematics,UniversityofSidiBelAbbes,SidiBelAbbes22000,Algeria3DepartmentofMathematicsandStatistics,KingFahdUniversityofPetroleumandMinerals,P.O.Box5046,Dhahran31261,SaudiArabiaCorrespondenceshouldbeaddressedtoA.Boucherif,aboucher@kfupm.edu.saReceiv
3、ed4November2011;Revised14January2012;Accepted19January2012AcademicEditor:JuanJ.NietoCopyrightq2012N.Daoudi-Merzaguietal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisprop
4、erlycited.Wediscusstheexistenceofsubharmonicsolutionsfornonautonomoussecondorderdifferentialequationswithsingularnonlinearities.Simplesufficientconditionsareprovidedenableustoobtaininfinitelymanydistinctsubharmonicsolutions.Ourapproachisbasedonavariationalmethod,inparticularthesaddlepointtheorem.1.
5、IntroductionandMainResultInthispaperwediscusstheproblemoftheexistenceofinfinitelymanysubharmonicsolu-tionsfornonautonomoussecondorderdifferentialequationswithsingularnonlinearitiesoftheformutft,utet,1.1wheref:R2→Riscontinuous,isT-periodic,initsfirstargumentwithT>0,andpresentsasingula
6、ritywithrespecttoitssecondargument.HerebyasubharmonicsolutionwemeanakT-periodicsolutionforanyintegerkifT>0istheminimalperiod.WhenthesolutionisnotT-periodicwecallitatruesubharmonic.Itwaspointedoutin1thatsingulardifferentialequationsoftheform1.1appearinthedescriptionofmanyphenomenaintheapplieds
7、cien-ces,suchastheBrillouinfocusingsystemandnonlinearelasticity.Severalauthorshave2AbstractandAppliedAnalysisinvestigatedtheproblemofexistenceofperiodicsolutionsforsecondorderdifferentialequationswithsingularnonlinearitiessee2–4a
