Fourth-Order Differential Equation with Deviating

HindawiPublishingCorporationAbstractandAppliedAnalysisVolume2012,ArticleID185242,17pagesdoi:10.1155/2012/185242ResearchArticleFourth-OrderDifferentialEquationwithDeviatingArgumentM.Bartuˇsek,1M.Cecchi,2Z.Doˇsl´a,1andM.Marini21DepartmentofMathematicsandStatistics,MasarykUniversity,61137Brno,CzechRepublic2DepartmentofElectronicsandTelecommunications,UniversityofFlorence,50139Florence,ItalyCorrespondenceshouldbeaddressedtoZ.Doˇsl´a,[e-mail]6December2011;Accepted13January2012AcademicEditor:PaulEloeCopyrightq2012M.Bartuˇseketal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.Weconsiderthefourth-orderdifferentialequationwithmiddle-termanddeviatingargumentx4tqtx2trtfxϕt0,incasewhenthecorrespondingsecond-orderequationhqth0isoscillatory.Necessaryandsufficientconditionsfortheexistenceofboundedandunboundedasymptoticallylinearsolutionsaregiven.Therolesofthedeviatingargumentandthenonlinearityareexplained,too.1.IntroductionTheaimofthispaperistoinvestigatethefourth-ordernonlineardifferentialequationwithmiddle-termanddeviatingargumentx4tqtx2trtfxϕt0.1.1Thefollowingassumptionswillbemade.iqisacontinuouslydifferentiableboundedawayfromzerofunction,thatis,qt≥q00forlargetsuchthat∞0qtdt∞.1.2iir,ϕarecontinuousfunctionsfort≥0,risnotidenticallyzeroforlarget,ϕt≥0,andϕ00,limt→∞ϕt∞.2AbstractandAppliedAnalysisiiifisacontinuousfunctionsuchthatfuu0foru/0.ObservethatiimpliesthatthereexistsapositiveconstantQsuchthatqt≤Qandthelinearsecond-orderequationhtqtht01.3isoscillatory.Moreover,solutionsof1.3areboundedtogetherwiththeirderivatives,seeforexample,1,Theorem2 .Byasolutionof1.1wemeanafunctionxdefinedonTx,∞,Tx≥0,whichisdifferentiableuptothefourthorderandsatisfies1.1onTx,∞andsup{|xt|:t≥T}0forT≥Tx.Asolutionxof1.1issaidtobeasymptoticallylinear(AL-solution)ifeitherlimt→∞xtcx/0,limt→∞xt0,1.4orlimt→∞|xt|∞,limt→∞xtdx/0,1.5forsomeconstantscx,dx.Fourth-ordernonlineardifferentialequationsnaturallyappearinmodelsconcerningphysical,biological,andchemicalphenomena,suchas,forinstance,problemsofelasticity,deformationofstructures,orsoilsettlement,see,forexample,2,3 .When1.3isnonoscillatoryandhisitseventuallypositivesolution,itisknownthat1.1canbewrittenasthetwo-termequationh2txtht htrtfxt0.1.6Inthiscase,thequestionofoscillationandasymptoticsofsuchclassofequationshasbeeninvestigatedwithsufficientthoroughness,see,forexample,thepapers3–10 orthemonographs11,12 andreferencestherein.Nevertheless,asfarweknown,thereareonlyfewresultsconcerning1.1when1.3isoscillatory.Forinstance,theequationwithoutdeviatingargumentxntqtxn−2trtfxt01.7hasbeeninvestigatedbyKiguradzein13 incaseqt≡1andbytheauthorsin14,15 whenqsatisfiesi.Inparticular,in14 theoscillationof1.1inthecasen3isstudied.In15 ,theexistenceofpositiveboundedandunboundedsolutionsaswellasofoscillatorysolutionsfor1.7hasbeenconsideredandthecasen4hasbeenanalyzedindetail.Otherresultscanbefoundin16 andreferencestherein,inwhichtheexistenceanduniquenessofalmostperiodicsolutionsforequationsoftype1.1withalmostperiodiccoefficientsq,rarestudied.AbstractandAppliedAnalysis3Motivatedby14,15 ,herewestudytheexistenceofAL-solutionsfor1.1.Theapproachiscompletelydifferentfromtheoneusedin15 ,inwhichaniterationprocess,jointlywithacomparisonwiththelinearequationy4qty20,isemployed.Ourtoolsarebasedonatopologicalmethod,certainintegralinequalities,andsomeauxiliaryfunctions.Inparticular,forprovingthecontinuityintheFr´echetspaceCt0,∞ofthefixedpointoperatorshereconsidered,weuseasimilarargumenttothatintheVitaliconvergencetheorem.Ourresultsextendtothecasewithdeviatingargumentanaloguesonesstatedin15 for1.7whenn4.WeobtainsharperconditionsfortheexistenceofunboundedAL-solutionsof1.1,and,inaddition,weshowthatunderadditionalassumptionsonq,r,theseconditionsbecomealsonecessaryfortheexistenceofAL-solutions,inboththeboundedandunboundedcases.Inthefinalpart,weconsidertheparticularcasefu|u|λsgnuλ01.8andwestudythepossiblecoexistenceofboundedandunboundedAL-solutions.Theroleofdeviatingargumentandtheoneofthegrowthofthenonlinearityarealsodiscussedandillustratedbysomeexamples.2.UnboundedSolutionsHerewestudytheexistenceofunboundedAL-solutionsof1.1.Ourfirstmainresultisthefollowing.Theorem2.1.Foranyc,0c∞,thereexistsanunboundedsolutionxof1.1suchthatlimt→∞xtc,limt→∞xit0,i2,3,2.1provided∞0|rt|Fϕtdt∞,2.2whereforu0Fumaxfv:|v−u|≤12u.2.3Proof.Withoutlossofgenerality,weprovetheexistenceofsolutionsof1.1satisfying2.1forc1.Letuandvbetwolinearlyindependentsolutionsof1.3withWronskiand1.Denotews,tusvt−utvs,zs,t∂∂tws,t.2.44AbstractandAppliedAnalysisAsclaimedbytheassumptionsonq,allsolutionsof1.3andtheirderivativesarebounded.Thus,putMsup{|ws,t||zs,t|:s≥0,t≥