Oscillation of a Linear Delay Impulsive Differenti

arXiv:funct-an/9502002v17Feb1995OscillationofaLinearDelayImpulsiveDifferentialEquationL.Berezansky,1Ben-GurionUniversityoftheNegev,DepartmentofMathematicsandComputerScience,Beer-Sheva84105,Israel,E.BravermanTechnion-IsraelInstituteofTechnology,IsraelInstituteofMetals,32000,Haifa,IsraelFebruary7,2008AbstractThemainresultofthepaperisthattheoscillation(non-oscillation)oftheimpulsivedelaydifferentialequation˙x(t)+mXk=1Ak(t)x[hk(t)]=0,t≥0,x(τj)=Bjx(τj−0),limτj=∞isequivalenttotheoscillation(non-oscillation)oftheequationwithoutimpulses˙x(t)=mXk=1Ak(t)Yhk(t)τj≤tB−1jx[hk(t)]=0,t≥0.Explicitoscillationresultsarepresented.1SupportedbytheIsraelMinistryofScienceandTechnologyandIsraelMinistryofAbsorption11IntroductionRecentlyresultsonoscillationofdelaydifferentialequationshavetakenshapeofadevelopedtheorypresentedinmonographs[1-4].Atthesametimeitisanintensivelydevelopingfieldwhichisanobjectiveofnumerouspublications.However,forimpulsivedifferentialequationsthereareonlyfewpublica-tionsdealingwithoscillationproblems[1,4,5,6].Thepurposeofthepresentpaperistofillupthisgap.Themainre-sultisthattheoscillation(non-oscillation)oftheimpulsivedelaydifferentialequationisequivalenttotheoscillation(non-oscillation)ofacertaindiffer-entialequationwithoutimpulseswhichcanbeconstructedexplicitlyfromanimpulsiveequation.Thustheoscillationproblems(inparticular,oscillationandnon-oscillationcriteria)foranimpulsiveequationcanbereducedtothesimilarproblemforacertainnon-impulsiveequation.Themethodproposedinthepresentpaperforoscillationisnewbothforimpulsiveandnon-impulsiveequations.Itisbasedonthesolutionrep-resentationformula.Recentlysuchformulasarewidelyusedinstabilityinvestigationsofnon-impulsive[7-9]andimpulsiveequations[5,10-12].Wedemonstratethattheexistenceofanonoscillatingsolutionisequiva-lenttothepositivenessofthefundamentalfunction.Atthesametimethisisequivalenttothesolvabilityofacertainnonlinearinequalitywhichissimilarto”thegeneralizedcharacteristicequation”fromthemonograph[2].Thepaperisorganizedasfollows.Theorems1and2areconcernedwiththeequivalenceofnon-oscillation,positivenessofafundamentalfunctionandsolvabilityofacertaininequality.Theyleadtoexplicitnon-oscillationresults(Theorem3).Theorem4comparesnon-oscillationconditionsfortwodifferentimpulsivedelaydifferentialequations.Theorems5and6givenewoscillationcriteriafordelaydifferentialequationswithoutimpulses.Theorem7containsthemainresultofthepaperconnectingoscillationofanimpulsiveandanon-impulsiveequation.Asacorollary(Theorem8)weobtainexplicitoscillationconditionsforanimpulsivedelayequation.WeareverygratefultoProf.YuryDomshlakforusefuldiscussionsoftheproblemsconsideredinthepaper.22PreliminariesWeconsiderascalardelaydifferentialequation˙x(t)+mXk=1Ak(t)x[hk(t)]=f(t),t≥0;(1)x(τj)=Bjx(τj−0),j=1,2,...,(2)underthefollowingassumptions(a1)0=τ0τ1τ2...arefixedpoints,limj→∞τj=∞;(a2)Ak,f,k=1,...,mareLebesguemeasurablefunctionsessentiallyboundedineachfiniteinterval[0,b],Bj∈R,j=1,...,Risarealaxis;(a3)hk:[0,∞)→RareLebesguemeasurablefunctions,hk(t)≤t.Togetherwith(1),(2)wewillconsiderforeacht0≥0aninitialvalueproblem˙x(t)+mXk=1Ak(t)x[hk(t)]=f(t),wheret≥t0,x(ξ)=ϕ(ξ),ξt0,(3)x(τj)=Bjx(τj−0),τjt0.(4)Weassumethatfortheinitialfunctionϕthefollowinghypothesisholds(a4)ϕ:(−∞,t0)→RisaBorelmeasurableboundedfunction.Definition.Anabsolutelycontinuousoneachinterval[τj,τj+1)functionx:[t0,∞)→Risasolutionoftheimpulsiveproblem(3),(4)if(3)issatisfiedforalmostallt∈[0,∞)andtheequalities(4)hold.Definition.Foreachs≥0thesolutionX(t,s)oftheproblem˙x(t)+mXk=1Ak(t)x[hk(t)]=0,wheret≥s;x(ξ)=0,ξs;x(τj)=Bjx(τj−0),τjs,x(s)=1,(5)isafundamentalfunctionoftheequation(1),(2).WeassumeX(t,s)=0,0≤ts.Lemma1[12]Let(a1)-(a4)hold.Thenthereexistoneandonlyonesolu-tionoftheproblem(3)withtheinitialconditionx(t0)=α0andimpulsiveconditionsx(τj)=Bjx(τj)+αj3thatcanbepresentedintheformx(t)=X(t,t0)x(t0)+Ztt0X(t,s)f(s)ds−−mXk=1Ztt0X(t,s)Ak(s)ϕ[hk(s)]ds+Xτjt0X(t,τj)αj,(6)whereϕ[hk(s)]=0,ifhk(s)t0.3Non-oscillationCriteriaforImpulsiveEqua-tionsDefinition.Theequation(1),(2)hasanon-oscillatingsolutionifthereexistt00,ϕ(t)satisfying(a4)suchthatforf≡0thesolutionof(3),(4)ispositivefort≥t0.Otherwise,allsolutionsof(1),(2)aresaidtobeoscillating.Insequelweacceptthatthefollowinghypothesisholds(a5)delaysarebounded:foreverys0μs=minkvraiinftshk(t)−∞andthereexistss′≥ssuchthathk(t)≥sift≥s′.DenoteforanysAsk(t)=(Ak(t),ift≥s,0,ifts,hsk(t)=(hk(t),ift≥s,s,ifts.(7)Thefollowingtheoremestablishesnon-oscillationcriteria.Theorem1Suppose(a1)-(a5)hold,Ak(t)≥0,k=1,...,m,andBj0,j=1,2,....Thenthefollowinghypothesesareequivalent1)Theequation(1),(2)hasanon-oscillatingsolution.2)Thereexistst0≥0suchthatX(t,s)0,t0≤st∞.3)Foracertaint1≥0thereexistsanon-negativeintegrableineachinterval[t1,b]solutionuofaninequalityu(t)≥mXk=1At1k(t)exp(Ztht1k(t)u(s)ds)Yht1k(t)τj≤tB−1j,t≥t1.(8)4Hereandinsequelweassumethataproductequalstounitifnumberoffactorsisequaltozero.Proof.Theschemeoftheproofis1)=⇒3)=⇒2)=⇒1).1)=⇒3).Letx(t)beapositivesolutionof(3),(4)(f≡0).By(a5)foracertaint1≥t0hk(t)t0,t≥t1,k=1,...,m.Letusdemonstratethatu(t)