www.elsevier.com/locate/aml
Ontheexistenceofperiodicsolutionsforaclassofgeneralized
forcedLi´enardequations✩
M.R.Pournakia,∗,A.Razanib,a
aSchoolofMathematics,InstituteforStudiesinTheoreticalPhysicsandMathematics,P.O.Box19395-5746,Tehran,Iran
bDepartmentofMathematics,FacultyofScience,ImamKhomeiniInternationalUniversity,P.O.Box34194-288,Qazvin,Iran
Received14July2005;receivedinrevisedform28May2006;accepted2June2006
Abstract
Inthisworkthesecond-ordergeneralizedforcedLi´enardequationx+f(x)+k(x)xx+g(x)=p(t)isconsideredandanewconditionforguaranteeingtheexistenceofatleastoneperiodicsolutionforthisequationisgiven.c2006ElsevierLtd.Allrightsreserved.
Keywords:Nonlinearboundaryvalueproblem;Li´enardequation;Periodicsolution;Banachspace;Schauder’sFixedPointTheorem
1.Introduction
Inthisworkweinvestigatetheexistenceofperiodicsolutionsforaclassofsecond-ordergeneralizedforcedLi´enardequations
(1.1)x+f(x)+k(x)xx+g(x)=p(t),wheref,k,andgarerealfunctionsonRandpisaT-periodicrealfunctionon[0,T],T>0.Generalizedforced
Li´enardequationsappearinanumberofphysicalmodelsandanimportantquestioniswhethertheseequationscansupportperiodicsolutions.Thisquestionhasbeenstudiedextensivelybyanumberofauthors;seeforexample[1–9].Inparticular,therearesomeexistenceandmultiplicityresultsforsuchequationswithnonconstantforcedterms;seeforexample[10–19].Inthisdirection,wewillobtainanewconditiontoguaranteetheexistenceofatleastoneperiodicsolutionfor(1.1)withanonconstantforcedterm.Themainpurposeofthisworkistoprovethefollowingresult:MainTheorem.Supposef,k,andgarerealfunctionsonRwhicharelocallyLipschitzandpisanonconstant,continuous,T-periodicrealfunctionon[0,T],T>0.Alsosupposeallsolutionsoftheinitialvalueproblem(1.1)canbeextendedto[0,T].Ifthereexistrealnumbersa1anda2forwhichg(a1)≤p(t)≤g(a2)holdsforeach0≤t≤T,thenEq.(1.1)hasatleastoneperiodicsolution.
✩ThisresearchwasinpartsupportedbyagrantfromIPM.∗Correspondingauthor.
E-mailaddresses:pournaki@ipm.ir(M.R.Pournaki),razani@ikiu.ac.ir(A.Razani).c2006ElsevierLtd.Allrightsreserved.0893-9659/$-seefrontmatter
doi:10.1016/j.aml.2006.06.004
M.R.Pournaki,A.Razani/AppliedMathematicsLetters20(2007)248–254249
Therestoftheworkisorganizedasfollows.InSection2,weprovethat(1.1)hasauniquesolutionsatisfyingcertainconditionsbyapplyingSchauder’sFixedPointTheorem.InSection3,theexistenceofatleastoneperiodicsolutionfor(1.1)whenghasthepropertymentionedintheMainTheoremisproved.2.Anexistenceanduniquenesstyperesult
WestartthissectionbyrecallingafamousfixedpointtheoremwhichwasoriginallyduetoSchauder:LetXbeaBanachspaceandΩbeaclosed,bounded,andconvexsubspaceofX.IfS:Ω→Ωisacompactoperator,thenShasatleastonefixedpointonΩ.
WenowstateandprovethefollowingexistenceanduniquenesstyperesultwhichisakeytoolforprovingtheMainTheorem.
Proposition2.1.Leta1 ,M,MaretheLipschitzconstants[0,T];M1,M2,M3arethemaximumvaluesof|f|,|k|,|g|on|x|≤A;andM123 off,k,gon|x|≤A,respectively.Consider M= 2 B2+(2M+M)B+M+M,M22113 √√1 N=,and0 (2.1) Thenforeacha1≤b≤a2,Eq.(1.1)hasauniquesolutionx(t),satisfying x(0)=x(T0)=b, forwhich|x(t)|≤Aand|x(t)|≤Bholdforeach0≤t≤T0. Proof.Considertheequationx=0withboundaryconditionx(0)=x(T0)=b.TheexistenceofaGreen’sfunctionforatypicaltwo-endpointproblemwassuggestedbyasimplephysicalexamplein[20]andisasfollows: s(t−T0)/T0:if0≤s≤t≤T0, G(t,s)= t(s−T0)/T0:if0≤t≤s≤T0.Ifwenowconsidertheintegralequation T0 x(t)=b+G(t,s)f(x(s))+k(x(s))x(s)x(s)+g(x(s))−p(s)ds, 0 (2.2) thenitiseasytoseethatthesolutionsof(2.2)areexactlythesolutionsof(1.1)satisfying(2.1).Hence,toprovetheproposition,itisenoughtoshowthat(2.2)hasauniquesolutionx(t)satisfying|x(t)|≤Aand|x(t)|≤Bforeach0≤t≤T0.Inordertodoso,supposeX=C1([0,T0],R),andforφ∈Xdefine φ=max|φ(t)|+max|φ(t)|. 0≤t≤T0 0≤t≤T0 ItisclearthatXisaBanachspace.Now,consider Ω=φ∈X:|φ(t)|≤Aand|φ(t)|≤Bholdforeach0≤t≤T0, whichisobviouslyaclosed,bounded,andconvexsubspaceofX.DefinetheoperatorS:Ω→XbymappingφtoS(φ),whereS(φ)isdefinedby T0 S(φ)(t)=b+G(t,s)f(φ(s))+k(φ(s))φ(s)φ(s)+g(φ(s))−p(s)ds. 0 250M.R.Pournaki,A.Razani/AppliedMathematicsLetters20(2007)248–254 First,weshowthatSmapsΩintoitself.Inordertodothis,notethatforeachx,x,andtsuchthat|x|≤A,|x|≤B,and0≤t≤T0wehave f(x)+k(x)xx+g(x)−p(t)≤M2B2+M1B+M3+M0 1 (2.3)=. NAlsoforeach0≤t≤T0wehave T0 1T02 ,|G(t,s)|ds=t(T0−t)≤ 280 and 0T0 ∂ G(t,s)ds=1t2−t+1T0≤T0. ∂tT022 Hence(2.3)impliesthatforeachφ∈Ωand0≤t≤T0, 1T0 |G(t,s)|ds|S(φ)(t)|≤|b|+ N0T02 ≤|b|+ 8NAA+≤22=A,and T0 ∂1G(t,s)ds|S(φ)(t)|≤N0∂t T0≤2N≤B. Thesemeanthatforeachφ∈Ω,S(φ)∈ΩandthereforeSisanoperatorfromΩtoΩ. Next,weshowthatSisacompactoperatoronΩ.Forthis,itisenoughtoshowthateachboundedsequence{φn}onΩhasasubsequence{φni}forwhich{S(φni)}isconvergentonΩ.Therefore,let{φn}beagivensequenceonΩwhichisautomaticallyboundedbydefinitionofΩ.Suppose>0isgiven.SinceGisauniformlycontinuousfunctionon[0,T0]×[0,T0],thereexistsδ,0<δ<N,suchthat(t1,s1),(t2,s2)∈[0,T0]×[0,T0]and (t1−t2)2+(s1−s2)2<δimplythat|G(t1,s1)−G(t2,s2)|<N/2T0.Byapplying(2.3)wenowconcludethatforeachnandforeacht1,t2∈[0,T0],if|t1−t2|<δ,then 1T0 |G(t1,s)−G(t2,s)|ds<,and|S(φn)(t1)−S(φn)(t2)|≤ N0 T0 1∂∂G(t1,s)−G(t2,s)ds=1|t1−t2|<.|S(φn)(t1)−S(φn)(t2)|≤N0∂t∂tNHence{S(φn)(t)}and{S(φn)(t)}areequicontinuousfamiliesoffunctionson[0,T0]andbytheclassical Ascoli–ArzelaTheorem,thereexistsasubsequence{φni(t)}of{φn(t)}forwhich{S(φni)(t)}and{S(φni)(t)}areuniformlyconvergenton[0,T0].Thisshowsthat{S(φni)}isconvergentonΩandsoSisacompactoperator. Therefore,bySchauder’sFixedPointTheorem,thereexistsφ∈ΩsuchthatS(φ)=φ.Soforeach0≤t≤T0,wehaveS(φ)(t)=φ(t)whichistosay T0 G(t,s)(f(φ(s))+k(φ(s))φ(s))φ(s)+g(φ(s))−p(s)ds.φ(t)=b+ 0 Thismeansthatφ∈Ωisasolutionof(2.2).Thereforeφisasolutionof(1.1)whichsatisfies(2.1)insuchawaythat |φ(t)|≤Aand|φ(t)|≤Bforeach0≤t≤T0. Wenowshowthatφistheuniquesolutionof(1.1)whichsatisfiestheaboveconditions.Supposeψisanothersolutionof(1.1)whichsatisfiestheboundarycondition(2.1)suchthat|ψ(t)|≤Aand|ψ(t)|≤Bholdforeach0≤t≤T0.Thismeansthatψ∈Ω,ψ=φ,andS(ψ)=ψ.BythelocallyLipschitzconditionforf,k,andg,note M.R.Pournaki,A.Razani/AppliedMathematicsLetters20(2007)248–254251 thatforeachx,y,x,y,andtsuchthat|x|≤A,|y|≤A,|x|≤B,|y|≤B,and0≤t≤T0wehave f(x)+k(x)xx+g(x)−p(t)−f(y)+k(y)yy+g(y)−p(t) =|(f(x)−f(y))x+f(y)(x−y)+(k(x)−k(y))x+k(y)(x−y)+g(x)−g(y)| 2 ≤(M2B+M1B+M3)|x−y|+(2M2B+M1)|x−y|. 2 2 2 Thereforebytheaboveinequality,foreach0≤t≤T0, T022 M2B+(2M2+M1)B+M3+M1φ−ψ|S(φ)(t)−S(ψ)(t)|≤8T022=φ−ψ 8MT02 φ−ψ,and= 4M T02 M2B+(2M2+M1)B+M3+M1φ−ψ|S(φ)(t)−S(ψ)(t)|≤2T02 φ−ψ= 2MT0=φ−ψ.M Hence, φ−ψ=S(φ)−S(ψ) =max|S(φ)(t)−S(ψ)(t)|+max|S(φ)(t)−S(ψ)(t)|0≤t≤T00≤t≤T0 T0T02+≤φ−ψ.4MM √ ThereforeweobtainT02+4T0≥4M,orT0≥2M+1−2whichiscontradictorywiththedefinitionofT0.Soφistheuniquesolutionof(1.1),satisfyingthegivenconditions.Theabovepropositionimpliesthefollowingexistenceresult. Corollary2.2.LetkbealocallyLipschitzrealfunctiononRwhichisnonconstantoneachcompactinterval.ThenforeachgivenT0>0andb,thefollowingboundaryvalueproblem: 2 x+k(x)x=0,x(0)=x(T0)=b,hasasolution. Proof.WeapplyProposition2.1withp=0,saydefinedon[0,T],T>0.Supposea1anda2aretworealnumberssuchthata1 istheLipschitzconstantofkon|x|≤A.Considervalueof|k|on|x|≤AandM2 2 B2+2MB,M221 N=, M2B2 andchooseBsmallenoughandalsoTlargeenoughsuchthat √ 22A2 T0 Proposition2.1nowimpliesthatthegivenboundaryvalueproblemhasasolution.Notethatthissolutionwithrestrictions|x(t)|≤Aand|x(t)|≤Bforeach0≤t≤T0isunique. 252M.R.Pournaki,A.Razani/AppliedMathematicsLetters20(2007)248–254 3.ProofoftheMainTheorem InthissectionweprovetheMainTheorem.Bytheassumptionweconcludea1=a2andsowithoutlossofgeneralitywecansupposethata1 g˜(x)= g(a1)+a1−x:ifx>a1,and gˆ(x)= g(x) g(a2)+a2−x :ifx≥a2,:ifx ,M,M,M˜,Mˆbethe˜3,Mˆ3bethemaximumvaluesof|f|,|k|,|g|,|g˜|,|gˆ|on|x|≤A;andM1M2,M3,M2333 Lipschitzconstantsoff,k,g,g˜,gˆon|x|≤A,respectively.Consider 2 B2+(2M+M)B+M+M,M22113 1 ,N= M2B2+M1B+M3+M0 2˜=M, ˜2MB+(2M2+M)B+M+M1M= 2 1 3 ˜=Nˆ=Mˆ=N 1 ˜3+M0M2B2+M1B+M 2 B2M2 , , ,and ˆ3+M0M2B2+M1B+M˜,Lˆ},where0 L=minT,2AN,2BN,2M+1−2, ˜=minT,2AN˜+1−2,˜,2BN˜,2ML ˆ,2BNˆ,2Mˆ+1−2.ˆ=minT,2ANL +(2M2+1 )BM1 ˆ+M1+M3 and Proposition2.1nowimpliesthatforeacha1≤b≤a2,theEq.(1.1)hasauniquesolution,sayxb(t),satisfying (t)|≤Bholdforeach0≤t≤T.xb(0)=xb(T0)=bforwhich|xb(t)|≤Aand|xb0Lemma3.1.Foreach0≤t≤T0,wehavexa1(t)≤a1 x+f(x)+k(x)xx+g˜(x)=p(t) hasauniquesolutionx(t)satisfyingx(0)=x(T0)=a1forwhich|x(t)|≤Aand|x(t)|≤Bholdforeach0≤t≤T0.Weclaimthatx(t)≤a1holdsforeach0≤t≤T0.Suppose,forthepurposeofacontradiction,there ˜)>a1.Thereforethefunctionx(t)−a1hasapositivemaximumontheinterval˜≤T0suchthatx(texistsapoint0≤t (0,T0),sayatt1.Hence(x(t)−a1)|t=t1=0,orx(t1)=0.Thereforewehaveestablished ˜(x(t1))+p(t1)x(t1)=−f(x(t1))+k(x(t1))x(t1)x(t1)−g =−g˜(x(t1))+p(t1) M.R.Pournaki,A.Razani/AppliedMathematicsLetters20(2007)248–254253 =−g(a1)−a1+x(t1)+p(t1)=(p(t1)−g(a1))+(x(t1)−a1)>0. Thisimpliesthat(x(t)−a1)|t=t1>0,whichisacontradictionsincex(t)−a1hasamaximumatt1.Thereforefor ˜,g˜(x(t))=g(x(t))holdsforeach0≤t≤T0.Thismeanseach0≤t≤T0,x(t)≤a1andsobythedefinitionofg thatx(t)isasolutionof(1.1)satisfyingx(0)=x(T0)=a1forwhich|x(t)|≤Aand|x(t)|≤Bholdforeach0≤t≤T0.Theuniquenesspropertynowimpliesthatforeach0≤t≤T0,x(t)=xa1(t)andsoxa1(t)≤a1holdsforeach0≤t≤T0. Next,weprovethata2≤xa2(t)holdsforeach0≤t≤T0.ByProposition2.1,theequation ˆ(x)=p(t)x+f(x)+k(x)xx+ghasauniquesolutionx(t)satisfyingx(0)=x(T0)=a2forwhich|x(t)|≤Aand|x(t)|≤Bholdforeach 0≤t≤T0.Weclaimthata2≤x(t)holdsforeach0≤t≤T0.Suppose,forthepurposeofacontradiction,there ˆ≤T0suchthata2>x(tˆ).Thereforethefunctionx(t)−a2hasanegativeminimumontheintervalexistsapoint0≤t (0,T0),sayatt2.Hence(x(t)−a2)|t=t2=0,orx(t2)=0.Thereforewehaveestablished x(t2)=−f(x(t2))+k(x(t2))x(t2)x(t2)−gˆ(x(t2))+p(t2) =−gˆ(x(t2))+p(t2) =−g(a2)−a2+x(t2)+p(t2)=(p(t2)−g(a2))+(x(t2)−a2)<0. Thisimpliesthat(x(t)−a2)|t=t2<0,whichisacontradictionsincex(t)−a2hasaminimumatt2.Thereforefor ˆ,gˆ(x(t))=g(x(t))holdsforeach0≤t≤T0.Thismeanseach0≤t≤T0,a2≤x(t)andsobythedefinitionofg thatx(t)isasolutionof(1.1)satisfyingx(0)=x(T0)=a2forwhich|x(t)|≤Aand|x(t)|≤Bholdforeach0≤t≤T0.Theuniquenesspropertynowimpliesthatforeach0≤t≤T0,x(t)=xa2(t)andsoa2≤xa2(t)holdsforeach0≤t≤T0. ˆa1≤bˆ≤a2,suchthatxˆ(0)=xˆ(T0).Lemma3.2.Thereexistsb,bbProof.Definethefunctionθon[a1,a2]by θ(b)=xb(0)−xb(T0). UsingtheAscoli–ArzelaTheorem,onemayeasilyverifythatbothxb(t)andxb(t)arecontinuouson[0,T0]×[a1,a2]. Thisimpliesthatθiscontinuousalso.Ontheotherhand,notethatfori∈{1,2}, (0)=limxai t→0+ xai(t)−ai ,t xa(T0)=limi t→0+ ai−xai(T0−t) , t andtherefore, θ(ai)=xai(0)−xai(T0) xai(t)+xai(T0−t)−2ai =lim. tt→0+ ˆ,a1≤bˆ≤a2,suchthatθ(bˆ)=0,orSobyLemma3.1,weobtainθ(a1)≤0andθ(a2)≥0.Hencethereexistsb (T).xbˆ(0)=xbˆ0 Thereforexbˆ(t)isasolutionof(1.1)satisfyingthefollowingperiodicboundaryconditions: xbˆ(0)=xbˆ(T0). xbˆ(0)=xbˆ(T0), Byamethodsimilartotheoneusedin[21],wenowextendxbˆ(t)periodicallywithperiodT0toobtainaperiodicsolutionoftheEq.(1.1).Notethatthisperiodicsolutionisnontrivial,sincepisanonconstantforcedfunction. 254M.R.Pournaki,A.Razani/AppliedMathematicsLetters20(2007)248–254 Acknowledgments ThisworkwasdonewhilethefirstauthorwasaPostdoctoralResearchAssociateattheSchoolofMathematics,InstituteforStudiesinTheoreticalPhysicsandMathematics(IPM).BothoftheauthorswouldliketothanktheIPMforfinancialsupport.Alsotheauthorswouldliketothanktherefereeforhis/herinterestinthesubjectandmakingusefulsuggestionsandcommentswhichledtoimprovementofthefirstdraft.References [1]A.Capietto,Z.Wang,PeriodicsolutionsofLi´enardequationswithasymmetricnonlinearitiesatresonance,J.LondonMath.Soc.(2)68(1) (2003)119–132. 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