On the existence of periodic solutions for a class of generalized forced Liénard equations

AppliedMathematicsLetters20(2007)

248–254

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)x󰀇󰀁

x󰀇+g(x)=p(t)isconsideredanda

newconditionforguaranteeingtheexistenceofatleastoneperiodicsolutionforthisequationisgiven.

c󰀃2006ElsevierLtd.Allrightsreserved.

Keywords:Nonlinearboundaryvalueproblem;Li´enardequation;Periodicsolution;Banachspace;Schauder’sFixedPointTheorem

1.Introduction

Inthisworkweinvestigatetheexistenceofperiodicsolutionsforaclassofsecond-ordergeneralizedforcedLi´enard

equations

x󰀇󰀇+󰀓

f(x)+k(x)x󰀇󰀁

x󰀇+g(x)=p(t),(1.1)

wheref,k,andgarerealfunctionsonRandpisaT-periodicrealfunctionon[0,T],T>0.Generalizedforced

Li´enardequationsappearinanumberofphysicalmodelsandanimportantquestioniswhethertheseequationscan

supportperiodicsolutions.Thisquestionhasbeenstudiedextensivelybyanumberofauthors;seeforexample[1–9].

Inparticular,therearesomeexistenceandmultiplicityresultsforsuchequationswithnonconstantforcedterms;see

forexample[10–19].Inthisdirection,wewillobtainanewconditiontoguaranteetheexistenceofatleastoneperiodic

solutionfor(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(a

2)holdsforeach

0≤t≤T,thenEq.(1.1)hasatleastoneperiodicsolution.

✩ThisresearchwasinpartsupportedbyagrantfromIPM.

∗Correspondingauthor.

E-mailaddresses:pournaki@ipm.ir(M.R.Pournaki),razani@ikiu.ac.ir(A.Razani).

0893-9659/$-seefrontmatterc󰀃2006ElsevierLtd.Allrightsreserved.

doi:10.1016/j.aml.2006.06.004M.R.Pournaki,A.Razani/AppliedMathematicsLetters20(2007)248–254249

Therestoftheworkisorganizedasfollows.InSection2,weprovethat(1.1)hasauniquesolutionsatisfying

certainconditionsbyapplyingSchauder’sFixedPointTheorem.InSection3,theexistenceofatleastoneperiodic

solutionfor(1.1)whenghasthepropertymentionedintheMainTheoremisproved.

2.Anexistenceanduniquenesstyperesult

WestartthissectionbyrecallingafamousfixedpointtheoremwhichwasoriginallyduetoSchauder:LetXbea

BanachspaceandΩbeaclosed,bounded,andconvexsubspaceofX.IfS:Ω→Ωisacompactoperator,thenS

hasatleastonefixedpointonΩ.

WenowstateandprovethefollowingexistenceanduniquenesstyperesultwhichisakeytoolforprovingtheMain

Theorem.

Proposition2.1.Leta1

2andB>0berealnumbersandconsiderA=max{2|a1|,2|a

2|}.Supposef,k,and

garerealfunctionsonRwhicharelocallyLipschitzandatleastoneofthef,k,orgisnonconstanton|x|≤A;

andpisacontinuousT-periodicrealfunctionon[0,T],T>0.AlsosupposeM0isthemaximumvalueof|p|on

[0,T];M

1,M

2,M

3arethemaximumvaluesof|f|,|k|,|g|on|x|≤A;andM󰀇

1,M󰀇

2,M󰀇

3aretheLipschitzconstants

off,k,gon|x|≤A,respectively.Consider

M=2

M󰀇

2B2+(2M

2+M󰀇

1)B+M󰀇

3+M

1,

N=1

M2B2+M

1B+M3+M

0,and0

0

T,2√

AN,2BN,2√

M+1−2󰀏

.

Thenforeacha1≤b≤a

2,Eq.(1.1)hasauniquesolutionx(t),satisfying

x(0)=x(T0)=b,(2.1)

forwhich|x(t)|≤Aand|x󰀇(t)|≤Bholdforeach0≤t≤T

0.

Proof.Considertheequationx󰀇󰀇=0withboundaryconditionx(0)=x(T

0)=b.TheexistenceofaGreen’sfunction

foratypicaltwo-endpointproblemwassuggestedbyasimplephysicalexamplein[20]andisasfollows:

G(t,s)=󰀇

s(t−T0)/T

0:if0≤s≤t≤T

0,

t(s−T0)/T

0:if0≤t≤s≤T

0.

Ifwenowconsidertheintegralequation

x(t)=b+󰀍

T0

0G(t,s)󰀓󰀓

f(x(s))+k(x(s))x󰀇(s)󰀁

x󰀇(s)+g(x(s))−p(s)󰀁

ds,(2.2)

thenitiseasytoseethatthesolutionsof(2.2)areexactlythesolutionsof(1.1)satisfying(2.1).Hence,toprovethe

proposition,itisenoughtoshowthat(2.2)hasauniquesolutionx(t)satisfying|x(t)|≤Aand|x󰀇(t)|≤Bforeach

0≤t≤T0.Inordertodoso,supposeX=C1([0,T

0],R),andforφ∈Xdefine

󰀖φ󰀖=max

0≤t≤T0|φ(t)|+max

0≤t≤T0|φ󰀇(t)|.

ItisclearthatXisaBanachspace.Now,consider

Ω=󰀂

φ∈X:|φ(t)|≤Aand|φ󰀇(t)|≤Bholdforeach0≤t≤T

0󰀃

,

whichisobviouslyaclosed,bounded,andconvexsubspaceofX.DefinetheoperatorS:Ω→Xbymappingφto

S(φ),whereS(φ)isdefinedby

S(φ)(t)=b+󰀍

T0

0G(t,s)󰀓󰀓

f(φ(s))+k(φ(s))φ󰀇(s)󰀁

φ󰀇(s)+g(φ(s))−p(s)󰀁

ds.