Periodic bifurcation from families of periodic solutions for(2)

Let A:D(A)\to E be an infinitesimal generator either of an analytic compact semigroup or of a contractive C_0-semigroup of linear operators acting in a Banach space E. In this paper we give both necessary and sufficient conditions for bifurcation of $T$-pe

1Introduction

Theaimofthispaperistogivebothnecessaryandsu cientconditionsforthebifurcationofT-periodicsolutionsofthesemi-lineardi erentialequation

x˙=Ax+f(t,x)+εg(t,x,ε)(1.1)

fromak-parameterizedfamilyofT-periodicsolutionsoftheunperturbedsystem,obtainedfrom(1.1)bylettingε=0.HereA:D(A)→E,D(A) E,isanin- nitesimalgeneratoreitherofananalyticcompactsemigrouporofacontractiveC0-semigroupoflinearoperatorsactingintheBanachspaceE,thenonlinearop-eratorsf∈C1(R×E,E)andg∈C0(R×E×[0,1],E)areT-periodicinthe rstvariable.

Inthecasewhentheunperturbedsystemisautonomoustheproblemwasstud-iedbyHenryin([7],Ch.8),whereitisassumedthatgisdi erentiableinthesecondvariable.Theauthorprovidedsu cientconditionsforbifurcationofT-periodicsolutionsfromaT-periodiccyclex0,themaintoolemployedinthatpaperistheclassicalLyapunov-Schmidtreduction,seeforinstanceChowandHale([4],Ch.2,§4).Theseconditionsareformulatedintermsoftheexistenceofnondegen-eratezerosofananalogueoftheMalkin’sbifurcationfunction[12]foranin nitedimensionalBanachspace.

Inthe nitedimensionalcase,usingtopologicaldegreearguments,FelmerandMan´asevichin[5]replacedtheassumptionoftheexistenceofnondegeneratezerosofthebifurcationfunctionbytherequestthatthetopologicaldegreeofthebifurcationfunctionisdi erentfromzerowithrespecttoasuitableset.Startingfrom[5]therehasbeenagreatamountofworkfordevelopingbifurcationresultsbyusingthetopologicaldegreetheory,seee.g.HenrardandZanolin[6]forbifurcationfromacycleofaHamiltoniansystemandKamenskii,MakarenkovandNistri[8]forbifurcationfromacycleofaself-oscillatingsystem.Inthepresentpaperweavoidtherequirementthatthezerosofthebifurcationfunctionarenondegenerate,insteadweformulatesuitableassumptionsonthebifurcationfunctionintermsofthetopologicaldegreetoobtainfor(1.1)resultssimilartothoseof([7],Ch.8).TothisendweproveanextensionoftheclassicalLyapunov-Schmidtreductionaspresentedin([4],Ch.2,§4)tothecasewhentheperturbationgisLipschitzian.Wementioninthesequelsomeproblemsinvolvingpartialdi erentialequationswhichreducetothesituationconsideredinthispaper.InChowandHale[4,Ch.8,§6]andSchae erandGolubitsky[14]theproblemofthedependanceofthesteadystatesinchemicalreactionmodelsontherelativedi usioncoe cientsleadstotheconsiderationofperturbedequationsinBanachspaceswiththepropertythatthecorrespondingunperturbedequationshaveafamilyofsolutions.

AnotherexampleofsuchasituationispresentedinBertiandBolle[2],wheretheproblemof ndingperiodicsolutionsofanonlinearwaveequationbyvariationalmethodsgivesrisetoanunperturbedequationwithafamilyofperiodicsolutions.Thepaperisorganizedasfollows.Amodi edLyapunov-SchmidtreductionforLipschitzianperturbationsofanoperatoroftheform(P I),withP∈C1(E,E),isobtainedinSection2.InordertoapplytheresultsofSection2somerelevant

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