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

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aPeriodicbifurcationfromfamiliesofperiodicsolutionsforsemilineardi erentialequationswithLipschitzianperturbationsinBanachspacesMikhailKamenskii DepartmentofMathematics,VoronezhStateUniversity,394006Voronezh,Russiae-mail:mikhailkamenski@mail.ruOlegMakarenkov ResearchInstituteofMathematics,VoronezhStateUniversity,394006Voronezh,Russiae-mail:omakarenkov@math.vsu.ruPaoloNistri DipartimentodiIngegneriadell’Informazione,Universit`adiSiena,53100Siena,Italye-mail:pnistri@dii.unisi.itAbstract.LetA:D(A)→E,D(A) E,beanin nitesimalgeneratoreitherofananalyticcompactsemigrouporofacontractiveC0-semigroupoflinearoperatorsactinginaBanachspaceE.Inthispaperwegivebothnecessaryandsu cientconditionsforbifurcationofT-periodicsolutionsfortheequationx˙=Ax+f(t,x)+εg(t,x,ε)fromak-parameterizedfamilyofT-periodicsolutionsoftheunperturbedequationcorrespondingtoε=0.Weshowthatbymeansofasuitablemodi cationoftheclassicalMel’nikovapproachwecanconstructabifurcationfunctionandto

formulatetheconditionsfortheexistenceofbifurcationintermsofthetopologicalindexofthebifurcationfunction.Todothis,sincetheperturbationtermgisonlyLipschitzianweneedtoextendtheclassicalLyapunov-Schmidtreductiontothepresentnonsmoothcase.

2000MathematicsSubjectClassi cation.34G05,37G15,47D05.

Keywords.Periodicbifurcation,Semigroups,Lipschitzperturbations,Lyapunov-Schmidtreduction.