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

Sincethefamily h∈VBRk(h,r(h))coverstheset

V.

Byassumption

3

thereisL>0suchthat (

1′Pξ(h1,0)

Finallythecontinuityassumptions1,2and3implythatξsatis es(5.2)andd′). (h1,ξ(h1,ε1),ε2) Q (h1,ξ(h1,ε1),ε1) .Q References

[1]R.R.Akhmerov,M.I.Kamenskii,A.S.Potapov,A.E.RodkinaandB.N.

Sadovskii,Measuresofnoncompactnessandcondensingoperators.Translatedfromthe1986RussianoriginalbyA.Iacob.OperatorTheory:AdvancesandApplications,55.BirkhauserVerlag,Basel,1992.

[2]M.BertiandP.Bolle,Multiplicityofperiodicsolutionsofnonlinearwave

equations,NonlinearAnal.56(2004),1011–1046.

[3]A.Buica,J.LlibreandO.Makarenkov,Lyapunov–Schmidtreductionfornon-

smoothfunctionswithapplicationtoMalkin’sproblemontheexistenceofpe-riodicsolutions,inprogress.

[4]S.N.ChowandJ.K.Hale,Methodsofbifurcationtheory.Grundlehren

derMathematischenWissenschaften,251,Springer-Verlag,NewYork-Berlin,1982.

17