Periodic bifurcation from families of periodic solutions for(3)

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

propertiesofthePoincar´emapforsystem(1.1)areestablishedinSection3.Bothnecessaryandsu cientconditionsforbifurcationofperiodicsolutionsto(1.1)areobtainedinSection4.Finally,intheappendixofSection5wegiveaproofofatechnicalresultneededinSection3.

2Lyapunov-Schmidtreduction

F(ξ,ε)=P(ξ) ξ+εQ(ξ,ε),LetEbeaBanachspaceandconsiderthefunctionF:E×[0,1]→Egivenby

whereP:E→EandQ:E×[0,1]→E.Assumethat

(A1)thereexisth0∈Rk,r0>0andafunctionS∈C1(BRk(h0,r0),E)suchthat

P(ξ)=ξforanyξ∈Z={S(h):h∈BRk(h0,r0)}.

HereandinwhatfollowsBX(c,r)denotestheballinthenormedspaceXcenteredatcwithradiusr>0.Itiswellknownthat,undertheassumption(A1)withP∈C1(E,E)andQ∈C1(E×[0,1],E),theLyapunov-Schmidtreduction([4],Ch.2,§4)allowstosolvetheequation

F(ξ,ε)=0,(2.1)

forε>0su cientlysmall.NexttheoremextendsthisresulttothecasewhenQsatis esthefollowingLipschitzcondition:

(L)ForanyR>0thereexistsL(R)>0suchthat

Q(ξ1,ε) Q(ξ2,ε) ≤L(R) ξ1 ξ2

wheneverξ1,ξ2∈BE(0,R)andε∈[0,1].

Theorem2.1LetP∈C1(E,E),Q∈C0(E×[0,1],E),whereEisaBanachspace.AssumethatQsatis es(L).Moreover,assume(A1)and

(A2)dimS′(h0)Rk=k.

LetE1,h=S′(h)Rk.LetE2,hbeanysubspaceofEsuchthatE=E1,h

assumethat E2,hand(A3)thereexistsr0>0suchthatboththeprojectorsπ1,hofEontoE1,halong

E2,handπ2,hofEontoE2,halongE1,harecontinuousinh∈BRk(h0,r0),(A4)forξ0=S(h0)wehave

π2,h0(P′(ξ0) I)π2,h0isinvertibleonE2,h0.

3(2.2)