Periodic bifurcation from families of periodic solutions for(7)

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

Therefore,

π1,h[P(β(h,ε)+S(h)) (β(h,ε)+S(h))+εQ(β(h,ε),ε)]=0(2.11)hasasolutionh=H(ξ).Sincer1>0hasbeenchoseninsuchawaythatS′(h)isinvertibleonE1,hforh∈BRk(h0,r1)then(2.11)canberewrittenas(2.5).Assume ∈Easnowthat(2.5)issatis edwithsome(h ,ε )∈BRk(h0,r1)×[0,r1].De neξ

) ξ +εQ(ξ, ε )]=0.OntheotherSince(S′(h )) 1isinvertiblethenπ1,h [P(ξhandfrom(2.12)wehave

π2,h [P(π2,h β(h ,ε )+S(h )) (π2,h β(h ,ε )+S(h ))+ ) ξ +εQ(ξ, ε)].+εQ(β(h ,ε )+S(h ),ε)]=π2,h [P(ξ =β(h ,ε )+S(h ).ξ(2.12)

Thus(ξ ,ε )solves(2.1)andsotheproofiscomplete.

ThefollowingtworesultsareconsequencesofTheorem2.1andtheyprovide,re-spectively,anecessaryandasu cientconditionfortheexistenceofsolutionsto(2.1)nearξ0whenε>0issu cientlysmall.Theseconditionsareexpressedintermsofthefollowingbifurcationfunction

M(h)=(S′(h)) 1π1,h[Q(S(h0),0)

1 (P′(S(h)) I)(π2,h(P′(S(h)) I)π2,h)π2,hQ(S(h),0)],

wherehvariesinasu cientlysmallneighborhoodofh0∈Rk.

Wecanprovethefollowing.

Theorem2.2LetalltheassumptionsofTheorem2.1besatis ed.Assumethatthereexistsequencesεn→0andξn→ξ0asn→∞suchthat(ξn,εn)solves(2.1).Then

M(h0)=0.(2.13)

Proof.ByTheorem2.1,forn≥n0,withn0∈Nsu cientlylarge,wehavethat

(S′(hn)) 1π1,hn[P(β(hn,εn)+S(hn))

(2.14) (β(hn,εn)+S(hn))+εnQ(β(hn,εn)+S(hn),εn)]=0

wherehn=H(ξn).Ontheotherhandn0canbechosensu cientlylargeinsuchawaythat

P(S(hn)) S(hn)=0forn≥n0

thus,forn≥n0,(2.14)canberewrittenas

(S′(hn)) 1π1,hn[(P′(S(hn)) I)β(hn,εn)

(2.15)εn+Q(β(hn,εn)+S(hn),εn)]=0.

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