Periodic bifurcation from families of periodic solutions for(5)

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

Thenthereexistr>0,M>0andafunctionβ:V×[0,r]→E,β(·,ε)∈C0(V,E)suchthat

a)β(h,ε)∈Ehforanyh∈V,ε∈[0,r],

b)Φh,ε(β(h,ε))=0foranyh∈V,ε∈[0,r],

c)β(h,ε)istheonlyzeroofΦh,εinBEh(0,r)foranyh∈V,ε∈[0,r],

d) β(h,ε) ≤Mεforanyh∈V,ε∈[0,r].

AlthoughLemma2.1lookswell-known,theauthorswereunableto ndaproofofitintheliterature,thusforthereaderconvenienceweprovideaproofofLemma2.1intheAppendixofSection5.

ProofofTheorem2.1.Inordertode nethefunctionβweconsiderthefollowingauxiliaryfunctionΦh,ε∈C0(E2,h,E2,h)givenby

Φh,ε(β)=π2,h[P(π2,hβ+S(h)) (π2,hβ+S(h))+εQ(β+S(h),ε)].

SinceP∈C1(E,E)andS∈C1(BRk(h0,r0),E)thenassumptions1and2ofLemma2.1aresatis ed.

Byourassumptionswehavethattheapplication(h,β,ε)→Φh,ε(β)isLipschitzianinβuniformlyonanyboundedsubsetofBRk(h0,r0)×E×[0,1]andtakingintoaccount(A1)wehave

1)Φh,0(0)=0foranyh∈BRk(h0,r0).

Byassumptions(A3)-(A4)r0>0canbediminishedinsuchawaythat

2)(Φh,0)(0)=π2,h(P′(S(h)) I)π2,hisaninvertibleoperatorfromE2,htoE2,hforh∈BRk(h0,r0).

Therefore,Lemma2.1applieswith

(h,β)=π2,h[P(π2,hβ+S(h)) (π2,hβ+S(h))],P′

Thusthereexistr1∈[0,r0],M>0andafunctionβ(·,ε)∈C0(BRk(h0,r1),E)satisfyingPropertiesa),b),c)andd)ofLemma2.1.Inparticular,fromPropertyb)wehave

π2,h[P(β(h,ε)+S(h)) (β(h,ε)+S(h))

(P(S(h)) S(h))+εQ(β(h,ε)+S(h),ε)]=0

orequivalently

π2,h[(P′(S(h)) I)π2,hβ(h,ε)+o(β(h,ε))+εQ(β(h,ε)+S(h),ε)]=0,foranyh∈BRk(h0,r1).

5 (h,β,ε)=π2,hQ(β+S(h),ε)andV=BRk(h0,r0).Q