Periodic bifurcation from families of periodic solutions for(16)

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

Observe,thatifthereexistsr>0,M>0andξ:Rk×[0,r]→Esatisfying

ξ(·,ε)∈C0(V,E),ξ(h,ε)→ξ(h,0)asε→0uniformlyinh∈V,(5.2)suchthat

b’)

Φh,εinBE(0,r),

d’) ξ(h,ε) ≤Mεforanyh∈V,ε∈[0,r],

thenβ(h,ε)=πhξ(h,ε)satis esa),b),c)andd).

Toprovethisassertionfromassumption4wehave

Φh,0)′(·)takingintoaccountthatP (h,·)actsonEhwehave

(

Φh,0)′(0)isinvertibleonEforh∈V,todothisweshowthat

givenb∈Ethereexistsauniqueab∈Esuchthat

(

Φh,0)′(0)) 1πhis

continuousinh.Now,introducing

Φh,ε(ξ)=

P(h,0)=0,

3’)P′

ξ(h,0) 1iscontinuousinh.

LetΦ h,ε(ξ)= Φh,ε(ξ).Since

Vthereexistsr(h)>0suchthat

I (Φ bh,0)′(ξ) ≤1/4

forany ξ ≤r(h)andany h∈BRk(h,r(h))∩