Periodic bifurcation from families of periodic solutions for(4)

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

Thenthereexist0<r2<r1<r0andfunctionsH:BE(ξ0,r1)→Rk,withH(ξ)→h0asξ→ξ0andβ:BRk(h0,r1)×[0,r1]→E,β(·,ε)∈C0(BRk(h0,r1),E), β(h,ε) ≤MεforsomeM>0,anyh∈BRk(h0,r1)andanyε∈[0,r1]with

β(h,ε)∈E2,h,(2.3)

andβ(h,ε)/ε→ (π2,h(P′(S(h)) I)π2,h) 1π2,hQ(S(h),0)asε→0,uniformlyinh∈B(2.4)

Rk(h0,r1)

suchthatthefollowingpropertieshold:

1)if(ξ,ε)∈BE(ξ0,r2)×[0,r2]isasolutiontoequation(2.1)then(h,ε),whereh=H(ξ),isasolutionto

(S′(h)) 1π1,h[P(β(h,ε)+S(h))

(β(h,ε)+S(h))+εQ(β(h,ε)+S(h),ε)]=0.(2.5)

2)if(h,ε)∈BRk(h0,r1)×[0,r1]solves(2.5)then(ξ,ε)solves(2.1),with

ξ=β(h,ε)+S(h)(2.6)

Note,thattheexistenceof(S′(h)) 1onE1,hforh∈Rksu cientlyclosetoh0isguaranteedby(A2)and(A3).ToproveTheorem2.1weneedthefollowingversionoftheimplicitfunctiontheorem.

Lemma2.1LetEbeaBanachspaceandV Rkbeanopenboundedset.Con-siderafamilyofprojectors{πh}h∈VonEcontinuousinhandletEh=πhEforanyh∈V.LetΦh,ε:Eh→Ehbede nedby

Φh,ε(β)=P

whereP (h,β)+εQ (h,β,ε),(2.7) ∈C0(Rk×E,E),Q ∈C0(Rk×E×[0,1],E),P (h,·),Q (h,·,ε):Eh→Ehforanyh∈V,ε∈[0,1].Assumethat

1.thecontinuityofP

×E, inthe rstvariableisuniformonanyboundedsubsetofV

2.P isdi erentiablewithrespecttothesecondvariableandthederivativeiscontinuousin