Periodic bifurcation from families of periodic solutions for(10)

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

Itisaclassicalresult(seee.g.[11])that(C1)and(C2)ensuresrespectivelythattheintegralequations

t

x(t)=eAtξ+ AαeA(t s)f(s,A αx(s))+εg(s,A αx(s),ε)ds,(3.1)

0

x(t)=eAtξ+ teA(t s)[f(s,x(s))+εg(s,x(s),ε)]ds(3.2)

haveauniquesolutionx(·)de nedonsomeinterval[0,d],d>0.Bymeansofthisfunctionxwecande netheshiftoperatorasfollows.

De nition3.1Letx:[0,d]×E×[0,1]→Ebede nedat(t,ξ,ε)asx(t,ξ,ε)=x(t)forallt∈[0,d].Ifforsomeξ∈Eandε∈[0,1]wehavethatx(·,ξ,ε)isde nedonthewholetimeinterval[0,T]thenforthesevaluesξandεwede nethePoincar´emapforsystem(1.1)as

Pε(ξ)=x(T,ξ,ε).

Acrucialroleinwhatfollowsisplayedbythefollowingtechnicallemma.

Lemma3.1Assumethateither(C1)or(C2)issatis ed.Assumethatforsomeξ0∈Etheshiftoperator(t,ξ,ε)→x(t,ξ,ε)iswellde nedfort=T,ξ=ξ0andε=0.Thenthereexistsr>0suchthatthisoperatoriswellde nedfort=T,anyξ∈BE(ξ0,r),anyε∈[0,r]andthefunction

u(t,ξ,ε)=x(t,ξ,ε) x(t,ξ,0)