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

Bymeansofproperty(2.4)wecanpasstothelimitasn→∞in(2.15)toobtain(2.13).

Theorem2.3LetalltheassumptionsofTheorem2.1besatis ed.Assumethat

h0isanisolatedzeroofM

and

ind(h0,M)=0.

Then,foranyε>0su cientlysmallthereexistsξε∈Esuchthat

F(ξε,ε)=0

and

ξε→ξ0asε→0.(2.18)

Proof.Letr1>0beasgivenbyTheorem2.1.Since

P(S(h))=S(h)foranyh∈BRk(h0,r1)

thenthezerosofthefunction

Φ(h,ε)=(S′(h)) 1(2.16)(2.17)(2.19)π1,h[P(β(h,ε)+S(h))

(β(h,ε)+S(h))+εQ(β(h,ε)+S(h),ε)]

coincidewiththezerosofthefunction

Mε(h)=(S′(h)) 1π1,h[(P′(S(h)) I)β(h,ε)

ε+Q(β(h,ε)+S(h),ε)].

InordertoapplyTheorem2.1weshownowthatr∈(0,r1]canbechoseninsuchawaythatthefunctionMεhaszerosinBRk(h0,r)foranyε>0su cientlysmall.Bycondition(2.16)r>0canbechosensu cientlysmallinsuchawaythat

theonlyzeroofMinBRk(h0,r)ish0.

Therefore,bycondition(2.17)wehave

d(M,BRk(h0,r))=ind(h0,M)=0.

Ontheotherhandfromproperty(2.4)wehavethat

Mε(h)→M(h)asε→0

uniformlywithrespecttoh∈BRk(h0,r).Thusweconcludethat

d(Mε,Br(h0))=0

8(2.21)(2.20)