Periodic bifurcation from families of periodic solutions for(13)

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

=| φ,Ψ′(ζ2+θ(ξ2 ζ2))(ξ2 ζ2) Ψ′(ζ1+θ(ξ1 ζ1))(ξ1 ζ1) |≤

≤| φ,Ψ′(ζ2+θ(ξ2 ζ2))(ξ2 ξ1 ζ2+ζ1) |+

+| φ,(Ψ′(ζ2+θ(ξ2 ζ2)) Ψ′(ζ1+θ(ξ1 ζ1)))(ξ1 ζ1) |≤

≤ Ψ′(ζ2+θ(ξ2 ζ2)) ξ2 ξ1 ζ2+ζ1 +

+ Ψ′(ζ2+θ(ξ2 ζ2)) Ψ′(ζ1+θ(ξ1 ζ1)) ξ1 ζ1 ≤

≤

θ∈[0,1]sup Ψ′(ζ2+θ(ξ2 ζ2) ξ2 ξ1 ζ2+ζ1 +

+L (1 θ)ζ2+θξ2 (1 θ)ζ1 θξ1 · ξ1 ξ2 =

=sup Ψ′(ζ2+θ(ξ2 ζ2) ξ2 ξ1 ζ2+ζ1 +

θ∈[0,1]

+L (1 θ)(ζ2 ζ1)+θ(ξ2 ξ1) ξ1 ξ2 ≤

≤

θ∈[0,1]sup Ψ′(ζ2+θ(ξ2 ζ2) ξ2 ξ1 ζ2+ζ1 +

+Lmax{ ξ2 ξ1 , ζ2 ζ1 } ξ1 ζ1 .

′ >0suchthatBytheLipschitzassumptiononfxthereexistsL

′′ ξ1 ξ2 (s,A 1ξ2) ≤L fx(s,A 1ξ1) fx

foranys∈[0,T],ξ1,ξ2∈x([0,T],BE(ξ0,r),[0,r]).

Considernow

u(t,ξ1,ε) u(t,ξ2,ε)

ε ξ1 ξ2

=1=

≤

≤ s∈[0,T],θ∈[0,1]t ξ1 ξ2 0′sup fx(s,A α(x(s,ξ2,0)+θ(x(s,ξ1,0) x(s,ξ2,0))))A α ≤c

ε ξ1 ξ2 x(s,ξ1,0) x(s,ξ1,ε)

ξ1 ξ2

ξ1 ξ2

13ds+ds.ds+ tAαeA(t s)(g(s,A αx(s,ξ1,ε)) g(s,A αx(s,ξ2,ε)))ds≤0+supmax· s∈[0,T]tε·+ 0t cL0 cL