Periodic bifurcation from families of periodic solutions for(12)

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

Dividingthelastinequalityby ξ1 ξ2 oneobtainsthat

x(t,ξ1,ε) x(t,ξ2,ε) x(s,ξ1,ε) x(s,ξ2,ε)

(t s)α·

ξ1 ξ2 ≤Mv

forany(t,ξ1,ξ2,ε)∈[0,T]×BE(ξ0,r)×BE(ξ0,r)×[0,r].

Forthefunctionu(t,ξ,ε)wehavethefollowinginequality

φ,u(t,ξ,ε) =

= φ,1

s)αds+ tcM

(t0 (3.5)