Periodic bifurcation from families of periodic solutions for(11)

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

Fromthecontinuousdi erentiabilityoffandtheLipschitzconditionongassumed >0suchthatin(C1)and(C2)wededucetheexistenceofM

foranyt∈[0,T],ξ∈x([0,T],BE(ξ0,r),[0,r])andε∈[0,r].

SinceA αx([0,T],BE(ξ0,r),[0,r])isboundedthenbyusingtheLipschitzcondition >0suchthatongweobtaintheexistenceofL

foranys∈[0,T],ξ1,ξ2∈x([0,T],BE(ξ0,r),[0,r])andε∈[0,r]. ξ1 ξ2 g(s,A αξ1,ε) g(s,A αξ2,ε) ≤L f(t,A αξ) + g(t,A αξ,ε) ≤M

Nowgivenanarbitraryφ∈BE (0,1),whereE denotesthedualspaceofE,weevaluate φ,x(t,ξ1,ε) x(t,ξ2,ε) asfollows

φ,x(t,ξ1,ε) x(t,ξ2,ε) =

t At αA(t s)′φ,Aefxs,A α{θ(s,ξ1,ξ2,ε)x(s,ξ1,ε)+=φ,e(ξ1 ξ2)+

+(1 θ(s,ξ1,ξ2,ε)x(s,ξ2,ε))}A αx(s,ξ1,ε) x(s,ξ2,ε)ds+

t 00 Furthermore,by[13,Theorem6.13]thereexistsc>0suchthatsup eAt <ct∈[0,T] αAt α andAe<c/t,whereeitherα=0orα>0.+εφ,AeαA(t s) α αgs,Ax(s,ξ2,ε),ε gs,Ax(s,ξ1,ε),εds≤

≤c ξ1 ξ2 + t

0 cM

(3.3)(t s)α x(s,ξ1,ε) x(s,ξ2,ε) ds.

Sinceφisarbitrarywehave

x(t,ξ1,ε) x(t,ξ2,ε) ≤c ξ1 ξ2 + t

0 cM

(3.4)(t s)α

11 x(s,ξ1,ε) x(s,ξ2,ε) ds.