Periodic bifurcation from families of periodic solutions for(9)

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

forε∈(0,ε0],whereε0>0issu cientlysmall.Thusforanyε∈(0,ε0]thereexistshεsuchthatMε(hε)=0.Moreover,wehavethat

hε→h0asε→0

otherwiseMwouldhavezerosinBRk(h0,r)di erentfromh0,contradicting(2.20).Finally,(2.18)followsfrom(2.6).

In nitedimensionalspacesresultssimilartopreviousTheorems2.2and2.3havebeenrecentlyobtainedbyBuica,LlibreandMakarenkov[3],wheretheuniquenessofthebifurcatingperiodicsolutionsisalsoproved.

3ThePoincar´emap

Sincethede nitionofthePoincar´emapforsystem(1.1)onthetimeinterval[0,T]dependsontheassumptionsonthelinearunboundedoperatorA,weprecisein(C1)and(C2)belowthetwocasesthatweconsiderforAinthepaper.

(C1)TheoperatorAisageneratorofananalyticcompactsemigroupeAtinE.The

operatorsf,garesubordinatedtosomeA α,0<α<1(seee.g.[11]),theoperatorf(·,A α·)isdi erentiableinthesecondvariableandtheoperators′f(2)(·,A α·),g(·,A α·,·)arecontinuousinR×EandtheysatisfyaLipschitzconditioninthesecondvariableuniformlywithrespecttotheothers.

(C2)TheoperatorAisageneratorofaC0-semigroupeAt.ThesemigroupeAtis

contractive,namely At e ≤e γt,

χ(f(t, ))≤kχ( ),χ(g(t, ,ε))≤kχ( ),

whereχistheHausdor measureofnoncompactness1inthespaceE,k≥0andq=k/γ<1.Theoperatorfisdi erentiableinthesecondvariableand′theoperatorsf(2)andgarecontinuousinR×EandtheysatisfyaLipschitzconditioninthesecondvariableuniformlywithrespecttotheothers.whereγ>0.TheoperatorsfandgarecontinuousfromR×E→Eandverifytheinequality