Periodic bifurcation from families of periodic solutions for semilinear differential equati

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

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aPeriodicbifurcationfromfamiliesofperiodicsolutionsforsemilineardi erentialequationswithLipschitzianperturbationsinBanachspacesMikhailKamenskii DepartmentofMathematics,VoronezhStateUniversity,394006Voronezh,Russiae-mail:mikhailkamenski@mail.ruOlegMakarenkov ResearchInstituteofMathematics,VoronezhStateUniversity,394006Voronezh,Russiae-mail:omakarenkov@math.vsu.ruPaoloNistri DipartimentodiIngegneriadell’Informazione,Universit`adiSiena,53100Siena,Italye-mail:pnistri@dii.unisi.itAbstract.LetA:D(A)→E,D(A) E,beanin nitesimalgeneratoreitherofananalyticcompactsemigrouporofacontractiveC0-semigroupoflinearoperatorsactinginaBanachspaceE.Inthispaperwegivebothnecessaryandsu cientconditionsforbifurcationofT-periodicsolutionsfortheequationx˙=Ax+f(t,x)+εg(t,x,ε)fromak-parameterizedfamilyofT-periodicsolutionsoftheunperturbedequationcorrespondingtoε=0.Weshowthatbymeansofasuitablemodi cationoftheclassicalMel’nikovapproachwecanconstructabifurcationfunctionandto

formulatetheconditionsfortheexistenceofbifurcationintermsofthetopologicalindexofthebifurcationfunction.Todothis,sincetheperturbationtermgisonlyLipschitzianweneedtoextendtheclassicalLyapunov-Schmidtreductiontothepresentnonsmoothcase.

2000MathematicsSubjectClassi cation.34G05,37G15,47D05.

Keywords.Periodicbifurcation,Semigroups,Lipschitzperturbations,Lyapunov-Schmidtreduction.

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

1Introduction

Theaimofthispaperistogivebothnecessaryandsu cientconditionsforthebifurcationofT-periodicsolutionsofthesemi-lineardi erentialequation

x˙=Ax+f(t,x)+εg(t,x,ε)(1.1)

fromak-parameterizedfamilyofT-periodicsolutionsoftheunperturbedsystem,obtainedfrom(1.1)bylettingε=0.HereA:D(A)→E,D(A) E,isanin- nitesimalgeneratoreitherofananalyticcompactsemigrouporofacontractiveC0-semigroupoflinearoperatorsactingintheBanachspaceE,thenonlinearop-eratorsf∈C1(R×E,E)andg∈C0(R×E×[0,1],E)areT-periodicinthe rstvariable.

Inthecasewhentheunperturbedsystemisautonomoustheproblemwasstud-iedbyHenryin([7],Ch.8),whereitisassumedthatgisdi erentiableinthesecondvariable.Theauthorprovidedsu cientconditionsforbifurcationofT-periodicsolutionsfromaT-periodiccyclex0,themaintoolemployedinthatpaperistheclassicalLyapunov-Schmidtreduction,seeforinstanceChowandHale([4],Ch.2,§4).Theseconditionsareformulatedintermsoftheexistenceofnondegen-eratezerosofananalogueoftheMalkin’sbifurcationfunction[12]foranin nitedimensionalBanachspace.

Inthe nitedimensionalcase,usingtopologicaldegreearguments,FelmerandMan´asevichin[5]replacedtheassumptionoftheexistenceofnondegeneratezerosofthebifurcationfunctionbytherequestthatthetopologicaldegreeofthebifurcationfunctionisdi erentfromzerowithrespecttoasuitableset.Startingfrom[5]therehasbeenagreatamountofworkfordevelopingbifurcationresultsbyusingthetopologicaldegreetheory,seee.g.HenrardandZanolin[6]forbifurcationfromacycleofaHamiltoniansystemandKamenskii,MakarenkovandNistri[8]forbifurcationfromacycleofaself-oscillatingsystem.Inthepresentpaperweavoidtherequirementthatthezerosofthebifurcationfunctionarenondegenerate,insteadweformulatesuitableassumptionsonthebifurcationfunctionintermsofthetopologicaldegreetoobtainfor(1.1)resultssimilartothoseof([7],Ch.8).TothisendweproveanextensionoftheclassicalLyapunov-Schmidtreductionaspresentedin([4],Ch.2,§4)tothecasewhentheperturbationgisLipschitzian.Wementioninthesequelsomeproblemsinvolvingpartialdi erentialequationswhichreducetothesituationconsideredinthispaper.InChowandHale[4,Ch.8,§6]andSchae erandGolubitsky[14]theproblemofthedependanceofthesteadystatesinchemicalreactionmodelsontherelativedi usioncoe cientsleadstotheconsiderationofperturbedequationsinBanachspaceswiththepropertythatthecorrespondingunperturbedequationshaveafamilyofsolutions.

AnotherexampleofsuchasituationispresentedinBertiandBolle[2],wheretheproblemof ndingperiodicsolutionsofanonlinearwaveequationbyvariationalmethodsgivesrisetoanunperturbedequationwithafamilyofperiodicsolutions.Thepaperisorganizedasfollows.Amodi edLyapunov-SchmidtreductionforLipschitzianperturbationsofanoperatoroftheform(P I),withP∈C1(E,E),isobtainedinSection2.InordertoapplytheresultsofSection2somerelevant

2

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

propertiesofthePoincar´emapforsystem(1.1)areestablishedinSection3.Bothnecessaryandsu cientconditionsforbifurcationofperiodicsolutionsto(1.1)areobtainedinSection4.Finally,intheappendixofSection5wegiveaproofofatechnicalresultneededinSection3.

2Lyapunov-Schmidtreduction

F(ξ,ε)=P(ξ) ξ+εQ(ξ,ε),LetEbeaBanachspaceandconsiderthefunctionF:E×[0,1]→Egivenby

whereP:E→EandQ:E×[0,1]→E.Assumethat

(A1)thereexisth0∈Rk,r0>0andafunctionS∈C1(BRk(h0,r0),E)suchthat

P(ξ)=ξforanyξ∈Z={S(h):h∈BRk(h0,r0)}.

HereandinwhatfollowsBX(c,r)denotestheballinthenormedspaceXcenteredatcwithradiusr>0.Itiswellknownthat,undertheassumption(A1)withP∈C1(E,E)andQ∈C1(E×[0,1],E),theLyapunov-Schmidtreduction([4],Ch.2,§4)allowstosolvetheequation

F(ξ,ε)=0,(2.1)

forε>0su cientlysmall.NexttheoremextendsthisresulttothecasewhenQsatis esthefollowingLipschitzcondition:

(L)ForanyR>0thereexistsL(R)>0suchthat

Q(ξ1,ε) Q(ξ2,ε) ≤L(R) ξ1 ξ2

wheneverξ1,ξ2∈BE(0,R)andε∈[0,1].

Theorem2.1LetP∈C1(E,E),Q∈C0(E×[0,1],E),whereEisaBanachspace.AssumethatQsatis es(L).Moreover,assume(A1)and

(A2)dimS′(h0)Rk=k.

LetE1,h=S′(h)Rk.LetE2,hbeanysubspaceofEsuchthatE=E1,h

assumethat E2,hand(A3)thereexistsr0>0suchthatboththeprojectorsπ1,hofEontoE1,halong

E2,handπ2,hofEontoE2,halongE1,harecontinuousinh∈BRk(h0,r0),(A4)forξ0=S(h0)wehave

π2,h0(P′(ξ0) I)π2,h0isinvertibleonE2,h0.

3(2.2)

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

Thenthereexist0<r2<r1<r0andfunctionsH:BE(ξ0,r1)→Rk,withH(ξ)→h0asξ→ξ0andβ:BRk(h0,r1)×[0,r1]→E,β(·,ε)∈C0(BRk(h0,r1),E), β(h,ε) ≤MεforsomeM>0,anyh∈BRk(h0,r1)andanyε∈[0,r1]with

β(h,ε)∈E2,h,(2.3)

andβ(h,ε)/ε→ (π2,h(P′(S(h)) I)π2,h) 1π2,hQ(S(h),0)asε→0,uniformlyinh∈B(2.4)

Rk(h0,r1)

suchthatthefollowingpropertieshold:

1)if(ξ,ε)∈BE(ξ0,r2)×[0,r2]isasolutiontoequation(2.1)then(h,ε),whereh=H(ξ),isasolutionto

(S′(h)) 1π1,h[P(β(h,ε)+S(h))

(β(h,ε)+S(h))+εQ(β(h,ε)+S(h),ε)]=0.(2.5)

2)if(h,ε)∈BRk(h0,r1)×[0,r1]solves(2.5)then(ξ,ε)solves(2.1),with

ξ=β(h,ε)+S(h)(2.6)

Note,thattheexistenceof(S′(h)) 1onE1,hforh∈Rksu cientlyclosetoh0isguaranteedby(A2)and(A3).ToproveTheorem2.1weneedthefollowingversionoftheimplicitfunctiontheorem.

Lemma2.1LetEbeaBanachspaceandV Rkbeanopenboundedset.Con-siderafamilyofprojectors{πh}h∈VonEcontinuousinhandletEh=πhEforanyh∈V.LetΦh,ε:Eh→Ehbede nedby

Φh,ε(β)=P

whereP (h,β)+εQ (h,β,ε),(2.7) ∈C0(Rk×E,E),Q ∈C0(Rk×E×[0,1],E),P (h,·),Q (h,·,ε):Eh→Ehforanyh∈V,ε∈[0,1].Assumethat

1.thecontinuityofP

×E, inthe rstvariableisuniformonanyboundedsubsetofV

2.P isdi erentiablewithrespecttothesecondvariableandthederivativeiscontinuousin

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

Thenthereexistr>0,M>0andafunctionβ:V×[0,r]→E,β(·,ε)∈C0(V,E)suchthat

a)β(h,ε)∈Ehforanyh∈V,ε∈[0,r],

b)Φh,ε(β(h,ε))=0foranyh∈V,ε∈[0,r],

c)β(h,ε)istheonlyzeroofΦh,εinBEh(0,r)foranyh∈V,ε∈[0,r],

d) β(h,ε) ≤Mεforanyh∈V,ε∈[0,r].

AlthoughLemma2.1lookswell-known,theauthorswereunableto ndaproofofitintheliterature,thusforthereaderconvenienceweprovideaproofofLemma2.1intheAppendixofSection5.

ProofofTheorem2.1.Inordertode nethefunctionβweconsiderthefollowingauxiliaryfunctionΦh,ε∈C0(E2,h,E2,h)givenby

Φh,ε(β)=π2,h[P(π2,hβ+S(h)) (π2,hβ+S(h))+εQ(β+S(h),ε)].

SinceP∈C1(E,E)andS∈C1(BRk(h0,r0),E)thenassumptions1and2ofLemma2.1aresatis ed.

Byourassumptionswehavethattheapplication(h,β,ε)→Φh,ε(β)isLipschitzianinβuniformlyonanyboundedsubsetofBRk(h0,r0)×E×[0,1]andtakingintoaccount(A1)wehave

1)Φh,0(0)=0foranyh∈BRk(h0,r0).

Byassumptions(A3)-(A4)r0>0canbediminishedinsuchawaythat

2)(Φh,0)(0)=π2,h(P′(S(h)) I)π2,hisaninvertibleoperatorfromE2,htoE2,hforh∈BRk(h0,r0).

Therefore,Lemma2.1applieswith

(h,β)=π2,h[P(π2,hβ+S(h)) (π2,hβ+S(h))],P′

Thusthereexistr1∈[0,r0],M>0andafunctionβ(·,ε)∈C0(BRk(h0,r1),E)satisfyingPropertiesa),b),c)andd)ofLemma2.1.Inparticular,fromPropertyb)wehave

π2,h[P(β(h,ε)+S(h)) (β(h,ε)+S(h))

(P(S(h)) S(h))+εQ(β(h,ε)+S(h),ε)]=0

orequivalently

π2,h[(P′(S(h)) I)π2,hβ(h,ε)+o(β(h,ε))+εQ(β(h,ε)+S(h),ε)]=0,foranyh∈BRk(h0,r1).

5 (h,β,ε)=π2,hQ(β+S(h),ε)andV=BRk(h0,r0).Q

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

Therefore

β(h,ε)= (π2,h(P′(S(h)) I)π2,h) 1(π2,ho(β(h,ε))+π2,hεQ(β(h,ε)+S(h),ε)).DuetoPropertyd)thelastequationimplies(2.4).Wenowproceedtode nethefunctionH.Forthisby(A2)wehavethatr1>0canbetakensu cientlysmallsuchthatS′(h):Rk→E1,hisinvertible.Thuswecande nethefunctionΦξ:Rk→Rk,ξ∈E,asfollows

Φξ(h)=(S′(h)) 1π1,h(ξ S(h)),h∈BRk(h0,r1).

WehavethefollowingpropertiesforΦξ.

1)Φξ0isdi erentiableath0.

2)(Φξ0)(h0)=(S′(h0))

vertiblek×k-matrix.′ 1π1,h0( S′(h0))= I,namely(Φξ0)(h0)isanin-′

Observethatproperty1)isadirectconsequenceofthefactthatξ0 S(h0)=0andthecontinuityofthefunctionh→S 1(h)πh,thereforethedi erentiabilityofπ1,hath=h0isnotnecessaryforthevalidityof1).

Letδ>0besuchthath0istheonlyzeroofΦξ0inBRk(h0,δ).By([10],Theorem6.3)wecanconsiderδ>0su cientlysmallinsuchawaythatd(Φξ0,BRk(h0,δ))=( 1)k.Bythecontinuitypropertyofthetopologicaldegreer1>0canbedi-minished,ifnecessary,insuchawaythatd(Φξ,BRk(h0,δ))=( 1)kforanyξ∈BE(ξ0,r1).Therefore,foranyξ∈BE(ξ0,r1)thereexistsH(ξ)∈BRk(h0,δ)suchthatΦξ(H(ξ))=0.LetusshowthatH(ξ)→h0asξ→ξ0.Indeed,arguingbycontradictionwewouldhaveasequence{ξn}n∈N BE(ξ0,r1),h ∈BRk(h0,δ)suchthatH(ξn)→h =h0asn→∞andthusΦξ0(h )=0contradictingthechoiceofδ>0.Therefore

π1,H(ξ)(ξ S(H(ξ)))=0,ξ∈BE(ξ0,r1).(2.8)

Moreover,weconsiderr2∈(0,r1]su cientlysmalltohave

ξ S(H(ξ)) ≤r1,ξ∈BE(ξ0,r2).(2.9)

Wearenowinthepositiontocompletetheproof.Forthislet(ξ,ε)∈BE(ξ0,r2)×

[0,r2]satisfying(2.1).Then(ξ,ε)alsosatis es

π[P(ξ S(H(ξ))+S(H(ξ))) 1,H(ξ) (ξ S(H(ξ))+S(H(ξ)))+εQ(ξ S(H(ξ))+S(H(ξ)),ε)]=0,

π[P(ξ S(H(ξ))+S(H(ξ))) 2,H(ξ) (ξ S(H(ξ))+S(H(ξ)))+εQ(ξ S(H(ξ))+S(H(ξ)),ε)]=0.

From(2.8),(2.9)andPropertyc)ofLemma2.1wehave

π1,H(ξ)[P(ξ S(H(ξ))+S(H(ξ)))

(ξ S(H(ξ))+S(H(ξ)))+εQ(ξ S(H(ξ))+S(H(ξ)),ε)]=0, β(H(ξ),ε)=ξ S(H(ξ)).

6(2.10)

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

Therefore,

π1,h[P(β(h,ε)+S(h)) (β(h,ε)+S(h))+εQ(β(h,ε),ε)]=0(2.11)hasasolutionh=H(ξ).Sincer1>0hasbeenchoseninsuchawaythatS′(h)isinvertibleonE1,hforh∈BRk(h0,r1)then(2.11)canberewrittenas(2.5).Assume ∈Easnowthat(2.5)issatis edwithsome(h ,ε )∈BRk(h0,r1)×[0,r1].De neξ

) ξ +εQ(ξ, ε )]=0.OntheotherSince(S′(h )) 1isinvertiblethenπ1,h [P(ξhandfrom(2.12)wehave

π2,h [P(π2,h β(h ,ε )+S(h )) (π2,h β(h ,ε )+S(h ))+ ) ξ +εQ(ξ, ε)].+εQ(β(h ,ε )+S(h ),ε)]=π2,h [P(ξ =β(h ,ε )+S(h ).ξ(2.12)

Thus(ξ ,ε )solves(2.1)andsotheproofiscomplete.

ThefollowingtworesultsareconsequencesofTheorem2.1andtheyprovide,re-spectively,anecessaryandasu cientconditionfortheexistenceofsolutionsto(2.1)nearξ0whenε>0issu cientlysmall.Theseconditionsareexpressedintermsofthefollowingbifurcationfunction

M(h)=(S′(h)) 1π1,h[Q(S(h0),0)

1 (P′(S(h)) I)(π2,h(P′(S(h)) I)π2,h)π2,hQ(S(h),0)],

wherehvariesinasu cientlysmallneighborhoodofh0∈Rk.

Wecanprovethefollowing.

Theorem2.2LetalltheassumptionsofTheorem2.1besatis ed.Assumethatthereexistsequencesεn→0andξn→ξ0asn→∞suchthat(ξn,εn)solves(2.1).Then

M(h0)=0.(2.13)

Proof.ByTheorem2.1,forn≥n0,withn0∈Nsu cientlylarge,wehavethat

(S′(hn)) 1π1,hn[P(β(hn,εn)+S(hn))

(2.14) (β(hn,εn)+S(hn))+εnQ(β(hn,εn)+S(hn),εn)]=0

wherehn=H(ξn).Ontheotherhandn0canbechosensu cientlylargeinsuchawaythat

P(S(hn)) S(hn)=0forn≥n0

thus,forn≥n0,(2.14)canberewrittenas

(S′(hn)) 1π1,hn[(P′(S(hn)) I)β(hn,εn)

(2.15)εn+Q(β(hn,εn)+S(hn),εn)]=0.

7

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)

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

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

Itisaclassicalresult(seee.g.[11])that(C1)and(C2)ensuresrespectivelythattheintegralequations

t

x(t)=eAtξ+ AαeA(t s)f(s,A αx(s))+εg(s,A αx(s),ε)ds,(3.1)

0

x(t)=eAtξ+ teA(t s)[f(s,x(s))+εg(s,x(s),ε)]ds(3.2)

haveauniquesolutionx(·)de nedonsomeinterval[0,d],d>0.Bymeansofthisfunctionxwecande netheshiftoperatorasfollows.

De nition3.1Letx:[0,d]×E×[0,1]→Ebede nedat(t,ξ,ε)asx(t,ξ,ε)=x(t)forallt∈[0,d].Ifforsomeξ∈Eandε∈[0,1]wehavethatx(·,ξ,ε)isde nedonthewholetimeinterval[0,T]thenforthesevaluesξandεwede nethePoincar´emapforsystem(1.1)as

Pε(ξ)=x(T,ξ,ε).

Acrucialroleinwhatfollowsisplayedbythefollowingtechnicallemma.

Lemma3.1Assumethateither(C1)or(C2)issatis ed.Assumethatforsomeξ0∈Etheshiftoperator(t,ξ,ε)→x(t,ξ,ε)iswellde nedfort=T,ξ=ξ0andε=0.Thenthereexistsr>0suchthatthisoperatoriswellde nedfort=T,anyξ∈BE(ξ0,r),anyε∈[0,r]andthefunction

u(t,ξ,ε)=x(t,ξ,ε) x(t,ξ,0)

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.

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)

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

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

By(3.6)and(3.5)thereexistsM>0suchthatthelastinequalitycanberewrittenas

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

(t s)α

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

Toseethis,observethatthefunctionuofLemma3.1satis esthefollowingintegralequation

t′u(t,ξ,ε)=AαeΛ(t s)fx(s,A αx(s,ξ,0))u(s,ξ,ε)ds+

0 to(x(s,ξ,ε) x(s,ξ,0))+AαeΛ(t s)

Φh,ε:E→Ebede nedby

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

Observe,thatifthereexistsr>0,M>0andξ:Rk×[0,r]→Esatisfying

ξ(·,ε)∈C0(V,E),ξ(h,ε)→ξ(h,0)asε→0uniformlyinh∈V,(5.2)suchthat

b’)

Φh,εinBE(0,r),

d’) ξ(h,ε) ≤Mεforanyh∈V,ε∈[0,r],

thenβ(h,ε)=πhξ(h,ε)satis esa),b),c)andd).

Toprovethisassertionfromassumption4wehave

Φh,0)′(·)takingintoaccountthatP (h,·)actsonEhwehave

(

Φh,0)′(0)isinvertibleonEforh∈V,todothisweshowthat

givenb∈Ethereexistsauniqueab∈Esuchthat

(

Φh,0)′(0)) 1πhis

continuousinh.Now,introducing

Φh,ε(ξ)=

P(h,0)=0,

3’)P′

ξ(h,0) 1iscontinuousinh.

LetΦ h,ε(ξ)= Φh,ε(ξ).Since

Vthereexistsr(h)>0suchthat

I (Φ bh,0)′(ξ) ≤1/4

forany ξ ≤r(h)andany h∈BRk(h,r(h))∩

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

Sincethefamily h∈VBRk(h,r(h))coverstheset

V.

Byassumption

3

thereisL>0suchthat (

1′Pξ(h1,0)

Finallythecontinuityassumptions1,2and3implythatξsatis es(5.2)andd′). (h1,ξ(h1,ε1),ε2) Q (h1,ξ(h1,ε1),ε1) .Q References

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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

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