Attraction property of local center-unstable manifolds for differential equations with state-dependent delay

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1 Elctronic Journal of Qualitativ Thory of Diffrntial Equation 2015, No. 4, 1 45; Attraction proprty of local cntr-untabl manifold for diffrntial quation with tat-dpndnt dlay Ddicatd to Profor Han-Otto Walthr on th occaion of hi 65th birthday Eugn Stumpf Dpartmnt of Mathmatic, Univrity of Hamburg, Bundtra 55, Hamburg, D 20146, Grmany Rcivd 14 Fbruary 2014, appard 13 Fbruary 2015 Communicatd by Tibor Kriztin Abtract. In th prnt papr w conidr local cntr-untabl manifold at a tationary point for a cla of functional diffrntial quation of th form ẋ(t) = f (x t ) undr aumption that ar dignd for application to diffrntial quation with tat-dpndnt dlay. Hr, w how an attraction proprty of th manifold. Mor prcily, w prov that, aftr fixing om local cntr-untabl manifold W cu of ẋ(t) = f (x t ) at om tationary point ϕ, ach olution of ẋ(t) = f (x t ) which xit and rmain ufficintly clo to ϕ for all t 0 and which do not blong to W cu convrg xponntially for t to a olution on W cu. Kyword: attraction, cntr-untabl manifold, functional diffrntial quation, tatdpndnt dlay Mathmatic Subjct Claification: 34K19, 34K25. 1 Introduction In th lat dcad th thory of diffrntial quation with tat-dpndnt dlay mad ignificant progr. Apart from othr rult, th framwork dvlopd by Walthr in [7, 8, 9] had a rmarkabl impact. Thi ri of work i concrnd with a cla of abtract functional diffrntial quation and contain a proof that undr crtain mild condition th olution of th aociatd Cauchy problm dfin a continuou miflow on a mooth ubmanifold of a function pac. In particular, th rulting miflow ha continuouly diffrntiabl olution oprator and th linarization of th miflow along a olution i dcribd by linar variational quation. Th vital point of that framwork with rpct to dlay diffrntial quation i th fact that it m to b typically applicabl in ca whr th functional diffrntial quation rprnt an autonomou diffrntial quation with tat-dpndnt dlay. Conquntly, undr th aumption of applicability, on obtain a gnral tting of mooth dynamical ytm for th tudy of diffrntial quation with tat-dpndnt dlay. ugn.tumpf@math.uni-hamburg.d

2 2 E. Stumpf Nowaday, th miflow mntiond abov i analyzd in variou articl and many of it dynamical apct ar wll-undrtood. For intanc, a gnral urvy of baic proprti togthr with th linarization proc at tationary point a wll a th principl of linarizd tability i prntd in [1]. In addition, [1] contain a proof of th xitnc of local tabl and local cntr manifold at tationary point. Th countrpart of th principl of linarizd tability, that i, th principl of linarizd intability i dicud in [4]. For th xitnc of continuouly diffrntiabl local untabl manifold at tationary point w rfr th radr to [2]. Th contruction of C 1 -mooth local cntr-untabl manifold i carrid out in [5], whra th author of [3] how th xitnc and moothn of local cntr-tabl manifold. In th prnt articl w addr anothr fatur from th dynamical ytm thory of miflow a laid out in th framwork [7, 8, 9]; namly, an attraction proprty of local cntruntabl manifold obtaind in [5]. W how that ach olution which tart and tay clo nough to a tationary point convrg xponntially for t to a olution on a local cntr-untabl manifold of th miflow. In particular, thi proprty provid an aymptotic dcription of th dynamic of uch olution: for all ufficintly larg t thy bhav lik olution on th conidrd local cntr-untabl manifold. Howvr, in ordr to formulat our main rult in dtail w hav to rcall om rlvant matrial. Thi i don blow without prnting proof. For a dpr dicuion of th thory and for th abnt proof w rfr th radr to [1, 7, 8, 9]. Throughout thi papr, lt h > 0, n N and lt R n dnot a norm in R n. Furthr, w writ C for th Banach pac of all continuou function from th intrval [ h, 0] into R n, providd with th uual norm ϕ C := up [ h,0] ϕ() R n of uniform convrgnc. Similarly, lt C 1 dnot th Banach pac of all continuouly diffrntiabl function ϕ : [ h, 0] R n with th norm ϕ C 1 := ϕ C ϕ C. Givn om function x : I R n dfind on om intrval I R, and om ral t R with [t h, t] I, th gmnt x t of x at t i dfind by x t () := x(t ), h 0. From now on, w conidr th functional diffrntial quation ẋ(t) = f (x t ) (1.1) givn by om function f : U R n dfind on om opn nighborhood U C 1 of th origin 0 C 1 and atifying f (0) = 0. A olution of Eq. (1.1) i ithr a continuouly diffrntiabl function x : [t 0 h, t ) R n with t 0 < t uch that x t U for all t 0 t < t and Eq. (1.1) hold for all t 0 < t < t, or a continuouly diffrntiabl function x : R R n atifying x t U and Eq. (1.1) for all t R, or a continuouly diffrntiabl function x : (, t r ] R n, t r R, uch that x t U for all t t r and Eq. (1.1) hold a t < t r. A f (0) = 0 by aumption, it i clar that x(t) = 0, t R, i a olution of Eq. (1.1) in th n abov. In particular, th ubt X f := {ϕ U ϕ (0) = f (ϕ)} of C 1 i not mpty. W impo that th function f additionally atifi th following condition: (S1) f i continuouly diffrntiabl, and (S2) for ach ϕ U th drivativ D f (ϕ) : C 1 R n xtnd to a linar map D f (ϕ) : C R n uch that th map U C (ϕ, χ) D f (ϕ)χ R n i continuou.

3 Attraction proprty of local cntr-untabl manifold 3 Thn th rult of th framwork [7, 8, 9] how that th ubt X f of U i a C 1 -mooth ubmanifold of codimnion n. Morovr, for ach ϕ X f thr i a uniqu ral t (ϕ) > 0 and a uniqu olution x ϕ : [ h, t (ϕ)) R n of Eq. (1.1) uch that x ϕ 0 = ϕ and xϕ i not continuabl in th forward tim dirction. For all ϕ X f and all 0 t < t (ϕ) th gmnt x ϕ t blong to X f, which i thrfor calld th olution manifold of Eq. (1.1). By aigning for all (t, ϕ) Ω whr F(t, ϕ) := x ϕ t Ω := {(, ψ) [0, ) X f 0 < t (ψ)}, w obtain a continuou miflow F : Ω X f with continuouly diffrntiabl tim-t-map. Sinc x(t) = 0, t R, i a olution of Eq. (1.1), it i clar that ϕ 0 := 0 U i a tationary point of th miflow F uch that F(t, 0) = 0 for all t 0. Th linarization of F at ϕ 0 = 0 i th trongly continuou migroup T = {T(t)} of boundd linar oprator T(t) := D 2 F(t, 0) on th Banach pac T 0 X f := {χ C 1 χ (0) = D f (0)χ} with th norm C 1 of C 1. Th action of an oprator T(t), t 0, on χ T 0 X f i givn by T(t)χ = v χ t, whr vχ : [ h, ) R n i th uniquly dtrmind olution of th variational quation v(t) = D f (0)v t with initial valu v 0 = χ. Th infinitimal gnrator G of th trongly continuou migroup T i givn by th linar oprator dfind on th ubt G : D(G) χ χ T 0 X f D(G) := {χ C 2 χ (0) = D f (0)χ, χ (0) = D f (0)χ } of th pac C 2 of all twic continuouly diffrntiabl function from [ h, 0] into R n. Th migroup T i cloly rlatd to anothr trongly continuou migroup. In ordr to clarify thi point, rcall that, du to aumption (S2) on f, th oprator D f (0) may b xtndd to a boundd linar oprator D f (0) : C R n on C. In particular, th oprator L := D f (0) dfin th linar rtardd functional diffrntial quation v (t) = L v t. Th olution of th aociatd initial valu problm { v (t) = L v t v 0 = χ C (1.2) induc a trongly continuou migroup T = {T (t)} on C. Th gnrator of T i dfind by G : D(G ) χ χ C on th domain D(G ) := {χ C 1 χ (0) = L χ}.

4 4 E. Stumpf W hav D(G ) = T 0 X f and T(t)ϕ = T (t)ϕ for all t 0 and all ϕ D(G ). Th rlation btwn th migroup T, T notably ha an ffct on th pctra σ(g), σ(g ) of th two gnrator G, G : thy coincid a hown in [1]. Th pctrum σ(g ) C of th gnrator G of T i givn by th zro of a familiar charactritic quation. In particular, it i dicrt and contain only ignvalu of finit rank, that i, all gnralizd ignpac ar finit-dimnional. Morovr, for ach β R th intrction σ(g ) {λ C R λ > β} i finit. Thrfor, th pctral part and σ c (G ) := {λ σ(g ) R λ = 0} σ u (G ) := {λ σ(g ) R λ > 0} of σ(g ) ar mpty or finit. Th aociatd ralifid gnralizd ignpac C c and C u ar calld th cntr and untabl pac of G, rpctivly, and ach of thm i a finit dimnional ubpac of C. In contrat, th tabl pac C of G, that i, th ralifid gnralizd ignpac aociatd with th pctral part σ (G ) := {λ σ(g ) R λ < 0}, i an infinit-dimnional ubpac of C. All th pac C u, C c, and C ar clod and invariant undr T (t), t 0, and provid th dcompoition C = C u C c C (1.3) of th Banach pac C. Th migroup T may b xtndd to a on-paramtr group on ach of th two finit dimnional pac C u, C c, and th dcompoition of C alo lad to a dcompoition of th mallr Banach pac C 1 : C 1 = C u C c C 1 (1.4) with th clod ubpac C 1 := C C 1 of C 1. With rpct to th migroup T and it gnrator G, it turn out that both C u and C blong to D(G ) = T 0 X f and coincid with th untabl and cntr pac of G, rpctivly. Th tabl pac of G i givn by th intrction C T 0 X f and w gt th dcompoition T 0 X f = C u C c (C T 0 X f ) of th Banach pac T 0 X f. All th pac C u, C c, and C T 0 X f ar clod in T 0 X f and invariant undr th action of th migroup T. In addition, T i xtndabl to a on-paramtr group on both C u and C c. Aftr th prparatory tp, w ar now in th poition to rcall th main rult from [5] about th xitnc of local cntr-untabl manifold for th miflow F at th tationary point ϕ 0 = 0. In doing o, w writ C cu for th o-calld cntr-untabl pac C c C u of G. Thorm 1.1 (Thorm 1 & 2 in [5]). Givn f : U R n, U C 1 opn, with f (0) = 0 and atifying aumption (S1) and (S2), uppo that {λ σ(g ) R λ 0} = or, quivalntly, C cu = {0}. Thn thr xit opn nighborhood C cu,0 of 0 in C cu and C,0 1 of 0 in C1 with N cu := C cu,0 C,0 1 containd in U, and a continuouly diffrntiabl map w cu : C cu,0 C,0 1 with w cu(0) = 0 and Dw cu (0) = 0 uch that W cu := {ϕ w cu (ϕ) ϕ C cu,0 } ha th following proprti.

5 Attraction proprty of local cntr-untabl manifold 5 (i) W cu i a continuouly diffrntiabl ubmanifold of th olution manifold X f of Eq. (1.1) and dim W cu = dim C cu. (ii) If x : (, 0] R n i a olution Eq. (1.1) and if x t N cu for all t 0, thn x t W cu a t 0. (iii) W cu i poitivly invariant with rpct to F rlativ to N cu ; that i, for all ϕ W cu and all t > 0 with {F(, ϕ) 0 t} N cu w hav {F(, ϕ) 0 t} W cu. Th goal of thi papr i to prov th following additional attraction proprty of local cntr-untabl manifold. Thorm 1.2. Undr th aumption of Thorm 1.1, thr xit an opn nighborhood U A of 0 in U, and ral K A > 0 and η A > 0 with th following proprty: If ϕ U A and if th olution x ϕ of Eq. (1.1) do xit for all t 0 and it gmnt x ϕ t blong to U A a long a t 0, thn thr i om ψ X f with x ψ t W cu for all t 0 uch that a t 0. x ϕ t x ψ t C 1 = F(t, ϕ) F(t, ψ) C 1 K A η At, In th nxt ction, w tablih thi tatmnt. Th main ida of th proof i to conidr th global cntr-untabl manifold of om mooth modification of Eq. (1.1) and to how an attraction proprty for th manifold compar Thorm 4.1 firt. Thi i don contructivly by adopting th ida containd in Vandrbauwhd [6], whr a imilar rult for ordinary diffrntial quation i givn. An ntial ingrdint of th mthod i to dduc crtain intgral quation and thn to olv th quation by th contraction principl on uitabl Banach pac. Having th attraction proprty of th global cntr-untabl manifold, th main rult aily follow by a cut-off tchniqu. Thi papr i organizd in dtail a follow. Th nxt ction contain om prliminari. Thr, w rcall th variation-of-contant formula and om intgral oprator for inhomognou linar functional diffrntial quation. Furthr, w introduc om mooth modification of Eq. (1.1) and dcrib th contruction of global cntr-untabl manifold. Th third ction i dvotd to th tudy of om global miflow of th modifid quation. Apart from th modification introducd in th cond ction, in thi ction w conidr furthr auxiliary modification of (1.1). Sction 4 bgin with a tatmnt about an attraction proprty of global cntr-untabl manifold. Thraftr, w dvlop tp by tp a tratgy for a proof of thi tatmnt. It turn out that th claimd attraction proprty may b charactrizd in an altrnativ way, which notably involv global olution of crtain paramtr-dpndnt intgral quation. In Sction 5 w prpar th lat argumnt for a proof of th attraction of global cntruntabl manifold: w contruct paramtr-dpndnt contraction on Banach pac to olv th paramtr-dpndnt intgral quation obtaind in Sction 4. In addition, w how that th rulting fixd point dpnd continuouly on th paramtr. At th nd of Sction 5, w finally giv a proof of th tatmnt claimd at th bginning of Sction 4. Th lat ction contain th proof of our main rult. 2 Prliminari In thi ction w rcapitulat om tandard fact on dlay diffrntial quation and dicu om baic rult ndd for a proof of th main tatmnt.

6 6 E. Stumpf 2.1 Sun-rflxifity For ach t 0, lt (t) dnot th adjoint oprator of th boundd linar oprator T (t) inducd by th olution of th initial valu problm (1.2). Th family = { (t)} form a migroup of boundd linar oprator on th dual pac C of C. But in gnral do not contitut a trongly continuou migroup on C with rpct to th topology inducd by th norm ϕ C := up ϕ C 1 ϕ (ϕ). Howvr, lt C dnot th t of all ϕ C with th proprty that th curv [0, ) t (t)ϕ C i continuou. Thn C i a clod ubpac of C and for all t 0 w hav (t)(c ) C. A a conqunc, th family = { (t)} of oprator (t) : C ϕ (t)ϕ C bcom a trongly continuou migroup on th Banach pac C. Similarly, carrying out th proc abov with th migroup on C intad of T on C, w firt obtain th family = { (t)} of adjoint oprator of on th dual pac C of C and thn th Banach pac C C, on which th rtriction of i trongly continuou. Th original Banach pac C and migroup T ar un-rflxiv: Thr i an iomtric linar map j : C C uch that j(c) = C and (t)(jϕ) = j(t (t)ϕ) for all t 0 and all ϕ C. For implicity, w idntify C with C and omit th mbdding oprator j in th following. For th pctrum σ(g ) of th infinitimal gnrator G of th migroup w hav σ(g ) = σ(g ). By analogy to th dcompoition (1.3) of C, C can b dcompod a C = C u C c C (2.1) and th ubpac C u, C u, and C ar clod and invariant undr. W hav continuou projction oprator Pu, Pc, and P of C onto C u, C c, and C, rpctivly. Furthr, thr xit ral contant K 1, c < 0 < c u, and 0 < c c < min{ c, c u } uch that T (t)ϕ C K c ut ϕ C, t 0, ϕ C u, T (t)ϕ C K c c t ϕ C, t R, ϕ C c, (t)ϕ C K c t ϕ C, t 0, ϕ C. (2.2) From th dcompoition (1.4) of C 1, w alo gt continuou projction oprator P u, P c, and P of C 1 onto ubpac C u, C c, and C 1, rpctivly. By th idntification of C and C it aily follow that C 1 = C 1 C. Finally, in analogy to (2.2), for th action of T on ubpac of T 0 X f w alo hav T(t)ϕ C 1 K c ut ϕ C 1, t 0, ϕ C u, T(t)ϕ C 1 K c c t ϕ C 1, t R, ϕ C c, T(t)ϕ C 1 K c t ϕ C 1, t 0, ϕ C D(G ). (2.3) 2.2 Variation-of-contant formula Nxt, w ar going to rcall th variation-of-contant formula for olution of th inhomognou linar diffrntial quation ẋ(t) = L x t q(t) (2.4) with a function q : I R n dfind on om intrval I R. Hr, a olution i a continuou function x : I [ h, 0] R n atifying (2.4) for all t I. In ordr to tat th variationof-contant formula and it proprti, w nd om prparation and notation. To bgin

7 Attraction proprty of local cntr-untabl manifold 7 with, lt L ([ h, 0], R n ) dnot th Banach pac of all maurabl and ntially boundd function from [ h, 0] into R n, quippd with th uual norm L of ntial lat uppr bound. Thn th product pac R n L ([ h, 0], R n ) quippd with th norm (α, ϕ) Rn L := max{ α R n, ϕ L } i iomtrically iomorphic to th Banach pac C. Aftr fixing a norm-prrving iomorphim k : C R n L ([ h, 0], R n ), lt for ach i = 1,..., n th lmnt r i C b dfind by r i := k 1 ( i, 0), whr i i th i-th canonical bai vctor of R n. Th family {r 1,..., r n } clarly form a bai of th ubpac Y := k 1 (R n {0}) of C and by claiming l( i ) = r i w find a uniqu linar bijctiv mapping l : R n Y with l = l 1 = 1. Givn ral a b c and a continuou function w : [a, b] C, dfin th wak-tarintgral b (c τ)w(τ) dτ C by ( b for all ϕ C. In addition, t a a ) b (c τ)w(τ) dτ (ϕ ( ) := (c τ)w(τ) ) (ϕ ) dτ a a b b (c τ)w(τ) dτ := (c τ)w(τ) dτ. a Thn it follow that th wak-tar-intgral li in C. W hav b b (t) (c τ)w(τ) dτ = a a a t 0, and for any of th continuou projction P λ P λ b a b (c τ)w(τ) dτ = a (t c τ)w(τ) dτ with λ {, c, u} th idntity (c τ)p λ w(τ) dτ hold. In addition, a uual th norm of th wak-tar-intgral i boundd by th intgral of th norm: b b (c τ)w(τ) dτ T (c τ)w(τ) C dτ. a C a W rturn to Eq. (2.4). If q : I R n i continuou and if x : I [ h, 0] R n i a olution of Eq. (2.4) thn th curv u : I τ x τ C atifi th abtract intgral quation u(t) = T (t )u() (t τ)q(τ) dτ (2.5) with Q : I τ l(q(τ)) Y for all, t I, t. Convrly, if Q : I Y i continuou and if u : I C i a olution of Eq. (2.5) thn thr xit a continuou x : I [ h, 0] R n uch that x t = u(t) for all t I and that x i a olution of Eq. (2.4) on I for q : I τ l 1 (Q(τ)) R n. In thi way w hav a on-to-on corrpondnc btwn olution of Eq. (2.4) and Eq. (2.5).

8 8 E. Stumpf 2.3 Prparatory rult on inhomognou linar quation Lt X dnot a Banach pac and X it norm. Thn for ach η 0 th linar pac { } and C η ((, 0], X) := C (R, X) := g C((, 0], X) providd with th wightd uprmum norm g Cη = up 0 { up η g() X < (,0] g C(R, X) up η g() X < R η g() X and g C = up η g() X, R rpctivly, bcom Banach pac a wll. Som of th pac w will conidr rpatdly in th qul. In ordr to implify notation, w hall u th following abbrviation throughout th papr: C 0 η := C η((, 0], C), C 1 η := C η((, 0], C 1 ), Y η := C η ((, 0], Y ), C 0 := C (R, C), C 1 := C (R, C 1 ), and Y := C (R, Y ). Morovr, w writ P cu := P c P u for th projction of C 1 along C 1 and P cu th projction of C along C. Dfinition 2.1. Givn Q : (, 0] Y and Q R : R Y, t for all t 0, and and for all t R. (K cu Q)(t) := 0 (t τ)pcu Q(τ) dτ (K 1 Q R )(t) := (K 2 Q R )(t) := } := P u Pc for (t τ)p Q(τ) dτ (2.6) (t τ)p Q R (τ) dτ (2.7) (t τ)pcu Q R (τ) dτ (2.8) Propoition 2.2 (compar Propoition 3.2 in [5]). Lt η R with c c < η < min{ c, c u } b givn. (i) Eq. (2.6) dfin a boundd linar map ˆK : Y η Q K cu Q C 0 η with ˆK K ( P ) c P u η c c c u η P. c η Morovr, u := ( ˆKQ), Q Y η, i a olution of th intgral quation u(t) = T (t )u() (t τ)q(τ) dτ (2.9) a < t < 0, and it i th only on in C 0 η with th proprty P cu u(0) = 0.

9 Attraction proprty of local cntr-untabl manifold 9 (ii) Eq. (2.7) dfin a boundd linar map ˆK 1 : Y Q (K 1 Q) C 0 with ˆK 1 K P c η. Morovr, ( ˆK 1 Q), Q Y, i a olution of th intgral quation u(t) = T (t )u() a < t <, and ( ˆK 1 Q)(t) C for all t R. (t τ)p Q(τ) dτ (2.10) (iii) Eq. (2.8) dfin a boundd linar map ˆK 2 : Y Q (K 2 Q) C 0 with ˆK 2 K ( P c P u η c c c u η ). Morovr, ( ˆK 2 Q), Q Y, i a olution of th intgral u(t) = T (t )u() a < t <, and ( ˆK 2 Q)(t) C cu for all t R. (t τ)pcu Q(τ) dτ (2.11) Proof. For th firt part of th tatmnt w rfr th radr to Propoition 3.2 in [5] whr th proof i carrid out in full dtail. Following th lin, on aily conclud part (ii) and (iii) of th tatmnt. Rmark 2.3. Undr th aumption on η from th lat propoition, it i not difficult to that (K 1 Q) i actually wll-dfind for all Q C(R, Y ) atifying Q (,0] Y η. Furthrmor, for uch Q C(R, Y ) th imag u := (K 1 Q) i a continuou map from R into C with u (,0] C 0 η, olv Eq. (2.10) for all < t <, and atifi u(t) C a t R. All th function ˆKQ, ˆK 1 Q and ˆK 2 Q ar not only continuou but continuouly diffrntiabl, a tablihd in our nxt rult. Propoition 2.4 (compar Corollary 3.4 in [5]). Conidr η R a in th lat rult. Thn th following hold. (i) Eq. (2.6) induc a boundd linar map K η : Y η Q K cu Q C 1 η with K η K(1 ηh L ) ( P ) c P u η c c c u η P ηh. c η Morovr, givn Q Y η, u := K η Q i th only olution of Eq. (2.9) in C 1 η with th proprty P cu u(0) = 0.

10 10 E. Stumpf (ii) Eq. (2.7) induc a boundd linar mapping K 1 η : Y Q K 1 Q C 1 with Kη 1 K(1 ηh L ) P c η ηh P. Morovr, givn Q Y η, all gmnt of th olution K 1 ηq of Eq. (2.10) blong to C 1. (iii) Eq. (2.8) induc a boundd linar mapping K 2 η : Y Q K 2 Q C 1 with K 2 η K(1 ηh L ) ( P c P u η c c c u η ) ηh P cu. Morovr, givn Q Y η, all gmnt of th olution K 2 ηq of Eq. (2.11) blong to C cu. Proof. For th proof of th firt artion compar Corollary 3.4 and it proof in [5], whra th proof of artion (ii) and (iii) immdiatly follow from Propoition 2.2 in combination with Propoition in Hartung t al. [1]. Rmark 2.5. An important ingrdint of th proof of th lat tatmnt i a moothing proprty of th intgral quation (2.5). For xampl, if Q C(R, Y ) and if u C(R, C) atifi th intgral quation (2.5) for all < t <, thn u C(R, C 1 ). For a proof, compar for intanc th proof of Propoition in Hartung t al. [1]. Corollary 2.6. For givn η R with c c < η < min{ c, c u }, lt K η : Y C 1 dnot th map Q (Kη 1 Kη)(Q), 2 whr th oprator Kη 1 and Kη 2 ar dfind a in th lat propoition. Thn K η form a boundd linar oprator with K η K(1 ηh L ) ( P u c u η P c P η c c c η and for ach Q Y th function u = (K 1 η K 2 η)(q) atifi for all < t <. u(t) = T (t )u() ) (t τ)q(τ) dτ ηh ( P Pcu ), Proof. In viw of Propoition 2.4, it i clar that th um K η of th two boundd linar oprator Kη 1 and Kη 2 from Y into C 1 i a boundd linar oprator from Y into C 1 a wll. Furthrmor, from th timat for Kη 1 and for Kη 2 w gt K η K 1 η K 2 η K(1 ηh L ) ( P u c u η P c P η c c c η ) ηh ( P Pcu ).

11 Attraction proprty of local cntr-untabl manifold 11 For th rmaining part of th artion, conidr u = K η Q for om fixd Q Y. Uing Propoition 2.4 again, it follow that u(t) = (K η Q)(t) = (K 1 ηq)(t) (K 2 ηq)(t) = (K 1 Q)(t) (K 2 Q)(t) = T (t )(K 1 Q)() (t τ)pcu Q(τ) dτ [ ] = T (t ) (K 1 Q)() (K 2 Q)() = T (t )(K η Q)() = T (t )u() (t τ)p Q(τ) dτ T (t )(K 2 Q)() (t τ)q(τ) dτ (t τ)q(τ) dτ for all < t <, which prov th corollary. (t τ) [ P Q(τ) Pcu Q(τ) ] dτ 2.4 Smooth modification of th nonlinarity and th global cntr-untabl manifold of th modifid quation Blow w rcapitulat om ntial ingrdint of th proof of Thorm 1.1. In particular, w dcrib th contruction of global cntr-untabl manifold for mooth modification of Eq. (1.1). For th dtail w rfr th radr to [5]. Compar alo th contruction of local cntr manifold containd in Hartung t al. [1]. Introducing th map w may rwrit Eq. (1.1) a L := D f (0) and r : U ϕ f (ϕ) Lϕ R n, ẋ(t) = Lx t r(x t ) whr th right-hand id i paratd into a linar and a nonlinar trm. It i aily n that r inhrit condition (S1) and (S2) from f. In particular, w hav r(0) = 0 and Dr(0) = 0. In viw of dim C cu <, thr xit a norm cu on C cu uch that it rtriction to C cu \ {0} i C -mooth. Uing thi norm, dfin ϕ 1 := max{ P cu ϕ cu, P ϕ C 1} for all ϕ C 1. In thi way w obtain anothr norm 1 on C 1, which i quivalnt to C 1. Choo nxt a C -mooth map ρ : [0, ) R atifying ρ(t) = 1 a 0 t 1, 0 < ρ(t) < 1 a 1 < t < 2, and ρ(t) = 0 for all t 2, and t { r(ϕ), for ϕ U, ˆr(ϕ) = 0, for ϕ U. For ach δ > 0 w introduc by ( ) ( ) r δ : C 1 ϕcu cu ϕ ϕ ρ ρ C 1 ˆr(ϕ) δ δ

12 12 E. Stumpf a modification of r that i dfind on all of C 1. Hr, ϕ cu dnot th componnt P cu ϕ of ϕ C 1, and analogouly ϕ th componnt P ϕ of ϕ C 1. For all ufficintly mall δ > 0 th rtriction of r δ to a mall nighborhood of 0 C 1 i continuouly diffrntiabl, boundd, and ha a boundd drivativ. Mor prcily, thr xit om δ 0 > 0 with {ψ C 1 ψ 1 < δ 0 } U uch that for ach 0 < δ < δ 0 th rtriction r δ {ψ C1 ψ 1 <δ} i a boundd and continuouly diffrntiabl function with a boundd drivativ. Furthrmor, for all ϕ {ψ C 1 ψ 1 < δ} w hav ( ) ϕcu cu r δ {ψ C1 ψ 1 <δ}(ϕ) = ˆr(ϕ) ρ. δ Nxt, thr i om 0 < δ 1 < δ 0 and a monoton incraing λ : [0, δ 1 ] [0, 1] atifying λ(0) = 0 and lim δ 0 λ(δ) = 0 uch that and r δ (ϕ) R n δ λ(δ) (2.12) r δ (ϕ) r δ (ψ) R n λ(δ) ϕ ψ C 1 (2.13) for all 0 < δ δ 1 and all ϕ, ψ C 1. Th proof for th xitnc part of Thorm 1.1 bgin with th contruction of global cntr-untabl manifold for th modifid quation ẋ(t) = Lx t r δ (x t ) (2.14) whr 0 < δ δ 1. Rcall that th quation ar cloly rlatd with th intgral quation u(t) = T (t )u() (t τ)l(r δ (u(τ))) dτ. (2.15) For intanc, if x : (, 0] R n i a continuouly diffrntiabl olution of Eq. (2.14), thn w obtain a olution of Eq. (2.15) by u : (, 0] t x t C 1. On th othr hand, if u : (, 0] C 1 atifi (2.15), thn x : (, 0] R n givn by x(t) := u(t)(0) a < t 0 i a C 1 - mooth olution of Eq. (2.14). Conidr now om fixd η R atifying Thr clarly xit om 0 < δ < δ 1 nuring c c < η < min{ c, c u }. λ(δ) K η < 1 2. With thi choic of δ, lt u tmporarily dnot by R : C((, 0], C 1 ) C((, 0], Y ) th ubtitution oprator of th map C 1 ϕ l(r δ (ϕ)) Y, that i, R(u)(t) = l(r δ (u(t))) for all u C((, 0], C 1 ) and all t 0. Thn R map C 1 η into Y η and thu induc a mapping which particularly atifi R δη : C 1 η u R(u) Y η, R δη (u) Yη δλ(δ) and R δη (u) R δη (v) Yη λ(δ) u v C 1 η (2.16)

13 Attraction proprty of local cntr-untabl manifold 13 for all u, v C 1 η. Givn om ϕ C cu, th curv (, 0] t T (t)ϕ C 1 blong to C 1 η. Thrfor, w may dfin a map S η : C 1 C cu C 1 η by (S η ϕ)(t) := T (t)ϕ a ϕ C cu and t 0. It aily follow that S η form a boundd linar oprator with S η K( Pc Pu ). (2.17) Uing th mapping K η from Propoition 2.4, R δη and S η, w introduc anothr map G η : C 1 η C cu C 1 η givn by G η (u, ϕ) := S η ϕ K η R δη (u). Undr th givn aumption, for ach ϕ C cu th inducd map G η (, ϕ) : C 1 η C 1 η ha an uniquly dtrmind fixd point u(ϕ) inc it form a contraction of a ufficintly larg ball about th origin into itlf. Th aociatd olution oprator ũ η : C cu ϕ u(ϕ) C 1 η i globally Lipchitz-continuou, and for ach ϕ C cu th function ũ η (ϕ) i a olution of Eq. (2.15) on (, 0] with vanihing C cu componnt at t = 0. Thu, in viw of th on-to-on corrpondnc of olution of Eq. (2.14) and Eq. (2.15), w that for ach ϕ C cu thr xit a continuouly diffrntiabl function x : (, 0] R n with x t = ũ(ϕ)(t) a t 0 uch that x olv Eq. (2.14) for all t 0. Th global cntr-untabl manifold of Eq. (2.14) at th tationary point 0 C 1 i now th t W hav W η := {ũ η (ϕ)(0) ϕ C cu }. W η = {ϕ w η (ϕ) ϕ C cu } with th map w η : C cu ϕ P (ũ η (ϕ)(0)) C 1 (2.18) and for ach olution v C 1 η of Eq. (2.15) w hav v(t) W η a t 0. 3 Global miflow of modifid quation Th firt tp toward a proof of our main rult Thorm 1.2 will b a imilar tatmnt for th modifid quation (2.14) and th aociatd global cntr-untabl manifold. A thi tatmnt will art an aymptotic bhaviour for t of crtain olution of Eq. (2.14), w nd om prparation containing, among othr thing, a dicuion about th xitnc of continuouly diffrntiabl olution for t 0. Thi i don blow. To bgin with, obrv that Eq. (2.14) may b writtn a ẋ(t) = f δ (x t ) (3.1) with th function f δ : C 1 ϕ Lϕ r δ (ϕ). By contruction, th t X δ := {ϕ C 1 ϕ (0) = f δ (ϕ)}

14 14 E. Stumpf i clarly not mpty inc w hav f δ (0) = f (0) = 0. Nvrthl, f δ do not nd to hav th proprti (S1) and (S2). For thi raon, w can not u th rult in Walthr [7, 8, 9] in ordr to conclud th xitnc of olution of th initial valu problm ẋ(t) = f δ (x t ) (= Lx t r δ (x t )), x 0 = ϕ X δ, (3.2) for t 0. Howvr, thi iu wa alrady addrd by Qmi and Walthr in [3] whr th author prov that for all ufficintly mall δ > 0 th initial valu problm hav uniquly dtrmind olution. Mor prcily, th following hold: Propoition 3.1. Lt 0 < δ < δ 1 with λ(δ) < 1/5 b givn. Thn for ach ϕ X δ thr xit a uniqu continuouly diffrntiabl olution x : [ h, ) R n of th initial valu problm (3.2), and x t X δ for all t 0. Morovr, th quation F δ (t, ϕ) = x ϕ t, ϕ X δ, t 0, dfin a continuou miflow F δ : [0, ) X δ X δ, and for ach 0 Lip 0 t F δ (t, ) <. Proof. Compar Corollary 6.2 and Propoition 6.3 in [3]. Now, rcall onc mor th on-to-on corrpondnc btwn olution of th diffrntial quation dfining th initial valu problm (3.2) and olution of th intgral quation (2.15). Fixing om appropriat δ > 0 and any ϕ X δ and tting u(t) := F δ (t, ϕ) for all t 0, w firt F δ (t, ϕ) = T (t )F δ (, ϕ) (t τ)l(r δ (F δ (τ, ϕ))) dτ (3.3) and thn aftr application of P cu that i, for all 0 t <. P cu F δ (t, ϕ) = P cu F δ (t, ϕ) = Pcu T (t )F δ (, ϕ) Pcu = T (t )P cu F δ (, ϕ) = T (t )P cu F δ (, ϕ) P cu F δ (t, ϕ) = T (t )P cu F δ (, ϕ) (t τ)l(r δ (F δ (τ, ϕ))) dτ (t τ)pcu l(r δ (F δ (τ, ϕ))) dτ (t τ)pcu l(r δ (F δ (τ, ϕ))) dτ, (t τ)pcu l(r δ (F δ (τ, ϕ))) dτ, (3.4) Apart from th global miflow F δ gnratd by olution of (3.2), w will nd othr auxiliary miflow. In ordr to dfin th miflow, w firt hav to tudy olution of th modification ẋ(t) = LP cu x t (l 1 P cu l)(r δ (x t P ϕ)) (3.5) of Eq. (3.1) for t 0 whr ϕ X δ. W how that, for ach η R with c c < η < min{ c, c u } and all ufficintly mall δ > 0, vry ϕ X δ uniquly dtrmin a continuouly diffrntiabl olution x : (, 0] R n of Eq. (3.5) with x 0 = P cu ϕ and [(, 0] t x t C 1 ] C 1 η.

15 Attraction proprty of local cntr-untabl manifold 15 Th proof of thi tatmnt i bad on a fixd-point argumnt compltly imilar to th on ud for th contruction of th global cntr-untabl manifold W η. Howvr, for th convninc of th radr, w carry out th dtail blow. For th rmaining part of thi ction fix om η R with and choo 0 < δ < δ 1 uch that c c < η < min{ c, c u } λ(δ) K η λ(δ) K η P cu < 1 2. (3.6) W bgin with a minor modification of Corollary 4.3 in [5]. Corollary 3.2. Lt R dnot th oprator which aign to u C((, 0], C 1 ) th mapping (, 0] P cu l(r δ (u())) Y in C((, 0], Y ). Thn R(C 1 η) Y η, and th inducd mapping R δη : C 1 η u R(u) Y η atifi and for all u, v C 1 η. R δη (u) Yη P cu δλ(δ) R δη (u) R δη (v) Yη λ(δ) P cu u v C 1 η Proof. Obrv that for ach u C((, 0], C 1 ) and all 0 w hav R(u)() = P cu l(r δ (u())) = P cu R(u)() whr R : C((, 0], C 1 ) C((, 0], Y ) dnot th ubtitution oprator of th map C 1 ϕ l(r δ (ϕ)) Y. A Pcu : C C cu i continuou and R(u) C((, 0], Y ), it bcom clar that R map C((, 0], C 1 ) into C((, 0], Y ). Nxt, conidr om u Cη. 1 Uing th firt inquality of (2.16) w infr up ηt R(u)(t) Y = up ηt Pcu R(u)(t) Y Pcu up ηt R(u)(t) Y P cu R δη (u) Yη P cu δλ(δ). Hnc, it follow that R(C 1 η) Y η and that R δη i boundd by P cu δλ(δ). Similarly, from th cond inquality of (2.16) w gt for all u, v C 1 η: Thi prov th artion. R δη (u) R δη (v) Yη = R(u) R(v) Yη = up ηt Pcu R(u)(t) Pcu R(v)(t) Y Pcu up ηt R(u)(t) R(v)(t) Y = P cu R δη (u) R δη (v) Yη λ(δ) P cu u v C 1 η.

16 16 E. Stumpf Nxt, w conidr a light modification of th boundd linar oprator S η ud for th contruction of W η. Corollary 3.3. For ach ϕ C 1 th curv (, 0] t T (t)p cu ϕ C 1 blong to Cη, 1 and th mapping S η : C 1 Cη 1 dfind by (S η ϕ)(t) = T (t)p cu ϕ for all ϕ C 1 and all t 0 i a boundd linar oprator with S η K P cu ( P c Pu ). Proof. Firt, by Corollary 4.5 in [5] it follow that for vry ϕ C 1 th continuou curv (, 0] t T (t)p cu ϕ C 1 i an lmnt of th Banach pac C 1 η. Morovr, w hav S η = S η P cu. Hnc, a a compoition of two boundd linar oprator, S η i a boundd linar oprator a wll, and th timat (2.17) finally lad to S η = S η P cu S η P cu K P cu ( P c Pu ). For givn ϕ C 1 conidr th contant map dfind by C(ϕ) : (, 0] C 1 C(ϕ)(t) := P ϕ. (3.7) Clarly, w hav C(ϕ) C 1 η. Uing Corollary 3.4 in [5] and th lat two corollari, w obtain a wll-dfind map G η : C 1 η C 1 C 1 η whr G η (u, ϕ) := S η ϕ K η R δη (u C(ϕ)) with th boundd linar oprator K η : Y η C 1 η from Propoition 2.4. Nxt, w how that for ach fixd ϕ C 1 th inducd map G η (, ϕ) : C 1 η C 1 η i a contraction uch that th quation u = G η (u, ϕ) ha xactly on olution in th Banach pac C 1 η. Propoition 3.4. For any ϕ C 1 th mapping G η (, ϕ) : Cη 1 u = u(ϕ) and th aociatd olution oprator C 1 η ha xactly on fixd point of u = G η (u, ϕ) i Lipchitz continuou. û η : C 1 ϕ u(ϕ) C 1 η Proof. W mimic th proof of Propoition 4.6 in [5]. 1. Lt ϕ C 1 b givn. Choo γ = γ(ϕ) > 0 uch that 2 S η ϕ C 1 < γ. Thn from Corollari 3.2 and 3.3 w infr G η (u, ϕ) C 1 η = S η ϕ K η R δη (u C(ϕ)) C 1 η S η ϕ C 1 η (K η R δη )(u P ϕ) C 1 η S η ϕ C 1 K η R δη (u) Yη S η ϕ C 1 λ(δ) K η P cu u C 1 η γ 2 γ 2 = γ ( (3.6))

17 Attraction proprty of local cntr-untabl manifold 17 for all u Cη 1 with u C 1 η γ. Conquntly, th rtriction of G η (, ϕ) to th clod ball {u Cη u 1 C 1 η γ} in Cη 1 i a lf-map. Morovr, G η (, ϕ) i a contraction. Indd, Corollary 3.2 in combination with condition (3.6) impli G η (u, ϕ) G η (v, ϕ) C 1 η = K η R δη (u C(ϕ)) K η R δη (v C(ϕ)) C 1 η K η R δη (u C(ϕ)) R δη (v C(ϕ)) Yη λ(δ) K η P cu u v C 1 η 1 2 u v C 1 η for all u, v C 1 η. W conclud that thr i a uniqu u(ϕ) {u C 1 η u C 1 η γ} C η 1 atifying u = G η (u, ϕ). 2. It rmain to how th global Lipchitz continuity of th map û η : C 1 ϕ u(ϕ) C 1 η. For thi purpo, conidr ϕ, ψ C 1. Uing Corollari 3.2 and 3.3 and th timat (3.6) onc mor, w infr û η (ϕ) û η (ψ) C 1 η = u(ϕ) u(ψ) C 1 η = G η (u(ϕ), ϕ) G η (u(ψ), ψ) C 1 η = S η (ϕ ψ) (K η R δη )(u(ϕ)) (K η R δη )(u(ψ)) C 1 η S η ϕ ψ C 1 K η R δη (u(ϕ)) R δη (u(ψ)) Yη S η ϕ ψ C 1 λ(δ) P cu K η u(ϕ) u(ψ) C 1 η S η ϕ ψ C u(ϕ) u(ψ) C 1 η = S η ϕ ψ C û η(ϕ) û η (ψ) C 1 η, that i, û η (ϕ) û η (ψ) C 1 η 2 S η ϕ ψ C 1. Conquntly, û ha a global Lipchitz contant a claimd. For vry ϕ C 1 th fixd point û η (ϕ) C 1 η of u = G η (u, ϕ) from th lat rult i a pcial olution of th abtract intgral quation aociatd with Eq. (3.5) by th variation-of-contant formula. Mor prcily, th following hold. Corollary 3.5. For all ϕ C 1 th mapping û η (ϕ) from th lat propoition i a olution of th abtract intgral quation u(t) = T (t )P cu u() for < t 0. In particular, û η (ϕ)(0) = P cu ϕ and û η (ϕ)(t) C cu for all t 0. (t τ)pcu l(r δ (u(τ) P ϕ))dτ (3.8)

18 18 E. Stumpf Proof. Following th proof of Corollary 4.7 in [5], conidr for givn ϕ C 1 th map By Corollary 3.4 in [5], w conclud that z := û η (ϕ) S η ϕ = G η (û η (ϕ), ϕ) C 1 η. z(t) = T (t )z() (t τ)r δη (û η (ϕ) C(ϕ))(τ)dτ a < t 0 and that P cu z(0) = P cu z(0) = 0. Hnc, it follow that that i, û η (ϕ)(t) T (t)p cu ϕ = û η (ϕ)(t) (S η ϕ)(t) = z(t) = T (t )z() (t τ)r δη (û η (ϕ) C(ϕ))(τ)dτ = T (t )û η (ϕ)() T (t )(S η ϕ)() (t τ)pcu l(r δ (û η (ϕ)(τ) C(ϕ)) dτ = T (t )û η (ϕ)() T (t )T ()P cu ϕ (t τ)pcu l(r δ (û η (ϕ)(τ) C(ϕ))) dτ, û η (ϕ)(t) = T (t )û η (ϕ)() for all < t 0. In addition, w gt (t τ)p cu l(r δ (û η (ϕ)(τ) C(ϕ)) dτ, (3.9) P cu (û η (ϕ)(0)) = P cu z(0) P cu ((S η ϕ)(0)) = 0 P cu T (0)ϕ = P cu ϕ. (3.10) Nxt, rcall that Pcu Pcu th migroup gt z(t) = G η (û(ϕ), ϕ)(t) = Pcu, Pcu P = 0, and that C cu i invariant undr th action of. Combining thi fact with th dfinition of K η from Propoition 2.4, w = [(K η R δη )(û η (ϕ) C(ϕ))](t) = = = 0 0 (t τ)pcu R(û η (ϕ) C(ϕ))(τ) dτ (t τ)pcu Pcu l(r δ (û η (ϕ)(τ) P ϕ)) dτ 0 = P cu for all t 0. Hnc, 0 (t τ)p Pcu l(r δ (û η (ϕ)(τ) P ϕ)) dτ (t τ)p cu l(r δ (û η (ϕ)(τ) P ϕ)) dτ (t τ)l(r δ (û η (ϕ)(τ) C(ϕ)) dτ C cu û η (ϕ)(t) = z(t) (S η ϕ)(t) = z(t) T (t)p cu ϕ C cu (t τ)p R(û η (ϕ) C(ϕ))(τ) dτ

19 Attraction proprty of local cntr-untabl manifold 19 a t 0 and thu û η (ϕ)(0) = P cu û η (ϕ)(0) = P cu ϕ du to Eq. (3.10). Morovr, w T (t )û η (ϕ)() = T (t )P cu û η (ϕ)() for all < t 0 uch that Eq. (3.9) tak th form û η (ϕ)(t) = T (t )P cu û η (ϕ)() a < t 0. Thi how th artion. (t )Pcu l(r δ (û η (ϕ)(τ) C(ϕ))) dτ From th on-to-on corrpondnc btwn olution of th abtract intgral quation (3.8) and olution of th diffrntial quation (3.5) w conclud that for ach ϕ C 1 thr i a continuouly diffrntiabl function x : (, 0] R n atifying Eq. (3.5) for all t 0 and having th proprti x 0 = P cu ϕ and [(, 0] t x t C 1 ] C 1 η. Morovr, all th gmnt of x ar containd in th cntr-untabl pac C cu, and in viw of th uniqun rult from Propoition 3.4 thr i no othr olution y : (, 0] R n of Eq. (3.5) having th two proprti y 0 = P cu ϕ and [(, 0] t y t C 1 ] C 1 η (compar alo th dtail in th proof of th nxt propoition). Uing th olution oprator û η, w dfin a map F cu η : (, 0] C 1 C 1 by F cu η (t, ϕ) := û η (ϕ)(t) C(ϕ)(t). Blow w prov that F cu η dfin a continuou dynamical ytm on C 1. Propoition 3.6. Th map Fη cu : (, 0] C 1 C 1 form a continuou miflow. Mor prcily, Fη cu i continuou and atifi for all, t (, 0] and all ϕ C 1. F cu η (0, ϕ) = ϕ and F cu η ( t, ϕ) = F cu η (, F cu η (t, ϕ)) Proof. 1. (Proof of th continuity of F cu η.) Lt t (, 0], ϕ C 1 and ε > 0 b givn. A û η (ϕ) C 1 η i continuou, thr i om δ 1 > 0 uch that for all (, 0] with t < δ 1 û η (ϕ)(t) û η (ϕ)() C 1 < ε 3 hold. In addition, w find δ 2 > 0 uch that for ach ψ C 1 with ϕ ψ C 1 < δ 2 w hav P ϕ ψ C 1 < ε 3 and η(t δ 1 ) Lip(û η ) ϕ ψ C 1 < ε 3 whr Lip(û η ) i a global Lipchitz contant of û η du to Propoition 3.4. St δ := min{ δ 1, δ 2 } and conidr arbitrary (, ψ) (, 0] C 1 with t < δ and ϕ ψ C 1 < δ. Thn w

20 20 E. Stumpf infr F cu η (t, ϕ) F cu η (, ψ) C 1 = û η (ϕ)(t) C(ϕ)(t) û η (ψ)() C(ψ)() C 1 û η (ϕ)(t) û η (ψ)() C 1 C(ϕ)(t) C(ψ)() C 1 û η (ϕ)(t) û η (ϕ)() C 1 û η (ϕ)() û η (ψ)() C 1 P ϕ P ψ C 1 ε 3 η η û η (ϕ)() û η (ψ)() C 1 P ϕ ψ C 1 ε 3 η(t δ) û η (ϕ) û η (ψ) C 1 η ε 3 ε 3 η(t δ) Lip(û η ) ϕ ψ C 1 ε 3 < ε. Thi prov th continuity of Fη cu at (t, ϕ). 2. (Proof of th algbraic proprti of a miflow.) To bgin with, obrv that from th dfinition of Fη cu and th lat rult it immdiatly follow that F cu η (0, ϕ) = û η (ϕ)(0) C(ϕ)(0) = P cu ϕ P ϕ = ϕ for all ϕ C 1. Thrfor, th only thing rmaining to prov i th additiv proprty of F cu η. For thi purpo, lt ˆt, ŝ (, 0] and ϕ C 1 b givn. W hav and F cu η (ŝ ˆt, ϕ) = û η (ϕ)(ŝ ˆt) C(ϕ)(ŝ ˆt) = û η (ϕ)(ŝ ˆt) P ϕ F cu η (ˆt, F cu η (ŝ, ϕ)) = û η (F cu η (ŝ, ϕ))(ˆt) C(F cu η (ŝ, ϕ))(ˆt) Thu, it uffic to prov = û η (F cu η (ŝ, ϕ))(ˆt) P F cu η (ŝ, ϕ) = û η (F cu η (ŝ, ϕ))(ˆt) P [û η (ϕ)(ŝ) C(ϕ)(ŝ)] = û η (F cu η (ŝ, ϕ))(ˆt) P [C(ϕ)(ŝ)] (du to Corollary 3.5) = û η (F cu η (ŝ, ϕ))(ˆt) P ϕ. û η (ϕ)(ŝ ˆt) = û η (F cu η (ŝ, ϕ))(ˆt) in ordr to F cu η (ŝ ˆt, ϕ) = F cu η (ˆt, F cu η (ŝ, ϕ)). To thi nd, dfin v(t) := û η (ϕ)(t ŝ) and w(t) := û η (F cu η (ŝ, ϕ))(t) for all t 0. Accordingly to th lat two rult, v, w C 1 η and v(t), w(t) C cu a t 0. Morovr, in viw of û η (ϕ)(ŝ) = P cu [û η (ϕ)(ŝ) P ϕ] = P cu [û η (ϕ)(ŝ) C(ϕ)(ŝ)] = P cu F cu η (ŝ, ϕ) = û(f cu η (ŝ, ϕ))(0), w hav v(0) = w(0) and both v and w atify u(t) = T (t )P cu u() for all < t 0 a P F cu η (ŝ, ϕ) = P ϕ. (t τ)pcu l(r δ (u(τ) P ϕ)) dτ

21 Attraction proprty of local cntr-untabl manifold 21 Conidr now th mapping z : (, 0] C 1 givn by Uing timat (2.2), w up z(t) := v(t) T (t)p cu w(0) = v(t) T (t)w(0). ηt T (t)w(0) C 1 = up ηt T (t)p cu w(0) C 1 up K up ηt T (t)p c w(0) C 1 up ηt T (t)p u w(0) C 1 (cc η)t P c w(0) C 1 K up (cuη)t P u w(0) C 1 K P c w(0) C 1 K P u w(0) C 1 <. Thu, z C 1 η. In addition, for all t 0 z(t) = v(t) T (t)w(0) = T (t )v() = T (t )z() (t τ)pcu l(r δ (v(τ) P ϕ))dτ T (t )T ()w(0) (t τ)pcu l(r δ (v(τ) C(ϕ)(τ))) dτ. Combining thi fact togthr with R δη (v C(ϕ)) Y η du to Corollary 3.2 and P cu z(0) = P cu z(0) = P cu v(0) T (0)P cu w(0) = v(0) w(0) = 0, w obtain z = K η R δη (v C(ϕ)) from Corollary 3.4 in [5]. Hnc, it follow that v(t) = z(t) T (t)w(0) = (K η R δη (v C(ϕ)))(t) T (t)p cu w(0) = (K η R δη )(v C(F cu η (ŝ, ϕ)))(t) T (t)p cu û η (F cu η (ŝ, ϕ))(0) = (K η R δη )(v C(F cu η (ŝ, ϕ)))(t) T (t)p cu F cu η (ŝ, ϕ) for all t 0. In th Banach pac C 1 η th lat quation rad v = K η R δη (v C(F cu η (ŝ, ϕ))) S η (F cu η (ŝ, ϕ)) = G η (v, F cu η (ŝ, ϕ)). Thrfor, Propoition 3.4 impli In particular, it follow that which complt th proof. v = û η (F cu η (ŝ, ϕ)) = w. û η (ϕ)(ˆt ŝ) = v(ˆt) = w(ˆt) = û η (F cu η (ŝ, ϕ))(ˆt),

22 22 E. Stumpf 4 An attraction proprty of th global cntr-untabl manifold of th modifid quation: th tatmnt and th main ida of th proof Aftr th prparation in th lat ction, w ar now in th poition to tat an attraction proprty of th global cntr-untabl manifold W η of th modifid quation (2.14). Thorm 4.1 (Attraction proprty of th global cntr-untabl manifold). Lt f : U R n, U C 1 opn, with f (0) = 0, atifying th proprti (S1) and (S2), and with C cu = {0} b givn. Furthr, for fixd η R with c c < η < min{ c, c u }, lt 0 < δ < δ 1 atify and Thn thr xit a continuou map Hcu η : X δ following hold: if and only if ψ = H η cu(ϕ). λ(δ) < 1 5, (4.1) λ(δ) K η < 1 2, (4.2) λ(δ) K η λ(δ) K η P cu < 1 2. (4.3) W η uch that for all (ψ, ϕ) W η X δ th up ηt F δ (t, ϕ) F δ (t, ψ) C 1 < (4.4) From now on and until th nd of th nxt ction, w uppo that th aumption of thi thorm ar atifid. For a proof w adopt th ida of Vandrbauwhd [6], whr th artion for th ca of ordinary diffrntial quation i dicud. Th initial point of thi tratgy i an altrnativ charactrization of proprty (4.4). Lmma 4.2. Suppo that F : R X δ C 1 i continuou and atifi (a) F(t, ϕ) = F δ (t, ϕ) for all t 0 and all ϕ X δ, and (b) F(, ϕ) (,0] C 1 η for ach ϕ X δ. Lt ϕ, ψ X δ b givn. Thn th following tatmnt ar quivalnt: (i) ψ W η and up ηt F δ (t, ψ) F δ (t, ϕ) C 1 <. (ii) Thr xit om z C 1 uch that F(, ϕ) z i a olution of a < t < and ψ = ϕ z(0). u(t) = T (t )u() (t τ)l(r δ (u(τ))) dτ (4.5)

23 Attraction proprty of local cntr-untabl manifold 23 Proof. W follow th proof of Lmma 5.6 in [6]. 1. To bgin with, aum that undr givn aumption, proprty (i) hold. Thn Corollary 4.7 in [5] in combination with th dfinition of W η how that ψ = ũ η (P cu ψ)(0). Morovr, ũ η (P cu ψ) C 1 η and ũ η (P cu ψ) i a olution of Eq. (4.5) a < t 0. Stting v(t) := {ũη (P cu ψ)(t), for t 0, F δ (t, ψ), for t 0, w obtain a continuou function v : R C 1, which atifi Eq. (4.5) for all < t <. Whil th lat point i clar for th ca < t 0 and 0 t <, in th ituation < < 0 < t < thi rult from th following traightforward calculation: v(t) = F δ (t, ψ) = T (t)f δ (0, ψ) = T (t)ũ η (P cu ψ)(0) 0 0 (t τ)l(r δ (F δ (τ, ϕ))) dτ = T (t)t ( )ũ η (P cu ψ)() T (t) 0 (t τ)l(r δ (v(τ))) dτ = T (t )ũ η (P cu ψ)() 0 = T (t )v() = T (t )v() 0 (t τ)l(r δ (v(τ))) dτ 0 (t τ)l(r δ (v(τ))) dτ 0 ( τ)l(r δ (ũ η (P cu ψ)(τ))) dτ (t τ)l(r δ (ũ η (P cu ψ)(τ))) dτ (t τ)l(r δ (v(τ))) dτ 0 (t τ)l(r δ (v(τ))) dτ. (t τ)l(r δ (v(τ))) dτ Conidr now z : R C 1 givn by z(t) := v(t) F(t, ϕ). In viw of proprty (b) and th abov, it follow that up ηt z(t) C 1 = up ηt v(t) F(t, ϕ) C 1 = up ηt ũ η (P cu ψ)(t) F(t, ϕ) C 1 up <, that i, z C 1 η. Morovr, combining (a) and (i) w up ηt ũ η (P cu ψ)(t) C 1 up F(t, ϕ) C 1 ηt z(t) C 1 = up ηt v(t) F(t, ϕ) C 1 = up ηt F δ (t, ψ) F(t, ϕ) C 1 = up ηt F δ (t, ψ) F δ (t, ϕ) C 1 <.

24 24 E. Stumpf Thrfor, z C 1. A, in addition, ψ = v(0) = F(0, ϕ) z(0) = ϕ z w conclud that proprty (i) indd impli (ii). 2. Suppo now that (ii) hold. Thn, in conidration of th on-to-on corrpondnc btwn olution of Eq. (2.14) and of Eq. (2.15) and in conidration of th uniqun rult containd in Propoition 3.1, w hav F δ (t, ψ) = F(t, ϕ) z(t) a t 0. Uing proprty (a) and th fact z C 1, w alo infr up ηt F δ (t, ψ) F δ (t, ϕ) C 1 = up ηt F δ (t, ψ) F(t, ϕ) C 1 = up ηt z(t) C 1<. Hnc, it rmain to prov ψ W η. For thi purpo, obrv that z (,0] C 1 η and F(, ϕ) (,0] C 1 η. Thu, for v : (, 0] C 1 givn by v(t) := F(t, ϕ) z(t) a t 0, w alo hav v C 1 η. A, in addition, v i a olution of Eq. (4.5) for all < t 0, Propoition 4.8 in [5] how v(0) = ψ W η and th artion follow. In viw of th aumption of th lat rult, it bcom clar that w nd om continuou map F : R X δ C 1 with proprti (a) and (b) in ordr to b abl to u proprty (ii) for a proof of Thorm 4.1. Blow w contruct uch a map. Th ky ingrdint hr ar th global miflow F δ and Fη cu dicud in th lat ction. Indd, dfining F : R X δ C 1 by F(t, ϕ) := { Fδ (t, ϕ), a t 0, F cu η (t, ϕ), th map F ha th dird proprti a hown nxt. othrwi, (4.6) Propoition 4.3. Th mapping F : R X δ C 1 dfind by Eq. (4.6) i continuou, po proprti (a) and (b) from Lmma 4.2, and atifi P cu F(t, ϕ) = T (t )P cu F(, ϕ) for all ϕ X δ and all < t <. (t τ)pcu l(r δ (F(τ, ϕ)) dτ (4.7) Proof. 1. Rcall that, by Propoition 3.1, F δ i continuou on [0, ) X δ, and that, by Propoition 3.6, Fη cu i continuou on (, 0] X δ. A for all ϕ X δ w hav F δ (0, ϕ) = ϕ = F cu η (0, ϕ), it i obviou that F i continuou on all of R X δ. Morovr, in conidration of th dfinition, it i alo clar that F ha proprty (a) from Lmma 4.2. Nxt, obrv that for ach ϕ X δ w hav F(, ϕ) (,0] = F cu η (, ϕ) = û(ϕ)( ) C(ϕ)( ) and both û(ϕ)( ) and C(ϕ)( ) blong to C 1 η du to Propoition 3.4 and th introduction in front of it. For thi raon, F(, ϕ) C 1 η, which how that F alo ha proprty (b) from Lmma 4.2.

25 Attraction proprty of local cntr-untabl manifold It rmain to prov that Eq. (4.7) hold. In ordr to how thi, conidr ϕ X δ and < t <. If t 0 thn Eq. (4.7) follow from Corollary 3.5: P cu F(t, ϕ) = P cu F cu η (t, ϕ) = P cu [ûη (ϕ)(t) C(ϕ)(t) ] = P cu û η (ϕ)(t) P cu P ϕ = P cu û η (ϕ)(t) = T (t )P cu û η (ϕ)() = T (t )P cu [ûη (ϕ)() P ϕ ] = T (t )P cu Fη cu (, ϕ) = T (t )P cu F(, ϕ) In ca 0 t <, Eq. (3.4) impli P cu F(t, ϕ) = P cu F δ (t, ϕ) = T (t )P cu F δ (, ϕ) = T (t )P cu F(, ϕ) (t τ)pcu l(r δ (û η (ϕ)(τ) P ϕ)) dτ (t τ)pcu l(r δ (Fη cu (τ, ϕ)) dτ (t τ)pcu l(r δ (F(τ, ϕ))) dτ (t τ)pcu l(r δ (F(τ, ϕ))) dτ. o formula (4.7) hold again. Finally, for < 0 < t w gt P cu F(t, ϕ) = T (t)p cu F(0, ϕ) 0 = T (t)t ( )P cu F(, ϕ) T (t) 0 (t τ)p = T (t )P cu F(, ϕ) (t τ)pcu l(r δ (F δ (τ, ϕ))) dτ (t τ)pcu l(r δ (F(τ, ϕ))) dτ, (t τ)pcu l(r δ (F(τ, ϕ))) dτ 0 cu l(r δ (F(τ, ϕ))) dτ ( τ)pcu l(r δ (F(τ, ϕ))) dτ (t τ)pcu l(r δ (F(τ, ϕ))) dτ by combining th two prcding ca. Thi tablih th formula. Givn ϕ X δ, w would lik to find om z C 1 uch that F(, ϕ) z i a olution of Eq. (4.5). For thi purpo, w dduc a ncary and ufficint condition for z C 1 to turn F(, ϕ) z into a olution of Eq. (4.5). Lmma 4.4. Lt ϕ X δ and z C 1 b givn. Thn F(, ϕ) z atifi u(t) = T (t )u() (t τ)l(r δ (u(τ))) dτ (4.8) for all < t < if and only if for all t R. z(t) = P F(t, ϕ) t (t τ)p cu (t τ)p l(r δ (F(τ, ϕ) z(τ))) dτ [ l(rδ (F(τ, ϕ) z(τ))) l(r δ (F(τ, ϕ))) ] dτ (4.9)

26 26 E. Stumpf Thi rult i motivatd by Vandrbauwhd [6, Lmma 5.7]. For it proof w nd two corollari, which both ar ay conqunc of th xponntial trichotomy (2.2) of th trongly continuou migroup T. Corollary 4.5. Lt z C 1 and t R b givn. Thn in C. lim T (t )P cu z() = 0 Proof. Rcall that T dfin a group on th cntr-untabl pac C cu C. In particular, R T (t )Pcu z() C cu i a continuou mapping from R into C. Furthrmor, combining th timat z() C η z C 0 and z C 0 z C 1 togthr with (2.2), w conclud T (t )Pcu z() C T (t )Pc z() C T (t )Pu z() C K c c t P c K c c( t) P c K c c( t) η P c (c c η) K c ct P c z() C K cu(t ) Pu z() C z() C K cu(t ) Pu z() C z C 0 K cu(t ) η P u z C 0 z C 0 (cuη) K cut P u z C 0 (cc η) K cct Pc z C 1 (cuη) K cut P u z C 1 }{{}}{{} < < for all, t R with t 0. A 0 < c c < η < c u, taking th limit for indd how lim T (t )P cu z() = 0 a claimd. Corollary 4.6. Lt ϕ X f, z C 1, and t R b givn. Thn in C. lim T (t ) [ P F(, ϕ) P z() ] = 0 Proof. Rcall that w hav F(, ϕ) (,0] C 1 η. Conquntly, uing th timat (2.2) w infr T (t )P (F(, ϕ) z()) C K c (t ) P K c(t ) P F(, ϕ) P z() C ) ( F(, ϕ) C z() C K c(t ) P η( ) η F(, ϕ) C η z() C ) (cη) K ct P ( F(, ϕ) (,0] C z 0η (,0] C 0η ) (cη) K ct P ( F(, ϕ) (,0] C z 0η C 1 for all min{0, t}. Sinc c < 0 < η < c it bcom clar that a. T (t )(P F(, ϕ) P z()) C 0

27 Attraction proprty of local cntr-untabl manifold 27 Having tablihd th auxiliary rult, w ar now in poition to prov Lmma 4.4. Proof of Lmma 4.4. W adopt th proof of Lmma 5.7 in Vandrbauwhd [6]. 1. Aum that, givn ϕ X δ and z C 1, F(, ϕ) z i a globally dfind olution of Eq. (4.8) for all < t <. Thn, in viw of Propoition 4.3 and Eq. (4.7), w gt that i, z(t) = T (t )[F(, ϕ) z()] (t τ)l(r δ (F(τ, ϕ) z(τ))) dτ F(t, ϕ) = P F(t, ϕ) P cu F(t, ϕ) T (t )[F(, ϕ) z()] (t τ)l(r δ (F(τ, ϕ) z(τ)))dτ = P F(t, ϕ) T (t )[P F(, ϕ) z()] P cu F(t, ϕ) T (t )P cu F(, ϕ) (t τ)pcu l(r δ (F(τ, ϕ) z(τ)))dτ (t τ)p l(r δ (F(τ, ϕ) z(τ))) dτ = P F(t, ϕ) T (t )[P F(, ϕ) z()] (t τ)pcu l(r δ (F(τ, ϕ) z(τ))) dτ (t τ)p l(r δ (F(, ϕ) z(τ))) dτ, (t τ)pcu l(r δ (F(τ, ϕ))) dτ z(t) = P F(t, ϕ) T (t )[P F(, ϕ) z()] (t τ)pcu [ l(rδ (F(τ, ϕ) z(τ))) l(r δ (F(τ, ϕ))) ] dτ (t τ)p l(r δ (F(τ, ϕ) z(τ))) dτ a < t <. Hnc, th application of th projction P cu and P how that and that P cu z(t) = T (t )Pcu z() P (t τ)pcu [l(r δ (F(τ, ϕ) z(τ))) l(r δ (F(τ, ϕ)))] dτ z(t) = P F(t, ϕ) T (t )[P F(, ϕ) P z()] (t τ)p l(r δ (F(τ, ϕ) z(τ))) dτ (4.10) (4.11) a < t <. Morovr, w claim that Eq. (4.10) hold for all, t R. Indd, a T dfin a group on th cntr-untabl pac C cu w may apply th oprator T ( t) to both id of Eq. (4.10) in ordr to that T ( t)pcu z(t) = T ( t)t (t )Pcu z() T ( t) = P cu z() (t τ)pcu [l(r δ (F(τ, ϕ) z(τ))) l(r δ (F(τ, ϕ)))] dτ ( τ)pcu [l(r δ (F(τ, ϕ) z(τ))) l(r δ (F(τ, ϕ)))] dτ,

28 28 E. Stumpf that i, P cu z() = T ( t)p cu z(t) ( τ)pcu [l(r δ (F(τ, ϕ) z(τ))) l(r δ (F(τ, ϕ)))] dτ, for all < t <. Thu, formula (4.10) hold for all, t R a claimd. In particular, thi prov that for fixd t R, w may tak th limit for in Eq. (4.10). Thn, in conidration of Corollary 4.5, w gt P cu z(t) = t (t τ)pcu [l(r δ (F(τ, ϕ) z(τ))) l(r δ (F(τ, ϕ)))] dτ. Similarly, carrying out th limit proc in Eq. (4.11) in combination with Corollary 4.6 lad to P z(t) = P F(t, ϕ) (t τ)p l(r δ (F(τ, ϕ) z(τ))) dτ. Hnc, it follow that z(t) = P z(t) Pcu z(t) = P F(t, ϕ) t = P F(t, ϕ) t (t τ)p cu (t τ)pcu (t τ)p l(r δ (F(τ, ϕ) z(τ))) dτ [ l(rδ (F(τ, ϕ) z(τ))) l(r δ (F(τ, ϕ))) ] (t τ)p l(r δ (F(τ, ϕ) z(τ))) dτ [ l(rδ (F(τ, ϕ) z(τ))) l(r δ (F(τ, ϕ))) ] for ach t R. Thi prov on dirction of th artion. 2. Suppo, convrly, that for givn ϕ X δ and z C 1 Eq. (4.9) hold, and lt t b givn. Obviouly, and P F(t, ϕ) z(t) = t z() = P F(, ϕ) t (t τ)p l(r δ (F(τ, ϕ) z(τ))) dτ (t τ)pcu l(r δ (F(τ, ϕ) z(τ))) dτ (t τ)pcu l(r δ (F(τ, ϕ))) dτ ( τ)p l(r δ (F(τ, ϕ) z(τ))) dτ ( τ)pcu l(r δ (F(τ, ϕ) z(τ))) dτ ( τ)pcu l(r δ (F(τ, ϕ))) dτ. Applying T (t ) on both id of th lat quation giv T (t )z() = T (t )P F(, ϕ) (t τ)pcu l(r δ (F(τ, ϕ) z(τ))) dτ (t τ)pcu l(r δ (F(τ, ϕ))) dτ, (t τ)p l(r δ (F(τ, ϕ) z(τ))) dτ (4.12)

29 Attraction proprty of local cntr-untabl manifold 29 that i, 0 = (t τ)pcu l(r δ (F(τ, ϕ))) dτ T (t )z() T (t )P F(, ϕ) (t τ)p l(r δ (F(τ, ϕ) z(τ))) dτ (t τ)pcu l(r δ (F(τ, ϕ) z(τ))) dτ. Nxt, aftr adding zro in th way rprntd abov to th right-hand id of Eq. (4.12), a impl calculation lad to P F(t, ϕ) z(t) = T (t )z() T (t )P F(, ϕ) (t τ)pcu l(r δ (F(τ, ϕ))) dτ. Hnc, by combining th lat quation with Eq. (4.7), w finally obtain F(t, ϕ) z(t) = P cu F(t, ϕ) P F(t, ϕ) z(t) = T (t )P cu F(, ϕ) T (t )z() T (t )P F(, ϕ) (t τ)p cu l(r δ (F(τ, ϕ))) dτ = T (t )F(, ϕ) T (t )z() = T (t )(F(, ϕ) z()) (t τ)l(r δ (F(τ, ϕ) z(τ))) dτ (t τ)pcu l(r δ (F(τ, ϕ))) dτ (t τ)l(r δ (F(τ, ϕ) z(τ))) dτ (t τ)l(r δ (F(τ, ϕ) z(τ))) dτ (t τ)l(r δ (F(τ, ϕ) z(τ))) dτ. A t wr arbitrary givn, w conclud that F(, ϕ) z i a olution of Eq. (4.8). Thi complt th proof. Now, conidr om fixd ϕ X δ. If w find om z C 1 atifying Eq. (4.9) thn Lmma 4.4 impli that F(, ϕ) z i a global olution of th abtract intgral quation (4.8). Hnc, in turn, by application of Lmma 4.2 it follow that ψ := F(0, ϕ) z(0) = ϕ z(0) blong to W η and that ϕ and ψ atify (4.4). Thrfor, in th nxt tp toward a proof of Thorm 4.1 w would lik to olv Eq. (4.9) in C 1 for ach givn ϕ X δ. Morovr, undr th aumption that th olution ar uniquly dtrmind, in thi way w alo would obtain a poibl choic for th map Hcu η from Thorm 4.1, namly, X δ ϕ ϕ z(0) = ψ W η. 5 Th rmaining part of th proof for th attraction proprty of th global cntr-untabl manifold Our nxt goal i to how that for ach fixd ϕ X δ Eq. (4.9) ha a uniquly dtrmind olution in C 1. Thi will b don by contruction of a paramtr-dpndnt contraction on th Banach pac C 1 blow.