Local invariant manifolds for delay differential equations with state space in C 1 ((− ∞ , 0 ] , R n )

Local invariant manifolds for delay differential equations with state space in C ¹ ((− _∞ , 0 ] , R ⁿ )

Dedicated to Professor Tibor Krisztin on the occasion of his 60th birthday

Hans-Otto Walther

Universität Gießen, Mathematisches Institut, Arndtstr. 2, Gießen, D 35392, Germany Received 20 January 2016, appeared 12 September 2016

Communicated by Ferenc Hartung

Abstract. Consider the delay differential equation x⁰(t) = f(xt) with the history xt:(−_{∞, 0}]→_Rⁿ of xat ‘time’tdefined byxt(s) = x(t+s). In order not to lose any possible entire solution of any example we work in the Fréchet spaceC¹((−∞, 0],Rⁿ), with the topology of uniform convergence of maps and their derivatives on compact sets. A previously obtained result, designed for the application to examples with un- bounded state-dependent delay, says that for maps f which are slightly better than continuously differentiable the delay differential equation defines a continuous semi- flow on a continuously differentiable submanifold X ⊂ C¹ of codimension n, with all time-t-maps continuously differentiable. Herecontinuously differentiablefor maps in Fréchet spaces is understood in the sense of Michal and Bastiani. It implies that f is of locally bounded delayin a certain sense. Using this property – and a related further mild smoothness hypothesis on f – we construct stable, unstable, and center manifolds of the semiflow at stationary points, by means of transversality and embeddings.

Keywords: delay differential equation, state-dependent delay, unbounded delay, Fréchet space, local invariant manifold.

2010 Mathematics Subject Classification: 34K05, 34K19, 37L05.

1 Introduction

Let Ube a set of maps (−_{∞, 0}] →_Rⁿand let a map f :U → _Rⁿbe given. A solution of the delay differential equation

x⁰(t) = f(x_t) (1.1)

is a map x : (−_{∞, 0}] +I → _Rⁿ, with I ⊂ _R an interval of positive length, such that all its segments

xt :(−_{∞, 0}]3s7→ x(t+s)∈_Rⁿ, t∈ I,

BEmail: Hans-Otto.Walther@math.uni-giessen.de

belong toU,xis differentiable onI, and satisfies (1.1) on I. In [18] we studied the initial value problem

x⁰(t) = f(x_t) for t>0 and x₀ =φ∈U (1.2) for a functional f on an open subsetUof the Fréchet spaceC¹ of continuously differentiable maps(−_{∞, 0}]→_Rⁿ, with the topology of uniform convergence of maps and their derivatives on compact sets. Let us briefly recall the motivation for working in the Fréchet space C¹, and not in a smaller Banach space of continuously differentiable maps (−_{∞, 0}] → _Rⁿ : we did not want to exclude any possible continuously differentiable map satisfying (1.1) on some interval, neither by growth conditions at−_∞nor by integrability conditions.

The main result of [18] says that if f : C¹ ⊃ U → _Rⁿ is continuously differentiable in the sense of Michal and Bastiani(we come back to this below) and if its derivatives satisfy a mild extension property then the initial value problem (1.2) defines a continuous semiflowSon the continuously differentiable submanifold

X={φ∈U:φ⁰(0) = f(φ)}, codimX =n,

with all solution operatorsS(t,·):φ7→ x_t continuously differentiable. The extension property is that

(e) each derivative D f(φ), φ ∈ U, extends to a linear map Def(φ) on the Fréchet space C of continuous maps(−_{∞, 0}]→_Rⁿ, and the map

U×C3 (φ,χ)7→ D_ef(φ)χ∈_Rⁿ is continuous.

The topology on C is given by uniform convergence on compact sets, of course. A first version of property (e) is the notion of beingalmost Fréchet differentiabledue to [10].

A toy example covered by the result from [18] is the state-dependent delay equation x⁰(t) = g(x(t−_∆)), ∆=δ(x(t)),

withg:R→_Randδ:R→[0,∞)continuously differentiable, not necessarily bounded.

Let us recall results on semiflows on submanifolds ofBanachspaces which will be used in the sequel. The Banach spaces are

C_d¹=C¹([−d, 0],Rⁿ), d>0, with the norm given by

|φ|_d,1= max

−d≤s≤0|φ(s)|+ max

−d≤s≤0|φ⁰(s)|, and

B¹_a =

φ∈C¹ : lim

s→−_∞φ(s)e^as =0, lim

s→−_∞φ⁰(s)e^as =0

, a>0, with the norm given by

|φ|_a,1=_sup

s≤0

|φ(s)|e^as+_sup

s≤0

|φ⁰(s)|e^as.

In [6,15,16] the initial value problem (1.2) was studied for f defined on an open subset of C¹_d, and the results apply to differential equations with bounded state-dependent delay. In [17] the initial value problem (1.2) was studied for f defined on an open subset of B¹_a, which covers differential equations with unbounded delay. The hypotheses are that fis continuously

differentiable and that the extension property holds, with an additional requirement in the second case: for a map f : B¹_a ⊃ U → _Rⁿ in (1.1) it is also assumed in [17] that f represents locally bounded delayin the sense that

(lbd)for everyφ∈U there are a neighbourhood N(φ)⊂U and some d>0so that f(ψ) = f(χ) for allψ,χin N(φ)withψ(s) =χ(s)on [−d, 0].

It may seem surprising that in case of maps f : U → _Rⁿ on open sets in Fréchet spaces property (lbd) is linked to smoothness. In fact, [18, Proposition 1.1] says that property (lbd) followsfromcontinuous differentiability in the sense of Michal and Bastiani[1,12]. The latter notion means for a continuous map

g:U→G, U ⊂F open, F andGFréchet spaces, that all directional derivatives

Dg(u)v= lim

06=h→0

1

h(g(u+hv)−g(u))∈G, u∈U,v ∈F, exist and that the map

U×F3 (u,v)7→ Dg(u)v∈G is continuous.

This notion of continuous differentiability avoids choosing a topology on the vector space L_c(F,G)of linear continuous mapsF →G. In caseF andGare Banach spaces it is obviously weaker than continuous differentiability in the sense of Fréchet (that is, there are derivatives Dg(u)∈ L_c(F,G),u∈U, in the sense of Fréchet andU3u7→ Dg(u)∈ L_c(F,G)is continuous with respect to the usual norm topology onL_c(F,G)).

In the sequel the labels (MB) and (F) are used in order to distinguish between both notions of continuous differentiability wherever confusion might arise.

In the present paper we find local invariant manifolds at a stationary point ¯φ ∈ X ⊂ C¹ of the semiflow S, for f continuously differentiable (MB), with property (e), and satisfying a further mild smoothness assumption (d) which requires that a map induced by f via property (lbd) is continuously differentiable (F). In order to state this precisely consider the restriction map R_d,1 : C¹ 3 φ 7→ φ|_[−d,0] ∈ C_d¹ and the prolongation map P_d,1 : C¹_d → C¹ given by (P_d,1φ)(s) = φ(s)for −d ≤ s ≤ 0 and (P_d,1φ)(s) = φ(−d) + (s+d)φ⁰(−d) fors < −d. Both maps are linear and continuous. Choose a neighbourhood N = N(φ^¯) ⊂ U of ¯φ andd > 0 according to property (lbd). Set ¯φ_d =R_d,1φ¯ and notice that

P_d,1φ¯_d=φ^¯ ∈ N

(because ¯φ is constant, see the preliminaries at the end of this introduction). By continuity there exist neighbourhoods V of ¯φ_d in C¹_d with P_d,1V ⊂ N, and due to the chain rule the composition f◦(P_d,1|_V)is continuously differentiable (MB), with

D(f◦(P_d,1|_V))(φ)χ=D f(P_d,1φ)P_d,1χ.

We assume that

(d)there is an open neighbourhood U_d ofφ¯_d in C¹_d with P_d,1U_d ⊂ N so that f_d = f◦(P_d,1|_Ud)is continuously differentiable (F).

Combining (e) and (d) we shall see in Proposition 2.2 below that the map f_dhas an exten- sion property analogous to (e). Then results from [15,16] apply and show that the equation

x⁰(t) = f_d(x_t) _(1.3)

(with segmentsx_t :[−d, 0]3s 7→ x(t+s)∈_Rⁿ) defines a semiflowS_dof continuously differ- entiable solution operatorsS_d(t,·)on domains in the continuously differentiable submanifold

X_d= {φ∈U_d:φ⁰(0) = f_d(φ)}, codimX_d =n,

of the Banach space C_d¹. The restriction ¯φ_d is a stationary point of S_d. From [6] we get local stable, center, and unstable manifolds ofS_dat ¯φ_d ∈X_d ⊂C_d¹.

We construct each local invariant manifold of S at ¯φ ∈ X ⊂ C¹ in a different way. For the local stable manifold of S at ¯φ ∈ X ⊂ C¹ we need the local stable manifold of S_d at φ¯_d ∈ X_d⊂C_d¹, and make use of a local transversality result in Fréchet spaces which is derived in the Appendix (Section 7). The local unstable manifold of S at ¯φ ∈ X ⊂ C¹ results from embedding the local unstable manifold obtained in [17], which sits in a Banach space B¹_a, a > 0. The construction of a local center manifold ofS at ¯φ∈ X ⊂ C¹ begins as in Krisztin’s Lyapunov–Perron type approach to a local center manifold ofS_d at ¯φ_d ∈ X_d ⊂ C¹_d from [6,8], and deviates at a certain point.

Section 3 below provides the tangent spaces of the local invariant manifolds of S at ¯φ ∈ X ⊂ C¹. Using the decomposition of the Banach space Y_d = Tφ¯_dX_d ⊂ C¹_d into stable, center and unstable spaces of the linearized solution operators

T_d,t:Y_d 3η7→ D₂S_d(t, ¯φ_d)η∈Y_d, t≥0,

from [6] we construct linear stable, center and unstable spaces in the tangent space Y=Tφ¯X={χ∈C¹ :χ⁰(0) =D f(φ¯)χ} ⊂C¹,

for the linearized solution operators

T_t :Y3χ7→ D₂S(t, ¯φ)χ∈Y, t ≥_0.

This is done without recourse to spectral properties of the operatorsT_t.

Returning to the hypotheses on f : C¹ ⊃ U → _Rⁿ it may be of interest to note that we could have started from another arrangement, in order to obtain the desired local invariant manifolds inC¹. As the objectives are local in nature it is possible to begin with property (lbd) of f where N(φ¯) =U. Next one can assume that an induced map like f_d on a neighbourhood of ¯φ_d in C¹_d is continuously differentiable (F) and that an analogue of the extension property (e) holds for the induced map. It would then follow that a restriction of f to a neighbourhood of ¯φin C¹ is continuously differentiable (MB) and has property (e), which means that we are back at the set of hypotheses which we prefer and actually use in this paper. – Arguing this way one finds in particular that for the toy example, where

f(φ) =g(φ(−δ(φ(0)))) for all φ∈C¹ (withn=1), all our hypotheses are satisfied.

Let us mention other recent work on invariant manifolds for equations with unbounded delay: in [11] Matsunaga et al. obtain local center manifolds for integral equations with un- bounded state- and time-invariant delay.

Preliminaries, notation. Banach spaces also are Fréchet spaces, that is, locally convex topological vector spaces which are complete and metrizable. For eachk ∈ _N0 the topology on the Fréchet spaceC^k ofktimes continuously differentiable maps(−_{∞, 0}]→_Rⁿis given by the seminorms| · |_k,j,j∈_{N, with}

|φ|_k,j=

∑

−maxj≤s≤0|φ^(κ)(s)|,

with the sets

V_k,j =

φ∈C^k :|φ|_k,j < ¹ j

forming a neighbourhood base at the origin. In C^k we haveφ_m → φas m→_∞ if and only if for every j∈_N,|φ_m−φ|_k,j →0 asm→_∞.

Continuously differentiable submanifolds of Fréchet spaces and continuously differen- tiable maps on such submanifolds are defined using continuous differentiability (MB). The reference for results on calculus in Fréchet spaces based on continuous differentiability (MB) which are freely used in the sequel is [5]. See also the survey [14]. For basic facts about topological vector spaces, see [13].

In the sequel also the vector spaceC^∞ =∩^∞_k=0C^k occurs, but without a topology on it.

It is convenient to denote the unique maximal solution to the initial value problem x⁰(t) = f(x_t) fort >0, x₀= φ∈ X,

byx^φ.

Stationary points of the semiflowSare constant. (Proof of this: supposeS(t,φ) =φfor all t ≥0. The solution x of (1.1) on [0,∞)with x0 = φsatisfiesx(t) =xt(0) =S(t,φ)(0) = φ(0) for all t≥0. For alls<0 we have x(s) =φ(s) =S(−s,φ)(s) =x−s(s) =x(0) =φ(0).)

For reals a < b andk ∈ _N0 letC^k([a,b],Rⁿ)denote the Banach space of k times continu- ously differentiable maps [a,b]→_Rⁿ, with the norm given by

|φ|_[a,b],k =

∑

amax≤s≤b|φ^(κ)(s)|.

In case a=−d<b=0 we abbreviateC_d^k =C^k([−d, 0],Rⁿ)and| · |_d,k =| · |_[−d,0],k. It is easy to see that the linear restriction maps

R_d,k :C^k →C^k_d, d>0 and k∈ _N0, and the linear prolongation maps

P_d,k :C_d^k →C^k, d>0 and k∈_N0, given by(P_d,kφ)(s) =φ(s)for−d≤s≤0 and

(P_d,kφ)(s) =

∑

φ^(κ)(−d)

κ! (s+d)^κ fors <−d are continuous, and for alld>0 andk∈_N0,

R_d,k◦P_d,k =id_Ck d. Solutions of equations

x⁰(t) =g(x_t), with g:C¹_d ⊃U→_Rⁿ or g: B¹_a ⊃U →_Rⁿ,

on some interval I ⊂ _R are defined as in case of (1.1): with J = [−d, 0] or J = (−_{∞, 0}], respectively, they are continuously differentiable maps x : J+I → _Rⁿ so that x_t ∈ U for all t ∈ I and the differential equation holds for all t ∈ I. Notice that x_t may denote a map on [−d, 0]or on(−_{∞, 0}], depending on the context.

The following statement on “globally bounded delay” for continuous linear maps corre- sponds to a special case of [18, Proposition 1.1].

Proposition 1.1. For every continuous linear map L: C⁰ → B, B a Banach space, there exists r> 0 with Lφ=0for allφ∈C⁰withφ(s) =0on[−r, 0].

Proof. Otherwise there are sequences r_m → _{∞ and} (φ_m)^∞_{1 in} C⁰ with φ_m(s) = _{0 on} [−r_m, 0] and 06= Lφm for allm∈_{N. For} cm =|Lφm|>0 we get _c¹

mφm→0 because for each j∈_Nand for all integersmwithrm ≥ j,

_c¹

mφm

0,j =0. By continuity, L

1 c_mφ_m

→0 asm→_∞, contradicting

L

1 c_mφm

=1 for allm∈ _N.

For results on strongly continuous semigroups given by solutions of linear autonomous retarded functional differential equations

x⁰(t) =_Λxt withΛ:C_d⁰→_Rⁿ linear and continuous, see [2,4].

2 On locally bounded delay, the extension property, and prolonga- tion and restriction

This section contains proofs of a few facts which were used already in Section 1, and further relations between the functionals f and f_d and between the semiflowsSandS_d. Recall that f is continuously differentiable (MB) and has property (lbd), withN= N(φ¯)_andd>_0.

Proposition 2.1. For everyφ∈ N we have

D f(φ)ψ=0 for allψ∈C¹withψ(s) =0on[−d, 0], and

Def(φ)χ=0 for allχ∈C⁰ withχ(s) =0on[−d, 0].

Proof. Let φ∈ Nandψ ∈ C¹ with ψ(s) = _{0 on}[−d, 0]be given. For h 6= 0 sufficiently small, φ+hψ ∈ N (due to continuity of multiplication with scalars), hence f(φ+hψ) = f(φ), and thereby,

D f(φ)ψ= lim

06=h→0

1

h(f(φ+hψ)− f(φ)) = lim

06=h→0

1

h(f(φ)− f(φ)) =0.

Letφ∈ Nandχ∈ C⁰withχ(s) =0 on[−d, 0]be given. Choose a sequence of pointsχ_m ∈C¹ withχ_m(s) =_{0 on}[−d, 0]which converges toχin the topology ofC⁰. (For example, letm≥ d and find ˆχm ∈C¹([−m, 0],Rⁿ)with

|χˆ_m(s)−χ(s)|< ¹

m on [−m, 0] and χˆ_m(s) =0 on[−d, 0].

Extend ˆχ_m to χ_m ∈ C¹ byχ_m(s) =χˆ_m(−m) + (χˆ_m)⁰(−m)(s+m)fors < −m. Conclude that for each j∈_N,|χm−χ|_0,j →0 as m→∞.) – We obtain

D_ef(φ)χ= lim

m→_∞D_ef(φ)χ_m = lim

m→_∞D f(φ)χ_m =0,

where the last equation follows from the first part of the assertion, with χ_m(s) =0 on[−d, 0].

With regard to the next result on the extension property of f_dobserve that each directional derivative of f_d, at a pointφ∈U_d in direction ofχ∈C¹_d, is given byD f_d(φ)χwith the Fréchet derivative D f_d(φ)∈L_c(C¹_d,Rⁿ)_.

Proposition 2.2. Each Fréchet derivative D f_d(φ) ∈ L_c(C_d¹,Rⁿ), φ ∈ U_d, extends to a linear map Def_d(φ):C_d⁰→_Rⁿand the map U_d×C_d⁰3 (φ,χ)7→ Def_d(φ)χ∈_Rⁿis continuous.

Proof. 1. Let φ ∈ U_d be given. By the chain rule for continuous differentiability (MB) in combination with the remark preceding Proposition 2.2 the Fréchet derivative of f_d at φ is given byD f_d(φ) =D f(P_d,1φ)◦P_d,1. DefineD_ef_d(φ)_:C⁰_d →_Rⁿ_byD_ef_d(φ)χ=D_ef(P_d,1φ)P_d,0χ.

The mapDef_d(φ)is linear. It also is a continuation of D f_d(φ)since forχ∈C¹_d we have D_ef_d(φ)χ= D_ef(P_d,1φ)P_d,0χ

= D_ef(P_d,1φ)P_d,1χ (with Proposition2.1andP_d,0χ(s) =P_d,1χ(s)on [−d, 0])

= D f(P_d,1φ)P_d,1χ=D f_d(φ)χ.

2. The continuity of the map

U_d×C⁰_d 3(φ,χ)7→ Def_d(φ)χ∈_Rⁿ

follows from its definition in combination with property (e) of f and the continuity ofP_d,1and P_d,0.

Proposition 2.3. X_d= R_d,1(X∩N∩R⁻_d,1¹(U_d))

Proof. 1. On X_d ⊂ R_d,1(X∩N∩R⁻_d,1¹(U_d)). For φ ∈ X_d ⊂ U_d we have P_d,1φ ∈ N. Using this andφ=R_d,1P_d,1φwe getP_d,1φ∈N∩R⁻_d,1¹(U_d)and

(P_d,1φ)⁰(₀) =φ⁰(₀) = f_d(φ) _(by φ∈ X_d)

= f(P_d,1φ),

which means P_d,1φ∈ X. It follows thatφ=R_d,1P_d,1φis inR_d,1(X∩N∩R⁻_d,1¹(U_d)).

2. OnR_d,1(X∩N∩R⁻_d,1¹(U_d))⊂ X_d. Considerφ= R_d,1ψwithψ∈ X∩N∩R⁻_d,1¹(U_d). Then φ= R_d,1ψ∈U_d ⊂P_d,1⁻¹(N),P_d,1R_d,1ψ= P_d,1φ∈ N,ψ∈X∩N⊂U, and

φ⁰(0) = (R_d,1ψ)⁰(0) =ψ⁰(0) = f(ψ) (sinceψ∈ X)

= f(P_d,1R_d,1ψ)

(with (lbd),ψ∈ N,P_d,1R_d,1ψ∈ N, and ψ(s) = P_d,1R_d,1ψ(s)on [−d, 0])

= f(P_d,1φ) = f_d(φ)_, which givesφ∈X_d.

Proposition 2.4. For everyφ∈ X∩N∩R⁻_d,1¹(U_d),

T_Rd,1φX_d =R_d,1T_φX.

Proof. Letφ∈X∩N∩R⁻_d,1¹(U_d)be given. Using Proposition2.3we inferR_d,1φ∈ X_dand R_d,1TφX= DR_d,1(φ)TφX ⊂TR_d,1φX_d,

hence codimR_d,1T_φX ≥codimT_Rd,1φX_d = n. As R_d,1 is surjective and codimT_φX = nwe also getn≥_codimR_d,1T_φX. It follows thatR_d,1T_φXandT_Rd,1φX_d have the same finite codimension n. Using the previous inclusion we obtain equality.

Let Ω ⊂ X×[0,∞) and Ωd ⊂ X_d×[0,∞) denote the domains of S and S_d, respectively.

The unique maximal solutions to the initial value problems

x⁰(t) = f_d(x_t) fort>0, x₀ =χ∈ X_d,

are denoted byx^χ(as in case of the initial value problem for (1.1) and data inX).

Proposition 2.5.

(i) For(t,φ)∈ _Ωwith S([0,t]× {φ})⊂ N∩R⁻_d,1¹(U_d),

(t,R_d,1φ)∈ _Ωd and S_d(t,R_d,1φ) =R_d,1S(t,φ).

(ii) If(t,χ)∈_Ωdand if x:(−_∞,t]→_Rⁿgiven by x(s) =x^χ(s)on[−d,t]and by x(s) = (P_d,1χ)(s) for s<−d satisfies{xs: 0≤s≤t} ⊂N then

(t,P_d,1χ)∈_Ω and R_d,1S(t,P_d,1χ) =S_d(t,χ).

Proof. On (i): Let x = x^φ and set y = x|_[−d,t]. Each segment ys ∈ C_d¹, 0 ≤ s ≤ t, equals R_d,1x_s ∈ R_d,1(X∩ N∩R⁻_d,1¹(U_d)) = X_d. In particular, y₀ = R_d,1x₀ = R_d,1φ ∈ U_d, and for 0≤s≤t,

y⁰(s) =x⁰(s) = f(xs)

= f(P_d,1R_d,1xs)

(by (lbd), usingx_s ∈N,R_d,1x_s ∈U_d ⊂ P_d,1⁻¹(N),P_d,1R_d,1x_s ∈N,x_s(v) =P_d,1R_d,1x_s(v) on[−d, 0])

= f_d(R_d,1x_s) = f(y_s),

which implies that the restrictiony= x|_[−d,t]satisfies (1.3) on[0,t]. Now the assertion becomes obvious.

On (ii): consider(t,χ)∈ _Ωd and the maximal solutionx^χ of (1.3) and x:(−_∞,t]→_Rⁿ as defined in assertion (ii) and assume the segmentsxs∈ C¹, 0≤ s ≤t, belong toN. For suchs we have

x⁰(s) = (x^χ)⁰(s) = f_d(x^χ_s) = f(P_d,1x^χ_s)

= f(x_s)

(by (lbd), useP_d,1x^χ_s ∈ N,x_s∈ N,(P_d,1x^χ_s)(v) =x_s^χ(v) =x^χ(s+v) =x(s+v) =x_s(v) for −d≤v≤0),

andx₀ = P_d,1χ∈ N. It follows that (t,P_d,1χ)∈ _Ωandx_s =S(s,P_d,1χ)for all s ∈[0,t]. Finally, observeR_d,1xs= x^χ_s =S_d(s,χ)for 0≤s≤ t.

Proposition2.5(i) shows that ¯φ_dis a stationary point of the semiflowS_d.

Fort ≥0 consider the operatorsT_t =D₂S(t, ¯φ)_onTφ¯XandT_d,t =D₂S_d(t, ¯φ_d)_on Tφ¯_dX_d.

Corollary 2.6.

(i) For(t,φ)∈_Ωas in Proposition 2.5 (i) and for allχ∈ TφX,

R_d,1χ∈T_Rd,1φX_d and R_d,1D₂S(t,φ)χ= D₂S_d(t,R_d,1φ)R_d,1χ.

(ii) For allχ∈ Tφ¯X and for all t≥0,

R_d,1χ∈Tφ¯dX_d and R_d,1T_tχ= T_d,tR_d,1χ.

Proof. On (i): forφ∈XwithS([0,t]× {φ})⊂ N∩R⁻_d,1¹(U_d)we haveS_d(t,R_d,1φ) =R_d,1S(t,φ), by Proposition 2.5. Let χ∈ T_φX be given. By Proposition2.4, R_d,1χ ∈ T_Rd,1φX_d. By the chain rule, D2S_d(t,R_d,1φ)R_d,1χ= R_d,1D2S(t,φ)χ, which yields the assertion.

On (ii): we have[0,∞)× {φ^¯} ⊂ Ω, and for allt ≥ 0, S(t, ¯φ) =φ^¯ ∈ N∩R⁻_d,1¹(U_d), because of ¯φ ∈ N and R_d,1φ¯ = φ^¯_d ∈ U_d. Using part (i) we conclude that for all t ≥ 0 and χ ∈ Tφ¯X, R_d,1χ∈Tφ¯_dX_d and

T_d,tR_d,1χ= D₂S_d(t,R_d,1φ¯)R_d,1χ=R_d,1D₂S(t, ¯φ)χ=R_d,1T_tχ.

3 Decompositions of tangent spaces

This section contains the decomposition of the Fréchet space Y = Tφ¯X ⊂ C¹ into the stable, center, and unstable spaces which in the subsequent sections will become the tangent spaces of the desired local invariant manifolds at ¯φ ∈ X. The construction does not make use of spectral properties of the operators T_t = D₂S(t, ¯φ), t ≥ 0, on Y, or of the generator of this semigroup, but exploits well-known properties of the strongly continuous semigroup on the Banach spaceC⁰_d which arises from linearizing the semiflowS_d at ¯φ_d ∈ X_das follows: in [6] it is shown that the derivatives T_d,t = D₂S_d(t, ¯φ_d), t≥ 0, form a strongly continuous semigroup on the Banach space

Y_d =Tφ¯dX_d ={χ∈C¹_d :χ⁰(0) =D f_d(φ^¯_d)χ}, and they are given by the equations

T_d,tχ=T_d,e,tχ fort≥0, χ∈Y

where T_d,e,tη=vtwith the continuous solution v:[−d,∞)→_Rⁿ of the initial value problem v⁰(t) =Def_d(φ^¯_d)vt fort >0, v0 =η∈C_d⁰.

Here the term continuous solution means that v is continuous, differentiable for t > 0, and satisfies the delay differential equation fort >0, as in [2,4]. – In the present section a symbol like v_t above always denotes a segment which is defined on[−d, 0].

The operatorsT_d,e,t: C_d⁰→ C_d⁰,t≥ 0, form a strongly continuous semigroup whose gener- ator has a discrete spectrum σ_d,e which consists of eigenvalues of finite algebraic multiplicity, with only a finite number of them in each halfplane {z ∈ _C : Rez > u}, u ∈ R. Then the stable, center, and unstable spaces of the semigroup are defined as the realified generalized eigenspacesC_d,s⁰ ,C⁰_d,c,C⁰_d,uwhich are given by the eigenvalues satisfying

Rez <0, Rez =0, Rez >0,

respectively. The operators T_d,e,t, t ≥ _{0, map} C_d,s⁰ into itself and act on C⁰_d,c and on C_d,u⁰ as isomorphisms. The center and unstable spaces are finite-dimensional. Initial data χ in C⁰_d,c

and inC⁰_d,uuniquely define analytic solutionsv= v^(χ)onRof the equationv⁰(t) =D_ef_d(φ^¯_d)v_t withv₀ = χand with all segmentsv_t :[−d, 0]3s 7→ v(t+s)∈_Rⁿ,t ∈ _{R, in}C_d,c⁰ and inC⁰_d,u, respectively. From χ ∈ C¹_d and χ⁰(₀) = D_ef_d(φ¯_d)χ = D f_d(φ¯_d)χwe have χ ∈ Y_d. This yields C⁰_d,c ⊂ Y_d, C⁰_d,u ⊂Y_d. For everyt ≥0 the operator T_d,t given byT_d,e,t acts as an ismorphism onY_d,c=C_d,c⁰ and onY_d,u=C⁰_d,u. With the closed spaceY_d,s =Y_d∩C_d,s⁰ ,

Y_d =Y_d,s⊕Y_d,c⊕Y_d,u and T_d,tY_d,s⊂Y_d,s for allt ≥0, see [6]. The injective linear maps

I_c :C_d,c⁰ 3χ7→ v^(χ)|_(−∞,0] ∈C¹ and I_u:C⁰_d,u3 χ7→ v^(χ)|_(−∞,0] ∈C¹ with finite-dimensional domains are continuous. Define

Y_c= I_cC_d,c⁰ = I_cY_d,c and Y_u= I_uC_d,u⁰ = I_uY_d,u. Notice that

φ= I_cR_d,1φ onY_c and φ= I_uR_d,1φ onY_u. The finite-dimensional spacesY_c andY_uare both contained inY, because of

(v^(χ)|_(−∞,0])⁰(0) =χ⁰(0) =D f_d(φ^¯_d)χ

= D f(P_d,1φ¯_d)P_d,1χ

= D f(φ^¯)(v^(χ)|_(−∞,0]) (by Proposition2.1).

The spacesY_c andY_userve as center and unstable spaces inY.

Proposition 3.1(Conjugacy, invariance). For every t≥0,

T_tI_cχ= I_cT_d,tχ for allχ∈Y_d,c=C_d,c⁰ and TtIuχ= IuT_d,tχ for allχ∈Y_d,u=C⁰_d,u,

and T_tY_c=Y_c and T_tY_u =Y_u.

Proof. Let χ ∈ C_d,c⁰ , v = v^(χ), t ≥ 0. Then vt = T_d,tχ ∈ C_d,c⁰ . The translate w = v(t+·) of v : R → _Rⁿ also is an analytic solution of the linear equation given by Def_d(φ^¯_d) : C⁰_d → _Rⁿ, with initial valuew₀=v_t ∈C⁰_d,c. Hencew|_(−∞,0] = I_cv_t. Next, I_cχ=v|_(−∞,0], and for alls >0,

v⁰(s) =D_ef_d(φ^¯_d)v_s =D f_d(φ^¯_d)v_s =D f(P_d,1φ¯_d)P_d,1v_s

=D f(φ^¯)P_d,1v_s

=D f(φ^¯)(v(s+·)|_(−∞,0])

(by Proposition2.1, with(P_d,1vs)(r) =vs(r) =v(s+r)for −d≤r≤0), which givesT_t(v|_(−∞,0]) =v(t+·)|_(−∞,0] =w|_(−∞,0]. Altogether,

TtIcχ=Tt(v|_(−∞,0]) =w|_(−∞,0] = Icvt= IcT_d,tχ.

The proof for χ ∈ C⁰_d,u is analogous. The last assertions follow from the first and second assertion, respectively.

Define the stable space inYas the closed space Y_s=Y∩R⁻_d,1¹Y_d,s. Proposition 3.2. Y=Ys⊕Yc⊕Yuand TtYs⊂Ysfor all t≥0.

Proof. 1. Proof ofY ⊂ Y_s⊕Y_c⊕Y_u: for φ ∈ Y, R_d,1φ ∈ Y_d, see Proposition 2.4. There exist χ_s∈Y_d,s,χ_c∈Y_d,c =C⁰_d,c,χ_u∈Y_d,u =C⁰_d,uso that R_d,1φ=χ_s+χ_c+χ_u. Hence

R_d,1(φ−I_cχ_c−I_uχ_u) =χ_s+χ_c+χ_u−R_d,1I_cχ_c−R_d,1I_uχ_u= χ_s∈Y_d,s, which in combination with φ−Icχ−Iuχu ∈Yyieldsφ−Icχc−Iuχu∈Ys.

2.1 Proof of Y_s∩Y_c ⊂ {0}: for φ ∈ Y_s∩Y_c = (Y∩R⁻_d,1¹Y_d,s)∩ I_cY_d.c we have R_d,1φ ∈ Y_d,s∩Y_d,c= {₀}. Consequently,R_d,1φ=0, and therebyφ= I_cR_d,1φ=_0.

2.2 The proof ofY_s∩Y_u ⊂ {0}is analogous.

2.3 Proof ofY_c∩Y_u⊂ {0}: forφ∈Y_c∩Y_u= I_cY_d.c∩I_uY_d,u, henceR_d,1φ∈Y_d,c∩Y_d,u= {0}. Consequently,R_d,1φ=0, and therebyφ= IcR_d,1φ=0.

3. Lett≥0, φ∈Y_s. ThenR_d,1φ∈Y_d,s. Using this and Corollary2.6one finds R_d,1T_tφ= T_d,tR_d,1φ∈Y_d,s,

which givesT_tφ∈Y_s.

What will be used from this section in the sequel are only the definitions of the spaces Ys,Yc,Yu and the inclusion

IuC_d,u⁰ =Yu⊂ B¹_a

which follows from v^(χ)(t)→0 and(v^(χ))⁰(t)→0 ast → −_∞for all χ∈C⁰_d,u.

4 The local stable manifold

We begin with the local stable manifold W_d^s ⊂ X_d of the semiflowS_d at the stationary point φ¯_d ∈ X_d ⊂ C¹_d as it was obtained in [6]. It is easy to see thatW_d^s is a continuously differen- tiable submanifold of the Banach spaceC¹_dwhich is locally positively invariant underS_d, with tangent space

Tφ¯dW_d^s=Y_d,s

at ¯φ_d, and that it has the following properties (I) and (II), for some β>0 chosen with

−β> Rez for allz∈ σ_d,e with Rez<0 and for someγ> β.

(I) There are an open neighbourhood ˜W_d^s of ¯φ_d in W_d^s such that [_0,∞)×W^˜_d^s ⊂ _Ωd and S_d([0,∞)×W^˜_d^s)⊂W_d^s, and a constant ˜c>0 such that for allψ∈W^˜_d^sand allt ≥0,

|S_d(t,ψ)−φ^¯_d|_d,1 ≤c e˜ ^−γt|ψ−φ^¯_d|_d,1.

(II) There exists a constant ¯c>0 such that each ψ∈ X_dwith [0,∞)× {ψ} ⊂_Ωdand e^βt|S_d(t,ψ)−φ¯_d|_d,1 <c¯ for allt ≥₀

belongs toW_d^s.

The codimension ofW_d^s inC¹_d is equal to

n+dimY_d,c+dimY_d,u =n+dimC_d,c⁰ +dimC_d,u⁰ .

As the continuous linear map R_d,1 : C¹ → C¹_d is surjective we can apply Proposition7.3 from the Appendix and obtain an open neighbourhoodV of ¯φinN ⊂U⊂C¹ so that

W^s=W^s(φ^¯) =V∩R⁻_d,1¹(W_d^s)

is a continuously differentiable submanifold ofC¹ with codimension n+dimC⁰_d,c+dimC_d,u⁰ and tangent space

Tφ¯W^s= R⁻_d,1¹(Tφ¯_dW_d^s) =R⁻_d,1¹(Y_d,s).

The next propositions show thatW^s is the desired local stable manifold ofSat ¯φ.

Proposition 4.1. W^s⊂ X and Tφ¯W^s=Y_s, and W^s is locally positively invariant.

Proof. 1. Let φ ∈ W^s. Then φ ∈ V ⊂ N and R_d,1φ ∈ W_d^s ⊂ X_d ⊂ U_d ⊂ P_d,1⁻¹(N) and φ(t) = P_d,1R_d,1φ(t) on [−d, 0]. Using R_d,1φ ∈ X_d, the definition of f_d, and property (lbd) we infer

φ⁰(0) = (R_d,1φ)⁰(0) = f_d(R_d,1φ) = f(P_d,1R_d,1φ) = f(φ) which meansφ∈X.

2. The first assertion yieldsTφ¯W^s ⊂Tφ¯X=Y⊂C¹. Hence Tφ¯W^s =Y∩R⁻_d,1¹(Y_d,s) =Y_s.

3. (On local positive invariance) Choose an open neighbourhoodV_dof ¯φ_daccording to local positive invariance ofW_d^s. Then choose an open neighbourhood ˆV⊂V of ¯φwith R_d,1Vˆ ⊂ V_d. Considert ≥ 0 and φ∈ W^s∩V^ˆ with S([0,t]× {φ}) ⊂ V. Then^ˆ R_d,1S([0,t]× {φ}) ⊂ V_d and R_d,1φ∈W_d^s∩V_d. For 0 ≤s≤t the solutionx:(−_∞,t]→_Rⁿof the initial value problem (1.2) satisfies

x⁰(s) = f(x_s) = f(P_d,1R_d,1x_s) (with (lbd); we have

R_d,1x_s∈U_d,P_d,1R_d,1x_s ∈N,x_s∈V^ˆ ⊂N,P_d,1R_d,1x_s =x_son[−d, 0])

= f_d(R_d,1x_s),

which shows that y = x|_[−d,t] is a solution of (1.3) on [0,t], with initial value y₀ = R_d,1φ ∈ W_d^s∩V_d and with the segments y_s = R_d,1x_s, 0 ≤ s ≤ t, in R_d,1Vˆ ⊂ V_d. By local positive invariance ofW_d^s,y_s =R_d,1x_s ∈W_d^sfor 0≤s≤ t. It follows that for suchs,x_s∈V^ˆ∩R⁻_d,1¹(W_d^s)⊂ V∩R⁻_d,1¹(W_d^s) =W^s.

Proposition 4.2.

(i) There are an open neighbourhoodV of˜ φ¯ in V with[0,∞)×(V^˜ ∩W^s)⊂ _Ωand a constantc˜> 0 such that for allφ∈V^˜ ∩W^sthe solution x :R→_Rⁿof the initial value problem(1.2)satisfies

|x(t)−φ¯(0)|+|x⁰(t)| ≤ce˜ ^−γt|R_d,1φ−φ¯_d|_d,1 for all t≥0.

(ii) There are an open neighbourhoodV ofˆ φ¯ in V and a constant cˆ > 0 such that for every solution x:R→_Rⁿ of the initial value problem(1.2)withφ∈V^ˆ ∩X and

|x(t)−φ^¯(0)|+|x⁰(t)| ≤c eˆ ^−βt for all t≥0 we haveφ∈W^s.

Proof. 1. Considerγ> β>0 and ˜W_d^s, ˜c, ¯cfrom statements (I) and (II) above. There is an open neighbourhood ˜V_d⊂U_dof ¯φ_d withW_d^s∩V^˜_d =W^˜_d^s.

2. On (i). Choose an integerj≥dso that for allχ∈C¹_dwith|χ−φ^¯_d|_d,1 < ¹_j we haveχ∈U_d, and for all ψ∈C¹with|ψ−φ^¯|_1,j< ¹_j we haveψ∈ N. Choose an open neighbourhood ˜V ⊂V of ¯φso that for allφ∈V^˜ we have

|φ−φ^¯|_1,j < ¹

2j(c˜+1) ^{and Rd,1V˜} ⊂V^˜_d.

For φ∈W^s∩V^˜ we obtain R_d,1φ∈W_d^s∩V^˜_d =W^˜_d^s. By statement (I),[0,∞)× {R_d,1φ} ⊂_Ωd and for all t≥0,

|S_d(t,R_d,1φ)−φ^¯_d|_d,1 ≤c e˜ ^−γt|R_d,1φ−φ^¯_d|_d,1

≤c e˜ ^−γt|φ−φ^¯|_1,j < ¹ 2j.

Then the solutiony :[−d,∞)→_Rⁿon [0,∞)of (1.3) with initial valuey0= R_d,1φ∈W^˜_d^s ⊂W_d^s satisfies

|y(s)−φ^¯(0)|+|y⁰(s)|< ¹

2j for alls≥ −d.

The map x : R → _Rⁿ given by x(t) = y(t) for t ≥ −d and x(t) = φ(t) for t < −d is continuously differentiable. Using x(s) =φ(s)fors≤0,φ∈V, and the previous estimate we^˜ infer

|x(s)−φ^¯(0)|+|x⁰(s)|< ¹

2j for alls≥ −j,

which yields |xt−φ^¯|_1,j < ¹_j for all segments xt : (−_{∞, 0}] 3 u 7→ x(t+u) ∈ _Rⁿ, t ≥ 0.

Consequently,x_t∈ Nfor all t≥0. Using

|R_d,1x_t−R_d,1φ¯|_d,1 ≤ |x_t−φ^¯|_1,j < ¹

j for allt≥0 we obtain for all t≥0 that R_d,1xt∈ U_d, henceP_d,1R_d,1xt ∈ N, and

x⁰(t) =y⁰(t) = f_d(yt) = f_d(R_d,1xt) = f(P_d,1R_d,1xt) = f(xt)(with (lbd)).

It follows thatx is the solution of the initial value problem (1.2), and for everyt≥0,

|x(t)−φ^¯(0)|+|x⁰(t)|=|y(t)−φ^¯(0)|+|y⁰(t)|

≤ |S_d(t,R_d,1φ)−φ^¯_d|_d,1 ≤c e˜ ^−γt|R_d,1φ−φ^¯_d|_d,1. 3. On (ii). Choose an integerj≥ dwith

1 j < ^c¯

2e^−βd so that

φ∈C¹:|φ−φ¯|_1,j < ² j

⊂V and

χ∈C¹_d :|χ−φ¯_d|_d,1< ² j

⊂U_d. Set

Vˆ =

φ∈ C¹:|φ−φ^¯|_1,j < ¹ j

and choose ˆc>0 with ˆc e^βd < ¹ j.