Integral manifolds for partial functional differential equations in admissible spaces on a half line

Integral manifolds for partial functional differential equations in admissible spaces on a half line tài liệu, giáo án,...

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Journal of Mathematical Analysis and

Applications

www.elsevier.com/locate/jmaa

Integral manifolds for partial functional differential equations

in admissible spaces on a half-line ✩

Nguyen Thieu Huya, ∗ , Trinh Viet Duocb

aSchool of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Viet Nam

bFaculty of Mathematics, Mechanics, and Informatics, Hanoi University of Science, 334 Nguyen Trai, Hanoi, Viet Nam

Article history:

Received 9 December 2012

Available online xxxx

Submitted by C.E Wayne

Keywords:

Exponential dichotomy and trichotomy

Partial functional differential equations

Stable and center-stable manifolds

Admissibility of function spaces

In this paper we investigate the existence of stable and center-stable manifolds for solutions to partial functional differential equations of the formu˙( =A ( u ( +f ( u t ),

t0, when its linear part, the family of operators( A ( )) t 0, generates the evolution family

( U ( s )) ts 0 having an exponential dichotomy or trichotomy on the half-line and the

nonlinear forcing term f satisfies the ϕ-Lipschitz condition, i.e., f ( u t )− f ( v t ) 

ϕ( u t−v tC where u t , v t∈C := C ([−r 0], X ), and ϕ( belongs to some admissible function space on the half-line Our main methods invoke Lyapunov–Perron methods and the use of admissible function spaces

©2013 Elsevier Inc All rights reserved

1 Introduction

Consider the partial functional differential equation

du

where A(t)is a (possibly unbounded) linear operator on a Banach space X for every fixed t; f: R+× C →X is a continuous

nonlinear operator withC :=C([−r,0],X), and ut is the history function defined by ut(θ ) :=u(t+ θ)forθ ∈ [−r,0] When the family of operators(A(t))t0generates the evolution family having an exponential dichotomy (or trichotomy), one tries

to find conditions on the nonlinear forcing term f such that Eq.(1.1)has an integral manifold (e.g., a stable, unstable or

cen-ter manifold) The most popular condition imposed on f is its uniform Lipchitz continuity with a sufficiently small Lipschitz

constant, i.e.,f(t, φ) −f(t, ψ)  qφ − ψC for q small enough (see[1,3,14]and references therein) However, for

equa-tions arising in complicated reaction–diffusion processes, the function f represents the source of material (or population)

which, in many contexts, depends on time in diversified manners (see[5, Chapt 11],[6,15]) Therefore, sometimes one may

not hope to have the uniform Lipschitz continuity of f Recently, for the case of partial differential equations without delay,

we have obtained exciting results in[9], where we have used the Lyapunov–Perron method and the characterization of the exponential dichotomy (obtained in[8]) of evolution equations in admissible function spaces to construct the structures of solutions in mild forms, which belong to some certain classes of admissible spaces on which we could employ some well-known principles in mathematical analysis such as the contraction mapping principle, the implicit function theorem, etc The use of admissible spaces has helped us to construct the invariant manifolds without using the smallness of Lipschitz constants of nonlinear forcing terms in classical sense (see[9,10])

✩ This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2011.25.

* Corresponding author.

E-mail addresses:huy.nguyenthieu@hust.vn (T.H Nguyen), tvduoc@gmail.com (V.D Trinh).

0022-247X/$ – see front matter ©2013 Elsevier Inc All rights reserved.

Another point we would like to mention is that in some applications the partial differential operator A(t)is defined only

for t0 (see e.g.,[4,11,13]and references therein) Therefore, the evolution family generated by(A(t))t0 is defined only

on a half-line.

The purpose of the present paper is to prove the existence of stable and center-stable manifolds for Eq (1.1)when its linear part (A(t))t0 generates the evolution family having an exponential dichotomy or trichotomy on the half-line under more general conditions on the nonlinear forcing term f , that is the ϕ-Lipschitz continuity of f , i.e.,f(t, φ) −f(t, ψ) 

ϕ (t)φ − ψC whereφ, ψ ∈ C, andϕ (t)is a real and positive function which belongs to admissible function spaces defined

in Definition 2.3below We will extend the methods in[9]to the case of partial functional differential equations (PFDE) The main difficulties that we face when passing to the case of PFDE are the following two features: Firstly, since the

nonlinear operator f is ϕ-Lipschitz, the existence and uniqueness theorem for solutions to(1.1)is not available Secondly, the evolution family generated by(A(t))t0 is defined only on a half-lineR+and doesn’t act on the same Banach space as

that the surfaces of the integral manifold belong to (in fact, the former acts on X , and the latter belongs to C) Therefore, the standard methods of nonlinear perturbations of an evolutionary process using graph transforms as formulated in [1,3]

cannot be applied here

To overcome such difficulties, we reformulate the definition of invariant manifolds such that it contains the existence and uniqueness theorem as a property of the manifold (see Definition 3.3below) Furthermore, we construct the structure

of the mild solutions to(1.1) using the Lyapunov–Perron equation (see Eq (3.5)) in such a way that it allows to combine the exponential dichotomy of the linear part of Eq (1.1)with the existence and uniqueness of its bounded solutions in the case ofϕ-Lipschitz nonlinear forcing terms Then, we use the admissible spaces to construct the integral manifolds for

Eq.(1.1)in the case of dichotomic linear part without using the smallness of Lipschitz constants of the nonlinear terms in classical sense Instead, the “smallness” is now understood as the sufficient smallness of supt0

t+1

t ϕ ( τ )dτ Consequently,

we obtain the existence of invariant stable manifolds for the case of dichotomic linear parts under very general conditions on

the nonlinear term f(t u t) Moreover, using these results and rescaling procedures we prove the existence of center-stable manifolds for the mild solutions to Eq (1.1) in the case of trichotomic linear parts under the same conditions on the

nonlinear term f as in the dichotomic case Our main results are contained inTheorems 3.7, 4.2

We now recall some notions

For a Banach space X (with a norm  · ) and a given r>0 we denote by C :=C( [−r,0],X) the Banach space of all continuous functions from [−r,0] into X , equipped with the normφC=supθ∈[−r,0]φ(θ)forφ ∈ C For a continuous

function v: [−r, ∞) →X the history function v t∈ Cis defined by vt(θ ) =v(t+ θ)for allθ ∈ [−r,0]

Definition 1.1 A family of bounded linear operators{U(t s}ts0on a Banach space X is a (strongly continuous, exponential

bounded) evolution family if

(i) U(t t) =Id and U(t r U( ,s =U(t s for all trs0,

(ii) the map(t s →U(t s x is continuous for every x∈X ,

(iii) there are constants K,c0 such thatU(t s x K e c t−sxfor all ts0 and x∈X

The notion of an evolution family arises naturally from the theory of non-autonomous evolution equations which are well-posed Meanwhile, if the abstract Cauchy problem

⎧

⎨

⎩

du

dt =A(t)u(t), ts0,

u( ) =x s∈X

(1.2)

is well-posed, there exists an evolution family {U(t s}ts0 such that the solution of the Cauchy problem (1.2) is given

by u(t) =U(t s u( ) For more details on the notion of evolution families, conditions for the existence of such families and applications to partial differential equations we refer the readers to Pazy[11] (see also Nagel and Nickel[7]for a detailed discussion of well-posedness for non-autonomous abstract Cauchy problems on the whole lineR)

2 Function spaces and admissibility

We recall some notions on function spaces and refer to Massera and Schäffer [2], Räbiger and Schnaubelt [12] for concrete applications

Denote by B the Borel algebra and by λ the Lebesgue measure on R+ The space L1, loc( R+) of real-valued locally integrable functions on R+ (modulo λ-nullfunctions) becomes a Fréchet space for the seminorms pn(f) := J n|f(t) |dt,

where Jn= [n,n+1]for each n∈ N(see[2, Chapt 2, §20])

We can now define Banach function spaces as follows

Definition 2.1 A vector space E of real-valued Borel-measurable functions onR+(moduloλ-nullfunctions) is called a Banach

function space (over( R+, B, λ)) if

1) E is Banach lattice with respect to a norm · E, i.e.,(E,  · E)is a Banach space, and ifϕ ∈E andψ is a real-valued Borel-measurable function such that|ψ(·)|  | ϕ (·)|, λ-a.e., thenψ ∈E andψE  ϕ E,

2) the characteristic functions χA belong to E for all A ∈ B of finite measure, and supt0 χ[t +1]E < ∞ and inft0 χ[t +1]E>0,

3) E →L1, loc(R+), i.e., for each seminorm p n of L1, loc(R+)there exists a numberβp n>0 such that p n(f)  βp nfE for

all f∈E.

We then define Banach spaces of vector-valued functions corresponding to Banach function spaces as follows

Definition 2.2 Let E be a Banach function space and X be a Banach space endowed with the norm ·  We set

E := E ( R+,X) = f : R+→X: f is strongly measurable andf( ·) ∈E

(moduloλ-nullfunctions) endowed with the normfE= f( ·)E One can easily see thatE is a Banach space We call

it the Banach space corresponding to the Banach function space E.

We now introduce the notion of admissibility in the following definition

Definition 2.3 The Banach function space E is called admissible if

(i) there is a constant M1 such that for every compact interval[a,b] ∈ R+we have

b

a

ϕ (t) dt M(b−a)

 χ[a , b]E ϕ E,

(ii) forϕ ∈E the functionΛ1ϕ defined byΛ1ϕ (t) = t+1

t ϕ ( τ )dτ belongs to E, (iii) E is T+

τ -invariant and T−

τ -invariant, where T+

τ and T−

τ are defined forτ ∈ R+ by

T+

τϕ (t) =

ϕ (t− τ ) for t τ 0,

0 for 0t τ ,

T−

τϕ (t) = ϕ (t+ τ ) for t0.

Moreover, there are constants N1,N2 such thatT+

τ N1,T−

τ N2for allτ ∈ R+

Example 2.4 Besides the spaces L p(R+), 1p ∞, and the space

M( R+) :=

f ∈L1, loc( R+): sup

t0

t+1

t

f( τ ) dτ < ∞

endowed with the normfM:=supt0

t+1

t |f( τ )|dτ, many other function spaces occurring in interpolation theory, e.g.,

the Lorentz spaces L p , q, 1<p< ∞, 1<q< ∞are admissible

Remark 2.5 One can easily see that if E is an admissible Banach function space then E →M(R+)

We now collect some properties of admissible Banach function spaces in the following proposition (see [9, Proposi-tion 2.6])

Proposition 2.6 Let E be an admissible Banach function space Then the following assertions hold.

(a) Letϕ ∈L1, loc( R+)such thatϕ 0 andΛ1ϕ ∈E, whereΛ1is defined as in Definition2.3(ii) Forσ >0 we define functionsΛ

σϕ

andΛ

σϕby

Λ

σϕ (t) =

t

0

e−σ ( t−sϕ ( )ds,

Λ

σϕ (t) =

∞

e−σ (−t )ϕ ( )ds.

σϕandΛ

σϕbelong to E In particular, if sup t0

t+1

t | ϕ ( τ ) |dτ < ∞(this will be satisfied ifϕ ∈E (see Remark 2.5))

thenΛ

σϕandΛ

σϕare bounded Moreover, denoted by · ∞for ess sup-norm, we have

 Λ

σϕ ∞ N1

1−e−σ Λ1T+

1ϕ ∞ and  Λ

σϕ ∞ N2

1−e−σΛ1ϕ ∞. (b) E contains exponentially decaying functionsψ(t) =e−α t for t0 and any fixed constantα >0.

(c) E does not contain exponentially growing functions f(t) =e bt for t0 and any constant b>0.

3 Exponential dichotomy and stable manifolds

In this section, we will find condition for the existence of stable manifolds Throughout this section we assume that the evolution family {U(t s}ts0 has an exponential dichotomy on R+ We now make precisely the notion of exponential dichotomies in the following definition

Definition 3.1 An evolution family{U(t s}ts0 on the Banach space X is said to have an exponential dichotomy on[0, ∞)

if there exist bounded linear projections P(t), t0, on X and positive constants N, ν such that

(a) U(t s P( ) =P(t)U(t s , ts0,

(b) the restriction U(t s|:Ker P( ) →Ker P(t), ts0, is an isomorphism, and we denote its inverse by U( , )|:=

(U(t s|)−1, 0st,

(c) U(t s x Ne−ν ( t−sxfor x∈P( )X , ts0,

(d) U( , )|x Ne−ν ( t−sxfor x∈Ker P(t), ts0

The projections P(t), t0, are called the dichotomy projections, and the constants N, ν – the dichotomy constants.

Using the projections(P(t))t0 on X , we can define the family of operators(P(t))t0onC as follows

P(t) : C → C ,

P(t)φ 

Then, we have that(P(t))2= P(t), and therefore the operatorsP(t), t0, are projections onC Moreover, ImP(t) = {φ ∈ C:

φ (θ ) =U(t− θ,t) ν0 ∀θ ∈ [−r,0]for someν0∈Im P(t)}

To obtain the existence of stable manifolds we need the following notion of theϕ-Lipschitz of the nonlinear term f

Definition 3.2 Let E be an admissible Banach function space and ϕ be a positive function belonging to E A function

f: [0, ∞) × C →X is said to beϕ-Lipschitz if f satisfies

(i) f(t 0)  ϕ (t)for all t∈ R+,

(ii) f(t, φ1) −f(t, φ2)   ϕ (t) φ1− φ2C for all t∈ R+and allφ1, φ2∈ C

Note that if f(t, φ)isϕ-Lipschitz thenf(t, φ)   ϕ (t)(1+ φC)for allφ ∈ C and t0

To prove the existence of a stable manifold, instead of Eq.(1.1)we consider the following integral equation

⎧

⎪

⎪

u(t) =U(t, )u( ) +

t

s

U(t, ξ )f(ξ,u ξ)dξ for ts0,

u s= φ ∈ C

(3.2)

We note that, if the evolution family {U(t s }ts0 arises from the well-posed Cauchy problem (1.2), then the function

u: [s−r, ∞) →X , which satisfies(3.2)for some given function f , is called a mild solution of the semilinear problem

⎧

⎨

⎩

du

dt =A(t)u(t) +f(t,u t), ts0,

u s= φ ∈ C

We refer the reader to J Wu [14] for more detailed treatments on the relations between classical and mild solutions of functional evolution equations

We now give the definition of a stable manifold for the solutions of the integral equation(3.2)

Definition 3.3 A set S⊂ R+× C is said to be an invariant stable manifold for the solutions to Eq.(3.2) if for every t∈ R+ the phase spaceC splits into a direct sum C = X0(t) ⊕ X1(t)with corresponding projectionsP(t)(i.e.,X0(t) =ImP(t)and

X1(t) =KerP(t)) such that

sup

t0

P(t)  < ∞,

and there exists a family of Lipschitz continuous mappings

Φt: X0(t) → X1(t), t∈ R+

with the Lipschitz constants independent of t such that

(i) S= {(t, ψ + Φt(ψ)) ∈ R+× (X0(t) ⊕ X1(t)) |t∈ R+, ψ ∈ X0(t) }, and we denote

S t:=  ψ + Φt(ψ ): 

t, ψ + Φt(ψ ) 

∈S

, (ii) St is homeomorphic toX0(t)for all t0,

(iii) to eachφ ∈S s there corresponds one and only one solution u(t)to Eq.(3.2)on[s−r, ∞)satisfying the conditions that

u s= φ and suptsu tC< ∞ Moreover, any two solutions u(t)and v(t) of Eq.(3.2)corresponding to different func-tionsφ1, φ2∈S sattract each other exponentially in the sense that, there exist positive constantsμand Cμ independent

of s0 such that

u t−v tCC μ e−μ ( t−sP( )φ1



(0) − P( )φ2



(iv) S is positively invariant under Eq.(3.2), i.e., if u(t), ts−r, is a solution to Eq.(3.2)satisfying conditions that us∈S s

and suptsu tC< ∞, then we have ut∈S t for all ts.

Note that if we identifyX0(t) ⊕ X1(t)withX0(t) × X1(t), then we can write S t=graph(Φt)

Let{U(t s}ts0 have an exponential dichotomy with the dichotomy projections P(t), t0, and constants N, ν >0 Note that the exponential dichotomy of {U(t s}ts0 implies that H:=supt0P(t)  < ∞ and the map t→P(t) is strongly continuous (see[4, Lemma 4.2]) We can then define the Green function on the half-line as follows

G (t, τ ) =

P(t)U(t, τ ) for t> τ 0,

It follows from the exponential dichotomy of{U(t s}ts0 that

 G (t, τ )  N(1+H)e−ν|t−τ| for all t= τ

The following lemma gives the form of bounded solutions to Eq.(3.2)

Lemma 3.4 Let the evolution family{U(t s}ts0have an exponential dichotomy with the dichotomy projections P(t), t0, and

constants N, ν >0 Suppose thatϕis a positive function which belongs to the admissible space E Let f : R+× C →X beϕ-Lipschitz and u(t)be a solution to Eq.(3.2)such that sup ts−ru(t) < ∞for fixed s0 Then, for ts we can rewrite u(t)in the form

⎧

⎪

⎪

u(t) =U(t, ) ν0+

∞

s

G (t, τ )f( τ ,u τ)dτ ,

u s= φ ∈ C

(3.5)

for someν0∈X0( ) =P( )X , whereG(t τ )is the Green function defined as in(3.4).

Proof Put y(t) = s∞G(t τ )f( τ ,uτ)dτ We have

y(t)   ∞

s

N(1+H)e−ν|t−τ|

1+ u τC

ϕ ( τ )dτ

N(1+H)



1+ sup

ξs−r

u(ξ )  ∞

0

e−ν|t−τ|ϕ ( τ )dτ .

UsingProposition 2.6we obtain

y(t)  N(1+H)



1+ sup

ξs−r

u(ξ ) (N1Λ1T+

1ϕ ∞+N2Λ1ϕ ∞)

On the other hand,

U(t, )y( ) = −

t

s

U(t, )U( , τ )|

I−P( τ ) 

f( τ ,u τ)dτ

−

∞

t

U(t, )U( , τ )|

I−P( τ ) 

f( τ ,u τ)dτ

= −

t

s

U(t, τ ) 

I−P( τ ) 

f( τ ,u τ)dτ −

∞

t

U(t, τ )|

I−P( τ ) 

f( τ ,u τ)dτ

Therefore,

y(t) =U(t, )y( ) +

t

s

U(t, τ )f( τ ,u τ)dτ

Since u(t)is a solution of Eq.(3.2)we obtain that u(t) −y(t) =U(t s)(u( ) −y( )) Put nowν0=u( ) −y( ) The

bounded-ness of u(t)and y(t)on[s−r, ∞)implies thatν0∈X0( )and P( )u( ) =P( )φ (0) = ν0 Therefore, u(t) =U(t sν0+y(t) for ts. 2

Remark 3.5 Eq. (3.5) is called the Lyapunov–Perron equation By computing directly, we can see that the converse of

Lemma 3.4is also true This means that, all solutions of the integral equation(3.5)satisfy Eq.(3.2)for ts.

Theorem 3.6 Let the evolution family{U(t s}ts0have an exponential dichotomy with the dichotomy projections P(t), t0, and

constants N, ν >0 Suppose thatϕis a positive function which belongs to E Let f: R+× C →X beϕ-Lipschitz, and let

k:=e ν r(1+H)N(N1Λ1T1+ϕ ∞+N2Λ1ϕ ∞)

Then, if k<1, there corresponds to eachφ ∈ImP( )one and only one solution u(t)of Eq.(3.5)on[s−r, ∞)satisfying the conditions thatP( )u s= φand sup tsu tC< ∞ Moreover, the following estimate is valid for any two solutions u(t),v(t)corresponding to different initial functionsφ1, φ2∈ImP( ):

u t−v tCC μ e−μ ( t−s φ1(0) − φ2(0)  for all ts0

whereμis a positive number satisfying

0< μ < ν +ln

1−N(1+H)e ν r

N1 Λ1T+

1ϕ ∞+N2Λ1ϕ ∞

1−N (1+H ) eν r

1−e −( ν−μ )(N1Λ1T+

1ϕ ∞+N2Λ1ϕ ∞) .

Proof Denote by C b([s−r, ∞),X)the Banach space of bounded, continuous and X -valued functions defined on[s−r, ∞), which is endowed with the sup-norm · ∞ Settingν0:= φ(0)we consider the transformation T defined by

(T u)(t) =

U(2s−t, ) ν0+ s∞G (2s−t, τ )f( τ ,u τ)dτ for s−rts

U(t, ) ν0+ s∞G (t, τ )f( τ ,u τ)dτ for ts.

Since ν0∈P(0)X , using the inequality(3.6) we can easily see that T acts from Cb( [s−r, ∞),X)into itself We next prove

that, if k<1, then T is a contraction mapping To do this, we estimate

 (T u)(t) − (T v)(t)   ∞

s

 G (t, τ ) 

f( τ ,u τ) − f( τ ,v τ) dτ

N(1+H)

∞

s

e−ν|t−τ|ϕ ( τ ) u τ−v τC dτ

ke−ν r sup

ts−r

u(t) −v(t)  for ts

 (T u)(t) − (T v)(t)   ∞

s

 G (2s−t, τ ) 

f( τ ,u τ) − f( τ ,v τ) dτ

N(1+H)

∞

s

e−ν|2s−t−τ|ϕ ( τ ) u τ−v τC dτ

N(1+H)e ν r

∞

s

e−ν|s−τ|ϕ ( τ ) u τ−v τC dτ

ts−r

u(t) −v(t)  for −r+sts.

Therefore, supts−r(T u)(t) − (T v)(t)  k sup ts−ru(t) −v(t) 

Hence, for k<1 the transformation T:C b([s−r, ∞),X) →C b([s−r, ∞),X)is a contraction mapping Thus, there exists

a unique u(·) ∈C([s−r, ∞),X)such that T u=u This yields that u(t), ts−r, is the unique solution of Eq. (3.5) with

u s(θ ) =U( − θ,sν0+ s∞G(s− θ, τ )f( τ ,uτ)dτ for allθ ∈ C, and P( )u( ) = ν0= φ(0) Therefore,P( )u s= φby definition

ofP( )(see equality(3.1))

Let u(t),v(t) be the two solutions to Eq.(3.5) corresponding to different initial functionsφ1, φ2∈ImP( ), respectively Puttingν1:= φ1(0),ν2:= φ2(0)we have that

u(t) −v(t)   Ne−ν ( t−s ν1− ν2 +N(1+H) ∞

s e−ν|t−τ|ϕ ( τ ) u τ−v τC dτ if ts

Ne−ν (−t) ν1− ν2 +N(1+H) ∞

s e−ν|2s−t−τ|ϕ ( τ ) u τ−v τC dτ if s−rts. Since t+ θ ∈ [−r+t t]for fixed t∈ [s, ∞)andθ ∈ [−r,0], we obtain

u t−v tCNe ν r e−ν ( t−s ν1− ν2 +N(1+H)e ν r

∞

s

e−ν|t−τ|ϕ ( τ ) u τ−v τC dτ , ts. Put h(t) = u t−v tC Then, supts h(t) < ∞and

h(t) Ne ν r e−ν ( t−s ν1− ν2 +N(1+H)e ν r

∞

s

e−ν|t−τ|ϕ ( τ )h( τ )dτ , ts. (3.8)

We will use the cone inequality theorem (see[9, Theorem 2.8]) applying to Banach space L∞[s, ∞) which is the space of real-valued functions defined and essentially bounded on[s, ∞)(endowed with the sup-norm denoted by · ∞) with the coneKbeing the set of all nonnegative functions We then consider the linear operator A defined for g∈L∞[s, ∞)by

(Ag)(t) =N(1+H)e ν r

∞

s

e−ν|t−τ|ϕ ( τ )g( τ )dτ , ts.

ByProposition 2.6we have that

sup

ts(Ag)(t) =sup

ts N(1+H)e ν r

∞

s

e−ν|t−τ|ϕ ( τ )g( τ )dτ kg∞.

Therefore, A∈ L(L∞[s, ∞)) andA k<1 Obviously, the coneKis invariant under the operator A The inequality(3.8)

can now be rewritten by

hAh+z for z(t) =Ne ν r e−ν ( t−s ν1− ν2.

By the cone inequality theorem[9, Theorem 2.8]we obtain that hg, where g is a solution in L∞[s, ∞)of the equation

g=Ag+z which can be rewritten as

g(t) =Ne ν r e−ν ( t−s ν1− ν2 +N(1+H)e ν r

∞

e−ν|t−τ|ϕ ( τ )g( τ )dτ , ts0.

To estimate g, we put w(t) =eμ ( t−s g(t)for ts0 Then, we obtain that

w(t) =Ne ν r e −( ν−μ )( t−s ν1− ν2 +N(1+H)e ν r

∞

s

e−ν|t−τ|+μ ( t−τ )ϕ ( τ )w( τ )dτ (3.9)

We next consider the linear operator D defined on L∞[s, ∞)as

(Dφ)(t) =N(1+H)e ν r

∞

s

e−ν|t−τ|+μ ( t−τ )ϕ ( τ )φ ( τ )dτ for all ts.

One can easily see that D∈ L(L∞[s, ∞))andD N (1+H) ν r

1−e−( ν−μ )(N1Λ1T+

1ϕ ∞+N2Λ1ϕ ∞) Eq.(3.9)can be rewritten as

w=D w+ ˜z for˜z(t) =Ne ν r e −( ν−μ )( t−s ν1− ν2.

We have D <1 if 0< μ < ν +ln(1−N(1+H)eν r(N1Λ1T+

1ϕ ∞+N2Λ1ϕ ∞)) Under this condition, the equation

w=D w+ ˜z has the unique solution w∈L∞[s, ∞) and w= (I−D)−1˜z Hence, we obtain that

w∞=  (I−D)−1˜z

∞ Ne ν r

1− D  ν1− ν2

 Ne ν r ν1− ν2

1−N (1+H) eν r

1−e −( ν−μ )(N1Λ1T+

1ϕ ∞+N2Λ1ϕ ∞) :=C μ ν1− ν2.

This yields that w(t) Cμ ν1− ν2for ts Hence,

h(t) = u t−v tCg(t) =e−μ ( t−s w(t) C μ e−μ ( t−s ν1− ν2 for ts. 2

We now prove our main result of this section

Theorem 3.7 Let the evolution family{U(t s}ts0have an exponential dichotomy with the dichotomy projections P(t), t0, and

constants N, ν >0 Suppose thatϕis a positive function which belongs to the admissible space E Let f: R+× C →X beϕ-Lipschitz satisfying k<1+Ne1 ν r where k is defined by(3.7) Then, there exists an invariant stable manifold S for the solutions to Eq.(3.2).

Proof Since {U(t s}ts0 has an exponential dichotomy, we have that for each t0 the phase space C splits into the direct sumC =ImP(t) ⊕KerP(t)where the projectionsP(t), t0, are defined as in equality(3.1) Clearly, supt0P(t) <

∞ We now construct a stable manifold S= {(t S t) }t0for the solutions to Eq.(3.2) To do this, we determine the surface St for t0 by the formula

S t:=  φ + Φt(φ): φ ∈ImP(t) 

⊂ C

where the operatorΦt0 is defined for each t00 by

Φt0(φ)(θ ) =

∞

t0

G (t0− θ, τ )f( τ ,u τ)dτ for allθ ∈ [−r 0],

here u( ·)is the unique solution of Eq.(3.2)on[−r+t0, ∞)satisfyingP(t0)u t0= φ(note that the existence and uniqueness

of u(·)is guaranteed byTheorem 3.6) On the other hand, by the definition of the Green functionG (see Eq.(3.4)) we have thatΦt0(φ) ∈KerP(t0) We next show that the stable manifold S satisfies the conditions ofDefinition 3.3

Firstly, we prove that Φt0 is Lipschitz continuity with Lipschitz constant independent of t0 Indeed, forφ1 andφ2 be-longing to ImP(t0)we have

 Φt0(φ1)(θ ) − Φt0(φ2)(θ )  N(1+H)

∞

t0

e−ν|t0−θ−τ|ϕ ( τ ) u τ−v τC dτ

N(1+H)e ν r

∞

t0

e−ν|t0−τ|ϕ ( τ ) u τ−v τC dτ

N(1+H)e ν rsup

τt0u τ−v τC

∞

t0

e−ν|t0−τ|ϕ ( τ )dτ

N(1+H)e ν r

1−e−ν



N1 Λ1T+

1ϕ 

∞+N2Λ1ϕ ∞

sup

τt u τ−v τC. (3.10)

Moreover, by the Lyapunov–Perron equation for u( ·)and v( ·)(see Eq.(3.5)) we have

sup

τt0u τ−v τCNe ν rφ1− φ2C+N(1+H)e ν r

1−e−ν



N1 Λ1T+

1ϕ 

∞+N2Λ1ϕ ∞

sup

τt0u τ−v τC.

It follows that

sup

τt0u τ−v τC Ne ν r

1−kφ1− φ2C.

Substituting this inequality into(3.10)we obtain that

 Φt0(φ1) − Φt0(φ2) C= sup

θ∈[−r0] Φt0(φ1)(θ ) − Φt0(φ2)(θ )  Nke ν r

1−kφ1− φ2C.

Therefore,Φt0 is Lipschitz continuous with the Lipschitz constant Nke1−ν k r independent of t0

To show that St0 is homeomorphic to ImP(t0) We define the transformation D:ImP(t0) →S t0 by Dφ := φ + Φt0(φ)

for all φ ∈ImP(t0) Then, applying the implicit function theorem for Lipschitz continuous mappings (see[3, Lemma 2.7])

we have that if the Lipschitz constant Nke1−kν r <1 (equivalently k< 1+Ne1 ν r) then D is a homeomorphism Therefore, the

condition (ii) inDefinition 3.3is satisfied

The condition (iii) inDefinition 3.3now follows fromTheorem 3.6 We now prove that the condition (iv) ofDefinition 3.3

is satisfied Indeed, let u(·)be solution of Eq.(3.2)such that the function u s(θ ) ∈S s Then, byLemma 3.4, the solution u(t) for t∈ [s, ∞)can be rewritten in the form

u(t) =U(t, ) ν0+

∞

s

G (t, τ )f( τ ,u τ)dτ for someν0∈Im P( ).

Thus, for ts andθ ∈ [−r,0]we have

u(t− θ) =U(t− θ,sν0+

∞

s

G (t− θ, τ )f( τ ,u τ)dτ

=U(t− θ,sν0+

t

s

G (t− θ, τ )f( τ ,u τ)dτ +

∞

t

G (t− θ, τ )f( τ ,u τ)dτ

=U(t− θ,sν0+

t

s

U(t− θ, τ )P( τ )f( τ ,u τ)dτ +

∞

t

G (t− θ, τ )f( τ ,u τ)dτ

=U(t− θ,t)



U(t, ) ν0+

t

s

U(t, τ )P( τ )f( τ ,u τ)dτ

 +

∞

t

G (t− θ, τ )f( τ ,u τ)dτ

Putμ0=U(t sν0+ t

s U(t τ )P( τ )f( τ ,uτ)dτ We have P(t) μ0= μ0, henceμ0∈Im P(t) We thus obtain that U(t−θ,t) μ0

belongs to ImP(t)and

u(t− θ) =U(t− θ,t) μ0+

∞

t

G (t− θ, τ )f( τ ,u τ)dτ

By the uniqueness of u( ·)on[s−r, ∞)as in the proof ofTheorem 3.6we have that Eq.(3.2)has a unique solution u( ·)on [−r+t ∞) satisfying(P(t)u t)(θ ) =U(t− θ,t) μ0 and

u(ξ ) =U(2t− ξ,t) μ0+

∞

t

G (2ξ −t, τ )f( τ ,u τ)dτ

forξ ∈ [−r+t t] Therefore, the history function ut can be viewed as

u t(θ ) =u(t+ θ) =U(t− θ,t) μ0+

∞

t

G (t− θ, τ )f( τ ,u τ)dτ = φ(θ) + Φt(φ)(θ ).

Hence, ut∈S for ts. 2

4 Exponential trichotomy and center-stable manifolds

In this section, we will generalize Theorem 3.7to the case that the evolution family{U(t s}ts0 has an exponential trichotomy on R+ and the nonlinear forcing term f is ϕ-Lipschitz In this case, under similar conditions as in the above section we will prove that there exists a center-stable manifold for the solutions to Eq.(3.2) We now recall the definition

of an exponential trichotomy

Definition 4.1 A given evolution family{U(t s}ts0is said to have an exponential trichotomy on the half-line if there are three families of projections {P j(t) }, t0, j=1,2,3, and positive constants N, α , β withα < β such that the following conditions are fulfilled:

(i) supt0P j(t)  < ∞, j=1,2,3

(ii) P1(t) +P2(t) +P3(t) =Id for t0 and P j(t)P i(t) =0 for all j=i.

(iii) P j(t)U(t s =U(t s P j( )for ts0 and j=1,2,3

(iv) U(t s|Im P j ( are homeomorphisms from Im P j( )onto Im P j(t), for all ts0 and j=2,3, respectively; we also

denote the inverse of U(t s|Im P j ( by U( , )|, 0st.

(v) For all ts0 and x∈X , the following estimates hold:

U(t, )P1( )x Ne −β( t−sP1( )x ,

U( ,t)|P2(t)x Ne −β( t−sP2(t)x ,

U(t, )P3( )x Ne α ( t−sP3( )x .

The projections {P j(t) }, t0, j=1,2,3, are called the trichotomy projections, and the constants N, α , β – the trichotomy

constants.

Using the trichotomy projections we can now construct three families of projections{P j(t) }, t0, j=1,2,3, onC as follows:

P j(t)φ 

(θ ) =U(t− θ,t)P j(t)φ (0) for allθ ∈ [−r 0]andφ ∈ C (4.1)

We come to our second main result It proves the existence of a center-stable manifold for solutions of Eq.(3.2)

Theorem 4.2 Let the evolution family{U(t s}ts0 have an exponential trichotomy with the trichotomy projections{P j(t) }t0,

j=1,2,3, and constants N, α , βgiven as in Definition 4.1 Suppose that f : R+× C →X isϕ-Lipschitz, whereϕ is a positive function which belongs to the admissible space E Set q:=sup{P j(t) : t0, j=1,3}, N0:=max{N,2Nq}, and

k:= (1+H)e ν r N0

1−e−ν



N1 Λ1T+

1ϕ 

∞+N2Λ1ϕ ∞

Then, if k<1+N0e1 ν r , for each fixedδ > αthere exists a center-stable manifold S= {(t S t)}t0⊂ R+× Cfor the solutions to Eq.(3.2), which is represented by a family of Lipschitz continuous mappings

Φt:ImP1(t) + P3(t) 

→ImP2(t)

with Lipschitz constants being independent of t such that S t=graph(Φt)has the following properties:

(i) St is homeomorphic to Im(P1(t) + P3(t))for all t0.

(ii) To eachφ ∈S s there corresponds one and only one solution u(t)to Eq.(3.2)on[s−r, ∞)satisfying e−γ ( +θ) u s(θ ) = φ(θ)

forθ ∈ [−r,0]and sup tse−γ ( t+·) u

t(·)C< ∞, whereγ =δ+α

2 Moreover, for any two solutions u(t)and v(t)to Eq.(3.2)

corresponding to different functionsφ1, φ2∈S s we have the estimate

u t−v tCC μ e ( γ−μ )( t−sP( )φ1



(0) − P( )φ2



whereμand Cμ are positive constants independent of s,u(·), and v(·).

(iii) S is positively invariant under Eq.(3.2)in the sense that, if u(t), ts−r, is the solution to Eq.(3.2)satisfying the conditions that the function e−γ ( +·) u s( ·) ∈S s and sup tse−γ ( t +·) u t( ·)C< ∞, then the function e−γ ( t +·) u t( ·) ∈S t for all ts.

Proof Set P(t) :=P1(t) +P3(t)and Q(t) :=P2(t) =Id−P(t)for t0 We have that P(t)and Q(t)are projections

com-plemented to each other on X We then define the families of projections{P j(t) }, t0, j=1,2,3, onCas in equality(4.1) SettingP(t) = P1(t) + P3(t)andQ(t) = P2(t), t0, we obtain thatP(t)andQ(t)are complemented projections onC for

each t0 We consider the following rescaling evolution family

U(t, ) =e−γ ( t−s U(t, ) for all ts0.

We now give the definition of a stable manifold for the solutions of the integral equation(3.2)