Startseite Approximations of center manifolds for delay stochastic differential equations with additive noise
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Approximations of center manifolds for delay stochastic differential equations with additive noise

  • Longyu Wu , Jiaxin Gong , Juan Yang und Ji Shu EMAIL logo
Veröffentlicht/Copyright: 17. März 2023
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Advances in Nonlinear Analysis
Aus der Zeitschrift Advances in Nonlinear Analysis Band 12 Heft 1

Abstract

This article deals with approximations of center manifolds for delay stochastic differential equations with additive noise. We first prove the existence and smoothness of random center manifolds for these approximation equations. Then we show that the C k invariant center manifolds of the system with colored noise approximate that of the original system.

Keywords: delay equation; center manifold; random dynamical system; additive noise
MSC 2010: Primary: 60H10; Secondary: 37D10; 37H10

1 Introduction

In this article, we investigate the approximations of center manifolds for the following delay stochastic equations in R n :

(1.1) d v = ( A v + H ( v ) + F ( v ( t − ρ ) ) ) d t + g d W , t > τ ,

where A is a n × n matrix, H is a Lipschitz continuous term, F is a nonlinearity with the time delay, ρ is a positive constant, g is an n × 1 matrix, and g ∈ D ( A ) , τ ∈ R . W ( t , ω ) is a two-sided real-valued Wiener process on a probability space.

Let ( Ω , ℱ , P ) be the classical Wiener probability space, where

Ω = C 0 ( R , R ) ≔ { ω ∈ C ( R , R ) : ω ( 0 ) = 0 }

with the open compact topology, ℱ is its Borel σ -algebra, and P is the Wiener measure. Then W has the form W ( t , ω ) = ω ( t ) . Consider the Wiener shift θ t defined on the probability space ( Ω , ℱ , P ) by

θ t ω ( ⋅ ) = ω ( t + ⋅ ) − ω ( t ) .

From Arnold [1], we know that P is an ergodic invariant measure for θ t . Then ( Ω , ℱ , P , { θ t } t ∈ R ) forms a metric dynamical system.

For each δ > 0 , we consider the following stochastic differential equation on R :

(1.2) d ζ δ = − 1 δ ζ δ d t + 1 δ d W .

This equation has a unique stationary solution given by

(1.3) ζ δ ( t , ω ) = 1 δ ∫ − ∞ t e s − t δ d W ( s ) ,

which is called an Ornstein-Uhlenbeck process or colored noise. For convenience, we denote by

ζ δ ( ω ) = 1 δ ∫ − ∞ 0 e s δ d W ( s ) .

Then we find that

(1.4) ζ δ ( t , ω ) = ζ δ ( θ t ω ) = − ∫ − ∞ 0 1 δ 2 e s δ θ t ω ( s ) d s .

The colored noise has been widely used to investigate the dynamical behavior of solutions of random systems, see, e.g., [13,20,21,30,36,39]. Recently, there have been some works on the attractors and invariant manifolds of random differential equations driven by additive or linear multiplicative colored noise, see, e.g., [14,15,18,19,33,35].

The colored noise can be regarded as an approximation to the white noise in the sense that

lim δ → 0 + sup t ∈ [ 0 , T ] ∫ 0 t ζ δ ( θ s ω ) d s − ω ( t ) = 0 , a.s.

for each T > 0 , see [14].

In this article, we consider the approximation of equation (1.1):

(1.5) v ˙ δ = A v δ + H ( v δ ) + F ( v δ ( t − ρ ) ) + g ζ δ ( θ t ω ) .

The study of invariant manifolds dates back to Hadamard [16], then, by Liapunov [22] and Perron [29] who used a different approach. In fact, Hadamard’s graph transform method is a geometric approach, while Lyapunov-Perron method is analytic in nature. Currently, the invariant manifolds have been extensively studied in the literature for both finite and infinite dimensional deterministic dynamical systems, see, e.g., [7] for finite dimensional dynamical systems, and [2–5,8,9,17,23,37] for infinite dimensional dynamical systems. When the system is given by stochastic or random ordinary differential equation (finite dimensional random dynamical systems), there are some results on invariant manifolds, see, e.g., [1,27,31,38], while for stochastic partial differential equation (infinite dimensional random dynamical systems), some results can be found, see, e.g., [6,10–12,24,28,32,34].

The differential equation with time delay has been widely used in various fields such as physics, biology, and financial mathematics. Recently, there are lots of publications on delay differential equations with either additive or multiplicative noise, see, e.g., [25,26,40,42,43].

As far as the authors are aware, there are no results on center manifolds of delay stochastic differential equations with additive noise. In this article, we study the center manifolds for equation (1.5), which is the approximation of equation (1.1). The main difficulty of this article is to deal with the approximations of center manifolds.

This article is organized as follows. In the next section, we introduce some basic concepts and assumptions. In Section 3, we show the existence of center manifolds for equations (1.5) and (1.1). In Section 4, we prove the smoothness of these invariant center manifolds. In Section 5, we show that the C k invariant center manifolds of the system with colored noise approximate that of original system as δ → 0 + .

2 Preliminaries

In this section, we first recall some basic concepts about nonautonomous random dynamical systems, which can be found in [41]. The reader is referred to [1] for such results for autonomous random dynamical systems. Then we introduce some assumptions about the equations.

2.1 Random dynamical system

Let ( Ω , ℱ , P ) be a probability space, and H be a separable Hilbert space. We denote the collections of Borel sets on R , R + and H by ℬ ( R ) , ℬ ( R + ) , and ℬ ( H ) .

Definition 2.1

( Ω , ℱ , P , { θ t } t ∈ R ) is called a metric dynamical system if the following conditions are satisfied:

  1. θ : R × Ω → Ω is ( ℬ ( R ) ⊗ ℱ , ℱ ) -measurable;

  2. θ 0 = i d Ω , the identity on Ω , θ t + s = θ t ∘ θ s for all t , s ∈ R ;

  3. θ t P = P for all t ∈ R .

Definition 2.2

A mapping

ϕ : R + × R × Ω × H → H , ( t , τ , ω , x ) ↦ ϕ ( t , τ , ω , x )

is called a nonautonomous random dynamical system over a metric dynamical system ( Ω , ℱ , P , { θ t } t ∈ R ) if the following properties are satisfied for all τ ∈ R :

  1. ϕ is ( ℬ ( R + ) ⊗ ℬ ( R ) ⊗ ℱ ⊗ ℬ ( H ) , ℬ ( H ) ) -measurable;

  2. the mapping ϕ ( t , τ , ω ) ≔ ϕ ( t , τ , ω , ⋅ ) : H → H forms a cocycle over θ t :

    ϕ ( 0 , τ , ω ) = i d H for all ω ∈ Ω , ϕ ( t + s , τ , ω ) = ϕ ( t , τ , θ s ω ) ∘ ϕ ( s , τ , ω ) for all s , t ∈ R , ω ∈ Ω .

ϕ is called a C k smooth random dynamical system if ϕ is a random dynamical system, and for each ( t , ω ) ∈ R × Ω , the mapping

ϕ ( t , ω ) : H → H , x ↦ ϕ ( t , ω ) x

is C k .

2.2 Some assumptions about the equations

In this article, we assume that A is a n × n matrix with eigenvalues of zero real parts. Now we write the spectrum σ ( A ) of matrix A as follows:

σ ( A ) = σ u ∪ σ c ∪ σ s ,

where σ u ( A ) ≔ { λ ∈ σ ( A ) ∣ Re λ > 0 } , σ c ( A ) ≔ { λ ∈ σ ( A ) ∣ Re λ = 0 } , and σ s ( A ) ≔ { λ ∈ σ ( A ) ∣ Re λ < 0 } . From the assumption, σ c ( A ) ≠ ∅ . Let E u , E c , and E s denote the corresponding eigenspaces of σ u , σ c , and σ s . Then we have

R n = E u ⊕ E c ⊕ E s

with corresponding projections P u : R n → E u , P c : R n → E c , and P s : R n → E s . Let

0 < β 1 < min { ∣ Re λ ∣ ∣ λ ∈ σ u ∪ σ s } .

It is well known that for each 0 < β 2 < β 1 , there is a constant K ≥ 1 such that

(2.1) ∣ e A t P c ∣ ≤ K e β 2 ∣ t ∣ , t ∈ R , ∣ e A t P u ∣ ≤ K e β 1 t , t ≤ 0 , ∣ e A t P s ∣ ≤ K e − β 1 t , t ≥ 0 .

For the nonlinear terms H and F , we assume that

(A1) H and F satisfy the Lipschitz condition, i.e., there exists two positive constants L H and L F such that for all u , v ∈ R n ,

(2.2) ∣ H ( u ) − H ( v ) ∣ ≤ L H ∣ u − v ∣ ,

(2.3) ∣ F ( u ) − F ( v ) ∣ ≤ L F ∣ u − v ∣

and H ( 0 ) = 0 , F ( 0 ) = 0 .

In order to study the smoothness of center manifolds, we assume that

(A2) H and F are C k in u and D u i H ( u ) , D u i F ( u ) ( 1 ≤ i ≤ k ) are bounded in ℒ i ( R n , R n ) , where ℒ i ( R n , R n ) is the Banach space of all bounded i -linear maps from R n to itself with the norm ‖ ⋅ ‖ ℒ i ( R n , R n ) .

Let C ( [ − ρ , 0 ] , R n ) with ρ > 0 be the space of all continuous functions from [ − ρ , 0 ] to R n with norm ‖ φ ‖ C ( [ − ρ , 0 ] , R n ) = sup { ∣ φ ( s ) ∣ : s ∈ [ − ρ , 0 ] for φ ∈ C ( [ − ρ , 0 ] , R n ) } . For convenience, we denote C ( [ − ρ , 0 ] , R n ) by C ρ .

3 Existence of the center manifolds

In this section, we consider the delay stochastic equation:

(3.1) d v = ( A v + H ( v ) + F ( v ( t − ρ ) ) ) d t + g d W , t > τ

and its approximation

(3.2) v ˙ δ = A v δ + H ( v δ ) + F ( v δ ( t − ρ ) ) + g ζ δ ( θ t ω ) , t > τ ,

where A , H , F , ρ , g , τ , W ( t , ω ) , and ζ δ ( θ t ω ) are defined in Section 1.

In order to show the solutions of equations (3.1) and (3.2) generate a random dynamical system, we consider a linear stochastic differential equation:

(3.3) d z = − z d t + d W .

A solution of this equation is called an Ornstein-Uhlenbeck process, which satisfies the following properties (see [11]).

Lemma 3.1

  1. There exists a { θ t } t ∈ R -invariant set Ω 1 ∈ ℬ ( C 0 ( R , R ) ) of full measure with sublinear growth:

    lim t → ± ∞ ∣ ω ( t ) ∣ ∣ t ∣ = 0 , ω ∈ Ω 1 .

  2. For ω ∈ Ω 1 , the random variable

    z ( ω ) = − ∫ − ∞ 0 e r ω ( r ) d r

    exists and generates a unique stationary solution of (3.3) given by

    Ω 1 × R ∋ ( ω , t ) → z ( θ t ω ) = − ∫ − ∞ 0 e r θ t ω ( r ) d r = − ∫ − ∞ 0 e r ω ( r + t ) d r + ω ( t ) .

    The mapping t → z ( θ t ω ) is continuous.

  3. In particular, we have

    lim t → ± ∞ ∣ z ( θ t ω ) ∣ ∣ t ∣ = 0 , ω ∈ Ω 1 , lim t → ± ∞ 1 t ∫ 0 t z ( θ r ω ) d r = 0 , ω ∈ Ω 1 .

By replacing the white noise in equation (3.3) by ζ δ ( θ t ω ) , we can obtain the following random differential equation:

(3.4) z ˙ δ = − z δ + ζ δ ( θ t ω ) .

For equation (3.4), we have the following results (see [33]).

Lemma 3.2

Let T 1 < T 2 and Ω 1 be given as mentioned earlier. Then the following statements hold:

  1. For δ > 0 , ω ∈ Ω 1 , the random variable

    z δ ( ω ) = ∫ − ∞ 0 e r ζ δ ( θ r ω ) d r

    exists and generates a stationary solution of (3.4) given by

    Ω 1 × R ∋ ( ω , t ) → z δ ( θ t ω ) = ∫ − ∞ 0 e r ζ δ ( θ r + t ω ) d r .

    The mapping t → z δ ( θ t ω ) is continuous.

  2. In particular, we have

    lim t → ± ∞ ∣ z δ ( θ t ω ) ∣ ∣ t ∣ = 0 , ω ∈ Ω 1 , lim t → ± ∞ 1 t ∫ 0 t z δ ( θ r ω ) d r = 0 , ω ∈ Ω 1 ,

    uniformly with respect to δ ∈ ( 0 , 1 2 ] .

  3. In addition, for ω ∈ Ω 1 ,

    lim δ → 0 + ‖ z δ ( θ ⋅ ω ) − z ( θ ⋅ ω ) ‖ C ( [ T 1 , T 2 ] ) = 0 ,

    where C ( [ T 1 , T 2 ] ) is the usual space of continuous functions defined on [ T 1 , T 2 ] endowed with the norm ‖ f ‖ C ( [ T 1 , T 2 ] ) ≔ max t ∈ [ T 1 , T 2 ] ∣ f ( t ) ∣ .

Now we consider the probability space ( Ω 1 , ℱ 1 , P ) , where ℱ 1 is the trace algebra of Ω 1 . For simplicity, we still denote it by ( Ω , ℱ , P ) .

Next, we prove the existence of center manifolds for equations (3.1) and (3.2). We first show that the solution of equation (3.1) defines a random dynamical system. To see this, we consider the random differential equation:

(3.5) d u d t = ( A u + H ˜ ( θ t ω , u ) + F ˜ ( θ t ω , u ( t − ρ ) ) + ( A + I ) g z ( θ t ω ) ) , t > τ

with the initial condition

(3.6) u τ ( s ) = u ( s + τ ) = x ( s ) , s ∈ [ − ρ , 0 ] ,

where H ˜ ( θ t ω , u ) = H ( u + g z ( θ t ω ) ) , F ˜ ( θ t − ρ ω , u ) = F ( u ( t − ρ ) + g z ( θ t − ρ ω ) ) , and x ( s ) ∈ C ρ . In contrast to the original stochastic differential equation, no stochastic integral appears here. By the usual theorem of the existence and uniqueness of solutions, this equation has a unique solution for each ω ∈ Ω . No exceptional sets appear. Hence, the solution mapping

( t , τ , ω , x ) ↦ u ( t , τ , ω , x )

generates a random dynamical system, i.e., u is ℬ ( R ) ⊗ ℱ ⊗ ℬ ( C ρ ) measurable and forms a cocycle:

u ( 0 , τ , ω , x ) = x , for all ω ∈ Ω , u ( t + s , τ , ω , x ) = u ( t , τ , θ s ω , ⋅ ) ∘ u ( s , τ , ω , x ) , for all t , s ∈ R , ω ∈ Ω , x ∈ C ρ .

Similarly, for equation (3.2), we consider

(3.7) u ˙ δ = A u δ + H ˜ δ ( θ t ω , u δ ) + F ˜ δ ( θ t ω , u δ ( t − ρ ) ) + ( A + I ) g z δ ( θ t ω ) ,

where H ˜ δ ( θ t ω , u δ ) = H ( u δ + g z δ ( θ t ω ) ) , F ˜ δ ( θ t − ρ ω , u δ ) = F ( u δ ( t − ρ ) + g z δ ( θ t − ρ ω ) ) . The same conclusions for equation (3.5) also hold for equation (3.7). Hence, the solution mapping

( t , τ , ω , x ) ↦ u δ ( t , τ , ω , x )

also generates a random dynamical system.

For each x ∈ C ρ and ω ∈ Ω , we introduce the following random transformations:

T ( ω , x ) ≔ x − g z ( ω ) and T δ ( ω , x ) ≔ x − g z δ ( ω ) ,

Obviously, for fixed ω ∈ Ω , their inverse transformations are as follows:

T − 1 ( ω , x ) = x + g z ( ω ) and T δ − 1 ( ω , x ) = x + g z δ ( ω ) .

For simplify, we set z 0 ( ω ) = z ( ω ) , T 0 ( ω , x ) = T ( ω , x ) , H 0 = H , F 0 = F , and u 0 ( t , τ , ω , x ) = u ( t , τ , ω , x ) .

The following proposition implies that the solution of equation (3.1) (resp. (3.2) as δ > 0 ) generates a random dynamical system.

Proposition 3.1

Suppose that u δ is the random dynamical system generated by equation (3.5) (resp. (3.7) as δ > 0 ). Then

(3.8) ( t , τ , ω , x ) ↦ T δ − 1 ( θ t ω , ⋅ ) ∘ u δ ( t , τ , ω , T δ ( ω , x ) ) = v δ ( t , τ , ω , x )

is a random dynamical system. For any x ∈ C ρ , this process is a solution of equation (3.1) (resp. (3.2) as δ > 0 ) and forms a random dynamical system.

Proof

This proof is similar to [33], so we omit it here.□

Now we recall the spectrum σ ( A ) of matrix A as follows:

σ ( A ) = σ u ∪ σ c ∪ σ s ,

where σ u ( A ) ≔ { λ ∈ σ ( A ) ∣ Re λ > 0 } , σ c ( A ) ≔ { λ ∈ σ ( A ) ∣ Re λ = 0 } , and σ s ( A ) ≔ { λ ∈ σ ( A ) ∣ Re λ < 0 } . Let 0 < β 1 < min { ∣ Re λ ∣ ∣ λ ∈ σ u ∪ σ s } . For each η ∈ ( β 2 , β 1 ) , let C η denote the following Banach space:

C η ≔ { φ ∈ C ( R , R n ) ∣ sup t ∈ R e − η ∣ t ∣ ∣ φ ( t ) ∣ < + ∞ }

with the norm

∣ φ ∣ C η = sup t ∈ R e − η ∣ t ∣ ∣ φ ( t ) ∣ .

Then, for each δ ≥ 0 , we define

M δ c ( ω ) ≔ { x ∈ C ρ ∣ u δ ( ⋅ , ω , x ) ∈ C η } ,

which is called a center manifold when it is a manifold. Clearly, it is nonempty and invariant. Next, we prove that M δ c ( ω ) is given by a graph of a Lipschitz (or C k ) function for all δ ≥ 0 . First, we need to show the following lemma.

Lemma 3.3

Assume that ( A 1 ) holds. For each η ∈ ( β 2 , β 1 ) , x ∈ M δ c ( ω ) if and only if there exists a function u δ ( ⋅ ) ∈ C η with the initial value u δ ( 0 ) = x and satisfies

(3.9) u δ ( t ) = e A t ξ + ∫ 0 t e A ( t − s ) P c [ H ˜ δ ( θ s ω , u δ ( s ) ) + F ˜ δ ( θ s − ρ ω , u δ ( s − ρ ) ) + ( A + I ) g z δ ( θ s ω ) ] d s + ∫ + ∞ t e A ( t − s ) P u [ H ˜ δ ( θ s ω , u δ ( s ) ) + F ˜ δ ( θ s − ρ ω , u δ ( s − ρ ) ) + ( A + I ) g z δ ( θ s ω ) ] d s + ∫ − ∞ t e A ( t − s ) P s [ H ˜ δ ( θ s ω , u δ ( s ) ) + F ˜ δ ( θ s − ρ ω , u δ ( s − ρ ) ) + ( A + I ) g z δ ( θ s ω ) ] d s ,

where ξ = P c x .

Proof

Let x ∈ M δ c ( ω ) , for r , t ∈ R , by the variation of constants formula, we have

(3.10) u δ ( t , τ , ω , x ) = e A ( t − r ) u δ ( r , τ , ω , x ) + ∫ r t e A ( t − s ) [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , x ) ) + F ˜ δ ( θ s − ρ ω , u δ ( s − ρ , τ , ω , x 0 ) ) + ( A + I ) g z δ ( θ s ω ) ] d s .

By taking r = 0 and applying projection P c , we obtain

(3.11) P c u δ ( t , τ , ω , x ) = e A t P c x + ∫ 0 t e A ( t − s ) P c [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , x ) ) + F ˜ δ ( θ s − ρ ω , u δ ( s − ρ , τ , ω , x ) ) + ( A + I ) g z δ ( θ s ω ) ] d s .

By using P u to (3.10), we obtain

(3.12) P u u δ ( t , τ , ω , x ) = e A ( t − r ) P u u δ ( r , τ , ω , x ) + ∫ r t e A ( t − s ) P u [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , x ) ) + F ˜ δ ( θ s − ρ ω , u δ ( s − ρ , τ , ω , x ) ) + ( A + I ) g z δ ( θ s ω ) ] d s .

By (2.1), for r > max { t , 0 } , we have

∣ e A ( t − r ) P u u δ ( r , τ , ω , x ) ∣ ≤ K e β 1 t e ( η − β 1 ) r ∣ u δ ( ⋅ , τ , ω , x ) ∣ C η → 0 , as r → + ∞ .

By taking the limit r → + ∞ in (3.12), we obtain

(3.13) P u u δ ( t , τ , ω , x ) = ∫ + ∞ t e A ( t − s ) P u [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , x ) ) + F ˜ δ ( θ s − ρ ω , u δ ( s − ρ , τ , ω , x ) ) + ( A + I ) g z δ ( θ s ω ) ] d s .

Similarly, we can obtain

(3.14) P s u δ ( t , τ , ω , x ) = ∫ − ∞ t e A ( t − s ) P s [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , x ) ) + F ˜ δ ( θ s − ρ ω , u δ ( s − ρ , τ , ω , x ) ) + ( A + I ) g z δ ( θ s ω ) ] d s .

Together with (3.11), (3.13), and (3.14), we obtain (3.9). The converse follows from a straight-forward computation.□

The following theorem shows the existence of center manifolds for equation (3.7) and (3.5).

Theorem 3.1

Assume that ( A 1 ) holds and

(3.15) K L 1 η − β 2 + 2 β 1 − η < 1 ,

where L ≔ ( L H + e η ρ L F ) . Then, we have:

  1. For each ξ ∈ E c , equation (3.9) has a unique solution u δ ( ⋅ , τ , ω , ξ ) ∈ C η , which satisfies

    (3.16) ∣ u δ ( t , τ , ω , ξ ) − u δ ( t , τ , ω , ξ ¯ ) ∣ C η ≤ K 1 − K L 1 η − β 2 + 2 β 1 − η ∣ ξ − ξ ¯ ∣ .

  2. There exists a Lipschitz center manifold for equation (3.7) as δ > 0 (resp. (3.5)), which is given by a graph of a Lipschitz mapping h δ c ( ω , ⋅ ) : E c → E u ⊕ E s , i.e.,

    M δ c ( ω ) = { ξ + h δ c ( ω , ξ ) ∣ ξ ∈ E c } ,

    where h δ c ( ω , ξ ) = P u u δ ( 0 , τ , ω , ξ ) + P s u δ ( 0 , τ , ω , ξ ) and h δ c ( ω , 0 ) ∈ C η .

Proof

First, we prove that equation (3.9) has a unique solution u δ = u δ ( ⋅ , ω , ξ ) in C η , which is Lipschitz continuous in ξ ∈ E c . For brevity, we denote the right-hand side of equation (3.9) by J δ c ( u δ , ω , ξ ) . Then, by (2.1), we have

e − η ∣ t ∣ ∣ J δ c ( u δ , ω , ξ ) ∣ ≤ ∣ e − η ∣ t ∣ + A t ξ ∣ + ∫ 0 t e − η ∣ t ∣ + A ( t − s ) P c [ H ˜ δ ( θ s ω , u δ ( s ) ) + ( A + I ) g z δ ( θ s ω ) ] d s + ∫ + ∞ t e − η ∣ t ∣ + A ( t − s ) P u [ H ˜ δ ( θ s ω , u δ ( s ) ) + ( A + I ) g z δ ( θ s ω ) ] d s + ∫ − ∞ t e − η ∣ t ∣ + A ( t − s ) P s [ H ˜ δ ( θ s ω , u δ ( s ) ) + ( A + I ) g z δ ( θ s ω ) ] d s + ∫ 0 t e − η ∣ t ∣ + A ( t − s ) P c F ˜ δ ( θ s − ρ ω , u δ ( s − ρ ) ) d s + ∫ + ∞ t e − η ∣ t ∣ + A ( t − s ) P u F ˜ δ ( θ s − ρ ω , u δ ( s − ρ ) ) d s + ∫ − ∞ t e − η ∣ t ∣ + A ( t − s ) P s F ˜ δ ( θ s − ρ ω , u δ ( s − ρ ) ) d s ≤ K ∣ ξ ∣ + K ∫ 0 t e β 2 ∣ t − s ∣ + η ∣ s ∣ − η ∣ t ∣ d s + ∫ + ∞ t e β 1 ( t − s ) + η ∣ s ∣ − η ∣ t ∣ d s + ∫ − ∞ t e − β 1 ( t − s ) + η ∣ s ∣ − η ∣ t ∣ d s × ( L H ∣ u δ + g z δ ( θ ⋅ ω ) ∣ C η + ∣ ( A + I ) g z δ ( θ ⋅ ω ) ∣ C η ) + K L F ∫ − ρ t − ρ e β 2 ∣ t − s − ρ ∣ + η ∣ s ∣ − η ∣ t ∣ d s + ∫ + ∞ t − ρ e β 1 ( t − s − ρ ) + η ∣ s ∣ − η ∣ t ∣ d s + ∫ − ∞ t − ρ e − β 1 ( t − s − ρ ) + η ∣ s ∣ − η ∣ t ∣ d s × ∣ u δ + g z δ ( θ ⋅ ω ) ∣ C η ≤ K ∣ ξ ∣ + K L 1 η − β 2 + 2 β 1 − η ∣ u δ + g z δ ( θ ⋅ ω ) ∣ C η + K 1 η − β 2 + 2 β 1 − η ∣ ( A + I ) g z δ ( θ ⋅ ω ) ∣ C η ,

where L ≔ ( L H + e η ρ L F ) . This implies that the operator J δ c ( ⋅ , ω , ξ ) maps from C η into itself.

For each u δ , u ¯ δ ∈ C η , we have that

∣ J δ c ( u δ , ω , ξ ) − J δ c ( u ¯ δ , ω , ξ ) ∣ ≤ ∫ 0 t e A ( t − s ) P c [ H ˜ δ ( θ s ω , u δ ( s ) ) − H ˜ δ ( θ s ω , u ¯ δ ( s ) ) ] d s + ∫ + ∞ t e A ( t − s ) P u [ H ˜ δ ( θ s ω , u δ ( s ) ) − H ˜ δ ( θ s ω , u ¯ δ ( s ) ) ] d s + ∫ − ∞ t e A ( t − s ) P s [ H ˜ δ ( θ s ω , u δ ( s ) ) − H ˜ δ ( θ s ω , u ¯ δ ( s ) ) ] d s + ∫ 0 t e A ( t − s ) P c [ F ˜ δ ( θ s − ρ ω , u δ ( s − ρ ) ) − F ˜ δ ( θ s − ρ ω , u ¯ δ ( s − ρ ) ) ] d s + ∫ + ∞ t e A ( t − s ) P u [ F ˜ δ ( θ s − ρ ω , u δ ( s − ρ ) ) − F ˜ δ ( θ s − ρ ω , u ¯ δ ( s − ρ ) ) ] d s + ∫ − ∞ t e A ( t − s ) P s [ F ˜ δ ( θ s − ρ ω , u δ ( s − ρ ) ) − F ˜ δ ( θ s − ρ ω , u ¯ δ ( s − ρ ) ) ] d s ,

which yields

∣ J δ c ( u δ , ω , ξ ) − J δ c ( u ¯ δ , ω , ξ ) ∣ C η ≤ K L 1 η − β 2 + 2 β 1 − η ∣ u δ − u ¯ δ ∣ C η .

This implies that J δ c ( ⋅ , ω , ξ ) is a uniform contraction with respect to ( ω , ξ ) . By the contraction mapping principle, the mapping J δ c ( ⋅ , ω , ξ ) has a unique fixed point u δ ( ⋅ , τ , ω , ξ ) ∈ C η for each ξ ∈ E c and ω ∈ Ω .

Similarly, for all ξ , ξ ¯ ∈ E c ,

∣ u δ ( t , τ , ω , ξ ) − u δ ( t , τ , ω , ξ ¯ ) ∣ C η ≤ K ∣ ξ − ξ ¯ ∣ + K L 1 η − β 2 + 2 β 1 − η ∣ u δ ( ⋅ , τ , ω , ξ ) − u δ ( ⋅ , τ , ω , ξ ¯ ) ∣ C η .

Thus, we have

∣ u δ ( t , τ , ω , ξ ) − u δ ( t , τ , ω , ξ ¯ ) ∣ C η ≤ K 1 − K L 1 η − β 2 + 2 β 1 − η ∣ ξ − ξ ¯ ∣ .

Moreover, since u δ ( ⋅ , τ , ω , ξ ) can be an ω -wise limit of the iteration of the contraction mapping J δ c starting at 0 and mapping a ℱ -measurable function to a measurable function, u δ ( ⋅ , τ , ω , ξ ) is ℱ -measurable in ω . On the other hand, since u δ ( ⋅ , τ , ω , ξ ) is Lipschitz continuous in ξ , u δ ( ⋅ , τ , ω , ξ ) is measurable with respect to ( ω , ξ ) .

Let h δ c ( ω , ξ ) = P u u δ ( 0 , τ , ω , ξ ) + P s u δ ( 0 , τ , ω , ξ ) , then

h δ c ( ω , ξ ) = ∫ + ∞ 0 e − A s P u [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) + F ˜ δ ( θ s − ρ ω , u δ ( s − ρ , τ , ω , ξ ) ) + ( A + I ) g z δ ( θ s ω ) ] d s + ∫ − ∞ 0 e − A s P s [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) + F ˜ δ ( θ s − ρ ω , u δ ( s − ρ , τ , ω , ξ ) ) + ( A + I ) g z δ ( θ s ω ) ] d s

and h δ c ( ω , 0 ) ∈ C η . By using (2.1) again, for any ξ , ξ ¯ ∈ E c , we obtain

(3.17) ∣ h δ c ( ω , ξ ) − h δ c ( ω , ξ ¯ ) ∣ ≤ 2 K 2 L ( β 1 − η ) 1 − K L 1 η − β 2 + 2 β 1 − η ∣ ξ − ξ ¯ ∣ .

Thus, h δ c is measurable with respect to ( ω , ξ ) .

From Lemma 3.3 and the definition of h δ c ( ω , ξ ) , we obtain

M δ c ( ω ) = { ξ + h δ c ( ω , ξ ) ∣ ξ ∈ E c } .

Then we show that M δ c ( ω ) is a random set, i.e., for any x ∈ C ρ ,

(3.18) ω ↦ inf y ∈ R n { ∣ x − ( P c y + h δ c ( ω , P c y ) ) ∣ }

is measurable. Note that Q n is a countable dense set of R n , so the right-hand side of (3.18) is equal to

inf y ∈ Q n { ∣ x − ( P c y + h δ c ( ω , P c y ) ) ∣ } ,

which follows immediately by the continuity of h δ c ( ω , ⋅ ) . The measurability of any expression under the infimum of (3.18) follows from that ω ↦ h δ c ( ω , P c y ) is measurable for any y ∈ R n .□

4 Smoothness of the center manifolds

In this section, we prove the C k smoothness of the center manifolds M δ c ( ω ) for equation (3.7) as δ > 0 (resp. (3.5)).

Theorem 4.1

Assume that ( A 1 ) and ( A 2 ) hold. If β 2 < k η < β 1 and

(4.1) K L 1 i η − β 2 + 2 β 1 − i η < 1 , 1 ≤ i ≤ k ,

u δ ( ⋅ , τ , ω , ξ ) ∈ C η is the solution of equation (3.9), then we have:

  1. u δ ( ⋅ , τ , ω , ξ ) is C k from E c to C k η , which implies that M δ c ( ω ) is a C k invariant center manifold for equation (3.7) as δ > 0 (resp. (3.5)).

  2. there exists a positive constant γ 0 such that for all 1 ≤ i ≤ k and 0 < γ ≤ γ 0 , we have

    ‖ D ξ i u δ ( ⋅ , τ , ω , ξ ) ‖ ℒ i ( E c , C i η − γ ) ≤ B i , γ ,

    where B i , γ is a positive constant.

Proof

We prove this theorem by induction. First, we consider the case k = 1 . Due to (4.1), there exists γ 0 > 0 , such that for 0 < γ ≤ γ 0 and β 2 < i η − 2 γ < i η − γ < β 1 , we have

K L 1 ( i η − j γ ) − β 2 + 2 β 1 − ( i η − j γ ) < 1 , 1 ≤ i ≤ k and j = 1 , 2 .

Fixing 0 < γ ≤ γ 0 , then we prove that u δ ( ⋅ , ω , ξ ) is differentiable from E c to C η − γ . using the same approach as mentioned earlier, we have that J δ c defined in the proof of Theorem 3.1 is a uniform contraction in C η − j γ ⊂ C η for any j = 1 , 2 . Hence, u δ ( ⋅ , τ , ω , ξ ) ∈ C η − j γ .

For ξ ¯ ∈ E c , v ∈ C η − γ , we define

T δ v = ∫ 0 t e A ( t − s ) P c D u δ [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) + ( A + I ) g z δ ( θ s ω ) ] v d s + ∫ + ∞ t e A ( t − s ) P u D u δ [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) + ( A + I ) g z δ ( θ s ω ) ] v d s + ∫ − ∞ t e A ( t − s ) P s D u δ [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) + ( A + I ) g z δ ( θ s ω ) ] v d s + ∫ − ρ t − ρ e A ( t − s − ρ ) P c D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) v d s + ∫ + ∞ t − ρ e A ( t − s − ρ ) P u D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) v d s + ∫ − ∞ t − ρ e A ( t − s − ρ ) P s D u δ F ˜ δ ( θ s ω , u δ ( s , ω , τ , ξ ¯ ) ) v d s .

It follows from the similar approach as the proof of J δ c , we have that T δ is a bounded linear operator from C η − ζ into itself with the norm

‖ T δ ‖ ≤ K L 1 ( η − γ ) − β 2 + 2 β 1 − ( η − γ ) < 1 ,

which implies that I d − T δ is invertible in C η − γ .

For ξ , ξ ¯ ∈ E c , let

ℐ δ = ∫ 0 t e A ( t − s ) P c [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) × ( u δ ( s , τ , ω , ξ ) − u δ ( s , τ , ω , ξ ¯ ) ) ] d s + ∫ − ρ t − ρ e A ( t − s − ρ ) P c [ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ( u δ ( s , τ , ω , ξ ) − u δ ( s , τ , ω , ξ ¯ ) ) ] d s + ∫ + ∞ t e A ( t − s ) P u [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ( u δ ( s , τ , ω , ξ ) − u δ ( s , τ , ω , ξ ¯ ) ) ] d s + ∫ + ∞ t − ρ e A ( t − s − ρ ) P u [ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ( u δ ( s , τ , ω , ξ ) − u δ ( s , τ , ω , ξ ¯ ) ) ] d s + ∫ − ∞ t e A ( t − s ) P s [ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ( u δ ( s , τ , ω , ξ ) − u δ ( s , τ , ω , ξ ¯ ) ) ] d s + ∫ − ∞ t − ρ e A ( t − s − ρ ) P s [ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ( u δ ( s , τ , ω , ξ ) − u δ ( s , τ , ω , ξ ¯ ) ) ] d s = ℐ δ , 1 + ℐ δ , 2 + ℐ δ , 3 .

We claim that ∣ ℐ δ ∣ C η − γ = o ( ∣ ξ − ξ ¯ ∣ ) as ξ → ξ ¯ , then we can obtain

u δ ( ⋅ , τ , ω , ξ ) − u δ ( ⋅ , τ , ω , ξ ¯ ) − T δ ( u δ ( ⋅ , τ , ω , ξ ) − u δ ( ⋅ , τ , ω , ξ ¯ ) ) = e A t ( ξ − ξ ¯ ) + ℐ δ = e A t ( ξ − ξ ¯ ) + o ( ∣ ξ − ξ ¯ ∣ ) , as ξ → ξ ¯ ,

which implies that

u δ ( ⋅ , τ , ω , ξ ) − u δ ( ⋅ , τ , ω , ξ ¯ ) = ( I d − T δ ) − 1 e A t ( ξ − ξ ¯ ) + o ( ∣ ξ − ξ ¯ ∣ ) .

Thus, we can obtain u δ ( ⋅ , τ , ω , ξ ) is differentiable in ξ .

Now, we prove the aforementioned claim. For the sake of convenience, we denote

Q ˜ δ , 1 ( s ) = H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ( u δ ( s , τ , ω , ξ ) − u δ ( s , τ , ω , ξ ¯ ) ) , Q ˜ δ , 2 ( s ) = F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ( u δ ( s , τ , ω , ξ ) − u δ ( s , τ , ω , ξ ¯ ) ) .

For the first integral ℐ δ , 1 , let N 1 be a large positive number to be chosen later and for t ≥ N 1 ,

ℐ δ , 11 = e − ( η − γ ) ∣ t ∣ ∫ 0 N 1 e A ( t − s ) P c Q ˜ δ , 1 ( s ) d s + ∫ − ρ N 1 − ρ e A ( t − s − ρ ) P c Q ˜ δ , 2 ( s ) d s

and

ℐ δ , 12 = e − ( η − γ ) ∣ t ∣ ∫ N 1 t e A ( t − s ) P c Q ˜ δ , 1 ( s ) d s + ∫ N 1 − ρ t − ρ e A ( t − s − ρ ) P c Q ˜ δ , 2 ( s ) d s ;

for t ≤ − N 1 ,

ℐ δ , 13 = e − ( η − γ ) ∣ t ∣ ∫ 0 − N 1 e A ( t − s ) P c Q ˜ δ , 1 ( s ) d s + ∫ − ρ − N 1 − ρ e A ( t − s − ρ ) P c Q ˜ δ , 2 ( s ) d s

and

ℐ δ , 14 = e − ( η − γ ) ∣ t ∣ ∫ − N 1 t e A ( t − s ) P c Q ˜ δ , 1 ( s ) d s + ∫ − N 1 − ρ t − ρ e A ( t − s − ρ ) P c Q ˜ δ , 2 ( s ) d s ;

for − N 1 ≤ t ≤ N 1 ,

ℐ δ , 15 = e − ( η − γ ) ∣ t ∣ ∫ − N 1 N 1 e A ( t − s ) P c Q ˜ δ , 1 ( s ) d s + ∫ − N 1 − ρ N 1 − ρ e A ( t − s − ρ ) P c Q ˜ δ , 2 ( s ) d s .

For the second integral ℐ δ , 2 , let N 2 be a large positive number to be chosen later and for t ≥ N 2 ,

ℐ δ , 21 = e − ( η − γ ) ∣ t ∣ ∫ + ∞ t e A ( t − s ) P u Q ˜ δ , 1 ( s ) d s + ∫ + ∞ t − ρ e A ( t − s − ρ ) P u Q ˜ δ , 2 ( s ) d s ;

for t ≤ − N 2 ,

ℐ δ , 22 = e − ( η − γ ) ∣ t ∣ ∫ + ∞ N 2 e A ( t − s ) P u Q ˜ δ , 1 ( s ) d s + ∫ + ∞ N 2 − ρ e A ( t − s − ρ ) P u Q ˜ δ , 2 ( s ) d s , ℐ δ , 23 = e − ( η − γ ) ∣ t ∣ ∫ N 2 − N 2 e A ( t − s ) P u Q ˜ δ , 1 ( s ) d s + ∫ N 2 − ρ − N 2 − ρ e A ( t − s − ρ ) P u Q ˜ δ , 2 ( s ) d s ,

and

ℐ δ , 24 = e − ( η − γ ) ∣ t ∣ ∫ − N 2 t e A ( t − s ) P u Q ˜ δ , 1 ( s ) d s + ∫ − N 2 − ρ t − ρ e A ( t − s − ρ ) P u Q ˜ δ , 2 ( s ) d s ;

for − N 2 ≤ t ≤ N 2 ,

ℐ δ , 25 = e − ( η − γ ) ∣ t ∣ ∫ + ∞ N 2 e A ( t − s ) P u Q ˜ δ , 1 ( s ) d s + ∫ + ∞ N 2 − ρ e A ( t − s − ρ ) P u Q ˜ δ , 2 ( s ) d s

and

ℐ δ , 26 = e − ( η − γ ) ∣ t ∣ ∫ N 2 t e A ( t − s ) P u Q ˜ δ , 1 ( s ) d s + ∫ N 2 − ρ t − ρ e A ( t − s − ρ ) P u Q ˜ δ , 2 ( s ) d s .

For the third integral ℐ δ , 3 , let N 3 be a large positive number to be chosen later and for t ≥ N 3 ,

ℐ δ , 31 = e − ( η − γ ) ∣ t ∣ ∫ − ∞ − N 3 e A ( t − s ) P s Q ˜ δ , 1 ( s ) d s + ∫ − ∞ − N 3 − ρ e A ( t − s − ρ ) P s Q ˜ δ , 2 ( s ) d s , ℐ δ , 32 = e − ( η − γ ) ∣ t ∣ ∫ − N 3 N 3 e A ( t − s ) P s Q ˜ δ , 1 ( s ) d s + ∫ − N 3 − ρ N 3 − ρ e A ( t − s − ρ ) P s Q ˜ δ , 2 ( s ) d s ,

and

ℐ δ , 33 = e − ( η − γ ) ∣ t ∣ ∫ N 3 t e A ( t − s ) P s Q ˜ δ , 1 ( s ) d s + ∫ N 3 − ρ t − ρ e A ( t − s − ρ ) P s Q ˜ δ , 2 ( s ) d s ;

for t ≤ − N 3 ,

ℐ δ , 34 = e − ( η − γ ) ∣ t ∣ ∫ − ∞ t e A ( t − s ) P s Q ˜ δ , 1 ( s ) d s + ∫ − ∞ t − ρ e A ( t − s − ρ ) P s Q ˜ δ , 2 ( s ) d s ;

for − N 3 ≤ t ≤ N 3 ,

ℐ δ , 35 = e − ( η − γ ) ∣ t ∣ ∫ − ∞ − N 3 e A ( t − s ) P s Q ˜ δ , 1 ( s ) d s + ∫ − ∞ − N 3 − ρ e A ( t − s − ρ ) P s Q ˜ δ , 2 ( s ) d s

and

ℐ δ , 36 = e − ( η − γ ) ∣ t ∣ ∫ − N 3 t e A ( t − s ) P s Q ˜ δ , 1 ( s ) d s + ∫ − N 3 − ρ t − ρ e A ( t − s − ρ ) P s Q ˜ δ , 2 ( s ) d s ;

By using the similar arguments of (3.16), we obtain

ℐ δ , 12 ≤ e − ( η − γ ) ∣ t ∣ ∫ N 1 t e A ( t − s ) P c H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ( u δ ( s , τ , ω , ξ ) − u δ ( s , τ , ω , ξ ¯ ) ) d s ∣ + ∫ N 1 − ρ t − ρ e A ( t − s − ρ ) P c F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ( u δ ( s , τ , ω , ξ ) − u δ ( s , τ , ω , ξ ¯ ) ) d s ∣ ≤ 2 K L H ∫ N 1 t e β 2 ∣ t − s ∣ + ( η − 2 γ ) ∣ s ∣ − ( η − γ ) ∣ t ∣ ∣ u δ ( ⋅ , τ , ω , ξ ) − u δ ( ⋅ , τ , ω , ξ ¯ ) ∣ C η − 2 γ d s + 2 K L F ∫ N 1 − ρ t − ρ e β 2 ∣ t − s − ρ ∣ + ( η − 2 γ ) ∣ s ∣ − ( η − γ ) ∣ t ∣ ∣ u δ ( ⋅ , τ , ω , ξ ) − u δ ( ⋅ , τ , ω , ξ ¯ ) ∣ C η − 2 γ d s ≤ 2 K L ∫ N 1 t e β 2 ∣ t − s ∣ + ( η − γ ) ∣ s ∣ − ( η − γ ) ∣ t ∣ e − γ ∣ s ∣ d s ∣ u δ ( ⋅ , τ , ω , ξ ) − u δ ( ⋅ , τ , ω , ξ ¯ ) ∣ C η − 2 γ ≤ 2 K 2 L e − γ N 1 ( η − γ − β 2 ) 1 − K L 1 ( η − 2 γ ) − β 2 + 2 β 1 − ( η − 2 γ ) ∣ ξ − ξ ¯ ∣ .

For any ε > 0 , choose N 1 so large that

2 K 2 L e − γ N 1 ( η − γ − β 2 ) 1 − K L 1 ( η − 2 γ ) − β 2 + 2 β 1 − ( η − 2 γ ) ≤ 1 15 ε .

Then we have that

sup t ≥ N 1 ℐ δ , 12 ≤ 1 15 ε ∣ ξ − ξ ¯ ∣ .

Fixing such N 1 , for ℐ δ , 11 , we have

ℐ δ , 11 ≤ K ∫ 0 N 1 e β 2 ( t − s ) + ( η − γ ) ∣ s ∣ − ( η − γ ) ∣ t ∣ ∫ 0 1 ∣ D u δ H ˜ δ ( θ s ω , r u δ ( s , τ , ω , ξ ) + ( 1 − r ) u δ ( s , τ , ω , ξ ¯ ) ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ∣ d r ∣ u δ ( ⋅ , τ , ω , ξ ) − u δ ( ⋅ , τ , ω , ξ ¯ ) ∣ C η − γ , δ d s + K ∫ − ρ N 1 − ρ e β 2 ( t − s − ρ ) + ( η − γ ) ∣ s ∣ − ( η − γ ) ∣ t ∣ ∫ 0 1 ∣ D u δ F ˜ δ ( θ s ω , r u δ ( s , τ , ω , ξ ) + ( 1 − r ) u δ ( s , τ , ω , ξ ¯ ) ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ∣ d r ∣ u δ ( ⋅ , τ , ω , ξ ) − u δ ( ⋅ , τ , ω , ξ ¯ ) ∣ C η − γ , δ d s ≤ K 2 ∣ ξ − ξ ¯ ∣ 1 − K L 1 ( η − γ ) − β 2 + 2 β 1 − ( η − γ ) ∫ 0 N 1 e − β 2 s + ( η − γ ) ∣ s ∣ ∫ 0 1 ∣ D u δ H ˜ δ ( θ s ω , r u δ ( s , τ , ω , ξ ) + ( 1 − r ) u δ ( s , τ , ω , ξ ¯ ) ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ∣ d r d s + ∫ − ρ N 1 − ρ e − β 2 s + ( η − γ ) ∣ s ∣ ∫ 0 1 ∣ D u δ F ˜ δ ( θ s ω , r u δ ( s , τ , ω , ξ ) + ( 1 − r ) u δ ( s , τ , ω , ξ ¯ ) ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ∣ d r d s .

The last two integrals are on the compact interval [ 0 , N 1 ] and [ − ρ , N 1 − ρ ] . Due to the continuity of the integrand in ( s , ξ ) , there exists σ 1 > 0 , such that if ∣ ξ − ξ ¯ ∣ ≤ σ 1 , then

sup t ≥ N 1 ℐ δ , 11 ≤ 1 15 ε ∣ ξ − ξ ¯ ∣ .

Moreover, if ∣ ξ − ξ ¯ ∣ ≤ σ 1 , then

sup t ≥ N 1 { ℐ δ , 11 + ℐ δ , 12 } + sup t ≤ − N 1 { ℐ δ , 13 + ℐ δ , 14 } + sup − N 1 ≤ t ≤ N 1 ℐ δ , 15 ≤ 1 3 ε ∣ ξ − ξ ¯ ∣ .

Thus,

(4.2) ∣ ℐ δ , 1 ∣ C η − γ ≤ 1 3 ε ∣ ξ − ξ ¯ ∣ .

By using the similar arguments of the proof to (4.2), we choose N 2 and N 3 to be sufficiently large, and there exists σ 2 > 0 , such that if ∣ ξ − ξ ¯ ∣ ≤ σ 2 , then

(4.3) ∣ ℐ δ , 2 ∣ C η − γ ≤ 1 3 ε ∣ ξ − ξ ¯ ∣ ,

(4.4) ∣ ℐ δ , 3 ∣ C η − γ ≤ 1 3 ε ∣ ξ − ξ ¯ ∣ .

Taking σ = min { σ 1 , σ 2 } and combining (4.2)–(4.4), we have that if ∣ ξ − ξ ¯ ∣ ≤ σ , then

∣ ℐ δ ∣ C η − γ ≤ ε ∣ ξ − ξ ¯ ∣ .

Thus, ∣ ℐ δ ∣ C η − γ = o ( ∣ ξ − ξ ¯ ∣ ) as ξ → ξ ¯ .

Therefore, u δ ( ⋅ , τ , ω , ξ ) is differentiable in ξ and its derivative satisfies D ξ u δ ( t , τ , ω , ξ ) ∈ ℒ ( E c , C η − γ ) ; thus, we have

D ξ u δ ( t , τ , ω , ξ ) = e A t P c + ∫ 0 t e A ( t − s ) P c D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ u δ ( s , τ , ω , ξ ) d s + ∫ + ∞ t e A ( t − s ) P u D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ u δ ( s , τ , ω , ξ ) d s + ∫ − ∞ t e A ( t − s ) P s D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ u δ ( s , τ , ω , ξ ) d s + ∫ − ρ t − ρ e A ( t − s − ρ ) P c D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ u δ ( s , τ , ω , ξ ) d s + ∫ + ∞ t − ρ e A ( t − s − ρ ) P u D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ u δ ( s , τ , ω , ξ ) d s + ∫ − ∞ t − ρ e A ( t − s − ρ ) P s D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ u δ ( s , τ , ω , ξ ) d s .

Furthermore,

‖ D ξ u δ ( ⋅ , τ , ω , ξ ) ‖ ℒ ( E c , C η − γ ) ≤ K 1 − K L 1 ( η − γ ) − β 2 + 2 β 1 − ( η − γ ) ≔ B 1 , γ .

For ξ , ξ ¯ ∈ E c , we have

D ξ u δ ( t , τ , ω , ξ ) − D ξ u δ ( t , τ , ω , ξ ¯ ) = ∫ 0 t e A ( t − s ) P c [ D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ u δ ( s , τ , ω , ξ ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) D ξ u δ ( s , τ , ω , ξ ¯ ) ] d s + ∫ + ∞ t e A ( t − s ) P u [ D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ u δ ( s , τ , ω , ξ ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) D ξ u δ ( s , τ , ω , ξ ¯ ) ] d s + ∫ − ∞ t e A ( t − s ) P s [ D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ u δ ( s , τ , ω , ξ ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) D ξ u δ ( s , τ , ω , ξ ¯ ) ] d s + ∫ − ρ t − ρ e A ( t − s − ρ ) P c [ D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ u δ ( s , τ , ω , ξ ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) D ξ u δ ( s , τ , ω , ξ ¯ ) ] d s + ∫ + ∞ t − ρ e A ( t − s − ρ ) P u [ D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ u δ ( s , τ , ω , ξ ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) D ξ u δ ( s , τ , ω , ξ ¯ ) ] d s + ∫ − ∞ t − ρ e A ( t − s − ρ ) P s [ D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ u δ ( s , τ , ω , ξ ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) D ξ u δ ( s , τ , ω , ξ ¯ ) ] d s

= ∫ 0 t e A ( t − s ) P c D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) [ D ξ u δ ( s , τ , ω , ξ ) − D ξ u δ ( s , τ , ω , ξ ¯ ) ] d s + ∫ + ∞ t e A ( t − s ) P u D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) [ D ξ u δ ( s , τ , ω , ξ ) − D ξ u δ ( s , τ , ω , ξ ¯ ) ] d s + ∫ − ∞ t e A ( t − s ) P s D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) [ D ξ u δ ( s , τ , ω , ξ ) − D ξ u δ ( s , ω , ξ ¯ ) ] d s + ∫ − ρ t − ρ e A ( t − s − ρ ) P c D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) [ D ξ u δ ( s , τ , ω , ξ ) − D ξ u δ ( s , τ , ω , ξ ¯ ) ] d s + ∫ + ∞ t − ρ e A ( t − s − ρ ) P u D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) [ D ξ u δ ( s , τ , ω , ξ ) − D ξ u δ ( s , τ , ω , ξ ¯ ) ] d s + ∫ − ∞ t − ρ e A ( t − s − ρ ) P s D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) [ D ξ u δ ( s , τ , ω , ξ ) − D ξ u δ ( s , τ , ω , ξ ¯ ) ] d s + ℐ ˜ δ ,

where

ℐ ˜ δ = ∫ 0 t e A ( t − s ) P c [ D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ] D ξ u δ ( s , τ , ω , ξ ¯ ) d s + ∫ + ∞ t e A ( t − s ) P u [ D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ] D ξ u δ ( s , τ , ω , ξ ¯ ) d s + ∫ − ∞ t e A ( t − s ) P s [ D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ] D ξ u δ ( s , τ , ω , ξ ¯ ) d s + ∫ − ρ t − ρ e A ( t − s − ρ ) P c [ D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ] D ξ u δ ( s , τ , ω , ξ ¯ ) d s + ∫ + ∞ t − ρ e A ( t − s − ρ ) P u [ D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ] D ξ u δ ( s , τ , ω , ξ ¯ ) d s + ∫ − ∞ t − ρ e A ( t − s − ρ ) P s [ D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) − D u δ F ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ¯ ) ) ] D ξ u δ ( s , τ , ω , ξ ¯ ) d s .

Then we obtain

‖ D ξ u δ ( t , τ , ω , ξ ) − D ξ u δ ( t , τ , ω , ξ ¯ ) ‖ ℒ ( E c , C η ) ≤ ‖ ℐ ˜ δ ‖ ℒ ( E c , C η ) 1 − K L 1 η − β 2 + 2 β 1 − η .

By utilizing the same argument that we prove the last claim, we obtain that ‖ ℐ ˜ δ ‖ ℒ ( E c , C η ) = o ( 1 ) as ξ → ξ ¯ . Thus, D ξ u δ ( ⋅ , τ , ω , ξ ) is continuous from E c to ℒ ( E c , C η ) . Hence, u δ ( ⋅ , τ , ω , ⋅ ) is C 1 from E c to C η .

Next, we show that u δ ( ⋅ , τ , ω , ⋅ ) is C k from E c to C k η . By inductive assumption, we have that u δ ( ⋅ , τ , ω , ⋅ ) is C m from E c to C m η for all 1 ≤ m ≤ k − 1 , and there exist constants B m , γ such that

‖ D ξ m u δ ( ⋅ , τ , ω , ξ ) ‖ L m ( E c , C m η − γ ) ≤ B m , γ .

By the calculation, we find that D ξ k − 1 u δ ( t , τ , ω , ξ ) satisfies the following equation:

D ξ k − 1 u δ ( t , τ , ω , ξ ) = ∫ 0 t e A ( t − s ) P c [ D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ k − 1 u δ ( s , τ , ω , ξ ) + D u δ F ˜ δ ( θ s − ρ ω , u δ ( s − ρ , τ , ω , ξ ) ) D ξ k − 1 u δ ( s − ρ , τ , ω , ξ ) ] d s + ∫ + ∞ t e A ( t − s ) P u [ D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ k − 1 u δ ( s , ω , ξ ) + D u δ F ˜ δ ( θ s − ρ ω , u δ ( s − ρ , τ , ω , ξ ) ) D ξ k − 1 u δ ( s − ρ , τ , ω , ξ ) ] d s + ∫ − ∞ t e A ( t − s ) P s [ D u δ H ˜ δ ( θ s ω , u δ ( s , ω , ξ ) ) D ξ k − 1 u δ ( s , τ , ω , ξ ) + D u δ F ˜ δ ( θ s − ρ ω , u δ ( s − ρ , τ , ω , ξ ) ) D ξ k − 1 u δ ( s − ρ , τ , ω , ξ ) ] d s + ∫ 0 t e A ( t − s ) P c [ R δ , k − 1 ( s , τ , ω , ξ ) + M δ , k − 1 ( s − ρ , τ , ω , ξ ) ] d s + ∫ + ∞ t e A ( t − s ) P u [ R δ , k − 1 ( s , τ , ω , ξ ) + M δ , k − 1 ( s − ρ , τ , ω , ξ ) ] d s + ∫ − ∞ t e A ( t − s ) P s [ R δ , k − 1 ( s , τ , ω , ξ ) + M δ , k − 1 ( s − ρ , τ , ω , ξ ) ] d s ,

where

R δ , k − 1 ( s , τ , ω , ξ ) = ∑ i = 0 k − 3 k − 2 i D ξ k − 2 − i D u δ H ˜ δ ( θ s ω , u δ ( s , τ , ω , ξ ) ) D ξ i + 1 u δ ( s , τ , ω , ξ ) , M δ , k − 1 ( s − ρ , τ , ω , ξ ) = ∑ i = 0 k − 3 k − 2 i D ξ k − 2 − i D u δ F ˜ δ ( θ s − ρ ω , u δ ( s − ρ , τ , ω , ξ ) ) D ξ i + 1 u δ ( s , τ , ω , ξ ) .

Note that D ξ i u δ ∈ C i η for i = 1 , … , k − 1 . Since H and F are C k , from the induction hypothesis, we have R δ , k − 1 ( ⋅ , τ , ω , ξ ) ∈ ℒ k − 1 ( E c , C ( k − 1 ) η ) , and M δ , k − 1 ( s , τ , ω , ξ ) ∈ ℒ k − 1 ( E c , C ( k − 1 ) η ) , and they are C 1 in ξ . ℒ k − 1 ( E c , C ( k − 1 ) η ) is the usual space of bounded k − 1 linear forms. Then we have

‖ D ξ R δ , k − 1 ( s , τ , ω , ξ ) ‖ ≤ T k , 1 e ( k η − γ ) s , ‖ D ξ M δ , k − 1 ( s , τ , ω , ξ ) ‖ ≤ T k , 2 e ( k η − γ ) s ,

where T k , 1 and T k , 2 are positive constants. By using the same argument, which we used in the case k = 1 , we can show that D ξ k − 1 u δ ( ⋅ , τ , ω , ξ ) is C 1 from E c to ℒ k ( E c , C k η ) and we can choose B i , γ for 1 ≤ i ≤ k such that ( ii ) holds. This completes the proof of this theorem.□

5 Approximations of the center manifolds

In this section, we show that the center manifold of equation (3.2) approximates the center manifold of equation (3.1).

Theorem 5.1

Assume that the same conditions in Theorem 4.1  hold. Then we have

lim δ → 0 + ∣ u δ ( ⋅ , τ , ω , ξ ) − u ( ⋅ , τ , ω , ξ ) ∣ C η = 0 ,

and

lim δ → 0 + ‖ D ξ i u δ ( ⋅ , τ , ω , ξ ) − D ξ i u ( ⋅ , τ , ω , ξ ) ‖ ℒ i ( E c , C i η ) = 0 , 1 ≤ i ≤ k ,

where ξ ∈ E c , u ( ⋅ , τ , ω , ξ ) = u 0 ( ⋅ , τ , ω , ξ ) .

Proof

Note that there exists 0 < γ ′ < γ 0 such that for all 0 < γ ≤ γ ′ , so we have β 2 < i η − γ < i η < β 1 and

K L 1 ( i η − γ ) − β 2 + 2 β 1 − ( i η − γ ) < 1 , 1 ≤ i ≤ k .

Following the proof of Theorem 3.1, the aforementioned conditions imply that J δ c is a uniform contraction from C i η − γ into itself with the contraction constant

K L 1 ( i η − γ ) − β 2 + 2 β 1 − ( i η − γ ) , 1 ≤ i ≤ k .

Thus, u δ , u ∈ C i η − γ . By Lemmas 3.1 and 3.2, there exists T 1 > 0 such that

(5.1) ∫ 0 t ( z δ ( θ r ω ) − z ( θ r ω ) ) d r < γ ∣ t ∣ and ∣ z δ ( θ t ω ) − z ( θ t ω ) ∣ < γ ∣ t ∣ ,

for any ∣ t ∣ ≥ T 1 , δ ∈ ( 0 , 1 2 ] . By using Lemma 3.2, there exists δ 0 ∈ ( 0 , 1 2 ] such that

(5.2) ‖ z δ ( θ ⋅ ω ) − z ( θ ⋅ ω ) ‖ C ( [ − T 1 , T 1 ] ) < γ , 0 < δ < δ 0 .

For any ∣ t ∣ ≤ T 1 and 0 < δ < δ 0 , we have

(5.3) ∫ 0 t ( z δ ( θ r ω ) − z ( θ r ω ) ) d r ≤ ‖ z δ ( θ ⋅ ω ) − z ( θ ⋅ ω ) ‖ C ( [ − T 1 , T 1 ] ) ∣ t ∣ < γ ∣ t ∣ .

Combining (5.1)–(5.3), for any t ∈ R and 0 < δ < δ 0 , we have

(5.4) ∫ 0 t ( z δ ( θ r ω ) − z ( θ r ω ) ) d r < γ ∣ t ∣ , ∣ z δ ( θ t ω ) − z ( θ t ω ) ∣ < γ ( ∣ t ∣ + 1 ) .

Let v δ ( t , τ , ω , ξ ) = u δ ( t , τ , ω , ξ ) − u ( t , τ , ω , ξ ) . Similar to Lemma 3.3, we can show that v δ ( t ) satisfies the following integral equation:

v δ ( t ) = ∫ 0 t e A ( t − s ) P c [ H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − H ˜ ( θ s ω , u ( s ) ) + F ˜ δ ( θ s − ρ ω , v δ ( s − ρ ) + u ( s − ρ ) ) − F ˜ ( θ s − ρ ω , u ( s − ρ ) ) + ( A + I ) g ( z δ ( θ s ω ) − z ( θ s ω ) ) ] d s + ∫ + ∞ t e A ( t − s ) P u [ H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − H ˜ ( θ s ω , u ( s ) ) + F ˜ δ ( θ s − ρ ω , v δ ( s − ρ ) + u ( s − ρ ) ) − F ˜ ( θ s − ρ ω , u ( s − ρ ) ) + ( A + I ) g ( z δ ( θ s ω ) − z ( θ s ω ) ) ] d s + ∫ − ∞ t e A ( t − s ) P s [ H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − H ˜ ( θ s ω , u ( s ) ) + F ˜ δ ( θ s − ρ ω , v δ ( s − ρ ) + u ( s − ρ ) ) − F ˜ ( θ s − ρ ω , u ( s − ρ ) ) + ( A + I ) g ( z δ ( θ s ω ) − z ( θ s ω ) ) ] d s = ∫ 0 t e A ( t − s ) P c [ H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − H ˜ ( θ s ω , u ( s ) ) + ( A + I ) g ( z δ ( θ s ω ) − z ( θ s ω ) ) ] d s + ∫ − ρ t − ρ e A ( t − s − ρ ) P c [ F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − F ˜ ( θ s ω , u ( s ) ) ] d s + ∫ + ∞ t e A ( t − s ) P u [ H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − H ˜ ( θ s ω , u ( s ) ) + ( A + I ) g ( z δ ( θ s ω ) − z ( θ s ω ) ) ] d s + ∫ + ∞ t − ρ e A ( t − s − ρ ) P u [ F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − F ˜ ( θ s ω , u ( s ) ) ] d s + ∫ − ∞ t e A ( t − s ) P s [ H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − H ˜ ( θ s ω , u ( s ) ) + ( A + I ) g ( z δ ( θ s ω ) − z ( θ s ω ) ) ] d s + ∫ − ∞ t − ρ e A ( t − s − ρ ) P s [ F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − F ˜ ( θ s ω , u ( s ) ) ] d s = P δ , 1 + P δ , 2 + P δ , 3 .

To deal with these integrals, we first estimate

(5.5) ∣ H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − H ˜ ( θ s ω , u ( s ) ) ∣ = ∣ H ( v δ ( s ) + u ( s ) + g z δ ( θ s ω ) ) − H ( u ( s ) + g z ( θ s ω ) ) ∣ ≤ L H ( e η ∣ s ∣ ∣ v δ ( ⋅ ) ∣ C η + ∣ g ∣ ∣ z δ ( θ s ω ) − z ( θ s ω ) ∣ ) .

Similarly,

(5.6) ∣ F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − F ˜ ( θ s ω , u ( s ) ) ∣ ≤ L F ( e η ∣ s ∣ ∣ v δ ( ⋅ ) ∣ C η + ∣ g ∣ ∣ z δ ( θ s ω ) − z ( θ s ω ) ∣ ) .

Thus, for the first integral P δ , 1 , we have

(5.7) e − η ∣ t ∣ ∣ P δ , 1 ∣ ≤ K L ∣ v δ ( ⋅ ) ∣ C η ∣ ∫ 0 t e β 2 ∣ t − s ∣ + η ∣ s ∣ − η ∣ t ∣ d s ∣ + K L H ∣ ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ ∣ g ∣ ∣ z δ ( θ s ω ) − z ( θ s ω ) ∣ d s ∣ + K L F ∣ ∫ − ρ t − ρ e β 2 ∣ t − s − ρ ∣ − η ∣ t ∣ ∣ g ∣ ∣ z δ ( θ s ω ) − z ( θ s ω ) ∣ d s ∣ + K ∣ ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ ∣ ( A + I ) g ∣ ∣ z δ ( θ s ω ) − z ( θ s ω ) ∣ d s ∣ ,

where L is defined in Theorem 3.1. We claim that

(5.8) ∣ P δ , 1 ∣ C η ≤ K L ∣ v δ ( ⋅ ) ∣ C η η − β 2 + o ( 1 ) , as δ → 0 + .

Next, we prove this claim. For the first term of (5.7), we have

(5.9) sup t ∈ R K L ∣ v δ ( ⋅ ) ∣ C η ∣ ∫ 0 t e β 2 ∣ t − s ∣ + η ∣ s ∣ − η ∣ t ∣ d s ∣ ≤ K L ∣ v δ ( ⋅ ) ∣ C η η − β 2 .

For the second term of (5.7), we show

(5.10) sup t ∈ R K L H ∣ ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ ∣ g ∣ ∣ z δ ( θ s ω ) − z ( θ s ω ) ∣ d s ∣ = o ( 1 ) .

By (5.4), we have ∣ z δ ( θ t ω ) − z ( θ t ω ) ∣ < γ ( ∣ t ∣ + 1 ) . Let

N = sup t ∈ R e − γ ∣ t ∣ ( ∣ t ∣ + 1 ) .

For any ε > 0 , let T 2 > 0 be large enough such that

K L H ∣ g ∣ γ N η − γ − β 2 e − γ T 2 < ε .

Then, for ∣ t ∣ ≥ T 2 and 0 < δ < δ 0 , we have

K L H ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ ∣ g ∣ ∣ z δ ( θ s ω ) − z ( θ s ω ) ∣ d s ≤ K L H ∣ g ∣ γ N ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ + γ ∣ s ∣ d s ≤ K L H ∣ g ∣ γ N ∫ 0 t e β 2 ∣ t − s ∣ − ( η − γ ) ∣ t ∣ + ( η − γ ) ∣ s ∣ e − γ ∣ t ∣ d s ≤ K L H ∣ g ∣ γ N η − γ − β 2 e − γ T 2 < ε .

For ∣ t ∣ ≤ T 2 , by Lemma 3.2, we have

K L H ∣ ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ ∣ g ∣ ∣ z δ ( θ s ω ) − z ( θ s ω ) ∣ d s ∣ ≤ K L H ∣ g ∣ ‖ z δ ( θ ⋅ ω ) − z ( θ ⋅ ω ) ‖ C ( [ − T 2 , T 2 ] ) ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ + η ∣ s ∣ d s ≤ K L H ∣ g ∣ ‖ z δ ( θ ⋅ ω ) − z ( θ ⋅ ω ) ‖ C ( [ − T 2 , T 2 ] ) η − β 2 < ε ,

when δ is sufficiently small. Hence, (5.10) holds. For the third term of (5.7), we show

(5.11) sup t ∈ R K L F ∫ − ρ t − ρ e β 2 ∣ t − s − ρ ∣ − η ∣ t ∣ ∣ g ∣ ∣ z δ ( θ s ω ) − z ( θ s ω ) ∣ d s = o ( 1 ) .

Choose T 3 > 0 be large enough such that

K L F ∣ g ∣ γ N e η ρ η − γ − β 2 e − γ T 3 < ε .

Then, for ∣ t ∣ ≥ T 3 and 0 < δ < δ 0 , we have

K L F ∣ ∫ − ρ t − ρ e β 2 ∣ t − s − ρ ∣ − η ∣ t ∣ ∣ g ∣ ∣ z δ ( θ s ω ) − z ( θ s ω ) ∣ d s ∣ ≤ K L F ∣ g ∣ γ N ∣ ∫ − ρ t − ρ e β 2 ∣ t − s − ρ ∣ − η ∣ t ∣ + γ ∣ s ∣ d s ∣ ≤ K L F ∣ g ∣ γ N e η ρ ∣ ∫ 0 t e β 2 ∣ t − s ∣ − ( η − γ ) ∣ t ∣ + ( η − γ ) ∣ s ∣ e − γ ∣ t ∣ d s ∣ ≤ K L F ∣ g ∣ γ N e η ρ η − γ − β 2 e − γ T 3 < ε .

For ∣ t ∣ ≤ T 3 , we can obtain

K L F ∣ ∫ − ρ t − ρ e β 2 ∣ t − s − ρ ∣ − η ∣ t ∣ ∣ g ∣ ∣ z δ ( θ s ω ) − z ( θ s ω ) ∣ d s ∣ ≤ K L F ∣ g ∣ ‖ z δ ( θ ⋅ ω ) − z ( θ ⋅ ω ) ‖ C ( [ − T 3 − ρ , T 3 + ρ ] ) ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ + η ∣ s ∣ d s ≤ K L F ∣ g ∣ ‖ z δ ( θ ⋅ ω ) − z ( θ ⋅ ω ) ‖ C ( [ − T 3 − ρ , T 3 + ρ ] ) η − β 2 < ε ,

when δ is sufficiently small. Hence, (5.11) holds. Finally, we show the last term of (5.7) can be given as:

(5.12) sup t ∈ R K ∣ ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ ∣ ( A + I ) g ∣ ∣ z δ ( θ s ω ) − z ( θ s ω ) ∣ d s ∣ = o ( 1 ) .

Similarly, we can choose T 4 > 0 be large enough such that

K ∣ ( A + I ) g ∣ γ N η − γ − β 2 e − γ T 4 < ε ,

and by using Lemma 3.2 again, we can obtain (5.12). Together with (5.9)–(5.12), we obtain (5.8).

By using the same argument of the aforementioned claim, we have

(5.13) ∣ P δ , 2 ∣ C η ≤ K L ∣ v δ ( ⋅ ) ∣ C η β 1 − η + o ( 1 ) , as δ → 0 + ,

and

(5.14) ∣ P δ , 3 ∣ C η ≤ K L ∣ v δ ( ⋅ ) ∣ C η β 1 − η + o ( 1 ) , as δ → 0 + .

Therefore, by combining (5.8), (5.13), and (5.14), we have

∣ v δ ( ⋅ ) ∣ C η ≤ K L 1 η − β 2 + 2 β 1 − η ∣ v δ ( ⋅ ) ∣ C η + o ( 1 ) .

Since

K L 1 η − β 2 + 2 β 1 − η < 1 ,

we have

∣ v δ ( ⋅ ) ∣ C η = o ( 1 ) , as δ → 0 + .

Namely,

(5.15) lim δ → 0 + ∣ u δ ( ⋅ , ω , ξ ) − u ( ⋅ , ω , ξ ) ∣ C η = 0 .

Next, we show

(5.16) lim δ → 0 + ‖ D ξ i u δ ( ⋅ , ω , ξ ) − D ξ i u ( ⋅ , ω , ξ ) ‖ ℒ i ( E c , C i η ) = 0 , 1 ≤ i ≤ k .

We first prove the case i = 1 . Note that

D ξ v δ ( t ) = ∫ 0 t e A ( t − s ) P c [ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ v δ ( s ) + ( D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u H ˜ δ ( θ s ω , u ( s ) ) ) D ξ u ( s ) ] d s + ∫ − ρ t − ρ e A ( t − s − ρ ) P c [ D u F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ v δ ( s ) + ( D u F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u F ˜ δ ( θ s ω , u ( s ) ) ) D ξ u ( s ) ] d s + ∫ + ∞ t e A ( t − s ) P u [ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ v δ ( s ) + ( D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u H ˜ δ ( θ s ω , u ( s ) ) ) D ξ u ( s ) ] d s + ∫ + ∞ t − ρ e A ( t − s − ρ ) P u [ D u F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ v δ ( s ) + ( D u F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u F ˜ δ ( θ s ω , u ( s ) ) ) D ξ u ( s ) ] d s + ∫ − ∞ t e A ( t − s ) P s [ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ v δ ( s ) + ( D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u H ˜ δ ( θ s ω , u ( s ) ) ) D ξ u ( s ) ] d s + ∫ − ∞ t − ρ e A ( t − s − ρ ) P s [ D u F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ v δ ( s ) + ( D u F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u F ˜ δ ( θ s ω , u ( s ) ) ) D ξ u ( s ) ] d s = S δ , 1 + S δ , 2 + S δ , 3 .

We first estimate S δ , 1 ,

e − η ∣ t ∣ ‖ S δ , 1 ‖ ≤ K L ‖ D ξ v δ ( ⋅ ) ‖ ℒ ( E c , C η ) ∣ ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ + η ∣ s ∣ d s ∣ + K ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∣ ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ + ( η − γ ) ∣ s ∣ ‖ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u H ˜ δ ( θ s ω , u ( s ) ) ‖ d s ∣ + K ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∣ ∫ − ρ t − ρ e β 2 ∣ t − s − ρ ∣ − η ∣ t ∣ + ( η − γ ) ∣ s ∣ ‖ D u F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u F ˜ δ ( θ s ω , u ( s ) ) ‖ d s ∣ = S δ , 11 + S δ , 12 + S δ , 13 .

We claim that

(5.17) ‖ S δ , 1 ‖ ℒ ( E c , C η ) ≤ K L ‖ D ξ v δ ( ⋅ ) ‖ ℒ ( E c , C η ) η − β 2 + o ( 1 ) , as δ → 0 + .

Now we prove this claim. First, we estimate S δ , 11 ,

(5.18) sup t ∈ R K L ‖ D ξ v δ ( ⋅ ) ‖ ℒ ( E c , C η ) ∣ ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ + η ∣ s ∣ d s ∣ ≤ K L ‖ D ξ v δ ( ⋅ ) ‖ ℒ ( E c , C η ) η − β 2 .

For the second term S δ , 12 , we show

(5.19) sup t ∈ R K ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∣ ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ + ( η − γ ) ∣ s ∣ ‖ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u H ˜ δ ( θ s ω , u ( s ) ) ‖ d s ∣ = o ( 1 ) .

For any ε > 0 , let T ˜ 1 > 0 be large enough such that

2 K L H ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) η − γ − β 2 e − γ T ˜ 1 < ε .

Then, for ∣ t ∣ ≥ T ˜ 1 , we have

K ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∣ ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ + ( η − γ ) ∣ s ∣ ‖ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u H ˜ δ ( θ s ω , u ( s ) ) ‖ d s ∣ ≤ 2 K L H ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∣ ∫ 0 t e β 2 ∣ t − s ∣ − η ∣ t ∣ + ( η − γ ) ∣ s ∣ d s ∣ ≤ 2 K L H ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) η − γ − β 2 e − γ T ˜ 1 < ε .

For ∣ t ∣ ≤ T ˜ 1 , by (5.15) and Lemma 3.2, we have

v δ ( s ) + u ( s ) + g z δ ( θ s ω ) → u ( s ) + g z ( θ s ω ) , as δ → 0 + .

Together with the continuity of D u H , we have

‖ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u H ˜ δ ( θ s ω , u ( s ) ) ‖ → 0 , as δ → 0 + ,

uniformly with respect to s ∈ [ − T ˜ 1 , T ˜ 1 ] . Thus, there exists δ ′ such that if 0 < δ < δ ′ ,

sup ∣ t ∣ ≤ T ˜ 1 S δ , 12 → 0 , as δ → 0 + .

Hence, we obtain (5.19). Similarly, we can obtain

sup t ∈ R S δ , 13 = o ( 1 ) , as δ → 0 + .

Therefore, (5.17) holds.

Now, we show the second integral S δ , 2 . First, we have

e − η ∣ t ∣ ‖ S δ , 2 ‖ ≤ K L ‖ D ξ v δ ( ⋅ ) ‖ ℒ ( E c , C η ) ∣ ∫ + ∞ t e β 1 ∣ t − s ∣ − η ∣ t ∣ + η ∣ s ∣ d s ∣ + K ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∣ ∫ + ∞ t e β 1 ∣ t − s ∣ − η ∣ t ∣ + ( η − γ ) ∣ s ∣ ‖ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u H ˜ δ ( θ s ω , u ( s ) ) ‖ d s ∣ + K ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∣ ∫ + ∞ t − ρ e β 1 ∣ t − s − ρ ∣ − η ∣ t ∣ + ( η − γ ) ∣ s ∣ ‖ D u F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u F ˜ δ ( θ s ω , u ( s ) ) ‖ d s ∣ = S δ , 21 + S δ , 22 + S δ , 23 .

We claim that

(5.20) ‖ S δ , 2 ‖ ℒ ( E c , C η ) ≤ K L ‖ D ξ v δ ( ⋅ ) ‖ ℒ ( E c , C η ) β 1 − η + o ( 1 ) , as δ → 0 + .

Next, we prove it holds. First, we estimate S δ , 21 ,

(5.21) sup t ∈ R K L ‖ D ξ v δ ( ⋅ ) ‖ ℒ ( E c , C η ) ∣ ∫ + ∞ t e β 1 ∣ t − s ∣ − η ∣ t ∣ + η ∣ s ∣ d s ∣ ≤ K L ‖ D ξ v δ ( ⋅ ) ‖ ℒ ( E c , C η ) β 1 − η .

For the second term S δ , 22 , we show

(5.22) sup t ∈ R K ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∣ ∫ + ∞ t e β 1 ∣ t − s ∣ − η ∣ t ∣ + ( η − γ ) ∣ s ∣ ‖ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u H ˜ δ ( θ s ω , u ( s ) ) ‖ d s ∣ = o ( 1 ) .

For any ε > 0 , let T ˜ 2 > 0 be large enough such that

2 K L H ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) β 1 − η e − γ T ˜ 2 < ε .

For ∣ t ∣ ≥ T ˜ 2 , we have

K ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∣ ∫ + ∞ t e β 1 ∣ t − s ∣ − η ∣ t ∣ + ( η − γ ) ∣ s ∣ ‖ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u H ˜ δ ( θ s ω , u ( s ) ) ‖ d s ∣ ≤ 2 K L H ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∣ ∫ + ∞ t e β 1 ∣ t − s ∣ − η ∣ t ∣ + ( η − γ ) ∣ s ∣ d s ∣ ≤ 2 K L H ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) β 1 − ( η − γ ) e − γ T ˜ 2 < 2 K L H ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) β 1 − η e − γ T ˜ 2 < ε .

For ∣ t ∣ ≤ T ˜ 2 ,

S δ , 22 ≤ K ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∫ + ∞ T ˜ 2 P ˜ ( s ) d s + ∫ T ˜ 2 t P ˜ ( s ) d s ,

where

P ˜ ( s ) = e β 1 ∣ t − s ∣ − η ∣ t ∣ + ( η − γ ) ∣ s ∣ ‖ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u H ˜ δ ( θ s ω , u ( s ) ) ‖ .

Then,

K ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∫ + ∞ T ˜ 2 P ˜ ( s ) d s ≤ 2 K L H ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∣ ∫ + ∞ t e β 1 ∣ t − s ∣ − η ∣ t ∣ + ( η − γ ) ∣ s ∣ d s ∣ ≤ 2 K L H ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) β 1 − η e − γ T ˜ 2 < ε .

By Lemma 3.2 and the continuity of D u H , there exists δ ″ such that if 0 < δ < δ ″ ,

sup ∣ t ∣ ≤ T ˜ 2 K ‖ D ξ u ( ⋅ ) ‖ ℒ ( E c , C η − γ ) ∣ ∫ T ˜ 2 t P ˜ ( s ) d s ∣ → 0 , as δ → 0 + .

By using the same procedure, we can obtain

sup t ∈ R S δ , 23 = o ( 1 ) , as δ → 0 + .

Thus, (5.20) holds. Similarly, we have

(5.23) ‖ S δ , 3 ‖ ℒ ( E c , C η ) ≤ K L ‖ D ξ v δ ( ⋅ ) ‖ ℒ ( E c , C η ) β 1 − η + o ( 1 ) , as δ → 0 + .

By combining (5.17), (5.20), and (5.23), we have

‖ D ξ v δ ( ⋅ ) ‖ ℒ ( E c , C η ) ≤ K L 1 η − β 2 + 2 β 1 − η ‖ D ξ v δ ( ⋅ ) ‖ ℒ ( E c , C η ) + o ( 1 ) .

Since

K L 1 η − β 2 + 2 β 1 − η < 1 ,

we have

‖ D ξ v δ ( ⋅ ) ‖ ℒ ( E c , C η ) = o ( 1 ) , as δ → 0 + .

Namely,

lim δ → 0 + ‖ D ξ u δ ( ⋅ , ω , ξ ) − D ξ u ( ⋅ , ω , ξ ) ‖ ℒ ( E c , C η ) = 0 .

Now we assume that (5.16) holds for i ≤ k − 1 , and then we prove that it holds for i = k .

D ξ k v δ ( t ) = ∫ 0 t e A ( t − s ) P c [ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ k v δ ( s ) + G δ , 1 k ( s ) ] d s + ∫ − ρ t − ρ e A ( t − s − ρ ) P c [ D u F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ k v δ ( s ) + G δ , 2 k ( s ) ] d s + ∫ + ∞ t e A ( t − s ) P u [ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ k v δ ( s ) + G δ , 1 k ( s ) ] d s + ∫ + ∞ t − ρ e A ( t − s − ρ ) P u [ D u F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ k v δ ( s ) + G δ , 2 k ( s ) ] d s + ∫ − ∞ t e A ( t − s ) P s [ D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ k v δ ( s ) + G δ , 1 k ( s ) ] d s + ∫ − ∞ t − ρ e A ( t − s − ρ ) P s [ D u F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ k v δ ( s ) + G δ , 2 k ( s ) ] d s ,

where

G δ , 1 k ( s ) = ∑ l = 0 k − 2 k − 1 l D ξ k − 1 − l D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) D ξ l + 1 v δ ( s ) + ∑ l = 0 k − 1 k − 1 l D ξ k − 1 − l ( D u H ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u H ˜ δ ( θ s ω , u ( s ) ) ) D ξ l + 1 u ( s ) , G δ , 2 k ( s ) = ∑ l = 0 k − 2 k − 1 l D ξ k − 1 − l D u F ˜ δ ( θ s ω , v δ ( s , ) + u ( s ) ) D ξ l + 1 v δ ( s ) + ∑ l = 0 k − 1 k − 1 l D ξ k − 1 − l ( D u F ˜ δ ( θ s ω , v δ ( s ) + u ( s ) ) − D u F ˜ δ ( θ s ω , u ( s ) ) ) D ξ l + 1 u ( s ) .

By using the same procedure as for D ξ v δ , we can obtain

‖ D ξ k v δ ( ⋅ ) ‖ ℒ k ( E c , C k η ) ≤ K L 1 η − β 2 + 2 β 1 − η ‖ D ξ k v δ ( ⋅ ) ‖ ℒ k ( E c , C k η ) + o ( 1 ) .

Since

K L 1 η − β 2 + 2 β 1 − η < 1 ,

we have

‖ D ξ k v δ ( ⋅ ) ‖ ℒ k ( E c , C k η ) = o ( 1 ) , as δ → 0 + .

Namely,

lim δ → 0 + ‖ D ξ k u δ ( ⋅ , ω , ξ ) − D ξ k u ( ⋅ , ω , ξ ) ‖ ℒ k ( E c , C k η ) = 0 .

This completes the proof of the theorem.□

Let M ˜ δ c ( ω ) = T δ − 1 ( ω , M δ c ( ω ) ) . Then M ˜ δ c ( ω ) is an invariant manifold for equation (3.2) as δ > 0 (resp. (3.1)). Moreover,

M ˜ δ c ( ω ) = T δ − 1 ( ω , M δ c ( ω ) ) = { T δ − 1 ( ω , ξ + h δ c ( ω , ξ ) ) ∣ ξ ∈ E c } = { ξ + h δ c ( ω , ξ ) + g z δ ( ω ) ∣ ξ ∈ E c } = { ξ + h ˜ δ c ( ω , ξ ) ∣ ξ ∈ E c } ,

where h ˜ δ c ( ω , ξ ) = h δ c ( ω , ξ ) + g z δ ( ω ) . We set h 0 c ( ω ) = h c ( ω ) , and M ˜ 0 c ( ω ) = M ˜ c ( ω ) .

The following theorem shows that M ˜ c ( ω ) can be approximated by M ˜ δ c ( ω ) as δ → 0 + .

Theorem 5.2

Assume that the same conditions in Theorem 5.1  hold. Then we have

lim δ → 0 + h ˜ δ c ( ω , ξ ) = h ˜ c ( ω , ξ ) ,

and

lim δ → 0 + D ξ i h ˜ δ c ( ω , ξ ) = D ξ i h ˜ c ( ω , ξ ) , 1 ≤ i ≤ k ,

where ξ ∈ E c , h ˜ 0 c ( ω , ξ ) = h ˜ c ( ω , ξ ) .

Proof

Note that

h δ c ( ω , ξ ) = P u u δ ( 0 , τ , ω , ξ ) + P s u δ ( 0 , τ , ω , ξ ) .

By Theorem 5.1, we have

∣ h δ c ( ω , ξ ) − h c ( ω , ξ ) ∣ → 0 , as δ → 0 + ,

and by using Lemma 3.2, we can obtain

∣ h ˜ δ c ( ω , ξ ) − h ˜ c ( ω , ξ ) ∣ ≤ ∣ h δ c ( ω , ξ ) − h δ c ( ω , ξ ) ∣ + ∣ g z δ ( ω ) − g z ( ω ) ∣ → 0 , as δ → 0 + .

By using the similar argument, for each 1 ≤ i ≤ k ,

‖ D ξ i h δ c ( ω , ξ ) − D ξ i h c ( ω , ξ ) ‖ ≤ ‖ P u ‖ ‖ D ξ i u δ ( ⋅ , τ , ω , ξ ) − D ξ i u ( ⋅ , τ , ω , ξ ) ‖ + ‖ P s ‖ ‖ D ξ i u δ ( ⋅ , τ , ω , ξ ) − D ξ i u ( ⋅ , τ , ω , ξ ) ‖ → 0 , as δ → 0 + .

Thus,

lim δ → 0 + D ξ i h ˜ δ c ( ω , ξ ) = D ξ i h ˜ c ( ω , ξ ) .

This completes the proof.□

Acknowledgement

The authors would like to thank the reviewers for their helpful comments. This work was partially supported by the National Natural Science Foundation of China (No. 11871138), Sichuan Science and Technology Program (No. 2023NSFSC0076).

  1. Conflict of interest: The authors state no conflict of interest.

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Received: 2022-05-23
Revised: 2023-01-30
Accepted: 2023-02-08
Published Online: 2023-03-17

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

  1. Regular Articles
  2. On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
  3. On the critical Choquard-Kirchhoff problem on the Heisenberg group
  4. On the local behavior of local weak solutions to some singular anisotropic elliptic equations
  5. Viscosity method for random homogenization of fully nonlinear elliptic equations with highly oscillating obstacles
  6. Double-phase parabolic equations with variable growth and nonlinear sources
  7. Logistic damping effect in chemotaxis models with density-suppressed motility
  8. Bifurcation diagrams of one-dimensional Kirchhoff-type equations
  9. Standing wave solution for the generalized Jackiw-Pi model
  10. Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
  11. Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order
  12. Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian
  13. Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
  14. Bautin bifurcation with additive noise
  15. Small solitons and multisolitons in the generalized Davey-Stewartson system
  16. Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
  17. A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
  18. On a strongly damped semilinear wave equation with time-varying source and singular dissipation
  19. Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals
  20. Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension
  21. Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects
  22. Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system
  23. Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices
  24. Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition
  25. Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity
  26. Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
  27. On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1
  28. Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications
  29. Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
  30. Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
  31. Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth
  32. Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model
  33. Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
  34. Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
  35. Approximations of center manifolds for delay stochastic differential equations with additive noise
  36. Periodic solutions to a class of distributed delay differential equations via variational methods
  37. Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
  38. Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
  39. Global Sobolev regular solution for Boussinesq system
  40. Normalized solutions for the p-Laplacian equation with a trapping potential
  41. Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
  42. Blow-up for compressible Euler system with space-dependent damping in 1-D
  43. High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
  44. On the dynamics of grounded shallow ice sheets: Modeling and analysis
  45. A survey on some vanishing viscosity limit results
  46. Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
  47. Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
  48. Front propagation in a double degenerate equation with delay
  49. Positive solutions for a class of singular (p, q)-equations
  50. Higher integrability for anisotropic parabolic systems of p-Laplace type
  51. The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
https://doi.org/10.1515/anona-2022-0301
Schlagwörter für diesen Artikel
delay equation; center manifold; random dynamical system; additive noise
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Regular Articles
  2. On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
  3. On the critical Choquard-Kirchhoff problem on the Heisenberg group
  4. On the local behavior of local weak solutions to some singular anisotropic elliptic equations
  5. Viscosity method for random homogenization of fully nonlinear elliptic equations with highly oscillating obstacles
  6. Double-phase parabolic equations with variable growth and nonlinear sources
  7. Logistic damping effect in chemotaxis models with density-suppressed motility
  8. Bifurcation diagrams of one-dimensional Kirchhoff-type equations
  9. Standing wave solution for the generalized Jackiw-Pi model
  10. Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
  11. Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order
  12. Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian
  13. Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
  14. Bautin bifurcation with additive noise
  15. Small solitons and multisolitons in the generalized Davey-Stewartson system
  16. Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
  17. A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
  18. On a strongly damped semilinear wave equation with time-varying source and singular dissipation
  19. Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals
  20. Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension
  21. Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects
  22. Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system
  23. Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices
  24. Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition
  25. Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity
  26. Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
  27. On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1
  28. Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications
  29. Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
  30. Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
  31. Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth
  32. Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model
  33. Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
  34. Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
  35. Approximations of center manifolds for delay stochastic differential equations with additive noise
  36. Periodic solutions to a class of distributed delay differential equations via variational methods
  37. Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
  38. Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
  39. Global Sobolev regular solution for Boussinesq system
  40. Normalized solutions for the p-Laplacian equation with a trapping potential
  41. Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
  42. Blow-up for compressible Euler system with space-dependent damping in 1-D
  43. High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
  44. On the dynamics of grounded shallow ice sheets: Modeling and analysis
  45. A survey on some vanishing viscosity limit results
  46. Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
  47. Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
  48. Front propagation in a double degenerate equation with delay
  49. Positive solutions for a class of singular (p, q)-equations
  50. Higher integrability for anisotropic parabolic systems of p-Laplace type
  51. The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
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