NEUTRAL FUNCTIONAL EQUATIONS WITH INFINITE DELAY. Sichuan University Chengdu , P.R. CHINA. Xidian University Xi an , P.R.

S ²

γ -WPAA FUNCTIONS AND APPLICATIONS TO STOCHASTIC

NEUTRAL FUNCTIONAL EQUATIONS WITH INFINITE DELAY

CHAO TANG ¹ AND YONG-KUI CHANG ²

1 College of Mathematics Sichuan University Chengdu 610064, P.R. CHINA

2 School of Mathematics and Statistics Xidian University

Xi’an 710071, P.R. CHINA

ABSTRACT: In this paper, we introduce the concept of S _γ ² -weighted pseudo almost automorphy. And then, we investigate some basic properties such as completeness of spaces, ergodic and composition theorems of such stochastic processes. Finally, by virtue of theories of evolution systems, fading phase spaces for infinite delay and the stochastic analysis techniques, we apply the results obtained to consider the existence and uniqueness results of weighted pseudo almost automorphic solutions in distribu- tion to a class of nonautonomous stochastic neutral functional equations with infinite delay under S _γ ² -weighted pseudo almost automorphic coefficients in real separable Hilbert spaces.

AMS Subject Classification: 34C27, 34D20, 34F05, 60H30

Key Words: S _γ ² -weighted pseudo almost automorphy, non-autonomous stochastic neutral functional equations, infinite delay

Received: March 31, 2017 ; Accepted: May 5, 2018 ; Published: June 7, 2018. doi: 10.12732/dsa.v27i3.4

Dynamic Publishers, Inc., Acad. Publishers, Ltd. https://acadsol.eu/dsa

1. INTRODUCTION

The concept of almost automorphy is an important generalization of the classical

almost periodicity. It was introduced by S. Bochner [4, 5], for more details about

this topics we refer the reader to [9, 22, 23]. In recent years, the existence of al- most periodic and almost automorphic solutions on different kinds of determinis- tic differential equations have been considerably investigated in lots of publications [1, 3, 11, 12, 21, 20, 24, 34, 35, 36] because of their significance and applications in physics, mechanics and mathematical biology. Recently, Diagana [10] presented the notion of S _γ ^p -pseudo almost automorphy(or generalized Stepanov-like pseudo al- most automorphy), which generalizes the well-known concept of S ^p -pseudo almost automorphy.

Stochastic differential equations arise in the mathematical modeling of many phe- nomena in real world problems (see [2, 18, 19, 25, 26, 27, 28, 30, 31, 33, 37]). More recently, Fu and Liu [14] generalized the almost automorphic theory from the de- terministic version to the stochastic one and studied the square-mean almost auto- morphic mild solution for the stochastic differential equations. Chen and Lin [7, 8]

investigated the existence, uniqueness and stability of the square-mean (weighted) pseudo almost automorphic solutions for some stochastic differential equations via the Banach fixed point theorem and the stochastic analysis techniques. Chang et al [6, 32] investigated the existence of S ² -almost automorphic and S ² -weighted pseudo almost automorphic mild solutions for a stochastic differential equation in a real sep- arable Hilbert space with the help of composition theorems together with fixed point theorems. As far as we know, however, there have been very few applicable results on S _γ ² -weighted pseudo almost automorphy for stochastic processes compared with the concept in [10] in deterministic sense.

From above mentioned works, in this paper we introduce a concept of S _γ ² -weighted pseudo almost automorphy for stochastic processes and investigate some basic proper- ties such as completeness of spaces, ergodic and composition theorems of S _γ ² -weighted pseudo almost automorphic stochastic processes. Finally, by virtue of theories of evolution systems, fading phase spaces for infinite delay and the stochastic analysis techniques, we establish the existence and uniqueness results of weighted pseudo al- most automorphic solutions in distribution to a class of non-autonomous stochastic neutral differential equations with infinite delay in the abstract form

d[u(t) + f (t, u t )] = [A(t)u(t) + g(t, u t )]dt + σ(t, u t )dW (t), t ∈ R, (1.1)

where A(t) : D(A(t)) ⊂ L ² (P, H) → L ² (P, H) is a family of densely defined closed lin-

ear operators satisfying the Acquistapace-Terrani conditions, the coefficients f, g, σ :

R × B → L ² (P, H) are appropriate functions, B is a uniform fading memory phase

space defined in the next section. Also, the history u t : (−∞, 0] → L ² (P, H), defined

by u t (θ) = u(t + θ) for each θ ∈ (−∞, 0]. Further, W (t) is a two-sided standard one

dimensional Brownian motion defined on the filtered probability space (Ω, F , (F t ), P )

where F t = σ{W (u) − W (v); u, v ≤ t}.

The rest of this paper is organized as follows. In Section 2, we present some basic definitions, lemmas and preliminary facts. In Section 3, we introduce the concept of S _γ ² -weighted pseudo almost automorphy for stochastic processes and prove some fundamental properties of such stochastic processes. In Section 4, we investigate the existence and uniqueness results of weighted pseudo almost automorphic solutions in distribution to the Eq (1.1) with S ² _γ -weighted pseudo almost automorphic coefficients.

2. PRELIMINARIES

In this section, we fix some basic definitions, lemmas and preliminary facts which will be used in the sequel. Throughout the paper, we assume that (H, k · k, h·, ·i) and (K, k · k ^K , h·, ·i K ) are two real and separable Hilbert spaces, (Ω, F , (F t ), P ) is supposed to be a probability space and L ² (P, H) stands for the space of all H-valued random variables x such that Ekxk ² = R

Ω kxk ² dP < ∞. Note that L ² (P, H) is a Hilbert space when it is equipped with the norm kxk 2 = (Ekxk ² )

¹²

. Let L(K, H) be the space of all bounded linear operators from K to H and this is denoted by L(H) when K = H.

We let C(R, L ² (P, H)) (respectively, C(R × L ² (P, H), L ² (P, H))) denote the collec- tion of all continuous stochastic processes from R into L ² (P, H) (respectively, the col- lection of all jointly continuous stochastic processes from R × L ² (P, H) into L ² (P, H)).

Furthermore, BC(R, L ² (P, H)) (respectively, BC(R × L ² (P, H), L ² (P, H)) stands for the class of all bounded continuous stochastic processes from R into L ² (P, H) (re- spectively, the class of all jointly bounded continuous stochastic processes from R × L ² (P, H) into L ² (P, H). Note that BC(R, L ² (P, H)) is a Banach space with the sup norm

kxk ∞ = sup

t∈R

(Ekx(t)k ² )

¹²

.

A Brownian motion plays a key role in the construction of stochastic integrals.

Definition 2.1. A (standard one-dimensional) Brownian motion is a continuous adapted real-valued process (W (t), t ≥ 0) such that

(i) W (0) = 0;

(ii) W (t) − W (s) is independent of F s for all 0 ≤ s < t;

(iii) W (t) − W (s) is N (0, t − s)-distributed for all 0 ≤ s ≤ t.

Note that the Brownian motion W has the following properties:

(a) W has independent increments, that is, for t 1 < t 2 < · · · < t n , W (t 1 ) −

W (0), W (t 2 ) − W (t 1 ), . . . W (t n ) − W (t n−1 ) are independent random variables;

(b) W has stationary increments, that is, W (t+s)−W (t) has the same distribution as W (s) − W (0).

Definition 2.2. [25] Let V = V(S, T ) be the class of functions f (t, ω) : [0, ∞) × Ω → R such that

(i) (t, ω) → f (t, ω) is B × F -measurable, where B denotes the Borel σ-algebra on [0, ∞).

(ii) f (t, ω) is F t -adapted.

(iii) E hR T

s f (t, ω) ² dt i

< ∞.

Definition 2.3. [25] Let f ∈ V(S, T ). Then the Itˆ o integral of f (from S to T ) is defined by

Z T S

f (t, ω)dW t (ω) = lim

n→∞

Z T S

f n (t, ω)dW t (ω) (limit in L ² (P)), where f n is a sequence of elementary functions such that

E

"Z T S

(f (t, ω) − f n (t, ω)) ² dt

#

→ 0 as n → ∞,

and W t is one-dimensional Brownian motion. Moreover, we have the following Itˆo isometry, for all f ∈ V(S, T ),

E



 Z T S

f (t, ω)dW t (ω)

! 2 

 = E "Z T S

f ² (t, ω)dt

# .

Definition 2.4. [14] A stochastic process x : R → L ² (P, H) is said to be stochasti- cally continuous if

t→s lim Ekx(t) − x(s)k ² = 0.

Definition 2.5. [14] A stochastically continuous stochastic process x : R → L ² (P, H) is said to be square-mean almost automorphic if for every sequence of real numbers there exists a subsequence {s n } and a stochastic process y : R → L ² (P, H) such that

n→∞ lim Ekx(t + s n ) − y(t)k ² = 0 and lim

n→∞ Eky(t − s n ) − x(t)k ² = 0,

hold for each t ∈ R. The collection of all square-mean almost automorphic stochastic processes x : R → L ² (P, H) is denoted by AA(R, L ² (P, H)).

Definition 2.6. [14] A stochastic process f : R × L ² (P, H) → L ² (P, H), (t, x) →

f (t, x), which is jointly continuous, is said to be square-mean almost automorphic in

t ∈ R for each x ∈ L ² (P, H) if for every sequence of real numbers {s ^′ _n }, there exists a subsequence {s n } and a stochastic process e f : R × L ² (P, H) → L ² (P, H) such that

n→∞ lim Ekf (t + s n , x) − e f (t, x)k ² = 0, lim

n→∞ Ek e f (t − s n , x) − f (t, x)k ² = 0, for each t ∈ R and each x ∈ L ² (P, H). The collection of such process is denoted by AA(R × L ² (P, H), L ² (P, H)).

Lemma 2.1. [14] (AA(R, L ² (P, H)), k · k ∞ ) is a Banach space when it is equipped with the norm k · k ∞ , for x ∈ AA(R, L ² (P, H)).

Let V denote the set of all functions ρ : R → (0, ∞), which are locally integrable over R such that ρ > 0 almost everywhere. For a given q > 0 and for each ρ ∈ V, we set m(q, ρ) := R q

−q ρ(t)dt.

Thus the space of weights V ∞ is defined by V _∞ := {ρ ∈ V : lim

r→∞ m(q, ρ) = ∞}.

Now for ρ ∈ V ∞ , we define a class of stochastic process P AA 0 (R, ρ)

:=



f ∈ BC(R, L ² (P, H)) : lim

q→∞

1 m(q, ρ)

Z q

−q

Ekf (t)k ² ρ(t)dt = 0

 ,

P AA 0 (R × L ² (P, H), ρ) :=



f ∈ C(R × L ² (P, H), L ² (P, H)) : f (·, x) is bounded for each x ∈ L ² (P, H) and lim

q→∞

1 m(q, ρ)

Z q

−q

Ekf (t, x)k ² ρ(t)dt = 0

 .

To study issues related to delay terms, we consider the new space of functions defined for each p > 0 by

P AA 0 (R, ρ, p) :=



f ∈ P AA 0 (R, ρ) : lim

q→∞

1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

(Ekf (θ)k ² )ρ(t)dt = 0

 ,

P AA 0 (R × L ² (P, H), ρ, p) :=



f ∈ P AA 0 (R × L ² (P, H), ρ) : f (·, x) is bounded for each x ∈ L ² (P, H) and lim

q→∞

1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

(Ekf (θ, x)k ² )ρ(t)dt = 0

 . In view of the previous definitions it is clear that P AA 0 (R, ρ, p) and P AA 0 (R × L ² (P, H), ρ, p) are continuously embedded and closed in the P AA 0 (R, ρ) and P AA 0 (R×

L ² (P, H), ρ), respectively.

Definition 2.7. [8] Let ρ ∈ V ∞ . A stochastically continuous process f : R → L ² (P, H) is said to be square-mean weighted pseudo almost automorphic provided that it can be decomposed as f = h + ϕ, where h ∈ AA(R, L ² (P, H)) and ϕ ∈ P AA 0 (R, ρ).

The collection of all such processes is denoted by W P AA(R, L ² (P, H)).

Definition 2.8. [8] Let ρ ∈ V ∞ . A stochastically continuous process f : R × L ² (P, H) → L ² (P, H) is said to be square-mean weighted pseudo almost automorphic in t for any x ∈ L ² (P, H) provided that it is decomposed as f = h + ϕ, where h ∈ AA(R × L ² (P, H), L ² (P, H)) and ϕ ∈ P AA 0 (R × L ² (P, H), ρ). Also denote by W P AA(R×L ² (P, H), L ² (P, H)) the set of all such stochastically continuous processes.

Lemma 2.2. [8] For some ρ ∈ V ∞ , P AA 0 (R, ρ) may be translation invariant. In fact, if ρ satisfies conditions:

t→∞ lim sup ρ(t + τ )

ρ(t) < ∞ and lim

r→∞ sup m(r + τ, ρ) m(r, ρ) < ∞,

for every τ ∈ R, one can validate that P AA 0 (R, ρ) may be translation invariant.

Lemma 2.3. [8] W P AA(R, L ² (P, H)) equipped with the norm k · k ∞ becomes a Banach space if P AA 0 (R, ρ) is translation invariant.

Lemma 2.4. [8] Suppose ρ satisfies the conditions in Lemma 2.2, f (t, x) ∈ W P AA(R×

L ² (P, H), L ² (P, H)) and there exists a number L > 0 such that for any x, y ∈ L ² (P, H), Ekf (t, x) − f (t, y)k ² ≤ LEkx − yk ² , t ∈ R.

Then for any x(·) ∈ W P AA(R, L ² (P, H)), then f (·, x(·)) ∈ W P AA(R, L ² (P, H)).

From [13] and [19], we define P(H) the space of all Borel probability measures on H with the β metric:

β(µ, η) := sup

    Z

f dµ − Z

f dη  

  : kf k ^BL ≤ 1



, µ, η ∈ P(H), where f are Lipschitz continuous real-valued functions on H with

kf k BL = kf k L + kf k ∞ , kf k L = sup

x6=y

|f (x) − f (y)|

kx − yk , kf k ∞ = sup

x∈H

|f (x)|.

Definition 2.9. [19] An H-valued stochastic process u(t) is said to be almost auto- morphic in distribution if its law µ(t) is a P(H)- valued almost automorphic mapping, i.e. for every sequence of real numbers {s ^′ _n }, there exist a subsequence {s n } and a P(H)- valued mapping e µ(t) such that

n→∞ lim β(µ(t + s n ), e µ(t))) = 0 and lim

n→∞ β (e µ(t − s n ), µ(t))) = 0,

hold for each t ∈ R.

Definition 2.10. [17] An H-valued stochastic process f (t) is said to be weighted pseudo almost automorphic in distribution with respect to ρ ∈ V ∞ , provided that it can be decomposed as f = h + ϕ, where h is almost automorphic in distribution and ϕ ∈ P AA 0 (R, ρ).

We now introduce positively bi-almost automorphic functions. For that, let T be the set defined by:

T := {(t, s) ∈ R × R : t ≥ s)}.

Definition 2.11. [10] A stochastically process L : T → L ² (P, H) is called positively square-mean bi-almost automorphic if for every sequence of real numbers (s ^′ _n ) n∈N , we can extract a subsequence (s n ) n∈N such that for some stochastic processes H : T → L ² (P, H)

n→∞ lim EkL(t + s n , s + s n ) − H(t, s)k ² = 0

n→∞ lim EkH(t − s n , s − s n ) − L(t, s)k ² = 0

for each (t, s) ∈ T. The collection of such processes will be denoted by bAA(T, L ² (P, H)).

Definition 2.12. [2] The Bochner transform x ^b (t, s), t ∈ R, s ∈ [0, 1] of a stochastic process x : R → L ² (P, H) is defined by

x ^b (t, s) := x(t + s).

Remark 2.1. (i) A function ϕ(t, s), t ∈ R, s ∈ [0, 1], is the Bochner transform of a certain function f , ϕ(t, s) = f ^b (t, s), if and only if ϕ(t + τ, s − τ ) = ϕ(t, s) for all t ∈ R, s ∈ [0, 1] and τ ∈ [s − 1, s].

(ii) Note that if f = h + ϕ, then f ^b = h ^b + ϕ ^b . Moreover, (λf ) ^b = λf ^b for each scalar λ.

Definition 2.13. [2] The Bochner transform f ^b (t, s, u), t ∈ R, s ∈ [0, 1], u ∈ L ² (P, H) of a stochastic process f : R × L ² (P, H) → L ² (P, H) is defined by

f ^b (t, s, u) := f (t + s, u) for each u ∈ L ² (P, H).

Now, we recall the definition of fading memory space (phase space) B axiomatically presented in [15, 21]. Let B denote the vector space of function x t : (−∞, 0] → L ² (P, H), defined as x t (s) = x(t + s) for s ∈ R ⁻ , endowed with a seminorm denoted by k · k B . A Banach space (B, k · k B ) which consists of such functions ψ : (−∞, 0] → L ² (P, H), is called a fading memory space, if it satisfies the following axioms.

(ai) If x : (−∞, r + a) → L ² (P, H) with a > 0, r ∈ R, is continuous on [r, r + a) and

x r ∈ B, then for each t ∈ [r, r + a) the following conditions hold:

(i) x t ∈ B,

(ii) kx(t)k ≤ Lkx t k B ,

(iii) kx t k B ≤ sup{G(t − r)kx(s)k : r ≤ s ≤ t} + N (t − r)kx r k B ,

where L > 0 is a constant, and G, N : [0, ∞) → [1, ∞) are functions such that G(·) and N (·) are respectively continuous and locally bounded, and L, G, N are independent of x(·).

(aii) If x(·) is a function as in (ai), then x t is a B valued continuous function on [r, r + a).

(aiii) The space B is complete.

(aiv) If (φ ⁿ ) n∈N is a sequence of continuous functions with compact support defined from (−∞, 0] into L ² (P, H), which converges to φ uniformly on compact subsets of (−∞, 0] and if {φ ⁿ } is a cauchy sequence in B, then φ ∈ B and φ ⁿ → φ in B.

Definition 2.14. [21] Let S(t) : B → B be a C 0 semigroup defined by S(t)φ(θ) = φ(0) on [−t, 0] and S(t)φ(θ) = φ(t + θ) on (−∞, −t]. The phase space B is called a fading memory space if kS(t)φk B → 0 as t → ∞ for each φ ∈ B with φ(0) = 0.

Also, by axiom (aiv), there exists a constant ℜ > 0 such that kφk B ≤ ℜsup

θ≤0

kφ(θ)k for every φ ∈ B bounded continuous. Moreover, if B is a fading memory, we assume that max{G(t), N (t))} ≤ K for t ≥ 0. Further, it should be mentioned that the phase B is a uniform fading memory space if and only if axiom (aiv) holds, the function G is bounded and lim

t→∞ N (t) = 0. For more details please refer the article [21].

Lemma 2.5. [18] Let x : (−∞, r + a) → L ² (P, H) be an F t -adapted measurable process such that the F 0 -adapted process x 0 = φ ∈ L ² ₀ (Ω, B), then

Ekx s k B ≤ DEkφk B + ℑsup

s∈R

Ekx(s)k, where D = sup

t∈R

{N (t)} and ℑ = sup

t∈R

{G(t)}.

3. GENERALIZED STEPANOV-LIKE WPAA STOCHASTIC PROCESSES

In this section, we shall introduce the concept of S _γ ² -weighted pseudo almost automor- phy for stochastic processes and give some fundamental properties of such stochastic processes.

Let U denote the collection of all measurable (weighted) functions γ : (0, ∞) → (0, ∞) satisfying

γ 0 := lim

ε→0

Z 1 ε

γ(σ)dσ = Z 1

0

γ(σ)dσ < ∞. (3.1)

Moreover, we let U ∞ be the collection of all functions γ ∈ U, which are differentiable.

Define the set of weights

U ⁺ _∞ := {γ ∈ U ∞ : dγ

dt > 0 for all t ∈ (0, ∞)},

U ⁻ _∞ := {γ ∈ U ∞ : dγ

dt < 0 for all t ∈ (0, ∞)}.

In addition to the above, we define the set of weights U _B := {γ ∈ U : sup

t∈(0,∞)

γ(t) < ∞}.

Throughout the rest of the paper, we suppose that γ satisfies

t∈(0,∞) inf γ(t) = m 0 > 0.

Definition 3.1. [10] Let µ, ν ∈ U ∞ . One says that µ is equivalent to ν and denote it µ ≺ ν, if ^µ _ν ∈ U B .

Remark 3.1. [10] Let µ, ν, γ ∈ U ∞ . Note that µ ≺ µ (reflexivity). If µ ≺ ν, then ν ≺ µ (symmetry). If µ ≺ ν and ν ≺ γ, then µ ≺ γ (transitivity). Therefore, ≺ is a binary equivalence relation on U ∞ .

We now introduce the space BS _γ ² (L ² (P, H)) of all generalized Stepanov spaces as follows.

Definition 3.2. [10] Let γ ∈ U. The space BS _γ ² (L ² (P, H)) of all generalized Stepanov bounded stochastic processes consists of all γds-measurable stochastic pro- cesses f : R → L ² (P, H) such that f ^b ∈ L ^∞ (R, L ² (0, 1; L ² (P, H), γds)). This is a Banach space with the norm

kf k _S

2

γ

: = sup

t∈R

Z t+1 t

γ(s − t)Ekf (s)k ² ds



¹2

= sup

t∈R

Z 1 0

γ(s)Ekf (s + t)k ² ds



¹₂

.

Definition 3.3. [10] Let γ ∈ U. A stochastic process f ∈ BS _γ ² (L ² (P, H)) is called S _γ ² -almost automorphic if f ^b ∈ AA(R, L ² (0, 1; L ² (P, H), γds)). In other words, a stochastic process f ∈ L ² _loc (R, γds) is said to be S _γ ² -almost automorphic if its Bochner transform f ^b : R → L ² (0, 1; L ² (P, H), γds) is square-mean almost automorphic in the sense that for every sequence of real numbers {s ^′ _n } there exists a subsequence {s n } and a stochastic process g ∈ L ² _loc (R, γds) such that

Z t+1 t

γ(s − t)Ekf (s + s n ) − g(s)k ² ds = Z 1

0

γ(s)Ekf (s + t + s n ) − g(s + t)k ² ds → 0,

Z t+1 t

γ(s − t)Ekg(s − s n ) − f (s)k ² ds = Z 1

0

γ(s)Ekg(s + t − s n ) − f (s + t)k ² ds → 0, as n → ∞ pointwise on R. The collection of all such processes will be denoted by AS _γ ² (R, L ² (P, H)).

Definition 3.4. [10] Let γ ∈ U. A process F : R × L ² (P, H) → L ² (P, H), (t, x) → F (t, x) with F (·, x) ∈ L ² _loc (R, γds) for each x ∈ L ² (P, H), is said to be S _γ ² -almost automorphic in t ∈ R uniformly in x ∈ L ² (P, H) if t → F (t, x) is S _γ ² -almost automorphic for each x ∈ L ² (P, H). That means, for every sequence of real numbers {s ^′ _n } there exist a subsequence {s n } and a process G(·, x) ∈ L ² _loc (R, γds) such that

Z t+1 t

γ(s − t)EkF (s + s n , x) − G(s, x)k ² ds → 0, Z t+1

t

γ(s − t)EkG(s − s n , x) − F (s, x)k ² ds → 0,

as n → ∞ pointwise on R and for each x ∈ L ² (P, H). We denote by AS _γ ² (R × L ² (P, H), L ² (P,

H )) the set of all such processes.

Similarly, as in Ding et al [12], for each K ⊂ L ² (P, H) compact subset, we denote by AS _γ,K ² (R × L ² (P, H), L ² (P, H)) the collection of all functions f ∈ AS _γ,K ² (R × L ² (P, H), L ² (P, H)) satisfying that for every sequence of real numbers (s ^′ _n ) n∈N , there exists a subsequence (s n ) n∈N and a function G : R × L ² (P, H) → L ² (P, H) with G(·, x) ∈ L ² _loc (R, γds) such that

Z 1 0

γ(s)

 sup

x∈K

EkF (s + t + s n , x) − G(s + t, x)k

 2

ds → 0, Z 1

0

γ(s)

 sup

x∈K

EkG(s + t − s _n , x) − F (s + t, x)k

 2

ds → 0, as n → ∞ for each t ∈ R.

Lemma 3.1. [10] Let f ∈ AS _γ ² (R × L ² (P, H), L ² (P, H)) and suppose f is Lipschitz, that is, there exists L > 0 such that for all x, y ∈ L ² (P, H) and t ∈ R

Ekf (t, x) − f (t, y)k ² ≤ LEkx − yk ² (3.2) Then for every K ⊂ L ² (P, H) a compact subset, the process

f ∈ AS _γ,K ² (R × L ² (P, H), L ² (P, H)).

Lemma 3.2. [10] Suppose ϕ ∈ AS _γ ² (R × L ² (P, H), L ² (P, H)) such that K = {ϕ(t) : t ∈ R} ⊂ L ² (P, H)) is a compact subset. If the process

F ∈ AS _γ ² (R × L ² (P, H), L ² (P, H))

and satisfies the Lipschitz condition (3.2), then it holds t → F (t, ϕ(t)) belongs to AS _γ ² (R × L ² (P, H), L ² (P, H)).

Let γ ∈ U and ρ ∈ V ∞ . We now introduce the new concept of S _γ ² -weighted pseudo almost automorphy for stochastic processes.

Definition 3.5. A stochastic process f ∈ BS _γ ² (L ² (P, H)) is called S _γ ² -weighted pseudo almost automorphic if it can be expressed as f = h+ϕ, where h ∈ AS _γ ² (R, L ² (P , H )) and ϕ ^b ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ). The collection of such processes will be denoted by W P AAS _γ ² (R, L ² (P, H)).

Definition 3.6. A stochastic process F : R × L ² (P, H) → L ² (P, H), (t, x) → F (t, x) with F (·, x) ∈ L ² _loc (R, γds) for each x ∈ L ² (P, H), is said to be S _γ ² -weighted pseudo almost automorphic in t ∈ R uniformly in x ∈ L ² (P, H) if it can be expressed as F = H + Φ, where H ∈ AS _γ ² (R × L ² (P, H), L ² (P, H)) and Φ ^b ∈ P AA 0 (R × L ² (P, H), L ² (0, 1; L ² (P, H), γds), ρ). The collection of such processes will be denoted by W P AAS _γ ² (R × L ² (P, H), L ² (P, H)).

Lemma 3.3. If f ∈ W P AA(R, L ² (P, H)), then f ∈ W P AAS _γ ² (R, L ² (P, H)). In other words, W P AA(R, L ² (P, H)) ⊂ W P AAS _γ ² (R, L ² (P, H)).

Lemma 3.4. Let γ, ν ∈ U. If γ ≺ ν, then it holds W P AAS _γ ² (R, L ² (P, H)) = W P AAS _ν ² (R, L ² (P, H)).

The proofs of the Lemmas 3.3 and 3.4 are straightforward and hence omitted.

Lemma 3.5. If φ ^b (·) ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ) and ρ satisfies the con- ditions in Lemma 2.2, then for any τ ∈ R, it holds that

φ ^b (· − τ ) ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ).

Proof. Let φ ^b (·) ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ), for the arbitrary τ ∈ R, one has

1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

Z θ+1 θ

γ(s − θ)Ekφ(s − τ )k ² ds

!!

ρ(t)dt

= 1

m(q, ρ) Z q

−q

sup

θ∈[t−p,t]

Z 1 0

γ(s)Ekφ(s + θ − τ )k ² ds

!

ρ(t)dt

= 1

m(q, ρ) Z q

−q

sup

θ∈[t−p−τ,t−τ ]

Z 1 0

γ(s)Ekφ(s + θ)k ² ds

!

ρ(t)dt

= 1

m(q, ρ) Z q−τ

−q−τ

sup

θ∈[t−p,t]

Z 1 0

γ(s)Ekφ(s + θ)k ² ds

!

ρ(t + τ )dt

≤ 1 m(q, ρ)

Z q−|τ |

−q+|τ |

sup

θ∈[t−p,t]

Z 1 0

γ(s)Ekφ(s + θ)k ² ds

!

ρ(t + τ )dt

≤ lim

t→∞ sup ρ(t + τ ) ρ(t) × lim

r→∞ sup m(q + |τ |, ρ) m(q, ρ)

× 1

m(q + |τ |, ρ) Z q−|τ |

−q+|τ |

sup

θ∈[t−p,t]

Z 1 0

γ(s)Ekφ(s + θ)k ² ds

!

ρ(t)dt,

which implies that

q→∞ lim 1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

Z θ+1 θ

γ(s − θ)Ekφ(s − τ )k ² ds

!!

ρ(t)dt = 0,

that is φ ^b (· − τ ) ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ). This completes the proof.

Theorem 3.1. Let γ ∈ U and P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ) be translation invariant. The space W P AAS _γ ² (R, L ² (P, H)) equipped with the norm k · k _S

2

γ

is a Banach space.

Proof. Let (f n ) n∈N be a Cauchy sequence in W P AAS _γ ² (R, L ² (P, H)). And let (h n ) n∈N , (ϕ n ) n∈N be sequences such that f n = h n +ϕ n , where (h n ) n∈N ⊂ AS _γ ² (R, L ² (P, H)) and (ϕ ^b _n ) n∈N ⊂ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ). Using similar ideas as in the proof of [35, Lemma 3.1] it can be shown that the following holds

kh n k _S

2

γ

≤ kf n k _S

2 γ

,

for all n ∈ N. Thus there exists a function h ∈ AS _γ ² (R, L ² (P, H)) such that kh n − hk _S

2

γ

→ 0 as n → ∞. Using the previous fact, it easily follows that there exists a function ϕ ∈ BS _γ ² (L ² (P, H)) such that kϕ n − ϕk _S

2

γ

→ 0 as n → ∞. Now, for q > 0, we obtain

1 m(q, ρ)

Z q

−q

Z t+1 t

γ(s − t)Ekϕ(s)k ² ds

 ρ(t)dt

≤ 1

m(q, ρ) Z q

−q

Z t+1 t

γ(s − t)Ekϕ n (s) − ϕ(s)k ² ds

 ρ(t)dt

+ 1

m(q, ρ) Z q

−q

Z t+1 t

γ(s − t)Ekϕ n (s)k ² ds

 ρ(t)dt

≤ kϕ _n − ϕk ² _S

2

γ

+ 1

m(q, ρ) Z q

−q

Z t+1 t

γ(s − t)Ekϕ n (s)k ² ds

 ρ(t)dt.

Letting q → ∞ and then n → ∞ in the previous inequality, it follows that ϕ ^b ∈ P AA ₀ (R, L ² (0, 1; L ² (P, H), γds), ρ). Above all we have f = h + ϕ ∈ W P AAS _γ ² (R, L ² (P, H)), which completes the proof.

Next, we prove a useful composition theorem.

Theorem 3.2. Let F = H + Φ ∈ W P AAS _γ ² (R × L ² (P, H), L ² (P, H)), φ = φ 1 + φ 2 ∈ W P AAS _γ ² (R, L ² (P, H)), where H ∈ AS _γ ² (R×L ² (P, H), L ² (P, H)) and Φ ^b ∈ P AA 0 (R×

L ² (P, H), L ² (0, 1; L ² (P, H), γds), ρ), φ 1 ∈ AS _γ ² (R, L ² (P, H)), φ 2 ∈ P AA 0 (R, L ² (0, 1;

L ² (P, H), γds), ρ). If K = {φ 1 (t) : t ∈ R} ⊂ L ² (P, H)) is a compact subset and there exists a continuous function L F (·) : R → [0, ∞) such that for all x, y ∈ L ² (P, H) and t ∈ R

Z t+1 t

γ(s − t)EkF (s, x) − F (s, y)k ² ds ≤ L _F (t)Ekx − yk ² . If for every χ ^b ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ) such that

lim sup

r→∞

1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

L F (θ)

!

ρ(t)dt < ∞, (3.3)

r→∞ lim 1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

L F (θ)

!

χ(t)ρ(t)dt = 0. (3.4)

(I) H(t, x) is uniformly continuous in any bounded subset K ^′ ⊂ L ² (P, H) uniformly for t ∈ R, then F (·, φ(·)) ∈ W P AAS _γ ² (R, L ² (P, H)).

Proof. Since F = H + Φ, φ = φ 1 + φ 2 , where H ∈ AS ² _γ (R × L ² (P, H), L ² (P, H)), Φ ^b ∈ P AA 0 (R × L ² (P, H), L ² (0, 1; L ² (P, H), γds), ρ), φ 1 ∈ AS _γ ² (R, L ² (P, H)), φ 2 ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ) and Q = {φ 1 (t) : t ∈ R} is compact. Now the function F can be decomposed as

F (t, φ(t)) = H(t, φ 1 (t)) + F (t, φ(t)) − F (t, φ 1 (t)) + Φ(t, φ 1 (t)).

denote

ζ(t) = H(t, φ 1 (t)), η(t) = F (t, φ(t)) − F (t, φ 1 (t)), ̟(t) = Φ(t, φ 1 (t)).

Then F (t, φ(t)) = ζ(t) + η(t) + ̟(t). Since the function H satisfies the condi- tion (I) and Q = {φ 1 (t) : t ∈ R} is compact, we have ζ(t) ∈ AS _γ ² (R, L ² (P, H)).

So it is sufficient to prove η ^b (t) ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ) and ̟ ^b (t) ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ). For p > 0 and q > 0, by using the conditions in the theorem above, we obtain

1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

Z θ+1 θ

γ(s − θ)Ekη(s)k ² ds

!!

ρ(t)dt

= 1

m(q, ρ) Z q

−q

 sup

θ∈[t−p,t]

 Z θ+1 θ

γ(s − θ)

×EkF (t, φ(t)) − F (t, φ 1 (t))k ² ds



ρ(t)dt

≤ 1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

L F (θ)Ekφ 2 (θ)k ²

! ρ(t)dt

≤ m ⁻¹ ₀ m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

L F (θ)

!

· sup

θ∈[t−p,t]

Ekφ 2 (θ)k ² γ(s − θ)
! ρ(t)dt

≤ m ⁻¹ ₀ m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

L _F (θ)

!

× sup

θ∈[t−p,t]

Z θ+1 θ

γ(s − θ)Ekφ 2 (s)k ² ds

!!

ρ(t)dt.

By using the (3.4), we obtain η ^b (t) ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ).

Next we prove ̟ ^b (t) ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ). Since the function H satisfies the condition (I), for the arbitrary ǫ > 0, there exists δ > 0 such that for each t ∈ R, EkH(t, x) − H(t, x)k ² < ǫ, whenever Ekx − xk ² ≤ δ, where x, x ∈ K ^′ . Furthermore,

Z t+1 t

γ(s − t)EkH(s, x) − H(s, x)k ² ds < ǫ, t ∈ R.

Now we fix x 1 , x 2 , . . . , x n ∈ φ 1 (R) such that φ 1 (R) ⊂ S n

i=1 B δ (x i , L ² (P, H)). Ob- viously, the set E i = φ ⁻¹ ₁ (B δ (x i )) generates an open covering on R, let B 1 = E 1 ,B k = E k \( S k

i=1 E i )(2 ≤ k ≤ m). Then R = S m

k=1 B k and B i ∩ B j = ∅, i 6= j, 1 ≤ i, j ≤ m.

When t ∈ R and φ 1 (t) ∈ B δ (x i ), it follows that Z t+1

t

γ(s − t)Ek̟(t)k ² ds

= Z t+1

t

γ(s − t)EkΦ(s, φ 1 (s))k ² ds

≤ Z t+1

t

γ(s − t)EkF (s, φ 1 (s)) − F (s, x i )k ² ds

+ Z t+1

t

γ(s − t)EkΦ(s, x i )k ² ds

+ Z t+1

t

γ(s − t)EkH(s, φ 1 (s)) − H(s, x i )k ² ds

≤ L _F (t)Ekφ 1 (t) − x i k ² + ǫ + Z t+1

t

γ(s − t)EkΦ(s, x i )k ² ds

≤ (L _F (t) + 1)ǫ + Z t+1

t

γ(s − t)EkΦ(s, x i )k ² ds.

For each q > 0, we obtain 1

m(q, ρ) Z q

−q

sup

θ∈[t−p,t]

Z θ+1 θ

γ(s − θ)EkΦ(s, φ 1 (s))k ² ds

!!

ρ(t)dt

≤ 1 m(q, ρ)

X n i=1

Z

B

i

∩[−q,q]

 sup

j=1,...,n

 sup

θ∈[t−p,t]∩B

j

 Z θ+1 θ

γ(s − θ)

×EkΦ(s, φ 1 (s))k ² ds



ρ(t)dt

≤ 1

m(q, ρ) X n i=1

Z

B

i

∩[−q,q]

 sup

j=1,...,n

 sup

θ∈[t−p,t]∩B

j

 Z θ+1 θ

γ(s − θ)

×EkF (s, φ 1 (s)) − F (s, x j )k ² ds



ρ(t)dt

+ 1

m(q, ρ) X n i=1

Z

B

i

∩[−q,q]

 sup

j=1,...,n

 sup

θ∈[t−p,t]∩B

j

 Z θ+1 θ

γ(s − θ)

×EkH(s, φ 1 (s)) − H(s, x j )k ² ds



ρ(t)dt

+ 1

m(q, ρ) X n i=1

Z

B

i

∩[−q,q]

 sup

j=1,...,n

 sup

θ∈[t−p,t]∩B

j

 Z θ+1 θ

γ(s − θ)

×EkΦ(s, x j )k ² ds



ρ(t)dt

≤ X n j=1

1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

Z θ+1 θ

γ(s − θ)EkΦ(s, x j )k ² ds

!!

ρ(t)dt

+ 1

m(q, ρ) sup

θ∈[t−p,t]

L _F (θ)ǫ + ǫ

! ρ(t)dt.

From the arbitrary ǫ, it follows that

q→∞ lim 1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

Z θ+1 θ

γ(s − θ)EkΦ(s, φ 1 (s)k ² ds

!!

ρ(t)dt = 0, which implies that F (·, φ(·)) ∈ W P AAS _γ ² (R, L ² (P, H)).

4. WEIGHTED PSEUDO ALMOST AUTOMORPHIC SOLUTIONS IN DISTRIBUTION

Fix γ ∈ U and ρ ∈ V ∞ . This section is devoted to the search of the existence and uniqueness results of weighted pseudo almost automorphic solutions in distribution to Eq. (1.1) with S _γ ² -weighted pseudo almost automorphic coefficients. For that, we assume that the following assumptions hold:

(H1) There exist constants λ 0 ≥ 0, θ ∈ ( ^π ₂ , π), K 1 , K 2 ≥ 0 and β 1 , β 2 ∈ (0, 1] with β 1 + β 2 > 1 such that

Σ θ ∪ {0} ⊂ ρ(A(t) − λ 0 ), kR(λ, A(t) − λ 0 )k ≤ K 1

1 + |λ| ,

and

k[A(t) − λ 0 ]R(λ, A(t) − λ 0 )[R(λ 0 , A(t)) − R(λ 0 , A(s))]k ≤ K 2 |t − s| ^β

¹

|λ| ^−β

²

for t, s ∈ R and λ ∈ Σ θ = {λ ∈ C {0}| | arg λ| ≤ θ}. Then there exists a unique evolution family {U (t, s)} −∞<s≤t<∞ , which governs the following linear equation

x ^′ (t) = A(t)x(t), t ≥ s, x(s) = ϕ ∈ X, where X is a Banach space.

(H2) U (t, s) is an exponentially stable evolution family on L ² (P, H), that is, there exist two numbers M, δ > 0 such that kU (t, s)k ≤ M e ^−δ(t−s) , for t ≥ s.

(H3) The function s → A(s)U (t, s) defined from (−∞, t] into L(L ² (P, H)) is strongly measurable and there exists a non-increasing function H : [0, ∞) → [0, ∞) with H ∈ L ¹ (0, ∞) and a constant ω > 0 such that

kA(s)U (t, s)k ≤ e ^−ω(t−s) H (t − s), t > s.

(H4) The function R × R → L ² (P, H), (t, s) → U (t, s)x ∈ bAA(T, L ² (P, H)) uniformly for all x in any bounded subset of L ² (P, H).

(H5) The function R × R → L ² (P, H), (t, s) → A(s)U (t, s)x ∈ bAA(T, L ² (P, H)) uniformly for all x in any bounded subset of L ² (P, H).

Definition 4.1. A continuous stochastic function u : R × Ω → L ² (P, H), a ∈ R, is called a mild solution of (1.1), provided that sup

t∈R

Eku(t)k ² < ∞, the function s → A(s)U (t, s)f (s, u _s ) is integrable on R and the following conditions hold:

(i) u s ∈ B for every s ∈ R.

(ii) for t ≥ a, a ∈ R, u(t) satisfies the following integral equation u(t) = U (t, a)[φ(a) + f (a, φ)] − f (t, u t ) −

Z t a

A(s)U (t, s)f (s, u s )ds

+ Z t

a

U(t, s)g(s, u s )ds + Z t

a

U (t, s)σ(s, u s )dW (s).

Under assumptions (H1)-(H3), it can be easily shown that the equation u(t) = −f (t, u t ) −

Z t

−∞

A(s)U (t, s)f (s, u s )ds + Z t

−∞

U(t, s)g(s, u s )ds

+ Z t

−∞

U (t, s)σ(s, u s )dW (s) for each t ∈ R, is a mild solution of (1.1).

Lemma 4.1. [29] Let u ∈ W P AA(R, L ² (P, H)) and assume that B is a uniform

fading memory space. Then the function t → u t belongs to W P AA(R, L ² (P, H)).

Lemma 4.2. Let u ∈ W P AAS _γ ² (R, L ² (P, H)) and assume that B is a uniform fading memory space. Then t → u t belongs to W P AAS _γ ² (R, L ² (P, H)).

Proof. Assume that u = h + ϕ with h ∈ AS _γ ² (R, L ² (P, H)) and also ϕ ^b ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ).

Clearly, in the light of the translation property of AS ² _γ (R, L ² (P, H)), we can obtain that h t ∈ AS _γ ² (R, L ² (P, H)). So it is sufficient to prove that

t → ϕ ^b _t ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ).

Let p > 0 and ǫ > 0. Since B is a uniform fading memory space, there exists τ ǫ > p such that N (τ ) < ǫ for every τ > τ ǫ and consequently D = sup N (τ ) < ǫ. Under these conditions, for q > 0 and τ > τ ǫ we find that

1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

Z θ+1 θ

γ(s − θ)Ekϕ θ k ² _B ds

!!

ρ(t)dt

≤ 1

m(q, ρ) Z q

−q

sup

θ∈[t−p,t]

D

Z θ+1 θ

γ(s − θ)Ekϕ θ−τ k ² _B ds

!

+ ℑ sup

s∈[θ−τ,θ]

Z s+1 s

γ(ξ − s)Ekϕ(ξ)k ² dξ

!

ρ(t)dt

≤ ℜkϕk _S

2

γ

ǫ + ℑ

m(q, ρ) Z q

−q

sup

s∈[t−2τ,t]

Z s+1 s

γ(ξ − s)Ekϕ(ξ)k ² dξ

 ρ(t)dt.

Next, we consider the second term of the inequality above ℑ

m(q, ρ) Z q

−q

sup

s∈[t−2τ,t]

Z s+1 s

γ(ξ − s)Ekϕ(ξ)k ² dξ

 ρ(t)dt

≤ ℑ

m(q, ρ) Z q−τ

−q−τ

sup

s∈[t−τ,t]

Z s+1 s

γ(ξ − s)Ekϕ(ξ)k ² dξ



+ sup

s∈[t,t+τ ]

Z s+1 s

γ(ξ − s)Ekϕ(ξ)k ² dξ

!

ρ(t)dt

≤ ℑ

m(q, ρ) Z q−τ

−q−τ

sup

s∈[t−τ,t]

Z s+1 s

γ(ξ − s)Ekϕ(ξ)k ² dξ

!

ρ(t)dt

+ ℑ

m(q, ρ) Z q−τ

−q−τ

sup

s∈[t,t+τ ]

Z s+1 s

γ(ξ − s)Ekϕ(ξ)k ² dξ

!

ρ(t)dt

≤ m(q + τ, ρ) m(q, ρ)

ℑ m(q + τ, ρ)

× Z q+τ

−q−τ

sup

s∈[t−τ,t]

Z s+1 s

γ(ξ − s)Ekϕ(ξ)k ² dξ

!

ρ(t)dt

+ ℑ m(q, ρ)

Z q

−q

sup

s∈[t−τ,t]

Z s+1 s

γ(ξ − s)Ekϕ(ξ)k ² dξ

 ρ(t)dt.

Consequently, 1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

Z θ+1 θ

γ(s − θ)Ekϕ θ k ² _B ds

!!

ρ(t)dt

≤ ℜkϕk S

_γ²

ǫ + ℑ m(q, ρ)

Z q

−q

sup

s∈[t−τ,t]

Z s+1 s

γ(ξ − s)Ekϕ(ξ)k ² dξ

 ρ(t)dt

+ m(q + τ, ρ) m(q, ρ)

ℑ m(q + τ, ρ)

× Z q+τ

−q−τ

sup

s∈[t−τ,t]

Z s+1 s

γ(ξ − s)Ekϕ(ξ)k ² dξ

!

ρ(t)dt,

which enables us to complete the proof, since ǫ is arbitrary and the conditions in Lemma 2.2.

Lemma 4.3. Let u ∈ W P AAS _γ ² (R, L ² (P, H)). Under assumptions (H1), (H2), (H4), the function v(·) is defined by

v(t) :=

Z t

−∞

U (t, s)u(s)ds

then v ∈ W P AA(R, L ² (P, H)).

Proof. Since u ∈ W P AAS _γ ² (R, L ² (P, H)), there exist h ∈ AS _γ ² and ϕ ^b ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ) such that u = h + ϕ. Therefore,

v(t) :=

Z t

−∞

U (t, s)h(s)ds + Z t

−∞

U (t, s)ϕ(s)ds = X(t) + Y (t).

Let us show that X(t) ∈ AA(R, L ² (P, H)). For n = 1, 2, · · · , consider the integral

X n (t) =

Z t−n+1 t−n

U (t, r)h(r)dr.

Now, by using the Cauchy-Schwartz inequality and the exponential dichotomy, it follows that

EkX _n (t)k ²

≤ E

Z t−n+1 t−n

kU (t, r)h(r)kdr

 ²

≤ M ² E

Z t−n+1 t−n

γ ⁻

¹²

(r − t + n)e ^−δ(t−r) kh(r)kγ

¹²

(r − t + n)dr

 ²

≤ M ²

Z t−n+1 t−n

γ ⁻¹ (r − t + n)e ^{−2δ(t−r)} dr



×

Z t−n+1 t−n

γ(r − t + n)Ekh(r)k ² dr



≤ M ² m ⁻¹ ₀

Z n n−1

e ^−2δs ds

 khk ² _S

γ2

≤



m ⁻¹ ₀ e ^−2δn M ² (1 + e ^2δ ) 2δ

 khk ² _S

2

γ

.

Since m ⁻¹ ₀ ^M

²

^(1+e _2δ

^2δ

⁾ Σ ^∞ _n=1 e ^−2δn < ∞, we deduce from the well-known Weierstrass theorem that the series Σ ^∞ _n=1 X n (t) is uniformly convergent on R. Furthermore,

X (t) :=

Z t

−∞

U (t, r)h(r)dr = X ∞ n=1

X _n (t),

X ∈ C(R, L ² (P, H)) and EkX(t)k ² ≤ Σ ^∞ _n=1 EkX n (t)k ² ≤ L(M, m 0 , δ)khk ² _S

γ2

, where L(M, m 0 , δ) > 0 is a constant. Since h ∈ AS _γ ² , then for every sequence of real numbers {s n } n∈N there exists a subsequence {s m } m∈N and functions U 1 and e h ∈ L ² _loc (R, γds) such that

m→∞ lim U (t + s m , s + s m )x = U 1 (t, s)x, t, s ∈ R, x ∈ L ² _loc (R, γds) (4.1)

m→∞ lim U 1 (t − s m , s − s m )x = U (t, s)x, t, s ∈ R, x ∈ L ² _loc (R, γds) (4.2)

m→∞ lim Z t+1

t

γ(s − t)Ekh(s + s m ) − e h(s)k ² ds = 0, (4.3)

m→∞ lim Z t+1

t

γ(s − t)Eke h(s − s m ) − h(s)k ² ds = 0. (4.4) Let e X _n (t) = R t−n+1

t−n U 1 (t, r)e h(r)dr. By using the Cauchy-Schwartz inequality and the exponential dichotomy, one has

EkX n (t + s m ) − e X n (t)k ²

= E

Z t+s

m

−n+1 t+s

m

−n

U (t + s m , r)h(r)dr −

Z t−n+1 t−n

U 1 (t, r)e h(r)dr

2

= E

Z t−n+1 t−n

[U (t + s m , r + s m )h(r + s m ) − U 1 (t, r)e h(r)]dr

2

≤ 2E

Z t−n+1 t−n

U (t + s m , r + s m )[h(r + s m ) − e h(r)]dr

2

+2E

Z t−n+1 t−n

[U (t + s m , r + s m ) − U 1 (t, r)]e h(r)dr

2

For the first term of the inequality above

2E

Z t−n+1 t−n

U (t + s m , r + s m )[h(r + s m ) − e h(r)]dr

2

≤ 2M ² E

 Z t−n+1 t−n

γ ⁻

¹²

(r − t + n)e ^−δ(t−r)

×kh(r + s _m ) − e h(r)kγ

¹²

(r − t + n)dr

 2

≤ 2M ²

Z t−n+1 t−n

γ ⁻¹ (r − t + n)e ^{−2δ(t−r)} dr



×

Z t−n+1 t−n

γ(r − t + n)Ekh(r + s m ) − e h(r)k ² dr



≤ 2L(M, m 0 , δ)

Z t−n+1 t−n

γ(r − t + n)Ekh(r + s m ) − e h(r)k ² dr.

Consequently,

EkX _n (t + s m ) − e X _n (t)k ²

≤ 2L(M, m 0 , δ)

Z t−n+1 t−n

γ(r − t + n)Ekh(r + s m ) − e h(r)k ² dr

+2E

Z t−n+1 t−n

[U (t + s m , r + s m ) − U 1 (t, r)]e h(r)dr

2

.

Now by using (4.1), (4.3) and also the Lebesgue Dominated Convergence theorem, for each t ∈ R, it follows that

m→∞ lim EkX n (t + s m ) − e X n (t)k ² = 0.

Similarly, by using (4.2) and (4.4) it can be shown that

m→∞ lim Ek e X _n (t − s m ) − X n (t)k ² = 0.

Therefore, for each X n ∈ AA(R, L ² (P, H)) for each n and their uniform limit X(t) ∈ AA(R, L ² (P, H)).

Next we prove Y (t) ∈ P AA 0 (R, ρ). Similarly, for n = 1, 2, · · · , consider the integral

Y n (t) =

Z t−n+1 t−n

U (t, r)ϕ(r)dr.

For p > 0, we can obtain sup

θ∈[t−p,t]

EkY n (θ)k ²

≤ sup

θ∈[t−p,t]

E

Z θ−n+1 θ−n

kU (θ, r)ϕ(r)kdr

! 2

≤ M ² sup

θ∈[t−p,t]

E

Z θ−n+1 θ−n

γ ⁻

¹²

(r − θ + n)e ^{−δ(θ−r)} kϕ(r)kγ

¹²

(r − θ + n)dr

! 2

≤ M ² m ⁻¹ ₀ e ^2δp

Z t−n+1 t−n

e ^{−2δ(t−r)} dr



× sup

θ∈[t−p,t]

Z θ−n+1 θ−n

γ(r − θ + n)Ekϕ(r)k ² dr

!!

≤



m ⁻¹ ₀ e ^{−2δ(n−p)} M ² (1 + e ^2δ ) 2δ



× sup

θ∈[t−p,t]

Z θ−n+1 θ−n

γ(r − θ + n)Ekϕ(r)k ² dr

!!

.

For q > 0, we have 1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

EkY n (θ)k ²

! ρ(t)dt

≤ m ⁻¹ ₀ e ^{−2δ(n−p)} M ² (1 + e ^2δ ) 2δ

× 1

m(q, ρ) Z q

−q

sup

θ∈[t−p,t]

Z θ−n+1 θ−n

γ(r − θ + n)Ekϕ(r)k ² dr

!!

ρ(t)dt.

Since ϕ ^b ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ), according to the inequality above, it follows that

EkY n (t)k ² ≤ m ⁻¹ ₀ e ^{−2δ(n−p)} M ² (1 + e ^2δ ) 2δ kϕk ² _S

γ2

.

Note that m ⁻¹ ₀ e ^{2δp M}

²

^(1+e _2δ

^2δ

⁾ Σ ^∞ _n=1 e ^−2δn < ∞, therefore, we deduce from the well- known Weierstrass theorem that the series Σ ^∞ _n=1 Y _n (t) is uniformly convergent on R.

Furthermore,

Y (t) :=

Z t

−∞

U (t, r)ϕ(r)dr = X ∞ n=1

Y _n (t),

Y ∈ C(R, L ² (P, H)) and EkY (t)k ² ≤ Σ ^∞ _n=1 EkY n (t)k ² ≤ L(M, m 0 , δ, p)kϕk ² _S

γ2

, where L(M, m 0 , δ, p) > 0 is a constant. Since Y n ∈ P AA 0 (R, ρ) and the following inequality

1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

EkY (θ)k ²

! ρ(t)dt

≤ 1

m(q, ρ) Z q

−q

sup

θ∈[t−p,t]

EkY (θ) − X n k=1

Y k (θ)k ²

!

ρ(t)dt

+ X n k=1

1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

EkY k (θ)k ²

! ρ(t)dt.

Consequently, the uniform limit Y (t) = Σ ^∞ _n=1 Y n (t) ∈ P AA 0 (R, ρ). Therefore, v(t) = X (t) + Y (t) ∈ W P AA(R, L ² (P, H)).

Lemma 4.4. Let u ^′ ∈ W P AAS _γ ² (R, L ² (P, H)). Under assumptions (H1), (H2), (H4), the function v ^′ (·) is defined by

v ^′ (t) :=

Z t

−∞

U (t, s)u ^′ (s)dW (s) then v ^′ ∈ W P AA(R, L ² (P, H)).

Proof. Since u ^′ ∈ W P AAS ² _γ (R, L ² (P, H)), there exist α ∈ AS _γ ² and β ^b ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ) such that u ^′ = α + β. Therefore,

v ^′ (t) :=

Z t

−∞

U (t, s)α(s)dW (s) + Z t

−∞

U (t, s)β(s)dW (s) = Υ(t) + Ψ(t).

Let us show that Υ(t) ∈ AA(R, L ² (P, H)). For n = 1, 2, · · · , consider the integral Υ n (t) =

Z t−n+1 t−n

U (t, r)α(r)dW (r).

Now using the Itˆo’s integral and H¨ older’s inequality, it follows that EkΥ n (t)k ² ≤ E

Z t−n+1 t−n

kU (t, r)α(r)k ² dr



≤ M ²

Z t−n+1 t−n

e ^{−2δ(t−r)} Ekα(r)k ² dr

= M ²

Z t−n+1 t−n

γ ⁻¹ (r − t + n)e ^{−2δ(t−r)} Ekα(r)k ² γ(r − t + n)dr

≤ M ²

Z t−n+1 t−n

γ ⁻¹ (r − t + n)e ^{−2δ(t−r)} dr



×

Z t−n+1 t−n

γ(r − t + n)Ekα(r)k ² dr



≤ M ² m ⁻¹ ₀

Z n n−1

e ^−2δs ds

 kαk ² _S

γ2

≤



m ⁻¹ ₀ e ^−2δn M ² (1 + e ^2δ ) 2δ

 kαk ² _S

2

γ

.

Since m ⁻¹ ₀ ^M

²

^(1+e _2δ

^2δ

⁾ Σ ^∞ _n=1 e ^−2δn < ∞, we deduce from the well-known Weierstrass theorem that the series Σ ^∞ _n=1 Υ n (t) is uniformly convergent on R. Furthermore,

Υ(t) :=

Z t

−∞

U (t, r)α(r)dr = X ∞ n=1

Υ n (t),

Υ ∈ C(R, L ² (P, H)) and EkΥ(t)k ² ≤ Σ ^∞ _n=1 EkΥ n (t)k ² ≤ L(M, m 0 , δ)kαk ² _S

2

γ

, where L(M, m 0 , δ) > 0 is a constant. Since α ∈ AS _γ ² , then for every sequence of real numbers {s n } n∈N there exists a subsequence {s m } m∈N and functions U 2 and e α ∈ L ² _loc (R, γds) such that

m→∞ lim U (t + s m , s + s m )x = U 2 (t, s)x, t, s ∈ R, x ∈ L ² _loc (R, γds) (4.5)

m→∞ lim U 2 (t − s m , s − s _m )x = U (t, s)x, t, s ∈ R, x ∈ L ² _loc (R, γds) (4.6)

m→∞ lim Z t+1

t

γ(s − t)Ekα(s + s m ) − e α(s)k ² ds = 0, (4.7)

m→∞ lim Z t+1

t

γ (s − t)Eke α(s − s m ) − α(s)k ² ds = 0. (4.8) Let e Υ n (t) = R t−n+1

t−n U 2 (t, r)e α(r)dr. By using H¨ older’s inequality, one has EkΥ n (t + s m ) − e Υ n (t)k ²

= E

Z t+s

m

−n+1 t+s

m

−n

U (t + s m , r)α(r)dr −

Z t−n+1 t−n

U 2 (t, r)e α(r)dW (r)

2

= E

Z t−n+1 t−n

[U (t + s m , r + s m )α(r + s m ) − U 2 (t, r)e α(r)]dW (r)

2

≤ 2E

Z t−n+1 t−n

U (t + s m , r + s m )[α(r + s m ) − e α(r)]dW (r)

2

+2E

Z t−n+1 t−n

[U (t + s m , r + s m ) − U 2 (t, r)]e α(r)dW (r)

2

≤ 2E

Z t−n+1 t−n

U (t + s m , r + s m )[α(r + s m ) − e α(r)]dW (r)

2

+2

Z t−n+1 t−n

kU (t + s m , r + s m ) − U 2 (t, r)kEke α(r)k ² dr For the first term of the inequality above

2E

Z t−n+1 t−n

U(t + s m , r + s m )[α(r + s m ) − e α(r)]dW (r)

2

≤ 2M ²

Z t−n+1 t−n

γ ⁻¹ (r − t + n)e ^{−2δ(t−r)}

×Ekα(r + s _m ) − e α(r)k ² γ(r − t + n)dr

≤ 2M ²

Z t−n+1 t−n

γ ⁻¹ (r − t + n)e ^{−2δ(t−r)} dr



×

Z t−n+1 t−n

γ(r − t + n)Ekα(r + s m ) − e α(r)k ² dr



≤ 2L(M, m 0 , δ)

Z t−n+1 t−n

γ(r − t + n)Ekα(r + s m ) − e α(r)k ² dr

Consequently,

EkΥ n (t + s m ) − e Υ n (t)k ²

≤ 2L(M, m 0 , δ)

Z t−n+1 t−n

γ(r − t + n)Ekα(r + s m ) − e α(r)k ² dr

+2

Z t−n+1 t−n

kU (t + s _m , r + s m ) − U 2 (t, r)kEke α(r)k ² dr

Now by using (4.5), (4.7) and also the Lebesgue Dominated Convergence theorem, for each t ∈ R, it follows that

m→∞ lim EkΥ n (t + s m ) − e Υ n (t)k ² = 0.

Similarly, by using (4.6) and (4.8) it can be shown that

m→∞ lim Ek e Υ n (t − s m ) − Υ n (t)k ² = 0.

Therefore, for each Υ n ∈ AA(R, L ² (P, H)) for each n and their uniform limit Υ(t) ∈ AA(R, L ² (P, H)).

Next we prove Ψ(t) ∈ P AA 0 (R, ρ). Similarly, for n = 1, 2, · · · , consider the integral

Ψ n (t) =

Z t−n+1 t−n

U (t, r)β(r)dW (r).

For p > 0, we can obtain sup

θ∈[t−p,t]

EkΨ n (θ)k ²

≤ sup

θ∈[t−p,t]

E

Z θ−n+1 θ−n

kU (θ, r)β(r)k ² dr

!

≤ M ² sup

θ∈[t−p,t]

Z θ−n+1 θ−n

e ^{−2δ(θ−r)} Ekβ(r)k ² dr

≤ M ² sup

θ∈[t−p,t]

e ^2δp

Z θ−n+1 θ−n

γ ⁻¹ (r − θ + n)e ^{−2δ(t−r)} Ekβ(r)k ² γ(r − θ + n)dr

≤ M ² m ⁻¹ ₀ e ^2δp

Z t−n+1 t−n

e ^{−2δ(t−r)} dr



× sup

θ∈[t−p,t]

Z θ−n+1 θ−n

γ(r − θ + n)Ekβ(r)k ² dr

!!

≤



m ⁻¹ ₀ e ^{−2δ(n−p)} M ² (1 + e ^2δ ) 2δ



× sup

θ∈[t−p,t]

Z θ−n+1 θ−n

γ(r − θ + n)Ekβ(r)k ² dr

!!

.

for q > 0, we have 1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

EkΨ n (θ)k ²

! ρ(t)dt

≤ m ⁻¹ ₀ e ^{−2δ(n−p)} M ² (1 + e ^2δ ) 2δ

× 1

m(q, ρ) Z q

−q

sup

θ∈[t−p,t]

Z θ−n+1 θ−n

γ(r − θ + n)Ekβ(r)k ² dr

!!

ρ(t)dt.

Since β ^b ∈ P AA 0 (R, L ² (0, 1; L ² (P, H), γds), ρ), according to the inequality above, it follows that

EkΨ n (t)k ² ≤ m ⁻¹ ₀ e ^{−2δ(n−p)} M ² (1 + e ^2δ ) 2δ kβk ² _S

2

γ

.

Note that m ⁻¹ ₀ e ^{2δp M}

²

^(1+e _2δ

^2δ

⁾ Σ ^∞ _n=1 e ^−2δn < ∞, therefore, we deduce from the well- known Weierstrass theorem that the series Σ ^∞ _n=1 Ψ n (t) is uniformly convergent on R.

Furthermore,

Ψ(t) :=

Z t

−∞

U (t, r)β(r)dr = X ∞ n=1

Ψ n (t),

Ψ ∈ C(R, L ² (P, H)) and EkΨ(t)k ² ≤ Σ ^∞ _n=1 EkΨ n (t)k ² ≤ L(M, m 0 , δ, p)kβk ² _S

2 γ

, where L(M, m 0 , δ, p) > 0 is a constant. Since Ψ n ∈ P AA 0 (R, ρ) and the following inequality

1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

EkΨ(θ)k ²

! ρ(t)dt

≤ 1

m(q, ρ) Z q

−q

sup

θ∈[t−p,t]

EkΨ(θ) − X n k=1

Ψ k (θ)k ²

! ρ(t)dt

+ X n k=1

1 m(q, ρ)

Z q

−q

sup

θ∈[t−p,t]

EkΨ k (θ)k ²

! ρ(t)dt.

Consequently, the uniform limit Ψ(t) = Σ ^∞ _n=1 Ψ n (t) ∈ P AA 0 (R, ρ). Therefore, v ^′ (t) = Υ(t) + Ψ(t) ∈ W P AA(R, L ² (P, H)).

Lemma 4.5. Let u ^′′ ∈ W P AAS _γ ² (R, L ² (P, H)). Under assumptions (H1), (H2), (H3), (H5) and provided that the series Σ ^∞ _n=1 hR n

n−1 e ^−2ωs H ² (s)ds i

converges and H _s = sup

s∈[0,∞)

H (s), the function v ^′′ (·) is defined by

v ^′′ (t) :=

Z t

−∞

A(s)U (t, s)u ^′′ (s)ds

then v ^′′ ∈ W P AA(R, L ² (P, H)).