Pseudo S-asymptotically Bloch type periodicity with applications to partial stochastic neutral evolution equations

Missaoui, Marwa

doi:10.1108/AJMS-01-2025-0014

Purpose

This paper introduces the concept of (µ, ν)-pseudo S-asymptotically Bloch type (ω, k)- periodic functions, aiming to extend the framework of periodicity in stochastic analysis and to investigate their role in neutral partial stochastic differential equations in separable Hilbert spaces.

Design/methodology/approach

Using stochastic analysis techniques together with Banach’s fixed point Theorem, we establish rigorous conditions that guarantee the existence and uniqueness of p-mean (µ, ν)-pseudo S-asymptotically Bloch type (ω, k)-periodic mild solutions.

Findings

Our results show that, under suitable assumptions, these solutions not only exist uniquely but also exhibit global exponential stability. To illustrate the practical applicability of the theoretical findings, a detailed example is provided.

Originality/value

This work provides a new perspective on analyzing periodicity and stability in stochastic differential equations.

1. Introduction

Neutral differential equations play a significant role in various scientific fields, including the study of oscillatory systems and the modeling of numerous physical phenomena. These equations have been extensively researched over recent decades, with significant contributions made in the context of abstract partial neutral differential equations (see Refs. [1–6] for a detailed discussion). The notion of Bloch-type periodicity, introduced by Hasler and N’Gurkata in Ref. [7], encompasses concepts such as ω periodicity, ω anti-periodicity as special cases. Additionally, the concept of asymptotically Bloch-type periodicity was proposed in Ref. [7] to address scenarios where Bloch-type periodicity diminishes at infinity. Subsequent studies, such as those by Oueama-Guengai and N’Guerekata in Ref. [8], have explored the existence and uniqueness of Bloc-type periodic mild solutions to semilinear fractional differential equations within Banach spaces. The concept of S-asymptotic ω-periodicity introduced in Ref. [9], represents an important extension of classical periodicity. It exhibits a more intricate form of ergodicity compared to traditional ω-periodicity and asymptotic ω-periodicity. For a detailed analysis and applications to various types of differential equations, one can refer to Refs. [10], [11], [8], [12] and related works. More recently, the idea of pseudo S-asymptotic ω-periodicity has been developed as a generalization of S-asymptotic ω-periodicity [13]. However, research on its counterpart, pseudo S-asymptotic ω-anti-periodicity remains limited, despite the fact that ω-anti-periodicity is a specific form of Bloch-type periodicity. Extending pseudo S-asymptotically ω-periodic functions through the framework of Bloch-type periodicity represents a natural and meaningful direction of study. In recent years, there has been growing interest in adapting deterministic results to the stochastic domain, as real-world phenomena modeled mathematically are often inherently stochastic. Many works have addressed the existence of solutions to stochastic differential equations, including almost periodic, asymptotically almost periodic, and pseudo almost periodic solutions (see Refs. [14–16], [11], [17–19] for further details).

The main purpose behind this work is to develop some results on (α, β)-pseudo S-asymptotically type periodic functions of class q ≥ 0 and give an application to differential equation with finite delay. In this work, we consider a generalization of the space of pseudo S-asymptotically Bloch type periodic functions by the space of (α, β)-Pseudo S-asymptotically periodic functions. In this paper, we are interested in the following stochastic differential equation

d [y (s) + ϕ (s, y (s))] = [A y (s) + φ (s, y (s))] d s + ψ (s, y (s)) d W (s),

(1.1)

where, A is an infinitesimal generator of a C₀ − semigroup {Q(s)}_s ≥ 0 on L^p(Ω, H) (p ≥ 2), $(W (s), s \in R)$ is two-sided standard Wiener process and ϕ, φ, ψ satisfy the hypothesis (H1), (H2) and (H3) recalled in Section 4.

The structure of this paper is as follows. Section 2, provides the necessary preliminaries, including key results, definitions, and notations. In Section 3, we present essential properties of the p-mean-value of pseudo S-asymptotically Bloch type (ω, k)-periodic processes. Using these properties, section 4 establishes the existence and uniqueness of pseudo S-asymptotically Bloch-type (ω, k)-periodic mild solutions to dynamic equations on time scales. In Section 5, we analyze the global exponential stability of these solutions for partial stochastic neutral differential equations in a separable Hilbert space F. Finally, we provide an example to illustrate the basic theory of this work.

2. Preliminaries

In this section, we present some preliminary results that will be utilized in the subsequent analysis. These findings are discussed with reference to Refs. [20–25]. In this paper, we will use the following notations.

(E, ‖⋅‖_E) and (F, ‖⋅‖_F) are real separable Hilbert spaces.
$(Ω, F, P)$ is a complete probability space.
Let L^p(P, F) be a Banach space defined by

L^{p} (P, F) ≔ {Y : F - valued random variable},

with the norm

‖ Y ‖_{p} = {(\int_{Ω} ‖ Y ‖^{p} d P)}^{\frac{1}{p}} .

$L (F, E) = {Y : F \to E linear bounded operators}$ ⁠. It is equipped with the usual operator norm ‖⋅‖.
The linear operator A defined by

A x = \lim_{t \to 0^{+}} \frac{Q (t) x - x}{t} = {\frac{d^{+} Q (t) x}{d t}|}_{t = 0} for x \in D (A)

is called the infinitesimal generator of the semigroup ${(Q (t))}_{t \geq 0}$ ⁠, where

D (A) = \{x \in L^{p} (P, F) : \lim_{t \to 0^{+}} \frac{Q (t) x - Q (0) x}{t} exists\}

is called the domain of A.

{Q(s)}_s ≥ 0 is an exponentially stable C₀ − semigroup on L^p(P, F) with the infinitesimal generator A, i.e, there exist $M, γ \in R_{+}^{*}$ such that

‖ Q (s) ‖ \leq M e^{- γ s}, for all s \geq 0 \cdot

$(Ω, F, {(F_{s})}_{s \geq 0}, P)$ is a filtered probability space where

F_{s} = σ {W (x) - W (y); x, y \leq s}

and $(W (s), s \in R)$ is two-sided standard Wiener process.

Let

Y : R \to L^{p} (P, F)

a stochastic process. If there exists $l \in R_{+}^{*}$ such that $E ‖ Y (t) ‖^{p} \leq l$ ⁠, then, Y is named to be stochastically bounded. It is named to be stochastically continuous if

\lim_{t \to s} E ‖ Y (t) - Y (s) ‖^{p} = 0 .

We denoted by $B C (R, L^{p} (P, F))$ the space of any stochastically bounded and continuous processes with the norm

‖ Y ‖_{\infty} = \sup_{s \in R} {(E ‖ Y (s) ‖^{p})}^{\frac{1}{p}} .

Definition 2.1.

For given $ω, k \in R$ , a function $f \in B C (R, L^{p} (P, E))$ is said to be Bloch (or (ω, k)) type periodic if for all $t \in R$ , f(t + ω) = e^ikωf(t).

We denote by ${B P}_{ω, k} (R, L^{p} (P, E))$ ⁠, the set of all Bloch type periodic function from $R$ to L^p(P, E).

Lemma 2.1.

Let $f, g \in {B P}_{ω, k} (R, L^{p} (P, E))$ , then, the following results hold

(i)
For each $α \in R$ , $α f + g \in {B P}_{ω, k} (R, L^{p} (P, E))$ ⁠.
(ii)
${B P}_{ω, k} (R, L^{p} (P, E))$ is a Banach space under the supremum norm.

The Definition 2.1 and Lemma 2.1 can be found in Ref. [7].

3. (α, β)-pseudo-S-asymptotically Bloch type periodic function

In this section, we collect some necessary lemmas and definitions which will be used in thereafter.

Definition 3.1.

A stochastic process $f \in B C (R, L^{p} (P, E))$ is said to be S-asymptotically (or (ω, k)) type periodic, if for given $ω, k \in R$ ,

\lim_{| s | \to + \infty} E ‖ f (t + s) - f (t) ‖^{p} d s = 0 .

We denote the set of all such processes by ${S A P}_{ω, k} (R, L^{p} (P, E))$ ⁠.

Definition 3.2.

A stochastic process $f \in B C (R, L^{p} (P, E))$ is said to be pseudo-S-asymptotically (or (ω, k)) type periodic, if for given $ω, k \in R$ ,

\lim_{x \to + \infty} \frac{1}{2 y} \int_{- y}^{y} E ‖ f (t + ω) - e^{i k ω} f (t) ‖^{p} d s = 0 .

The space of such processes will be denoted by ${P S A B P}_{ω, k} (R, L^{p} (P, E))$ ⁠.

Denoting $B$ the Lebesgue σ − field of $R$ and by

\begin{matrix} M & = & \{μ : positive measure on B; μ (R) = + \infty and \\ μ ([x, y]) < + \infty for all x, y \in R, x < y\} \cdot \end{matrix}

Definition 3.3.

Let $μ, ν \in M$ , the measures μ and ν are said to be equivalent (μ ∼ ν) if there exist constants a, λ and a bounded interval $I \subset R$ (eventually ∅) such that

a μ (A) \leq ν (A) \leq λ μ (A), f o r a l l A \in B s a t i s f y i n g A \cap I = \emptyset \cdot

For $ν \in M, τ \in R$ ⁠, we denote ν_τ the measure on $(R, B)$ defined by

ν_{τ} (A) = ν (a + τ, a \in A), for all a \in B \cdot

Let $μ, ν \in M$ and y > 0, now we need the following hypothesis.

(M₁): Let $μ, ν \in M$ such that

\underset{y \to + \infty}{lim sup} \frac{ν ([- y, y])}{μ ([- y, y])} = \underset{y \to + \infty}{lim sup} M_{y} < + \infty \cdot

(M₂): Let $μ, ν \in M$ such that for all $τ \in R$ ⁠, there exist b > 0, and a bounded interval I such that

ν_{τ} (A) \leq b ν (A), μ_{τ} (A) \leq b μ (A), if A \in B satisfies A \cap I = \emptyset \cdot

Lemma 3.1.

ν satisfies (M₂) if and only if, ν ∼ ν_τ for any $τ \in R$ ⁠.

Lemma 3.2.

If (M₂) holds, then, for all λ > 0

\underset{y \to + \infty}{lim sup} \frac{μ ([- y - λ, y + λ])}{μ ([- y, y])} < + \infty \cdot

Definition 3.4.

Let $μ, ν \in M$ ⁠. A stochastic process $f \in B C (R, L^{p} (P, E))$ is said (μ, ν) − pseudo-S-asymptotically (or (ω, k)) type periodic, if for given $ω, k \in R$ , we have

\lim_{y \to + \infty} \frac{1}{μ ([- y, y])} \int_{- y}^{y} E ‖ Φ (t + ω) - e^{i k ω} Φ (t) ‖^{p} d ν (t) = 0 \cdot

The set of all such processes will be denoted by ${P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν)$ ⁠.

Definition 3.5.

Let $μ, ν \in M$ ⁠. A stochastic process $ϕ \in B C (R, L^{p} (P, E))$ is said (μ, ν) − pseudo-S-asymptotically (or (ω, k)) type periodic of a class q > 0, if for given $ω, k \in R$ , we have

\lim_{y \to + \infty} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) = 0 \cdot

The set of all such processes will be denoted by ${P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ ⁠.

Lemma 3.3.

Let μ₁, μ₂ ∈ M satisfy (M₁) and (M₂), $φ, ϕ, ψ \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ ⁠. Then we have the following results:

(i)
$φ + ψ \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ and $a ϕ \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ for each $a \in R$ ⁠.
(ii)
The function $φ_{b} \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ for any $b \in R$ ⁠.
(iii)
${P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q, ‖ \cdot ‖_{\infty})$ is a Banach space.

Proof: (i) We have, for y > 0

\begin{matrix} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ (φ + ψ) (λ + ω) - e^{i k ω} (φ + ψ)) (λ) ‖^{p} d ν (λ) \\ \leq & \frac{2^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ φ (λ + ω) - e^{i k ω} φ (λ) ‖^{p} d ν (λ) \\ + & \frac{2^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ ψ (λ + ω) - e^{i k ω} ψ (λ) ‖^{p} d ν (λ), \end{matrix}

and

\frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ (a ϕ) (λ + ω) - e^{i k ω} (a ϕ) (λ) ‖^{p} d ν (λ) \leq \frac{| a |^{p}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ ϕ (λ + ω) - e^{i k ω} ϕ (λ) ‖^{p} d ν (λ) .

Then we can write $φ + ψ, a ϕ \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ ⁠.

(ii) For each $b \in R$ ⁠, we have

\begin{matrix} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ φ (λ + b + ω) - e^{i k ω} φ (λ + b) ‖^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y + b}^{y + b} \sup_{λ \in [t - q, t]} E ‖ φ (λ + ω) - e^{i k ω} φ (λ) ‖^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y - | b |}^{y + | b |} \sup_{λ \in [t - q, t]} E ‖ φ (λ + ω) - e^{i k ω} φ (λ) ‖^{p} d ν (λ) \\ = & \frac{μ ([- y - | b |, y + | b |])}{μ ([- y, y])} (\frac{1}{μ ([- y - | b |, y + | b |])} \int_{- y - | b |}^{y + | b |} \sup_{λ \in [t - q, t]} E ‖ φ (λ + ω) - e^{i k ω} φ (λ) ‖^{p} d ν (λ)) \end{matrix}

For y sufficiently large, we obtain. $φ_{b} \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$

(iii) Let ${(φ_{n})}_{n} \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ converge to φ as n → ∞. Then for any ɛ > 0, we can choose suitable constants N > 0 and θ_ɛ such that

\frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ φ_{n} (λ + ω) - e^{i k ω} φ_{n} (λ) ‖^{p} d ν (λ) \leq \frac{ε}{2 \times 3^{p - 1}}, ‖ φ_{n} - φ ‖_{\infty}^{p} \leq \frac{ε}{4 M_{y} 3^{p - 1}}, for n > N, r > θ_{ε}

\begin{matrix} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ φ (λ + ω) - e^{i k ω} φ (λ) ‖^{p} d ν (λ) \\ \leq & \frac{3^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ φ (λ + ω) - φ_{n} (λ + ω) ‖^{p} d ν (λ) \\ + & \frac{3^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ e^{i k ω} φ_{n} (λ) - e^{i k ω} φ (λ) ‖^{p} d ν (λ) \\ + & \frac{3^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ φ_{n} (λ + ω) - e^{i k ω} φ_{n} (t) ‖^{p} d ν (λ) \\ \leq & 2 \times 3^{p - 1} M_{y} ‖ φ_{n} - φ ‖_{\infty}^{p} + \frac{3^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ φ_{n} (λ + ω) - e^{i k ω} φ_{n} (t) ‖^{p} d ν (λ) \\ \leq & \frac{ε}{2} + \frac{ε}{2} = ε . \end{matrix}

■

Theorem 3.1.

Let $μ, ν \in M$ , satisfy (M₁), q > 0 and let I be a bounded interval (eventually I ≠ ∅). Suppose that $Φ \in B C (R, L^{p} (P, E))$ ⁠. Then, the following assertions are equivalent.

(i)
(ii) $Φ \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q) \cdot$

\lim_{y \to + \infty} \frac{1}{μ ([- y, y] \ I)} \int_{[- y, y] \ I} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) = 0 \cdot

(iii)
For any ɛ > 0, we have

\lim_{y \to + \infty} \frac{ν (t \in [- y, y] \ I; \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p}) > ε)}{μ ([- y, y] \ I)} = 0 \cdot

Proof: (i) ⇔ (ii) A = μ(I); B = ν(I) and

C = \int_{I} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) .

I is bounded and $Φ \in B C (R, L^{P} (P, E))$ ⁠, then A, B and C are finite. Let y > 0 such as I ⊂ [−y, y] and μ([−y, y]\I) > 0. We have

\begin{matrix} \frac{1}{μ ([- y, y] \ I)} \int_{[- y, y] \ I} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \\ = & \frac{1}{μ ([- y, y]) - A} (\int_{[- y, y]} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) - C) \\ = & \frac{μ ([- y, y]}{μ ([- y, y]) - A} \frac{1}{μ ([- y, y])} (\int_{[- y, y]} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) - C) \\ = & \frac{μ ([- y, y]}{μ ([- y, y]) - A} (\frac{1}{μ ([- y, y])} \int_{[- y, y]} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) - \frac{C}{μ ([- y, y])}) . \end{matrix}

Since $μ (R) = + \infty$ ⁠, we deduce that (ii) ⇔ (i).

(ii) ⇒ (iii) Given ɛ > 0,

A_{y}^{ε} = {t \in [- y, y] \ I : \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} > ε} \cdot

and

C_{y}^{ε} = {t \in [- y, y] \ I : \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} \leq ε} \cdot

Therefore, one can get

\begin{matrix} \int_{[- y, y] \ I} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) & = & \int_{A_{y}^{ε}} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \\ + & \int_{C_{y}^{ε}} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \cdot \end{matrix}

Then,

\int_{[- y, y] \ I} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \geq ε ν (A_{y}^{ε})

and

\frac{1}{μ ([- y, y] \ I)} \int_{[- y, y] \ I} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \geq \frac{ε ν (A_{y}^{ε})}{μ ([- y, y] \ I)} \cdot

For y large enough, we obtain (iii).

(iii) ⇒ (ii) For y sufficiently large and ɛ > 0, we have

\begin{matrix} \frac{1}{μ ([- y, y] \ I)} \int_{[- y, y] \ I} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y] \ I)} \int_{A_{r}^{ε}} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \\ + \frac{1}{μ ([- y, y] \ I)} \int_{C_{r}^{ε}} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \cdot \end{matrix}

Then,

\begin{matrix} \frac{1}{μ ([- y, y] \ I)} \int_{[- y, y] \ I} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \\ \leq & \frac{2 ν (A_{y}^{ε}) ‖ Φ ‖_{\infty}}{μ ([- y, y] \ I)} + \frac{ε ν (C_{y}^{ε})}{μ ([- y, y] \ I)} \\ \leq & \frac{2 ν (A_{y}^{ε}) ‖ Φ ‖_{\infty}}{μ ([- y, y] \ I)} + \frac{ε ν ([- y, y] \ I)}{μ ([- y, y] \ I)} \\ \leq & \frac{2 ν (A_{y}^{ε}) ‖ Φ ‖_{\infty}}{μ ([- y, y] \ I)} + \frac{ε (ν ([- y, y]) - ν (I))}{μ ([- y, y] - A)} \\ \leq & \frac{2 ν (A_{y}^{ε}) ‖ Φ ‖_{\infty}}{μ ([- y, y] \ I)} + ε \cdot \frac{ν ([- y, y])}{μ ([- y, y])} \cdot \frac{1 - \frac{B}{ν ([- y, y])}}{1 - \frac{A}{μ ([- y, y])}} \cdot \end{matrix}

Since $μ (R) = ν (R) = 0$ ⁠, then for any ɛ > 0, we have

\lim_{y \to + \infty} \frac{1}{μ ([- y, y] \ I)} \int_{[- y, y] \ I} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \leq K \cdot ε

therefore, (ii) holds. ■

Proposition 3.1.

Assume that (M₁) holds. If $μ_{i}, ν_{i} \in M (i = 1,2)$ and ν₁ ∼ ν₂, μ₁ ∼ μ₂, then

{P S A B P}_{ω, k} (R, L^{p} (P, E), μ_{1}, ν_{1}, q) = {P S A B P}_{ω, k} (R, L^{p} (P, E), μ_{2}, ν_{2}, q) \cdot

Proof. Since ν₁ ∼ ν₂, μ₁ ∼ μ₂ and $B$ is the Lebesgue σ field, then for all $A \in B$ satisfying A ∩ I ≠ ∅, by Definition 3.3 there exist k_j > 0 and γ_j > 0 (j = 1, 2) such that

ρ_{1} ν_{2} (A) \leq ν_{1} (A) \leq γ_{1} ν_{2} (A)

and

ρ_{2} μ_{2} (A) \leq μ_{1} (A) \leq γ_{2} μ_{2} (A) \cdot

For y sufficiently large, one has

\begin{matrix} \frac{ρ_{1}}{γ_{2}} \frac{ν_{2} (t \in [- y, y] \ I; \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p}) > ε)}{μ_{2} ([- y, y] \ I)} \\ \leq & \frac{ρ_{1}}{γ_{2}} \frac{ν_{1} (t \in [- y, y] \ I; \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p}) > ε)}{μ_{1} ([- y, y] \ I)} \\ \leq & \frac{ρ_{1}}{γ_{2}} \frac{ν_{2} (t \in [- y, y] \ I; \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p}) > ε)}{μ_{2} ([- y, y] \ I)} \cdot \end{matrix}

Therefore, by Theorem 3.1, we have

{P S A B P}_{ω, k} (R,, L^{p} (P, E), μ_{1}, ν_{1}, q) = {P S A B P}_{ω, k} (R,, L^{p} (P, E), μ_{2}, ν_{2}, q) \cdot

Proposition 3.2.

If (M₁), (M₂) hold and $Φ \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ ,

then $Φ (\cdot - ξ) \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ for all. $ξ \in R \cdot$

Proof: For $μ \in M$ such that $μ (R) = + \infty$ ⁠, there exists y₀ > 0 such that μ([−y − |ξ|, y + |ξ|]) > 0, for all y > y₀. We consider

ξ^{+} = \max (ξ, 0) and ξ^{-} = \max (- ξ, 0) \cdot

Then, one has

\{\begin{cases} | ξ | + ξ = 2 ξ^{+} \\ | ξ | - ξ = 2 ξ^{-} \cdot \end{cases}

So

[- y - | ξ | + ξ, y + | ξ | + ξ] = [- y - 2 ξ^{-}, y + 2 ξ^{+}] .

For y > y₀ and $ξ \in R$ ⁠, we have

\begin{matrix} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ (λ - ξ + ω) - e^{i k ω} Φ (λ - ξ) ‖^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y - 2 ξ^{-}}^{y + 2 ξ^{+}} \sup_{λ \in [t - q, t]} E ‖ Φ (λ - ξ + ω) - e^{i k ω} Φ (λ - ξ) ‖^{p} d ν (λ) \\ \leq & \frac{μ ([- y - 2 ξ^{-}, y + 2 ξ^{+}])}{μ ([- y, y])} \cdot ϒ_{ξ} (y), \end{matrix}

where

\begin{matrix} ϒ_{ξ} (y) & ≔ & \frac{1}{μ ([- y - 2 ξ^{-}, y + 2 ξ^{+}])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ (λ - ξ + ω) - e^{i k ω} Φ (λ - ξ) ‖^{p} d ν (λ) \\ = & \frac{1}{μ ([- y - | ξ | + ξ, y + | ξ | + ξ])} \int_{- y - | ξ | + ξ}^{y + | ξ | + ξ} \sup_{λ \in [t - q, t]} E ‖ Φ (λ - ξ + ω) - e^{i k ω} Φ (λ - ξ) ‖^{p} d ν (λ) \\ = & \frac{1}{μ_{ξ} ([- y - | ξ |, y + | ξ |])} \int_{- y - | ξ | + ξ}^{y + | ξ | + ξ} \sup_{λ \in [t - ξ - q, t - ξ]} E ‖ Φ (λ - ξ + ω) - e^{i k ω} Φ (λ - ξ) ‖^{p} d ν (λ) \\ = & \frac{1}{μ_{ξ} ([- y - | ξ |, y + | ξ |])} \int_{- y - | ξ |}^{y + | ξ |} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \cdot \end{matrix}

Note that ν ∼ ν_ξ and μ ∼ μ_ξ. By Lemma 3.1, we have $Φ \in {P S A B P}_{ω, k} (R, L^{P} (P, E), μ, ν, q)$ ⁠. By Proposition 3.1, we get

\lim_{w \to + \infty} ϒ_{ξ} (x) = 0 \cdot

Using Lemma 3.2, one has

\lim_{x \to + \infty} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ (λ - ξ + ω) - e^{i k ω} Φ (λ - ξ) ‖^{p} d ν (λ) = 0 \cdot

Hence, $Φ (\cdot - ξ) \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ for any $ξ \in R \cdot$ ■

Proposition 3.3.

If (M₁) and (M₂) hold one can show that

(i)
${P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q) \subset {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν) \cdot$
(ii)
${P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ is closed subspace of $B C (R, L^{p} (P, E)) \cdot$
(iii)
${P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ is Banach space under the supremum norm ‖ ⋅‖_∞⋅

Remark 3.1.

${P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν) = {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, 0)$ ⁠.

Proof:

(i)
According to this inequality

\frac{1}{μ ([- y, y])} \int_{- y}^{y} E ‖ Φ (λ - ξ + ω) - e^{i k ω} Φ (λ - ξ) ‖^{p} d ν (λ)

\leq \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ (λ - ξ + ω) - e^{i k ω} Φ (λ - ξ) ‖^{p} d ν (λ)

We see that (i) holds.

(ii) and (iii) Let $Φ_{n} \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ and Φ_n → Φ in $B C (R, L^{p} (P, E))$ ⁠. Then from (M₁), it follows that

\begin{matrix} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \\ \leq & \frac{3^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - Φ_{n} (λ + ω) ‖^{p} d ν (λ) \\ + \frac{3^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ_{n} (λ + ω) - e^{i k ω} Φ_{n} (λ) ‖^{p} d ν (λ) \\ + \frac{3^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ e^{i k ω} Φ_{n} (λ) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \\ \leq & \frac{3^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - Φ_{n} (λ + ω) ‖^{p} d ν (λ) \\ + \frac{3^{p} - 1}{μ ([- y, y])} \int_{- y}^{y} E ‖ Φ_{n} (λ + ω) - e^{i k ω} Φ_{n} (λ) ‖^{p} d ν (λ) \\ + \frac{3^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ_{n} (λ) - Φ (λ) ‖^{p} d ν (λ) \\ \leq & 3^{p - 1} M_{y} ‖ Φ - Φ_{n} ‖^{p} + 3^{p - 1} M_{y} ‖ Φ - Φ_{n} ‖^{p} \\ + \frac{3^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ_{n} (λ + ω) - e^{i k ω} Φ_{n} (λ) ‖^{p} d ν (λ) \\ \leq & 2 3^{p - 1} M_{y} ‖ Φ - Φ_{n} ‖^{p} + \frac{3^{p - 1}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} ‖ Φ_{n} (λ + ω) - e^{i k ω} Φ_{n} (λ) ‖^{p} d ν (λ) \cdot \end{matrix}

For each $n \in N$ ⁠, since ‖Φ − Φ_n‖_∞ → 0 and the fact that $Φ_{n} \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ ⁠, we deduce that

\lim_{y \to + \infty} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) = 0 \cdot

Which ends the proof.

■

Proposition 3.4.

Assume that (M₁) and (M₂) hold. Let q₁ > 0 and q₂ > 0. Then,

{P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q_{1}) = {P S A B P}_{ω, k} (R,, L^{p} (P, E), μ, ν, q_{2}) \cdot

Proof: Let q > 0. First of all, we want to prove that

{P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q) \subset {P S A B P}_{ω, k} (R,, L^{p} (P, E), μ, ν, 2 q) \cdot

For $Φ \in {P S A B P}_{ω, k} (R,, L^{p} (P, E), μ, ν, q)$ ⁠, one has

\begin{matrix} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - 2 q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - 2 q, t - q]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \\ + \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω - q) - e^{i k ω} Φ (λ - q) ‖^{p} d ν (λ) \\ + \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - 2 q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \cdot \end{matrix}

By Proposition 3.2, we have

\lim_{y \to + \infty} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - 2 q, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) = 0 \cdot

Therefore, we have $Φ \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, 2 q)$ ⁠.

Now let q₁ > q₂, if $Φ \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q_{1})$ ⁠, then

\frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q_{2}, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \leq \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q_{1}, t]} E ‖ Φ (λ + ω) - e^{i k ω} Φ (λ) ‖^{p} d ν (λ) \cdot

Therefore, $Φ \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q_{2})$ ⁠. Hence,

{P S A B P}_{ω, k} (R, L^{P} (P, E), μ, ν, q_{1}) \subset {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q_{2}) \cdot

On the other hand, since q₁ > q₂, there exists $d \in N$ such that 2^dq₂ > q₁, from the above, we have

{P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q_{2}) \subset {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, 2^{d} q_{2}) \subset {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q_{1}) \cdot

This complete the proof. ■

Theorem 3.2.

Let $Φ \in B C (R \times L^{p} (P, E), L^{p} (P, F))$ satisfies the following conditions

For any $(t, z) \in R \times L^{P} (P, E)$ , Φ(t + ω, z) = e^ikωΦ(t, e^−ikωz)⋅
There exists a constant L > 0 such that for all z, y ∈ L^P(P, E) and $t \in R$ ,

E ‖ Φ (t, z) - Φ (t, y) ‖^{p} \leq L E ‖ z - y ‖^{p} .

Then, for each $χ \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ ⁠, we have. $t \mapsto Φ (t, χ (t)) \in {P S A B P}_{ω, k} (R, L^{p} (P, F), μ, ν, q) \cdot$

Proof: Since $χ \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ ⁠, we have

\lim_{y \to \infty} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ χ (λ + ω) - e^{- i k ω} χ (λ) ‖^{p} d ν (λ) = 0 .

On the other hand

\begin{matrix} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω, χ (λ + ω)) - e^{i k ω} Φ (λ, χ (λ)) ‖^{p} d ν (λ) \\ = & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ e^{i k ω} Φ (λ + ω, e^{- i k ω} χ (λ + ω)) - e^{i k ω} Φ (λ, χ (λ)) ‖^{p} d ν (λ) \\ \leq & \frac{L}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ e^{i k ω} χ (λ + ω) - χ (λ) ‖^{p} d ν (λ) \\ \leq & \frac{L}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ χ (λ + ω) - e^{- i k ω} χ (λ) ‖^{p} d ν (λ) . \end{matrix}

Then

\lim_{y \to \infty} \frac{1}{μ ([- y, q])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ Φ (λ + ω, χ (λ + ω)) - e^{i k ω} Φ (λ, χ (λ)) ‖^{p} d ν (λ) = 0 .

This complete the proof. ■

In the following, we assume that μ, ν satisfy the two hypotheses (M₁), (M₂).

4. Partial functional differential equations with finite delay

This section is devoted to prove the uniqueness and existence of the (μ, ν) − pseudo-S-asymptotically Bloch type periodic mild solution to partial evolution equation (1.1). For this purpose, we introduce the following hypotheses.

H1.

The functions

φ, ϕ \in {P S A B P}_{ω, k} (R \times L^{p} (P, F), L^{p} (P, E), μ, ν) \cap C (R \times L^{p} (P, F), L^{p} (P, E)),

ψ \in {P S A B P}_{ω, k} (R \times L^{p} (P, F), L^{p} (P, L_{0}^{2}), μ, ν) \cap C (R \times L^{p} (P, F), L^{p} (P, L_{0}^{2})) \cdot

H2.

There exist positive constants $C_{1}, C_{2}, C_{3} \in R_{+}^{*}$ such that for each stochastic process Y₁, Y₂ ∈ L^p(P, E), the functions ϕ, φ, ψ satisfy

\begin{matrix} E ‖ ϕ (s, Y_{1}) - ϕ (s, Y_{2}) ‖^{p} & \leq & C_{1} E ‖ Y_{1} - Y_{2} ‖^{p} \\ E ‖ φ (s, Y_{1}) - φ (s, Y_{2}) ‖^{p} & \leq & C_{2} E ‖ Y_{1} - Y_{2} ‖^{p} \\ E ‖ ψ (s, Y_{1}) - ψ (s, Y_{2}) ‖^{p} & \leq & C_{3} E ‖ Y_{1} - Y_{2} ‖^{p} \cdot \end{matrix}

H3.

The function

\begin{matrix} R_{+} & \to & L (Y) \\ ρ & \mapsto & A Q (ρ) \end{matrix}

is strong (Lebesgue) measure and there exists a decreasing function J(s) with $η ≔ \int_{0}^{\infty} J^{p} (s) d s < \infty$ such that

‖ A Q (s) ‖ \leq J (s), s \in R_{+} \cdot

Definition 4.1.

$F -$ progressively measurable stochastic Y is called a mild solution of the equation (1.1) if it satisfies the following equation

\begin{matrix} X (t) + ϕ (t, X (t)) & = & Q (t - s) [X (s) + ϕ (s, X (s))] + \int_{s}^{t} A Q (t - ξ) ϕ (ξ, X (ξ)) d ξ \\ + & \int_{s}^{t} Q (t - ξ) φ (ξ, X (ξ)) d ξ + \int_{s}^{t} Q (t - ξ) ψ (ξ, X (ξ)) d W (ξ) \cdot \end{matrix}

Lemma 4.1.

Let $ϕ \in {P S A B P}_{ω, k} (R \times L^{p} (P, F), L^{p} (P, E), μ, ν, q) \cap C (R \times L^{p} (P, F), L^{p} (P, E))$ and f the following function defined by

f (t) ≔ \int_{- \infty}^{t} A Q (t - ξ) ϕ (ξ, X (ξ)) d (ξ), f o r a l l t \in R \cdot

Then, $f \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q) \cdot$

Proof: Let $ϕ \in {P S A B P}_{ω, k} (R \times L^{p} (P, F), L^{p} (P, E), μ, ν, q) \cap C (R \times L^{p} (P, F), L^{p} (P, E))$ ⁠. For any y > 0 we have

\begin{matrix} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ f (λ + ω) - e^{i k ω} f (λ) ‖^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E {‖\int_{- \infty}^{λ + ω} A Q (λ + ω - ξ) ϕ (ξ, X (ξ)) d ξ - e^{i k ω} \int_{- \infty}^{λ} A Q (λ - ξ) ϕ (ξ, X (ξ)) d ξ‖}^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} {(\int_{- \infty}^{λ} E ‖A Q (λ - ξ) [ϕ (ω + ξ, X (ω + ξ)) - e^{i k ω} ϕ (ξ, X (ξ))]‖ d ξ)}^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} {(\int_{0}^{+ \infty} E ‖Q (n) [ϕ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ϕ (λ - n, X (λ - n))]‖ d n)}^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} {(\int_{0}^{+ \infty} E ‖ Q (n) ‖ E ‖ϕ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ϕ (λ - n, X (λ - n))‖ d n)}^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} \\ (\int_{0}^{+ \infty} J^{p} (n) d n) {(\int_{0}^{+ \infty} E ‖ϕ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ϕ (λ - n, X (λ - n))‖ d n)}^{p} d ν (λ) \\ \leq & \frac{η}{μ ([- y, y])} \int_{- y}^{y} \int_{0}^{+ \infty} \sup_{λ \in [t - q, t]} E {‖ϕ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ϕ (λ - n, X (λ - n))‖}^{p} d n d ν (λ) \\ \leq & \int_{0}^{+ \infty} \frac{η}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E {‖ϕ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ϕ (λ - n, X (λ - n))‖}^{p} d ν (λ) d n \cdot \end{matrix}

Since

\lim_{y \to + \infty} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E {‖ϕ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ϕ (λ - n, X (λ - n))‖}^{p} d ν (λ) = 0 \cdot

Then,

\frac{η}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E {‖φ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ϕ (λ - n, X (λ - n))‖}^{p} d ν (λ) \leq 2 C η ‖ ϕ ‖_{\infty}^{p} ≔ ρ (n)

and we have, $ρ \in L^{1} (R_{+})$ by the Lebesgue's Dominated Convergence, we obtain $f \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q) \cdot$ ■

Lemma 4.2.

Let $φ \in {P S A B P}_{ω, k} (R \times L^{p} (P, F), L^{p} (P, E), μ, ν, q) \cap C (R \times L^{p} (P, F), L^{p} (P, E))$ and g the following function defined by

g (t) ≔ \int_{- \infty}^{t} Q (t - ξ) φ (ξ, y (ξ)) d ξ, f o r a l l t \in R \cdot

Then, $g \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q) \cdot$

Proof: Let $φ \in {P S A B P}_{ω, k} (R \times L^{p} (P, F), L^{p} (P, E), μ, ν, q) \cap C (R \times L^{p} (P, F), L^{p} (P, E))$ then for any y > 0 we have

\begin{matrix} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ g (λ + ω) - e^{i k ω} g (λ) ‖^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E {‖\int_{- \infty}^{λ + ω} Q (λ + ω - ξ) φ (ξ, X (ξ)) d ξ - e^{i k ω} \int_{- \infty}^{λ} Q (λ - ξ) φ (ξ, X (ξ)) d ξ‖}^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} \int_{- \infty}^{λ} E {‖Q (λ - ξ) [φ (ω + ξ, X (ω + ξ)) - e^{i k ω} φ (ξ, X (ξ))]‖}^{p} d ξ d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} \int_{0}^{+ \infty} E {‖Q (n) [φ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} φ (λ - n, X (λ - n))]‖}^{p} d n d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} \int_{0}^{+ \infty} E ‖ Q (n) ‖^{p} E {‖φ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} φ (λ - n, X (λ - n))‖}^{p} d n d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \int_{0}^{+ \infty} \sup_{λ \in [t - q, t]} M^{p} e^{- p γ n} E {‖φ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} φ (λ - n, X (λ - n))‖}^{p} d n d ν (λ) \\ \leq & \int_{0}^{+ \infty} \frac{M^{p}}{μ ([- y, y])} \int_{- y}^{y} e^{- p γ n} \sup_{λ \in [t - q, t]} E {‖φ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} φ (λ - n, X (λ - n))‖}^{p} d ν (λ) d n \cdot \end{matrix}

Since

\lim_{y \to + \infty} \frac{1}{μ ([- y, y])} \int_{- y}^{y} e^{- p γ n} \sup_{λ \in [t - q, t]} E {‖φ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} φ (λ - n, X (λ - n))‖}^{p} d ν (λ) = 0 \cdot

Then,

\frac{M^{p}}{μ ([- y, y])} \int_{- y}^{y} e^{- p γ n} \sup_{λ \in [t - q, t]} E {‖φ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} φ (λ - n, X (λ - n))‖}^{p} d ν (λ) \leq 2 M^{p} \cdot C \cdot ‖ φ ‖_{\infty}^{p} e^{- p γ n} ≔ ρ_{1} (n)

and we have, $ρ_{1} \in L^{1} (R_{+})$ by the Lebesgue's Dominated Convergence, we obtain $g \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q) \cdot$ ■

Lemma 4.3.

Let $ψ \in {P S A B P}_{ω, k} (R \times L^{p} (P, F), L^{p} (P, L_{2}^{0}), μ, ν, q)) \cap C (R \times L^{p} (P, F), L^{p} (P, L_{2}^{0}))$ and h the following function defined by

h (t) ≔ \int_{- \infty}^{t} Q (t - ξ) ψ (ξ, y (ξ)) d W (ξ), f o r a l l t \in R \cdot

Then, $h \in {P S A B P}_{ω, k} (R, L^{p} (P, L_{2}^{0}), μ, ν, q) \cdot$

Proof: Let $ψ \in {P S A B P}_{ω, k} (R \times L^{p} (P, F), L^{p} (P, L_{2}^{0}), μ, ν, q)) \cap C (R \times L^{p} (P, F), L^{p} (P, L_{2}^{0}))$ then for any y > 0 we have

\begin{matrix} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E ‖ h (λ + ω) - e^{i k ω} h (λ) ‖^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E {‖\int_{- \infty}^{λ + ω} Q (λ + ω - ξ) ψ (ξ, X (ξ)) d W (ξ) - e^{i k ω} \int_{- \infty}^{λ} Q (λ - ξ) ψ (ξ, X (ξ)) d W (ξ)‖}^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E {‖\int_{- \infty}^{λ} Q (λ - ξ) [ψ (ω + ξ, X (ω + ξ)) - e^{i k ω} ψ (ξ, X (ξ))] d W (ξ)‖}^{p} d ν (λ) \\ \leq & \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E {‖\int_{0}^{+ \infty} Q (n) [ψ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ψ (λ - n, X (λ - n))] d W (n)‖}^{p} d ν (λ) \\ \leq & \frac{C_{p}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} {[\int_{0}^{+ \infty} E ‖ Q (n) ‖^{p} E {‖ψ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ψ (λ - n, X (λ - n))‖}^{p} d n]}^{p / 2} d ν (λ) \\ \leq & \frac{C_{p} M^{p}}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} {[\int_{0}^{+ \infty} e^{- \frac{p}{p - 2} γ n} d n]}^{\frac{p - 2}{2}} \int_{0}^{+ \infty} E {‖ψ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ψ (λ - n, X (λ - n))‖}^{p} d n d ν (λ) \\ \leq & \int_{0}^{+ \infty} \frac{C_{p} M^{p}}{μ ([- y, y])} {(\frac{2 - p}{p γ})}^{\frac{p - 2}{2}} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E {‖ψ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ψ (λ - n, X (λ - n))‖}^{p} d ν (λ) d n \cdot \end{matrix}

Since

\lim_{x \to + \infty} \frac{1}{μ ([- y, y])} \int_{- y}^{y} \sup_{λ \in [t - q, t]} E {‖ψ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ψ (λ - n, X (λ - n))‖}^{p} d ν (λ) = 0 \cdot

Then,

\begin{matrix} \frac{M^{p} C_{p}}{μ ([- y, y])} {(\frac{2 - p}{p γ})}^{\frac{p - 2}{2}} \int_{- y}^{y} e^{- p γ n} \sup_{λ \in [t - q, t]} E {‖ψ (ω + λ - n, X (ω + λ - n)) - e^{i k ω} ψ (λ - n, X (λ - n))‖}^{p} d ν (λ) \\ \leq & 2 M^{p} C^{p} C_{1} ‖ ψ ‖_{\infty}^{p} ≔ ρ_{2} (n), \end{matrix}

and we have, $ρ_{2} \in L^{1} (R_{+})$ by the Lebesgue's dominated convergence, we obtain $h \in {P S A B P}_{ω, k} (R, L^{p} (P, L_{2}^{0}), μ, ν, q) \cdot$ ■

Theorem 4.1.

Suppose that (H1)-(H3) hold. Further assume that

R = 4^{p - 1} [(1 + η^{p}) C_{1} + M^{p} \frac{1}{γ^{p}} C_{2} + C_{p} C_{3} M^{p} {(\frac{1}{2 γ})}^{\frac{p}{2}}] < 1 .

Then the equation (1.1) has a unique mild solution $y \in {P S A B P}_{ω, k} (R, L^{p} (P, E), μ, ν, q)$ ⁠.

Proof: Define the operator $Λ : {P S A B P}_{ω, k} (R, L^{p} (P, F), μ, ν, q) \to {P S A B P}_{ω, k} (R, L^{p} (P, F), μ, ν, q)$ by

\begin{matrix} (Λ y) (t) & = & - ϕ (t, y (t)) + \int_{- \infty}^{t} A Q (t - s) ϕ (s, y (s)) d s \\ + \int_{- \infty}^{t} Q (t - s) φ (s, y (s)) d s + \int_{- \infty}^{t} Q (t - s) ψ (s, y (s)) d W (s) \cdot \end{matrix}

In order to show that the equation (1.1) has a p-mean (μ, ν)- pseudo-S-asymptotically Bloch type periodic mild solution, we only need to prove the operator Λ has a fixed point in ${P S A B P}_{ω, k} (R, L^{p} (P, F), μ, ν, q)$ ⁠. Let $x, y \in {P S A B P}_{ω, k} (R, L^{p} (P, F), μ, ν, q)$ ⁠, one has

\begin{array}{l} E ‖ (Λ y) (t) - (Λ x) (t) ‖^{p} & \leq E ‖ ϕ (s, x (s)) - ϕ (s, y (s)) \\ + \int_{- \infty}^{t} A Q (t - s) [ϕ (s, y (s)) - ϕ (s, x (s))] d s \\ + \int_{- \infty}^{t} Q (t - s) [φ (s, y (s)) - φ (s, x (s))] d s \\ + {\int_{- \infty}^{t} Q (t - s) [ψ (s, y (s)) - ψ (s, x (s))] d W (s)‖}^{p} . \end{array}

Therefore, from the identity

{(a + b + c + d)}^{p} \leq 4^{p - 1} (a^{p} + b^{p} + c^{p} + d^{p})

(4.1)

it follows that

\begin{matrix} E ‖ (Λ y) (t) - (Λ x) (t) ‖^{p} & \leq & 4^{p - 1} E ‖ ϕ (s, y (s)) - ϕ (s, x (s)) ‖^{p} \\ + & 4^{p - 1} E {‖\int_{- \infty}^{t} A Q (t - s) [ϕ (s, y (s)) - ϕ (s, x (s))] d s‖}^{p} \\ + & 4^{p - 1} E {‖\int_{- \infty}^{t} Q (t - s) [φ (s, y (s)) - φ (s, x (s))] d s‖}^{p} \\ + & 4^{p - 1} E {‖\int_{- \infty}^{t} Q (t - s) [ψ (s, y (s)) - ψ (s, x (s))] d W (s)‖}^{p} \cdot \end{matrix}

For the first term on the right-hand side, we have

\begin{matrix} E ‖ ϕ (t, y (t)) - ϕ (t, x (t)) ‖^{p} & \leq & C_{1} E ‖ y (t) - x (t) ‖^{p} \\ \leq & C_{1} \sup_{s \in R} E ‖ y (s) - x (s) ‖^{p} \\ \leq & C_{1} ‖ y - x ‖_{\infty}^{p} . \end{matrix}

(4.2)

Taking the second term on the right-hand side and using the Hölder inequality, it easily follows that

\begin{matrix} E {‖\int_{- \infty}^{t} A Q (t - s) [ϕ (s, y (s)) - ϕ (s, x (s))] d s‖}^{p} \\ \leq & E {(\int_{- \infty}^{t} ‖ A Q (t - s) ‖^{\frac{p - 1}{p}} ‖ A Q (t - s) ‖^{\frac{1}{p}} ‖ ϕ (s, y (s)) - ϕ (s, x (s)) ‖ d s)}^{p} \\ \leq & {(\int_{- \infty}^{t} ‖ A Q (t - s) ‖ d s)}^{p - 1} \int_{- \infty}^{t} ‖ A Q (t - s) ‖ E ‖ ϕ (s, y (s)) - ϕ (s, x (s)) ‖^{p} d s \\ \leq & C_{1} {(\int_{- \infty}^{t} J (t - s) d s)}^{p} (\sup_{s \in R} E ‖ y (s) - x (s) ‖^{p}) \\ \leq & C_{1} η^{p} ‖ y - x ‖_{\infty}^{p} . \end{matrix}

(4.3)

Using the Hölder inequality in the third term on the right-hand side, we obtain

\begin{matrix} E {‖\int_{- \infty}^{t} Q (t - s) [φ (s, y (s)) - φ (s, x (s))] d s‖}^{p} \\ \leq & {(\int_{- \infty}^{t} E ‖ Q (t - s) ‖ E ‖ φ (s, y (s)) - φ (s, x (s)) ‖ d s)}^{p} \\ \leq & {(\int_{- \infty}^{t} M e^{γ (t - s)} E ‖ φ (s, y (s)) - φ (s, x (s)) ‖ d s)}^{p} \\ \leq & M^{p} {[\int_{- \infty}^{t} e^{γ (t - s)} d s]}^{p - 1} \int_{- \infty}^{t} e^{γ (t - s)} E ‖ φ (s, y (s)) - φ (s, x (s)) ‖^{p} d s \\ \leq & M^{p} {[\int_{- \infty}^{t} e^{γ (t - s)} d s]}^{p - 1} \int_{- \infty}^{t} e^{γ (t - s)} E ‖ φ (s, y (s)) - φ (s, x (s)) ‖^{p} d s \\ \leq & M^{p} \frac{1}{γ^{p}} C_{2} ‖ y - x ‖_{\infty}^{p} \cdot \end{matrix}

(4.5)

Up taking an estimate on the Itô integral in the last term, we find

\begin{matrix} E {‖\int_{- \infty}^{t} Q (t - s) [ψ (s, y (s)) - ψ (s, x (s))] d W (s)‖}^{p} \\ \leq & C_{p} E {(\int_{0}^{+ \infty} ‖ Q (s) ‖^{2} ‖ ψ (t - s, y (t - s)) - ψ (t - s, x (t - s))] ‖^{2} d s)}^{\frac{p}{2}} \\ \leq & C_{p} M^{p} {(\int_{0}^{+ \infty} e^{- 2 γ s} E ‖ ψ (t - s, y (t - s)) - ψ (t - s, x (t - s)) ‖^{2} d s)}^{\frac{p}{2}} \\ \leq & C_{p} M^{p} C_{3} {(\int_{0}^{+ \infty} e^{- 2 γ s} d s)}^{\frac{p - 2}{2}} (\int_{0}^{+ \infty} e^{- 2 γ s} E ‖ y (s) - x (s) ‖^{p} d s) \\ \leq & C_{p} M^{p} C_{3} {(\frac{1}{2 γ})}^{\frac{p}{2}} ‖ y - x ‖_{\infty}^{p} \cdot \end{matrix}

(4.6)

Putting all evaluations (4.2), (4.3), (4.5) and (4.6) together, we get

E ‖ (Λ y) (t) - (Λ x) (t) ‖^{p} \leq 4^{p - 1} [(1 + η^{p}) C_{1} + M^{p} \frac{1}{γ^{p}} C_{2} + C_{p} C_{3} M^{p} {(\frac{1}{2 γ})}^{\frac{p}{2}}] ‖ y - x ‖_{\infty}^{p} \cdot

This implies that,

E ‖ (Λ y) (t) - (Λ x) (t) ‖^{p} \leq R ‖ y - x ‖_{\infty}^{p} \cdot

Since by construction R < 1, we have the strict contraction of operator Λ and by the Banach contraction mapping principle, Λ has a unique fixed point.

Finally, to prove that X satisfies (3.3) for all τ ≤ t and $τ \in R$ ⁠.

Let

\begin{matrix} y (s) + ϕ (s, y (s)) & = & \int_{- \infty}^{s} A Q (s - ξ) ϕ (ξ, y (ξ)) d ξ \\ + \int_{- \infty}^{s} Q (s - ξ) φ (ξ, y (ξ)) d ξ + \int_{- \infty}^{s} Q (s - ξ) ψ (ξ, y (ξ)) d W (ξ) \cdot \end{matrix}

Multiplying by Q(t − s), one can show that

\begin{matrix} Q (t - s) [y (s) + ϕ (s, y (s))] = Q (t - s) \int_{- \infty}^{s} A Q (s - ξ) ϕ (ξ, y (ξ)) d ξ \\ + Q (t - s) \int_{- \infty}^{s} Q (s - ξ) φ (ξ, y (ξ)) d ξ + Q (t - s) \int_{- \infty}^{s} Q (s - ξ) φ (ξ, y (ξ)) d W (ξ) \\ = & \int_{- \infty}^{s} A Q (t - ξ) ϕ (ξ, y (ξ)) d ξ + \int_{- \infty}^{s} Q (t - ξ) φ (ξ, y (ξ)) d ξ + \int_{- \infty}^{s} Q (t - ξ) ψ (ξ, y (ξ)) d W (ξ) \\ = & \int_{- \infty}^{t} A Q (t - ξ) ϕ (ξ, y (ξ)) d ξ + \int_{- \infty}^{t} Q (t - ξ) φ (ξ, y (ξ)) d ξ + \int_{- \infty}^{t} Q (t - ξ) ψ (ξ, y (ξ)) d W (ξ) \\ - \int_{s}^{t} A Q (t - ξ) G (ξ, y (ξ)) d ξ - \int_{s}^{t} Q (t - ξ) φ (ξ, y (ξ)) d ξ - \int_{s}^{t} Q (t - ξ) F (ξ, y (ξ)) d W (ξ) \\ = & y (t) + ϕ (t, y (t)) + \int_{s}^{t} A Q (t - ξ) ϕ (ξ, y (ξ)) d ξ \\ + \int_{s}^{t} Q (t - ξ) φ (ξ, y (ξ)) d ξ + \int_{s}^{t} Q (t - ξ) ψ (ξ, y (ξ)) d W (ξ) \cdot \end{matrix}

Then,

\begin{matrix} Q (t - s) [y (s) + ϕ (s, y (s))] & = & y (t) + ϕ (t, y (t)) + \int_{s}^{t} A Q (t - ξ) ϕ (ξ, y (ξ)) d ξ \\ + \int_{s}^{t} Q (t - ξ) φ (ξ, y (ξ)) d ξ + \int_{s}^{t} Q (t - ξ) ψ (ξ, y (ξ)) d W (ξ) . \end{matrix}

Hence $y \in {P S A B P}_{ω, k} (R, L^{p} (P, F), μ, ν, q)$ is the unique solution to equation (1.1). The proof is complete. ■

5. Global exponential stability

In this section, we need an assumptions to present the exponential stability of a (μ, ν)-pseudo-S-asymptotically Bloch type periodic mild solution of (1.1).

H4.

There exists constant C₄, such that

E ‖ ϕ (t, x (t)) - ϕ (t, y (t)) ‖^{p} \leq C_{4} e^{- γ t} E ‖ x (t) - y (t) ‖^{p}, t \in R, x, y \in L^{p} (P, F) .

Theorem 5.1.

Under assumptions of Theorem 4.1 and (H4). We also assume that

L = - γ + 4^{p - 1} M^{p} [\frac{C_{2}}{γ^{p - 1}} + C_{p} C_{3} {(\frac{1}{2 γ})}^{\frac{p - 2}{2}}] < 0 .

Then, the mild solution of (1.1) is exponentially stable.

Proof: Let y(t) be a fixed point of Λ in ${P S A B P}_{ω, k} (R, L^{p} (P, F), μ, ν, q)$ by Theorem 4.1, any fixed point of F is a mild solution of the equation (1.1). Now let x(t) be an arbitrary solution of system (1.1) with initial value x(0), then

\begin{matrix} E ‖ x (t) - y (t) ‖^{p} & = & E ‖ - ϕ (t, x (0)) + ϕ (t, y (0)) + \int_{0}^{t} A Q (t - s) [ϕ (s, x (s)) - ϕ (s, y (s))] d s \\ + \int_{0}^{t} Q (t - s) [φ (s, x (s)) - φ (s, y (s))] d s \\ + {\int_{0}^{t} Q (t - s) [ψ (s, x (s)) - ψ (s, y (s))] d B (s)‖}^{p} . \end{matrix}

Using an estimate on the Itô integral, we get

\begin{matrix} E ‖ x (t) - y (t) ‖^{p} & \leq & 4^{p - 1} E {‖ϕ (t, x (t)) - ϕ (t, y (t))‖}^{p} \\ + 4^{p - 1} E {(\int_{0}^{t} ‖A Q (t - s) [ϕ (s, x (s)) - ϕ (s, y (s))]‖ d s)}^{p} \\ + 4^{p - 1} E {(\int_{0}^{t} ‖Q (t - s) [φ (s, x (s)) - φ (s, y (s))]‖ d s)}^{p} \\ + 4^{p - 1} C_{p} E {(\int_{0}^{t} {‖Q (t - s) [ψ (s, x (s)) - ψ (s, y (s))]‖}^{2} d s)}^{p / 2} . \end{matrix}

Using the Hölder inequality, we obtain

\begin{matrix} E ‖ x (t) - y (t) ‖^{p} & \leq & 4^{p - 1} C_{4} e^{- γ t} E {‖x (t) - y (t)‖}^{p} \\ + 4^{p - 1} C_{4} {(\int_{0}^{t} ‖A Q (t - s)‖ d s)}^{p} E ‖ x (s) - y (s) ‖^{p} \\ + 4^{p - 1} {(\int_{0}^{t} ‖ Q (t - s) ‖ d s)}^{p - 1} \int_{0}^{t} {‖Q (t - s) ‖ E ‖ φ (s, x (s)) - φ (s, y (s))‖}^{p} d s \\ + 4^{p - 1} C_{p} {(\int_{0}^{t} {‖Q (t - s)‖}^{2} d s)}^{\frac{p - 2}{2}} \int_{0}^{t} {‖Q (t - s)‖}^{2} E ‖ ψ (s, x (s)) - ψ (s, y (s)) ‖^{p} d s \end{matrix}

which implies that

\begin{matrix} E ‖ x (t) - y (t) ‖^{p} & \leq & 4^{p - 1} C_{4} e^{- γ t} E {‖x (t) - y (t)‖}^{p} \\ + 4^{p - 1} C_{4} e^{- γ t} η^{p} E ‖ x (s) - y (s) ‖^{p} \\ + 4^{p - 1} M^{p} {(\int_{0}^{t} e^{- γ (t - s)} d s)}^{p - 1} \int_{0}^{t} e^{- γ (t - s)} E ‖ φ (s, x (s)) - φ (s, y (s)) ‖^{p} d s \\ + 4^{p - 1} C_{p} {(\int_{0}^{t} {‖Q (t - s)‖}^{2} d s)}^{\frac{p - 2}{2}} \int_{0}^{t} {‖Q (t - s)‖}^{2} E ‖ ψ (s, x (s)) - ψ (s, y (s)) ‖^{p} d s \\ \leq & 4^{p - 1} C_{4} e^{- γ t} [E {‖x (t) - y (t)‖}^{p} + η^{p} E ‖ x (s) - y (s) ‖^{p}] \\ + 4^{p - 1} M^{p} C_{2} \frac{1}{γ^{p - 1}} \int_{0}^{t} e^{- γ (t - s)} E ‖ x (s) - y (s) ‖^{p} d s \\ + 4^{p - 1} C_{p} C_{3} M^{p} {(\frac{1}{2 γ})}^{\frac{p - 2}{2}} \int_{0}^{t} e^{- 2 γ (t - s)} E ‖ x (s) - y (s) ‖^{p} d s \\ \leq & 4^{p - 1} C_{4} e^{- γ t} [E {‖x (t) - y (t)‖}^{p} + η^{p} E ‖ x (s) - y (s) ‖^{p}] \\ + 4^{p - 1} M^{p} [C_{2} \frac{1}{γ^{p - 1}} + C_{p} C_{3} {(\frac{1}{2 γ})}^{\frac{p - 2}{2}}] \\ \times \int_{0}^{t} e^{- γ (t - s)} E ‖ x (s) - y (s) ‖^{p} d s . \end{matrix}

Then

\begin{matrix} e^{γ t} E ‖ x (t) - y (t) ‖^{p} & \leq & 4^{p - 1} C_{4} [E {‖x (t) - y (t)‖}^{p} + η^{p} E ‖ x (s) - y (s) ‖^{p}] \\ + 4^{p - 1} M^{p} [C_{2} \frac{1}{γ^{p - 1}} + C_{p} C_{3} {(\frac{1}{2 γ})}^{\frac{p - 2}{2}}] \\ \times \int_{0}^{t} e^{γ s} E ‖ x (s) - y (s) ‖^{p} d s . \end{matrix}

Using Gronwall-Bellman lemma, we obtain

\begin{matrix} e^{γ t} E ‖ x (t) - y (t) ‖^{p} & \leq & 4^{p - 1} C_{4} [E {‖x (t) - y (t)‖}^{p} + η^{p} E ‖ x (s) - y (s) ‖^{p}] \\ \times e^{\int_{0}^{t} 4^{p - 1} M^{p} [\frac{C_{2}}{γ^{p - 1}} + C_{p} C_{3} {(\frac{1}{2 γ})}^{\frac{p - 2}{2}}] d s} \\ \leq & 4^{p - 1} C_{4} [E {‖x (t) - y (t)‖}^{p} + η^{p} E ‖ x (s) - y (s) ‖^{p}] \\ \times e^{4^{p - 1} M^{p} [\frac{C_{2}}{γ^{p - 1}} + C_{p} C_{3} {(\frac{1}{2 γ})}^{\frac{p - 2}{2}}] t} . \end{matrix}

Then

\begin{matrix} E ‖ x (t) - y (t) ‖^{p} & \leq & 4^{p - 1} C_{4} [E {‖x (t) - y (t)‖}^{p} + η^{p} E ‖ x (s) - y (s) ‖^{p}] \\ \times e^{[- γ + 4^{p - 1} M^{p} [\frac{C_{2}}{γ^{p - 1}} + C_{p} C_{3} {(\frac{1}{2 γ})}^{\frac{p - 2}{2}}] t} . \end{matrix}

Since

- γ + 4^{p - 1} M^{p} [\frac{C_{2}}{γ^{p - 1}} + C_{p} C_{3} {(\frac{1}{2 γ})}^{\frac{p - 2}{2}}] < 0

then, the mild solution of (1.1) is exponentially stable ■

6. Example

The aim of this section, we will apply our theoretical results, we discuss the existence and uniqueness of (μ, ν) pseudo S − asymptotically Bloch solution of the scalar reaction-diffusion equation given by

\{\begin{cases} d y (t, x) = \frac{\partial^{2}}{\partial x^{2}} y (t, x) d t + φ (t, x) d t + ψ (t, x) d W (t), (t, x) \in R \times (0, π), \\ y (t, 0) = y (t, π), \end{cases}

(6.1)

where

φ (t, x) = υ (t) θ_{1} (x),

ψ (t, x) = υ (t) θ_{2} (x) .

Let H = L²[0, π], assume that γ(t) is bounded, continuous, and ω-periodic, i.e., υ(t + ω) = υ(t), and that the function θ_j, (j = 1, 2) satisfies the conditions

θ_{j} (e^{i k ω} x) = e^{i k ω} θ_{j} (x), {‖θ_{j} (x) - θ_{j} (y)‖}_{H}^{2} \leq L_{θ_{j}} {‖x - y‖}_{H}^{2}, for all x, y \in H, with L_{θ_{j}} > 0 .

Let $(P, F, P, G_{t})$ be the filtered probability space. W(t) is a two-sided and standard one-dimensional Brownian motion. To apply our theoretical results, we consider the measure ν where its Radon-Nikodym derivative is defined by

ρ (t) = \{\begin{cases} e^{t} if t \leq 0 \\ 1 otherwise \end{cases}

and μ = ν. Then, μ and ν satisfy (M₁) and (M₂).

We consider the linear operator A: ⊂ L²(0, 1) → L²(0, 1), given by

\begin{matrix} D (A) & = & H^{2} (0,1) \cap H_{0}^{1} (0,1) \\ A y (ξ) & = & y^{″} (ξ) for ξ \in (0,1) and y \in D (A) . \end{matrix}

It is well known that A generates a C₀ − semigroup (Q(t))_t ≥ 0 on L²(0, 1) defined by

(Q (t) y) (r) = \sum_{n = 1}^{\infty} e^{- n^{2} π^{2} t} {〈 y, e_{n} 〉}_{L^{2}} e_{n} (r),

where $e_{n} (r) = \sqrt{2} \sin (n π r)$ for n = 1, 2, ⋯ , and $‖ Q (t) ‖ \leq e^{- π^{2} t}$ for all t ≥ 0.

Since,

φ (t + ω, x) = υ (t + ω) θ_{1} (x) = υ (t) e^{i k ω} θ_{1} (e^{- i k ω} x) = e^{i k ω} φ (t, e^{- i k ω} x)

ψ (t + ω, x) = υ (t + ω) θ_{2} (x) = υ (t) e^{i k ω} θ_{2} (e^{- i k ω} x) = e^{i k ω} ψ (t, e^{- i k ω} x)

\begin{matrix} E ‖ φ (t, x) - φ (t, y) ‖_{H}^{2} & = & E ‖ υ (t) θ_{1} (x) - υ (t) θ_{1} (y) ‖_{H}^{2} \\ = & E ‖ υ (t) (θ_{1} (x) - θ_{1} (y)) ‖_{H}^{2} \\ \leq & ‖ υ ‖_{\infty}^{2} L_{θ_{1}} E ‖ x - y ‖_{H}^{2} . \end{matrix}

And

E ‖ ψ (t, x) - ψ (t, y) ‖_{H}^{2} \leq ‖ υ ‖_{\infty}^{2} L_{θ_{2}} E ‖ x - y ‖_{H}^{2} .

Then by Theorem 3.2, $φ, ψ \in {P S A B P}_{ω, k} (R, L^{2} (0, π), μ, ν, q)$ ⁠.

Therefore, by Theorems 4.1 and 5.1, the system (6.1) has a unique square-mean pseudo S-asymptotically Bloch type periodic mild solution on $R$ provided that $‖ υ ‖_{\infty}^{2} < {(\frac{L_{θ_{1}}}{π^{2}} + \frac{2 L_{θ_{2}}}{π^{2}})}^{- 1}$ and globally exponential stable if $4 L_{θ_{2}} + 2 L_{θ_{1}} - π^{4} < π^{2}$

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Pseudo S-asymptotically Bloch type periodicity with applications to partial stochastic neutral evolution equations

1. Introduction

2. Preliminaries

3. (α, β)-pseudo-S-asymptotically Bloch type periodic function

4. Partial functional differential equations with finite delay

5. Global exponential stability

6. Example

References

Further reading

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Pseudo S-asymptotically Bloch type periodicity with applications to partial stochastic neutral evolution equations

1. Introduction

2. Preliminaries

3. (α, β)-pseudo-S-asymptotically Bloch type periodic function

4. Partial functional differential equations with finite delay

5. Global exponential stability

6. Example

References

Further reading

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Email Alerts

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