Asymptotic Almost Periodicity of Stochastic Evolution Equations

Abstract

In this paper, we establish a new existence theorem for asymptotically almost periodic mild solutions to a class of semilinear stochastic evolution equations. As one would expect, our result would generalize and improve some related results in this area.

1 Introduction

The theory of almost periodic functions was introduced in the literature around 1924–1926 with the pioneering works of the Danish mathematician Bohr [1, 2]. Loosely speaking, almost periodic functions are those functions which come arbitrarily close to being periodic when one looks over long enough time scales, they play an important role in describing the phenomena that are similar to the periodic oscillations which can be observed frequently in many fields, such as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma physics, engineering, ecosphere and so on [3,4,5]. In the last several decades, the almost periodic functions have been investigated by many mathematicians, the basic theories on almost periodic functions have been well developed and have led to strong development of harmonic analysis on groups and compact topological semigroups of linear operators, etc. (see for instance [6,7,8,9,10,11]). Moreover, the almost periodic functions have been applied successfully to the investigation of almost periodic dynamics produced by many different kinds of differential equations [10,11,12,13,14,15,16,17,18,19,20,21].

As a natural extension of almost periodicity, the concept of asymptotic almost periodicity was introduced in the literature [22, 23] by Fr$\acute{\hbox {e}}$chet in the early 1940s. Since then, this notion has generated lots of developments and applications. In particular, many authors apply the asymptotic property of asymptotically almost periodic functions to determine the existence of almost periodic solutions to various ordinary differential equations, partial differential equations, functional differential equations, integro-differential equations as well as fractional differential equations (see for instance [24,25,26,27,28,29,30,31,32] and the references therein), and due to the significance and applications of almost periodic functions as well as asymptotically almost periodic functions in control theory, mathematical biology and physics, etc., the study of almost periodic as well as asymptotically almost periodic solutions to various differential equations becomes an attractive topic in the qualitative theory of differential equations.

Besides, noise or stochastic perturbation is unavoidable and omnipresent in nature as well as in man-made systems. Therefore, it is of great significance to import the stochastic effects into the investigation of differential systems [33, 34]. For the elementary theories and analysis methods for stochastic differential systems, we refer the reader to the monographs of $\emptyset$sendal [35].

The concepts of almost periodicity and asymptotic almost periodicity are also important in probability for investigating stochastic processes, and they are interesting for applications arising in mathematical physics and statistics. From the 1980s, almost periodic solutions to stochastic differential equations driven by Gaussian noise have been studied extensively, e.g., see Halanay [36] for the connection between the existence of almost periodic solutions and the exponential stability or exponential dichotomy for finite-dimensional linear inhomogeneous stochastic systems, Tudor [37] as well as Da Prato and Tudor [38] for the periodic and almost periodic solutions in distribution sense for stochastic evolution equations, and Arnold and Tudor [39] for almost periodic solutions in distribution sense for stochastic ordinary affine equations.

Recently, in [40, 41], the concept of p-mean almost periodicity was introduced and studied by Bezandry and Diagana. In particular, such a concept was utilized to study the existence and uniqueness of quadratic-mean almost periodic solution to a class of stochastic differential equations. And in [42], Huang and Yang presented some criteria ensuring the existence and uniqueness of quadratic-mean almost periodic solution as well as global exponential stability of the quadratic-mean almost periodic solution for stochastic cellular neural networks with delay. Motivated by [40,41,42], in [43], Cao, Yang and Huang investigated the existence and stability of quadratic-mean almost periodic mild solutions for a class of stochastic functional differential equations. Some sufficient conditions ensuring the existence and stability of quadratic-mean almost periodic mild solutions were presented. More recently, in [44], Cao, Yang, Huang and Liu furthermore introduced and developed the notion of p-mean asymptotic almost periodicity for stochastic processes. They showed that each p-mean asymptotically almost periodic stochastic process is stochastically bounded, and the collection of all p-mean asymptotically almost periodic stochastic processes is a Banach space when it is equipped with some norm. They subsequently applied the basic results to study the existence, uniqueness as well as global exponential stability of quadratic-mean asymptotically almost periodic mild solution to a class of stochastic functional differential equations. These works generalized the almost periodic and asymptotically almost periodic theory from the deterministic version to the stochastic one. For more details about this topic and the related works, one may further refer to [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68].

Stimulated by the works above, this paper is concerned with the existence of asymptotically almost periodic mild solutions to the following semilinear stochastic evolution equations

$$\begin{aligned} \hbox {d}x(t)=[Ax(t)+F(t,x(t))]\hbox {d}t+G(t,x(t))\hbox {d}W(t), \quad t\in {\mathbb {R}}. \end{aligned}$$

(1.1)

A new existence theorem of p-mean asymptotically almost periodic mild solutions to Eq. (1.1) is established. In our result, F(t, x) and G(t, x) do not have to meet the (locally) Lipschitz conditions with respect to x (see Remark 3.1). However, many papers (such as [44, 45]) on asymptotically almost periodic solutions need the nonlinearities of corresponding differential equations to satisfy the (locally) Lipschitz conditions and hence the Banach contraction principle becomes the key tool in the study of the corresponding problems. As can be seen, the hypotheses in our result are reasonably weak (see Remarks 3.2 and 3.6). Moreover, in [44, 45], the authors studied the existence of square-mean asymptotically almost periodic mild solutions, which is a special case ($p=2$) of the p-mean asymptotically almost periodic mild solutions; the present paper deals with the p-mean asymptotically almost periodic mild solutions for any $p\ge 2$. So our result generalizes those as well as related research and has more broad applications.

The rest of this paper is organized as follows. In Sect. 2, some concepts, the related notations and some useful lemmas are introduced. In Sect. 3, we present some criteria ensuring the existence of asymptotically almost periodic mild solutions. An example is given to illustrate our result in Sect. 4.

2 Preliminaries

This section is concerned with some notations, definitions, lemmas and preliminary facts which are used in what follows.

Let $(\varOmega , {\mathbb {F}}, {\mathbb {P}})$ be a probability space. For a real separable Hilbert space $({\mathbb {H}},\Vert \cdot \Vert )$ and $p\ge 2$, denote by $L^{p}({\mathbb {P}},{\mathbb {H}})$ the space of all ${\mathbb {H}}$-value random variables Y such that

$$\begin{aligned} {\mathbb {E}}\Vert Y\Vert ^{p}=\int _{\varOmega } \Vert Y\Vert ^{p}\mathrm{d}{\mathbb {P}}<\infty . \end{aligned}$$

For $Y\in L^{p}({\mathbb {P}},{\mathbb {H}})$, let

$$\begin{aligned} \Vert Y\Vert _{p}:=\left( \int _{\varOmega } \Vert Y\Vert ^{p}d{\mathbb {P}}\right) ^{\frac{1}{p}}<\infty . \end{aligned}$$

Then $L^{p}({\mathbb {P}},{\mathbb {H}})$ is a Hilbert space equipped with the norm $\Vert \cdot \Vert _{p}$.

The following useful Definitions 2.1–2.4 and Lemma 2.1 come from [40, 41].

Definition 2.1

[40, 41] A stochastic process $Y: {\mathbb {R}}\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}})$ is said to be $L^{p}$-bounded if there exists a constant $M>0$ such that ${\mathbb {E}}\Vert Y\Vert ^{p}<M$.

Definition 2.2

[40, 41] A stochastic process $Y: {\mathbb {R}}\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}})$ is said to be $L^{p}$-continuous provided that for any $s\in {\mathbb {R}}$,

$$\begin{aligned} \lim \limits _{t\rightarrow s}\mathbb { E}\Vert Y(t)-Y(s)\Vert ^{p}=0. \end{aligned}$$

Denote by $SBC({\mathbb {R}}, L^{p}({\mathbb {P}}, {\mathbb {H}}))$ the collection of all $L^{p}$-bounded and $L^{p}$-continuous stochastic processes. It is easy to check that $SBC({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ is a Banach space when it is equipped with the norm

$$\begin{aligned} \Vert Y\Vert _{\infty }=\sup \limits _{t\in {\mathbb {R}}}\Big \{\Big ({\mathbb {E}}\Vert X(t)\Vert ^{p}\Big )^{\frac{1}{p}}\Big \}, \end{aligned}$$

see e.g. [40, 41].

Definition 2.3

[40, 41] An $L^{p}$-continuous stochastic process $X: {\mathbb {R}}\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}})$ is said to be p-mean almost periodic if for each $\varepsilon >0$ there exists $l(\varepsilon )>0$ such that any interval of length $l(\varepsilon )$ contains at least a number $\tau$ for which

$$\begin{aligned} \sup \limits _{t\in {\mathbb {R}}}{\mathbb {E}}\Vert X(t+\tau )-X(t)\Vert ^{p}<\varepsilon . \end{aligned}$$

The almost periodicity in Definition 2.3 is also called Bochner almost periodicity. Note that an almost periodic stochastic process is actually a usual $L^{p}({\mathbb {P}}, {\mathbb {H}})$-valued almost periodic function.

The collection of all almost periodic stochastic processes $X: {\mathbb {R}}\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}})$ is denoted by $\mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$. It is easy to check that $\mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ is a closed subspace of $SBC({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$. Therefore, $\mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ is also a Banach space when it is equipped with the norm $\Vert \cdot \Vert _{\infty }$ (see e.g.[40, 41]).

Let $(\mathbb {U},\Vert \cdot \Vert _{\mathbb {U}})$ be another real separable Hilbert space, and $L^{p}({\mathbb {P}}, \mathbb {U})$ be its corresponding $L^{p}$-space.

Definition 2.4

[40, 41] A function

$$\begin{aligned} F(t, x):{\mathbb {R}}\times L^{p}({\mathbb {P}},\mathbb {U})\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}}), (t, x)\rightarrow F(t, x), \end{aligned}$$

which is jointly continuous, is called p-mean almost periodicity in $t\in {\mathbb {R}}$ for each $x\in L^{p}({\mathbb {P}},\mathbb {U})$, provided that for each $\varepsilon >0$ there exists $l=l(\varepsilon )>0$ such that any interval of length l contains at least a number $\tau$ for which

$$\begin{aligned} \sup \limits _{t\in {\mathbb {R}}}{\mathbb {E}}\Vert F(t+\tau , x)-F(t, x)\Vert ^{p}<\varepsilon \end{aligned}$$

for each stochastic process $x\in L^{p}({\mathbb {P}},\mathbb {U})$.

The collection of such functions will be denoted by $\mathrm{AP}({\mathbb {R}}\times L^{p}({\mathbb {P}},\mathbb {U}), L^{p}({\mathbb {P}},{\mathbb {H}}))$.

Lemma 2.1

[40, 41] Let

$$\begin{aligned} F:{\mathbb {R}}\times L^{p}({\mathbb {P}},\mathbb {U})\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}}), \quad (t, x)\rightarrow F(t, x) \end{aligned}$$

be p-mean almost periodic in $t\in {\mathbb {R}}$ for each $x\in L^{p}({\mathbb {P}},\mathbb {U})$. Suppose that F(t, x) is Lipschitzian in the following sense:

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert F(t, x)-F(t, y)\Vert ^{p}\le L{\mathbb {E}}\Vert x-y\Vert ^{p}_{U} \quad \hbox {for all} \ x, y\in L^{p}({\mathbb {P}},\mathbb {U}) \ \hbox {and each} \ t\in {\mathbb {R}}, \end{aligned} \end{aligned}$$

where $L>0$ is a constant. Then for any p-mean almost periodic stochastic process $\varPhi : {\mathbb {R}}\rightarrow L^{p}({\mathbb {P}},\mathbb {U})$, the stochastic process $t\rightarrow F(t, \varPhi (t))$ is p-mean almost periodic.

Denote by $C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ the space of all $L^{p}$-continuous stochastic processes $Z:{\mathbb {R}}\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}})$ such that

$$\begin{aligned} \lim \limits _{|t|\rightarrow +\infty }{\mathbb {E}}\Vert Z(t)\Vert ^{p}=0, \end{aligned}$$

and denote by $C_{0}({\mathbb {R}}\times L^{p}({\mathbb {P}},\mathbb {U}), L^{p}({\mathbb {P}},{\mathbb {H}}))$ the space of all jointly continuous functions $Z:{\mathbb {R}}\times L^{p}({\mathbb {P}},\mathbb {U})\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}})$ such that

$$\begin{aligned} \lim \limits _{|t|\rightarrow +\infty }{\mathbb {E}}\Vert Z(t,x)\Vert ^{p}=0 \ \hbox {uniformly for} \ x\in L^{p}({\mathbb {P}},\mathbb {U}). \end{aligned}$$

Remark 2.1

[44] $C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ is a linear closed subspace of $SBC({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$. Therefore, $C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ is also a Banach space when it is equipped with the norm $\Vert \cdot \Vert _{\infty }$.

Now, we give the concept of p-mean asymptotically almost periodic stochastic process.

Definition 2.5

[44] An $L^{p}$-continuous stochastic process $X:{\mathbb {R}}\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}})$ is said to be p-mean asymptotically almost periodic if there exist two $L^{p}$-continuous stochastic processes

$$\begin{aligned} Y(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})), \quad Z(t)\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})) \end{aligned}$$

such that

$$\begin{aligned} X(t)=Y(t)+Z(t). \end{aligned}$$

By $\mathrm{AAP}({\mathbb {R}},L^{p}({\mathbb {P}},{\mathbb {H}}))$ we denote the collection of all p-mean asymptotically almost periodic stochastic processes.

Remark 2.2

[44] It is easy to see that

$$\begin{aligned} \mathrm{AAP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))=\mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))\oplus C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Then the space $\mathrm{AAP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ of p-mean asymptotically almost periodic stochastic processes is also a Banach space when it is equipped with the norm $\Vert \cdot \Vert _{\infty }$.

Definition 2.6

[44] A function

$$\begin{aligned} F(t, x):{\mathbb {R}}\times L^{p}({\mathbb {P}},\mathbb {U})\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}}), (t, x)\rightarrow F(t, x), \end{aligned}$$

which is jointly continuous, is called p-mean asymptotic almost periodicity in $t\in {\mathbb {R}}$ for each $x\in L^{p}({\mathbb {P}},\mathbb {U})$, if there exist two jointly continuous functions

$$\begin{aligned} G(t, x)\in \mathrm{AP}({\mathbb {R}}\times L^{p}({\mathbb {P}},\mathbb {U}), L^{p}({\mathbb {P}},{\mathbb {H}})), \quad H(t, x)\in C_{0}({\mathbb {R}}\times L^{p}({\mathbb {P}},\mathbb {U}), L^{p}({\mathbb {P}},{\mathbb {H}})) \end{aligned}$$

such that

$$\begin{aligned} F(t, x)=G(t, x)+H(t, x). \end{aligned}$$

By $\mathrm{AAP}({\mathbb {R}}\times L^{p}({\mathbb {P}},\mathbb {U}), L^{p}({\mathbb {P}},{\mathbb {H}}))$, we denote the collection of such functions.

In the following, we present the following compactness criterion, which is a special case of the general compactness result of Theorem 2.1 in [28].

Lemma 2.2

[28] Let S be a Banach space, and $C_{0}({\mathbb {R}}, S)$ be the closed subspace of the Banach space of all bounded, continuous functions from ${\mathbb {R}}$ to S, consisting of functions vanishing at infinity. A set $D\subset C_{0}({\mathbb {R}}, S)$ is relatively compact if

• (1) D is equicontinuous;

• (2) $\lim \limits _{|t|\rightarrow +\infty }u(t)=0$ uniformly for $u\in D$;

• (3) the set $D(t):=\{u(t): u\in D\}$ is relatively compact in S for each $t\in {\mathbb {R}}$.

The following Krasnoselskii’s fixed point theorem plays a key role in the proofs of our main results, which can be found in many books.

Lemma 2.3

[69] Let B be a bounded closed and convex subset of X, and $J_{1}, J_{2}$ be maps of B into X such that $J_{1}x+J_{2}y\in B$ for every pair $x, y\in B$. If $J_{1}$ is a contraction and $J_{2}$ is completely continuous, then the equation $J_{1}x+J_{2}x=x$ has a solution on B.

3 Existence of Asymptotically Almost Periodic Mild Solutions

In this section, we study the existence of p-mean asymptotically almost periodic mild solutions to the following semilinear stochastic evolution equations

$$\begin{aligned} \hbox {d}x(t)=[Ax(t)+F(t,x(t))]\hbox {d}t+G(t,x(t))\hbox {d}W(t), \quad t\in {\mathbb {R}}, \end{aligned}$$

(3.1)

where the operator $A: D(A)\subset L^{p}({\mathbb {P}},{\mathbb {H}})\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}})$ is the infinitesimal generator of a compact and uniformly exponentially stable $C_{0}$-semigroup $\{T(t)\}_{t\ge 0}$, i.e. there exist two constants $M, \delta >0$ such that

$$\begin{aligned} \Vert T(t)\Vert \le M e^{-\delta t} \quad \hbox {for all} \ t>0, \end{aligned}$$

W(t) is a two-sided and standard one-dimensional Brownian motion, which is defined on the filtered probability space $(\varOmega ,{\mathbb {F}}, {\mathbb {P}}, {\mathbb {F}}_{t})$ with

$$\begin{aligned} {\mathbb {F}}_{t}=\sigma \{W(u)-W(v): u, v\le t\}, \end{aligned}$$

and $F, G:{\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}})\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}})$ are given jointly continuous functions satisfying the following assumptions:

($H_{1}$) $F(t, x)=F_{1}(t, x)+F_{2}(t, x)\in \mathrm{AAP}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}}))$ with

$$\begin{aligned} F_{1}(t, x)\in \mathrm{AP}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}})), \quad F_{2}(t, x)\in C_{0}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}})) \end{aligned}$$

and there exists a constant $L_{1}>0$ such that

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert F_{1}(t, x)-F_{1}(t, y)\Vert ^{p}\le L_{1}{\mathbb {E}}\Vert x-y\Vert ^{p} \quad \hbox {for all} \ t\in {\mathbb {R}}, \quad x, y\in L^{p}({\mathbb {P}},{\mathbb {H}}). \end{aligned} \end{aligned}$$

(3.2)

Moreover, there exist a function $\beta _{1}(t)\in C_{0}({\mathbb {R}}, {\mathbb {R}}^{+})$ and a nondecreasing function $\varPhi _{1}: {\mathbb {R}}^{+}\rightarrow {\mathbb {R}}^{+}$ such that for all $t\in {\mathbb {R}}$ and $x\in L^{p}({\mathbb {P}},{\mathbb {H}})$ with ${\mathbb {E}}\Vert x\Vert ^{p}\le r$,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert F_{2}(t, x)\Vert ^{p}\le \beta _{1}(t)\varPhi _{1}(r) \ \ \hbox {and} \ \ \liminf \limits _{r\rightarrow +\infty }\frac{\varPhi _{1}(r)}{r}=\rho _{1}. \end{aligned} \end{aligned}$$

(3.3)

($H_{2}$) $G(t, x)=G_{1}(t, x)+G_{2}(t, x)\in \mathrm{AAP}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}}))$ with

$$\begin{aligned} G_{1}(t, x)\in \mathrm{AP}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}})), \quad G_{2}(t, x)\in C_{0}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}})) \end{aligned}$$

and there exists a constant $L_{2}>0$ such that

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert G_{1}(t, x)-G_{1}(t, y)\Vert ^{p}\le L_{2}{\mathbb {E}}\Vert x-y\Vert ^{p} \quad \hbox {for all} \ t\in {\mathbb {R}}, \quad x, y\in L^{p}({\mathbb {P}},{\mathbb {H}}). \end{aligned} \end{aligned}$$

(3.4)

Moreover, there exist a function $\beta _{2}(t)\in C_{0}({\mathbb {R}}, {\mathbb {R}}^{+})$ and a nondecreasing function $\varPhi _{2}: {\mathbb {R}}^{+}\rightarrow {\mathbb {R}}^{+}$ such that for all $t\in {\mathbb {R}}$ and $x\in L^{p}({\mathbb {P}},{\mathbb {H}})$ with ${\mathbb {E}}\Vert x\Vert ^{p}\le r$,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert G_{2}(t, x)\Vert ^{p}\le \beta _{2}(t)\varPhi _{2}(r) \ \ \hbox {and} \ \ \liminf \limits _{r\rightarrow +\infty }\frac{\varPhi _{2}(r)}{r}=\rho _{2}. \end{aligned} \end{aligned}$$

(3.5)

Remark 3.1

Assume that F(t, x) and G(t, x) satisfy the assumptions ($H_{1}$) and ($H_{2}$), respectively, it is noted that F(t, x) as well as G(t, x) as a whole does not have to meet the Lipschitz continuity with respect to x in the following sense:

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert F(t, x)-F(t, y)\Vert ^{p}\le L_{1}'{\mathbb {E}}\Vert x-y\Vert ^{p}, \quad {\mathbb {E}}\Vert G(t, x)-G(t, y)\Vert ^{p}\le L_{2}'{\mathbb {E}}\Vert x-y\Vert ^{p} \end{aligned} \end{aligned}$$

for all $x, y\in L^{p}({\mathbb {P}}, {\mathbb {H}})$ and each $t\in {\mathbb {R}}$, where $L_{1}', L_{2}'>0$ are two constants, which are important conditions in [44, 45]. The class of p-mean asymptotically almost periodic functions F(t, x) satisfying the assumption ($H_{1}$) are more complicated than those satisfying Lipschitz continuity with respect to x and little is known about them.

Lemma 3.1

Given

$$\begin{aligned} F(t, x)=F_{1}(t, x)+F_{2}(t, x)\in \mathrm{AAP}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}})) \end{aligned}$$

with

$$\begin{aligned} F_{1}(t, x)\in \mathrm{AP}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}})), \quad F_{2}(t, x)\in C_{0}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Then it yields that

$$\begin{aligned} \begin{aligned} \sup \limits _{t\in {\mathbb {R}}}{\mathbb {E}}\Vert F_{1}(t, x)-F_{1}(t, y)\Vert ^{p}\le \sup \limits _{t\in {\mathbb {R}}}{\mathbb {E}}\Vert F(t, x)-F(t, y)\Vert ^{p}, \quad x, y\in L^{p}({\mathbb {P}},{\mathbb {H}}). \end{aligned} \end{aligned}$$

(3.6)

Proof

To show our result, it suffices to verify that

$$\begin{aligned} \{F_{1}(t, x)-F_{1}(t, y): t\in {\mathbb {R}}\}\subset \overline{\{F(t, x)-F(t, y): t\in {\mathbb {R}}\}}, \quad x, y\in L^{p}({\mathbb {P}},{\mathbb {H}}). \end{aligned}$$

In fact, if this is not the case, then for fixed $x, y\in L^{p}({\mathbb {P}},{\mathbb {H}})$, there exist some $t_{0}\in {\mathbb {R}}$ and $\varepsilon >0$ such that

$$\begin{aligned} {\mathbb {E}}\Vert (F_{1}(t_{0}, x)-F_{1}(t_{0}, y))-(F(t, x)-F(t, y))\Vert ^{p}\ge 3^{p}\varepsilon \quad \hbox {for all} \ t\in {\mathbb {R}}. \end{aligned}$$

Assume, without loss of generality, that $t_{0}\ge 0$, since the case when $t_{0}\le 0$ can be treated in a similar way.

It is clear that

$$\begin{aligned} \lim \limits _{|t|\rightarrow +\infty }{\mathbb {E}}\Vert F_{2}(t, x)-F_{2}(t, y)\Vert ^{p}=0, \end{aligned}$$

which implies that there exists a positive number T such that

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert F_{2}(t, x)-F_{2}(t, y)\Vert ^{p}<\varepsilon \end{aligned} \end{aligned}$$

(3.7)

whenever $t\ge T$. Since $F_{1}(t, x)\in \mathrm{AP}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}}))$ one can take $l=l(\varepsilon )>0$ such that $[T, T + l]$ of length l contains at least a $\tau$ with the properties

$$\begin{aligned} {\mathbb {E}}\Vert F_{1}(t_{0}+\tau , x)-F_{1}(t_{0}, x)\Vert ^{p}<\varepsilon , \quad \ {\mathbb {E}}\Vert F_{1}(t_{0}+\tau , y)-F_{1}(t_{0}, y)\Vert ^{p}<\varepsilon , \end{aligned}$$

which together with

$$\begin{aligned} a^{p}\le 3^{p-1}[(a-b-c)^{p}+b^{p}+c^{p}]\Longrightarrow (a-b-c)^{p}\ge 3^{-p+1}a^{p}-b^{p}-c^{p}, \end{aligned}$$

enable us to find that

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert F_{2}(t_{0}+\tau , x)-F_{2}(t_{0}+\tau , y)\Vert ^{p}\\&\quad \ge {\mathbb {E}}\Big (\Vert F(t_{0}+\tau , x)-F(t_{0}+\tau , y)-F_{1}(t_{0}, X)+F_{1}(t_{0}, y)\Vert \\&\qquad -\,\Vert F_{1}(t_{0}+\tau , x)-F_{1}(t_{0}, x)\Vert -\Vert F_{1}(t_{0}+\tau , y)-F_{1}(t_{0}, y)\Vert \Big )^{p}\\&\quad \ge 3^{-p+1}{\mathbb {E}}\Vert F(t_{0}+\tau , x)-F(t_{0}+\tau , y)-F_{1}(t_{0}, x)+F_{1}(t_{0}, y)\Vert ^{p}\\&\qquad -\,{\mathbb {E}}\Vert F_{1}(t_{0}+\tau , x)-F_{1}(t_{0}, x)\Vert ^{p}-{\mathbb {E}}\Vert F_{1}(t_{0}+\tau , y)-F_{1}(t_{0}, y)\Vert ^{p}\ge \varepsilon , \end{aligned} \end{aligned}$$

which contradicts (3.7) (noticing $t_{0}+\tau \ge T$), completing the proof. $\square$

Remark 3.2

In Lemma 3.1, (3.6) implies that when F(t, x) meets the Lipschitz continuity with respect to x in the following sense:

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert F(t, x)-F(t, y)\Vert ^{p}\le L_{1}{\mathbb {E}}\Vert x-y\Vert ^{p} \quad \hbox {for all} \ x, y\in L^{p}({\mathbb {P}}, {\mathbb {H}})\ \hbox {and each} \ t\in {\mathbb {R}}, \end{aligned} \end{aligned}$$

(3.8)

where $L_{1}>0$ is a constant, then $F_{1}(t, x)$ satisfies (3.2) with the same Lipschitz constant $L_{1}>0$. Note that in many papers (such as [44, 45]) on asymptotically almost periodic solutions, to be able to apply the well-known Banach contraction principle, the (locally) Lipschitz conditions like (3.8) for the nonlinearities of corresponding differential equations are needed. Thus, our conditions in the assumptions ($H_{1}$) and ($H_{2}$) are weaker than those of [44, 45].

To prove our result, we also need the following stochastic integral inequality and composition result concerning p-mean asymptotically almost periodic functions.

Lemma 3.2

[68] Let $F:{\mathbb {R}}\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}})$ be a measurable stochastic process. Then for any $p\in [2, +\infty )$ and $s<t$,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\left\| \int _{s}^{t}F(\tau )\hbox {d}W(\tau )\right\| ^{p}\le C_{p}\left( {\mathbb {E}}\int _{s}^{t}\Vert F(\tau )\Vert ^{2}\hbox {d}\tau \right) ^{\frac{p}{2}}, \end{aligned} \end{aligned}$$

(3.9)

where

$$\begin{aligned} \begin{aligned}C_{p}=\left\{ \begin{array}{l@{\quad }l} 2 &{} \hbox {when} \ p=2,\\ \left[ \frac{p(p-1)}{2}\left( \frac{p}{p-1}\right) ^{p-2}\right] ^{\frac{p}{2}} &{} \hbox {when} \ p>2. \end{array}\right. \end{aligned} \end{aligned}$$

Lemma 3.3

Given $F(t, x)=F_{1}(t, x)+F_{2}(t, x)\in \mathrm{AAP}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}}))$ with

$$\begin{aligned} F_{1}(t, x)\in \mathrm{AP}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}})), \quad F_{2}(t, x)\in C_{0}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}})) \end{aligned}$$

satisfying the assumption ($H_{1}$) and $X(t)=Y(t)+Z(t)\in \mathrm{AAP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ with

$$\begin{aligned} Y(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})), \quad Z(t)\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Then $F(t, X(t))\in \mathrm{AAP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ with $F_{1}(t, Y(t))\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ and

$$\begin{aligned} F_{1}(t, X(t))-F_{1}(t, Y(t))\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})), \quad F_{2}(t, X(t))\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Proof

Note that

$$\begin{aligned} \begin{aligned} F(t, X(t))&=F_{1}(t, Y(t))+[F(t, X(t))-F_{1}(t, Y(t))]\\&=F_{1}(t, Y(t))+[F_{1}(t, X(t))-F_{1}(t, Y(t))]+F_{2}(t, X(t)). \end{aligned} \end{aligned}$$

Since $F_{1}(t, x)\in \mathrm{AP}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}}))$ and $Y(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$, from Lemma 2.1, together with (3.2), it follows that

$$\begin{aligned} F_{1}(t, Y(t))\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

As $F_{2}(t, x)\in C_{0}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}}))$, by (3.3), one has

$$\begin{aligned} F_{2}(t, X(t))\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Moreover, (3.2) implies that

$$\begin{aligned} F_{1}(t, X(t))-F_{1}(t, Y(t))\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})) \end{aligned}$$

due to

$$\begin{aligned} Z(t)=X(t)-Y(t)\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Before stating the existence theorem, we first prove the following auxiliary results.$\square$

Lemma 3.4

Given $Y(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ and $Z(t)\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$. Write

$$\begin{aligned} \varPhi _{1}(t):=\int ^{t}_{-\infty }T(t-s)Y(s)\hbox {d}s, \quad \varPhi _{2}(t):=\int ^{t}_{-\infty }T(t-s)Z(s)\hbox {d}s, \quad t\in {\mathbb {R}}. \end{aligned}$$

Then

$$\begin{aligned} \varPhi _{1}(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})), \quad \varPhi _{2}(t)\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Proof

From the exponential stability of $\{T(t)\}_{t\ge 0}$, it is clear that $\varPhi _{1}(t)$ and $\varPhi _{2}(t)$ are well defined and $L^{p}$-continuous on ${\mathbb {R}}$. Since $Y(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$, then for each $\varepsilon >0$, there exists a constant $l(\varepsilon )>0$ such that any interval of length $l(\varepsilon )$ contains at least $\tau$ satisfying

$$\begin{aligned} {\mathbb {E}}\Vert Y(s+\tau )-Y(s)\Vert ^{p}<\varepsilon \quad \hbox {for each} \ s\in {\mathbb {R}}. \end{aligned}$$

Using Hölder’s inequality with exponent $(p, \frac{p}{p-1})$, one has

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert \varPhi _{1}(t+\tau )-\varPhi _{1}(t)\Vert ^{p}&={\mathbb {E}}\Big \Vert \int ^{t+\tau }_{-\infty }T(t+\tau -s)Y(s)\hbox {d}s-\int ^{t}_{-\infty }T(t-s)Y(s)\hbox {d}s\Big \Vert ^{p}\\&={\mathbb {E}}\left\| \int ^{t}_{-\infty }T(t-s)[Y(s+\tau )-Y(s)]\hbox {d}s\right\| ^{p}\\&\le {\mathbb {E}}\left( \int ^{t}_{-\infty }M e^{-\delta (t-s)}\Vert Y(s+\tau )-Y(s)\Vert \hbox {d}s\right) ^{p}\\&\le M^{p}\left( \int ^{t}_{-\infty }e^{-\frac{1}{2}\delta \frac{p}{p-1}(t-s)}\hbox {d}s\right) ^{p-1} \\&\quad {\mathbb {E}}\int ^{t}_{-\infty } e^{-\frac{1}{2}p\delta (t-s)}\Vert Y(s+\tau )-Y(s)\Vert ^{p}\hbox {d}s\\&\le M^{p}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\frac{2}{p\delta }\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert Y(s+\tau )-Y(s)\Vert ^{p}\\&\le M^{p}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\frac{2\varepsilon }{p\delta } \end{aligned} \end{aligned}$$

is responsible for the fact that $\varPhi _{1}(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$.

Since $Z(t)\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$, one can choose a $T>0$ such that

$$\begin{aligned} {\mathbb {E}}\Vert Z(t)\Vert ^{p}<\varepsilon \quad \hbox {for all} \ t>T. \end{aligned}$$

This, together with Hölder’s inequality with exponent $(p, \frac{p}{p-1})$, enables us to conclude that for all $t>T$,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert \varPhi _{2}(t)\Vert ^{p}&\le 2^{p-1}{\mathbb {E}}\Big \Vert \int ^{N}_{-\infty }T(t-s)Z(s)\hbox {d}s\Big \Vert ^{p}+2^{p-1}{\mathbb {E}}\Big \Vert \int ^{t}_{N}T(t-s)Z(s)\hbox {d}s\Big \Vert ^{p}\\&\le 2^{p-1}{\mathbb {E}}\left( \int ^{N}_{-\infty }M e^{-\delta (t-s)}\Vert Z(s)\Vert \hbox {d}s\right) ^{p}\\&\quad +\,2^{p-1}{\mathbb {E}}\left( \int ^{t}_{N}M e^{-\delta (t-s)}\Vert Z(s)\Vert \hbox {d}s\right) ^{p}\\&\le 2^{p-1}M^{p}\left( \int ^{N}_{-\infty }e^{-\frac{1}{2}\delta \frac{p}{p-1}(t-s)}\hbox {d}s\right) ^{p-1}{\mathbb {E}}\int ^{N}_{-\infty } e^{-\frac{1}{2}p\delta (t-s)}\Vert Z(s)\Vert ^{p}\hbox {d}s\\&\quad +\,2^{p-1}M^{p}\left( \int ^{t}_{N}e^{-\frac{1}{2}\delta \frac{p}{p-1}(t-s)}\hbox {d}s\right) ^{p-1}{\mathbb {E}}\int ^{t}_{N} e^{-\frac{1}{2}p\delta (t-s)}\Vert Z(s)\Vert ^{p}\hbox {d}s\\&\le 2^{p-1}M^{p}e^{-[\frac{1}{2}p\delta +1](t-N)}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\frac{2}{p\delta }\Vert Z\Vert _{\infty }^{p}\\&\quad +\,2^{p-1}M^{p}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\frac{2\varepsilon }{p\delta }, \end{aligned} \end{aligned}$$

which implies that

$$\begin{aligned} \lim \limits _{t\rightarrow +\infty }{\mathbb {E}}\Vert \varPhi _{2}(t)\Vert ^{p}=0. \end{aligned}$$

By a similar argument, it follows readily that

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }{\mathbb {E}}\Vert \varPhi _{2}(t)\Vert ^{p}=0. \end{aligned}$$

The proof is then completed. $\square$

Lemma 3.5

Given $Y(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$, $Z(t)\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ and $p>2$. Let

$$\begin{aligned} \varPsi _{1}(t):=\int ^{t}_{-\infty }T(t-s)Y(s)\hbox {d}W(s), \quad \varPsi _{2}(t):=\int ^{t}_{-\infty }T(t-s)Z(s)\hbox {d}W(s), \quad t\in {\mathbb {R}}. \end{aligned}$$

Then

$$\begin{aligned} \varPsi _{1}(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})), \quad \varPsi _{2}(t)\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Proof

From the exponential stability of $\{T(t)\}_{t\ge 0}$ and Lemma 3.2, together with Hölder’s inequality with exponent $(\frac{p}{2}, \frac{p}{p-2})$, it is clear that

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert \varPsi _{1}(t)\Vert ^{p}&={\mathbb {E}}\Big \Vert \int ^{t}_{-\infty }T(t-s)Y(s)\hbox {d}W(s)\Big \Vert ^{p} \le C_{p}\left( {\mathbb {E}}\int ^{t}_{-\infty }\Vert T(t-s)Y(s)\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\le C_{p}M^{p}\left( {\mathbb {E}}\int ^{t}_{-\infty }e^{-2\delta (t-s)}\Vert Y(s)\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\le C_{p}M^{p}\left( \int ^{t}_{-\infty }e^{-\frac{p\delta (t-s)}{p-2}}\hbox {d}s\right) ^{\frac{p}{2}-1}{\mathbb {E}}\int ^{t}_{-\infty }e^{-\frac{1}{2}p\delta (t-s)}\Vert Y(s)\Vert ^{p}\hbox {d}s\\&\le C_{p}M^{p}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\frac{2}{p\delta }\Vert Y\Vert _{\infty }^{p}, \end{aligned} \end{aligned}$$

which implies $\varPsi _{1}(t)$ is well defined. Moreover let

$$\begin{aligned} \widetilde{W}(\eta ):=W(\eta +t-s)-W(t-s) \quad \hbox {for each} \ \eta \in {\mathbb {R}}. \end{aligned}$$

Then it is easy to verify that $\widetilde{W}$ is also a Wiener process and that it obeys the same distribution as W. Letting $\eta =\nu -t+s$, and using Lemma 3.2, together with Hölder’s inequality with exponent $(\frac{p}{2}, \frac{p}{p-2})$, one has

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert \varPsi _{1}(t)-\varPsi _{1}(s)\Vert ^{p}&={\mathbb {E}}\Big \Vert \int ^{t}_{-\infty }T(t-\eta )Y(\eta )\hbox {d}W(\eta )-\int ^{s}_{-\infty }T(s-\nu )Y(\nu )\hbox {d}W(\nu )\Big \Vert ^{p}\\&={\mathbb {E}}\Big \Vert \int ^{s}_{-\infty }T(s-\eta )Y(\eta +t-s)\hbox {d}W(\eta +t-s)\\&\quad -\,\int ^{s}_{-\infty }T(s-\nu )Y(\nu )\hbox {d}W(\nu )\Big \Vert ^{p}\\&={\mathbb {E}}\Big \Vert \int ^{s}_{-\infty }T(s-\eta )Y(\eta +t-s)\hbox {d}\widetilde{W}(\eta )\\&\quad \int ^{s}_{-\infty }T(s-\nu )Y(\nu )\hbox {d}\widetilde{W}(\nu )\Big \Vert ^{p}\\&={\mathbb {E}}\Big \Vert \int ^{s}_{-\infty }T(s-\eta )[Y(\eta +t-s)-Y(\eta )]\hbox {d}\widetilde{W}(\eta )\Big \Vert ^{p}\\&\le C_{p}\left( {\mathbb {E}}\int ^{s}_{-\infty }\Vert T(s-\eta )[Y(\eta +t-s)-Y(\eta )]\Vert ^{2}\hbox {d}\eta \right) ^{\frac{p}{2}}\\&\le C_{p}M^{p}\left( {\mathbb {E}}\int ^{s}_{-\infty }e^{-2\delta (s-\eta )}\Vert Y(\eta +t-s)-Y(\eta )\Vert ^{2}\hbox {d}\eta \right) ^{\frac{p}{2}}\\&\le C_{p}M^{p}\left( \int ^{s}_{-\infty }e^{-\frac{p\delta (s-\eta )}{p-2}}\hbox {d}\eta \right) ^{\frac{p}{2}-1} {\mathbb {E}}\\&\quad \int ^{s}_{-\infty }e^{-\frac{1}{2}p\delta (s-\eta )}\Vert Y(\eta +t-s)-Y(\eta )\Vert ^{p}\hbox {d}\eta \\&\le C_{p}M^{p}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\frac{2}{p\delta }\sup \limits _{\eta \in {\mathbb {R}}}{\mathbb {E}}\Vert Y(\eta +t-s)-Y(\eta )\Vert ^{p}. \end{aligned} \end{aligned}$$

That is

$$\begin{aligned} \lim \limits _{t\rightarrow s}{\mathbb {E}}\Vert \varPsi _{1}(t)-\varPsi _{1}(s)\Vert ^{p}=0, \end{aligned}$$

which implies that $\varPsi _{1}(t)$ is $L^{p}$-continuous on ${\mathbb {R}}$.

On the other hand, let

$$\begin{aligned} \widetilde{W}(s):=W(s+\tau )-W(\tau ) \quad \hbox {for each} \ s\in {\mathbb {R}}, \quad \mu =s-\tau , \end{aligned}$$

and using Lemma 3.2, together with Hölder’s inequality with exponent $(\frac{p}{2}, \frac{p}{p-2})$, one has

$$\begin{aligned}&{\mathbb {E}}\Vert \varPsi _{1}(t+\tau )-\varPsi _{1}(t)\Vert ^{p}\\&\quad ={\mathbb {E}}\Big \Vert \int ^{t+\tau }_{-\infty }T(t+\tau -s)Y(s)\hbox {d}W(s)-\int ^{t}_{-\infty }T(t-s)Y(s)\hbox {d}W(s)\Big \Vert ^{p}\\&\quad ={\mathbb {E}}\Big \Vert \int ^{t}_{-\infty }T(t-\mu )Y(\mu +\tau )\hbox {d}W(\mu +\tau )-\int ^{t}_{-\infty }T(t-s)Y(s)\hbox {d}W(s)\Big \Vert ^{p}\\&\quad ={\mathbb {E}}\Big \Vert \int ^{t}_{-\infty }T(t-\mu )Y(\mu +\tau )\hbox {d}\widetilde{W}(\mu )-\int ^{t}_{-\infty }T(t-s)Y(s)\hbox {d}\widetilde{W}(s)\Big \Vert ^{p}\\&\quad ={\mathbb {E}}\Big \Vert \int ^{t}_{-\infty }T(t-\mu )[Y(\mu +\tau )-Y(\mu )]\hbox {d}\widetilde{W}(\mu )\Big \Vert ^{p}\\&\quad \le C_{p}\left( {\mathbb {E}}\int ^{t}_{-\infty }\Vert T(t-\mu )[Y(\mu +\tau )-Y(\mu )]\Vert ^{2}\hbox {d}\mu \right) ^{\frac{p}{2}}\\&\quad \le C_{p}M^{p}\left( {\mathbb {E}}\int ^{t}_{-\infty }e^{-2\delta (t-\mu )}\Vert Y(\mu +\tau )-Y(\mu )\Vert ^{2}\hbox {d}\mu \right) ^{\frac{p}{2}}\\&\quad \le C_{p}M^{p}\left( \int ^{t}_{-\infty }e^{-\frac{p\delta (t-\mu )}{p-2}}\hbox {d}\mu \right) ^{\frac{p}{2}-1}{\mathbb {E}}\int ^{t}_{-\infty }e^{-\frac{1}{2}p\delta (t-\mu )}\Vert Y(\mu +\tau )-Y(\mu )\Vert ^{p}\hbox {d}\mu \\&\quad \le C_{p}M^{p}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\frac{2}{p\delta }\sup \limits _{\mu \in {\mathbb {R}}}{\mathbb {E}}\Vert Y(\mu +\tau )-Y(\mu )\Vert ^{p}. \end{aligned}$$

This implies that

$$\begin{aligned} \varPsi _{1}(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})) \ \hbox {due to} \ Y(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Similarly, from the exponential stability of $\{T(t)\}_{t\ge 0}$, it is clear that $\varPsi _{2}(t)$ is well defined and $L^{p}$-continuous on ${\mathbb {R}}$. Moreover since $Z(t)\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$, one can choose a $T>0$ such that

$$\begin{aligned} {\mathbb {E}}\Vert Z(t)\Vert ^{p}<\varepsilon \quad \hbox {for all} \ t>T. \end{aligned}$$

This enables us to conclude that for all $t>T$,

$$\begin{aligned}&{\mathbb {E}}\Vert \varPsi _{2}(t)\Vert ^{p}\\&\quad \le 2^{p-1}{\mathbb {E}}\left\| \int ^{N}_{-\infty }T(t-s)Z(s)\hbox {d}W(s)\right\| ^{p}+2^{p-1}{\mathbb {E}}\left\| \int ^{t}_{N}T(t-s)PZ(s)\hbox {d}W(s)\right\| ^{p}\\&\quad \le 2^{p-1}C_{p}\left( {\mathbb {E}}\int ^{N}_{-\infty }\Vert T(t-s)Z(s)\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\qquad +\,2^{p-1}C_{p}\left( {\mathbb {E}}\int ^{t}_{N}\Vert T(t-s)Z(s)\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\quad \le 2^{p-1}C_{p}M^{p}\left( {\mathbb {E}}\int ^{N}_{-\infty }e^{-2\delta (t-s)}\Vert Z(s)\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\qquad +\,2^{p-1}C_{p}M^{p}\left( {\mathbb {E}}\int ^{t}_{N}e^{-2\delta (t-s)}\Vert Y(s)\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\quad \le 2^{p-1}C_{p}M^{p}\left( \int ^{N}_{-\infty }e^{-\frac{p\delta (t-s)}{p-2}}\hbox {d}s\right) ^{\frac{p}{2}-1}{\mathbb {E}}\int ^{N}_{-\infty }e^{-\frac{1}{2}p\delta (t-s)}\Vert Z(s)\Vert ^{p}\hbox {d}s\\&\qquad +\,2^{p-1}C_{p}M^{p}\left( \int ^{t}_{N}e^{-\frac{p\delta (t-s)}{p-2}}\hbox {d}s\right) ^{\frac{p}{2}-1}{\mathbb {E}}\int ^{t}_{N}e^{-\frac{1}{2}p\delta (t-s)}\Vert Z(s)\Vert ^{p}\hbox {d}s\\&\quad \le 2^{p-1}C_{p}M^{p}e^{-\left[ \frac{p\delta }{2}+1\right] (t-N)} \left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\frac{2}{p\delta }\Vert Z\Vert _{\infty }^{p}\\&\qquad +\,2^{p-1}C_{p}M^{p}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\frac{2\varepsilon }{p\delta }, \end{aligned}$$

which implies that

$$\begin{aligned} \lim \limits _{t\rightarrow +\infty }{\mathbb {E}}\Vert \varPsi _{2}(t)\Vert ^{p}=0. \end{aligned}$$

By a similar argument, it follows readily that

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }{\mathbb {E}}\Vert \varPsi _{2}(t)\Vert ^{p}=0. \end{aligned}$$

The proof is then completed. $\square$

Lemma 3.5 is still valid for $p=2$.

Lemma 3.6

Given $U(t)\in \mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$, $V(t)\in C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$. Let

$$\begin{aligned} \Upsilon _{1}(t):=\int ^{t}_{-\infty }T(t-s)U(s)\hbox {d}W(s), \quad \Upsilon _{2}(t):=\int ^{t}_{-\infty }T(t-s)V(s)\hbox {d}W(s), \quad t\in {\mathbb {R}}. \end{aligned}$$

Then

$$\begin{aligned} \Upsilon _{1}(t)\in \mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}})), \quad \Upsilon _{2}(t)\in C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Proof

From the exponential stability of $\{T(t)\}_{t\ge 0}$, together with the It$\hat{o}$’s isometry property of the stochastic integral, it is clear that

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert \Upsilon _{1}(t)\Vert ^{2}&={\mathbb {E}}\int ^{t}_{-\infty }\Vert T(t-s)U(s)\Vert ^{2}\hbox {d}s\\&\le M^{2}\int ^{t}_{-\infty }e^{-2\delta (t-s)}{\mathbb {E}}\Vert U(s)\Vert ^{2}\hbox {d}s\le \frac{M^{2}}{2\delta }\Vert U\Vert _{\infty }^{2}, \end{aligned} \end{aligned}$$

which implies $\Upsilon _{1}(t)$ is well defined. Moreover let

$$\begin{aligned} \widetilde{W}(\eta ):=W(\eta +t-s)-W(t-s) \quad \hbox {for each} \ \eta \in {\mathbb {R}}, \quad \eta =\nu -t+s. \end{aligned}$$

Using the It$\hat{o}$’s isometry property of the stochastic integral again, one has

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert \Upsilon _{1}(t)-\Upsilon _{1}(s)\Vert ^{2}&={\mathbb {E}}\Big \Vert \int ^{t}_{-\infty }T(t-\eta )U(\eta )\hbox {d}W(\eta )\\&\quad -\,\int ^{s}_{-\infty }T(s-\nu )U(\nu )\hbox {d}W(\nu )\Big \Vert ^{2}\\&={\mathbb {E}}\Big \Vert \int ^{s}_{-\infty }T(s-\eta )U(\eta +t-s)\hbox {d}W(\eta +t-s)\\&\quad -\,\int ^{s}_{-\infty }T(s-\nu )U(\nu )\hbox {d}W(\nu )\Big \Vert ^{2}\\&={\mathbb {E}}\Big \Vert \int ^{s}_{-\infty }T(s-\eta )U(\eta +t-s)\hbox {d}\widetilde{W}(\eta )\\&\quad -\,\int ^{s}_{-\infty }T(s-\nu )U(\nu )\hbox {d}\widetilde{W}(\nu )\Big \Vert ^{2}\\&={\mathbb {E}}\Big \Vert \int ^{s}_{-\infty }T(s-\eta )[U(\eta +t-s)-U(\eta )]\hbox {d}\widetilde{W}(\eta )\Big \Vert ^{2}\\&={\mathbb {E}}\int ^{s}_{-\infty }\Vert T(s-\eta )[U(\eta +t-s)-U(\eta )]\Vert ^{2}\hbox {d}\eta \\&\le M^{2}\int ^{s}_{-\infty }e^{-2\delta (s-\eta )}{\mathbb {E}}\Vert U(\eta +t-s)-U(\eta )\Vert ^{2}\hbox {d}\eta \\&\le \frac{M^{2}}{2\delta }\sup \limits _{\eta \in {\mathbb {R}}}{\mathbb {E}}\Vert U(\eta +t-s)-U(\eta )\Vert ^{2}. \end{aligned} \end{aligned}$$

That is

$$\begin{aligned} \lim \limits _{t\rightarrow s}{\mathbb {E}}\Vert \Upsilon _{1}(t)-\Upsilon _{1}(s)\Vert ^{2}=0, \end{aligned}$$

which implies that $\Upsilon _{1}(t)$ is $L^{2}$-continuous on ${\mathbb {R}}$.

On the other hand, let

$$\begin{aligned} \widetilde{W}(s):=W(s+\tau )-W(\tau ) \quad \hbox {for each} \ s\in {\mathbb {R}}, \quad \mu =s-\tau . \end{aligned}$$

From the It$\hat{o}$’s isometry property of the stochastic integral, it follows that

$$\begin{aligned} {\mathbb {E}}\Vert \Upsilon _{1}(t+\tau )-\Upsilon _{1}(t)\Vert ^{2}&={\mathbb {E}}\Big \Vert \int ^{t+\tau }_{-\infty }T(t+\tau -s)U(s)\hbox {d}W(s)\\&\quad -\,\int ^{t}_{-\infty }T(t-s)U(s)\hbox {d}W(s)\Big \Vert ^{2}\\&={\mathbb {E}}\Big \Vert \int ^{t}_{-\infty }T(t-\mu )U(\mu +\tau )\hbox {d}W(\mu +\tau )\\&\quad -\,\int ^{t}_{-\infty }T(t-s)U(s)\hbox {d}W(s)\Big \Vert ^{2}\\&={\mathbb {E}}\Big \Vert \int ^{t}_{-\infty }T(t-\mu )U(\mu +\tau )\hbox {d}\widetilde{W}(\mu )\\&\quad -\,\int ^{t}_{-\infty }T(t-s)U(s)\hbox {d}\widetilde{W}(s)\Big \Vert ^{2}\\&={\mathbb {E}}\Big \Vert \int ^{t}_{-\infty }T(t-\mu )[U(\mu +\tau )-U(\mu )]\hbox {d}\widetilde{W}(\mu )\Big \Vert ^{2}\\&={\mathbb {E}}\int ^{t}_{-\infty }\Vert T(t-\mu )[U(\mu +\tau )-U(\mu )]\Vert ^{2}\hbox {d}\mu \\&\le M^{2}\int ^{t}_{-\infty }e^{-2\delta (t-\mu )}{\mathbb {E}}\Vert U(\mu +\tau )-U(\mu )\Vert ^{2}\hbox {d}\mu \\&\le \frac{M^{2}}{2\delta }\sup \limits _{\mu \in {\mathbb {R}}}{\mathbb {E}}\Vert U(\mu +\tau )-U(\mu )\Vert ^{2}. \end{aligned}$$

This implies that

$$\begin{aligned} \Upsilon _{1}(t)\in \mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}})) \ \hbox {due to} \ U(t)\in \mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Similarly, from the exponential stability of $\{T(t)\}_{t\ge 0}$ it is clear that $\Upsilon _{2}(t)$ is well defined and $L^{2}$-continuous on ${\mathbb {R}}$. Moreover since $V(t)\in C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$, one can choose a $T>0$ such that

$$\begin{aligned} {\mathbb {E}}\Vert V(t)\Vert ^{2}<\varepsilon \quad \hbox {for all} \ t>T. \end{aligned}$$

This enables us to conclude that for all $t>T$,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert \Upsilon _{2}(t)\Vert ^{2}&\le 2{\mathbb {E}}\left\| \int ^{N}_{-\infty }T(t-s)V(s)\hbox {d}W(s)\right\| ^{2}+2{\mathbb {E}}\left\| \int ^{t}_{N}T(t-s)V(s)\hbox {d}W(s)\right\| ^{2}\\&=2{\mathbb {E}}\int ^{N}_{-\infty }\Vert T(t-s)V(s)\Vert ^{2}\hbox {d}s+2{\mathbb {E}}\int ^{t}_{N}\Vert T(t-s)V(s)\Vert ^{2}\hbox {d}s\\&\le 2M^{2}{\mathbb {E}}\int ^{N}_{-\infty }e^{-2\delta (t-s)}\Vert V(s)\Vert ^{2}\hbox {d}s+2M^{2}{\mathbb {E}}\int ^{t}_{N}e^{-2\delta (t-s)}\Vert V(s)\Vert ^{2}\hbox {d}s\\&\le \frac{M^{2}e^{-2\delta (t-N)}}{\delta }\Vert V\Vert _{\infty }^{2}+\frac{M^{2}\varepsilon }{\delta }, \end{aligned} \end{aligned}$$

which implies that

$$\begin{aligned} \lim \limits _{t\rightarrow +\infty }{\mathbb {E}}\Vert \Upsilon _{2}(t)\Vert ^{2}=0. \end{aligned}$$

By a similar argument, it follows readily that

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }{\mathbb {E}}\Vert \Upsilon _{2}(t)\Vert ^{2}=0. \end{aligned}$$

The proof is then completed. $\square$

We give the following definition of p-mean asymptotically almost periodic mild solutions to Eq. (3.1).

Definition 3.1

An ${\mathbb {F}}_{t}$-progressively measurable process $x:{\mathbb {R}}\rightarrow L^{p}({\mathbb {P}},{\mathbb {H}})$ is called a p-mean asymptotically almost periodic mild solution to Eq. (3.1) on ${\mathbb {R}}$ if $x\in \mathrm{AAP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$, and it satisfies the corresponding stochastic integral equation

$$\begin{aligned} \begin{aligned} x(t)=T(t-\tau )x(\tau )+\int _{\tau }^{t}T(t-s)F(s, x(s))\hbox {d}s+\int _{\tau }^{t}T(t-s)G(s, x(s))\hbox {d}W(s) \end{aligned} \end{aligned}$$

for all $t>\tau$ and each $t\in {\mathbb {R}}$.

Let $\beta _{1}(t), \beta _{2}(t)$ be the functions involved in the assumptions ($H_{1}$) and ($H_{2}$). Define

$$\begin{aligned} \sigma _{1}(t):=\int _{-\infty }^{t}e^{-\delta (t-s)}\beta _{1}(s)\hbox {d}s, \quad \sigma _{2}(t):=\int _{-\infty }^{t}e^{-\delta (t-s)}\beta _{2}(s)\hbox {d}s, \quad t\in {\mathbb {R}}. \end{aligned}$$

Then $\sigma _{1}(t), \sigma _{2}(t)\in C_{0}({\mathbb {R}}, {\mathbb {R}}^{+})$. Put

$$\begin{aligned} \rho _{1}':=\sup \limits _{t\in {\mathbb {R}}}\sigma _{1}(t), \quad \rho _{2}':=\sup \limits _{t\in {\mathbb {R}}}\sigma _{2}(t). \end{aligned}$$

Now we are in a position to present our main result:

Theorem 3.1

Assume that ($H_{1}$) and ($H_{2}$) hold with $p>2$, then Eq. (3.1) has at least one p-mean asymptotically almost periodic mild solution whenever

$$\begin{aligned} \begin{aligned}&4^{p-1}M^{p}\left[ L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}+C_{p}L_{2}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\right] \frac{2}{p\delta }\\&\quad +\,4^{p-1}M^{p}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\frac{2}{p\delta }\rho _{1}\rho _{1}'+4^{p-1}C_{p}M^{p}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\frac{2}{p\delta }\rho _{2}\rho _{2}'<1. \end{aligned} \end{aligned}$$

(3.10)

Proof

Consider the following coupled system of integral equations

$$\begin{aligned} \begin{aligned}\left\{ \begin{aligned} v(t)&=\int _{-\infty }^{t}T(t-s)F_{1}(s, v(s))\hbox {d}s+\int _{-\infty }^{t}T(t-s)G_{1}(s, v(s))\hbox {d}W(s)\ t\in {\mathbb {R}},\\ \omega (t)&=\int _{-\infty }^{t}T(t-s)[F_{1}(s, v(s)+\omega (s))-F_{1}(s, v(s))]\hbox {d}s\\&\quad +\,\int _{-\infty }^{t}T(t-s)F_{2}(s, v(s)+\omega (s))\hbox {d}s\\&\quad +\,\int _{-\infty }^{t}T(t-s)[G_{1}(s, v(s)+\omega (s))-G_{1}(s, v(s))]\hbox {d}W(s)\\&\quad +\,\int _{-\infty }^{t}T(t-s)G_{2}(s, v(s)+\omega (s))\hbox {d}W(s) \ t\in {\mathbb {R}}. \end{aligned} \right. \end{aligned} \end{aligned}$$

(3.11)

If $(v(t), \omega (t))\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))\times C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ is a solution to system (3.11), then

$$\begin{aligned} x(t):=v(t)+\omega (t)\in \mathrm{AAP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})) \end{aligned}$$

and it is a solution to the following integral equation

$$\begin{aligned} \begin{aligned} x(\tau )&=\int _{-\infty }^{\tau }T(\tau -s)F(s, x(s))\hbox {d}s+\int _{-\infty }^{\tau }T(\tau -s)G(s, x(s))\hbox {d}W(s),\ \tau \in {\mathbb {R}}. \end{aligned} \end{aligned}$$

(3.12)

Multiplying both sides above by $T(t-\tau )$ with $t>\tau$, one obtains that x(t) is a p-mean asymptotically almost periodic mild solution to Eq. (3.1). Hence the problem has shifted to show that system (3.11) has at least a solution in $\mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))\times C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$.

Defining $\varLambda$ on $\mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ as follows:

$$\begin{aligned} \begin{aligned} (\varLambda v)(t)&=\int _{-\infty }^{t}T(t-s)F_{1}(s, v(s))\hbox {d}s+\int _{-\infty }^{t}T(t-s)G_{1}(s, v(s))\hbox {d}W(s),\ t\in {\mathbb {R}}. \end{aligned} \end{aligned}$$

(3.13)

From $F_{1}(t, x)\in \mathrm{AP}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}}))$ satisfying (3.2) and $G_{1}(t, x)\in \mathrm{AP}({\mathbb {R}}\times L^{p}({\mathbb {P}},{\mathbb {H}}), L^{p}({\mathbb {P}},{\mathbb {H}}))$ satisfying (3.4), together with Lemma 2.1, it follows that

$$\begin{aligned} F_{1}(\cdot , v(\cdot )), G_{1}(\cdot , v(\cdot ))\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))\quad \hbox {for each} \ v(\cdot )\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Those, together with Lemmas 3.4 and 3.5, imply that $\varLambda$ is well defined and maps $\mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ into itself. Moreover, for any $v_{1}(t), v_{2}(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$, by (3.2), (3.4) and Lemma 3.2, together with Hölder’s inequality with exponents $(p, \frac{p}{p-1})$ and $(\frac{p}{2}, \frac{p}{p-2})$, respectively, one has

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert (\varLambda v_{1})(t)-(\varLambda v_{2})(t)\Vert ^{p}\\&\quad \le 2^{p-1}{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)[F_{1}(s, v_{1}(s))-F_{1}(s, v_{2}(s))]\hbox {d}s\Big \Vert ^{p}\\&\qquad +\,2^{p-1}{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)[G_{1}(s, v_{1}(s))-G_{1}(s, v_{2}(s))]\hbox {d}W(s)\Big \Vert ^{p}\\&\quad \le 2^{p-1}{\mathbb {E}}\left( \int _{-\infty }^{t}Me^{-\delta (t-s)}\Vert F_{1}(s, v_{1}(s))-F_{1}(s, v_{2}(s))\Vert \hbox {d}s\right) ^{p}\\&\qquad +\,2^{p-1}C_{p}\left( {\mathbb {E}}\int _{-\infty }^{t}\Vert T(t-s)[G_{1}(s, v_{1}(s))-G_{1}(s, v_{2}(s))]\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\quad \le 2^{p-1}M^{p}\left( \int ^{t}_{-\infty }e^{-\frac{1}{2}\delta \frac{p}{p-1}(t-s)}\hbox {d}s\right) ^{p-1}{\mathbb {E}}\int ^{t}_{-\infty } e^{-\frac{1}{2}p\delta (t-s)}\Vert F_{1}(s, v_{1}(s))\\&\qquad -\,F_{1}(s, v_{2}(s))\Vert ^{p}\hbox {d}s\\&\qquad +\,2^{p-1}C_{p}M^{p}\left( {\mathbb {E}}\int ^{t}_{-\infty }e^{-2\delta (t-s)}\Vert G_{1}(s, v_{1}(s))-G_{1}(s, v_{2}(s))\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\quad \le 2^{p-1}M^{p}\left[ L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}+C_{p}L_{2}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\right] \frac{2}{p\delta }\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\\&\qquad \times \Vert v_{1}(s)-v_{2}(s)\Vert ^{p}\\&\quad \le 2^{p-1}M^{p}\left[ L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}+C_{p}L_{2}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\right] \frac{2}{p\delta }\Vert v_{1}-v_{2}\Vert _{\infty }^{p}, \end{aligned} \end{aligned}$$

which implies

$$\begin{aligned} \begin{aligned}&\Vert (\varLambda v_{1})-(\varLambda v_{2})\Vert _{\infty }\\&\quad \le \root p \of {2^{p-1}M^{p}\left[ L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}+C_{p}L_{2}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\right] \frac{2}{p\delta }}\Vert v_{1}-v_{2}\Vert _{\infty }. \end{aligned} \end{aligned}$$

Together with (3.10), this proves that $\varLambda$ is a contraction on $\mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$. Thus, the Banach’s fixed point theorem implies that $\varLambda$ has a unique fixed point $v(t)\in \mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$.

For the above v(t), define $\varGamma :=\varGamma ^{1}+\varGamma ^{2}$ on $C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ as

$$\begin{aligned} \begin{aligned} (\varGamma ^{1} \omega )(t)&=\int _{-\infty }^{t}T(t-s)J_{v}'(s, \omega (s))\hbox {d}s+\int _{-\infty }^{t}T(t-s)J_{v}''(s, \omega (s))\hbox {d}W(s) \ t\in {\mathbb {R}},\\ (\varGamma ^{2} \omega )(t)&=\int _{-\infty }^{t}T(t-s)K_{v}'(s, \omega (s))\hbox {d}s+\int _{-\infty }^{t}T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s) \ t\in {\mathbb {R}}, \end{aligned} \end{aligned}$$

(3.14)

where

$$\begin{aligned} \begin{aligned} J_{v}'(s, \omega (s))&=F_{1}(s, v(s)+\omega (s))-F_{1}(s, v(s)), \quad K_{v}'(s, \omega (s))=F_{2}(s, v(s)+\omega (s)), \\ J_{v}''(s, \omega (s))&=G_{1}(s, v(s)+\omega (s))-G_{1}(s, v(s)), \quad K_{v}''(s, \omega (s))=G_{2}(s, v(s)\\&\quad +\,\omega (s)). \end{aligned} \end{aligned}$$

From (3.2) and (3.4), it follows that for all $s\in {\mathbb {R}}$, $\omega (s)\in L^{p}({\mathbb {P}},{\mathbb {H}})$,

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert J_{v}'(s, \omega (s))\Vert ^{p}\le L_{1}{\mathbb {E}}\Vert \omega (s)\Vert ^{p},\ {\mathbb {E}}\Vert J_{v}''(s, \omega (s))\Vert ^{p}\le L_{2}{\mathbb {E}}\Vert \omega (s)\Vert ^{p}, \end{aligned} \end{aligned}$$

which imply that

$$\begin{aligned} \begin{aligned} J_{v}'(\cdot , \omega (\cdot )), J_{v}''(\cdot , \omega (\cdot ))\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})) \quad \hbox {for each} \ \omega (\cdot )\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})), \end{aligned} \end{aligned}$$

and by (3.3) as well as (3.5), one has for all $s\in {\mathbb {R}}$ and $\omega (s)\in L^{p}({\mathbb {P}},{\mathbb {H}})$ with ${\mathbb {E}}\Vert \omega (s)\Vert ^{p}\le r$,

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert K_{v}'(s, \omega (s))\Vert ^{p}\le \beta _{1}(s)\varPhi _{1}\Big (r+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{p}\Big ),\\&{\mathbb {E}}\Vert K_{v}''(s, \omega (s))\Vert ^{p}\le \beta _{2}(s)\varPhi _{2}\Big (r+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{p}\Big ), \end{aligned} \end{aligned}$$

(3.15)

which imply that

$$\begin{aligned} \begin{aligned} K_{v}'(\cdot , \omega (\cdot )), K_{v}''(\cdot , \omega (\cdot ))\in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})) \ \hbox {due to} \ \beta _{1}(s), \beta _{2}(s)\in C_{0}({\mathbb {R}}, {\mathbb {R}}^{+}). \end{aligned} \end{aligned}$$

Those, together with Lemmas 3.4 and 3.5, yield that $\varGamma$ is well defined and maps $C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$ into itself. To complete the proof, it suffices to prove that $\varGamma$ possesses at least one fixed point in $C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$.

Set

$$\begin{aligned} \varOmega _{r}:=\{\omega \in C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}})): \Vert \omega \Vert _{\infty }^{p}\le r\}. \end{aligned}$$

In view of (3.3), (3.5) and (3.10), it is not difficult to see that there exists a constant $k_{0}>0$ such that

$$\begin{aligned} \begin{aligned}&4^{p-1}M^{p}\left[ L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}+C_{p}L_{2}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\right] \frac{2}{p\delta }k_{0}\\&\quad +\,4^{p-1}M^{p}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\\&\quad \times \frac{2\rho _{1}'}{p\delta }\varPhi _{1}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}\Vert v(s)\Vert \Big ) +4^{p-1}C_{p}M^{p}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\frac{2\rho _{2}'}{p\delta }\varPhi _{2}\\&\quad \Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}\Vert v(s)\Vert \Big )\le k_{0}. \end{aligned} \end{aligned}$$

This enables us to conclude that for any $t\in {\mathbb {R}}$ and $\omega _{1}(t), \omega _{2}(t)\in \varOmega _{k_{0}}$,

$$\begin{aligned}&{\mathbb {E}}\Vert (\varGamma ^{1}\omega _{1})(t)+(\varGamma ^{2}\omega _{2})(t)\Vert ^{p}\\&\quad \le 4^{p-1}{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)J_{v}'(s, \omega _{1}(s))\hbox {d}s\Big \Vert ^{p}\\&\qquad +\,4^{p-1}{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)J_{v}''(s, \omega _{1}(s))\hbox {dd}W(s)\Big \Vert ^{p}\\&\qquad +\,4^{p-1}{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)K_{v}'(s, \omega _{2}(s))\hbox {d}s\Big \Vert ^{p}\\&\qquad +\,4^{p-1}{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)K_{v}''(s, \omega _{2}(s))\hbox {dd}W(s)\Big \Vert ^{p}\\&\quad \le 4^{p-1}{\mathbb {E}}\left( \int _{-\infty }^{t}Me^{-\delta (t-s)}\Vert J_{v}'(s, \omega _{1}(s))\Vert \hbox {d}s\right) ^{p}\\&\qquad +\,4^{p-1}C_{p}\left( {\mathbb {E}}\int _{-\infty }^{t}\Vert T(t-s)J_{v}''(s, \omega _{1}(s))\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\qquad +\,4^{p-1}{\mathbb {E}}\left( \int _{-\infty }^{t}Me^{-\delta (t-s)}\Vert K_{v}'(s, \omega _{2}(s))\Vert \hbox {d}s\right) ^{p}\\&\qquad +\,4^{p-1}C_{p}\left( {\mathbb {E}}\int _{-\infty }^{t}\Vert T(t-s)K_{v}''(s, \omega _{2}(s))\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\quad \le 4^{p-1}M^{p}\left( \int ^{t}_{-\infty }e^{-\frac{1}{2}\delta \frac{p}{p-1}(t-s)}\hbox {d}s\right) ^{p-1}{\mathbb {E}}\int ^{t}_{-\infty } e^{-\frac{1}{2}p\delta (t-s)}\Vert J_{v}'(s, \omega _{1}(s))\Vert ^{p}\hbox {d}s\\&\qquad +\,4^{p-1}C_{p}M^{p}\left( {\mathbb {E}}\int ^{t}_{-\infty }e^{-2\delta (t-s)}\Vert J_{v}''(s, \omega _{1}(s))\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\qquad +\,4^{p-1}M^{p}\left( \int ^{t}_{-\infty }e^{-\frac{1}{2}\delta \frac{p}{p-1}(t-s)}\hbox {d}s\right) ^{p-1}{\mathbb {E}}\int ^{t}_{-\infty } e^{-\frac{1}{2}p\delta (t-s)}\Vert K_{v}'(s, \omega _{2}(s))\Vert ^{p}\hbox {d}s\\&\qquad +\,4^{p-1}C_{p}M^{p}\left( {\mathbb {E}}\int ^{t}_{-\infty }e^{-2\delta (t-s)}\Vert K_{v}''(s, \omega _{2}(s))\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\quad \le 4^{p-1}M^{p}L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\frac{2}{p\delta }\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert \omega _{1}(s)\Vert ^{p}\\&\qquad +\,4^{p-1}C_{p}M^{p}\left( \int ^{t}_{-\infty }e^{-\frac{p\delta (t-s)}{p-2}}\hbox {d}s\right) ^{\frac{p}{2}-1}{\mathbb {E}}\int ^{t}_{-\infty }e^{-\frac{1}{2}p\delta (t-s)}\Vert J_{v}''(s, \omega _{1}(s))\Vert ^{p}\hbox {d}s\\&\qquad +\,4^{p-1}M^{p}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\frac{2}{p\delta }\rho _{1}'\varPhi _{1}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{p}\Big )\\&\qquad +\,4^{p-1}C_{p}M^{p}\left( \int ^{t}_{-\infty }e^{-\frac{p\delta (t-s)}{p-2}}\hbox {d}s\right) ^{\frac{p}{2}-1}{\mathbb {E}}\int ^{t}_{-\infty }e^{-\frac{1}{2}p\delta (t-s)}\Vert K_{v}''(s, \omega _{2}(s))\Vert ^{p}\hbox {d}s \end{aligned}$$

$$\begin{aligned}&\quad \le 4^{p-1}M^{p}\left[ L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}+C_{p}L_{2}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\right] \frac{2}{p\delta }k_{0}\\&\qquad +\,4^{p-1}M^{p}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\frac{2}{p\delta }\rho _{1}'\varPhi _{1}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{p}\Big )\\&\qquad +\,4^{p-1}C_{p}M^{p}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\frac{2}{p\delta }\rho _{2}'\varPhi _{2}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{p}\Big )\le k_{0}, \end{aligned}$$

which implies that for every pair $\omega _{1}(t), \omega _{2}(t)\in \varOmega _{k_{0}}$,

$$\begin{aligned} (\varGamma ^{1}\omega _{1})(t)+(\varGamma ^{2}\omega _{2})(t)\in \varOmega _{k_{0}}. \end{aligned}$$

Thus $\varGamma$ maps $\varOmega _{k_{0}}$ into itself.

In the following, we show that $\varGamma ^{1}$ is a contraction on $\varOmega _{k_{0}}$.

For $\omega _{1}(t), \omega _{2}(t)\in \varOmega _{k_{0}}$ and $t\in {\mathbb {R}}$, from (3.2) and (3.4), it follows that

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert J_{v}'(s, \omega _{1}(s))-J_{v}'(s, \omega _{2}(s))\Vert ^{p}\le L_{1}{\mathbb {E}}\Vert \omega _{1}(s)-\omega _{2}(s)\Vert ^{p},\\&{\mathbb {E}}\Vert J_{v}''(s, \omega _{1}(s))-J_{v}''(s, \omega _{2}(s))\Vert ^{p}\le L_{2}{\mathbb {E}}\Vert \omega _{1}(s)-\omega _{2}(s)\Vert ^{p}. \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned}&{\mathbb {E}}\Vert (\varGamma ^{1}\omega _{1})(t)-(\varGamma ^{1}\omega _{2})(t)\Vert ^{p}\\&\quad \le 2^{p-1}{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)(J_{v}'(s, \omega _{1}(s))-J_{v}'(s, \omega _{2}(s)))\hbox {d}s\Big \Vert ^{p}\\&\qquad +\,2^{p-1}{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)(J_{v}''(s, \omega _{1}(s))-J_{v}''(s, \omega _{2}(s)))\hbox {d}W(s)\Big \Vert ^{p}\\&\quad \le 2^{p-1}{\mathbb {E}}\left( \int _{-\infty }^{t}Me^{-\delta (t-s)}\Vert J_{v}'(s, \omega _{1}(s))-J_{v}'(s, \omega _{2}(s))\Vert \hbox {d}s\right) ^{p}\\&\qquad +\,2^{p-1}C_{p}\left( {\mathbb {E}}\int _{-\infty }^{t}\Vert T(t-s)(J_{v}''(s, \omega _{1}(s))-J_{v}''(s, \omega _{2}(s)))\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\quad \le 2^{p-1}M^{p}\left( \int ^{t}_{-\infty }e^{-\frac{1}{2}\delta \frac{p}{p-1}(t-s)}\hbox {d}s\right) ^{p-1}{\mathbb {E}}\\&\qquad \int ^{t}_{-\infty } e^{-\frac{1}{2}p\delta (t-s)}\Vert J_{v}'(s, \omega _{1}(s))-J_{v}'(s, \omega _{2}(s))\Vert ^{p}\hbox {d}s\\&\qquad +\,2^{p-1}C_{p}M^{p}\Big ({\mathbb {E}}\int ^{t}_{-\infty }e^{-2\delta (t-s)}\Vert J_{v}''(s, \omega _{1}(s))-J_{v}''(s, \omega _{2}(s))\Vert ^{2}\hbox {d}s\Big )^{\frac{p}{2}}\\&\quad \le 2^{p-1}M^{p}L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\frac{2}{p\delta }\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert \omega _{1}(s)\\&\qquad -\,\omega _{2}(s)\Vert ^{p} +2^{p-1}C_{p}M^{p}\left( \int ^{t}_{-\infty }e^{-\frac{p\delta (t-s)}{p-2}}\hbox {d}s\right) ^{\frac{p}{2}-1}\\&\qquad {\mathbb {E}}\int ^{t}_{-\infty }e^{-\frac{1}{2}p\delta (t-s)}\Vert J_{v}''(s, \omega _{1}(s))-J_{v}''(s, \omega _{2}(s))\Vert ^{p}\hbox {d}s\\&\quad \le 2^{p-1}M^{p}\left[ L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}+C_{p}L_{2}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\right] \frac{2}{p\delta }\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\\&\qquad \times \Vert \omega _{1}(s)-\omega _{2}(s)\Vert ^{p}\\&\quad \le 2^{p-1}M^{p}\left[ L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}+C_{p}L_{2}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\right] \frac{2}{p\delta }\Vert \omega _{1}-\omega _{2}\Vert _{\infty }^{p}, \end{aligned}$$

which implies

$$\begin{aligned} \begin{aligned}&\Vert (\varGamma ^{1}\omega _{1})(t)-(\varGamma ^{1}\omega _{2})(t)\Vert _{\infty }\\&\quad \le \root p \of {2^{p-1}M^{p}\left[ L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}+C_{p}L_{2}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\right] \frac{2}{p\delta }}\Vert \omega _{1}-\omega _{2}\Vert _{\infty }. \end{aligned} \end{aligned}$$

Thus, in view of (3.10), one obtains the conclusion.

From our assumptions, it is clear that $\varGamma ^{2}$ is a $L^{p}$-continuous mapping from $\varOmega _{k_{0}}$ to $\varOmega _{k_{0}}$. Thus, in order to apply the well-known Krasnoselskii’s fixed point theorem to obtain a fixed point of $\varGamma$, one needs to verify that $\varGamma ^{2}$ is compact on $\varOmega _{k_{0}}$.

First, for all $\omega (t)\in \varOmega _{k_{0}}$ and $t\in {\mathbb {R}}$,

$$\begin{aligned}&{\mathbb {E}}\Vert (\varGamma ^{2} \omega )(t)\Vert ^{p}\\&\quad \le 2^{p-1}{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{p}+2^{p-1}{\mathbb {E}}\Big \Vert \\&\qquad \int _{-\infty }^{t}T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{p}\\&\quad \le 2^{p-1}{\mathbb {E}}\left( \int _{-\infty }^{t}Me^{-\delta (t-s)}\Vert K_{v}'(s, \omega (s))\Vert \hbox {d}s\right) ^{p}\\&\qquad +\,2^{p-1}C_{p}\left( {\mathbb {E}}\int _{-\infty }^{t}\Vert T(t-s)K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\quad \le 2^{p-1}M^{p}\left( \int ^{t}_{-\infty }e^{-\frac{1}{2}\delta \frac{p}{p-1}(t-s)}\hbox {d}s\right) ^{p-1}{\mathbb {E}}\int ^{t}_{-\infty } e^{-\frac{1}{2}p\delta (t-s)}\Vert K_{v}'(s, \omega (s))\Vert ^{p}\hbox {d}s\\&\qquad +\,2^{p-1}C_{p}M^{p}\left( {\mathbb {E}}\int ^{t}_{-\infty }e^{-2\delta (t-s)}\Vert K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\quad \le 2^{p-1}M^{p}L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\frac{2}{p\delta }\sigma _{1}\varPhi _{1}\Big (r+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{p}\Big )\\&\qquad +\,2^{p-1}C_{p}M^{p}\left( \int ^{t}_{-\infty }e^{-\frac{p\delta (t-s)}{p-2}}\hbox {d}s\right) ^{\frac{p}{2}-1}{\mathbb {E}}\int ^{t}_{-\infty }e^{-\frac{1}{2}p\delta (t-s)}\Vert K_{v}''(s, \omega (s))\Vert ^{p}\hbox {d}s\\&\quad \le 2^{p-1}M^{p}L_{1}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\frac{2}{p\delta }\sigma _{1}\varPhi _{1}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{p}\Big )\\&\qquad +\,2^{p-1}M^{p}C_{p}L_{2}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\frac{2}{p\delta }\sigma _{2}\varPhi _{2}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{p}\Big ), \end{aligned}$$

in view of $\sigma _{1}(t), \sigma _{2}(t)\in C_{0}({\mathbb {R}}, {\mathbb {R}}^{+})$, one concludes that

$$\begin{aligned} \lim \limits _{|t|\rightarrow +\infty }{\mathbb {E}}\Vert (\varGamma ^{2} \omega )(t)\Vert ^{p}=0 \ \hbox {uniformly for} \ \omega (t)\in \varOmega _{k_{0}}. \end{aligned}$$

Let $t\in {\mathbb {R}}$ be fixed. For given $\varepsilon _{0}>0$, from (3.15), it follows that

$$\begin{aligned} \begin{aligned} (\varGamma ^{2}_{\varepsilon _{0}}\omega )(t)&=\int _{-\infty }^{t-\varepsilon _{0}}T(t-\varepsilon _{0}-s)K_{v}'(s, \omega (s))\hbox {d}s\\&\quad +\,\int _{-\infty }^{t-\varepsilon _{0}}T(t-\varepsilon _{0}-s)K_{v}''(s, \omega (s))\hbox {d}W(s) \end{aligned} \end{aligned}$$

is uniformly bounded for $\omega (t)\in \varOmega _{k_{0}}$. This, together with the compactness of $T(\varepsilon _{0})$, yields that the set $\{T(\varepsilon _{0})(\varGamma ^{2}_{\varepsilon _{0}}\omega )(t): \omega (t)\in \varOmega _{k_{0}}\}$ is relatively compact in $L^{p}({\mathbb {P}},{\mathbb {H}})$. On the other hand

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert (\varGamma ^{2}\omega )(t)-T(\varepsilon _{0})(\varGamma ^{2}_{\varepsilon _{0}}\omega )(t)\Vert ^{p}\\&\quad \le 2^{p-1}{\mathbb {E}}\Big \Vert \int _{t-\varepsilon _{0}}^{t}T(t-s)K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{p}\\&\qquad +\,2^{p-1}{\mathbb {E}}\Big \Vert \int _{t-\varepsilon _{0}}^{t}T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{p}\\&\quad \le 2^{p-1}{\mathbb {E}}\Big (\int _{t-\varepsilon _{0}}^{t}Me^{-\delta (t-s)}\Vert K_{v}'(s, \omega (s))\Vert \hbox {d}s\Big )^{p}\\&\qquad +\,2^{p-1}C_{p}\Big ({\mathbb {E}}\int _{t-\varepsilon _{0}}^{t}\Vert T(t-s)K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\Big )^{\frac{p}{2}}\\&\quad \le 2^{p-1}M^{p}\left( \int _{t-\varepsilon _{0}}^{t}e^{-\frac{1}{2}\delta \frac{p}{p-1}(t-s)}\hbox {d}s\right) ^{p-1}{\mathbb {E}}\int ^{t}_{-\infty } e^{-\frac{1}{2}p\delta (t-s)}\Vert K_{v}'(s, \omega (s))\Vert ^{p}\hbox {d}s\\&\qquad +\,2^{p-1}C_{p}M^{p}\left( {\mathbb {E}}\int _{t-\varepsilon _{0}}^{t}e^{-2\delta (t-s)}\Vert K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\\&\quad \le 2^{p-1}M^{p}\left( \int _{t-\varepsilon _{0}}^{t}e^{-\frac{1}{2}\delta \frac{p}{p-1}(t-s)}\hbox {d}s\right) ^{p-1}{\mathbb {E}}\int ^{t}_{-\infty } e^{-\frac{1}{2}p\delta (t-s)}\Vert K_{v}'(s, \omega (s))\Vert ^{p}\hbox {d}s\\&\qquad +\,2^{p-1}C_{p}M^{p}\left( \int _{t-\varepsilon _{0}}^{t}e^{-\frac{p\delta (t-s)}{p-2}}\hbox {d}s\right) ^{\frac{p}{2}-1}{\mathbb {E}}\\&\qquad \int ^{t}_{-\infty }e^{-\frac{1}{2}p\delta (t-s)}\Vert K_{v}''(s, \omega (s))\Vert ^{p}\hbox {d}s \rightarrow 0 \ \ \hbox {as} \ \ \varepsilon _{0}\rightarrow 0^{+}, \end{aligned} \end{aligned}$$

this, together with the total boundedness, yields that the set $\{(\varGamma ^{2}\omega )(t): \omega (t)\in \varOmega _{k_{0}}\}$ is relatively compact in $L^{p}({\mathbb {P}},{\mathbb {H}})$ for each $t\in {\mathbb {R}}$.

Next, we consider the equicontinuity of the set $\{(\varGamma ^{2}\omega )(t): \omega (t)\in \varOmega _{k_{0}}\}$.

Given $\varepsilon _{1}>0$, in view of (3.15), there exists an $\eta >0$ such that for all $\omega (t)\in \varOmega _{k_{0}}$ and $t\ge \tau$ with $t-\tau <\eta$,

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Big \Vert \int _{\tau }^{t}T(t-s)K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{p}\le \frac{\varepsilon _{1}}{8^{p}},\\&\qquad {\mathbb {E}}\Big \Vert \int _{\tau -\eta }^{\tau }[T(t-s)-T(\tau -s)]K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{p}\le \frac{\varepsilon _{1}}{8^{p}},\\&\qquad {\mathbb {E}}\Big \Vert \int _{\tau }^{t}T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{p}\\&\quad \le C_{p}\left( {\mathbb {E}}\int _{\tau }^{t}\Vert T(t-s)K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\le \frac{\varepsilon _{1}}{8^{p}},\\&\qquad {\mathbb {E}}\Big \Vert \int _{\tau -\eta }^{\tau }[T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{p}\\&\quad \le C_{p}\left( {\mathbb {E}}\int _{\tau -\eta }^{\tau }\Vert [T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\le \frac{\varepsilon _{1}}{8^{p}}. \end{aligned} \end{aligned}$$

Also, one can choose a $k>0$ such that

$$\begin{aligned} \begin{aligned}&2^{p-1}M^{p}\left( \frac{2(p-1)}{p\delta }\right) ^{p-1}\frac{2}{p\delta }\varPhi _{1}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}\Vert v(s)\Vert \Big )e^{-p\delta k}\sup \limits _{s\in {\mathbb {R}}}\beta _{1}(s)\le \frac{\varepsilon _{1}}{8^{p}},\\&2^{p-1}M^{p}C_{p}\left( \frac{p-2}{p\delta }\right) ^{\frac{p}{2}-1}\frac{2}{p\delta }\varPhi _{2}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}\Vert v(s)\Vert \Big )e^{-p\delta k}\sup \limits _{s\in {\mathbb {R}}}\beta _{2}(s)\le \frac{\varepsilon _{1}}{8^{p}}, \end{aligned} \end{aligned}$$

which yield that for all $\omega \in \varOmega _{k_{0}}$ and $t\ge \tau$,

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Big \Vert \int _{-\infty }^{\tau -k}[T(t-s)-T(\tau -s)]K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{p}\le \frac{\varepsilon _{1}}{8^{p}},\\&{\mathbb {E}}\Big \Vert \int _{-\infty }^{\tau -k}[T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{p}\\&\quad \le C_{p}\left( {\mathbb {E}}\int _{-\infty }^{\tau -k}\Vert [T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\right) ^{\frac{p}{2}}\le \frac{\varepsilon _{1}}{8^{p}}. \end{aligned} \end{aligned}$$

Those, together with the fact that $\{T(t)\}_{t\ge 0}$ is compact implies its norm continuity, yield that there exists an $\eta '\in (0, \eta )$ such that for every $\omega (t)\in \varOmega _{k_{0}}$ and $t\ge \tau$ with $t-\tau <\eta ^{'}$,

$$\begin{aligned}&{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)K_{v}'(s, \omega (s))\hbox {d}s+\int _{-\infty }^{t}T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s)\\&\qquad -\,\int _{-\infty }^{\tau }T(t-s)K_{v}'(s, \omega (s))\hbox {d}s-\int _{-\infty }^{\tau }T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{p}\\&\quad \le 8^{p-1}{\mathbb {E}}\Big \Vert \int _{\tau }^{t}T(t-s)K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{p} \\&\qquad +\,8^{p-1}{\mathbb {E}}\Big \Vert \int _{\tau -\eta '}^{\tau }[T(t-s)-T(\tau -s)]K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{p}\\&\qquad +\,8^{p-1}{\mathbb {E}}\Big \Vert \int _{-\infty }^{\tau -k}[T(t-s)-T(\tau -s)]K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{p}\\&\qquad +\,8^{p-1}{\mathbb {E}}\Big \Vert \int _{\tau -k}^{\tau -\eta '}[T(t-s)-T(\tau -s)]K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{p}\\&\qquad +\,8^{p-1}{\mathbb {E}}\Big \Vert \int _{\tau }^{t}T(t-s)-T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{p}\\&\qquad +\,8^{p-1}{\mathbb {E}}\Big \Vert \int _{\tau -\eta '}^{\tau }[T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{p}\\&\qquad +\,8^{p-1}{\mathbb {E}}\Big \Vert \int _{-\infty }^{\tau -k}[T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{p}\\&\qquad +\,8^{p-1}{\mathbb {E}}\Big \Vert \int _{\tau -k}^{\tau -\eta '}[T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{p} <\varepsilon _{1}, \end{aligned}$$

which implies the equicontinuity of the set $\{(\varGamma ^{2}\omega ): \omega (t)\in \varOmega _{k_{0}}\}$. Applying Lemma 2.2 yields the compactness of $\varGamma ^{2}$ on $\varOmega _{k_{0}}$.

Finally, from Lemma 2.3, it follows that $\varGamma$ has at least one fixed point in $\varOmega _{k_{0}}$. This proves that system (3.11) has at least one solution in $\mathrm{AP}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))\times C_{0}({\mathbb {R}}, L^{p}({\mathbb {P}},{\mathbb {H}}))$. This completes this proof.$\square$

Remark 3.3

Note that the conditions (3.2) and (3.4) in ($H_{1}$) and ($H_{2}$) of Theorem 3.1 can be easily extended to the case of $F_{1}(t, x)$ and $G_{1}(t, x)$ being locally Lipschitz continuous in the following sense:

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert F_{1}(t, x)-F_{1}(t, y)\Vert ^{p}\le L_{1}(r){\mathbb {E}}\Vert X-Y\Vert ^{p}, \\&{\mathbb {E}}\Vert G_{1}(t, x)-G_{1}(t, y)\Vert ^{p}\le L_{2}(r){\mathbb {E}}\Vert X-Y\Vert ^{p} \end{aligned} \end{aligned}$$

for all $t\in {\mathbb {R}}$ and $x, y\in L^{p}({\mathbb {P}},{\mathbb {H}})$ satisfying $\Vert x\Vert _{\infty }^{p}, \Vert y\Vert _{\infty }^{p}\le r$, where $L_{1}(t), L_{2}(t)\in BC({\mathbb {R}}^{+}, {\mathbb {R}}^{+})$ are two functions.

Remark 3.4

The Theorem 3.1 is still valid for $p=2$. That is, let ($H_{1}$) as well as ($H_{2}$) hold with $p=2$, then Eq. (3.1) has at least one square-mean asymptotically almost periodic mild solution whenever

$$\begin{aligned} \begin{aligned} 4M^{2}\left[ \frac{L_{1}+\rho _{1}\rho _{1}'}{\delta ^{2}}+\frac{L_{2}+\rho _{2}\rho _{2}'}{2\delta }\right] <1. \end{aligned} \end{aligned}$$

(3.16)

Proof

Consider the coupled system of integral equations (3.11) again. If $(v(t), \omega (t))\in \mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))\times C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$ is a solution to system (3.11), then

$$\begin{aligned} x(t):=v(t)+\omega (t)\in \mathrm{AAP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}})) \end{aligned}$$

and it is a solution to the integral Eq. (3.12). Then multiplying both sides of (3.12) by $T(t-\tau )$ with $t>\tau$, one obtains that x(t) is a square-mean asymptotically almost periodic mild solution to Eq. (3.1). Hence the problem has shifted to show that system (3.11) has at least a solution in $\mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))\times C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$.

Defining $\varLambda$ on $\mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$ as (3.13). From $F_{1}(t, x)\in AA({\mathbb {R}}\times L^{2}({\mathbb {P}},{\mathbb {H}}), L^{2}({\mathbb {P}},{\mathbb {H}}))$ satisfying (3.2) with $p=2$ and $G_{1}(t, x)\in AA({\mathbb {R}}\times L^{2}({\mathbb {P}},{\mathbb {H}}), L^{2}({\mathbb {P}},{\mathbb {H}}))$ satisfying (3.4) with $p=2$, together with Lemma 2.1, it follows that

$$\begin{aligned} F_{1}(\cdot , v(\cdot )), G_{1}(\cdot , v(\cdot ))\in \mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}})) \quad \hbox {for each} \ v(\cdot )\in \mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}})). \end{aligned}$$

Those, together with Lemmas 3.4 and 3.6, imply that $\varLambda$ is well defined and maps $\mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$ into itself. Moreover, for any $v_{1}(t), v_{2}(t)\in \mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$, by (3.2) and (3.4) with $p=2$, together with the It$\hat{o}$’s isometry property of the stochastic integral, one has

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert (\varLambda v_{1})(t)-(\varLambda v_{2})(t)\Vert ^{2}\\&\quad \le 2{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)[F_{1}(s, v_{1}(s))-F_{1}(s, v_{2}(s))]\hbox {d}s\Big \Vert ^{2}\\&\qquad +\,2{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)[G_{1}(s, v_{1}(s))-G_{1}(s, v_{2}(s))]\hbox {d}W(s)\Big \Vert ^{2}\\&\quad \le 2{\mathbb {E}}\left( \int _{-\infty }^{t}Me^{-\delta (t-s)}\Vert F_{1}(s, v_{1}(s))-F_{1}(s, v_{2}(s))\Vert \hbox {d}s\right) ^{2}\\&\qquad +\,2{\mathbb {E}}\int _{-\infty }^{t}\Vert T(t-s)[G_{1}(s, v_{1}(s))-G_{1}(s, v_{2}(s))]\Vert ^{2}\hbox {d}s\\&\quad \le 2M^{2}\int ^{t}_{-\infty }e^{-\delta (t-s)}\hbox {d}s{\mathbb {E}}\int ^{t}_{-\infty } e^{-\delta (t-s)}\Vert F_{1}(s, v_{1}(s))-F_{1}(s, v_{2}(s))\Vert ^{2}\hbox {d}s\\&\qquad +\,2M^{2}{\mathbb {E}}\int ^{t}_{-\infty }e^{-2\delta (t-s)}\Vert G_{1}(s, v_{1}(s))-G_{1}(s, v_{2}(s))\Vert ^{2}\hbox {d}s\\&\quad \le \frac{2M^{2}L_{1}}{\delta ^{2}}\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v_{1}(s)-v_{2}(s)\Vert ^{2}+\frac{2M^{2}L_{2}}{2\delta }\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v_{1}(s)-v_{2}(s)\Vert ^{2}\\&\quad \le 2M^{2}\left[ \frac{L_{1}}{\delta ^{2}}+\frac{L_{2}}{2\delta }\right] \Vert v_{1}-v_{2}\Vert _{\infty }^{p}, \end{aligned} \end{aligned}$$

which implies

$$\begin{aligned} \begin{aligned} \Vert (\varLambda v_{1})-(\varLambda v_{2})\Vert _{\infty }\le \sqrt{2M^{2}\left[ \frac{L_{1}}{\delta ^{2}}+\frac{L_{2}}{2\delta }\right] }\Vert v_{1}-v_{2}\Vert _{\infty }. \end{aligned} \end{aligned}$$

Together with (3.16), this proves that $\varLambda$ is a contraction on $\mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$. Thus, the Banach’s fixed point theorem implies that $\varLambda$ has a unique fixed point $v(t)\in \mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$.

For the above v(t), define $\varGamma :=\varGamma ^{1}+\varGamma ^{2}$ on $C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$ as (3.14). From (3.2) and (3.4) with $p=2$, it follows that for all $s\in {\mathbb {R}}$, $\omega (s)\in L^{2}({\mathbb {P}},{\mathbb {H}})$,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Vert J_{v}'(s, \omega (s))\Vert ^{2}\le L_{1}{\mathbb {E}}\Vert \omega (s)\Vert ^{2},\ {\mathbb {E}}\Vert J_{v}''(s, \omega (s))\Vert ^{2}\le L_{2}{\mathbb {E}}\Vert \omega (s)\Vert ^{2}, \end{aligned} \end{aligned}$$

which imply that

$$\begin{aligned} J_{v}'(\cdot , \omega (\cdot )), J_{v}''(\cdot , \omega (\cdot ))\in C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))\quad \hbox {for each} \ \omega (\cdot )\in C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}})) \end{aligned}$$

and by (3.3) and (3.5) with $p=2$, one has for all $s\in {\mathbb {R}}$ and $\omega (s)\in L^{2}({\mathbb {P}},{\mathbb {H}})$ with ${\mathbb {E}}\Vert \omega (s)\Vert ^{2}\le r$,

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert K_{v}'(s, \omega (s))\Vert ^{2}\le \beta _{1}(s)\varPhi _{1}\Big (r+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{2}\Big ),\\&{\mathbb {E}}\Vert K_{v}''(s, \omega (s))\Vert ^{2}\le \beta _{2}(s)\varPhi _{2}\Big (r+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{2}\Big ), \end{aligned} \end{aligned}$$

(3.17)

which imply that

$$\begin{aligned} K_{v}'(\cdot , \omega (\cdot )), K_{v}''(\cdot , \omega (\cdot ))\in C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))\ \hbox {due to} \ \beta _{1}(s), \beta _{2}(s)\in C_{0}({\mathbb {R}}, {\mathbb {R}}^{+}). \end{aligned}$$

Those, together with Lemmas 3.4 and 3.6, yield that $\varGamma$ is well defined and maps $C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$ into itself. To complete the proof, it suffices to prove that $\varGamma$ possesses at least one fixed point in $C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$.

Set

$$\begin{aligned} \varOmega _{r}:=\{\omega \in C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}})): \Vert \omega \Vert _{\infty }^{2}\le r\}. \end{aligned}$$

In view of (3.3), (3.5) with $p=2$ and (3.16), it is not difficult to see that there exists a constant $k_{0}>0$ such that

$$\begin{aligned} \begin{aligned}&4M^{2}\left[ \frac{L_{1}}{\delta ^{2}}+\frac{L_{2}}{2\delta }\right] k_{0}\\&\quad +\frac{4M^{2\rho _{1}'}}{\delta ^{2}}\varPhi _{1} \left( k_{0}+\sup \limits _{s\in {\mathbb {R}}}\Vert v(s)\Vert \right) + \frac{4M^{2}\rho _{2}'}{2\delta }\varPhi _{2}\left( k_{0}+\sup \limits _{s\in {\mathbb {R}}}\Vert v(s) \Vert \right) \le k_{0}. \end{aligned} \end{aligned}$$

This enables us to conclude that for any $t\in {\mathbb {R}}$ and $\omega _{1}(t), \omega _{2}(t)\in \varTheta _{k_{0}}$,

$$\begin{aligned}&{\mathbb {E}}\Vert (\varGamma ^{1}\omega _{1})(t)+(\varGamma ^{2}\omega _{2})(t)\Vert ^{2}\\&\quad \le 4{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)J_{v}'(s, \omega _{1}(s))\hbox {d}s\Big \Vert ^{2}\\&\qquad +\,4{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)J_{v}''(s, \omega _{1}(s))\hbox {dd}W(s)\Big \Vert ^{2}\\&\qquad +\,4{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)K_{v}'(s, \omega _{2}(s))\hbox {d}s\Big \Vert ^{2} \\&\qquad +\,4{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)K_{v}''(s, \omega _{2}(s))\hbox {dd}W(s)\Big \Vert ^{2}\\&\quad \le 4{\mathbb {E}}\left( \int _{-\infty }^{t}Me^{-\delta (t-s)}\Vert J_{v}'(s, \omega _{1}(s))\Vert \hbox {d}s\right) ^{2}\\&\qquad +\,4{\mathbb {E}}\int _{-\infty }^{t}\Vert T(t-s)J_{v}''(s, \omega _{1}(s))\Vert ^{2}\hbox {d}s\\&\qquad +\,4{\mathbb {E}}\left( \int _{-\infty }^{t}Me^{-\delta (t-s)}\Vert K_{v}'(s, \omega _{2}(s))\Vert \hbox {d}s\right) ^{2}\\&\qquad +\,4{\mathbb {E}}\int _{-\infty }^{t}\Vert T(t-s)K_{v}''(s, \omega _{2}(s))\Vert ^{2}\hbox {d}s\\&\quad \le 4M^{2}\int ^{t}_{-\infty }e^{-\delta (t-s)}\hbox {d}s{\mathbb {E}}\int ^{t}_{-\infty } e^{-\delta (t-s)}\Vert J_{v}'(s, \omega _{1}(s))\Vert ^{2}\hbox {d}s\\&\qquad +\,4M^{2}{\mathbb {E}}\int ^{t}_{-\infty }e^{-2\delta (t-s)}\Vert J_{v}''(s, \omega _{1}(s))\Vert ^{2}\hbox {d}s\\&\qquad +\,4M^{2}\int ^{t}_{-\infty }e^{-\delta (t-s)}{\mathbb {E}}\int ^{t}_{-\infty } e^{-\delta (t-s)}\Vert K_{v}'(s, \omega _{2}(s))\Vert ^{2}\hbox {d}s\\&\qquad +\,4M^{2}{\mathbb {E}}\int ^{t}_{-\infty }e^{-2\delta (t-s)}\Vert K_{v}''(s, \omega _{2}(s))\Vert ^{2}\hbox {d}s\\&\quad \le \frac{4M^{2}L_{1}}{\delta ^{2}}\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert \omega _{1}(s)\Vert ^{2}+\frac{4M^{2}L_{2}}{2\delta }\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert \omega _{1}(s)\Vert ^{2}\\&\qquad +\,\frac{4M^{2}\rho _{1}'}{\delta ^{2}}\varPhi _{1}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{2}\Big )+\frac{4M^{2}\rho _{2}'}{2\delta }\varPhi _{2}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{2}\Big )\\&\quad \le 4M^{2}\left[ \frac{L_{1}}{\delta ^{2}}+\frac{L_{2}}{2\delta }\right] k_{0}\\&\qquad +\,\frac{4M^{2}\rho _{1}'}{\delta ^{2}}\varPhi _{1}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{2}\Big )+\frac{4M^{2}\rho _{2}'}{2\delta }\varPhi _{2}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{2}\Big )\le k_{0}, \end{aligned}$$

which implies that for every pair $\omega _{1}(t), \omega _{2}(t)\in \varTheta _{k_{0}}$,

$$\begin{aligned} (\varGamma ^{1}\omega _{1})(t)+(\varGamma ^{2}\omega _{2})(t)\in \varTheta _{k_{0}}. \end{aligned}$$

Thus $\varGamma$ maps $\varTheta _{k_{0}}$ into itself.

In the following, we show that $\varGamma ^{1}$ is a contraction on $\varTheta _{k_{0}}$.

For $\omega _{1}(t), \omega _{2}(t)\in \varTheta _{k_{0}}$ and $t\in {\mathbb {R}}$, from (3.2) and (3.4) with $p=2$ it follows that

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert J_{v}'(s, \omega _{1}(s))-J_{v}'(s, \omega _{2}(s))\Vert ^{2}\le L_{1}{\mathbb {E}}\Vert \omega _{1}(s)-\omega _{2}(s)\Vert ^{2},\\&{\mathbb {E}}\Vert J_{v}''(s, \omega _{1}(s))-J_{v}''(s, \omega _{2}(s))\Vert ^{2}\le L_{2}{\mathbb {E}}\Vert \omega _{1}(s)-\omega _{2}(s)\Vert ^{2}. \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert (\varGamma ^{1}\omega _{1})(t)-(\varGamma ^{1}\omega _{2})(t)\Vert ^{2}\\&\quad \le 2{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)(J_{v}'(s, \omega _{1}(s))-J_{v}'(s, \omega _{2}(s)))\hbox {d}s\Big \Vert ^{2}\\&\qquad +\,2{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)(J_{v}''(s, \omega _{1}(s))-J_{v}''(s, \omega _{2}(s)))\hbox {d}W(s)\Big \Vert ^{2}\\&\quad \le 2{\mathbb {E}}\left( \int _{-\infty }^{t}Me^{-\delta (t-s)}\Vert J_{v}'(s, \omega _{1}(s))-J_{v}'(s, \omega _{2}(s))\Vert \hbox {d}s\right) ^{2}\\&\qquad +\,2{\mathbb {E}}\int _{-\infty }^{t}\Vert T(t-s)(J_{v}''(s, \omega _{1}(s))-J_{v}''(s, \omega _{2}(s)))\Vert ^{2}\hbox {d}s\\&\quad \le 2M^{2}\int ^{t}_{-\infty }e^{-\delta (t-s)}{\mathbb {E}}\int ^{t}_{-\infty } e^{-\delta (t-s)}\Vert J_{v}'(s, \omega _{1}(s))-J_{v}'(s, \omega _{2}(s))\Vert ^{2}\hbox {d}s\\&\qquad +\,2M^{2}{\mathbb {E}}\int ^{t}_{-\infty }e^{-2\delta (t-s)}\Vert J_{v}''(s, \omega _{1}(s))-J_{v}''(s, \omega _{2}(s))\Vert ^{2}\hbox {d}s\\&\quad \le \frac{2M^{2}L_{1}}{\delta ^{2}}\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert \omega _{1}(s)-\omega _{2}(s)\Vert ^{2}+\frac{2M^{2}L_{2}}{2\delta }\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert \omega _{1}(s)-\omega _{2}(s)\Vert ^{2}\\&\quad \le 2M^{2}\left[ \frac{L_{1}}{\delta ^{2}}+\frac{L_{2}}{2\delta }\right] \Vert \omega _{1}-\omega _{2}\Vert _{\infty }^{2}, \end{aligned} \end{aligned}$$

which implies

$$\begin{aligned} \begin{aligned} \Vert (\varGamma ^{1}\omega _{1})(t)-(\varGamma ^{1}\omega _{2})(t)\Vert _{\infty } \le \sqrt{2M^{2}\left[ \frac{L_{1}}{\delta ^{2}}+\frac{L_{2}}{2\delta }\right] \Vert }\omega _{1}-\omega _{2}\Vert _{\infty }. \end{aligned} \end{aligned}$$

Thus, in view of (3.16), one obtains the conclusion.

From our assumptions, it is clear that $\varGamma ^{2}$ is a continuous mapping from $\varTheta _{k_{0}}$ to $\varTheta _{k_{0}}$. Thus, in order to apply the well-known Krasnoselskii’s fixed point theorem to obtain a fixed point of $\varGamma$, one needs to verify that $\varGamma ^{2}$ is compact on $\varTheta _{k_{0}}$.

First, for all $\omega (t)\in \varTheta _{k_{0}}$ and $t\in {\mathbb {R}}$,

$$\begin{aligned}&{\mathbb {E}}\Vert (\varGamma ^{2} \omega )(t)\Vert ^{2}\\&\quad \le 2{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{2}+2{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{2}\\&\quad \le 2{\mathbb {E}}\left( \int _{-\infty }^{t}Me^{-\delta (t-s)}\Vert K_{v}'(s, \omega (s))\Vert \hbox {d}s\right) ^{2}+2{\mathbb {E}}\int _{-\infty }^{t}\Vert T(t-s)K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\\&\quad \le 2M^{2}\int ^{t}_{-\infty }e^{-\delta (t-s)}\hbox {d}s{\mathbb {E}}\int ^{t}_{-\infty } e^{-\delta (t-s)}\Vert K_{v}'(s, \omega (s))\Vert ^{2}\hbox {d}s\\&\qquad +\,2M^{2}{\mathbb {E}}\int ^{t}_{-\infty }e^{-2\delta (t-s)}\Vert K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\\&\quad \le \frac{2M^{2}L_{1}\sigma _{1}}{\delta ^{2}}\varPhi _{1}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{2}\Big )+\frac{2M^{2}L_{2}\sigma _{2}}{2\delta }\varPhi _{2}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}{\mathbb {E}}\Vert v(s)\Vert ^{2}\Big ), \end{aligned}$$

in view of $\sigma _{1}(t), \sigma _{2}(t)\in C_{0}({\mathbb {R}}, {\mathbb {R}}^{+})$, one concludes that

$$\begin{aligned} \lim \limits _{|t|\rightarrow +\infty }{\mathbb {E}}\Vert (\varGamma ^{2} \omega )(t)\Vert ^{2}=0 \ \hbox {uniformly for} \ \omega (t)\in \varTheta _{k_{0}}. \end{aligned}$$

Let $t\in {\mathbb {R}}$ be fixed. For given $\varepsilon _{0}>0$, from (3.17) it follows that

$$\begin{aligned} \begin{aligned} (\varGamma ^{2}_{\varepsilon _{0}}\omega )(t)&=\int _{-\infty }^{t-\varepsilon _{0}}T(t-\varepsilon _{0}-s)K_{v}'(s, \omega (s))\hbox {d}s\\&\quad +\,\int _{-\infty }^{t-\varepsilon _{0}}T(t-\varepsilon _{0}-s)K_{v}''(s, \omega (s))\hbox {d}W(s) \end{aligned} \end{aligned}$$

is uniformly bounded for $\omega (t)\in \varTheta _{k_{0}}$. This, together with the compactness of $T(\varepsilon _{0})$, yields that the set $\{T(\varepsilon _{0})(\varGamma ^{2}_{\varepsilon _{0}}\omega )(t): \omega (t)\in \varTheta _{k_{0}}\}$ is relatively compact in $L^{2}({\mathbb {P}},{\mathbb {H}})$. On the other hand

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert (\varGamma ^{2}\omega )(t)-T(\varepsilon _{0})(\varGamma ^{2}_{\varepsilon _{0}}\omega )(t)\Vert ^{2}\\&\quad \le 2{\mathbb {E}}\Big \Vert \int _{t-\varepsilon _{0}}^{t}T(t-s)K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{2} +2{\mathbb {E}}\Big \Vert \int _{t-\varepsilon _{0}}^{t}T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{2}\\&\quad \le 2{\mathbb {E}}\left( \int _{t-\varepsilon _{0}}^{t}Me^{-\delta (t-s)}\Vert K_{v}'(s, \omega (s))\Vert \hbox {d}s\right) ^{2}+2{\mathbb {E}}\int _{t-\varepsilon _{0}}^{t}\Vert T(t-s)K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\\&\quad \le 2M^{2}\int _{t-\varepsilon _{0}}^{t}e^{-\delta (t-s)}\hbox {d}s{\mathbb {E}}\int ^{t}_{-\infty }e^{-\delta (t-s)}\Vert K_{v}'(s, \omega (s))\Vert ^{2}\hbox {d}s\\&\qquad +\,2M^{2}{\mathbb {E}}\int _{t-\varepsilon _{0}}^{t}e^{-2\delta (t-s)}\Vert K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\\&\qquad \rightarrow 0 \ \ \hbox {as} \ \ \varepsilon _{0}\rightarrow 0^{+}, \end{aligned} \end{aligned}$$

this, together with the total boundedness, yields that the set $\{(\varGamma ^{2}\omega )(t): \omega (t)\in \varTheta _{k_{0}}\}$ is relatively compact in $L^{2}({\mathbb {P}},{\mathbb {H}})$ for each $t\in {\mathbb {R}}$.

Next, we consider the equicontinuity of the set $\{(\varGamma ^{2}\omega )(t): \omega (t)\in \varTheta _{k_{0}}\}$. Given $\varepsilon _{1}>0$. In view of (3.17), there exists an $\eta >0$ such that for all $\omega (t)\in \varTheta _{k_{0}}$ and $t\ge \tau$ with $t-\tau <\eta$,

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Big \Vert \int _{\tau }^{t}T(t-s)K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{2}\le \frac{\varepsilon _{1}}{64},\\&{\mathbb {E}}\Big \Vert \int _{\tau -\eta }^{\tau }[T(t-s)-T(\tau -s)]K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{2}\le \frac{\varepsilon _{1}}{64},\\&{\mathbb {E}}\Big \Vert \int _{\tau }^{t}T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{2}={\mathbb {E}}\int _{\tau }^{t}\Vert T(t-s)K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\le \frac{\varepsilon _{1}}{64},\\&{\mathbb {E}}\Big \Vert \int _{\tau -\eta }^{\tau }[T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{2}\\&\quad ={\mathbb {E}}\int _{\tau -\eta }^{\tau }\Vert [T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\le \frac{\varepsilon _{1}}{64}. \end{aligned} \end{aligned}$$

Also, one can choose a $k>0$ such that

$$\begin{aligned} \begin{aligned}&\frac{2M^{2}}{\delta ^{2}}\varPhi _{1}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}\Vert v(s)\Vert \Big )e^{-2\delta k}\sup \limits _{s\in {\mathbb {R}}}\beta _{1}(s)\le \frac{\varepsilon _{1}}{64},\\&\frac{2M^{2}}{2\delta }\varPhi _{2}\Big (k_{0}+\sup \limits _{s\in {\mathbb {R}}}\Vert v(s)\Vert \Big )e^{-2\delta k}\sup \limits _{s\in {\mathbb {R}}}\beta _{2}(s)\le \frac{\varepsilon _{1}}{64}, \end{aligned} \end{aligned}$$

which yield that for all $\omega \in \varOmega _{k_{0}}$ and $t\ge \tau$,

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Big \Vert \int _{-\infty }^{\tau -k}[T(t-s)-T(\tau -s)]K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{p}\le \frac{\varepsilon _{1}}{64},\\&{\mathbb {E}}\Big \Vert \int _{-\infty }^{\tau -k}[T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{2}\\&\quad ={\mathbb {E}}\int _{-\infty }^{\tau -k}\Vert [T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\Vert ^{2}\hbox {d}s\le \frac{\varepsilon _{1}}{64}. \end{aligned} \end{aligned}$$

Those, together with the fact that $\{T(t)\}_{t\ge 0}$ is compact implies its norm continuity, yield that there exists an $\eta '\in (0, \eta )$ such that for every $\omega (t)\in \varTheta _{k_{0}}$ and $t\ge \tau$ with $t-\tau <\eta ^{'}$,

$$\begin{aligned}&{\mathbb {E}}\Big \Vert \int _{-\infty }^{t}T(t-s)K_{v}'(s, \omega (s))\hbox {d}s+\int _{-\infty }^{t}T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s)\\&\qquad -\,\int _{-\infty }^{\tau }T(t-s)K_{v}'(s, \omega (s))\hbox {d}s-\int _{-\infty }^{\tau }T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{2}\\&\quad \le 8{\mathbb {E}}\Big \Vert \int _{\tau }^{t}T(t-s)K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{2}\\&\qquad +\,8{\mathbb {E}}\Big \Vert \int _{\tau -\eta '}^{\tau }[T(t-s)-T(\tau -s)]K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{2}\\&\qquad +\,8{\mathbb {E}}\Big \Vert \int _{-\infty }^{\tau -k}[T(t-s)-T(\tau -s)]K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{2}\\&\qquad +\,8{\mathbb {E}}\Big \Vert \int _{\tau -k}^{\tau -\eta '}[T(t-s)-T(\tau -s)]K_{v}'(s, \omega (s))\hbox {d}s\Big \Vert ^{2}\\&\qquad +\,8{\mathbb {E}}\Big \Vert \int _{\tau }^{t}T(t-s)-T(t-s)K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{p}\\&\qquad +\,8{\mathbb {E}}\Big \Vert \int _{\tau -\eta '}^{\tau }[T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{2}\\&\qquad +\,8{\mathbb {E}}\Big \Vert \int _{-\infty }^{\tau -k}[T(t-s)-T(\tau -s)]K_{v}''(s, \omega (s))\hbox {d}W(s)\Big \Vert ^{2}\\&\qquad +\,8{\mathbb {E}}\Big \Vert \int _{\tau -k}^{\tau -\eta '}[T(t-s)-T(\tau -s)]K_{v}'(s, \omega (s))'\hbox {d}W(s)\Big \Vert ^{2}<\varepsilon _{1}, \end{aligned}$$

which implies the equicontinuity of the set $\{(\varGamma ^{2}\omega ): \omega (t)\in \varTheta _{k_{0}}\}$. Applying Lemma 2.2 yields the compactness of $\varGamma ^{2}$ on $\varTheta _{k_{0}}$.

Finally, from Lemma 2.3 it follows that $\varGamma$ has at least one fixed point in $\varTheta _{k_{0}}$. This proves that Eq. (3.11) has at least one solution in $\mathrm{AP}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))\times C_{0}({\mathbb {R}}, L^{2}({\mathbb {P}},{\mathbb {H}}))$. This completes this proof. $\square$

Remark 3.5

Note that the conditions (3.2) and (3.4) in ($H_{1}$) and ($H_{2}$) with $p=2$ of Remark 3.4 can be also easily extended to the case of $F_{1}(t, x)$ and $G_{1}(t, x)$ being locally Lipschitz continuous in the following sense:

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\Vert F_{1}(t, x)-F_{1}(t, y)\Vert ^{2}\le L_{1}(r){\mathbb {E}}\Vert X-Y\Vert ^{2}, \\&{\mathbb {E}}\Vert G_{1}(t, x)-G_{1}(t, y)\Vert ^{2}\le L_{2}(r){\mathbb {E}}\Vert X-Y\Vert ^{2} \end{aligned} \end{aligned}$$

for all $t\in {\mathbb {R}}$ and $x, y\in L^{2}({\mathbb {P}},{\mathbb {H}})$ satisfying $\Vert x\Vert _{\infty }^{2}, \Vert y\Vert _{\infty }^{2}\le r$, where $L_{1}(t), L_{2}(t)\in BC({\mathbb {R}}^{+}, {\mathbb {R}}^{+})$ are two functions.

Remark 3.6

Note that in many papers (for instance [44, 45]) on square-mean asymptotically almost periodic mild solutions, to be able to apply the well-known Banach contraction principle, (locally) Lipschitz conditions for the nonlinearities of corresponding stochastic equations are needed. As the readers see, in Remark 3.4, we established a new existence result of square-mean asymptotically almost periodic mild solutions when the nonlinearities F(t, x) and G(t, x) as a whole lose the Lipschitz continuity with respect to x. Thus, our conditions in the assumptions ($H_{1}$) and ($H_{2}$) with $p=2$ are weaker than those of [44, 45] for the square-mean asymptotically almost periodic mild solutions.

4 Applications

In this section, we provide with an example to illustrate the practical usefulness of the theoretical result established in the preceding section.

Consider the stochastic partial differential equation with Dirichlet boundary conditions of the form

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&\hbox {d}u(t,\xi )=\Bigg [\frac{\partial ^{2}}{\partial \xi ^{2}}u(t, \xi )+\alpha _{1}(\sin t+\sin \sqrt{2}t)\sin u(t, \xi )\\&\qquad +\nu _{1} e^{-|t|}u(t, \xi )\sin u^{2}(t, \xi )\Bigg ]\hbox {d}t\\&\qquad +\Big [\alpha _{2}(\cos t+\cos \pi t)\cos u(t, \xi )+\nu _{2} e^{-|t|}u(t, \xi )\cos u^{2}(t, \xi )\Big ]\hbox {d}W(t), \quad \\&\qquad t\in {\mathbb {R}}, \quad \xi \in [0, \pi ],\\&u(t,0)=u(t,\pi )=0,\quad t\in {\mathbb {R}}, \end{aligned} \right. \end{aligned} \end{aligned}$$

(4.1)

where $\alpha _{1}, \alpha _{2}$ and $\nu _{1}, \nu _{2}$ are constants, W(t) is a two-sided and standard one-dimensional Brownian motion defined on the filtered probability space $(\varOmega , {\mathbb {F}}, {\mathbb {P}}, {\mathbb {F}}_{t})$.

Let $p=2$, take $H=L^{2}[0,\pi ]$ with norm $\Vert \cdot \Vert$ and define $A:D(A)\subset L^{2}({\mathbb {P}}, H)\rightarrow L^{2}({\mathbb {P}}, H)$ given by $Ax=\frac{\partial ^{2} x(\xi )}{\partial \xi ^{2}}$ with the domain

$$\begin{aligned} \begin{aligned} D(A)&=\Big \{x(\cdot )\in L^{2}({\mathbb {P}}, H): x''\in L^{2}({\mathbb {P}}, H), x'\in L^{2}({\mathbb {P}}, H)\\&\quad \hbox {is absolutely continuous on}\ [0,\pi ], x(0)=x(\pi )=0\Big \}. \end{aligned} \end{aligned}$$

It is well known that A is self-adjoint, with compact resolvent, and is the infinitesimal generator of an analytic as well as compact semigroup $\{T(t)\}_{t\ge 0}$ satisfying

$$\begin{aligned} \Vert T(t)\Vert \le e^{-t} \quad \hbox {for} \ t>0. \end{aligned}$$

Let

$$\begin{aligned} F_{1}(t, x(\xi )):= & {} \alpha _{1}(\sin t+\sin \sqrt{2}t)\sin x(\xi ), \quad F_{2}(t, x(\xi )):=\nu _{1} e^{-|t|}x(\xi )\sin x^{2}(\xi ).\\ G_{1}(t, x(\xi )):= & {} \alpha _{2}(\cos t+\cos \pi t)\cos x(\xi ), \quad G_{2}(t, x(\xi )):=\nu _{2} e^{-|t|}u(t, \xi )\cos x^{2}(\xi ). \end{aligned}$$

Then it is easy to verify that $F_{1}, F_{2}, G_{1}, G_{2}: {\mathbb {R}}\times L^{2}({\mathbb {P}}, H)\rightarrow L^{2}({\mathbb {P}}, H)$ are $L^{2}$-continuous, $F_{1}, G_{1}\in \mathrm{AP}({\mathbb {R}}\times L^{2}({\mathbb {P}}, H), L^{2}({\mathbb {P}}, H))$ and

$$\begin{aligned} {\mathbb {E}}\Vert F_{1}(t, x)-F_{1}(t, y)\Vert ^{2}\le & {} 4\alpha _{1}^{2}{\mathbb {E}}\Vert x-y\Vert ^{2}, \quad {\mathbb {E}}\Vert F_{2}(t, x)\Vert ^{2}\le \nu _{1}^{2}e^{-2|t|}{\mathbb {E}}\Vert x\Vert ^{2},\\ {\mathbb {E}}\Vert G_{1}(t, x)-F_{1}(t, y)\Vert ^{2}\le & {} 4\alpha _{1}^{2}{\mathbb {E}}\Vert x-y\Vert ^{2}, \quad {\mathbb {E}}\Vert G_{2}(t, x)\Vert ^{2}\le \nu _{2}^{2}e^{-2|t|}{\mathbb {E}}\Vert x\Vert ^{2}, \end{aligned}$$

which imply

$$\begin{aligned} F_{2}(t, x), G_{2}(t, x)\in C_{0}({\mathbb {R}}\times L^{2}({\mathbb {P}}, H), L^{2}({\mathbb {P}}, H)). \end{aligned}$$

Then further one has

$$\begin{aligned} F(t, x)= & {} F_{1}(t, x)+F_{2}(t, x)\in \mathrm{AAP}({\mathbb {R}}\times L^{2}({\mathbb {P}}, H), L^{2}({\mathbb {P}}, H)),\\ G(t, x)= & {} G_{1}(t, x)+G_{2}(t, x)\in \mathrm{AAP}({\mathbb {R}}\times L^{2}({\mathbb {P}}, H), L^{2}({\mathbb {P}}, H)). \end{aligned}$$

Thus, Eq. (4.1) can be reformulated as the abstract problem (3.1) and the assumptions $(H_{1})$ and $(H_{2})$ hold with

$$\begin{aligned} \begin{aligned}&L_{1}=4\alpha _{1}^{2}, \quad L_{2}=4\alpha _{2}^{2}, \quad \varPhi _{1}(r)=\varPhi _{2}(r)=r, \quad \beta _{1}(t)=\nu _{1}^{2}e^{-2|t|},\\&\beta _{2}(t)=\nu _{2}^{2}e^{-2|t|}, \quad \rho _{1}=\rho _{2}=1, \quad \rho _{1}'\le \nu _{1}^{2}, \quad \rho _{2}'\le \nu _{2}^{2}. \end{aligned} \end{aligned}$$

Then from Remark 3.4 it follows that Eq. (4.1) at least has one square-mean asymptotically almost periodic mild solutions whenever

$$\begin{aligned} 8\alpha _{1}^{2}+4\alpha _{2}^{2}+2\nu _{1}^{2}+\nu _{2}^{2}<\frac{1}{2}. \end{aligned}$$