Square-mean $\mu$-pseudo almost periodic and automophic solutions for a class of semilinear integro-differential stochastic evolution equations

Abstract

In this work, we use the concept of $\mu$-square-mean pseudo almost periodic and automorphic processes introduced by Diop et al. (Afr Mat 26(5):779–812, 2015) to discuss the existence and uniqueness of solutions for some semilinear integro-differential stochastic evolution equations. We provide an example to illustrate ours results.

1 Introduction

Integro-differential equations arose naturally in mechanics, electromagnetic theory, heat flow, nuclear reactor dynamics and population dynamics. Ding et al. [14] investigated the existence of pseudo almost periodic solutions for an equation arising in the study of heat conduction in materials with memory, which could be transformed into the following abstract integro-differential equation

$$\begin{aligned} x'(t) = Ax(t) + \int _{0}^{t}B(t-s)x(s)ds + h(t,x(t))\quad \text{ for } \text{ all }\quad t \ge 0. \end{aligned}$$

(1.1)

Furthermore, noise or stochastic perturbation is unavoidable and omnipresent in nature as well as in man-made systems. This paper is mainly focused on the existence and uniqueness of $\mu$-pseudo almost periodic and automorphic solutions to the semilinear integro-differential stochastic evolution equations in a Hilbert space $\mathbb {H}$

$$\begin{aligned} x'(t)= & {} Ax(t) + g(t,x(t)) + \int _{-\infty }^{t}B_{1}(t-s)f(s,x(s))ds\nonumber \\&+\, \int _{-\infty }^{t}B_{2}(t-s)h(s,x(s))dW(s) \quad \text{ for } \text{ all }\quad t \in \mathbb {R}, \end{aligned}$$

(1.2)

where $A :D(A)\subset \mathbb {H} \rightarrow \mathbb {H}$ is densely defined closed operator (possibly unbounded). $B_{1}$ and $B_{2}$ are convolution-type kernels in $L^{1}(0,\infty )$ and $L^{2}(0,\infty )$ respectively. $f,g,h: \mathbb {R}\times L^{2}(\Omega ,\mathbb {H})\rightarrow L^{2}(\Omega ,\mathbb {H})$ are two stochastic processes and W(t) is a two-sided and standard one-dimensional Brownian motion defined on the filtered probability space $(\Omega ,\mathcal {F},P,\mathcal {F}_{t})$ with $\mathcal {F}_{t}=\sigma \{W(u) - W(v) | u,v\le t \}.$

We assume $(\mathbb {H}, \Vert \cdot \Vert )$ is real separable Hilbert space and $L^{2}(\Omega ,\mathbb {H})$ represents the space of all $\mathbb {H}$-valued random variables x such that

$$\begin{aligned} \mathbb {E}\Vert x\Vert ^{2}=\int _{\Omega }\Vert x\Vert ^{2}dP < + \infty . \end{aligned}$$

The concept of almost automorphic is a natural generalization of the almost periodicity that was introduced by Bochner [8]. For more details about the almost automorphic functions we refer the reader to the book [27] where the author gave an important overview about the theory of almost automorphic functions and their applications to differential equations. In the last decade, many authors have produced extensive literature on the theory of almost automorphy and its applications to differential equations, more details can be found in [9, 17,18,19, 22,23,24, 26, 31, 32] and the references therein. Then a generalization of almost automorphic functions gives pseudo almost automorphic functions. Also weighted pseudo almost automorphic functions which are more general than weighted pseudo almost periodic functions in [2, 12, 13] and the references therein.

In recent years, the study of almost periodic and almost automorphic solutions to some stochastic differential equations have been considerably investigated in lots of publications [4, 6, 7, 10, 20, 21, 28, 30] because of its significance and applications in physics, mechanics and mathematical biology. The concept of square-mean almost pseudo automorphic stochastic processes was introduced by Chen and Lin in [11]. One say that a continuous stochastic process (see Definition 2.1) x is $\mu$-pseudo almost periodic (resp. automorphic) in the square-mean sense if

$$\begin{aligned} x=x_{1}+x_{2}, \end{aligned}$$

where $x_{1}$ is almost periodic (resp. automorphic) and $x_{2}$ is $\mu$-ergodic in the sense that

$$\begin{aligned} \lim _{r \rightarrow \infty } \frac{1}{\mu ([-r,r])}\int _{-r}^{r}\mathbb {E}\Vert x_{2}(t)\Vert ^{2}d\mu (t)= 0, \end{aligned}$$

where $\mu$ is a positive measure on $\mathbb {R}$, $\mu ([-r,r])$ is the measure of the set $[-r,r]$. One can observe that a square-mean pseudo almost periodic (resp. automorphic) process is a square-mean $\mu$-pseudo almost periodic (resp. automorphic) process in the particular case where the measure $\mu$ is the Lebesgue measure on $\mathbb {R}$. For more details about the $\mu$-pseudo almost periodic (resp. automorphic) processes, one can refer to [15, 16]. Note that in [3, 25], a new concept of almost periodic (resp. automorphic) process in a distribution sense was introduced in the literature. The authors use this concept to study some stochastic differential equations.

Recently, Bezandry in [5] investigate the existence and uniqueness of the square-mean almost periodic solution of the Eq. (1.2). However, to the best of the author knowledge, the existence of $\mu$-square-mean pseudo almost periodic and automorphic solutions of the Eq. (1.2) remains an untreated question, which is the mean motivation of this paper.

The organization of this work is as follows. In Sect. 2, we make a recalling on $\mu$-pseudo almost periodic and automorphic processes. In Sect. 3, using the results obtain in Sect. 2 and some suitable conditions, we prove the existence and the uniqueness of the square-mean $\mu$-pseudo almost periodic and automorphic mild solution of the Eq. (1.2). In Sect. 4, we provided an example to illustrate our results.

2 Preliminaries

2.1 Square-mean $\mu$-ergodic process

Denote by $\mathcal {B}$ the Lebesgue $\sigma$-field $\mathbb {R}$ of and by $\mathcal {M}$ the set of all positive measures $\mu$ on $\mathcal {B}$ satisfying $\mu (\mathbb {R}) = +\infty$ and $\mu ([a, b])< + \infty$ for all $a, b \in \mathbb {R}$ $(a\le b)$.

$L^{2}(\Omega ,\mathbb {H})$ is Hilbert space equipped with the following norm

$$\begin{aligned} \Vert x\Vert _{L^{2}}=\left( \int _{\Omega }\Vert x\Vert ^{2}dP\right) ^{\frac{1}{2}}. \end{aligned}$$

Definition 2.1

Let $x : \mathbb {R} \rightarrow L^{2}(\Omega ,\mathbb {H})$ be a stochastic process.

• (1) x is said to be stochastically bounded if there exists $M > 0$ such that

  $$\begin{aligned} \mathbb {E}\Vert x(t)\Vert ^{2} \le M \quad \text{ for } \text{ all }\quad t \in \mathbb {R}. \end{aligned}$$

• (2) x is said to be stochastically continuous if

  $$\begin{aligned} \lim _{t \rightarrow s}\mathbb {E}\Vert x(t) - x(s)\Vert ^{2} = 0 \quad \text{ for } \text{ all }\quad s \in \mathbb {R}. \end{aligned}$$

We denote by $SBC(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$ the space of all the stochastically bounded and continuous process. Clearly, the space $SBC(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$ is a Banach space equipped with the following norm

$$\begin{aligned} \Vert x\Vert _{\infty }=\sup _{t\in \mathbb {R}}(\mathbb {E}\Vert x(t)\Vert ^{2})^{\frac{1}{2}}. \end{aligned}$$

Definition 2.2

[15] Let $\mu \in \mathcal {M}$. A stochastic process x is said to be square-mean $\mu$-ergodic if $x \in SBC(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$ and satisfied

$$\begin{aligned} \lim _{r \rightarrow \infty } \frac{1}{\mu ([-r,r])}\int _{-r}^{r}\mathbb {E} \Vert x(t)\Vert ^{2} d\mu (t) = 0. \end{aligned}$$

We denote the space of all such process by $\mathcal {E}(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$.

For $\mu \in \mathcal {M}$, we denote

$$\begin{aligned} \mathcal {E}(\mathbb {R} \times L^{2}(\Omega ,\mathbb {H}), L^{2}(\Omega ,\mathbb {H}),\mu ) =\{\varphi (\cdot ,x) \in \mathcal {E}(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu ) \quad \text{ for } \text{ any }\quad x \in L^{2}(\Omega ,\mathbb {H})\}. \end{aligned}$$

Proposition 2.1

[15] Let $\mu \in \mathcal {M}$. Then $\mathcal {E}(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$ is a Banach space with the norm $\Vert \cdot \Vert _{\infty }$.

Example 2.1

Let $\rho$ be a nonnegative $\mathcal {B}$-measure function. Denote by $\mu$ the positive measure defined by

$$\begin{aligned} \mu (A)=\int _A\rho (t)dt \quad \text{ for } \quad A \in \mathcal {B}, \end{aligned}$$

(2.1)

where dt denotes the Lebesgue measure on $\mathbb {R}$. the function $\rho$ which occurs in (2.1) is called the Radon–Nikodym derivative of $\mu$ with respect to the Lebesgue measure on $\mathbb {R}$ [29, p. 130]. In this case, $\mu \in \mathcal {M}$ if and only if its Radon–Nikodym derivative $\rho$ is locally Lebesgue-integrable on $\mathbb {R}$ and it satisfies

$$\begin{aligned} \int ^{+\infty }_{-\infty }\rho (t)dt=+\infty . \end{aligned}$$

Definition 2.3

[1] Let $\mu _{1}$ and $\mu _{2} \in \mathcal {M}$. $\mu _{1}$ is said to be equivalent to $\mu _{2}$ ($\mu _{1}\sim \mu _{2}$) if there exist constants $\alpha$ and $\beta >0$ and a bounded interval I (eventually $I=\emptyset$) such that

$$\begin{aligned} \alpha \mu _{1}(A)\le \mu _{2}(A)\le \beta \mu _{1}(A) \end{aligned}$$

for $A \in \mathcal {B}$    satisfying    $A \cap I={\varnothing }.$

Theorem 2.1

[15] Let $\mu _{1}$ and $\mu _{2} \in \mathcal {M}$. If $\mu _{1}$ and $\mu _{2}$ are equivalent, then $\mathcal {E}(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu _{1})=\mathcal {E}(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu _{2}).$

For $\mu \in \mathcal {M}$ and $\tau \in \mathbb {R}$, we denote by $\mu _{\tau }$ the positive measure on $(\mathbb {R},\mathcal {B})$ defined by

$$\begin{aligned} \mu _{\tau }(A)=\mu ({a+\tau :a\in A}) \mathrm{for} A \in \mathcal {B}. \end{aligned}$$

From $\mu \in \mathcal {M}$, we formulate the following hypothesis.

(H) For all $\tau \in \mathbb {R}$, there exist $\beta >0$ and a bounded interval I such that

$$\begin{aligned} \mu _{\tau }(A)\le \beta \mu (A)\quad \mathrm{when}\,\, A \in {\mathcal {B}}\, \text {satisfies}\, A \cap I={\varnothing }. \end{aligned}$$

Lemma 2.1

[1] Let $\mu \in \mathcal {M}$. Then $\mu$ satisfies (H) if and only if $\mu$ and $\mu _{\tau }$ are equivalent for all $\tau \in \mathbb {R}$.

Lemma 2.2

[1] Hypothesis (H) implies that for all $\sigma >0$,

$$\begin{aligned} \limsup _{r\rightarrow +\infty } \frac{\mu ([-r-\sigma ,r+\sigma ])}{\mu ([-r,r])}< +\infty . \end{aligned}$$

Remark 2.1

[1] For Example 2.1, Hypothesis (H) holds if and only if for all $\tau \in \mathbb {R}$, there exist a constant $\beta > 0$ and a bounded interval I such that

$$\begin{aligned} \rho (t+\tau )\le \beta \rho (t) \quad \text{ a.e } \quad \mathbb {R}\backslash I. \end{aligned}$$

When $\mu$ is given by a density as follows

$$\begin{aligned} \mu (t)=\rho (t)dt, \end{aligned}$$

where $\rho$ satisfies the condition of Example 2.1, then Hypothesis (H) is equivalent to say

$$\begin{aligned} \text{ for } \text{ all }\quad \quad \tau \in \mathbb {R}, \quad \quad \limsup _{|t|\rightarrow +\infty }\frac{\rho (t+\tau )}{\rho (t)}<+\infty . \end{aligned}$$

Example 2.2

$$\begin{aligned} \rho (t)= \left\{ \begin{array}{ll} \exp (t)\quad if \quad t\le 0\\ 1 \quad \text{ if } \quad t>0. \end{array} \right. \end{aligned}$$

In fact

$$\begin{aligned} \lim _{t\rightarrow +\infty }\frac{\rho (t+\tau )}{\rho (t)}=1 \quad \text{ and } \quad \lim _{t\rightarrow -\infty }\frac{\rho (t+\tau )}{\rho (t)}=\exp (\tau )\quad \text{ for }\quad \tau \in \mathbb {R}. \end{aligned}$$

Let $f \in SBC(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$ and $\tau \in \mathbb {R}$. We denote by $f_{\tau }$ the function defined by $f_{\tau }(t)=f(t + \tau )$ for $t \in \mathbb {R}$.

A subset $\mathfrak {F}$ of $SBC(\mathbb {R},L^{2}(\Omega ,\mathbb {H})$ is said to translation invariant if for all $f \in \mathfrak {F}$ we have $f_{\tau } \in \mathfrak {F}$ for all $\tau \in \mathbb {R}$.

Theorem 2.2

[15] Let $\mu \in \mathcal {M}$ satisfy (H). Then $\mathcal {E}(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$ is translation invariant.

2.2 $\mu$-Pseudo almost periodic process

Definition 2.4

[6] Let $x : \mathbb {R} \rightarrow L^{2}(\Omega ,\mathbb {H})$ be a continuous stochastic process. x is said be square-mean almost periodic process if for each $\epsilon >0$ there exists $l>0$ such that for all $\alpha \in \mathbb {R}$, there exists $\tau \in [\alpha ,\alpha +l]$ satisfying

$$\begin{aligned} \sup _{t \in \mathbb {R}}\mathbb {E}\Vert x(t + \tau ) - x(t)\Vert ^{2}<\epsilon . \end{aligned}$$

We denote the space of all such stochastic processes by $SAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}))$.

Theorem 2.3

[6] $SAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}))$ equipped with the norm $\Vert \cdot \Vert _{\infty }$ is a Banach space.

Definition 2.5

[15] Let $\mu \in \mathcal {M}$ and $f : \mathbb {R} \rightarrow L^{2}(\Omega ,\mathbb {H})$ be a continuous stochastic process. f is said be $\mu$-square-mean pseudo almost periodic process if it can be decomposed as follows

$$\begin{aligned} f = g + \varphi , \end{aligned}$$

where $g \in SAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}))$ and $\varphi \in \mathcal {E}(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$.

We denote the space of all such stochastic processes by $SPAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu ).$ Then we have

$$\begin{aligned} SPAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu ) \subset SBC(\mathbb {R},L^{2}(\Omega ,\mathbb {H})). \end{aligned}$$

Theorem 2.4

[15] Let $\mu \in \mathcal {M}$ satisfy (H). Then $SPAP(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$ is translation invariant.

Theorem 2.5

[15] Let $\mu \in \mathcal {M}$ and $f \in SPAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$ be such that

$$\begin{aligned} f=g + \varphi , \end{aligned}$$

where $g \in SAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}))$ and $\varphi \in \mathcal {E}(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$. If $SPAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$ is translation invariant, then

$$\begin{aligned} \overline{\{f(t) | t \in \mathbb {R}\}}\supset \{g(t) | t \in \mathbb {R}\}. \end{aligned}$$

Theorem 2.6

[15] Let $\mu \in \mathcal {M}$. Assume that $SPAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$ is translation invariant. Then $SPAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$ is a Banach space with the norm $\Vert \cdot \Vert _{\infty }$.

Definition 2.6

[6] Let $f : \mathbb {R}\times L^{2}(\Omega ,\mathbb {H}) \rightarrow L^{2}(\Omega ,\mathbb {H}), (t,x)\mapsto f(t,x)$ be continuous.

f is said be square-mean almost periodic in $t \in \mathbb {R}$ uniformly in $x \in K$ where $K\subset L^{2}(\Omega ,\mathbb {H})$ is a compact if for any $\epsilon >0$, there exists $l(\epsilon ,K)>0$ such that for all $\alpha \in \mathbb {R}$, there exists $\tau \in [\alpha ,\alpha +l(\epsilon ,K)]$ satisfying

$$\begin{aligned} \sup _{t \in \mathbb {R}}\mathbb {E}\Vert f(t + \tau ,x) - f(t,x)\Vert ^{2}<\epsilon \end{aligned}$$

for each stochastic process $x: \mathbb {R} \rightarrow K$.

We denote the space of such stochastic processes by

$$\begin{aligned} SAP(\mathbb {R} \times L^{2}(\Omega ,\mathbb {H}), L^{2}(\Omega ,\mathbb {H})) = \{g(\cdot ,x) \in SAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H})) \quad \text{ for } \text{ any }\quad x \in L^{2}(\Omega ,\mathbb {H})\}. \end{aligned}$$

Definition 2.7

[15] Let $\mu \in \mathcal {M}$. A continuous function $f(t,x) : \mathbb {R}\times L^{2}(\Omega ,\mathbb {H}) \rightarrow L^{2}(\Omega ,\mathbb {H})$ is said to be square-mean $\mu -$pseudo almost periodic in $t \in \mathbb {R}$ for any $x \in L^{2}(\Omega ,\mathbb {H})$ if it can be decomposed as $f=g+\varphi$, where $g \in SAP(\mathbb {R} \times L^{2}(\Omega ,\mathbb {H}), L^{2}(\Omega ,\mathbb {H}))$ and $\varphi \in \mathcal {E}(\mathbb {R} \times L^{2}(\Omega ,\mathbb {H}), L^{2}(\Omega ,\mathbb {H}),\mu )$. Denote the set of all such stochastically continuous processes by $SPAP(\mathbb {R} \times L^{2}(\Omega ,\mathbb {H}), L^{2}(\Omega ,\mathbb {H}),\mu )$.

Theorem 2.7

[15] Let $\mu \in \mathcal {M}$ satisfy (H). Suppose that $f \in SPAP(\mathbb {R} \times L^{2}(\Omega ,\mathbb {H}), L^{2}(\Omega ,\mathbb {H}),\mu )$ and that there exists a positive number L such that, for any $x,y \in L^{2}(\Omega ,\mathbb {H})$,

$$\begin{aligned} \mathbb {E}\Vert f(t,x) - f(t,y)\Vert ^{2} \le L\cdot \mathbb {E}\Vert x - y\Vert ^{2} \end{aligned}$$

for $t \in \mathbb {R}$. Then $t \mapsto f(t,x(t)) \in SPAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$ for any $x \in SPAP(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$.

2.3 $\mu$-Pseudo almost automorphic process

Definition 2.8

[21] Let $x : \mathbb {R} \rightarrow L^{2}(\Omega ,\mathbb {H})$ be a continuous stochastic process. x is said be square-mean almost automorphic process if for every sequence of real numbers $(t^{'}_n)_n$ we can extract a subsequence $(t_n)_n$ such that, for some stochastic process $y : \mathbb {R} \rightarrow L^{2}(\Omega ,\mathbb {H})$, we have

$$\begin{aligned} \lim _{n\rightarrow +\infty }\mathbb {E}\Vert x(t + t_{n}) - y(t)\Vert ^{2}=0 \quad \text{ for } \text{ all }\quad t \in \mathbb {R} \end{aligned}$$

and

$$\begin{aligned} \lim _{n\rightarrow +\infty }\mathbb {E}\Vert y(t - t_{n}) - x(t)\Vert ^{2}=0 \quad \text{ for } \text{ all }\quad t \in \mathbb {R}. \end{aligned}$$

We denote the space of all such stochastic processes by $SAA(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}))$.

Theorem 2.8

[21] $SAA(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}))$ equipped with the norm $\Vert \cdot \Vert _{\infty }$ is a Banach space.

Definition 2.9

Let $\mu \in \mathcal {M}$ and $f : \mathbb {R} \rightarrow L^{2}(\Omega ,\mathbb {H})$ be a continuous stochastic process. f is said be $\mu$-square-mean pseudo almost automorphic process if it can be decomposed as follows

$$\begin{aligned} f = g + \varphi , \end{aligned}$$

where $g \in SAA(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}))$ and $\varphi \in \mathcal {E}(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$.

We denote the space of all such stochastic processes by $SPAA(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu ).$ Then we have

$$\begin{aligned} SPAA(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu ) \subset SBC(\mathbb {R},L^{2}(\Omega ,\mathbb {H})). \end{aligned}$$

Hence, together with Theorem 2.2 and Definition 2.9, we arrive at the following conclusion.

Theorem 2.9

Let $\mu \in \mathcal {M}$ satisfy (H). Then $SPAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$ is translation invariant.

Theorem 2.10

Let $\mu \in \mathcal {M}$ and $f \in SPAA(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$ be such that

$$\begin{aligned} f=g + \varphi , \end{aligned}$$

where $g \in SAA(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}))$ and $\varphi \in \mathcal {E}(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$. If $SPAA(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$ is translation invariant, then

$$\begin{aligned} \overline{\{f(t) | t \in \mathbb {R}\}}\supset \{g(t) | t \in \mathbb {R}\}. \end{aligned}$$

The proof of Theorem 2.10 is similar to the proof of Theorem 4.1 in [1].

Theorem 2.11

Let $\mu \in \mathcal {M}$. Assume that $SPAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$ is translation invariant. Then $SPAA(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$ is a Banach space with the norm $\Vert \cdot \Vert _{\infty }$.

The proof of the theorem above is similar to the proof of Theorem 4.9 in [1].

For $\mu \in \mathcal {M}$, we denote

$$\begin{aligned} SAA(\mathbb {R} \times L^{2}(\Omega ,\mathbb {H}), L^{2}(\Omega ,\mathbb {H})) = \{g(\cdot ,x) \in SAA(\mathbb {R}, L^{2}(\Omega ,\mathbb {H})) \quad \text{ for } \text{ any }\quad x \in L^{2}(\Omega ,\mathbb {H})\}. \end{aligned}$$

Definition 2.10

Let $\mu \in \mathcal {M}$. A continuous function $f(t,x) : \mathbb {R}\times L^{2}(\Omega ,\mathbb {H}) \rightarrow L^{2}(\Omega ,\mathbb {H})$ is said to be square-mean $\mu -$pseudo almost automorphic in $t \in \mathbb {R}$ for any $x \in L^{2}(\Omega ,\mathbb {H})$ if it can be decomposed as $f=g+\varphi$, where $g \in SAA(\mathbb {R} \times L^{2}(\Omega ,\mathbb {H}), L^{2}(\Omega ,\mathbb {H})$ and $\varphi \in \mathcal {E}(\mathbb {R} \times L^{2}(\Omega ,\mathbb {H}), L^{2}(\Omega ,\mathbb {H}),\mu )$.

Denote the set of all such stochastically continuous processes by $SPAA(\mathbb {R} \times L^{2}(\Omega ,\mathbb {H}), L^{2}(\Omega ,\mathbb {H}),\mu )$.

Theorem 2.12

Let $\mu \in \mathcal {M}$ satisfy (H). Suppose that $f \in SPAA(\mathbb {R} \times L^{2}(\Omega ,\mathbb {H}), L^{2}(\Omega ,\mathbb {H}),\mu )$ and that there exists a positive number L such that, for any $x,y \in L^{2}(\Omega ,\mathbb {H})$,

$$\begin{aligned} \mathbb {E}\Vert f(t,x) - f(t,y)\Vert ^{2} \le L\cdot \mathbb {E}\Vert x - y\Vert ^{2} \end{aligned}$$

for $t \in \mathbb {R}$. Then $t \mapsto f(t,x(t)) \in SPAA(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$ for any $x \in SPAA(\mathbb {R}, L^{2}(\Omega ,\mathbb {H}),\mu )$.

Proof

Let $f=g + \varphi$ with $g \in SAA(\mathbb {R}\times L^{2}(\Omega ,\mathbb {H}),L^{2}(\Omega ,\mathbb {H}))$ and $\varphi \in \mathcal {E}(\mathbb {R}\times L^{2}(\Omega ,\mathbb {H}),L^{2}(\Omega ,\mathbb {H}),\mu )$. Similarly, let $x=x_{1}+x_{2}$ with $x_{1} \in SAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$ and $x_{2}\in \mathcal {E}(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$.

The function f can be decomposed as follows

$$\begin{aligned} f(t,x(t))= & {} g(t,x_{1}(t))+[f(t,x(t))-f(t,x_{1}(t))]+[ f(t,x_{1}(t))-g(t,x_{1}(t))]\\= & {} g(t,x_{1}(t))+[f(t,x(t))-f(t,x_{1}(t))] +\varphi (t,x_{1}(t)). \end{aligned}$$

From Theorem 2.10, we deduce that g is Lipschitz. Then using the theorem of composition of almost automorphic process (Theorem 2.6 in [21]), we obtain that $g(\cdot ,x_{1}(\cdot )) \in SAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$. With the same argument used in the steps 2 and 3 of the proof of Theorem 5.7 in [15], gives as that $[ f(\cdot ,x(\cdot ))-f(\cdot ,x_{1}(\cdot ))], \varphi (\cdot ,x_{1}(\cdot )) \in \mathcal {E}(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu ).$ $\square$

3 Main results

To discuss the existence and uniqueness of the mild solution of the Eq. (1.2), we suppose that the following assumptions hold:

• ($\mathbf H _{1}$) The operator $A :D(A)\subset \mathbb {H} \rightarrow \mathbb {H}$ is the infinitesimal generator of a uniformly exponentially stable semigroup $(T(t))_{t\ge 0}$ such that there exist constants $M\ge 1$ and $\delta >0$ with

  $$\begin{aligned} \Vert T(t)\Vert \le Me^{-\delta t}\quad \text{ for } \text{ all } \quad t\ge 0. \end{aligned}$$

• ($\mathbf H _{2}$) $B_{1}$ and $B_{2}$ are convolution-type kernels in $L^{1}(0,\infty )$ and $L^{2}(0,\infty )$ respectively.

• ($\mathbf H _{3}$) Let $f,g,h: \mathbb {R}\times L^{2}(\Omega ,\mathbb {H}) \rightarrow L^{2}(\Omega ,\mathbb {H})$ be stochastic processes such that there existsonstant $L>0$ such that

  $$\begin{aligned}&\mathbb {E}\parallel f(t,x)-f(t,y) \parallel ^{2}\le L\cdot \mathbb {E}\parallel x-y \parallel ^{2},\\&\mathbb {E}\parallel g(t,x)-g(t,y) \parallel ^{2}\le L\cdot \mathbb {E}\parallel x-y \parallel ^{2},\\&\mathbb {E}\parallel h(t,x)-h(t,y) \parallel ^{2}\le L\cdot \mathbb {E}\parallel x-y \parallel ^{2} \end{aligned}$$

  for all $t \in \mathbb {R}$ and for any $x,y \in L^{2}(\Omega ,\mathbb {H})$.

Definition 3.1

[5] An $\mathcal {F}_{t}$-progressively measurable process $\{x(t)\}_{t\in \mathbb {R}}$ is called a mild solution on $\mathbb {R}$ of the Eq. (1.2) provided that it satisfies the corresponding stochastic integral equation

$$\begin{aligned} x(t)= & {} T(t-a)x(a)+ \int _{a}^{t}T(t-s)g(s,x(s))ds + \int _{a}^{t}T(t-\sigma )\int _{a}^{\sigma }B_{1}(\sigma -s)f(s,x(s))dsd\sigma \nonumber \\&+\int _{a}^{t}T(t-\sigma )\int _{a}^{\sigma }B_{2}(\sigma -s)h(s,x(s))dW(s)d\sigma \end{aligned}$$

(3.1)

for all $t\ge a$ and each $a\in \mathbb {R}.$

Remark 3.1

If we let $a\rightarrow -\infty$ in the stochastic integral equation (3.1), by the exponential dissipation condition of $(T(t))_{t\ge 0}$, then we obtain the stochastic process $x : \mathbb {R} \rightarrow L^{2}(\Omega ,\mathbb {H})$ is a mild solution of the Eq. (1.2) if and only if x satisfies the stochastic integral equation

$$\begin{aligned} x(t)= & {} \int _{-\infty }^{t}T(t-s)g(s,x(s))ds + \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)f(s,x(s))dsd\sigma \nonumber \\&+\int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)h(s,x(s))dW(s)d\sigma . \end{aligned}$$

(3.2)

Theorem 3.1

Let $\mu \in \mathcal {M}$ satisfy (H) and ($\mathbf H _{1}$)–($\mathbf H _{2}$)–($\mathbf H _{3}$) hold. If f, g and h are square-mean $\mu$-pseudo almost periodic, then the Eq. (1.2) has a unique square-mean $\mu$-pseudo almost periodic mild solution on $\mathbb {R}$, whenever

$$\begin{aligned} \Theta := 3\frac{M^{2}}{\delta ^{2}}L(1 + \Vert B_{1}\Vert ^{2}_{L^{1}(0,\infty )} + \Vert B_{2}\Vert ^{2}_{L^{2}(0,\infty )})<1. \end{aligned}$$

Proof

We define the nonlinear operator $\Lambda$ by

$$\begin{aligned} (\Lambda x)(t)= & {} \int _{-\infty }^{t}T(t-s)g(s,x(s))ds + \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)f(s,x(s))dsd\sigma \\&+\int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)h(s,x(s))dW(s)d\sigma . \end{aligned}$$

for any $x \in SPAP(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$.

We claim to prove that $\Lambda$ is a strict contraction mapping from $SPAP(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$ to itself.

Step 1 Firstly, we have to show that $\Lambda$ is well defined. Let $x \in SPAP(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$, we have

$$\begin{aligned} \mathbb {E}\Vert (\Lambda x)(t)\Vert ^{2}&=\mathbb {E}\Bigg \Vert \int _{-\infty }^{t}T(t-s)g(s,x(s))ds + \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)f(s,x(s))dsd\sigma \\&\quad +\int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)h(s,x(s))dW(s)d\sigma \Bigg \Vert ^{2}\\&\le 3\mathbb {E}\Bigg \Vert \int _{-\infty }^{t}T(t-s)g(s,x(s))ds\Bigg \Vert ^{2} + 3\mathbb {E}\Bigg \Vert \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)f(s,x(s))dsd\sigma \Bigg \Vert ^{2}\\&\quad +3\mathbb {E}\Bigg \Vert \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)h(s,x(s))dW(s)d\sigma \Bigg \Vert ^{2}. \end{aligned}$$

Since (H) and ($\mathbf H _{3}$) hold. Then by Theorem 2.7, we deduce that $g(\cdot ,x(\cdot )), f(\cdot ,x(\cdot )), h(\cdot ,x(\cdot )) \in SPAP(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$. Using Cauchy-Schwartz’s inequality and Ito’s isometry property, we obtain that

$$\begin{aligned} \mathbb {E}\Vert (\Lambda x)(t)\Vert ^{2}&\le 3M^{2}\int _{-\infty }^{t}e^{-\delta (t-s)}ds\int _{-\infty }^{t}e^{-\delta (t-s)}\mathbb {E}\Vert g(s,x(s))\Vert ^{2}ds\\&\qquad + 3M^{2}\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}d\sigma \Bigg )\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )} \mathbb {E}\Bigg \Vert \int _{-\infty }^{\sigma }B_{1}(\sigma -s)f(s,x(s))ds\Bigg \Vert ^{2}d\sigma \Bigg )\\&\qquad + 3M^{2}\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}d\sigma \Bigg )\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )} \mathbb {E}\Bigg \Vert \int _{-\infty }^{\sigma }B_{2}(\sigma -s)h(s,x(s))dW(s)\Bigg \Vert ^{2}d\sigma \Bigg )\\&\le \frac{3M^{2}}{\delta ^{2}}\sup _{s \in \mathbb {R}}\mathbb {E}\Vert g(s,x(s))\Vert ^{2}\\&\qquad + \frac{3M^{2}}{\delta }\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\Bigg (\int _{-\infty }^{\sigma }\Vert B_{1}(\sigma -s)\Vert ds\Bigg )\\&\qquad \times \Bigg ( \int _{-\infty }^{\sigma }\Vert B_{1}(\sigma -s)\Vert \mathbb {E}\Bigg \Vert f(s,x(s))\Bigg \Vert ^{2}ds\Bigg )d\sigma \Bigg )\\&\qquad + \frac{3M^{2}}{\delta }\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )} \mathbb {E}\Bigg \Vert \int _{-\infty }^{\sigma }\Vert B_{2}(\sigma -s)\Vert ^{2}\mathbb {E}\Bigg \Vert h(s,x(s))\Bigg \Vert ^{2}dsd\sigma \Bigg )\\&\le \frac{3M^{2}}{\delta ^{2}}\sup _{s \in \mathbb {R}}\mathbb {E}\Vert g(s,x(s))\Vert ^{2} + \frac{3M^{2}}{\delta ^{2}}\Vert B_{1}\Vert ^{2}_{L^{1}(0,\infty )}\sup _{s \in \mathbb {R}}\mathbb {E}\Vert f(s,x(s))\Vert ^{2}\\&\qquad + \frac{3M^{2}}{\delta ^{2}}\Vert B_{2}\Vert ^{2}_{L^{1}(0,\infty )}\sup _{s \in \mathbb {R}}\mathbb {E}\Vert h(s,x(s))\Vert ^{2}\\&<\infty . \end{aligned}$$

Step 2 Now, we verify that $\Lambda$ is a self-mapping. Let $x \in SPAP(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$. Then

$$\begin{aligned} x=x_{1}+x_{2} \end{aligned}$$

where $x_{1} \in AP(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$ and $x_{2}\in \mathcal {E}(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$. Since

• $f \in SPAP(\mathbb {R}\times L^{2}(\Omega ,\mathbb {H}),L^{2}(\Omega ,\mathbb {H}),\mu )$ then there exist $f_{1}\in AP(\mathbb {R}\times L^{2}(\Omega ,\mathbb {H}),L^{2}(\Omega ,\mathbb {H}))$ and $f_{2} \in \mathcal {E}(\mathbb {R}\times L^{2}(\Omega ,\mathbb {H}),L^{2}(\Omega ,\mathbb {H}),\mu )$ such that $f=f_{1}+f_{2}$.

• $g\in SPAP(\mathbb {R}\times L^{2}(\Omega ,\mathbb {H}),L^{2}(\Omega ,\mathbb {H}),\mu )$ then there exist $g_{1} \in AP(\mathbb {R}\times L^{2}(\Omega ,\mathbb {H}),L^{2}(\Omega ,\mathbb {H}))$ and $g_{2} \in \mathcal {E}(\mathbb {R}\times L^{2}(\Omega ,\mathbb {H}),L^{2}(\Omega ,\mathbb {H}),\mu )$ such that $g=g_{1}+g_{2}$.

• $h \in PAP(\mathbb {R}\times L^{2}(\Omega ,\mathbb {H}),L^{2}(\Omega ,\mathbb {H}),\mu )$ then there exist $h_{1} \in AP(\mathbb {R}\times L^{2}(\Omega ,\mathbb {H}),L^{2}(\Omega ,\mathbb {H})$ and $h_{2} \in \mathcal {E}(\mathbb {R}\times L^{2}(\Omega ,\mathbb {H}),L^{2}(\Omega ,\mathbb {H}),\mu )$ such that $h=h_{1}+h_{2}$.

Hence the functions f, g and h are decomposed as follows:

$$\begin{aligned} f(t,x(t))= & {} f_{1}(t,x_{1}(t))+[f(t,x(t))-f(t,x_{1}(t))]+[ f(t,x_{1}(t))-f_{1}(t,x_{1}(t))]\\= & {} f_{1}(t,x_{1}(t))+[f(t,x(t))-f(t,x_{1}(t))]+f_{2}(t,x_{1}(t)), \end{aligned}$$

$$\begin{aligned} g(t,x(t))= & {} g_{1}(t,x_{1}(t))+[g(t,x(t))-g(t,x_{1}(t))]+[ g(t,x_{1}(t))-g_{1}(t,x_{1}(t))]\\= & {} g_{1}(t,x_{1}(t))+[g(t,x(t))-g(t,x_{1}(t))]+g_{2}(t,x_{1}(t)), \end{aligned}$$

and

$$\begin{aligned} h(t,x(t))= & {} h_{1}(t,x_{1}(t))+[h(t,x(t))-h(t,x_{1}(t))]+[ h(t,x_{1}(t))-h_{1}(t,x_{1}(t))]\\= & {} h_{1}(t,x_{1}(t))+[h(t,x(t))-h(t,x_{1}(t))]+h_{2}(t,x_{1}(t)). \end{aligned}$$

It follows that

$$\begin{aligned} (\Lambda x)(t)=(\Lambda _{1}x_{1})(t) + \Psi (t), \end{aligned}$$

where

$$\begin{aligned} (\Lambda _{1}x_{1})(t)&=\int _{-\infty }^{t}T(t-s)g_{1}(s,x_{1}(s))ds + \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)f_{1}(s,x_{1}(s))dsd\sigma \\&\quad +\int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)h_{1}(s,x_{1}(s))dW(s)d\sigma \, \end{aligned}$$

and

$$\begin{aligned} \Psi (t)&=\int _{-\infty }^{t}T(t-s)[[g(s,x(s))-g(s,x_{1}(s))]+g_{2}(s,x_{1}(s))]ds\\&\quad + \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)[[f(s,x(s))-f(s,x_{1}(s))]+f_{2}(s,x_{1}(s))]dsd\sigma \\&\quad +\int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)[[h(s,x(s))-h(s,x_{1}(s))]+ h_{2}(s,x_{1}(s))]dW(s)d\sigma . \end{aligned}$$

Using Theorem 2.5 and ($\mathbf H _{3}$), we obtain that

$f_{1},g_{1},h_{1}\in AP(\mathbb {R}\times L^{2}(\Omega ,\mathbb {H}),L^{2}(\Omega ,\mathbb {H}))$ are Lipschitzian in the following sense

$$\begin{aligned} \mathbb {E}\parallel f_{1}(t,x)-f_{1}(t,y) \parallel ^{2}\le L\cdot \mathbb {E}\parallel x-y \parallel ^{2}, \end{aligned}$$

(3.3)

$$\begin{aligned} \mathbb {E}\parallel g_{1}(t,x)-g_{1}(t,y) \parallel ^{2}\le L\cdot \mathbb {E}\parallel x-y \parallel ^{2} \end{aligned}$$

(3.4)

and

$$\begin{aligned} \mathbb {E}\parallel h_{1}(t,x)-h_{1}(t,y) \parallel ^{2}\le L\cdot \mathbb {E}\parallel x-y \parallel ^{2}. \end{aligned}$$

(3.5)

Hence, using (3.3), (3.4) and (3.5), it follows from Theorem 3.2 in [5] that $\Lambda _{1}x_{1}$ is almost periodic in the square-mean sense.

Next, we have to check that

$$\begin{aligned} \lim _{r \rightarrow \infty } \frac{1}{\mu ([-r,r])}\int _{-r}^{r}\mathbb {E} \Vert \Psi (t)\Vert ^{2} d\mu (t) = 0. \end{aligned}$$

Let

$$\begin{aligned} \Psi (t)&=\int _{-\infty }^{t}T(t-s)G(s)ds + \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)F(s)dsd\sigma \\&\quad +\int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)H(s)dW(s)d\sigma , \end{aligned}$$

where

$$\begin{aligned} G(s)&= [[g(s,x(s))-g(s,x_{1}(s))]+g_{2}(s,x_{1}(s))],\\ F(s)&= [[f(s,x(s))-f(s,x_{1}(s))]+f_{2}(s,x_{1}(s))] \end{aligned}$$

and

$$\begin{aligned} H(s)= [[h(s,x(s))-h(s,x_{1}(s))]+ h_{2}(s,x_{1}(s))]. \end{aligned}$$

Using the similar arguments performs in the steps 2 and 3 of the proof of Theorem 5.7 in [15], we obtain that $G,F,H \in \mathcal {E}(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$. Then, we have

$$\begin{aligned} \mathbb {E} \Vert \Psi (t)\Vert ^{2}&\le 3 \mathbb {E} \Bigg \Vert \int _{-\infty }^{t}T(t-s)G(s)ds\Bigg \Vert ^2 + 3 \mathbb {E} \Bigg \Vert \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)F(s)dsd\sigma \Bigg \Vert ^2 \\&\quad + 3 \mathbb {E} \Bigg \Vert \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)H(s)dW(s)d\sigma \Bigg \Vert ^2. \end{aligned}$$

That implies for $r>0$, that

$$\begin{aligned}&\frac{1}{\mu ([-r,r])}\int _{-r}^{r}\mathbb {E} \Vert \Psi (t)\Vert ^{2}d\mu (t)\\&\quad \le \frac{3}{\mu ([-r,r])}\Bigg \{\int _{-r}^{r} \mathbb {E} \Bigg \Vert \int _{-\infty }^{t}T(t-s)G(s)ds\Bigg \Vert ^2d\mu (t)\\&\quad \quad + \int _{-r}^{r} \mathbb {E}\Bigg \Vert \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)F(s)dsd\sigma \Bigg \Vert ^2d\mu (t) \\&\quad \quad +\int _{-r}^{r} \mathbb {E} \Bigg \Vert \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)H(s)dW(s)d\sigma \Bigg \Vert ^2 d\mu (t)\Bigg \}\\&\quad \le \frac{3}{\mu ([-r,r])}\Bigg \{I_{1} + I_{2} + I_{3}\Bigg \}. \end{aligned}$$

Firstly, using Cauchy–Schwartz’s inequality and Fubini’s theorem, we obtain the following estimation of $I_{1}$

$$\begin{aligned} I_{1}\le&M^{2}\int _{-r}^{r}d\mu (t)\int _{-\infty }^te^{-\delta (t-s)}\,ds \int _{-\infty }^te^{-\delta (t-s)} \mathbb {E}\Vert G(s)\Vert ^2\,ds\\ \le&\frac{M^{2}}{\delta }\int _{-r}^{r} d\mu (t)\int _{0}^{\infty }e^{-\delta u}\mathbb {E}\Vert G(t-u)\Vert ^2\,du \quad \text{(setting }\quad u=t-s)\\ \le&\frac{M^{2}}{\delta }\int _{0}^{\infty } \Bigg \{e^{-\delta u} \int _{-r}^{r} \mathbb {E}\Vert G(t-u)\Vert ^2\,d\mu (t)\Bigg \}du.\\ \end{aligned}$$

It follows that

$$\begin{aligned} \frac{1}{\mu ([-r,r])} I_{1}\le \frac{M^{2}}{\delta }\int _{0}^{\infty } \Bigg \{ \frac{e^{-\delta u}}{\mu ([-r,r])}\int _{-r}^{r} \mathbb {E}\Vert G(t-u)\Vert ^2\,d\mu (t)\Bigg \}du. \end{aligned}$$

Since

$$\begin{aligned} \Bigg |\frac{e^{-\delta u}}{\mu ([-r,r])}\int _{-r}^{r} \mathbb {E}\Vert G(s)\Vert ^2\,d\mu (t)\Bigg |\le e^{-\delta u}\Vert F\Vert _{\infty }^{2}. \end{aligned}$$

Then, using the Lebesgue dominated convergence theorem and the fact that $\mathcal {E}(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$ is translation invariant, we get that

$$\begin{aligned} \lim _{r \rightarrow \infty } \frac{1}{\mu ([-r,r])} I_{1}\le \frac{M^{2}}{\delta }\int _{0}^{\infty }\Bigg \{e^{-\delta u} \lim _{r \rightarrow \infty } \frac{1}{\mu ([-r,r])}\int _{-r}^{r} \mathbb {E}\Vert G(t-u)\Vert ^2\,d\mu (t)\Bigg \}du=0. \end{aligned}$$

(3.6)

Next, we have to estimate $I_{2}$. We have

$$\begin{aligned} I_{2}&=\int _{-r}^{r} \mathbb {E}\Bigg \Vert \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)F(s)dsd\sigma \Bigg \Vert ^2d\mu (t)\\&\le M^{2}\int _{-r}^{r} \mathbb {E}\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\Bigg \Vert \int _{-\infty }^{\sigma }B_{1}(\sigma -s)F(s)ds\Bigg \Vert d\sigma \Bigg )^2d\mu (t). \end{aligned}$$

Using Cauchy–Schwartz’s inequality and Fubini’s theorem, we obtain that

$$\begin{aligned} I_{2}&\le M^{2}\int _{-r}^{r} \Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}d\sigma \Bigg )\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\mathbb {E} \Bigg \Vert \int _{-\infty }^{\sigma }B_{1}(\sigma -s)F(s)ds\Bigg \Vert ^2d\sigma \Bigg )d\mu (t)\\&\le \frac{M^{2}}{\delta }\int _{-r}^{r}\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\mathbb {E} \Bigg \Vert \int _{-\infty }^{\sigma }B_{1}(\sigma -s)F(s)ds\Bigg \Vert ^2d\sigma \Bigg )d\mu (t)\\&\le \frac{M^{2}}{\delta }\int _{-r}^{r}\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\Bigg \{\int _{-\infty }^{\sigma }\Bigg \Vert B_{1}(\sigma -s)\Bigg \Vert ds\Bigg \} \Bigg \{\int _{-\infty }^{\sigma }\Bigg \Vert B_{1}(\sigma -s)\Bigg \Vert \mathbb {E}\Bigg \Vert F(s)\Bigg \Vert ^2ds\Bigg \}d\sigma \Bigg )d\mu (t)\\&\le \frac{M^{2}}{\delta }\Vert B_{1}\Vert _{L^{1}(0,\infty )}\int _{-r}^{r}\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )} \int _{-\infty }^{\sigma }\Bigg \Vert B_{1}(\sigma -s)\Bigg \Vert \mathbb {E}\Bigg \Vert F(s)\Bigg \Vert ^2dsd\sigma \Bigg )d\mu (t)\\&\le \frac{M^{2}}{\delta }\Vert B_{1}\Vert _{L^{1}(0,\infty )}\int _{-r}^{r}\Bigg (\int _{0}^{\infty }e^{-\delta u} \int _{0}^{\infty }\Bigg \Vert B_{1}(v)\Bigg \Vert \mathbb {E}\Bigg \Vert F(t-u-v)\Bigg \Vert ^2dvdu\Bigg )d\mu (t) ~~\quad (\text{ setting }\quad u=t-\sigma ,~ v= \sigma -s)\\&\le \frac{M^{2}}{\delta }\Vert B_{1}\Vert _{L^{1}(0,\infty )}\int _{0}^{\infty }e^{-\delta u} \int _{0}^{\infty }\Bigg \Vert B_{1}(v)\Bigg \Vert \Bigg (\int _{-r}^{r}\mathbb {E}\Bigg \Vert F(t-u-v)\Bigg \Vert ^2d\mu (t)\Bigg )dvdu. \end{aligned}$$

It follows that

$$\begin{aligned} \frac{1}{\mu ([-r,r])} I_{1}&\le \frac{M^{2}}{\delta }\Vert B_{1}\Vert _{L^{1}(0,\infty )}\int _{0}^{\infty }e^{-\delta u} \int _{0}^{\infty }\Vert B_{1}(v)\Vert \\&\quad \times \Bigg (\frac{1}{\mu ([-r,r])}\int _{-r}^{r}\mathbb {E}\Vert F(t-u-v)\Vert ^2d\mu (t)\Bigg )dvdu. \end{aligned}$$

Since

$$\begin{aligned} \Bigg |e^{-\delta u} \int _{0}^{\infty }\Vert B_{1}(v)\Vert \Bigg (\frac{1}{\mu ([-r,r])}\int _{-r}^{r}\mathbb {E}\Vert F(t-u-v)\Vert ^2d\mu (t)\Bigg )dv\Bigg |\le e^{-\delta u}\Vert B_{1}\Vert _{L^{1}(0,\infty )}\Vert F\Vert _{\infty }^{2} \end{aligned}$$

and

$$\begin{aligned} \Bigg |\Vert B_{1}(v)\Vert \Bigg (\frac{1}{\mu ([-r,r])}\int _{-r}^{r}\mathbb {E}\Vert F(t-u-v)\Vert ^2d\mu (t)\Bigg )\Bigg |\le \Vert B_{1}(v)\Vert \Vert F\Vert _{\infty }^{2}. \end{aligned}$$

Then, using the Lebesgue dominated convergence theorem and the fact that $\mathcal {E}(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$ is translation invariant, we get that

$$\begin{aligned}&\lim _{r \rightarrow \infty }\frac{1}{\mu ([-r,r])} I_{1}\le \frac{M^{2}}{\delta }\Vert B_{1}\Vert _{L^{1}(0,\infty )}\int _{0}^{\infty }e^{-\delta u} \int _{0}^{\infty }\Vert B_{1}(v)\Vert \nonumber \\&\quad \times \Bigg (\lim _{r \rightarrow \infty }\frac{1}{\mu ([-r,r])}\int _{-r}^{r}\mathbb {E}\Vert F(t-u-v)\Vert ^2d\mu (t)\Bigg )dvdu=0. \end{aligned}$$

(3.7)

Next, we need to estimate $I_{3}$. In fact using Cauchy–Schwartz’s inequality and Ito’s isometry property, we obtain that

$$\begin{aligned} I_{3}&=\int _{-r}^{r} \mathbb {E} \Bigg \Vert \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)H(s)dW(s)d\sigma \Bigg \Vert ^2 d\mu (t)\Bigg \}\\&\le M^{2}\int _{-r}^{r} \Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}d\sigma \Bigg )\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\mathbb {E} \Bigg \Vert \int _{-\infty }^{\sigma }B_{2}(\sigma -s)H(s)dW(s)\Bigg \Vert ^2d\sigma \Bigg )d\mu (t)\\&\le \frac{M^{2}}{\delta }\int _{-r}^{r}\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )} \int _{-\infty }^{\sigma }\Vert B_{2}(\sigma -s)\Vert ^2\mathbb {E} \Vert H(s)\Vert ^2dsd\sigma \Bigg )d\mu (t). \end{aligned}$$

Using Fubini’s theorem, Lebesgue dominated convergence theorem and the same calculus techniques used above, one can obtain that

$$\begin{aligned} \lim _{r \rightarrow \infty } \frac{1}{\mu ([-r,r])} I_{3}=0. \end{aligned}$$

(3.8)

Consequently, combining (3.6)–(3.8), we obtain that

$$\begin{aligned} \lim _{r \rightarrow \infty } \frac{1}{\mu ([-r,r])}\int _{-r}^{r}\mathbb {E} \Vert \Psi (t)\Vert ^{2} d\mu (t) = 0. \end{aligned}$$

Hence $\Lambda$ is a self-mapping from $SPAP(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$ to itself. Next, we have to check that $\Lambda$ is a strict contraction. For $x_{1},x_{2} \in SPAP(\mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$ and each $t \in \mathbb {R}$, we have

$$\begin{aligned}&\mathbb {E}\Vert (\Lambda x_{1})(t) - (\Lambda x_{2})(t)\Vert ^{2}\\&\quad =\mathbb {E}\Bigg \Vert \int _{-\infty }^{t}T(t-s)[g(s,x_{1}(s)) -g(s,x_{2}(s))]ds\\&\quad \quad + \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)[f(s,x_{1}(s)) - f(s,x_{2}(s))]dsd\sigma \\&\quad \quad + \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)[h(s,x_{1}(s)) - h(s,x_{2}(s))]dW(s)d\sigma \Bigg \Vert ^{2}\\&\quad \le 3\mathbb {E}\Bigg \Vert \int _{-\infty }^{t}T(t-s)[g(s,x_{1}(s)) -g(s,x_{2}(s))]ds\Bigg \Vert ^{2}\\&\quad \quad +3 \mathbb {E}\Bigg \Vert \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)[f(s,x_{1}(s)) - f(s,x_{2}(s))]dsd\sigma \Bigg \Vert ^{2}\\&\quad \quad +3\mathbb {E}\Bigg \Vert \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)[h(s,x_{1}(s)) - h(s,x_{2}(s))]dW(s)d\sigma \Bigg \Vert ^{2} \end{aligned}$$

Using Cauchy–Schwartz’s inequality and Ito’s isometry property, we obtain that

$$\begin{aligned}&\mathbb {E}\Vert (\Lambda x_{1})(t) - (\Lambda x_{2})(t)\Vert ^{2}\\&\quad \le \frac{3M^{2}}{\delta } \int _{-\infty }^{t}e^{-\delta (t-s)}\mathbb {E}\Vert g(s,x_{1}(s)) -g(s,x_{2}(s))\Vert ^{2}ds\\&~~~~~~+\frac{3M^{2}}{\delta }\Vert B_{1}\Vert _{L^{1}(0,\infty )}\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\Bigg (\int _{-\infty }^{\sigma }\Vert B_{1}(\sigma -s)\Vert \mathbb {E}\Vert f(s,x_{1}(s)) - f(s,x_{2}(s))\Vert ^{2}ds\Bigg )d\sigma \\&~~~~~~+\frac{3M^{2}}{\delta }\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\Bigg (\int _{-\infty }^{\sigma }\Vert B_{2}(\sigma -s)\Vert ^{2} \mathbb {E}\Vert h(s,x_{1}(s)) - h(s,x_{2}(s))\Vert ^{2}ds\Bigg )d\sigma \\&\quad \le \frac{3M^{2}L}{\delta } \int _{-\infty }^{t}e^{-\delta (t-s)}\mathbb {E}\Vert x_{1}(s) -x_{2}(s)\Vert ^{2}ds\\&\quad \quad +\frac{3M^{2}L}{\delta }\Vert B_{1}\Vert _{L^{1}(0,\infty )}\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\Bigg (\int _{-\infty }^{\sigma }\Vert B_{1}(\sigma -s)\Vert \mathbb {E}\Vert x_{1}(s) -x_{2}(s)\Vert ^{2}ds\Bigg )d\sigma \\&\quad \quad +\frac{3M^{2}L}{\delta }\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\Bigg (\int _{-\infty }^{\sigma }\Vert B_{2}(\sigma -s)\Vert ^{2} \mathbb {E}\Vert x_{1}(s) -x_{2}(s)\Vert ^{2}ds\Bigg )d\sigma \\&\quad \le \frac{3M^{2}L}{\delta ^{2}} \sup _{s\in \mathbb {R}}\mathbb {E}\Vert x_{1}(s) -x_{2}(s)\Vert ^{2} +\frac{3M^{2}L}{\delta ^{2}}\Vert B_{1}\Vert ^{2}_{L^{1}(0,\infty )}\sup _{s\in \mathbb {R}}\mathbb {E}\Vert x_{1}(s) -x_{2}(s)\Vert ^{2}\\&~~~~~~~+\frac{3M^{2}L}{\delta ^{2}}\Vert B_{2}\Vert ^{2}_{L^{2}(0,\infty )}\sup _{s\in \mathbb {R}}\mathbb {E}\Vert x_{1}(s) -x_{2}(s)\Vert ^{2}\\&\quad \le \Theta \sup _{s\in \mathbb {R}}\mathbb {E}\Vert x_{1}(s) -x_{2}(s)\Vert ^{2}. \end{aligned}$$

Thus, it follows that for each $t \in \mathbb {R}$

$$\begin{aligned} \Vert (\Lambda x_{1})(t) - (\Lambda x_{2})(t)\Vert _{L^{2}}^{2}\le \Theta \sup _{s\in \mathbb {R}}\Vert x_{1}(s) - x_{2}(s)\Vert _{L^{2}}^{2}\le \Theta (\sup _{s\in \mathbb {R}}\Vert x_{1}(s) - x_{2}(s)\Vert _{L^{2}})^{2}. \end{aligned}$$

Hence

$$\begin{aligned} \Vert (\Lambda x_{1})(t) - (\Lambda x_{2})(t)\Vert _{\infty }=\sup _{t\in \mathbb {R}}\Vert (\Lambda x_{1})(t) - (\Lambda x_{2})(t)\Vert _{L^{2}}\le \sqrt{\Theta }\Vert x_{1} - x_{2}\Vert _{\infty }. \end{aligned}$$

Since $\Theta <1$, it follows $\Lambda$ is a strict contraction mapping on $SPAP( \mathbb {R},L^{2}(\Omega ,\mathbb {H}),\mu )$. We deduce that $\Lambda$ has a unique fixed point, which gives a unique $\mu -$pseudo almost periodic mild solution on $\mathbb {R}$ of Eq. (1.2). The proof is complete. $\square$

Theorem 3.2

Let $\mu \in \mathcal {M}$ satisfy (H) and ($\mathbf H _{1}$)–($\mathbf H _{2}$)–($\mathbf H _{3}$) hold. If f, g and h are square-mean $\mu -$pseudo almost automorphic, then the Eq. (1.2) has a unique square-mean $\mu -$pseudo almost automorphic mild solution on $\mathbb {R}$, whenever

$$\begin{aligned} 3\frac{M^{2}}{\delta ^{2}}L(1 + \Vert B_{1}\Vert ^{2}_{L^{1}(0,\infty )} + \Vert B_{2}\Vert ^{2}_{L^{2}(0,\infty )})<1. \end{aligned}$$

Proof

We reproduce the proof of Theorem 3.1 in what we take $x_{1}, f_{1}, g_{1}, h_{1} \in SAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$. Then, we only need to verify that $\Lambda _{1}x_{1}\in SAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$. We have

$$\begin{aligned} (\Lambda _{1}x_{1})(t)&=\int _{-\infty }^{t}T(t-s)g_{1}(s,x_{1}(s))ds + \int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)f_{1}(s,x_{1}(s))dsd\sigma \\&\quad +\int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)h_{1}(s,x_{1}(s))dW(s)d\sigma \,\\&:= (\Gamma _{1}x_{1})(t) + (\Gamma _{2}x_{1})(t) + (\Gamma _{3}x_{1})(t). \end{aligned}$$

Similarly as the proof of Theorem 4.2 in [21], $\Gamma _{1}x_{1} \in SAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$. Next, we have to check that $\Gamma _{2}x_{1}, \Gamma _{3}x_{1} \in SAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$. Since $f_{1}(\cdot ,x_{1}(\cdot )) \in SAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$, then for every sequence of real numbers $(t^{^{\prime }}_n)_n$ we can extract a subsequence $(t_n)_n$ such that, for some stochastic process $\tilde{f_{1}} \in L^{2}(\Omega ,\mathbb {H})$, we have

$$\begin{aligned} \lim _{n\rightarrow +\infty }\mathbb {E}\Vert f_{1}(s- t_{n},x_{1}(s- t_{n})) - \tilde{f_{1}}(s)\Vert ^{2}=0 \quad \text{ for } \text{ all }\quad s \in \mathbb {R} \end{aligned}$$

and

$$\begin{aligned} \lim _{n\rightarrow +\infty }\mathbb {E}\Vert \tilde{f_{1}}(s - t_{n}) - f_{1}(s,x_{1}(s))\Vert ^{2}=0 \quad \text{ for } \text{ all }\quad s \in \mathbb {R}. \end{aligned}$$

For $t\in \mathbb {R}$, we define $\widetilde{\Gamma _{2}}x_{1}(t) =\int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{1}(\sigma -s)\tilde{f_{1}}(s)dsd\sigma$. Using Cauchy–Schwartz’s inequality, we have

$$\begin{aligned}&\mathbb {E}\Vert \Gamma _{2}x_{1}(t + t_{n}) - \widetilde{\Gamma _{2}}x_{1}(t)\Vert ^{2}\\&\quad =\mathbb {E}\Bigg \Vert \int _{-\infty }^{t}T(t-\sigma ) \int _{-\infty }^{\sigma }B_{1}(\sigma -s)(f_{1}(s- t_{n},x_{1}(s- t_{n})) - \tilde{f_{1}}(s))dsd\sigma \Bigg \Vert ^{2}\\&\quad \le \frac{M^{2}}{\delta }\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\mathbb {E} \Bigg \Vert \int _{-\infty }^{\sigma }B_{1}(\sigma -s)(f_{1}(s- t_{n},x_{1}(s- t_{n})) - \tilde{f_{1}}(s))ds\Bigg \Vert ^2d\sigma \Bigg )\\&\quad \le \frac{M^{2}}{\delta }\Vert B_{1}\Vert _{L^{1}(0,\infty )}\int _{-\infty }^{t}e^{-\delta (t-\sigma )} \int _{-\infty }^{\sigma }\Vert B_{1}(\sigma -s)\Vert \mathbb {E}\Vert f_{1}(s- t_{n},x_{1}(s- t_{n})) - \tilde{f_{1}}(s)\Vert ^2dsd\sigma . \end{aligned}$$

Using the Lebesgue dominated convergence theorem, one has

$$\begin{aligned} \lim _{n\rightarrow +\infty }\mathbb {E}\Vert \Gamma _{2}x_{1}(t + t_{n}) - \widetilde{\Gamma _{2}}x_{1}(t)\Vert ^{2}=0. \end{aligned}$$

Similarly, $\lim \nolimits _{n\rightarrow +\infty }\mathbb {E}\Vert \widetilde{\Gamma _{2}}x_{1}(t-t_{n}) - \Gamma _{2}x_{1}(t)\Vert ^{2}=0.$ Hence $\Gamma _{2}x_{1} \in SAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$.

To end the proof, we need check that $\Gamma _{3}x_{1} \in SAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$. Since $h_{1}(\cdot ,x_{1}(\cdot )) \in SAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$, then for every sequence of real numbers $(t^{^{\prime }}_n)_n$ we can extract a subsequence $(t_n)_n$ such that, for some stochastic process $\tilde{h_{1}} \in L^{2}(\Omega ,\mathbb {H})$, we have

$$\begin{aligned} \lim _{n\rightarrow +\infty }\mathbb {E}\Vert h_{1}(s- t_{n},x_{1}(s- t_{n})) - \tilde{h_{1}}(s)\Vert ^{2}=0 \quad \text{ for } \text{ all }\quad s \in \mathbb {R} \end{aligned}$$

and

$$\begin{aligned} \lim _{n\rightarrow +\infty }\mathbb {E}\Vert \tilde{h_{1}}(s - t_{n}) - h_{1}(s,x_{1}(s))\Vert ^{2}=0 \quad \text{ for } \text{ all }\quad s \in \mathbb {R}. \end{aligned}$$

For $t\in \mathbb {R}$, we define $\widetilde{\Gamma _{3}}x_{1}(t) =\int _{-\infty }^{t}T(t-\sigma )\int _{-\infty }^{\sigma }B_{2}(\sigma -s)\tilde{h_{1}}(s)dW(s)d\sigma$. Let $\tilde{W}(s) = W(s+ t_{n}) - W(t_{n})$ for each $s \in \mathbb {R}$. Then $\tilde{W}$ is also a Wiener process having the same distribution as W. Using Cauchy–Schwartz’s inequality and the Ito’s isometry, we obtain the following estimation

$$\begin{aligned}&\mathbb {E}\Vert \Gamma _{3}x_{1}(t + t_{n}) - \widetilde{\Gamma _{3}}x_{1}(t)\Vert ^{2}\\&\quad =\mathbb {E}\Bigg \Vert \int _{-\infty }^{t}T(t-\sigma ) \int _{-\infty }^{\sigma }B_{2}(\sigma -s)(h_{1}(s- t_{n},x_{1}(s- t_{n})) - \tilde{h_{1}}(s))dW(s)d\sigma \Bigg \Vert ^{2}\\&\quad \le \frac{M^{2}}{\delta }\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\mathbb {E} \Bigg \Vert \int _{-\infty }^{\sigma }B_{2}(\sigma -s)(h_{1}(s- t_{n},x_{1}(s- t_{n})) - \tilde{h_{1}}(s))dW(s)\Bigg \Vert ^2d\sigma \Bigg )\\&\quad \le \frac{M^{2}}{\delta }\Bigg (\int _{-\infty }^{t}e^{-\delta (t-\sigma )}\int _{-\infty }^{\sigma }\Vert B_{2}(\sigma -s)\Vert ^2\mathbb {E} \Vert h_{1}(s- t_{n},x_{1}(s- t_{n})) - \tilde{h_{1}}(s)\Vert ^2dsd\sigma \Bigg ) \end{aligned}$$

By the Lebesgue dominated convergence theorem, one can obtain that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\mathbb {E}\Vert \Gamma _{3}x_{1}(t + t_{n}) - \widetilde{\Gamma _{3}}x_{1}(t)\Vert ^{2}=0. \end{aligned}$$

Similarly, $\lim \nolimits _{n\rightarrow +\infty }\mathbb {E}\Vert \widetilde{\Gamma _{3}}x_{1}(t-t_{n}) - \Gamma _{3}x_{1}(t)\Vert ^{2}=0.$ Hence $\Gamma _{3}x_{1} \in SAA(\mathbb {R},L^{2}(\Omega ,\mathbb {H}))$. This completes the proof. $\square$

4 Example

Consider the following stochastic integro-differential equations:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial u(t,x)}{\partial t}= \frac{\partial ^{2}u(t,x)}{\partial x^{2}} + g(t,u(t,x))+ \int _{-\infty }^{t}e^{-\omega (t-s)}f(s,u(s,x))ds\\ \quad + \int _{-\infty }^{t}e^{-\omega (t-s)}h(s,u(s,x))dW(s),\\ (t,x) \in \mathbb {R}\times (0,1),\\ u(t,0)=u(t,1)=0, ~~~~\quad t \in \mathbb {R},\\ \end{array} \right. \end{aligned}$$

(4.1)

where W(t) is a two-sided and standard one-dimensional Brownian motion defined on the filtered probability space $(\Omega ,\mathcal {F},P,\mathcal {F}_{t})$ with $\mathcal {F}_{t}=\sigma \{W(w) - W(v) | w,v\le t \}$ and $\omega >0$.

To apply our theoretical results, we consider the measure $\mu$ where its Radon–Nikodym derivative is

$$\begin{aligned}\rho (s)= \left\{ \begin{array}{ll} \exp (s)\quad \text{ if } \quad s\le 0\\ 1 \quad \text{ if } \quad s>0. \end{array} \right. \end{aligned}$$

Then $\mu$ satisfies (H) (cf. Example 2.2). The forcing terms as follows:

$$\begin{aligned} g(s,u)= \frac{L}{2}u\sin \frac{1}{2+\cos s + \cos \sqrt{2}s} + e^{-|s|} \cos u, \end{aligned}$$

$$\begin{aligned} f(s,u)= \frac{L}{2}u\cos \frac{1}{2+\sin s + \sin \pi s}+ e^{-2|s|} \sin u, \end{aligned}$$

$$\begin{aligned} h(s,u)= \frac{L}{2}u\sin \frac{1}{2+\cos s + \cos \sqrt{3}s}+ e^{-3|s|}\sin u. \end{aligned}$$

In order to write the system (4.1) on the abstract form (1.2), we consider the linear operator $A: D(A)\subset L^{2}(0,1)\rightarrow L^{2}(0,1)$, given by

$$\begin{aligned} D(A)&=H^{2}(0,1)\cap H^{1}_{0}(0,1),\\ Ax(\xi )&=x''(\xi ) \quad \text{ for }\quad \xi \in (0,1)\quad \text{ and }\quad x \in D(A). \end{aligned}$$

It is well-known that A generates a $C_0$-semigroup $(T(t))_{t\ge 0}$ on $L^{2}(0,1)$ defined by

$$\begin{aligned} (T(t)x)(r)=\sum _{n=1}^{\infty }e^{-n^{2}\pi ^{2}t} \langle x ,e_{n}\rangle _{L^{2}}e_{n}(r), \end{aligned}$$

where $e_{n}(r)=\sqrt{2}\sin (n \pi r)$ for $n=1,2,....,$ and $\Vert T(t)\Vert \le e^{-\pi ^{2}t}$ for all $t\ge 0$. Hence ($\mathbf H _{1}$) hold.

Then the system (4.1) takes the following abstract form

$$\begin{aligned} u'(t)= & {} Au(t) + g(t,u(t)) + \int _{-\infty }^{t}e^{-\omega (t-s)}f(s,u(s))ds\nonumber \\&+ \int _{-\infty }^{t}e^{-\omega (t-s)}h(s,u(s))dW(s) \quad \text{ for } \text{ all }\quad t \in \mathbb {R}. \end{aligned}$$

(4.2)

$u \sin \frac{1}{2+\cos s + \cos \sqrt{2}s}+ e^{-|s|}\cos u$ belongs to $SPAA(\mathbb {R}\times L^{2}(\Omega ,L^{2}[0,1]),L^{2}(\Omega ,L^{2}[0,1]),\mu )$ where $u\sin \frac{1}{2+\cos s + \cos \sqrt{2}s}$ is the almost automorphic component and $e^{-|s|}\cos u$ is the $\mu$-ergodic perturbation, since

$$\begin{aligned} \frac{1}{\mu ([-r,r])}\int _{-r}^{0}\mathbb {E}\Vert e^{-|s|}\cos u\Vert ^{2}d\mu (s)&\le \frac{1}{1-e^{-r}+r}\int _{-r}^{0}e^{-2|s|}e^{s}ds =\frac{1}{1-e^{-r}+r}\int _{-r}^{0}e^{3s}ds\\&=\frac{1}{3}\frac{1-e^{-3r}}{1-e^{-r}+r}\rightarrow 0 \quad \text{ as } \quad r\rightarrow +\infty \end{aligned}$$

and

$$\begin{aligned} \frac{1}{\mu ([-r,r])}\int _{0}^{r}\mathbb {E}\Vert e^{-|s|}\cos u\Vert ^{2}d\mu (s)&\le \frac{1}{1-e^{-r}+r}\int _{0}^{r}e^{-2|s|}dt= \frac{1}{1-e^{-r}+r}\int _{0}^{r}e^{-2s}ds\\&=\frac{1}{2}\frac{1-e^{-2r}}{1-e^{-r}+r}\rightarrow 0 \quad \text{ as } \quad r\rightarrow +\infty . \end{aligned}$$

By analogous argument performed above, we have also that $f,h \in SPAA(\mathbb {R}\times L^{2}(\Omega ,L^{2}[0,1]),L^{2}(\Omega ,L^{2}[0,1]),\mu )$.

Clearly, g, f and h satisfy the Lipschitz conditions ($\mathbf H _{3}$). Moreover, it is easy to see that ($\mathbf H _{2}$) hold. Therefore, by Theorem 3.2, the Eq. (4.2) has a unique square-mean $\mu$-pseudo almost automorphic mild solution on $\mathbb {R}$ whenever L is small enough.