Stepanov-like Pseudo S-Asymptotically (ω, c)-Periodic Solutions of a Class of Stochastic Integro-Differential Equations

Abstract

The study of long-term behavior in stochastic systems is critical for understanding the dynamics of complex processes influenced by randomness. This paper addresses the existence and uniqueness of Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic solutions for a class of stochastic integro-differential equations. These equations model systems where the interplay between deterministic and stochastic components dictates the overall dynamics, making periodic analysis essential. The problem addressed in this study is the lack of a comprehensive framework to describe the periodic behavior of such systems in noisy environments. To tackle this, we employ advanced techniques in stochastic analysis, fixed-point theorems and the properties of $L^1$- and $L^2$-convolution kernels to establish conditions for the existence and uniqueness of mild solutions under these extended periodicity settings. The methodology involves leveraging the decay properties of the operator kernels and the boundedness of stochastic integrals to ensure well-posedness. The major outputs of this study include novel results on the existence, uniqueness and stability of Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic solutions, along with illustrative example demonstrating their applicability in real-world stochastic systems.

1. Introduction and Preliminaries

The study of periodicity and asymptotic behavior in solutions of differential equations, especially within stochastic environments, has become a crucial subject of investigation due to its wide range of applications in various scientific and engineering fields [1,2,3,4,5,6]. Particularly, pseudo S-asymptotic periodicity and $(\omega ,c)$-periodic solutions have attracted significant attention in the recent literature, providing a general framework to understand complex dynamic systems with both periodic and stochastic components (cf. [7,8,9,10]). This paper focuses on Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic solutions of stochastic integro-differential equations, a class that extends classical results and explores new behaviors in stochastic environments.

The class of $(\omega ,c)$-periodic functions, an extension of Bloch periodic functions, has been introduced and studied by Alvarez, Gómez and Pinto [11], as well as Alvarez, Castillo and Pinto [12] and Khalladi, Kostić, Rahmani and Velinov [13]. These functions play a significant role in the qualitative analysis of solutions to the Mathieu linear differential equation $y^{\prime \prime }\left(t\right)+[a-2qcos2t]y\left(t\right)=0$, which models seasonally forced population dynamics. Linear delayed equations are also known to admit $(\omega ,c)$-periodic solutions (see, e.g., [11], Example 2.5). Additionally, the work in [12] establishes the existence of positive $(\omega ,c)$-pseudo periodic solutions for the Lasota–Wazewska equation with $(\omega ,c)$-pseudo periodic coefficients $y^{\prime }\left(t\right)=-\delta y\left(t\right)+h\left(t\right)e^{-a\left(t\right)y(t-\tau )},t\ge 0$. This equation describes the survival of red blood cells in an animal’s bloodstream (see, e.g., Wazewska-Czyzewska and Lasota [14]).

Previous research has extensively explored S-asymptotically periodic solutions in both deterministic and stochastic systems. Bezandry [1] and Bezandry and Diagana [2] analyzed almost periodic and $S^2$-almost periodic solutions for stochastic evolution equations driven by fractional Brownian motion. Their work laid the groundwork for studying periodic behaviors in the stochastic context and subsequent research has built on these results to investigate more general forms of periodicity.

The S-asymptotically $\omega$-periodic framework has been explored by other researchers in various contexts. Dimbour and Manou-Abi [15] studied S-asymptotically $\omega$-periodic solutions with piecewise constant arguments, Nicola and Pierri [16] studied S-asymptotically $\omega$-periodic functions, while Cuevas and de Souza [17,18] provided significant insights into S-asymptotically $\omega$-periodic solutions for semilinear fractional equations. De Andrade and Cuevas [19] extended these results to semilinear Cauchy problems with non-dense domains. The existence of S-asymptotically $\omega$-periodic solutions of fractional differential equations was considered in [20,21]. Henríquez, Pierri and Taboas explored the existence of S-asymptotically $\omega$-periodic solutions for abstract neutral differential equations, addressing the unique challenges posed by the presence of delay terms in the equations [22]. In a related study, the same authors investigated the properties of S-asymptotically $\omega$-periodic functions in Banach spaces, providing foundational results that can be applied to a wide range of differential equations [23]. These studies demonstrate the broad applicability of S-asymptotically periodic solutions in various types of integro-differential equations.

In recent studies, Brindle and N’ Guérékata have explored S-asymptotically periodic solutions to fractional differential equations [24,25], with extension to difference equations in [26]. Their work highlighted the importance of understanding asymptotic periodicity in the analysis of stochastic and non-autonomous systems. Also in [3], the S-asymptotically $\omega$-periodic solutions to stochastic fractional differential equations were studied. Meanwhile, Chang and Wei [7,27] introduced the notion of pseudo S-asymptotically Bloch-type periodicity, further extending the theory to cover more generalized periodic functions. This paper adopts their approach in the stochastic integro-differential equation setting.

Recent advancements in stochastic systems, particularly those driven by fractional Brownian motions, have expanded the scope of periodicity analysis. Chang, Diop and their collaborators [4,5,8] explored Stepanov-like weighted pseudo S-asymptotically Bloch-type periodicity and its applications to stochastic evolution equations. Their results emphasize the importance of such periodic behaviors in practical applications and our work builds on their findings to explore more general stochastic systems.

For instance, Bezandry established the existence of almost periodic solutions for functional integro-differential stochastic equations by utilizing fixed-point theorems in Banach spaces, highlighting the impact of stochastic perturbations on long-term behavior [6]. Mbaye extended this line of research by investigating square-mean $\mu$-pseudo almost periodic and automorphic solutions, providing a comprehensive approach to handle stochastic evolution equations with more general forms of periodicity [9]. Dos Santos and Henríquez studied S-asymptotically $\omega$-periodic solutions for abstract integro-differential equations, highlighting the asymptotic behaviors that can arise in more general settings [28]. Wang and Liu addressed the uniqueness and existence of positive solutions for fractional integro-differential equations, highlighting their significance in applied mathematics [29]. Similarly, Lassoued et al. provided a foundational understanding of almost periodic and asymptotically almost periodic functions, which are essential for extending classical periodicity to generalized settings [30]. Furthermore, Wang et al. explored the quasilinearization method for impulsive integro-differential equations, establishing a framework for solving first-order problems [31]. Additional advancements by Wang et al. investigated the convergence of solutions for functional integro-differential equations with nonlinear boundary conditions, emphasizing the interplay between functional and boundary conditions in achieving stable solutions [32]. Additionally, Xia contributed by examining pseudo almost automorphic solutions in distribution using measure theory, further expanding the types of solution behaviors that can be addressed within the framework of stochastic integro-differential equations [33].

Our work also relates to the theoretical developments in almost periodicity and almost automorphic-type solutions, as described in the comprehensive monographs by Kostić [34,35,36], where the author presents detailed results on almost periodic and asymptotically periodic solutions to integro-differential equations. These monographs provide a critical background to the periodicity framework used in this paper, especially when dealing with the interplay between stochasticity and periodicity.

The motivation for this study arises from the need to develop a comprehensive theoretical framework to describe the long-term periodic behavior of stochastic systems with both deterministic and stochastic components. Various applications, such as those in physics, biology and economics, exhibit periodic characteristics in noisy environments, making the analysis of stochastic periodicity crucial. In this paper, we aim to extend the classical results of periodicity and asymptotic periodicity to a more generalized setting, focusing on Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic solutions for stochastic integro-differential equations. This work draws on recent contributions by several researchers, including Chang and Zhao [10], Chang and Wei [37], Li, Liu and Wei [38], and Pierri and O’Regan [39], among others.

In summary, this paper aims to establish the existence and uniqueness of Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic solutions for a class of stochastic integro-differential equations. This extends the results on stochastic periodicity to a broader class of equations, offering new insights into the behavior of complex stochastic systems with periodic components. The results obtained in this work have significant implications for both the theory and practical applications of stochastic differential equations.

The structure of this paper can be succinctly outlined as follows. Section 2 provides the essential background, including the definitions and fundamental properties related to Stepanov-like pseudo S-asymptotic $(\omega ,c)$-periodic processes. In Section 3, the primary results regarding the existence and uniqueness of Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic mild solutions for the stochastic integro-differential problem under consideration are presented. Finally, Section 4 offers an example that demonstrates the practical application of the theoretical findings.

1.1. Problem Formulation

The investigated problem in this paper is the following stochastic integro-differential equation

$${\mathsf{\Psi }}^{\prime }\left(\tau \right)=A\mathsf{\Psi }\left(\tau \right)+g(\tau ,\mathsf{\Psi }\left(\tau \right))+\underset{-\infty }{\overset{\tau }{\int }}{A}_1(\tau -s)f(s,\mathsf{\Psi }\left(s\right))ds+\underset{-\infty }{\overset{\tau }{\int }}{A}_2(\tau -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right),$$

(1)

where $A:D\left(A\right)\subset H$ is an infinitesimal generator of a $C_0$-semigroup ${\left(T\left(\tau \right)\right)}_{\tau \ge 0}$, $A_1,A_2$ are convolution-type kernels in $L^1(0,\infty )$ and $L^2(0,\infty )$, respectively, $g,f:\mathbb{R}\times L^2(\mathsf{\Omega },H)\to L^2(\mathsf{\Omega },H)$, $h:\mathbb{R}\times L^2(\mathsf{\Omega },H)\to L^2(\mathbb{R},L^2(\mathsf{\Omega },H))$ and W is a Brownian motion, with covariance operator Q, such that $trQ<\infty$.

The kernel $A_1(\tau -s)$ ensures that the deterministic part of the system evolves smoothly over time. Its membership in $L^1(0,\infty )$ guarantees integrability, making the convolution with $f(s,\mathsf{\Psi }(s\left)\right)$ well-defined for square-integrable stochastic processes. The kernel $A_2(\tau -s)$, used in the stochastic integral, is square-integrable. This ensures that the stochastic term ${\int }_{\infty }^{\tau }{A}_2(\tau -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)$ remains well-posed within Itô’s framework. The square-integrability aligns with the energy constraints of stochastic processes, maintaining the mean-square boundedness of the solution. The stochastic integrals involving $dW\left(s\right)$ adhere to Itô’s isometry, where the variance (or energy) of the integral is proportional to the $L^2$-norm of the integrand. The convolution properties ensure that the stochastic nature of W propagates through the system without violating integrability constraints.

1.2. Preliminaries

We denote by $\mathbb{R}$ and $\mathbb{C}$ the sets of real and complex numbers, respectively. Here, we assume that $(\mathsf{\Omega },\mathcal{F},p)$ represents a probability space, where H is a complex separable Hilbert space. We will use the notation $\Vert ·\Vert$ to represent the norm on H. Let $L(K,H)$ denote the Banach space of all bounded linear operators from K to H equipped with the topology induced by the operator norm. Additionally, $L^2(\mathsf{\Omega },H)$ refers to the set of all strongly measurable, square-integrable H-valued random variables, which forms a complex Hilbert space with the norm ${\Vert \mathsf{\Psi }\Vert }_{L^2}={(E\Vert \mathsf{\Psi }\Vert }^2{)}^{{\displaystyle \frac{1}{2}}}$, for $\mathsf{\Psi }\in L^2(\mathsf{\Omega },H)$, where the expectation $E(·)$ is defined by ${E\Vert \mathsf{\Psi }\Vert }^2={\int }_{\mathsf{\Omega }}{\Vert \mathsf{\Psi }\Vert }^{2}dp$.

The stochastic process $\mathsf{\Psi }:\mathbb{R}\to L^2(\mathsf{\Omega },H)$ is said to be stochastically bounded if

$$\begin{array}{c}\hfill {E\Vert \mathsf{\Psi }\left(\tau \right)\Vert }^2=\underset{\mathsf{\Omega }}{\int }{\Vert \mathsf{\Psi }\left(\tau \right)\Vert }^{2}dp<M,\end{array}$$

for some positive constant M and all $\tau \in \mathbb{R}$.

The stochastic process $\mathsf{\Psi }:\mathbb{R}\to L^2(\mathsf{\Omega },H)$ is said to be stochastically continuous if

$$\begin{array}{c}\hfill \underset{\tau \to s}{lim}E\Vert \mathsf{\Psi }\left(\tau \right)-\mathsf{\Psi }\left(s\right)\Vert =0.\end{array}$$

The space of all bounded and continuous stochastic processes $\mathsf{\Psi }:\mathbb{R}\to L^2(\mathsf{\Omega },H)$ will be denoted by $\mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H))$. The space $\mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H))$ equipped with the norm $\Vert \mathsf{\Psi }\Vert ={\left(\underset{\tau \in \mathbb{R}}{sup}E{\Vert \mathsf{\Psi }\left(\tau \right)\Vert }^2\right)}^{{\displaystyle \frac{1}{2}}}$ becomes a Banach space.

The space of all Stepanov bounded stochastic processes $\mathsf{\Psi }:\mathbb{R}\to L^2(\mathsf{\Omega },H)$, such that ${\mathsf{\Psi }}^{*}(t,s)=\mathsf{\Psi }(t+s)\in L^{\infty }(\mathbb{R},L^2([0,1];L^2(\mathsf{\Omega },H))$, will be denoted by $BSS(\mathbb{R},L^2(\mathsf{\Omega },H))$. This space, endowed with the norm

$$\begin{array}{c}\hfill {\Vert \mathsf{\Psi }\Vert }_{BSS}=\underset{\tau \in \mathbb{R}}{sup}{\left(\underset{0}{\overset{1}{\int }}E{\Vert \mathsf{\Psi }(\tau +s)\Vert }^{2}ds\right)}^{{\displaystyle \frac{1}{2}}}=\underset{\tau \in \mathbb{R}}{sup}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert \mathsf{\Psi }\left(r\right)\Vert }^{2}dr\right)}^{{\displaystyle \frac{1}{2}}},\end{array}$$

becomes a Banach space.

Let $c\in \mathbb{C}\setminus \left\{0\right\}$, $\omega >0$. For a continuous function $u:\mathbb{R}\to X$, where X is a Banach space, it is said to be $(\omega ,c)$-periodic if $u(s+\omega )=u\left(s\right)$, for all $s\in \mathbb{R}$. The number $\omega$ is called the c-period of u. Using the principal branch of the complex logarithm, $c^{{\displaystyle \frac{s}{\omega }}}:=exp\left({\displaystyle \frac{s}{\omega }}\mathrm{Log}\left(c\right)\right)=c^{^}\left(s\right)$.

2. Stepanov-like Pseudo $\mathit{S}$-Asymptotically $(\mathit{\omega },\mathit{c})$-Periodic Processes

We start this section with the definition of Stepanov pseudo S-asymptotically $(\omega ,c)$-periodic processes and basic facts about the space of Stepanov pseudo S-asymptotically $(\omega ,c)$-periodic processes.

Definition 1.

For a stochastic process $\mathsf{\Psi }\in \mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H))$, it is said to be mean-square pseudo-S-asymptotically $(\omega ,c)$-periodic if

$$\begin{array}{c}\hfill \underset{\mu \to \infty }{lim}{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{2}ds=0,s\in \mathbb{R}.\end{array}$$

The set of all square-mean pseudo $\mathcal{S}$-asymptotically $(\omega ,c)$-periodic stochastic processes will be denoted by ${\mathcal{P}SAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$.

Like in [4], we can give the following theorem:

Theorem 1.

Let $\mathsf{\Psi },{\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2\in {\mathcal{P}SAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$. Then

(i) 

${\mathsf{\Psi }}_1+{\mathsf{\Psi }}_2\in {\mathcal{P}SAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$;

(ii) 

$a\mathsf{\Psi }\in {\mathcal{P}SAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H)),$ for every $a\in \mathbb{C}$;

(iii) 

${\mathsf{\Psi }}_a\left(\tau \right)=\mathsf{\Psi }(\tau +a)\in {\mathcal{P}SAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$, for each $a,\tau \in \mathbb{R}$;

(iv) 

${\mathcal{P}SAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$ endowed with the norm ${\Vert \mathsf{\Psi }\Vert }_{\infty }={\left(\underset{\tau \in \mathbb{R}}{sup}E{\Vert \mathsf{\Psi }\left(\tau \right)\Vert }^2\right)}^{{\displaystyle \frac{1}{2}}}$,

$\mathsf{\Psi }\in {\mathcal{P}SAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$ is a Banach space.

Definition 2.

For a stochastic process $\mathsf{\Psi }\in \mathcal{B}SS(\mathbb{R},L^2(\mathsf{\Omega },H))$, it is said to be mean-square Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic if

$$\begin{array}{c}\hfill \underset{\mu \to \infty }{lim}{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau =0,\tau \in \mathbb{R}.\end{array}$$

The set of all square-mean Stepanov-like pseudo $\mathcal{S}$-asymptotically $(\omega ,c)$-periodic stochastic processes will be denoted by ${\mathcal{S}PSAP}_{\omega ,c}$.

Theorem 2.

The following hold:

(i) 

${\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$ is a Banach space with the norm

${\Vert \mathsf{\Psi }\Vert }_{BSS}=\underset{\tau \in \mathbb{R}}{sup}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert \mathsf{\Psi }\left(s\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}$, for $\mathsf{\Psi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$;

(ii) 

${\mathcal{P}SAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))\subset {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$.

Proof. 

(i) Let ${\left(\mathsf{\Psi }\right) }_n$ be a sequence in ${\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$ and ${\mathsf{\Psi }}_n\to \mathsf{\Psi }$, when $n\to \infty$. Hence, for every $\epsilon >0$, there exist constants $n_0$ and $M>0$ such that for $n\ge n_0$, $\Vert {\mathsf{\Psi }}_n{-\mathsf{\Psi }\Vert }_{BSS}\le {\displaystyle \frac{\epsilon }{3\left|c\right|}}$ and for $\mu >M$,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E\Vert c^{^}(-s)\left({\mathsf{\Psi }}_n(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \le {\displaystyle \frac{\epsilon }{3}}.\end{array}$$

Now,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & ={\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}(\underset{\tau }{\overset{\tau +1}{\int }}{E\Vert c}^{^}(-s)(\mathsf{\Psi }(s+\omega )-{\mathsf{\Psi }}_n(s+\omega )\hfill \\ \hfill & +{\mathsf{\Psi }}_n(s+\omega )-c{\mathsf{\Psi }}_n\left(s\right)+c{\mathsf{\Psi }}_n{\left(s\right)-c\mathsf{\Psi }\left(s\right))\Vert }^{ 2}ds{)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{\left|c\right|}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s-\omega )\left(\mathsf{\Psi }(s+\omega )-{\mathsf{\Psi }}_n(s+\omega )\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left({\mathsf{\Psi }}_n(s+\omega )-c{\mathsf{\Psi }}_n\left(s\right)\right)\Vert }^{2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(c{\mathsf{\Psi }}_n\left(s\right)-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & \le \left|c\right| \Vert {\mathsf{\Psi }}_n{-\mathsf{\Psi }\Vert }_{BSS}+{\displaystyle \frac{\epsilon }{3}}+\left|c\right| \Vert {\mathsf{\Psi }}_n{-\mathsf{\Psi }\Vert }_{BSS}\le {\displaystyle \frac{\epsilon }{3}}+{\displaystyle \frac{\epsilon }{3}}+{\displaystyle \frac{\epsilon }{3}}=\epsilon .\hfill \end{array}$$

Therefore, the space ${\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$ is a closed subspace of $BSS(\mathbb{R},L^2(\mathsf{\Omega },H))$; so, this space is a Banach space, with the ${\Vert ·\Vert }_{BSS}$ norm.

(ii) Let $\mathsf{\Psi }\in {\mathcal{P}SAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$. Hence $\mathsf{\Psi }(·+s)\in {\mathcal{P}SAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$ for $s\in [0,1]$. Now,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{{\left(2\mu \right)}^{\frac{1}{2}}}{2\mu }}{\left(\underset{-\mu }{\overset{\mu }{\int }}\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}dsd\tau \right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & {\displaystyle \frac{1}{{\left(2\mu \right)}^{\frac{1}{2}}}}{\left(\underset{-\mu }{\overset{\mu }{\int }}\underset{0}{\overset{1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(\tau +s+\omega )-c\mathsf{\Psi }(\tau +s)\right)\Vert }^{ 2}dsd\tau \right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & =\left(\underset{0}{\overset{1}{\int }}{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(\tau +s+\omega )-c\mathsf{\Psi }(\tau +s)\right)\Vert }^{ 2}d\tau ds\right)\to 0,\hfill \end{array}$$

when $\mu \to +\infty$. Hence, $\mathsf{\Psi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$. □

Theorem 3.

Let $\mathsf{\Psi }\in \mathcal{B}SS(\mathbb{R},L^2(\mathsf{\Omega },H))$. The following statements are equivalent:

(i) 

$\underset{\mu \to \infty }{lim}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau =0$.

(ii) 

For every $\epsilon >0$,

$$\begin{array}{c}\hfill \underset{\mu \to \infty }{lim}{\displaystyle \frac{1}{2\mu }}\underset{T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }d\tau =0,\end{array}$$

where

$$\begin{array}{c}\hfill T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)=\left\{\tau \in [-\mu ,\mu ]:{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}\ge \epsilon \right\}.\end{array}$$

Proof. 

Let (i) hold and $\epsilon >0$. Then,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & ={\displaystyle \frac{1}{2\mu }}\underset{[-\mu ,\mu ]\setminus T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\displaystyle \frac{1}{2\mu }}\underset{T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & \ge {\displaystyle \frac{1}{2\mu }}\underset{T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \ge {\displaystyle \frac{\epsilon }{2\mu }}\underset{T_{\mu ,\epsilon \left(\mathsf{\Psi }\right)}}{\int }d\tau \ge 0,\hfill \end{array}$$

so, (ii) is true.

Now, let (ii) hold. By $\mathsf{\Psi }\in \mathcal{B}SS(\mathbb{R},L^2(\mathsf{\Omega },H))$, we obtain the existence of $K\ge 0$ such that ${\Vert \mathsf{\Psi }\Vert }_{bwc}:=\underset{\tau \in \mathbb{R}}{sup}E\Vert c^{^}(-\tau )\mathsf{\Psi }\left(\tau \right)\Vert \le K$. Also, by (ii), we have that for every $\epsilon >0$, there exists $T>0$ such that for $\tau \ge T$, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2\mu }}\underset{T_{\mu ,\epsilon \left(\mathsf{\Psi }\right)}}{\int }d\tau \le {\displaystyle \frac{\epsilon }{2K\left|c\right|}}.\end{array}$$

So,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & ={\displaystyle \frac{1}{2\mu }}\underset{T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{2\mu }}\underset{[-\mu ,\mu ]\setminus T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{1}{2\mu }}\underset{T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }{2\left|c\right|\Vert \mathsf{\Psi }\Vert }_{bwc}d\tau +{\displaystyle \frac{\epsilon }{2\mu }}\underset{[-\mu ,\mu ]\setminus T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }d\tau \hfill \\ \hfill & \le {\displaystyle \frac{2\left|c\right|K}{2\mu }}\underset{T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }d\tau +\left(1-{\displaystyle \frac{1}{2\mu }}\underset{T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }d\tau \right)\epsilon \hfill \\ \hfill & \le {\displaystyle \frac{\epsilon }{2\left|c\right|K}}·2\left|c\right|K-{\displaystyle \frac{\epsilon }{2\mu }}\underset{T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }d\tau +\epsilon \le 2\epsilon ,\hfill \end{array}$$

We conclude that,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau =0,\end{array}$$

implying that $\mathsf{\Psi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$, i.e., (i) holds. □

Let $f,g,h\in \mathcal{B}SS(\mathbb{R}\times L^2(\mathsf{\Omega },H),L^2(\mathsf{\Omega },H))$. We give a list of assumptions that are going to be used in the sequel:

(A0)

Let $(\tau ,x)\in \mathbb{R}\times L^2(\mathsf{\Omega },H)$.

$$\begin{array}{c}\hfill \underset{\mu \to \infty }{lim}{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(f(s+\omega ,x)-cf(s,c^{-1}x)\right)\Vert }^{2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau =0,\end{array}$$

$$\begin{array}{c}\hfill \underset{\mu \to \infty }{lim}{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(g(s+\omega ,x)-cg(s,c^{-1}x)\right)\Vert }^{2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau =0,\end{array}$$

$$\begin{array}{c}\hfill \underset{\mu \to \infty }{lim}{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(h(s+\omega ,x)-ch(s,c^{-1}x)\right)\Vert }^{2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau =0,\end{array}$$

converge uniformly on any bounded set of $L^2(\mathsf{\Omega },H)$;

(A1)

There exist constants $L_1,L_2,L_3>0$ such that

$$\begin{array}{cc}\hfill & {E\Vert g(\tau ,\mathsf{\Phi })-g(\tau ,\mathsf{\Psi })\Vert }^{ 2}\le L_{1}E{\Vert \mathsf{\Phi }-\mathsf{\Psi }\Vert }^{ 2},\hfill \\ \hfill & {E\Vert f(\tau ,\mathsf{\Phi })-f(\tau ,\mathsf{\Psi })\Vert }^{ 2}\le L_{2}E{\Vert \mathsf{\Phi }-\mathsf{\Psi }\Vert }^{ 2},\hfill \\ \hfill & E\Vert h(\tau ,\mathsf{\Phi })-h(\tau ,\mathsf{\Psi })Q^{{\displaystyle \frac{1}{2}}}{\Vert }_{L(\mathbb{R},L^2(\mathsf{\Omega },H))}^2\le L_{3}E{\Vert \mathsf{\Phi }-\mathsf{\Psi }\Vert }^{ 2},\hfill \end{array}$$

for all $\tau \in \mathbb{R}$ and every $\mathsf{\Phi },\mathsf{\Psi }\in L^2(\mathsf{\Omega },H)$;

(A2)

The semigroup ${\left(T\left(\tau \right)\right)}_{\tau >0}$ is compact and exponentially stable, meaning that there are constants $K,\theta >0$ such that

$$\begin{array}{c}\hfill \Vert T\left(\tau \right)\Vert \le K{e}^{-\theta t},\mathrm{for}\mathrm{all}\tau \ge 0;\end{array}$$

(A3)

The functions $f,g,h$ are uniformly continuous on every bounded set $B\subset L^2(\mathsf{\Omega },H)$ for every $\tau \in \mathbb{R}$. Additionally, for every bounded set $B\subset L^2(\mathsf{\Omega },H)$, $g(\mathbb{R},B)$, $f(\mathbb{R},B)$ and $h(\mathbb{R},B)$ are bounded. There exists $b>0$ such that ${\Lambda }_b\le {\displaystyle \frac{\theta b}{20\nu K^2}}$, where

$${\Lambda }_b=max\{\underset{\tau \in \mathbb{R},\Vert \mathsf{\Psi }\Vert \le b}{sup}\Vert g(\tau ,\mathsf{\Psi })\Vert ,\underset{\tau \in \mathbb{R},\Vert \mathsf{\Psi }\Vert \le b}{sup}\Vert f(\tau ,\mathsf{\Psi })\Vert ,\underset{\tau \in \mathbb{R},\Vert \mathsf{\Psi }\Vert \le b}{sup}\Vert h(\tau ,\mathsf{\Psi })\Vert \},$$

and $\nu =max\{1,\Vert A_1{\Vert }_{L^1(0,\infty )}^2,4\Vert A_2{\Vert }_{L^2(0,\infty )}^2\}$;

(A4)

There exist measurable functions $m_f,m_g$ and $m_h$ from $\mathbb{R}$ to $[0,\infty )$ such that

$$\begin{array}{cc}\hfill & {E\Vert g(\tau ,\mathsf{\Phi })-g(\tau ,\mathsf{\Psi })\Vert }^{ 2}\le m_g\left(\tau \right)·E{\Vert \mathsf{\Phi }-\mathsf{\Psi }\Vert }^{ 2},\hfill \\ \hfill & {E\Vert f(\tau ,\mathsf{\Phi })-f(\tau ,\mathsf{\Psi })\Vert }^{ 2}\le m_f\left(\tau \right)·E{\Vert \mathsf{\Phi }-\mathsf{\Psi }\Vert }^{ 2},\hfill \\ \hfill & E\Vert h(\tau ,\mathsf{\Phi })-h(\tau ,\mathsf{\Psi })Q^{{\displaystyle \frac{1}{2}}}{\Vert }_{L(\mathbb{R},L^2(\mathsf{\Omega },H))}^2\le m_h\left(\tau \right)·E{\Vert \mathsf{\Phi }-\mathsf{\Psi }\Vert }^{ 2},\hfill \end{array}$$

for all $\tau \in \mathbb{R}$ and every $\mathsf{\Phi },\mathsf{\Psi }\in L^2(\mathsf{\Omega },H)$;

(A5)

Let ${\left({\mathsf{\Psi }}_n\right)}_n\in \mathcal{B}SS(\mathbb{R},L^2(\mathsf{\Omega },H))$ be uniformly bounded and uniformly convergent on every compact subset of $\mathbb{R}$. Then, $g(·,{\mathsf{\Psi }}_n(·))$, $f(·,{\mathsf{\Psi }}_n(·))$ and $h(·,{\mathsf{\Psi }}_n)$ are relatively compact in $\mathcal{B}SS(\mathbb{R},L^2(\mathsf{\Omega },H))$.

Theorem 4.

Let $u\in \mathcal{B}SS(\mathbb{R}\times L^2(\mathsf{\Omega },H),L^2(\mathsf{\Omega },H))$ fulfill the assumption (A0). We suppose the existence of a positive real number L such that for any ${\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2\in L^2(\mathsf{\Omega },H)$ holds $E\Vert u(\tau ,{\mathsf{\Psi }}_1)-u(\tau ,{\mathsf{\Psi }}_2){\Vert }^2\le L{\Vert {\mathsf{\Psi }}_1-{\mathsf{\Psi }}_2\Vert }^2$ uniformly for all $\tau \in \mathbb{R}$, and let $\mathsf{\Psi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$. Then, $u(·,\mathsf{\Psi }(·))\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$.

Proof. 

Let $\mathsf{\Psi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$. Then,

$$\begin{array}{c}\hfill \underset{\mu \to \infty }{lim}{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(\mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau =0,\end{array}$$

for every $\tau \in \mathbb{R}$. Now, using the assumptions of the theorem, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(u(s+\omega ,\mathsf{\Psi }(s+\omega \left)\right)-cu(s,\mathsf{\Psi }(s\left)\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(u(s+\omega ,\mathsf{\Psi }(s+\omega ))-cu(s,c^{-1}\mathsf{\Psi }(s+\omega ))\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(cu(s,c^{-1})\mathsf{\Psi }(s+\omega ))-cu(s,\mathsf{\Psi }\left(s\right))\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(u(s+\omega ,\mathsf{\Psi }(s+\omega ))-cu(s,c^{-1}\mathsf{\Psi }(s+\omega ))\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & +{\displaystyle \frac{L\left|c\right|}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(c^{-1}\mathsf{\Psi }(s+\omega )-\mathsf{\Psi }\left(s\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau .\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill \underset{\mu \to \infty }{lim}{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(u(s+\omega ,\mathsf{\Psi }(s+\omega \left)\right)-cu(s,\mathsf{\Psi }(s\left)\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau =0,\end{array}$$

meaning that $u(·,\mathsf{\Psi }(·))\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$. □

• A similar result holds, under different assumptions, as can be seen in the next theorem.

Theorem 5.

Let for every $\epsilon >0$ and bounded subset $B\subset L^2(\mathsf{\Omega },H)$, there are constants $C_{\epsilon ,B}$ and ${\delta }_{\epsilon ,B}>0$ such that

$$\begin{array}{c}\hfill E\Vert w(\tau ,{\mathsf{\Psi }}_1)-w(\tau ,{\mathsf{\Psi }}_2){\Vert }^2\le \epsilon ,\end{array}$$

for all ${\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2\in B$ with $E\Vert {\mathsf{\Psi }}_1-{\mathsf{\Psi }}_2{\Vert }^2\le {\delta }_{\epsilon ,B}$ and $\tau \ge C_{\epsilon ,B}$. Let $u\in \mathcal{B}SS(\mathbb{R}\times L^2(\mathsf{\Omega },H),L^2(\mathsf{\Omega },H))$ fulfill the assumption (A0) and let $\mathsf{\Psi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$.

• Then, $u(·,\mathsf{\Psi }(·))\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$.

Proof. 

By (A0), we have that

$$\begin{array}{cc}\hfill & {E\Vert c}^{^}(-\tau )\left(u(\tau +\omega ,\mathsf{\Psi }(\tau +\omega \left)\right)-cu(\tau ,\mathsf{\Psi }(\tau \left)\right)\right){\Vert }^2\hfill \\ \hfill & {=E\Vert c}^{^}(-\tau )\left(u(\tau ,c^{-1}\mathsf{\Psi }(\tau +\omega ))-u(\tau ,\mathsf{\Psi }\left(\left(\tau \right)\right))\right){\Vert }^2.\hfill \end{array}$$

Also, using (A0) and Theorem 3

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(u(s+\omega ,\mathsf{\Psi }(s+\omega ))-cu(s,c^{-1}\mathsf{\Psi }(s+\omega ))\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau <\epsilon ,\end{array}$$

and for every $\epsilon >0$, there exits ${\mu }_{\epsilon }>0$ such that for all $\mu \le {\mu }_{\epsilon }$,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2\mu }}\underset{T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }d\tau \le {\displaystyle \frac{\epsilon }{\left|c\right|(2\Vert u{\Vert }_{bwc}+1)}},\end{array}$$

for $\mathsf{\Psi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$. By (A2), for every $\epsilon >0$, there exists ${\delta }_{\epsilon ,B}:=\epsilon$ and $C_{\epsilon ,B}:={\mu }_{\epsilon }$ such that

$$\begin{array}{c}\hfill E\Vert u(\tau ,c^{^}\left(\tau \right)\mathsf{\Psi }(\tau +\omega ))-u(\tau ,\mathsf{\Psi }\left(\tau \right)){\Vert }^2\le \epsilon ,\end{array}$$

for ${E\Vert \mathsf{\Psi }(\tau +\omega )-c}^{^}{\left(\tau \right)\mathsf{\Psi }\left(\tau \right)\Vert }^2\le \epsilon$ and $\left|\tau \right|\ge {\mu }_{\epsilon }$. Now,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(u(s+\omega ,\mathsf{\Psi }(s+\omega \left)\right)-cu(s,\mathsf{\Psi }(s\left)\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(u(s+\omega ,\mathsf{\Psi }(s+\omega ))-cu(s,c^{-1}\mathsf{\Psi }(s+\omega ))\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}\left(cu(s,c^{-1}\mathsf{\Psi }(s+\omega ))-cu(s,\mathsf{\Psi }\left(s\right))\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & \le \epsilon +{\displaystyle \frac{\left|c\right|}{2\mu }}\underset{[-\mu ,\mu ]\setminus T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}\left(u(s,c^{-1}\mathsf{\Psi }(s+\omega ))-u(s,\mathsf{\Psi }\left(s\right))\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & +{\displaystyle \frac{\left|c\right|}{2\mu }}\underset{T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(u(s,c^{-1}\left(s\right)\mathsf{\Psi }(s+\omega ))-u(s,\mathsf{\Psi }\left(s\right))\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & \le \epsilon +{\displaystyle \frac{\epsilon \left|c\right|}{2\mu }}\underset{[-\mu ,\mu ]\setminus T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }d\tau +{\displaystyle \frac{\left|c\right|}{2\mu }}\underset{T_{\mu ,\epsilon \left(\mathsf{\Psi }\right)}}{\int }2{\Vert u\Vert }_{bwc}d\tau \hfill \\ \hfill & \le \epsilon +{\displaystyle \frac{\epsilon \left|c\right|}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}d\tau +{\displaystyle \frac{1}{2\mu }}\underset{T_{\mu ,\epsilon }\left(\mathsf{\Psi }\right)}{\int }2{\Vert u\Vert }_{bwc}d\tau \le \epsilon +\epsilon +\epsilon =3\epsilon ,\hfill \end{array}$$

for every $\mu \ge {\mu }_{\epsilon }$. Hence,

$$\begin{array}{c}\hfill \underset{\mu \to \infty }{lim}{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(u(s+\omega ,\mathsf{\Psi }(s+\omega \left)\right)-cu(s,\mathsf{\Psi }(s\left)\right)\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau =0,\end{array}$$

so $u(·,\mathsf{\Psi }(·))\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$. □

Theorem 6.

Let ${\left(T\left(\tau \right)\right)}_{\tau \ge 0}$ be a strongly continuous stochastically bounded operator family of operators, $\phi :[0,\infty )\to [0,\infty )$ be a non-increasing function with ${\sum }_{n=1}^{\infty }\phi \left(n\right)<\infty$ such that ${\Vert T\left(\tau \right)\Vert }^2\le \phi \left(\tau \right)$ for all $\tau \in [0,\infty )$ and $\mathsf{\Psi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))\cap \mathcal{C}(\mathbb{R},L^2(\mathsf{\Omega },H))$. Then,

$$\begin{array}{c}\hfill \mathsf{\Phi }\left(\tau \right)=\underset{-\infty }{\overset{\tau }{\int }}T(\tau -s)\mathsf{\Psi }\left(s\right)ds+\underset{-\infty }{\overset{\tau }{\int }}T(\tau -s)\mathsf{\Psi }\left(s\right)dW\left(s\right)\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H)).\end{array}$$

Proof. 

We consider

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_n^1\left(\tau \right)=\underset{\tau -n}{\overset{\tau -n+1}{\int }}T(\tau -s)\mathsf{\Psi }\left(s\right)ds=\underset{n-1}{\overset{n}{\int }}T\left(s\right)\mathsf{\Psi }(t-s)ds\hfill \\ \hfill & {\mathsf{\Phi }}_n^2\left(\tau \right)=\underset{\tau -n}{\overset{\tau -n+1}{\int }}T(\tau -s)\mathsf{\Psi }\left(s\right)dW\left(s\right)=\underset{n-1}{\overset{n}{\int }}T\left(s\right)\mathsf{\Psi }(\tau -s)d\overline{W}\left(s\right),\hfill \end{array}$$

for every n positive integer, where $\overline{W}\left(s\right)=W\left(\tau \right)-W(\tau -s)$. As a remark here we can say that $\overline{W}$ is a Brownian motion with the same distribution as W.

First, we are going to show that ${\mathsf{\Phi }}_n^1,{\mathsf{\Phi }}_n^2\in \mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H))$, for all $n=1,2,\dots$.

• By the principle of uniform boundedness, we have

$$\begin{array}{c}\hfill K_n=\underset{n-1\le \tau \le n}{sup}\Vert T\left(\tau \right)\Vert <\infty ,\end{array}$$

for $n=1,2,\dots$. Let n and $t_0\in \mathbb{R}$ be fixed. So,

$$\begin{array}{cc}\hfill & E\Vert {\mathsf{\Phi }}_n^1\left(\tau \right)-{\mathsf{\Phi }}_n^1{\left(\tau \right)\Vert }^2=E{‖\underset{n-1}{\overset{n}{\int }}T\left(s\right)\mathsf{\Psi }(\tau -s)ds-\underset{n-1}{\overset{n}{\int }}T\left(s\right)\mathsf{\Psi }({\tau }_0-s)ds‖}^{ 2}\hfill \\ \hfill & =E{‖\underset{n-1}{\overset{n}{\int }}T\left(s\right)\left(\mathsf{\Psi }(\tau -s)-\mathsf{\Psi }({\tau }_0-s)\right)ds‖}^{ 2}\hfill \\ \hfill & \le {\left(\underset{n-1}{\overset{n}{\int }}\Vert T\left(s\right)\Vert ·\Vert \mathsf{\Psi }(\tau -s)-\mathsf{\Psi }({\tau }_0-s)\Vert ds\right)}^{ 2}\hfill \\ \hfill & \le K_n^{2}E{\left(\underset{n-1}{\overset{n}{\int }}\Vert \mathsf{\Psi }(\tau -s)-\mathsf{\Psi }({\tau }_0-s)\Vert ds\right)}^{ 2}\hfill \\ \hfill & \le K_n^2\underset{n-1}{\overset{n}{\int }}E{\Vert \mathsf{\Psi }(\tau -s)-\mathsf{\Psi }({\tau }_0-s)\Vert }^{ 2}ds\hfill \\ \hfill & \le K_n^2\underset{0}{\overset{1}{\int }}E{\Vert \mathsf{\Psi }(\tau -n+s)-\mathsf{\Psi }({\tau }_0-n+s)\Vert }^{ 2}ds.\hfill \end{array}$$

From $\mathsf{\Psi }\in \mathcal{B}SS(\mathbb{R},L^2(\mathsf{\Omega },H))\cap \mathcal{C}(\mathbb{R},L^2(\mathsf{\Omega },H))$, we have

$$\begin{array}{c}\hfill \underset{\tau \to {\tau }_0}{lim}\underset{0}{\overset{1}{\int }}E{\Vert \mathsf{\Psi }(\tau -n+s)-\mathsf{\Psi }({\tau }_0-n+s)\Vert }^{ 2}ds=0,\end{array}$$

so,

$$\begin{array}{c}\hfill \underset{\tau \to {\tau }_0}{lim}E{\Vert {\mathsf{\Phi }}_n^1\left(\tau \right)-{\mathsf{\Phi }}_n^1\left({\tau }_0\right)\Vert }^2=0.\end{array}$$

Thus, ${\mathsf{\Phi }}_n^1\in \mathcal{C}(\mathbb{R},L^2(\mathsf{\Omega },H))$, for any $n=1,2,\dots$.

Using Itô’s isometry property on the stochastic integral, we have

$$\begin{array}{cc}\hfill & E\Vert {\mathsf{\Phi }}_n^2\left(\tau \right)-{\mathsf{\Phi }}_n^2\left({\tau }_0\right){\Vert }^2=E{‖\underset{n-1}{\overset{n}{\int }}T\left(s\right)\mathsf{\Psi }(\tau -s)d\overline{W}\left(s\right)-\underset{n-1}{\overset{n}{\int }}T\left(s\right)\mathsf{\Psi }({\tau }_0-s)d\overline{W}\left(s\right)‖}^{ 2}\hfill \\ \hfill & =E{‖\underset{n-1}{\overset{n}{\int }}T\left(s\right)\left(\mathsf{\Psi }(\tau -s)-\mathsf{\Psi }({\tau }_0-s)\right)d\overline{W}\left(s\right)‖}^{ 2}\hfill \\ \hfill & =\underset{n-1}{\overset{n}{\int }}{\Vert T\left(s\right)\Vert }^2·E{\Vert \mathsf{\Psi }(\tau -s)-\mathsf{\Psi }({\tau }_0-s)\Vert }^{ 2}ds.\hfill \end{array}$$

Proceeding like in the case ${\mathsf{\Phi }}_n^1$ above, we obtain that ${\mathsf{\Phi }}_n^2\in \mathcal{C}(\mathbb{R},L^2(\mathsf{\Omega },H))$, for any $n=1,2,\dots$. Since, $\mathsf{\Psi }\in \mathcal{B}SS(\mathbb{R},L^2(\mathsf{\Omega },H))\cap \mathcal{C}(\mathbb{R},L^2(\mathsf{\Omega },H))$, we have

$$\begin{array}{cc}\hfill & E\Vert {\mathsf{\Phi }}_n^1{\left(\tau \right)\Vert }^2+E{\Vert {\mathsf{\Phi }}_n^2\left(\tau \right)\Vert }^2=E{‖\underset{n-1}{\overset{n}{\int }}T\left(s\right)\mathsf{\Psi }(\tau -s)ds‖}^{ 2}+E{‖\underset{n-1}{\overset{n}{\int }}T\left(s\right)\mathsf{\Psi }(\tau -s)d\overline{W}\left(s\right)‖}^{ 2}\hfill \\ \hfill & \le 2K_n^2\underset{n-1}{\overset{n}{\int }}{E\Vert \mathsf{\Psi }(\tau -s)\Vert }^{ 2}ds\le 2K_n^2\underset{0}{\overset{1}{\int }}{E\Vert \mathsf{\Psi }(\tau -n+s)\Vert }^{ 2}ds\le 2K_n^2{\Vert \mathsf{\Psi }\Vert }_{BSS}^2<\infty ,\hfill \end{array}$$

so, ${\mathsf{\Phi }}_n^1,{\mathsf{\Phi }}_n^2\in \mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H))$, for any $n=1,2,\dots$.

Next, we show that the series ${\sum }_{n=1}^{\infty }{\mathsf{\Phi }}_n^1\left(\tau \right)$ and ${\sum }_{n=1}^{\infty }{\mathsf{\Phi }}_n^2\left(\tau \right)$ are convergent in the norm ${\Vert ·\Vert }_{\infty }$ uniformly on $\mathbb{R}$. Using the Cauchy–Schwartz inequality and Itô’s isometry property on the stochastic integral, we obtain

$$\begin{array}{cc}\hfill & E\Vert {\mathsf{\Phi }}_n^1{\left(\tau \right)\Vert }^2+E{\Vert {\mathsf{\Phi }}_n^2\left(\tau \right)\Vert }^2=E{‖\underset{n-1}{\overset{n}{\int }}T\left(s\right)\mathsf{\Psi }(\tau -s)ds‖}^{ 2}+E{‖\underset{n-1}{\overset{n}{\int }}T\left(s\right)\mathsf{\Psi }(\tau -s)d\overline{W}\left(s\right)‖}^{ 2}\hfill \\ \hfill & \le {\left(\underset{n-1}{\overset{n}{\int }}\Vert T\left(s\right)\Vert ·\Vert \mathsf{\Psi }(\tau -s)\Vert ds\right)}^{ 2}+\underset{n-1}{\overset{n}{\int }}{\Vert T\left(s\right)\Vert }^{2}E{\Vert \mathsf{\Psi }(\tau -s)\Vert }^{2}ds\hfill \\ \hfill & \le 2\phi (n-1)\underset{n-1}{\overset{n}{\int }}{E\Vert \psi (\tau -s)\Vert }^{2}ds\le 2\phi (n-1)\underset{0}{\overset{1}{\int }}{E\Vert \mathsf{\Psi }(\tau -n+s)\Vert }^{2}ds\le 2\phi (n-1){\Vert \mathsf{\Psi }\Vert }_{BSS}^2.\hfill \end{array}$$

Since, ${2\Vert \mathsf{\Psi }\Vert }_{BSS}^2{\sum }_{n=1}^{\infty }\phi \left(n\right)<\infty$, we conclude that the series ${\sum }_{n=1}^{\infty }{\mathsf{\Phi }}_n^1\left(\tau \right)$ and ${\sum }_{n=1}^{\infty }{\mathsf{\Phi }}_n^2\left(\tau \right)$ are convergent uniformly on $\mathbb{R}$. Additionally, the following hold

$$\begin{array}{c}\hfill \underset{-\infty }{\overset{\tau }{\int }}T(\tau -s)\mathsf{\Psi }\left(s\right)ds=\sum _{n=1}^{\infty }{\mathsf{\Phi }}_n^1\left(\tau \right)\in \mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H)),\end{array}$$

$$\begin{array}{c}\hfill \underset{-\infty }{\overset{\tau }{\int }}T(\tau -s)\mathsf{\Psi }\left(s\right)d\overline{W}s=\sum _{n=1}^{\infty }{\mathsf{\Phi }}_n^2\left(\tau \right)\in \mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H)).\end{array}$$

Next, we are going to prove that ${\mathsf{\Phi }}_n^1,{\mathsf{\Phi }}_n^2\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$, for any $n=1,2,\dots$. Let $\mathsf{\Psi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$ and $\mu >0$. Then,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}E{\Vert c^{^}(-\tau )\left({\mathsf{\Phi }}_n^1(\tau +\omega )-c{\mathsf{\Phi }}_n^1\left(\tau \right)\right)\Vert }^{2}d\tau \hfill \\ \hfill & ={\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}E{‖c^{^}(-\tau )\left(\underset{n-1}{\overset{n}{\int }}T\left(s\right)\mathsf{\Psi }(s+\omega -t)ds-c\underset{n-1}{\overset{n}{\int }}T\left(s\right)\mathsf{\Psi }(\tau -s)ds\right)‖}^{ 2}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{{|c}^{^}(-n+1)|}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}E{\left(\underset{n-1}{\overset{n}{\int }}\Vert T\left(s\right)\Vert ·\Vert \mathsf{\Psi }(\tau +\omega -s)-c\mathsf{\Psi }(\tau -s)\Vert \right)}^{ 2}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{K_n^2\left|c^{^}(-n+1)\right|}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}\underset{n-1}{\overset{n}{\int }}E{\Vert \mathsf{\Psi }(\tau +\omega -s)-c\mathsf{\Psi }(\tau -s)\Vert }^{ 2}dsd\tau \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\displaystyle \frac{K_n^2\left|c^{^}(-n+1)\right|}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}({\left(\underset{n-1}{\overset{n}{\int }}E{\Vert \mathsf{\Psi }(\tau +\omega -s)-c\mathsf{\Psi }(\tau -s)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \times {\left(\underset{n-1}{\overset{n}{\int }}E{\Vert \mathsf{\Psi }(\tau +\omega -s)-c\mathsf{\Psi }(\tau -s)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}})d\tau \hfill \\ \hfill & \le {\displaystyle \frac{{2|c}^{^}(-n+1)|K_n^2{\Vert \mathsf{\Psi }\Vert }_{BSS}}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{n-1}{\overset{n}{\int }}E{\Vert \mathsf{\Psi }(\tau +\omega -s)-c\mathsf{\Psi }(\tau -s)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{{2|c}^{^}(-n+1)|K_n^2{\Vert \mathsf{\Psi }\Vert }_{BSS}}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau -1}{\overset{\tau -n+1}{\int }}E{\Vert \mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{{2|c}^{^}(-n+1)|K_n^2{\Vert \mathsf{\Psi }\Vert }_{BSS}}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert \mathsf{\Psi }(s+\omega )-c\mathsf{\Psi }\left(s\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau \to 0,\hfill \end{array}$$

when $\mu \to \infty$. Therefore, ${\mathsf{\Phi }}_n^1\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$, for any $n=1,2,\dots$. Using that ${\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$ is a Banach space with sup-norm, we have that

$$\begin{array}{c}\hfill \underset{-\infty }{\overset{\tau }{\int }}T(\tau -s)\mathsf{\Psi }\left(s\right)ds=\sum _{n=1}^{\infty }{\mathsf{\Phi }}_n^1\left(\tau \right)\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H)).\end{array}$$

Similar arguments and Itô’s isometry property of stochastic integral lead us to the conclusion that

$$\begin{array}{c}\hfill \underset{-\infty }{\overset{\tau }{\int }}T(\tau -s)\mathsf{\Psi }\left(s\right)dW\left(s\right)=\sum _{n=1}^{\infty }{\mathsf{\Phi }}_n^2\left(\tau \right)\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H)).\end{array}$$

Now, from Theorem 1 (i), we deduce that $\mathsf{\Phi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$. □

3. Existence of Stepanov-like Pseudo $\mathit{S}$-Asymptotically $(\mathit{\omega },\mathit{c})$-Periodic Mild Solutions of a Class of Stochastic Integro-Differential Equations

This section is devoted to establishing the existence and uniqueness of the Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic mild solutions to Equation (1).

Definition 3.

For a ${\mathcal{F}}_{\tau }$-progressively measurable process ${\left(\mathsf{\Psi }\left(\tau \right)\right)}_{\tau \in \mathbb{R}}$ it is said to be a mild solution of (1) if it satisfies the following stochastic integral equation

$$\begin{array}{c}\hfill \mathsf{\Psi }\left(\tau \right)=T(\tau -{\tau }_0)\mathsf{\Psi }\left({\tau }_0\right)+\underset{{\tau }_0}{\overset{\tau }{\int }}T(\tau -s)g(s,\mathsf{\Psi }\left(s\right))ds\end{array}$$

(2)

$$\begin{array}{cc}\hfill & +\underset{{\tau }_0}{\overset{\tau }{\int }}T(\tau -\sigma )\underset{{\tau }_0}{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma \hfill \\ \hfill & +\underset{{\tau }_0}{\overset{\tau }{\int }}T(\tau -\sigma )\underset{{\tau }_0}{\overset{\sigma }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma ,\hfill \end{array}$$

for all $\tau \ge {\tau }_0$.

Remark 1.

If we put ${\tau }_0\to -\infty$ in the above equation, by the exponential dissipation condition imposed on ${\left(T\left(\tau \right)\right)}_{\tau \ge 0}$, the stochastic process $\mathsf{\Psi }:\mathbb{R}\to L^2(\mathsf{\Omega },H)$ is a mild solution of Equation (2) if and only if Ψ satisfies the stochastic integral equation

$$\mathsf{\Psi }\left(\tau \right)=\underset{-\infty }{\overset{\tau }{\int }}T(\tau -s)g(s,\mathsf{\Psi }\left(s\right))ds+\underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma$$

(3)

$$\begin{array}{c}\hfill +\underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma .\end{array}$$

Note that the first and second integral on the right-hand integral in (3) are taken in the Bochner sense, while the third integral is interpreted in Itô’s sense.

We define the integral operator $\mathsf{\Theta }$ as

$$\begin{array}{cc}\hfill & \left(\mathsf{\Theta }\mathsf{\Psi }\right)\left(\tau \right)=\underset{-\infty }{\overset{\tau }{\int }}T(\tau -s)g(s,\mathsf{\Psi }\left(s\right))ds+\underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma \hfill \\ \hfill & +\underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma .\hfill \end{array}$$

Lemma 1.

Let (A2) and (A3) be satisfied. Then $\mathsf{\Theta }:\mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H))\to \mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H))$ is well-defined and continuous.

Proof. 

It is straightforward to see that the operator $\mathsf{\Theta }$ is well-defined. It remains to show that the operator $\mathsf{\Theta }$ is continuous. Let ${\left({\mathsf{\Psi }}_n\right)}_n\in \mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H))$ be an arbitrary sequence converging uniformly to $\mathsf{\Psi }\in \mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H))$, i.e., $\Vert {\mathsf{\Psi }}_n{-\mathsf{\Psi }\Vert }_{\infty }\to 0$, when $n\to \infty$. There exists a bounded set $B\subset L^2(\mathsf{\Omega },H)$ such that ${\mathsf{\Psi }}_n,\mathsf{\Psi }\in B$ for every $\tau \in \mathbb{R}$ and $n=1,2,\dots$. By the assumption of the lemma, for every $\epsilon >0$, there exist $\delta >0$ and $n_0>0$ such that $E\Vert {\mathsf{\Psi }}_n{\left(\tau \right)-\mathsf{\Psi }\left(\tau \right)\Vert }^2<\delta$, implying

$$\begin{array}{c}\hfill E\Vert g(\tau ,{\mathsf{\Psi }}_n\left(\tau \right))-g(\tau ,\mathsf{\Psi }\left(\tau \right)){\Vert }^2<{\displaystyle \frac{{\theta }^2\epsilon }{9K^2}},\end{array}$$

$$\begin{array}{c}\hfill E\Vert f(\tau ,{\mathsf{\Psi }}_n\left(\tau \right))-f(\tau ,\mathsf{\Psi }\left(\tau \right)){\Vert }^2<{\displaystyle \frac{{\theta }^2\epsilon }{9K^2(\Vert A_1{\Vert }_{L^1(0,\infty )}^2+1)}},\end{array}$$

$$\begin{array}{c}\hfill E\Vert h(\tau ,{\mathsf{\Psi }}_n\left(\tau \right))-h(\tau ,\mathsf{\Psi }\left(\tau \right))Q^{{\displaystyle \frac{1}{2}}}{\Vert }^2<{\displaystyle \frac{{\theta }^2\epsilon }{9K^2(\Vert A_2{\Vert }_{L^2(0,\infty )}^2+1)}}.\end{array}$$

Thus,

$$\begin{array}{cc}\hfill & E\Vert \left(\mathsf{\Theta }{\mathsf{\Psi }}_n\right)\left(\tau \right)-\left(\mathsf{\Theta }\mathsf{\Psi }\right)\left(\tau \right){\Vert }^2=\Vert \underset{-\infty }{\overset{\tau }{\int }}T(\tau -s)\left(g(s,{\mathsf{\Psi }}_n\left(s\right))-g(s,\mathsf{\Psi }\left(s\right))\right)ds\hfill \\ \hfill & +\underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)\left(f(s,{\mathsf{\Psi }}_n\left(s\right))-f(s,\mathsf{\Psi }\left(s\right))\right)dsd\sigma \hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)\left(h(s,{\mathsf{\Psi }}_n\left(s\right))-h(s,\mathsf{\Psi }\left(s\right))\right)dW\left(s\right)d\sigma {\Vert }^2\hfill \\ \hfill & \le 3E{‖\underset{-\infty }{\overset{\tau }{\int }}T(\tau -s)\left(g(s,{\mathsf{\Psi }}_n\left(s\right))-g(s,\mathsf{\Psi }\left(s\right))\right)ds‖}^{ 2}\hfill \\ \hfill & +3E{‖\underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)\left(f(s,{\mathsf{\Psi }}_n\left(s\right))-f(s,\mathsf{\Psi }\left(s\right))\right)dsd\sigma ‖}^{ 2}\hfill \\ \hfill & +3E{‖\underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma ){\int }_{\infty }^{\sigma }{A}_2(\sigma -s)\left(h(s,{\mathsf{\Psi }}_n\left(s\right))-h(s,\mathsf{\Psi }\left(s\right))\right)dW\left(s\right)d\sigma ‖}^{ 2}\hfill \\ \hfill & \le 3(I_1+I_2+I_3).\hfill \end{array}$$

Now, using Cauchy–Schwartz’s inequality,

$$\left|\underset{a}{\overset{b}{\int }}f\left(x\right)g\left(x\right)dx\right|\le {\left(\underset{a}{\overset{b}{\int }}{\left|f\left(x\right)\right|}^{2}dx\right)}^{{\displaystyle \frac{1}{2}}}{\left(\underset{a}{\overset{b}{\int }}{\left|g\left(x\right)\right|}^{2}dx\right)}^{{\displaystyle \frac{1}{2}}},$$

$f,g\in L^2(a,b)$, we obtain

$$\begin{array}{c}\hfill I_1<{\displaystyle \frac{\epsilon }{9}},I_2<{\displaystyle \frac{\epsilon }{9}}.\end{array}$$

For the integral $I_3$, we use Cauchy–Schwartz’s inequality and Itô’s isometry property to obtain

$$\begin{array}{c}\hfill I_3\le {\displaystyle \frac{K^2}{\theta }}\underset{-\infty }{\overset{\tau }{\int }}{e}^{-\theta (\tau -\sigma )}\underset{-\infty }{\overset{\sigma }{\int }}E{‖A_2(\sigma -s)\left(h(s,{\mathsf{\Psi }}_n\left(s\right))-h(s,\mathsf{\Psi }\left(s\right))\right)Q^{{\displaystyle \frac{1}{2}}}‖}^{ 2}dsd\sigma <{\displaystyle \frac{\epsilon }{9}}.\end{array}$$

Combining these three inequalities, we obtain that for each $\tau \in \mathbb{R}$ and $n>n_0$

$$\begin{array}{c}\hfill E\Vert \left(\mathsf{\Theta }{\mathsf{\Psi }}_n\right)\left(\tau \right)-\left(\mathsf{\Theta }\mathsf{\Psi }\right)\left(\tau \right){\Vert }^2<\epsilon ,\end{array}$$

implying that the integral operator $\mathsf{\Theta }$ is continuous. □

Theorem 7.

Let (A0) and (A2)–(A5) hold and let $\phi :[0,\infty )\to [0,\infty )$ be a non-increasing function with ${\sum }_{n=1}^{\infty }\phi \left(n\right)<\infty$ such that ${\Vert T\left(\tau \right)\Vert }^2\le \phi \left(\tau \right)$ for all $\tau \in [0,\infty )$. If $f,g\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R}\times L^2(\mathsf{\Omega },H),L^2(\mathsf{\Omega },H))$ and $h\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R}\times L^2(\mathsf{\Omega },H),L(\mathbb{R},L^2(\mathsf{\Omega },H))$, then the stochastic Equation (1) has at least one Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic solution mild solution.

Proof. 

Let $\mathcal{T}=\{\mathsf{\Psi }\in \mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H)):\Vert \mathsf{\Psi }{\Vert }_{\infty }^2\le b\}$. Using Lemma 1, we conclude that the set $\mathcal{T}$ is convex and a closed set satisfying $\mathsf{\Theta }\mathcal{T}\subset \mathcal{T}$. From Theorems 4 and 6, it follows that the mild solution of (1) is a Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic solution, i.e., $\mathsf{\Theta }:{\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))\to {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$.

Step 1. We are going to show that the set $V=\left\{\right(\mathsf{\Theta }\mathsf{\Psi }\left)\right(\tau ):\mathsf{\Psi }\in \mathcal{T}\}$ is a relatively compact subset of $L^2(\mathsf{\Omega },H)$ for every $\tau \in \mathbb{R}$. Let $\epsilon >0$ be arbitrary and $\tau \in \mathbb{R}$ be fixed. We have,

$$\begin{array}{cc}\hfill & \left({\mathsf{\Theta }}_{\epsilon }\mathsf{\Psi }\right)\left(\tau \right)=\underset{-\infty }{\overset{\tau -\epsilon }{\int }}T(\tau -s)g(s,\mathsf{\Psi }\left(s\right))ds+\underset{-\infty }{\overset{\tau -\epsilon }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma \hfill \\ \hfill & +\underset{-\infty }{\overset{\tau -\epsilon }{\int }}t(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma \hfill \\ \hfill & =T\left(\epsilon \right)\underset{-\infty }{\overset{\tau -\epsilon }{\int }}T(\tau -\epsilon -s)g(s,\mathsf{\Psi }\left(s\right))ds\hfill \\ \hfill & +\underset{-\infty }{\overset{\tau -\epsilon }{\int }}T(\tau -\epsilon -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma \hfill \\ \hfill & +\underset{-\infty }{\overset{\tau -\epsilon }{\int }}T(\tau -\epsilon -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma \hfill \\ \hfill & =T\left(\epsilon \right)\left(\mathsf{\Theta }\mathsf{\Psi }\right)(\tau -\epsilon ),\hfill \end{array}$$

so $V_{\epsilon }:=\{\left({\mathsf{\Theta }}_{\epsilon }\mathsf{\Psi }\right)\left(\tau \right):\mathsf{\Psi }\in \mathcal{T}\}$ is relatively compact in $L^2(\mathsf{\Omega },H)$, by the compactness of the semigroup $T\left(\epsilon \right)$. Now,

$$\begin{array}{cc}\hfill & E\Vert \left(\mathsf{\Theta }\mathsf{\Psi }\right)\left(\tau \right)-\left({\mathsf{\Theta }}_{\epsilon }\mathsf{\Psi }\right)\left(\tau \right){\Vert }^2\hfill \\ \hfill & =E\Vert \underset{\tau -\epsilon }{\overset{\tau }{\int }}T(\tau -s)g(s,\mathsf{\Psi }\left(s\right))ds+\underset{\tau -\epsilon }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma \hfill \\ \hfill & +\underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma {\Vert }^2\hfill \\ \hfill & \le 3E{‖\underset{\tau -\epsilon }{\overset{\tau }{\int }}T(\tau -s)g(s,\mathsf{\Psi }\left(s\right))ds ‖}^2\hfill \\ \hfill & +3E{‖\underset{\tau -\epsilon }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma ‖}^2\hfill \\ \hfill & +3E{‖\underset{\tau -\epsilon }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma ‖}^2\hfill \\ \hfill & \le 3{\Lambda }_b{K}^2{\left(\underset{\tau -\epsilon }{\overset{\tau }{\int }}{e}^{-\theta (\tau -\sigma )}d\sigma \right)}^2+3{\Lambda }_b{K}^2{\Vert A_1\Vert }_{L^1(0,\infty )}^2{\left(\underset{\tau -\epsilon }{\overset{\tau }{\int }}{e}^{-\theta (\tau -\sigma )}d\sigma \right)}^2\hfill \\ \hfill & +3{\Lambda }_b{K}^2{\Vert A_2\Vert }_{L^2(0,\infty )}^2{\left(\underset{\tau -\epsilon }{\overset{\tau }{\int }}{e}^{-\theta (\tau -\sigma )}d\sigma \right)}^2\hfill \\ \hfill & +3{\Lambda }_b{K}^2{\Vert A_2\Vert }_{L^2(0,\infty )}^2{\left(\underset{\tau -\epsilon }{\overset{\tau }{\int }}{e}^{-\theta (\tau -\sigma )}d\sigma \right)}^2\hfill \\ \hfill & \le 3{\Lambda }_b{K}^2\left(1+\Vert A_1{\Vert }_{L^1(0,\infty )}^2+4{\Vert A_2\Vert }_{L^2(0,\infty )}^2\right){\epsilon }^2.\hfill \end{array}$$

Therefore, $V=\left\{\right(\mathsf{\Theta }\mathsf{\Psi }\left)\right(\tau ):\mathsf{\Psi }\in \mathcal{T}\}$ is a relatively compact subset of $L^2(\mathsf{\Omega },H)$, for every $\tau \in \mathbb{R}$.

Step 2. We are going to show that $U=\{\left(\mathsf{\Theta }\mathsf{\Psi }\right):\mathsf{\Psi }\in \mathcal{T}\}\subseteq \mathcal{B}C(\mathbb{R},L^2(\mathsf{\Omega },H))$ is equicontinuous. Let $\mathsf{\Psi }\in \mathcal{T}$ and ${\tau }_1,{\tau }_2\in \mathbb{R}$ and ${\tau }_1<{\tau }_2$. Now,

$$\begin{array}{cc}\hfill & E\Vert \left(\mathsf{\Theta }\mathsf{\Psi }\right)\left({\tau }_1\right)-\left(\mathsf{\Theta }\mathsf{\Psi }\right)\left({\tau }_2\right){\Vert }^2\hfill \\ \hfill & =E\Vert \underset{-\infty }{\overset{{\tau }_2}{\int }}T({\tau }_2-s)g(s,\mathsf{\Psi }\left(s\right))ds+\underset{-\infty }{\overset{{\tau }_2}{\int }}T({\tau }_2-\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma \hfill \\ \hfill & +\underset{-\infty }{\overset{{\tau }_2}{\int }}T({\tau }_2-\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma \hfill \\ \hfill & -(\underset{-\infty }{\overset{{\tau }_1}{\int }}T({\tau }_1-s)g(s,\mathsf{\Psi }\left(s\right))ds+\underset{-\infty }{\overset{{\tau }_1}{\int }}T({\tau }_1-\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma \hfill \\ \hfill & +\underset{-\infty }{\overset{{\tau }_1}{\int }}T({\tau }_1-\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma ){\Vert }^2\hfill \\ \hfill & =E\Vert \left(T({\tau }_2-{\tau }_1)-I\right)(\underset{-\infty }{\overset{{\tau }_1}{\int }}T({\tau }_1-s)g(s,\mathsf{\Psi }\left(s\right))ds\hfill \\ \hfill & +\underset{-\infty }{\overset{{\tau }_1}{\int }}T({\tau }_1-\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma \hfill \\ \hfill & +\underset{-\infty }{\overset{{\tau }_1}{\int }}T({\tau }_1-\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma )\hfill \\ \hfill & +\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}T({\tau }_2-s)g(s,\mathsf{\Psi }\left(s\right))ds+\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}T({\tau }_2-\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma \hfill \\ \hfill & +\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}T({\tau }_2-\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma {\Vert }^2\hfill \\ \hfill & \le 4\underset{\mathsf{\Phi }\in V}{sup}E{‖\left(T({\tau }_2-{\tau }_1)-I\right)\mathsf{\Phi }‖}^{ 2}+4{‖\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}T({\tau }_2-s)g(s,\mathsf{\Psi }\left(s\right))ds‖}^{ 2}\hfill \\ \hfill & +4E{‖\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}T({\tau }_2-\sigma )\underset{\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma ‖}^{ 2}\hfill \\ \hfill & +4E{‖\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}T({\tau }_2-\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma ‖}^{ 2}\hfill \\ \hfill & \le 4\underset{\mathsf{\Phi }\in V}{sup}E{‖\left(T({\tau }_2-{\tau }_1)-I\right)\mathsf{\Phi }‖}^{ 2}+4{\Lambda }_c{K}^2{\left(\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}{e}^{-\theta ({\tau }_2-\sigma )}d\sigma \right)}^2\hfill \\ \hfill & +4{\Lambda }_b{K}^2{\Vert A_1\Vert }_{L^1(0,\infty )}^2{\left(\underset{\tau -\epsilon }{\overset{\tau }{\int }}{e}^{-\theta ({\tau }_2-\sigma )}d\sigma \right)}^2\hfill \\ \hfill & +4{\Lambda }_b{K}^2{\Vert A_2\Vert }_{L^2(0,\infty )}^2{\left(\underset{\tau -\epsilon }{\overset{\tau }{\int }}{e}^{-\theta ({\tau }_2-\sigma )}d\sigma \right)}^2\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +4{\Lambda }_b{K}^2{\Vert A_2\Vert }_{L^2(0,\infty )}^2{\left(\underset{\tau -\epsilon }{\overset{\tau }{\int }}{e}^{-\theta ({\tau }_2-\sigma )}d\sigma \right)}^2\hfill \\ \hfill & \le 4\underset{\mathsf{\Phi }\in V}{sup}E{‖\left(T({\tau }_2-{\tau }_1)-I\right)\mathsf{\Phi }‖}^{ 2}\hfill \\ \hfill & +4{\Lambda }_b{K}^2\left(1+\Vert A_1{\Vert }_{L^1(0,\infty )}^2+4{\Vert A_2\Vert }_{L^2(0,\infty )}^2\right){\left(\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}{e}^{-\theta ({\tau }_2-\sigma )}\right)}^2.\hfill \end{array}$$

So, the right-hand side tends to 0, when ${\tau }_2\to {\tau }_1$, independently on $\mathsf{\Psi }\in \mathcal{T}$. Hence, U is right equicontinuous at $\tau$. Similarly U is left equicontinuous at $\tau$.

Let the convex hull of $\mathsf{\Theta }\mathcal{T}$ be denoted by $\overline{co}\mathsf{\Theta }\mathcal{T}$. Since $\overline{co}\mathsf{\Theta }\mathcal{T}\subset \mathcal{T}$ and $\mathcal{T}$ is a convex closed set, so $\mathsf{\Theta }\left(\overline{co}\mathsf{\Theta }\mathcal{T}\right)\subset \mathsf{\Theta }\mathcal{T}\subset \overline{co}\mathsf{\Theta }\mathcal{T}$. Next, it is clear that $\overline{co}\mathsf{\Theta }\mathcal{T}$ is relatively compact in $L^2(\mathsf{\Omega },H)$. By the Arzelá–Ascoli theorem, we conclude that the restriction of $\overline{co}\mathsf{\Theta }\mathcal{T}$ to any compact subset J of $\mathbb{R}$ is relatively compact in $\mathcal{C}(J,L^2(\mathsf{\Omega },H))$. Hence, by the assumption (A5), we have that $\mathsf{\Theta }:\overline{co}\mathsf{\Theta }\mathcal{T}\to \overline{co}\mathsf{\Theta }\mathcal{T}$ is a compact operator, meaning that $\mathsf{\Theta }:\overline{co}\mathsf{\Theta }\mathcal{T}\to \overline{co}\mathsf{\Theta }\mathcal{T}$ is continuous and compact. Now, using the Schauder fixed point theorem, the operator $\mathsf{\Theta }$ has a fixed point, meaning that Equation (1) has at least one Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic solution mild solution. □

Theorem 8.

Let (A0)–(A2) hold and $f,g,h\in {SPSAP}_{\omega ,c}(\mathbb{R}\times L^2(\mathsf{\Omega },H),L^2(\mathsf{\Omega },H))\cap \mathcal{C}(\mathbb{R}\times L^2(\mathsf{\Omega },H),L^2(\mathsf{\Omega },H))$. Then, Equation (1) has a unique mild solution

• $\mathsf{\Psi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$, provided that there exist $K,\theta >0$, such that $\Vert A_1\left(\tau \right)\Vert \le K{e}^{-\theta \tau }$, $\Vert A_2\left(\tau \right)\Vert \le K{e}^{-\theta \tau }$ and

  $$\begin{array}{c}\hfill {\displaystyle \frac{3K^2}{{\theta }^2}}\left({\displaystyle \frac{L_1\theta }{K}}+L_2+L_3\right)<1.\end{array}$$

Proof. 

From Theorem 4, $\tau \mapsto g(\tau ,\mathsf{\Psi }(\tau \left)\right)$, $\tau \mapsto f(\tau ,\mathsf{\Psi }(\tau \left)\right)$, $\tau \mapsto h(\tau ,\mathsf{\Psi }(\tau \left)\right)$ are in ${\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$, for $\mathsf{\Psi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$. We define the operator $\mathsf{\Theta }:{\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))\to {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$ by

$$\begin{array}{cc}\hfill & \left(\mathsf{\Theta }\mathsf{\Psi }\right)\left(\tau \right)=\underset{-\infty }{\overset{\tau }{\int }}T(\tau -s)g(s,\mathsf{\Psi }\left(s\right))ds+\underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)f(s,\mathsf{\Psi }\left(s\right))dsd\sigma \hfill \\ \hfill & +\underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\tau }{\int }}{A}_2(\sigma -s)h(s,\mathsf{\Psi }\left(s\right))dW\left(s\right)d\sigma .\hfill \end{array}$$

Let ${\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$. Thus,

$$\begin{array}{cc}\hfill & E\Vert \left(\mathsf{\Theta }{\mathsf{\Psi }}_1\right)\left(\tau \right)-\left(\mathsf{\Theta }{\mathsf{\Psi }}_2\right)\left(\tau \right){\Vert }^2\hfill \\ \hfill & \le 3E{‖\underset{-\infty }{\overset{\tau }{\int }}T(\tau -s)\left(g(s,{\mathsf{\Psi }}_1\left(s\right))-g(s,{\mathsf{\Psi }}_2\left(s\right))\right)ds‖}^2\hfill \\ \hfill & +3E\Vert \underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_1(\sigma -s)\left(f(s,{\mathsf{\Psi }}_1\left(s\right))-f(s,{\mathsf{\Psi }}_2\left(s\right))\right)dsd\sigma {\Vert }^2\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +3E\Vert \underset{-\infty }{\overset{\tau }{\int }}T(\tau -\sigma )\underset{-\infty }{\overset{\sigma }{\int }}{A}_2(\sigma -s)\left(h(s,{\mathsf{\Psi }}_1\left(s\right))-h(s,{\mathsf{\Psi }}_2\left(s\right))\right)dW\left(s\right)d\sigma {\Vert }^2\hfill \\ \hfill & \le 3\underset{-\infty }{\overset{\tau }{\int }}K{e}^{-\theta (\tau -\sigma )}E{\Vert g(s,{\mathsf{\Psi }}_1\left(s\right))-g(s,{\mathsf{\Psi }}_2\left(s\right))\Vert }^{2}ds\hfill \\ \hfill & +3\underset{-\infty }{\overset{\tau }{\int }}K{e}^{-\theta (\tau -\sigma )}\underset{\infty }{\overset{\sigma }{\int }}K{e}^{-\theta (\sigma -s)}E{\Vert f(s,{\mathsf{\Psi }}_1\left(s\right))-f(s,{\mathsf{\Psi }}_2\left(s\right))\Vert }^{2}dsd\sigma \hfill \\ \hfill & +3\underset{-\infty }{\overset{\tau }{\int }}K{e}^{-\theta (\tau -\sigma )}\underset{\infty }{\overset{\sigma }{\int }}K{e}^{-\theta (\sigma -s)}E{\Vert h(s,{\mathsf{\Psi }}_1\left(s\right))-h(s,{\mathsf{\Psi }}_2\left(s\right))\Vert }^{2}dsd\sigma \hfill \\ \hfill & \le 3K{L}_1\underset{-\infty }{\overset{\tau }{\int }}{e}^{-\theta (\tau -s)}\underset{s\in \mathbb{R}}{sup}{\Vert {\mathsf{\Psi }}_1\left(s\right)-{\mathsf{\Psi }}_2\left(s\right)\Vert }^{2}ds\hfill \\ \hfill & +3K^2{L}_2\underset{\infty }{\overset{\tau }{\int }}{e}^{-\theta (\tau -\sigma )}\underset{\infty }{\overset{\sigma }{\int }}{e}^{-\theta (\sigma -s)}\underset{s\in \mathbb{R}}{sup}{\Vert {\mathsf{\Psi }}_1\left(s\right)-{\mathsf{\Psi }}_2\left(s\right)\Vert }^{2}dsd\sigma \hfill \\ \hfill & +3K^2{L}_2\underset{-\infty }{\overset{\tau }{\int }}{e}^{-\theta (\tau -\sigma )}\underset{-\infty }{\overset{\tau }{\int }}{e}^{-\theta (\sigma -s)}\underset{s\in \mathbb{R}}{sup}{\Vert {\mathsf{\Psi }}_1\left(s\right)-{\mathsf{\Psi }}_2\left(s\right)\Vert }^{2}dsd\sigma \hfill \\ \hfill & ={\displaystyle \frac{3K{L}_1}{\theta }}\underset{s\in \mathbb{R}}{sup}\Vert {\mathsf{\Psi }}_1\left(s\right)-{\mathsf{\Psi }}_2{\left(s\right)\Vert }^2+{\displaystyle \frac{3K^2{L}_2}{{\theta }^2}}\underset{s\in \mathbb{R}}{sup}{\Vert {\mathsf{\Psi }}_1\left(s\right)-{\mathsf{\Psi }}_2\left(s\right)\Vert }^2\hfill \\ \hfill & +{\displaystyle \frac{3K^2{L}_2}{{\theta }^2}}\underset{s\in \mathbb{R}}{sup}{\Vert {\mathsf{\Psi }}_1\left(s\right)-{\mathsf{\Psi }}_2\left(s\right)\Vert }^2\hfill \\ \hfill & \le {\displaystyle \frac{3K^2}{{\theta }^2}}\left({\displaystyle \frac{L_1\theta }{K}}+L_2+L_3\right){\Vert {\mathsf{\Psi }}_1-{\mathsf{\Psi }}_2\Vert }_{\infty }^2.\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill \Vert \mathsf{\Theta }{\mathsf{\Psi }}_1-\mathsf{\Theta }{\mathsf{\Psi }}_2{\Vert }_{\infty }^2\le {\displaystyle \frac{3K^2}{{\theta }^2}}\left({\displaystyle \frac{L_1\theta }{K}}+L_2+L_3\right){\Vert {\mathsf{\Psi }}_1-{\mathsf{\Psi }}_2\Vert }_{\infty }^2,\end{array}$$

so $\mathsf{\Theta }$ is a contraction, since ${\displaystyle \frac{3K^2}{{\theta }^2}}\left({\displaystyle \frac{L_1\theta }{K}}+L_2+L_3\right)<1$. Using the Banach contraction mapping principle, there exists a unique mild solution $\mathsf{\Psi }\in {\mathcal{S}PSAP}_{\omega ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$ of Equation (1). □

4. Applications

Consider the following example:

Example 1.

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathsf{\partial }\mathsf{\Psi }(\tau ,x)}{\mathsf{\partial }\tau }}={\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{x}^2}}\mathsf{\Psi }(\tau ,x)+a\left(\tau \right){\sigma }_1\left(\mathsf{\Psi }(\tau ,x)\right)\end{array}$$

$$\begin{array}{c}\hfill +\underset{-\infty }{\overset{\tau }{\int }}{e}^{-{\pi }^2(\tau -s)}a\left(\tau \right){\sigma }_2\left(\mathsf{\Psi }(s.x)\right)ds+\underset{-\infty }{\overset{\tau }{\int }}{e}^{-{\pi }^2(\tau -s)}a\left(\tau \right){\sigma }_3\left(\mathsf{\Psi }(s.x)\right)dW\left(s\right),\end{array}$$

(4)

with $\mathsf{\Psi }(t,0)=\mathsf{\Psi }(t,1)=0$, $t\in \mathbb{R}$ and W is a Brownian motion with covariance operator Q, such that $trQ<\infty$. The forcing terms are given as follows

$$\begin{array}{cc}\hfill & g(\tau ,\mathsf{\Psi })\left(s\right)=a\left(\tau \right){\sigma }_1\left(\mathsf{\Psi }\left(s\right)\right),\hfill \\ \hfill & f(\tau ,\mathsf{\Psi })\left(s\right)=a\left(\tau \right){\sigma }_2\left(\mathsf{\Psi }\left(s\right)\right),\hfill \\ \hfill & h(\tau ,\mathsf{\Psi })\left(s\right)=a\left(\tau \right){\sigma }_3\left(\mathsf{\Psi }\left(s\right)\right),\hfill \end{array}$$

where the following conditions are verified: $a\left(\tau \right)$ is bounded, continuous and $2\pi$-periodic function,

$$\begin{array}{c}\hfill {\sigma }_i\left(c\mathsf{\Psi }\right)=c{\sigma }_i\left(\mathsf{\Psi }\left(s\right)\right),\end{array}$$

$$\begin{array}{c}\hfill E\Vert {\sigma }_i\left({\mathsf{\Psi }}_1\right)-{\sigma }_i\left({\mathsf{\Psi }}_2\right){\Vert }^2\le L_{i}E{\Vert {\mathsf{\Psi }}_1-{\mathsf{\Psi }}_2\Vert }^2\end{array}$$

for $i=1,2,3$, $c\in \mathbb{C}\setminus \left\{0\right\}$ and ${\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2\in {\mathcal{S}PSAP}_{2\pi ,c}(\mathbb{R},L^2(\mathsf{\Omega },H))$.

Now,

$$\begin{array}{cc}\hfill & f(\tau +2\pi ,\mathsf{\Psi })\left(s\right)=a(\tau +2\pi ){\sigma }_1\left(\mathsf{\Psi }\right)=a\left(\tau \right)c{\sigma }_1\left(c^{-1}\mathsf{\Psi }\right)=cf(\tau ,c^{-1}\mathsf{\Psi })\left(s\right),\hfill \\ \hfill & g(\tau +2\pi ,\mathsf{\Psi })\left(s\right)=a(\tau +2\pi ){\sigma }_2\left(\mathsf{\Psi }\right)=a\left(\tau \right)c{\sigma }_2\left(c^{-1}\mathsf{\Psi }\right)=cg(\tau ,c^{-1}\mathsf{\Psi })\left(s\right),\hfill \\ \hfill & h(\tau +2\pi ,\mathsf{\Psi })\left(s\right)=a(\tau +2\pi ){\sigma }_3\left(\mathsf{\Psi }\right)=a\left(\tau \right)c{\sigma }_3\left(c^{-1}\mathsf{\Psi }\right)=ch(\tau ,c^{-1}\mathsf{\Psi })\left(s\right),\hfill \end{array}$$

• so by using the Lebesgue dominated convergence theorem, we have

  $$\begin{array}{cc}\hfill & \underset{\mu \to \infty }{lim}{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(f(s+2\pi ,\mathsf{\Psi }(s+2\pi ))-cf(s,c^{-1}\mathsf{\Psi }(s+2\pi ))\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau =0,\hfill \\ \hfill & \underset{\mu \to \infty }{lim}{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(g(s+2\pi ,\mathsf{\Psi }(s+2\pi ))-cg(s,c^{-1}\mathsf{\Psi }(s+2\pi ))\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau =0,\hfill \\ \hfill & \underset{\mu \to \infty }{lim}{\displaystyle \frac{1}{2\mu }}\underset{-\mu }{\overset{\mu }{\int }}{\left(\underset{\tau }{\overset{\tau +1}{\int }}E{\Vert c^{^}(-s)\left(h(s+2\pi ,\mathsf{\Psi }(s+2\pi ))-ch(s,c^{-1}\mathsf{\Psi }(s+2\pi ))\right)\Vert }^{ 2}ds\right)}^{{\displaystyle \frac{1}{2}}}d\tau =0,\hfill \end{array}$$

  so (A0) is satisfied.

Denote $H=L^2(0,1)$, and let us consider the linear operator $A:D\left(A\right)\subset L^2(0,1)\to L^2(0,1)$ defined by $D\left(A\right)=H^2(0,1)\cap H_0^1(0,1)$ and $A\varphi \left(s\right):={\displaystyle \frac{{\mathsf{\partial }}^2\varphi }{\mathsf{\partial }{s}^2}}$, for $s\in (0,1)$ and $\varphi \in D\left(A\right)$. Note that $H^1(0,1)=\{u\in L^2(0,1):u^{\prime }\in L^2(0,1)\}$, $H^2(0,1)=\{u\in L^2(0,1):u^{\prime },u^{\prime \prime }\in L^2(0,1)\}$ and $H_0^1(0,1)=\{u\in H^1(0,1):u\left(0\right)=u\left(1\right)=0\}$, where the derivatives are taken in the weak sense. The operator A is generating a strongly continuous semigroup ${\left(T\left(\tau \right)\right)}_{\tau \ge 0}$ given by

$$\begin{array}{c}\hfill \left(T\left(\tau \right)\varphi \right)\left(z\right)=\sum _{n=1}^{\infty }{e}^{-n^2{\pi }^2\tau }{\langle \varphi ,e_n\rangle }_{L^2}{e}_n\left(z\right),\end{array}$$

where $e_n\left(z\right)=\sqrt{2}sin\left(n\pi z\right)$, for $n=1,2,...$ and $\Vert T\left(\tau \right)\Vert \le e^{-{\pi }^2\tau }$, for all $\tau \ge 0$.

• Clearly, conditions (A0)–(A2) are satisfied. We choose the functions ${\sigma }_i$, such that $L_i$, $i=1,2,3$ are such that $\rho <1$. Thus, by Theorem 8, we can conclude the existence of a unique Stepanov-like pseudo S-asymptotically $(2\pi ,c)$-periodic mild solution of (4).

5. Conclusions

In this paper, we have successfully established the existence and uniqueness of Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic solutions for a class of stochastic integro-differential equations. By extending the notions of $(\omega ,c)$-periodicity and S-asymptotically periodic solutions to stochastic settings, we provided a new framework for understanding the long-term behavior of systems affected by random perturbations. A detailed analysis of the results reveals the critical role of operator kernels and decay properties in ensuring well-posedness. The results demonstrate how $L^1$- and $L^2$-convolution kernels, combined with specific decay rates and boundedness conditions, govern the stability and asymptotic behavior of solutions. This contributes to the understanding of the interplay between deterministic and stochastic components in such systems.

The results obtained here significantly generalize classical periodicity concepts and offer valuable insights into the dynamics of stochastic evolution equations. The original contributions of this work are twofold. First, we extend the theoretical boundaries of Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic solutions to include stochastic systems, providing a novel approach to studying their long-term behavior. Second, we establish new stability criteria and uniqueness conditions that generalize and unify existing results in the literature, bridging gaps in the theory of stochastic periodic systems. The major outputs of this study include the following:

1.

A set of sufficient conditions for the existence and uniqueness of mild solutions under the extended periodicity framework.

2.

A demonstration of how decay and Lipschitz parameters interact to control the stability of solutions.

3.

An example that illustrates the applicability of the theoretical findings in real-world stochastic systems.

For future research, several directions can be explored. One possibility is to study the stability properties of these solutions under different types of stochastic perturbations, such as Lévy processes or Poisson noise, to broaden the applicability of the theory to more general systems. Another direction is to investigate the impact of memory effects by incorporating fractional derivatives into the stochastic integro-differential equations. Additionally, applying the theoretical results to concrete physical, biological and financial models could provide practical insights and highlight new challenges. Further refinement of the conditions under which Stepanov-like pseudo S-asymptotically $(\omega ,c)$-periodic solutions exist could also lead to a deeper understanding of the interplay between periodicity, randomness and system stability.