A Study on Square-Mean S-Asymptotically Bloch Type Periodic Solutions for Some Stochastic Evolution Systems with Piecewise Constant Argument

Abstract

This work is mainly focused on square-mean S-asymptotically Bloch type periodicity and its applications. The main aim of the paper is to introduce the definition of square-mean S-asymptotically Bloch type periodic processes with values in complex Hilbert spaces and systematically analyze some qualitative properties of this type of processes. These properties, combined with the inequality technique, evolution operator theory, fixed-point theory, and stochastic analysis approach, allow us to establish conditions for the existence and uniqueness of square-mean S-asymptotically Bloch type periodicity of bounded mild solutions for a class of stochastic evolution equations with infinite delay and piecewise constant argument. In the end, examples are given to illustrate the feasibility of our results.

1. Introduction

As it is known, nonlinear differential equations with piecewise constant argument (in short, DEPCA) have properties of both nonlinear differential equations and nonlinear difference equations. The study of such equations has been initially described in [1,2] in the 1980s, and the importance of such equations is justified by their various applications in mathematical models appearing in disease dynamics, biomedical models, physical phenomena, and discretization problems. They also have invaluable importance in mathematical economy. As a direct result of this fact, there has been a lot of interest in the investigation of the various properties that are associated with this class of equations [3,4,5,6]. Note that in practice, random or stochastic factors like accidental events or the effect noise are unavoidable and may greatly impact a system. Thus, many scholars have studied stochastic differential equations involving piecewise constant argument (SDEPCAs). For the well-posedness, mean-square stability, almost sure stability, and the numerical approximations of such equations, we refer to [7,8,9] and the references therein.

On the one hand, due to the flaws of strict periodicity in describing certain mathematical models, many authors have introduced other definitions of generalized periodicity, such as almost periodic, asymptotically almost periodic, pseudo-almost periodic, asymptotically pseudo-almost periodic, etc. For more details on these subjects, see the articles [10,11,12,13] and the references therein. We note that some of these notions generalize others. Especially, in [11], the authors proved that any asymptotic $\omega$-periodic function is not S-asymptotically $\omega$-periodic. Moreover, the concept of S-asymptotically $\omega$-periodicity introduced recently in [10,11] has a greater advantage of allowing us to take into account certain distortions of the phenomena that some of periodic extensions do not take in account.

On the other hand, the theory of Bloch-periodic phenomena has its origin in the publication [14] by F. Bloch, while working on the conductivity of crystalline solids. Recently, the concept of Bloch type periodic functions was formally studied by N’guérékata and Hasler in [15], and thus, the classical $\omega$-periodicity and $\omega$-anti-periodicity become particular cases of this notion. To characterize the influence of perturbations on Bloch-periodic functions, some quasi-Bloch periodicity concepts are presented in some publications. For example, asymptotically Bloch-periodic solutions of various deterministic differential systems without piecewise constant argument with their applications have been investigated in [15,16], while notions of (pseudo-) S-asymptotically Bloch-periodic functions with applications have been studied in [17,18].

The previous quasi-Bloch-periodic functions actually extend the notions of asymptotically $\omega$-periodic and (pseudo-) S-asymptotically $\omega$-periodic functions in a natural fashion. Therefore, it is a natural and meaningful work to extend S-asymptotically $\omega$-periodicity from the perspective of Bloch type periodicity. However, it is not so easily to compare the class of S-asymptotically Bloch-periodic function with the class of almost periodic functions as an S-asymptotically $\omega$-periodic function (which is a special case of the S-asymptotically Bloch-periodic function) is not necessarily uniformly continuous. Additionally, thanks to [12], one can derive that the assertion that an S-asymptotically Bloch-periodic function is asymptotically $\omega$-periodic is false.

Before proceeding further, it should be noted that some related works concerning S-asymptotically $\omega$-periodic solutions for DEPCAs in Banach spaces can be founded in [6,19]. The paper [20] deals with SDEPCAs, and sufficient conditions for the existence of the square-mean S-asymptotically $\omega$-periodic solutions are derived where $\omega$ is an integer. To the best of the authors’ knowledge, the study of (quasi-) Bloch-periodic solutions for SDEPCAs can rarely be found in the literature. It should be mentioned that in the paper [16], asymptotic Bloch periodicity of solutions for certain classes of abstract fractional nonlinear DEPCAs are considered whereas pseudo-asymptotically Bloch-periodic solution for equation models with piecewise constant argument are studied in [21]. Due to the great importance and increasing interest in extending certain classical results to stochastic cases, the notions of (weighted) pseudo-S-asymptotically Bloch type periodic processes in the square-mean sense S-asymptotically Bloch type periodic processes in the square-mean sense have been introduced and formally studied in [22,23]. These generalizations are not just mathematical problems but are caused by numerous physical phenomena.

Motivated by the above considerations and recent developments on stochastic Bloch type periodicity, the first objective of our work is to introduce a new class of stochastic processes, named square-mean S-asymptotically Bloch type periodic stochastic process, and to prove some results on the completeness, composition, and convolution of such stochastic processes. Secondly, we are also looking to make more contributions to the literature by considering certain applications of our results to the following class of abstract delayed stochastic evolution equations involving piecewise constant argument

$$\left\{\begin{array}{cc}d\mathrm{q}\left(t\right)\hfill & =\left(\mathbb{A}\left(t\right)\mathrm{q}\left(t\right)+{\mathbb{A}}_0\mathrm{q}\left(\lfloor t\rfloor \right)+\mathrm{f}(t,{\mathrm{q}}_t)\right)dt+\mathrm{g}(t,{\mathrm{q}}_t)d\mathrm{W}\left(t\right),t>0,\hfill \\ {\mathrm{q}}_0\hfill & =\varphi \in \mathrm{\Xi },\hfill \end{array}\right.$$

(1)

where the state $\mathrm{q}(·)$ takes values in a complex separable Hilbert $\mathcal{H}$; ${\left\{\mathbb{A}\left(t\right)\right\}}_{t\in \mathbb{R}}$ is a family of densely defined closed linear operators satisfying Acquistapace–Terreni conditions. The history function ${\mathrm{q}}_t:(-\infty ,0]\to \mathcal{H}$ given by ${\mathrm{q}}_t\left(\varsigma \right)=\mathrm{q}(t+\varsigma )$ for $\varsigma \le 0$ belongs to the phase space $\mathrm{\Xi }$; ${\mathbb{A}}_0$ is a bounded linear operator, $\lfloor ·\rfloor$ is the largest integer function, $\mathrm{f}$ and $\mathrm{g}$ are given functions to be specified later (see Section 4), and $\mathrm{W}\left(t\right)$ is a one-sided and standard one-dimensional Brownian motion on a separable real Hilbert space $\mathbb{K}$. The initial datum $\varphi$ is an ${\mathcal{F}}_0$-adapted process, $\mathrm{\Xi }$-valued random variable independent of the Wiener process W with finite second moments. Throughout the paper, we use the phase space $\left(\mathrm{\Xi },\parallel · {\parallel }_{\mathrm{\Xi }}\right)$ in the sense given by Hale and Kato [24] and (axiomatically) defined in [25], introduced in Section 2. It is noticed that the interest in this type of non-autonomous differential equations lies in the fact that a system subjected to external inputs can include periodic ones. Examples are included in the Floquet theory, which is used to study the stability of linear periodic systems in continuous time. When ${\mathbb{A}}_0\equiv \mathrm{g}(t,{\mathrm{q}}_t)\equiv 0$ and $\mathrm{f}(t,{\mathrm{q}}_t)=\mathrm{f}(t,\mathrm{q}\left(t\right))$, some important regularity results (such as asymptotic periodicity, almost periodicity, almost automorphy, $(\omega ,c)$-periodicity, etc.) for mild solutions to problem (1) have been examined in [26,27,28,29] and the references therein. However, for the piecewise constant argument case and with delayed effects, i.e., Equation (1), the study of the asymptotic behavior of the solution is rare; particularly for the square-mean S-asymptotically Bloch periodicity of Equation (1), it is an untreated topic, and this is the main motivation of this paper. We will make use of the Acquistapace–Terreni conditions for $\mathbb{A}\left(t\right)$ and the exponential dichotomy of the corresponding homogeneous system and global and local Lipschitz conditions on the drift and diffusion terms (see Theorem 4 and 5), we shall show some existence results for problem (1) as applications of square-mean S-asymptotically Bloch-periodic processes introduced in Section 3.

In this work, at first, we will discuss some basic concepts of stochastic analysis, fading memory spaces, and the formulation of the mild solution. In the next section, square-mean S-asymptotically Bloch type periodic processes are discussed. The existence results for problem (1) are given in Section 4. In the last section, we provide an example illustrating the feasibility of the theory discussed in this paper.

2. Preliminaries

This section gives the background material and basic definitions. It also defines the notation. Assume that $\mathcal{H}$ and $\mathbb{K}$ are two complex separable Hilbert spaces. For convenience, the same notations $\parallel ·\parallel$ and $(·,·)$ are applied to denote the norms and the inner products in $\mathcal{H}$ and $\mathbb{K}$. We denote using $\mathcal{L}(\mathbb{K},\mathcal{H})$ the Banach space of all bounded linear operators from $\mathbb{K}$ to $\mathcal{H}$ endowed with the topology defined by the operator norm. We suppose that $(\mathrm{\Omega },\mathcal{F},P)$ represents a probability space, and ${\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})$ stands for the collection of all strongly measurable, square-integrable $\mathcal{H}$-valued random variables, which is a Banach space endowed with the norm ${\parallel \mathrm{q}(·)\parallel }_{L^2}={(\mathsf{E}\parallel \mathrm{q}(·,\omega )\parallel }^2{)}^{1/2},$ where $\mathsf{E}(·)$ is the expectation defined by ${\displaystyle \mathsf{E}\left(\mathrm{q}\right)={\int }_{\mathrm{\Omega }}\mathrm{q}\left(\omega \right)d\mathbb{P}}$.

2.1. On Stochastic Process and Fading Memory Space

Definition 1

([30]). A stochastic process $\mathrm{q}:{\mathbb{R}}^{+}\to {\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})$ is said to be

(i) 

Stochastically bounded if there exists a constant $M>0$ such that

$${\displaystyle {\mathsf{E}\parallel \mathrm{q}\left(t\right)\parallel }^2={\int }_{\mathrm{\Omega }}{\parallel \mathrm{q}\left(t\right)\parallel }^{2}d\mathbb{P}<M for all t\in {\mathbb{R}}^{+};}$$

(ii) 

Stochastically continuous if $\underset{t\to \varsigma }{lim}\mathsf{E}{\parallel \mathrm{q}\left(t\right)-\mathrm{q}\left(\varsigma \right)\parallel }^2=0for all \varsigma \in {\mathbb{R}}^{+}.$

We denote using $\mathfrak{BC}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ the Banach space of all bounded and continuous stochastic processes $\mathrm{q}$ from ${\mathbb{R}}^{+}$ into ${\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})$ with the norm ${\parallel \mathrm{q}\parallel }_{\infty }={\left({\displaystyle \underset{\varsigma \in {\mathbb{R}}^{+}}{sup}\mathsf{E}{\parallel \mathrm{q}\left(\varsigma \right)\parallel }^2}\right)}^{1/2}$.

We can now introduce one of the most commonly used phase space $\mathrm{\Xi }$ introduced in [25], which will be a linear space of functions mapping $(-\infty ,0]$ into ${\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})$ endowed with a seminorm ${\parallel ·\parallel }_{\mathrm{\Xi }}$ and verifying the following axioms:

(A)

If $\mathrm{q}:(-\infty ,b+a)\to {\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H}),a>0,b\in \mathbb{R}$, is continuous on $[b,b+a)$ and ${\mathrm{q}}_b\in \mathrm{\Xi }$, then for every $\varsigma \in [b,b+a)$, the following holds:

(a)

${\mathrm{q}}_{\varsigma }$ is in $\mathrm{\Xi }$;

(b)

${\mathsf{E}\parallel \mathrm{q}\left(\varsigma \right)\parallel }^2\le H{\parallel {\mathrm{q}}_{\varsigma }\parallel }_{\mathrm{\Xi }}^2$;

(c)

$\parallel {\mathrm{q}}_{\varsigma }{\parallel }_{\mathrm{\Xi }}\le K(\varsigma -b){sup}_{b\le s\le \varsigma }{\parallel \mathrm{q}\left(\varsigma \right)\parallel }_{L^2}+M(\varsigma -b){\parallel {\mathrm{q}}_b\parallel }_{\mathrm{\Xi }}$, where $H>0$ is a constant, K; $M:[0,\infty )\to [0,\infty ),K$ is continuous, M is locally bounded, and $H,K,M$ are independent of $\mathrm{q}(·).$

(A1)

For the function $\mathrm{q}$ in $\left(A\right)$, the function $\varsigma \to {\mathrm{q}}_{\varsigma }$ is continuous from $[b,b+a)$ into $\mathrm{\Xi }.$

(B)

  The space $\mathrm{\Xi }$ is complete.

(C2)

If ${\left({\psi }^n\right)}_{n\in \mathbb{N}}$ is a uniformly bounded sequence of continuous functions with compact support and converges to $\psi$ compactly on $(\infty ,0]$, then $\psi \in \mathrm{\Xi }$ and $\parallel {\psi }^n{-\psi \parallel }_{\mathrm{\Xi }}$ as $n\to \infty$.

Definition 2

([25] Definition 2.1). The phase space Ξ is called a fading memory space if $\parallel S_0{\left(\varsigma \right)\left(\psi \right)\parallel }_{\mathrm{\Xi }}\to 0$ as $\varsigma \to \infty$ for every $\psi \in {\mathrm{\Xi }}_0$, where ${\mathrm{\Xi }}_0=\{\psi \in \mathrm{\Xi }|\psi \left(0\right)=0\}$ and for $\varsigma \ge 0$, the operator $S_0\left(\varsigma \right):{\mathrm{\Xi }}_0\to {\mathrm{\Xi }}_0$ is given by

$$\left[S_0\left(\varsigma \right)\left(\psi \right)\right]\left(s\right)=\left\{\begin{array}{c}0,-\varsigma \le s\le 0\hfill \\ \psi (\varsigma +s),\infty <s\le -\varsigma .\hfill \end{array}\right.$$

Remark 1.

From ([25] Proposition 7.1.5), if Ξ is a fading memory space, then K, M (defined in condition (A)–(c)) are bounded functions.

Example 1.

Let z be a non-negative continuous map defined on $(-\infty ,0]$ and satisfying, in the terminology of [25], the following conditions:

(g-5) 

${\displaystyle {\int }_u^{0}z\left(\theta \right)d\theta <\infty }$ for every $u\in (\infty ,0)$;

(g-6) 

$z(u+\theta )\le G\left(u\right)z\left(\theta \right)$, for all $u\le 0$ and $\theta \in (\infty ,0)N_u$, for a set $N_u\subset (\infty ,0)$ with Lebesgue measure zero and for a non-negative function G which is locally bounded on $(\infty ,0]$;

(g-7) 

${\displaystyle {\int }_{\infty }^{0}z\left(\theta \right)d\theta <\infty }$.

For λ, the positive Borel measure on $(-\infty ,0]$ defined by ${\displaystyle \lambda \left(J\right)={\int }_{J}z\left(\varsigma \right)d\varsigma }$ for $J\in \mathfrak{B}$ where $\mathfrak{B}$ is the family of Borel sets of $(-\infty ,0]$; the space ${\mathcal{D}}_z$ defined by the set

$$\left\{\begin{array}{cc}\psi :(-\infty ,0]\to {\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H}):for every c>0,\mathbb{E}{\parallel \psi \left(\gamma \right)\parallel }^2 is a bounded and\hfill & \\ {\displaystyle integrable with respect to \lambda on[-c,0]with{\left({\int }_{-\infty }^{0}z\left(\varsigma \right)\underset{\varsigma \le d\le 0}{sup}\mathbb{E}{\parallel \psi \left(d\right)\parallel }^{2}d\varsigma \right)}^{\frac{1}{2}}<\infty }\hfill \end{array}\right\}$$

is a fading memory space (see ([25] Example 7.1.8)). By taking $z\left(\varsigma \right)=e^{\theta \varsigma }$ for $\theta \in \mathbb{R}$, it is known that $({\mathcal{D}}_z,{\parallel .\parallel }_{{\mathcal{D}}_z})$ is a Banach space, where

$${\displaystyle {\parallel \psi \parallel }_{{\mathcal{D}}_z}={\left({\int }_{-\infty }^{0}z\left(\varsigma \right)\underset{\varsigma \le d\le 0}{sup}\mathbb{E}{\parallel \psi \left(d\right)\parallel }^{2}d\varsigma \right)}^{\frac{1}{2}},\psi \in {\mathcal{D}}_z.}$$

Remark 2.

From ([25] page 188, Proposition 1.1), we derive that if the condition $\left(C2\right)$ holds, then there exists a constant $L>0$ such that

$$\begin{array}{c}\hfill {\parallel \mathrm{q}\parallel }_{\mathrm{\Xi }}^2\le L{\parallel \mathrm{q}\parallel }_{\infty }^{2}forany\mathrm{q}\in \mathfrak{BC}\left((-\infty ,0],{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right).\end{array}$$

2.2. On Evolution Families

From this point onward, we suppose that $(\mathfrak{X},\parallel ·\parallel )$ is a Banach space and denote using $\mathcal{L}\left(\mathfrak{X}\right)$ the space of all bounded linear operators from $\mathfrak{X}$ to $\mathfrak{X}$. We recall the following definition introduced by Acquistapace and Terreni in [31].

Definition 3

([31]). A family of closed linear operators $\mathbb{A}\left(t\right)$ for $t\in \mathbb{R}$ on $\mathfrak{X}$ with domain $\mathrm{D}\left(\mathbb{A}\right(t\left)\right)$ (possibly not densely defined) satisfy the so-called Acquistapace–Terreni condition; if there exist constants $\beta \ge 0$, $\theta \in (\pi /2,\pi )$, $C_1,C_2>0$, $\mu ,\nu \in (0,1]$ with $\mu +\nu >1$ such that

$$S_{\theta }\cup \left\{0\right\}\subset \rho (\mathbb{A}\left(t\right)-\beta )\ni \lambda ,\parallel R(\lambda ,\mathbb{A}\left(t\right)-\beta )\parallel \le {\displaystyle \frac{C_1}{1+\left|\lambda \right|}}\mathit{for}\mathit{all}t\in \mathbb{R}$$

(2)

and

$$\parallel (\mathbb{A}\left(t\right)-\beta )R(\lambda ,\mathbb{A}\left(t\right)-\beta )[R(\beta ,\mathbb{A}\left(t\right))-R(\beta ,\mathbb{A}\left(\varsigma \right))]\parallel \le C_2{\displaystyle \frac{{|t-\varsigma |}^{\mu }}{{\left|\lambda \right|}^{\nu }}}$$

(3)

for $t,\varsigma \in \mathbb{R}$, where $\lambda \in S_{\theta }:=\{\lambda \in \mathbb{C}\setminus \left\{0\right\}:|arg\lambda |\le \theta \}$, $R(\lambda ,\mathbb{A}\left(t\right)-\beta ):={\left(\lambda I-\mathbb{A}\left(t\right)+\beta \right)}^{-1}$ and $\rho \left(\mathbb{A}\right(t)-\beta )$ represent the resolvent set of $\mathbb{A}\left(t\right)-\beta$.

When $\mathbb{A}\left(t\right)$ has a constant domain $\mathrm{D}=\mathrm{D}\left(\mathbb{A}\right(t\left)\right)$, then condition (3) can be replaced with the following one: there exist constants $C_3>0$ and $0<\ell \le 1$ such that

$$\begin{array}{c}\hfill \parallel (\mathbb{A}\left(t\right)-\mathbb{A}\left(\varsigma \right))R(\beta ,\mathbb{A}\left(r\right))\parallel \le C_3{\parallel t-\varsigma \parallel }^{\ell }\mathrm{for} \mathrm{all}t,\varsigma ,r\in \mathbb{R}.\end{array}$$

(4)

More details can be found in [32].

Theorem 1

([30]). Let $\mathbb{A}\left(t\right)$ be a family of closed linear operators that satisfies Acquistapace–Terreni conditions. Then, there exists a unique evolution family

$$\mathcal{Q}=\left\{\mathcal{Q}\right(t,\varsigma ):t,\varsigma \in \mathbb{R}suchthatt>\varsigma \}$$

on $\mathfrak{X}$ such that

(a) 

$\mathcal{Q}(t,\varsigma )\mathfrak{X}\subseteq D\left(\mathbb{A}\right(t\left)\right)$ for all $t,\varsigma \in \mathbb{R}$ with $t>\varsigma$;

(b) 

$\mathcal{Q}(t,\varsigma )\mathcal{Q}(\varsigma ,r)=\mathcal{Q}(t,r)$ for $t,\varsigma \in \mathbb{R}$ such that $t\ge \varsigma \ge r$;

(c) 

$\mathcal{Q}(t,t)=I$ for $t\in \mathbb{R}$ where I is the identity operator of $\mathfrak{X}$;

(d) 

$(t,\varsigma )\to \mathcal{Q}(t,\varsigma )\in \mathcal{L}\left(\mathfrak{X}\right)$ is continuous for $t>\varsigma$;

(e) 

$\mathcal{Q}(·,\varsigma )\in C^1((\varsigma ,\infty ),\mathcal{L}\left(\mathfrak{X}\right))$, ${\displaystyle \frac{\partial \mathcal{Q}}{\partial t}}(t,\varsigma )=\mathbb{A}\left(t\right)\mathcal{Q}(t,\varsigma )$ and

$${\parallel \mathbb{A}\left(t\right)}^k{\mathcal{Q}(t,\varsigma )\parallel \le K(t-\varsigma )}^{-k}$$

for $0<t-\varsigma \le 1$ and $k=0,1$.

Definition 4.

An evolution family $\mathcal{Q}=\left\{\mathcal{Q}\right(t,\varsigma ):t,\varsigma \in \mathbb{R}suchthatt\ge \varsigma \}$ is said to have an exponential dichotomy (or is hyperbolic) if there are projections $P\left(t\right)$ ($t\in \mathbb{R}$) that are uniformly bounded (i.e, there exist $M\le 1$ such that $\parallel P\left(t\right)\parallel \le M$ for all $t\in \mathbb{R}$), strongly continuous in t, and there are constants $\delta >0$ and $N\ge 1$ such that

(f) 

$\mathcal{Q}(t,\varsigma )P\left(\varsigma \right)=P\left(t\right)\mathcal{Q}(t,\varsigma )$;

(g) 

the restriction ${\mathcal{Q}}_V(t,\varsigma ):V\left(\varsigma \right)\mathfrak{X}\to V\left(t\right)\mathfrak{X}$ of $\mathcal{Q}(t,\varsigma )$ is invertible;

(h) 

$\parallel \mathcal{Q}(t,\varsigma )P\left(\varsigma \right)\parallel \le N{e}^{-\delta (t-\varsigma )}$ and $\parallel {\tilde{\mathcal{Q}}}_V(\varsigma ,t)V\left(t\right)\parallel \le N{e}^{-\delta (t-\varsigma )}$ for $t\ge \varsigma$ and $t,\varsigma \in \mathbb{R}$

where $V(·)=I-P(·)$ and ${\tilde{\mathcal{Q}}}_V(\varsigma ,t):={\mathcal{Q}}_V{(t,\varsigma )}^{-1}$.

Note that if $\mathcal{Q}(t,\varsigma )$ is exponentially stable, then $\mathcal{Q}$ is hyperbolic with $P\left(t\right)=I$. More details about the evolutions families can be found in [33].

3. Square-Mean S-Asymptotically Bloch Type Periodic Process

In this part, we introduce the notion and some properties of S-asymptotically Bloch type periodic stochastic processes.

Definition 5.

A stochastic process $\mathrm{q}\in \mathfrak{BC}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ is said to be mean-square S-asymptotically Bloch-periodic (or $(\varpi ,\mathbb{k})$-periodic) if, for a given $\varpi \in {\mathbb{R}}^{+},\mathbb{k}\in \mathbb{R}$,

$$\underset{t\to +\infty }{lim}\mathbb{E}{\parallel \mathrm{q}(t+\varpi )-e^{i\mathbb{k}\varpi }\mathrm{q}\left(t\right)\parallel }^2=0.$$

We denote by ${\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ the collection of all the square-mean S-asymptotically $(\varpi ,\mathbb{k})$-periodic stochastic processes and $\mathfrak{BC}({\mathbb{R}}^{+}\times L^2(\mathrm{\Omega },\mathcal{H}),L^2(\mathrm{\Omega },\mathcal{H}))$ the set

$$\left\{h(·,\mathrm{q})\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)|\mathrm{for}\mathrm{any}\mathrm{q}\in {\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right\}.$$

Lemma 1.

Let ${\mathrm{q}}_1,{\mathrm{q}}_2\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ and $\kappa \in \mathbb{C}$. The following results hold:

(i) 

${\mathrm{q}}_1+\kappa {\mathrm{q}}_2\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$.

(ii) 

${\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ is a Banach space endowed with the norm ${\parallel ·\parallel }_{\infty }$.

Proof. 

(i) Using Definition 5, we have

$$\begin{array}{cc}\hfill & \mathsf{E}\parallel ({\mathrm{q}}_1+\kappa {\mathrm{q}}_2)(t+\varpi )-e^{i\mathbb{k}\varpi }({\mathrm{q}}_1+\kappa {\mathrm{q}}_2)\left(t\right){\parallel }^2\hfill \\ \hfill & \le 2\mathsf{E}\parallel {\mathrm{q}}_1(t+\varpi )-e^{i\mathbb{k}\varpi }{\mathrm{q}}_1{\left(t\right)\parallel }^2+2{\left|\kappa \right|}^2{\parallel {\mathrm{q}}_2(t+\varpi )-e^{i\mathbb{k}\varpi }{\mathrm{q}}_2\left(t\right)\parallel }^2\hfill \\ \hfill & \to 0\mathrm{as}t\to \infty .\hfill \end{array}$$

Thus, ${\mathrm{q}}_1+\kappa {\mathrm{q}}_2\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$.

(ii) 

Let ${\left\{{\mathrm{q}}_n\right\}}_n\subseteq {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ such that ${\mathrm{q}}_n\to \mathrm{q}$ as $n\to \infty$. Then, for any $ϵ>0$, there exists a constants $N>0$ and $R>0$ such that:

for $n\ge N$, we have $\parallel {\mathrm{q}}_n{-\mathrm{q}\parallel }_{\infty }\le \left(\sqrt{ϵ}/3\right)$ and

for $t\ge A$, we have $\mathsf{E}\parallel {\mathrm{q}}_n(t+\varpi )-e^{i\mathbb{k}\varpi }{\mathrm{q}}_n{\left(t\right)\parallel }^2\le (ϵ/9)$. We obtain

$$\begin{array}{cc}\hfill \mathsf{E}\parallel \mathrm{q}& (t+\varpi )-e^{i\mathbb{k}\varpi }{\mathrm{q}\left(t\right)\parallel }^2\hfill \\ \hfill =& \mathsf{E}\parallel \mathrm{q}(t+\varpi )-{\mathrm{q}}_n(t+\varpi )+{\mathrm{q}}_n(t+\varpi )-e^{i\mathbb{k}\varpi }{\mathrm{q}}_n\left(t\right)+e^{i\mathbb{k}\varpi }{\mathrm{q}}_n\left(t\right)-e^{i\mathbb{k}\varpi }{\mathrm{q}\left(t\right)\parallel }^2\hfill \\ \hfill \le & \mathsf{E}\parallel \mathrm{q}(t+\varpi )-{\mathrm{q}}_n{(t+\varpi )\parallel }^2+\mathsf{E}{\parallel {\mathrm{q}}_n(t+\varpi )-e^{i\mathbb{k}\varpi }{\mathrm{q}}_n\left(t\right)\parallel }^2\hfill \\ \hfill & +\mathsf{E}\parallel e^{i\mathbb{k}\varpi }{\mathrm{q}}_n\left(t\right)-e^{i\mathbb{k}\varpi }{\mathrm{q}\left(t\right)\parallel }^2\hfill \\ \hfill \le & 3\parallel {\mathrm{q}}_n{-\mathrm{q}\parallel }_{\infty }^2+3(ϵ/9)+3{\parallel {\mathrm{q}}_n-\mathrm{q}\parallel }_{\infty }^2\hfill \\ \hfill \le & (ϵ/3)+(ϵ/3)+(ϵ/3)=ϵ.\hfill \end{array}$$

This implies that the space ${\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ is a closed subspace of $\mathfrak{BC}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$, so it is a Banach space equipped with the sup-norm.

□

We state the following useful lemma.

Lemma 2.

Let Ξ be a fading memory space and $\psi :\mathbb{R}\to {\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})$ be a process with ${\psi }_0\in \mathrm{\Xi }$. If ${\psi |}_{[0,\infty )}\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$, then $t\to {\psi }_t$ is S-asymptotically Bloch-periodic such that ${\psi }_t\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},\mathrm{\Xi }\right)$.

Proof. 

Because K, M, and ${\psi |}_{[0,\infty )}$ are bounded, it follows from (A) to (c) that $t\to {\psi }_t$ is bounded on $[0,\infty )$. Next, we define for $t\in J$

$$\phi \left(t\right)=\psi (t+\varpi )-e^{i\mathbb{k}\varpi }\psi \left(t\right)={\psi }_{\varpi }\left(t\right)-e^{i\mathbb{k}\varpi }\psi \left(t\right).$$

Then, we have ${\phi }_0={\psi }_{\varpi }-{\psi }_0\in \mathrm{\Xi }$. Because ${\psi |}_{[0,\infty )}\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$, then $\phi :{\mathbb{R}}^{+}\to {\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})$ is continuous and $\phi \left(t\right)\to 0$ in ${\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})$. Therefore, it follows from ([25] Proposition 7.1.3) that

$$\parallel {\phi }_t{\parallel }_{\mathrm{\Xi }}={\parallel {\psi }_{t+\varpi }-e^{i\mathbb{k}\varpi }{\psi }_t\parallel }_{\mathrm{\Xi }}\to 0\mathrm{as}t\to \infty .$$

□

Before proceeding further, we assume the following assumptions:

(A0) 

The family of operators $\mathbb{A}\left(t\right)$ on ${\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})$ satisfies the Acquistapace–Terreni condition and the evolution family $\mathcal{Q}=\left\{\mathcal{Q}(t,\varsigma ),t\ge \varsigma \right\}$ associated with $\mathbb{A}\left(t\right)$ is exponentially stable such that there exists the constant M, $\delta >0$ such that

$$\parallel \mathcal{Q}(t,\varsigma )\parallel \le M{e}^{-\delta (t-\varsigma )}\mathrm{for}\mathrm{all}t\ge \varsigma .$$

That implies $\mathcal{Q}(t,\varsigma )$ is hyperbolic, where $P\left(t\right)=I$.

Let $\mathrm{\Xi }$ be a fading memory space and $h\in \mathfrak{BC}({\mathbb{R}}^{+}\times \mathrm{\Xi },{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H}))$ and consider the following assumptions:

(H0) 

$\underset{t\to +\infty }{lim}\mathbb{E}{\parallel h(t+\varpi ,x)-e^{i\mathbb{k}\varpi }h(t,e^{-i\mathbb{k}\varpi }x)\parallel }^2=0$ for x in any uniformly bounded subset of $\mathrm{\Xi }$.

(H1) 

There exists a number $L>0$ such that for any $u,v\in \mathrm{\Xi }$,

$${\mathsf{E}\parallel h(t,u)-h(t,v)\parallel }^2\le L.{\parallel u-v\parallel }_{\mathrm{\Xi }}^2,$$

uniformly for all $t\in {\mathbb{R}}^{+}$.

3.1. Composition Results

The following compositions results are highly significant in proving the existence of mild solutions of many evolution systems.

Theorem 2.

If $h\in \mathfrak{BC}({\mathbb{R}}^{+}\times \mathrm{\Xi },{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H}))$ satisfies $\left(\mathbf{H}\mathbf{0}\right)$–$\left(\mathbf{H}\mathbf{1}\right)$, then we have $h(·,\mathrm{q}(·))\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ for every $\mathrm{q}\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},\mathrm{\Xi }\right)$.

Proof. 

Because $\mathrm{q}\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},\mathrm{\Xi }\right)$, then we have ${\displaystyle \underset{t\to +\infty }{lim}{\parallel \mathrm{q}(t+\varpi )-e^{i\mathbb{k}\varpi }\mathrm{q}\left(t\right)\parallel }_{\mathrm{\Xi }}=0.}$ We have

$$\begin{array}{cc}\hfill \mathsf{E}\parallel h(t+\varpi ,\mathrm{q}(t+\varpi \left)\right)& -e^{i\mathbb{k}\varpi }{h(t,\mathrm{q}\left(t\right))\parallel }^2\hfill \\ & \le 2\mathsf{E}\parallel h(t+\varpi ,\mathrm{q}(t+\varpi ))-e^{i\mathbb{k}\varpi }h(t,e^{-i\mathbb{k}\varpi }\mathrm{q}(t+\varpi )){\parallel }^2\hfill \\ & +2\mathsf{E}\parallel e^{i\mathbb{k}\varpi }h(t,e^{-i\mathbb{k}\varpi }\mathrm{q}(t+\varpi ))-e^{i\mathbb{k}\varpi }h(t,\mathrm{q}\left(t\right)){\parallel }^2\hfill \\ & \le 2\mathsf{E}\parallel h(t+\varpi ,\mathrm{q}(t+\varpi )-e^{i\mathbb{k}\varpi }h(t,e^{-i\mathbb{k}\varpi }\mathrm{q}(t+\varpi )){\parallel }^2\hfill \\ & +2L\parallel e^{-i\mathbb{k}\varpi }\mathrm{q}(t+\varpi ))-\mathrm{q}\left(t\right){\parallel }_{\mathrm{\Xi }}^2\hfill \\ & \to 0 \mathrm{as} t\to +\infty .\hfill \end{array}$$

i.e., $h(·,\mathrm{q}(·))\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. □

Theorem 3.

If $h\in \mathfrak{BC}({\mathbb{R}}^{+}\times \mathrm{\Xi },{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H}))$ satisfies $\left(\mathbf{H}\mathbf{0}\right)$ and the following condition:

$\left(\mathbf{H}\mathbf{2}\right)$: for any $ϵ>0$ and any bounded subset $B\subset \mathrm{\Xi }$, there exist constants $T_{ϵ,B}$ and ${\delta }_{ϵ,B}>0$ such that

$$\mathsf{E}\parallel h(t,{\mathrm{q}}_1)-h(t,{\mathrm{q}}_2){\parallel }^2\le ϵ$$

for all ${\mathrm{q}}_1,{\mathrm{q}}_2\in B$ with $\parallel {\mathrm{q}}_1-{\mathrm{q}}_2{\parallel }_{\mathrm{\Xi }}^2\le {\delta }_{ϵ,B}$ and $t\ge T_{ϵ,B}$, then we have $h(·,\mathrm{q}(·))\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ whenever $\mathrm{q}\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},\mathrm{\Xi }\right)$.

Proof. 

Because $\mathrm{q}\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},\mathrm{\Xi }\right)$, then $B:=\{\mathrm{q}\left(t\right):t\in {\mathbb{R}}^{+}\}\subset \mathrm{\Xi }$ is bounded. Let $ϵ>0$. By assumption $\left(\mathbf{H}\mathbf{0}\right)$ and the fact that $\mathrm{q}\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},\mathrm{\Xi }\right)$, we have that there exists $T_{ϵ}>0$ such that for $t\ge T_{ϵ}$

$$\mathsf{E}\parallel h(t+\varpi ,\mathrm{q}(t+\varpi ))-e^{i\mathbb{k}\varpi }h(t,e^{-i\mathbb{k}\varpi }\mathrm{q}\left(t\right)){\parallel }^2<{\displaystyle \frac{ϵ}{4}}\mathrm{and}{\parallel \mathrm{q}(t+\varpi )-e^{i\mathbb{k}\varpi }\mathrm{q}\left(t\right)\parallel }_{\mathrm{\Xi }}^2\le ϵ.$$

Thanks to condition $\left(\mathbf{H}\mathbf{2}\right)$, we have that there exists ${\delta }_{ϵ,B}:=ϵ$ and $T_{ϵ,B}:=T_{ϵ}$ such that

$$\mathsf{E}\parallel h(t,e^{-i\mathbb{k}\varpi }\mathrm{q}(t+\varpi ))-h(t,\mathrm{q}\left(t\right)){\parallel }^2\le {\displaystyle \frac{ϵ}{4}}$$

whenever $\parallel \mathrm{q}(t+\varpi )-e^{i\mathbb{k}\varpi }{\mathrm{q}\left(t\right)\parallel }_{\mathrm{\Xi }}^2\le ϵ$ and $t\ge T_{ϵ}$. We have

$$\begin{array}{cc}\hfill \mathsf{E}\parallel h(t+\varpi ,\mathrm{q}(t+\varpi \left)\right)& -e^{i\mathbb{k}\varpi }{h(t,\mathrm{q}\left(t\right))\parallel }^2\hfill \\ & \le 2\mathsf{E}\parallel h(t+\varpi ,\mathrm{q}(t+\varpi ))-e^{i\mathbb{k}\varpi }h(t,e^{-i\mathbb{k}\varpi }\mathrm{q}(t+\varpi )){\parallel }^2\hfill \\ & +2\mathsf{E}\parallel e^{i\mathbb{k}\varpi }h(t,e^{-i\mathbb{k}\varpi }\mathrm{q}(t+\varpi ))-e^{i\mathbb{k}\varpi }h(t,\mathrm{q}\left(t\right)){\parallel }^2\hfill \\ & \le 2\mathsf{E}\parallel h(t+\varpi ,\mathrm{q}(t+\varpi ))-e^{i\mathbb{k}\varpi }h(t,e^{-i\mathbb{k}\varpi }\mathrm{q}(t+\varpi )){\parallel }^2\hfill \\ & +2\mathsf{E}\parallel h(t,e^{-i\mathbb{k}\varpi }\mathrm{q}(t+\varpi ))-h(t,\mathrm{q}\left(t\right)){\parallel }^2\hfill \\ & <ϵ\mathrm{for}t\ge T_{ϵ}.\hfill \end{array}$$

Thus,

$$\underset{t\to \infty }{lim}\mathsf{E}{\parallel h(t+\varpi ,\mathrm{q}(t+\varpi ))-e^{i\mathbb{k}\varpi }h(t,\mathrm{q}\left(t\right))\parallel }^2=0.$$

The proof is completed. □

Corollary 1.

If $h\in \mathfrak{BC}({\mathbb{R}}^{+}\times \mathrm{\Xi },{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H}))$ satisfies $\left(\mathbf{H}\mathbf{0}\right)$ and the following condition:

$\left(\mathbf{H}\mathbf{3}\right)$: there exist a function $L_{\mathrm{g}}:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that for each $r>0$ and for all ${\mathrm{q}}_1,{\mathrm{q}}_2\in {\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})$ with $\mathsf{E}\parallel {\mathrm{q}}_1{\parallel }_{\mathrm{\Xi }}^2\le r$, $\mathsf{E}\parallel {\mathrm{q}}_2{\parallel }_{\mathrm{\Xi }}^2\le r$,

$$\mathsf{E}\parallel h(t,{\mathrm{q}}_1)-h(t,{\mathrm{q}}_2){\parallel }_{\mathcal{L}(\mathbb{K},\mathcal{H})}^2\le L_{\mathrm{g}}\left(r\right)\parallel {\mathrm{q}}_1-{\mathrm{q}}_2{\parallel }_{\mathrm{\Xi }}^2,$$

uniformly for all $t\in {\mathbb{R}}_{+}$, then for each $\mathrm{q}\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},\mathrm{\Xi }\right)$, we have $h(·,\mathrm{q}(·))\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$.

Proof. 

It is easy to show that $\left(\mathbf{H}\mathbf{2}\right)$ in Theorem 3 is satisfied if $\left(\mathbf{H}\mathbf{3}\right)$ holds. So, the conclusion is a direct consequence of Theorem 3. □

3.2. Convolution Product

The convolution product is greatly important for defining the integral and (mild) solutions of many evolution systems.

Lemma 3.

Let $\left(\mathbf{A}\mathbf{0}\right)$ holds. Then, for each $\mathsf{X}\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$, ${\displaystyle \mathrm{\Phi }:t\mapsto \mathrm{\Phi }\left(t\right)={\int }_0^t\mathcal{Q}(t,r){\mathbb{A}}_0\mathrm{X}\left(\left[r\right]\right)dr\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)}$.

Proof. 

It is easy to show that $\mathrm{\Phi }\in \mathfrak{BC}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. Now note that we derive by condition $\mathbf{\left(}\mathbf{A}\mathbf{0}\mathbf{\right)}$ that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{e}^{{\displaystyle \frac{\delta }{2}}(t-r)}\parallel \mathcal{Q}(t+\varpi ,r+\varpi )-\mathcal{Q}(t,r){\parallel }^2=0\mathrm{for}\mathrm{each}r\in \mathbb{R}.\end{array}$$

Now let $p\in (0,{\displaystyle \frac{\delta }{2}})$ and $X\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. Then, for $ϵ>0$, there exists $T_{ϵ}^1>0$, such that

$$\begin{array}{c}\hfill e^{{\displaystyle \frac{\delta }{2}}(t-r)}\parallel \mathcal{Q}(t+\varpi ,r+\varpi )-\mathcal{Q}(t,r){\parallel }^2<({\displaystyle \frac{\delta }{2}}-p)ϵ\mathrm{and}\\ \hfill \mathsf{E}{∥\mathrm{X}(t+\varpi )-e^{i\mathbb{k}\varpi }\mathrm{X}\left(t\right)∥}^2<\delta ϵ\mathrm{for}t>T_{ϵ}^1.\end{array}$$

Let $T_{ϵ}=\left[T_{ϵ}^1\right]+1$ then for $t>T_{ϵ}$, we have $\lfloor t\rfloor >T_{ϵ}$. Therefore

$$\begin{array}{c}\hfill \mathsf{E}{∥\mathrm{X}\left([t+\varpi ]\right)-e^{i\mathbb{k}\varpi }\mathrm{X}\left(\lfloor t\rfloor \right)∥}^2=\mathsf{E}{∥\mathrm{X}(\lfloor t\rfloor +\varpi )-e^{i\mathbb{k}\varpi }\mathrm{X}\left(\lfloor t\rfloor \right)∥}^2<\delta ϵ.\end{array}$$

It is clear that $\mathrm{\Phi }\in \mathfrak{BC}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. For $t>T_{ϵ}$, we have

$$\begin{array}{cc}\hfill \mathsf{E}\parallel & \mathrm{\Phi }(t+\varpi )-e^{i\mathbb{k}\varpi }{\mathrm{\Phi }\left(t\right)\parallel }^2\hfill \\ \hfill =& \mathsf{E}{∥{\int }_0^{t+\varpi }\mathcal{Q}(t+\varpi ,r){\mathbb{A}}_0\mathrm{X}\left(\left[r\right]\right)dr-e^{i\mathbb{k}\varpi }{\int }_0^t\mathcal{Q}(t,r){\mathbb{A}}_0\mathrm{X}\left(\left[r\right]\right)dr∥}^2\hfill \\ \hfill \le & 2\mathsf{E}{∥{\int }_{\omega }^{t+\varpi }\mathcal{Q}(t+\varpi ,r){\mathbb{A}}_0\mathrm{X}\left(\left[r\right]\right)dr-e^{i\mathbb{k}\varpi }{\int }_0^t\mathcal{Q}(t,r){\mathbb{A}}_0\mathrm{X}\left(\left[r\right]\right)dr∥}^2\hfill \\ \hfill & +2\mathsf{E}{∥{\int }_0^{\omega }\mathcal{Q}(t+\varpi ,r){\mathbb{A}}_0\mathrm{X}\left(\left[r\right]\right)dr∥}^2\hfill \\ \hfill \le & 2\mathsf{E}{∥{\int }_0^t\mathcal{Q}(t+\varpi ,r+\varpi ){\mathbb{A}}_0\mathrm{X}\left([r+\varpi ]\right)dr-e^{i\mathbb{k}\varpi }{\int }_0^t\mathcal{Q}(t,r){\mathbb{A}}_0\mathrm{X}\left(\left[r\right]\right)dr∥}^2\hfill \\ \hfill & +2\mathsf{E}{∥{\int }_0^{\omega }\mathcal{Q}(t+\varpi ,r){\mathbb{A}}_0\mathrm{X}\left(\left[r\right]\right)dr∥}^2\hfill \\ \hfill \le & 4\mathsf{E}{∥{\int }_0^t\left(\mathcal{Q}(t+\varpi ,r+\varpi )-\mathcal{Q}(t,r)\right){\mathbb{A}}_0\mathrm{X}\left([r+\varpi ]\right)dr∥}^2\hfill \\ \hfill & +4\mathsf{E}{∥{\int }_0^t\mathcal{Q}(t,r){\mathbb{A}}_0\left(\mathrm{X}\left([r+\varpi ]\right)-e^{i\mathbb{k}\varpi }\mathrm{X}\left(\left[r\right]\right)\right)dr∥}^2+2\mathsf{E}{∥{\int }_0^{\omega }\mathcal{Q}(t+\varpi ,r){\mathbb{A}}_0\mathrm{X}\left(\left[r\right]\right)dr∥}^2\hfill \\ \hfill \le & 4I_1\left(t\right)+4I_2\left(t\right)+2I_3\left(t\right).\hfill \end{array}$$

For $I_1\left(t\right)$, by using the Cauchy–Schwarz inequality, we obtain

$$\begin{array}{cc}\hfill I_1\left(t\right)\le & {\int }_0^t{e}^{-p(t-r)}dr{\int }_0^t{e}^{p(t-r)}\parallel \mathcal{Q}(t+\varpi ,r+\varpi )-\mathcal{Q}(t,r){\parallel }^2\parallel {\mathbb{A}}_0{\parallel }^2\mathsf{E}{\parallel \mathrm{X}\left([r+\varpi ]\right)\parallel }^{2}dr\hfill \\ \hfill \le & {\displaystyle \frac{{\parallel X\parallel }_{\infty }^2}{p}}\left(4M^2{\int }_0^{T_{ϵ}}{e}^{(p-2\delta )(t-r)}dr+{\int }_{T_{ϵ}}^t{e}^{p(t-r)}\parallel \mathcal{Q}(t+\varpi ,r+\varpi )-\mathcal{Q}(t,r){\parallel }^{2}dr\right)\hfill \\ \hfill \le & {\displaystyle \frac{{\parallel X\parallel }_{\infty }^2}{p}}\left(4M^2{\int }_0^{T_{ϵ}}{e}^{(p-2\delta )(t-r)}dr+({\displaystyle \frac{\delta }{2}}-p)ϵ{\int }_{T_{ϵ}}^t{e}^{(p-{\displaystyle \frac{\delta }{2}})(t-r)}dr\right)\hfill \\ \hfill \le & {\displaystyle \frac{{\parallel X\parallel }_{\infty }^2}{p}}\left(4M^2{\int }_0^{T_{ϵ}}{e}^{(p-2\delta )(t-r)}dr+ϵ\right).\hfill \end{array}$$

(5)

Because

$$\begin{array}{c}\hfill {\int }_0^{T_{ϵ}}{e}^{(p-\delta )(t-r)}dr\to 0\mathrm{as}t\to +\infty \mathrm{therefore}\underset{r\to +\infty }{lim}{I}_1\left(t\right)=0.\end{array}$$

Applying a similar idea used for the term $I_1\left(t\right)$, we obtain:

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}{I}_2\left(t\right)=\underset{t\to +\infty }{lim}{I}_3\left(t\right)=0.\end{array}$$

Thus, $\mathrm{\Phi }\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. The proof is completed. □

Lemma 4.

Let $\left(\mathbf{A}\mathbf{0}\right)$ holds. Then, for each $Y\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ and $X\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}({\mathbb{R}}^{+},$$\mathcal{L}(\mathbb{K},\mathcal{H}))$,

$${\displaystyle \mathrm{\Phi }:t\mapsto \mathrm{\Phi }\left(t\right)={\int }_0^t\mathcal{Q}(t,r)Y\left(r\right)dr+{\int }_0^t\mathcal{Q}(t,r)\mathrm{X}\left(r\right)d\mathrm{W}\left(r\right)\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right).}$$

Proof. 

For $Y\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ and $X\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}({\mathbb{R}}^{+},\mathcal{L}(\mathbb{K},\mathcal{H}))$, it is easy to show that $\mathrm{\Phi }\in \mathfrak{BC}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. Now let

$${\displaystyle \mathrm{\Phi }\left(t\right)={\int }_0^t\mathcal{Q}(t,r)Y\left(r\right)dr+{\int }_0^t\mathcal{Q}(t,r)\mathrm{X}\left(r\right)d\mathrm{W}\left(r\right)={\mathrm{\Phi }}_1\left(t\right)+{\mathrm{\Phi }}_2\left(t\right).}$$

By analogous arguments performed in the proof of Lemma 3, we obtain ${\mathrm{\Phi }}_1\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. Now, for ${\mathrm{\Phi }}_2$, let $p\in (0,{\displaystyle \frac{\delta }{2}})$ and $X\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}({\mathbb{R}}^{+},\mathcal{L}(\mathbb{K},\mathcal{H}))$. By using the condition $\mathbf{\left(}\mathbf{A}\mathbf{0}\mathbf{\right)}$ similarly to the proof of Lemma 3, we obtain for $ϵ>0$ that there exists $T_{ϵ}>0$ such that

$$\begin{array}{c}\hfill e^{{\displaystyle \frac{\delta }{2}}(t-r)}\parallel \mathcal{Q}(t+\varpi ,r+\varpi )-\mathcal{Q}(t,r){\parallel }^2<{\displaystyle \frac{\delta }{2}}ϵ\mathrm{and}\\ \hfill \mathsf{E}{∥\mathrm{X}(t+\varpi )-e^{i\mathbb{k}\varpi }\mathrm{X}\left(t\right)∥}^2<\delta ϵ\mathrm{for}t>T_{ϵ}.\end{array}$$

Therefore, for $t>T_{ϵ}$, we have

$$\begin{array}{cc}\hfill \mathsf{E}\parallel & {\mathrm{\Phi }}_2(t+\varpi )-e^{i\mathbb{k}\varpi }{\mathrm{\Phi }}_2{\left(t\right)\parallel }^2\hfill \\ \hfill =& \mathsf{E}{∥{\int }_0^{t+\varpi }\mathcal{Q}(t+\varpi ,r)\mathrm{X}\left(r\right)d\mathrm{W}\left(r\right)-e^{i\mathbb{k}\varpi }{\int }_0^t\mathcal{Q}(t,r)\mathrm{X}\left(r\right)d\mathrm{W}\left(r\right)∥}^2\hfill \\ \hfill \le & 2\mathsf{E}{∥{\int }_0^{\omega }\mathcal{Q}(t+\varpi ,r)\mathrm{X}\left(r\right)d\mathrm{W}\left(r\right)∥}^2\hfill \\ \hfill & +2\mathsf{E}{∥{\int }_0^t\mathcal{Q}(t+\varpi ,r+\varpi )\mathrm{X}(r+\varpi )d\mathrm{W}(r+\varpi )-e^{i\mathbb{k}\varpi }{\int }_0^t\mathcal{Q}(t,r)\mathrm{X}\left(r\right)d\mathrm{W}\left(r\right)∥}^2\hfill \\ \hfill \le & 2I_1\left(t\right)+2I_2\left(t\right).\hfill \end{array}$$

For $I_1\left(t\right)$, combining with Ito’s isometry property of stochastic integral, we obtain

$$\begin{array}{cc}\hfill I_1\left(t\right)=\mathsf{E}{∥{\int }_0^{\omega }\mathcal{Q}(t+\varpi ,r)\mathrm{X}\left(r\right)d\mathrm{W}\left(r\right)dr∥}^2\le & M^2{\int }_0^{\omega }{e}^{-2\delta (t+\varpi -r)}\mathsf{E}{∥\mathrm{X}\left(r\right)∥}^{2}dr\hfill \\ \hfill \le & M^2{\parallel X\parallel }_{\infty }^2{\int }_0^{\omega }{e}^{-2\delta (t+\varpi -r)}dr\hfill \\ \hfill & \to 0\mathrm{as}t\to +\infty ,\hfill \end{array}$$

therefore $\underset{t\to +\infty }{lim}{I}_1\left(t\right)=0$. For $I_2\left(t\right)$, let $\tilde{W}\left(r\right)=\mathrm{W}(r+\varpi )-\mathrm{W}\left(\omega \right)$. We know that $\tilde{W}$ is a Brownian motion and has the same distribution as W. Then, we obtain, by using Ito’s isometry property of stochastic integral

$$\begin{array}{cc}\hfill I_2\left(t\right)& =\mathsf{E}{∥{\int }_0^t\mathcal{Q}(t+\varpi ,r+\varpi )\mathrm{X}(r+\varpi )d\tilde{W}\left(r\right)-e^{i\mathbb{k}\varpi }{\int }_0^t\mathcal{Q}(t,r)\mathrm{X}\left(r\right)d\tilde{W}\left(r\right)∥}^2\hfill \\ \hfill \le & \mathsf{E}{∥{\int }_0^t\left(\mathcal{Q}(t+\varpi ,r+\varpi )-\mathcal{Q}(t,r)\right)\mathrm{X}(r+\varpi )d\tilde{W}\left(r\right)∥}^2\hfill \\ \hfill & +\mathsf{E}{∥{\int }_0^t\mathcal{Q}(t,r)\left(\mathrm{X}(r+\varpi )-e^{i\mathbb{k}\varpi }\mathrm{X}\left(r\right)\right)d\tilde{W}\left(r\right)∥}^2\hfill \\ \hfill \le & {\int }_0^t\parallel \mathcal{Q}(t+\varpi ,r+\varpi )-\mathcal{Q}(t,r){\parallel }^2\mathsf{E}{\parallel \mathrm{X}(r+\varpi )\parallel }^{2}dr\hfill \\ \hfill & +{\int }_0^t\parallel \mathcal{Q}(t,r){\parallel }^2\mathsf{E}{\parallel \mathrm{X}(r+\varpi )-e^{i\mathbb{k}\varpi }\mathrm{X}\left(r\right)\parallel }^{2}dr\hfill \\ \hfill \le & 2I_{21}\left(t\right)+2I_{22}\left(t\right).\hfill \end{array}$$

Then, by a calculation similar to that used in (5), it is easy to check that

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}{I}_{21}\left(t\right)=\underset{t\to +\infty }{lim}{I}_{22}\left(t\right)=0.\end{array}$$

As a result of these evaluations, it follows that ${\mathrm{\Phi }}_2\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. Thanks to Lemma 1-(i), we derive that $\mathrm{\Phi }\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. □

4. On Square-Mean S-Asymptotically Bloch-Type Periodic Mild Solutions

The main problem we tackle in this section is to find the conditions under which the dynamical system expressed by (1) possesses an unique solution which is a S-asymptotically Bloch type periodic.

Definition 6.

A stochastic variable $\mathrm{q}:\mathbb{R}\to \mathcal{H}$ is a mild solution of Equation (1) if for any $t\in [0,\infty )$ the following conditions hold:

1. 

$\mathrm{q}\left(t\right)$ is measurable and ${\mathcal{F}}_t$- adapted, and ${\mathrm{q}}_t$ is a Ξ-valued stochastic process.

2. 

$\mathrm{q}\left(t\right)\in {\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})$ has cádlág paths on $t\in [0,\infty )$ almost surely and

$$\mathrm{q}\left(t\right)=\left\{\begin{array}{c}{\displaystyle \mathcal{Q}(t,0)\varphi \left(0\right)+{\int }_0^t\mathcal{Q}(t,\varsigma ){\mathbb{A}}_0\mathrm{q}\left(\lfloor \varsigma \rfloor \right))d\varsigma +{\int }_0^t\mathcal{Q}(t,\varsigma )\mathrm{f}(\varsigma ,{\mathrm{q}}_{\varsigma })d\varsigma }\hfill \\ {\displaystyle +{\int }_0^t\mathcal{Q}(t,\varsigma )\mathrm{g}(\varsigma ,{\mathrm{q}}_{\varsigma })d\mathrm{W}\left(\varsigma \right),t\in [0,\infty ),}\hfill \\ \varphi \left(t\right),t\in (-\infty ,0].\hfill \end{array}\right.$$

(6)

Moreover, if ${\mathrm{q}|}_{[0,\infty )}\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$, then $\mathrm{q}$ is referred as an S-asymptotically Bloch type periodic mild solution.

Our first main outcome on the existence and uniqueness of square-mean S-asymptotically Bloch type periodic mild solution for problem (1) involving global Lipschitz conditions is presented in the following theorem.

Theorem 4.

Assume that Ξ is a fading memory space and $\varpi \in \mathbb{N}$. Let $\mathrm{f}\in \mathfrak{BC}({\mathbb{R}}^{+}\times \mathrm{\Xi },{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H}))$, $\mathrm{g}\in \mathfrak{BC}({\mathbb{R}}^{+},\mathcal{L}(\mathbb{K},\mathcal{H}))$ and verify $\left(\mathbf{H}\mathbf{0}\right)$. Suppose that there exist constants $L_{\mathrm{f}},L_{\mathrm{g}}>0$ such that for any ${\mathrm{v}}_1,{\mathrm{v}}_2\in {\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})$,

$$\begin{array}{c}\hfill \mathsf{E}\parallel \mathrm{f}(t,{\mathrm{v}}_1)-\mathrm{f}(t,{\mathrm{v}}_2){\parallel }^2\le L_{\mathrm{f}}.\parallel {\mathrm{v}}_1-{\mathrm{v}}_2{\parallel }_{\mathrm{\Xi }}^2 and \mathsf{E}\parallel \mathrm{g}(t,{\mathrm{v}}_1)-\mathrm{g}(t,{\mathrm{v}}_2){\parallel }_{\mathcal{L}(\mathbb{K},\mathcal{H})}^2\le L_{\mathrm{g}}.{\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel }_{\mathrm{\Xi }}^2,\end{array}$$

uniformly for all $t\in \mathbb{R}$. Then, problem (1) admits a unique S-asymptotically Bloch type periodic mild solution provided

$$\begin{array}{c}\hfill 3M^2\left({\displaystyle \frac{\parallel {\mathbb{A}}_0{\parallel }^2}{{\delta }^2}}+{\displaystyle \frac{L_{\mathrm{f}}L}{{\delta }^2}}+{\displaystyle \frac{L_{\mathrm{g}}L}{2\delta }}\right)<1,\end{array}$$

where L is the constant in Remark 2.

Proof. 

We set

$${\mathsf{SABP}}_{\varpi ,\mathbb{k}}^0\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)=\left\{\mathrm{q}\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right):\mathrm{q}\left(0\right)=0\right\}$$

It is clear that ${\mathsf{SABP}}_{\varpi ,\mathbb{k}}^0\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ is a closed subspace of ${\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. We next identify the elements $v\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ with its extension to $\mathbb{R}$ given by $v\left(\theta \right)=0$ for $\theta \le 0$. Moreover, we denote by $y(·)$ the process defined by $y_0=\varphi$ and $y\left(t\right)=\mathcal{Q}(t,0)\varphi \left(0\right)$ for $t\ge 0$. It follows by using ($\mathbf{A}\mathbf{0}$) that ${y|}_{[0,\infty )}\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. Then, by Lemma 2, we have $t\mapsto y_t\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},\mathrm{\Xi }\right)$.

Now, define the operator $\mathcal{S}$ on ${\mathsf{SABP}}_{\varpi ,\mathbb{k}}^0\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)\to {\mathsf{SABP}}_{\varpi ,\mathbb{k}}^0\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ by

$$\begin{array}{cc}\hfill \mathcal{S}\left(v\right)\left(t\right)& {\displaystyle ={\int }_0^t\mathcal{Q}(t,\varsigma ){\mathbb{A}}_0\left(v(\lfloor \varsigma \rfloor )+y(\lfloor \varsigma \rfloor )\right)d\varsigma +{\int }_0^t\mathcal{Q}(t,\varsigma )\mathrm{f}(\varsigma ,{\mathrm{v}}_{\varsigma }+y_{\varsigma })d\varsigma }\hfill \\ \hfill & {\displaystyle +{\int }_0^t\mathcal{Q}(t,\varsigma )\mathrm{g}(\varsigma ,{\mathrm{v}}_{\varsigma }+y_{\varsigma })d\mathrm{W}\left(\varsigma \right).}\hfill \end{array}$$

(7)

From Theorem 2, we have for each $v\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}^0\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ the stochastic processes $s\mapsto \mathrm{f}(\varsigma ,v(\varsigma \left)\right)$ and $s\mapsto \mathrm{g}(\varsigma ,v(\varsigma \left)\right)$ belong to ${\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. Then, it follows from Lemmas 3, 4, and 1-(i) that $\mathcal{S}v\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}^0\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. Using the Cauchy–Schwarz inequality and Ito’s isometry property of stochastic integral, we have

$$\begin{array}{cc}\hfill {\mathsf{E}\parallel \left(\mathcal{S}u\right)\left(t\right)-\left(\mathcal{S}v\right)\left(t\right)\parallel }^2\le & {\displaystyle 3\mathsf{E}\parallel {\int }_{-\infty }^t\mathcal{Q}(t,\varsigma ){\mathbb{A}}_0{[u\left(\lfloor \varsigma \rfloor \right)-v\left(\lfloor \varsigma \rfloor \right)]d\varsigma \parallel }^2}\hfill \\ \hfill & {\displaystyle +3\mathsf{E}\parallel {\int }_{-\infty }^t\mathcal{Q}(t,\varsigma )[\mathrm{f}(\varsigma ,u_{\varsigma }+y_{\varsigma })-\mathrm{f}(\varsigma ,{\mathrm{v}}_{\varsigma }+y_{\varsigma })]{d\varsigma \parallel }^2}\hfill \\ \hfill & {\displaystyle +3\mathsf{E}\parallel {\int }_{-\infty }^t\mathcal{Q}(t,\varsigma )[\mathrm{g}(\varsigma ,u_{\varsigma }+y_{\varsigma })-\mathrm{g}(\varsigma ,{\mathrm{v}}_{\varsigma }+y_{\varsigma })]d\mathrm{W}\left(\varsigma \right){\parallel }^2}\hfill \\ \hfill \le & 3M^2\left({\displaystyle \frac{\parallel {\mathbb{A}}_0{\parallel }^2}{{\delta }^2}}+{\displaystyle \frac{L_{\mathrm{f}}L}{{\delta }^2}}+{\displaystyle \frac{L_{\mathrm{g}}L}{2\delta }}\right){\parallel u-v\parallel }_{\infty }^2,\hfill \end{array}$$

Therefore,

$${\parallel \mathcal{S}u-\mathcal{S}v\parallel }_{\infty }^2\le 3M^2\left({\displaystyle \frac{\parallel {\mathbb{A}}_0{\parallel }^2}{{\delta }^2}}+{\displaystyle \frac{L_{\mathrm{f}}L}{{\delta }^2}}+{\displaystyle \frac{L_{\mathrm{g}}L}{2\delta }}\right){\parallel u-v\parallel }_{\infty }^2,$$

which proves that $\mathcal{S}$ is a contraction. Thus, there exists a unique $v\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}^0\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ such that $\mathcal{S}v=v$ via a Banach fixed-point theorem. By defining $\mathrm{q}\left(t\right)=y\left(t\right)+v\left(t\right)$ for $t\in \mathbb{R}$, it follows that $\mathrm{q}$ is the solution of Equation (1) and ${\mathrm{q}|}_{[0,\infty )}{=y|}_{[0,\infty )}+v\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. □

Our second main outcome on the existence and uniqueness of square-mean S-asymptotically Bloch type periodic mild solution for problem (1) involving local Lipschitz type conditions on f and g is presented in the following theorem.

Theorem 5.

Assume that Ξ is a fading memory space and $\varpi \in \mathbb{N}$. Let $\mathrm{f}\in \mathfrak{BC}({\mathbb{R}}^{+}\times \mathrm{\Xi },{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H}))$, $\mathrm{g}\in \mathfrak{BC}({\mathbb{R}}^{+}\times \mathrm{\Xi },\mathcal{L}(\mathbb{K},\mathcal{H}))$, and verify $\left(\mathbf{H}\mathbf{0}\right)$. In addition, assume that there exist functions $L_{\mathrm{f}},L_{\mathrm{g}}:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that for each $r>0$ and for all $u_1,u_2\in \mathrm{\Xi }$ with $\mathsf{E}\parallel u_1{\parallel }_{\mathrm{\Xi }}^2\le r$, $\mathsf{E}\parallel u_2{\parallel }_{\mathrm{\Xi }}^2\le r$,

$$\begin{array}{c}\hfill \mathsf{E}\parallel \mathrm{f}(t,u_1)-\mathrm{f}(t,u_2){\parallel }^2\le L_{\mathrm{f}}\left(r\right)\parallel u_1-u_2{\parallel }_{\mathrm{\Xi }}^2,\end{array}$$

(8)

$$\begin{array}{c}\hfill \mathsf{E}\parallel \mathrm{g}(t,u_1)-\mathrm{g}(t,u_2){\parallel }_{\mathcal{L}(\mathbb{K},\mathcal{H})}^2\le L_{\mathrm{g}}\left(r\right)\parallel u_1-u_2{\parallel }_{\mathrm{\Xi }}^2,\end{array}$$

(9)

uniformly for all $t\in {\mathbb{R}}^{+}$. Then, problem (1) admits a unique S-asymptotically Bloch type periodic mild solution on an arbitrarily closed ball ${\mathbb{B}}_r$ with center 0 and radius r satisfying the following condition

$$\begin{array}{c}\hfill \underset{r>0}{sup}Z\left(r\right)>6M^2\left[{\displaystyle \frac{1}{{\delta }^2}}\underset{t\in {\mathbb{R}}^{+}}{sup}\mathsf{E}\parallel \mathrm{f}(t,y_t){\parallel }^2+{\displaystyle \frac{1}{2\delta }}\underset{t\in {\mathbb{R}}^{+}}{sup}\mathsf{E}{\parallel \mathrm{g}(t,y_t)\parallel }_{\mathcal{L}(\mathbb{K},\mathcal{H})}^2\right].\end{array}$$

(10)

where $Z\left(r\right)=\left[r-3M^2\left({\displaystyle \frac{\parallel {\mathbb{A}}_0{\parallel }^2}{{\delta }^2}}+{\displaystyle \frac{2L_{\mathrm{f}}\left(r\right)L}{{\delta }^2}}+{\displaystyle \frac{2L_{\mathrm{g}}\left(r\right)L}{2\delta }}\right)r\right]$, $y(·)$ is the process defined by $y_0=\varphi$ and for $t\ge 0$ $y\left(t\right)=\mathcal{Q}(t,0)\varphi \left(0\right)$ and L the constant in Remark 2.

Proof. 

From (10), we deduce that there exists $r>0$ such that

$$\begin{array}{c}\hfill Z\left(r\right)>6M^2\left[{\displaystyle \frac{1}{{\delta }^2}}\underset{t\in {\mathbb{R}}^{+}}{sup}\mathsf{E}\parallel \mathrm{f}(t,y_t){\parallel }^2+{\displaystyle \frac{1}{2\delta }}\underset{t\in {\mathbb{R}}^{+}}{sup}\mathsf{E}{\parallel \mathrm{g}(t,y_t)\parallel }_{\mathcal{L}(\mathbb{K},\mathcal{H})}^2\right].\end{array}$$

(11)

Now similarly to the proof of Theorem 4, let $\mathcal{S}v$ define in (7) and ${\displaystyle {\mathrm{v}}_r=\{v\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}^0\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right){:\parallel v\parallel }_{\infty }^2<r\}.}$ To complete the proof, we just have to show that $\mathcal{S}$ is a contraction in ${\displaystyle {\mathrm{v}}_r}$. We start by showing that the operator $\mathcal{S}$ maps the set ${\mathrm{v}}_r$ to ${\mathrm{v}}_r$. From Corollary 1, combined with conditions (8) and (9), for each $v\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}^0\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$, the stochastic processes $\varsigma \mapsto \mathrm{f}(\varsigma ,v\left(\varsigma \right))\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ and $\varsigma \mapsto g(\varsigma ,v(\varsigma \left)\right)$ belong to $\mathsf{SABP}({\mathbb{R}}^{+},\mathcal{L}(\mathbb{K},\mathcal{H}))$. It follows from Lemmas 3, 4, and 1-(i) that $\mathcal{S}v\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}^0\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$. Let $v\in {\mathsf{SABP}}_{\varpi ,\mathbb{k}}^0\left({\mathbb{R}}^{+},{\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H})\right)$ and $t\in \mathbb{R}$; we have

$$\begin{array}{cc}\hfill {\mathsf{E}\parallel \left(\mathcal{S}v\right)\left(t\right)\parallel }^2& {\displaystyle \le 3\mathsf{E}\parallel {\int }_{-\infty }^t\mathcal{Q}(t,\varsigma ){\mathbb{A}}_0{v\left(\lfloor \varsigma \rfloor \right)d\varsigma \parallel }^2}\hfill \\ \hfill +& {\displaystyle 3\mathsf{E}\parallel {\int }_{-\infty }^t\mathcal{Q}(t,\varsigma )[\mathrm{f}(\varsigma ,{\mathrm{v}}_{\varsigma }+y_{\varsigma })-\mathrm{f}(\varsigma ,y_{\varsigma })+\mathrm{f}(\varsigma ,y_{\varsigma })]{d\varsigma \parallel }^2}\hfill \\ \hfill & {\displaystyle +3\mathsf{E}\parallel {\int }_{-\infty }^t\mathcal{Q}(t,\varsigma )[g(\varsigma ,{\mathrm{v}}_{\varsigma }+y_{\varsigma })-g(\varsigma ,y_{\varsigma })+g(\varsigma ,y_{\varsigma })]d\mathrm{W}\left(\varsigma \right){\parallel }^2}\hfill \\ \hfill \le & {\mathrm{\Delta }}_0+3M^2\left({\displaystyle \frac{\parallel {\mathbb{A}}_0{\parallel }^2}{{\delta }^2}}+{\displaystyle \frac{2L_{\mathrm{f}}\left(r\right)L}{{\delta }^2}}+{\displaystyle \frac{2L_{\mathrm{g}}\left(r\right)L}{2\delta }}\right){\parallel v\parallel }_{\infty }^2,\hfill \end{array}$$

where

$${\mathrm{\Delta }}_0:=6M^2\left[{\displaystyle \frac{1}{{\delta }^2}}\underset{t\in {\mathbb{R}}^{+}}{sup}\mathsf{E}{\parallel \mathrm{f}(t,y_t)\parallel }^2+{\displaystyle \frac{1}{2\delta }}\underset{t\in {\mathbb{R}}^{+}}{sup}\mathsf{E}{\parallel \mathrm{g}(t,y_t)\parallel }_{\mathcal{L}(\mathbb{K},\mathcal{H})}^2\right].$$

Then, for each $v\in {\mathrm{v}}_r$ and all $t\in \mathbb{R}$, we have from (11) that ${\mathsf{E}\parallel \left(\mathcal{S}v\right)\left(t\right)\parallel }^2\le r$. Thus, $\mathcal{S}\left({\mathrm{v}}_r\right)\subset {\mathrm{v}}_r$. Because for each ${\mathrm{v}}_1,{\mathrm{v}}_2\in {\mathrm{v}}_r$ and for all $t\in \mathbb{R}$, we have

$$\begin{array}{cc}\hfill & \mathsf{E}\parallel \left(\mathcal{S}{\mathrm{v}}_1\right)\left(t\right)-\left(\mathcal{S}{\mathrm{v}}_2\right)\left(t\right){\parallel }^2\le 3M^2\left({\displaystyle \frac{\parallel {\mathbb{A}}_0{\parallel }^2}{{\delta }^2}}+{\displaystyle \frac{L_{\mathrm{f}}\left(r\right)L}{{\delta }^2}}+{\displaystyle \frac{L_{\mathrm{g}}\left(r\right)L}{2\delta }}\right){\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel }_{\infty }^2,\hfill \end{array}$$

Therefore,

$$\parallel \mathcal{S}{\mathrm{v}}_1-\mathcal{S}{\mathrm{v}}_2{\parallel }_{\infty }^2\le 3M^2\left({\displaystyle \frac{\parallel {\mathbb{A}}_0{\parallel }^2}{{\delta }^2}}+{\displaystyle \frac{L_{\mathrm{f}}\left(r\right)L}{{\delta }^2}}+{\displaystyle \frac{L_{\mathrm{g}}\left(r\right)L}{2\delta }}\right){\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel }_{\infty }^2,$$

which proves that $\mathcal{S}$ is a contraction owing to the condition (11). Thus, there exists a unique $v\in {\mathrm{v}}_r$ such that $\mathcal{S}v=v$ via a Banach fixed-point theorem. This completes the proof. □

5. Examples

To show the applicability of the above obtained results, we provide the following examples.

5.1. Example 1

Let $\mathcal{H}=L^2\left((0,\pi )\right)$, $\mathbb{k}\in \mathbb{R}$ and $\gamma :t\in (0,+\infty )\mapsto \gamma \left(t\right)\in \mathbb{R}$ be a bounded continuous and $\gamma (t+1)=\gamma \left(t\right)$. Choosing $z\left(\varsigma \right)=e^{\varsigma }$ for $\varsigma <0$, then ${\displaystyle \beta ={\int }_{-\infty }^{0}z\left(\varsigma \right)d\varsigma =1}$, $\mathrm{\Xi }={\mathcal{D}}_z$ is a fading memory space and $({\mathcal{D}}_z,\parallel ·{\parallel }_z)$ is a Banach space with the norm. Choosing $z\left(\varsigma \right)=e^{\varsigma }$ for $\varsigma <0$, then ${\displaystyle \beta ={\int }_{-\infty }^{0}z\left(\varsigma \right)d\varsigma =1}$, $\mathrm{\Xi }={\mathcal{D}}_z$ is a fading memory space and $({\mathcal{D}}_z,\parallel ·{\parallel }_z)$ is a Banach space with the norm

$${\displaystyle {\parallel \psi \parallel }_{{\mathcal{D}}_z}={\left({\int }_{-\infty }^0{e}^t\underset{t\le \varsigma \le 0}{sup}\mathsf{E}{\parallel \psi \left(\varsigma \right)\parallel }^{2}dt\right)}^{\frac{1}{2}},\psi \in {\mathcal{D}}_z.}$$

We consider the following stochastic problem

$$\left\{\begin{array}{c}d\mathrm{v}(t,x)=[\left({\displaystyle \frac{{\partial }^2}{\partial x^2}}+sint\right)\mathrm{v}(t,x)+{\parallel \gamma \parallel }_{\infty }\mathrm{v}(\lfloor t\rfloor ,x)\hfill \\ {\displaystyle +\gamma \left(t\right){\int }_{-\infty }^t{\beta }_1(\varsigma -t)\left({\mathrm{f}}_1\left(\mathrm{v}(\varsigma ,x)\right)+{\displaystyle \frac{1}{1+t^2}}cos\left(\mathrm{v}(\varsigma ,x)\right)\right)d\varsigma ]dt}\hfill \\ +\left[{\displaystyle \gamma \left(t\right){\int }_{-\infty }^t{\beta }_2(\varsigma -t)\left({\mathrm{f}}_2\left(\mathrm{v}(\varsigma ,x)\right)+{\displaystyle \frac{1}{1+t^2}}sin\left(\mathrm{v}(\varsigma ,x)\right)\right)d\varsigma }\right]d\mathrm{W}\left(t\right),t>0,x\in [0,\pi ]\hfill \\ \mathrm{v}(t,0)=\mathrm{v}(t,\pi )=0,\hfill \\ \mathrm{v}(\tau ,x)={\varphi }_0(\tau ,x),fort\le 0,x\in [0,\pi ],\hfill \end{array}\right.$$

(12)

where $\mathrm{W}\left(t\right)$ is a one-sided and standard one-dimensional Brownian motion defined on the filtered probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P},{\mathcal{F}}_t)$. ${\varphi }_0\in {\mathcal{D}}_z$, ${\beta }_i(i=1,2):{\mathbb{R}}^{+}\to \mathbb{R}$ given by ${\beta }_1\left(t\right)=e^{-t}$ and ${\beta }_2\left(t\right)=e^{-2t}$ for all $t\in \mathbb{R})$ with

$$\begin{array}{c}\hfill {\displaystyle {\int }_{-\infty }^0\frac{{\beta }_1^2(-\varsigma )}{e^{-\varsigma }}d\varsigma ={\displaystyle \frac{1}{3}}and{\int }_{-\infty }^0\frac{{\beta }_2^2(-\varsigma )}{e^{-\varsigma }}d\varsigma ={\displaystyle \frac{1}{5}},}\end{array}$$

and there exist constants $L_i>0(i=1,2)$ such that ${\sigma }_i$ satisfies the conditions

$${\mathrm{f}}_i\left(e^{i\mathbb{k}}z\right)=e^{i\mathbb{k}}{\mathrm{f}}_i\left(z\right),\mathsf{E}\parallel {\mathrm{f}}_i\left(u\right)-{\mathrm{f}}_i\left(v\right){\parallel }^2\le L_i\mathsf{E}{\parallel u-v\parallel }^2.$$

In order to write the system (12) on the abstract form (1), we consider the linear operator $A:D\left(A\right)\subset L^2\left((0,\pi )\right)\to L^2\left((0,\pi )\right)$, given by

$$\begin{array}{cc}\hfill D\left(A\right)& =H^2(0,\pi )\cap H_0^1(0,\pi ),\hfill \\ \hfill A\mathrm{v}\left(\xi \right)& ={\mathrm{v}}^{\prime \prime }\left(\xi \right)for\xi \in (0,\pi )\mathrm{and}\mathrm{v}\in D\left(A\right).\hfill \end{array}$$

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

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

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

Define a family of linear operator $\mathbb{A}\left(t\right)$ as follows:

$$\left\{\begin{array}{cc}D\left(\mathbb{A}\right(t\left)\right)\hfill & =D\left(A\right),\hfill \\ \mathbb{A}\left(t\right)u\hfill & =\left[A+sint\right]u\mathrm{for}u\in D\left(A\right).\hfill \end{array}\right.$$

Hence, ${\left\{\mathbb{A}\left(t\right)\right\}}_{t\in \mathbb{R}}$ generates an evolution family $\left\{\mathcal{Q}\right(t,\varsigma ),t\ge s\}$ such that

$$\begin{array}{c}\hfill \mathcal{Q}(t,\varsigma )u=T(t-\varsigma )exp\left[{\int }_{\varsigma }^{t}sinrdr\right]u.\end{array}$$

Because $\parallel \mathcal{Q}(t,\varsigma )\parallel \le e^{-({\pi }^2-1)(t-\varsigma )}$ for $t\ge \varsigma$ and $\varsigma ,t\in \mathbb{R}$, it follows that (H0) holds.

Let $q\in [0,\pi ]$; we set

$$\begin{array}{cc}\hfill \mathrm{v}\left(t\right)\left(x\right)& =\mathrm{v}(t,x),{\mathbb{A}}_0={\parallel \gamma \parallel }_{\infty }I\left(I\mathrm{the}\mathrm{identity}\mathrm{in}\mathcal{H}\right),\varphi \left(t\right)\left(x\right)={\mathbb{A}}_0(t,x)\mathrm{for}t\le 0,\hfill \\ \hfill & {\mathrm{f}}_i\left(\mathrm{v}\right)\left(x\right)={\mathrm{f}}_i\left(\mathrm{v}\left(x\right)\right)sin\left(\mathrm{v}\right)\left(x\right)=sin\left(\mathrm{v}\left(x\right)\right),cos\left(\mathrm{v}\right)\left(x\right)=cos\left(\mathrm{v}\left(x\right)\right)\hfill \\ \hfill & {\displaystyle \mathrm{f}(t,\mathrm{v})\left(x\right)=\gamma \left(t\right){\int }_{-\infty }^0{\beta }_1(-\varsigma )\left({\mathrm{f}}_1\left(\mathrm{v}\left(\varsigma \right)\right)+{\displaystyle \frac{1}{1+t^2}}cos\left(\mathrm{v}\left(\varsigma \right)\right)\right)\left(x\right)d\varsigma \mathrm{and}}\hfill \\ \hfill & {\displaystyle \mathrm{g}(t,\mathrm{v})\left(x\right)=\gamma \left(t\right){\int }_{-\infty }^0{\beta }_2(-\varsigma )\left({\mathrm{f}}_2\left(\mathrm{v}\left(\varsigma \right)\right)+{\displaystyle \frac{1}{1+t^2}}sin\left(\mathrm{v}\left(\varsigma \right)\right)\right)\left(x\right)d\varsigma .}\hfill \end{array}$$

Therefore, the above problem (12) can be reformulated as the abstract (1).

Because for $(i=1,2)$, we have

$$\begin{array}{c}\hfill \gamma (t+1){\mathrm{f}}_i\left(z\left(\varsigma \right)\right)=\gamma \left(t\right)e^{i\mathbb{k}}{\mathrm{f}}_i\left(e^{-i\mathbb{k}}z\left(\varsigma \right)\right),\end{array}$$

then we obtain the following estimation:

$$\begin{array}{cc}\hfill & \mathsf{E}\parallel \mathrm{f}(t+1,\mathrm{v})-e^{i\mathbb{k}}\mathrm{f}(t,e^{-i\mathbb{k}}\mathrm{v}){\parallel }^2\hfill \\ \hfill & {\displaystyle =\mathsf{E}\parallel \gamma \left(t\right){\int }_{-\infty }^0{\beta }_1(-\varsigma )\left({\displaystyle \frac{1}{1+{(t+1)}^2}}cos\left(\mathrm{v}\left(\varsigma \right)\right)-{\displaystyle \frac{e^{i\mathbb{k}}}{1+t^2}}cos\left(e^{-i\mathbb{k}}\mathrm{v}\left(\varsigma \right)\right)\right)d\varsigma {\parallel }^2}\hfill \\ \hfill & {\displaystyle \le {\parallel \gamma \parallel }_{\infty }^2\left({\int }_{-\infty }^0\frac{{\beta }_1^2(-\varsigma )}{e^{-\varsigma }}d\varsigma \right)\left({\int }_{-\infty }^0{e}^{-\varsigma }\mathsf{E}\parallel {\displaystyle \frac{1}{1+{(t+1)}^2}}cos\left(\mathrm{v}\left(\varsigma \right)\right)-{\displaystyle \frac{e^{i\mathbb{k}}}{1+t^2}}cos\left(e^{-i\mathbb{k}}\mathrm{v}\left(\varsigma \right)\right){\parallel }^{2}d\varsigma \right)}\hfill \\ \hfill & {\displaystyle \le {\parallel \gamma \parallel }_{\infty }^2\left({\int }_{-\infty }^0\frac{{\beta }_1^2(-\varsigma )}{e^{-\varsigma }}d\varsigma \right)\left({\left|{\displaystyle \frac{1}{1+{(t+1)}^2}}\right|}^2+{\left|{\displaystyle \frac{1}{1+t^2}}\right|}^2\right)\to 0\mathrm{as}t\to \infty .}\hfill \end{array}$$

Then, it follows that

$$\underset{t\to \infty }{lim}\mathsf{E}{\parallel \mathrm{f}(t+1,\mathrm{v})-e^{i\mathbb{k}}\mathrm{f}(t,e^{-i\mathbb{k}}\mathrm{v})\parallel }^2=0.$$

Similarly, we have

$$\underset{t\to \infty }{lim}\mathsf{E}{\parallel \mathrm{g}(t+1,\mathrm{v})-e^{i\mathbb{k}}\mathrm{g}(t,e^{-i\mathbb{k}}\mathrm{v})\parallel }^2=0.$$

Now, let $u,v\in {\mathcal{D}}_z$ and $t\in \mathbb{R}$; then, we have following estimation:

$$\begin{array}{cc}\hfill \mathsf{E}{\parallel \mathrm{f}(t,u)-\mathrm{f}(t,v)\parallel }^2& {\displaystyle \le 2{\parallel \gamma \parallel }_{\infty }^2(\mathsf{E}\parallel {\int }_{-\infty }^0{\beta }_1(-\varsigma )\left({\mathrm{f}}_1\left(u\left(\varsigma \right)\right)-{\mathrm{f}}_1\left(v\left(\varsigma \right)\right)\right)d\varsigma {\parallel }^2}\hfill \\ \hfill & {\displaystyle +\mathsf{E}\parallel {\int }_{-\infty }^0{\beta }_1(-\varsigma )\left(cos\left(u\right(\varsigma \left)\right)-cos\left(v\right(\varsigma \left)\right)\right)d\varsigma {\parallel }^2)}\hfill \\ \hfill & {\displaystyle \le {2\parallel \gamma \parallel }_{\infty }^2[\left({\int }_{-\infty }^0\frac{{\beta }_1^2(-\varsigma )}{e^{-\varsigma }}d\varsigma \right)\left({\int }_{-\infty }^0{e}^s\mathsf{E}{\parallel {\mathrm{f}}_1\left(u\left(\varsigma \right)\right)-{\mathrm{f}}_1\left(v\left(\varsigma \right)\right)\parallel }^{2}d\varsigma \right)}\hfill \\ \hfill & {\displaystyle +\left({\int }_{-\infty }^0\frac{{\beta }_1^2(-\varsigma )}{e^{-\varsigma }}d\varsigma \right)\left({\int }_{-\infty }^0{e}^s\mathsf{E}{\parallel cos\left(u\left(\varsigma \right)\right)-cos\left(v\left(\varsigma \right)\right)\parallel }^{2}d\varsigma \right)]}\hfill \\ \hfill & {\displaystyle \le {2\parallel \gamma \parallel }_{\infty }^2\left({\int }_{-\infty }^0\frac{{\beta }_1^2(-\varsigma )}{e^{-\varsigma }}d\varsigma \right)\left[L_1+1\right]{\parallel u-v\parallel }_{{\mathcal{D}}_z}^2,}\hfill \end{array}$$

Similarly, we have

$$\begin{array}{c}\hfill {\displaystyle {\mathsf{E}\parallel \mathrm{g}(t,u)-\mathrm{g}(t,v)\parallel }^2\le {2\parallel \gamma \parallel }_{\infty }^2\left({\int }_{-\infty }^0\frac{{\beta }_2^2(-\varsigma )}{e^{-\varsigma }}d\varsigma \right)\left[L_2+1\right]{\parallel u-v\parallel }_{{\mathcal{D}}_z}^2.}\end{array}$$

Therefore, by Theorem 4, the problem (12) has a unique square-mean S-asymptotically Bloch type periodic mild solution provided that ${\parallel \gamma \parallel }_{\infty }$ is small enough.

5.2. Example 2

Let $W\left(t\right)$ be a standard one-sided and one-dimensional Brownian motion defined on the filtered probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P},{\mathcal{F}}_t)$. A stochastic two-dimensional complex Hopfield neural networks with infinite delay is considered as follows:

$$\left\{\begin{array}{cc}dy\left(t\right)\hfill & =[-a_0\left(t\right)y\left(t\right)+ay\left(\lceil t\rceil \right)+a_1\left(t\right)f_1(t,y_t)]dt+a_1\left(t\right){\sigma }_1(t,y_t)dW\left(t\right),\hfill \\ dz\left(t\right)\hfill & =[-b_0\left(t\right)z\left(t\right)+az\left(\lceil t\rceil \right)+a_2\left(t\right)f_2(t,z_t)]dt+a_2\left(t\right){\sigma }_1(t,z_t)dW\left(t\right),\hfill \end{array}\right.$$

(13)

where

-

$y\left(t\right)$, $z\left(t\right)$ the state variables of the first and second neuron at time instant t;

-

$a_0\left(t\right),b_0\left(t\right),a$ are the rate with which the neurons resets its potential to the resting state in isolation when disconnected from the network and external inputs;

-

$a_1\left(t\right)$, $a_2\left(t\right)$ are the connection weight of the neural networks at time instant t;

-

$f_1,f_2$ represent the activation functions of the incoming potentials of of the neurons;

-

${\sigma }_1,{\sigma }_2$ are a noise intensity functions.

These functions are given by $a_0\left(t\right)=2+sin\left(\pi t\right),b_0\left(t\right)=3+sin\left(\sqrt{2}t\right),a_1\left(t\right)={\displaystyle \frac{sin\left(2\pi t\right)}{40}},a_2\left(t\right)={\displaystyle \frac{sin\left(2\pi t\right)}{10}},a={\displaystyle \frac{e^{-2}}{4}},$

$$\begin{array}{cc}\hfill f_1(t,y_t)& {\displaystyle ={\int }_{-\infty }^t{e}^{-(t-3s/2)}\left({\displaystyle \frac{tanh\left(\right|y\left(s\right)\left|\right)}{\left|y\right(s\left)\right|}}ds+{\displaystyle \frac{1}{\sqrt{2}+t^2}}\right){\displaystyle \frac{y_1\left(s\right)}{\sqrt{|y_1{\left(s\right)|}^2+1}}}ds}\hfill \\ \hfill & {\displaystyle ={\int }_{-\infty }^t{e}^{3s/2}{F}_t^1\left(\right|y\left(s\right)\left|\right)G\left(y\left(s\right)\right)ds,}\hfill \\ \hfill {\sigma }_1(t,y_t)& {\displaystyle ={\int }_{-\infty }^0{e}^{-(t-3s/2)}\left({\displaystyle \frac{1}{max(1,|y\left(s\right)\left|\right)}}-{\displaystyle \frac{1}{\sqrt{2}+t^2}}\right){\displaystyle \frac{y_1\left(s\right)}{\sqrt{|y_1{\left(s\right)|}^2+1}}}ds}\hfill \\ \hfill & {\displaystyle ={\int }_{-\infty }^0{e}^{3s/2}{\sigma }_t^1\left(\right|y\left(s\right)\left|\right)G\left(y\left(s\right)\right)ds,}\hfill \\ \hfill f_2(t,y\left(t\right))& {\displaystyle ={\int }_{-\infty }^t{e}^{-(t-3s/2)}\left({\displaystyle \frac{tanh\left(\right|y\left(s\right)\left|\right)}{\left|y\right(s\left)\right|}}ds+{\displaystyle \frac{1}{2(1+t^2)}}\right){\displaystyle \frac{y\left(s\right)}{\sqrt{{\left|y\left(s\right)\right|}^2+1}}}ds}\hfill \\ \hfill & {\displaystyle ={\int }_{-\infty }^0{e}^{3s/2}{F}_t^2\left(\right|y\left(s\right)\left|\right)G\left(y\left(s\right)\right)ds,}\hfill \\ \hfill {\sigma }_2(t,y_t)& {\displaystyle ={\int }_{-\infty }^t{e}^{-(t-3s/2)}\left({\displaystyle \frac{1}{max(1,|y\left(s\right)\left|\right)}}-{\displaystyle \frac{1}{2{(1+t)}^2}}\right){\displaystyle \frac{y_1\left(s\right)}{\sqrt{|y_1{\left(s\right)|}^2+1}}}ds}\hfill \\ \hfill & {\displaystyle ={\int }_{-\infty }^0{e}^{3s/2}{\sigma }_t^2\left(\right|y\left(s\right)\left|\right)G\left(y\left(s\right)\right)ds.}\hfill \end{array}$$

with $F_t^1\left(\right|y\left(s\right)\left|\right)={\displaystyle \frac{tanh\left(\right|y_1\left(s\right)\left|\right)}{|y_1\left(s\right)|}}+{\displaystyle \frac{1}{\sqrt{2}+t^2}},F_t^2\left(\right|y\left(s\right)\left|\right)={\displaystyle \frac{tanh\left(\right|y\left(s\right)\left|\right)}{\left|y\right(s\left)\right|}}ds+{\displaystyle \frac{1}{2(1+t^2)}}$, ${\sigma }_t^1\left(y\left(s\right)\right)={\displaystyle \frac{1}{max(1,|y\left(s\right)\left|\right)}}-{\displaystyle \frac{1}{\sqrt{2}+t^2}}$, ${\sigma }_t^2\left(y\left(s\right)\right)={\displaystyle \frac{1}{max(1,|y\left(s\right)\left|\right)}}-{\displaystyle \frac{1}{2{(1+t)}^2}}$ and $G\left(y\left(s\right)\right)={\displaystyle \frac{y_1\left(s\right)}{\sqrt{|y_1{\left(s\right)|}^2+1}}}$.

The approach that we shall follow is to rewrite system (13) in the form of Equation (1), which we study using the Theorem 4. Let consider $\mathcal{H}={\mathbb{C}}^2$,

$$\mathrm{\Xi }=\left\{\begin{array}{cc}\psi :(-\infty ,0]\to {\mathbb{L}}^2(\mathrm{\Omega },\mathcal{H}):\mathrm{for} \mathrm{every}c>0,\mathbb{E}{\parallel \psi \left(\gamma \right)\parallel }^2\mathrm{is} a \mathrm{bounded} \mathrm{and}\hfill & \\ {\displaystyle \mathrm{integrable} \mathrm{with} \mathrm{respect} \mathrm{to}\lambda \mathrm{on}[-c,0]\mathrm{with}{\left({\int }_{-\infty }^0{e}^{\varsigma }\underset{\varsigma \le d\le 0}{sup}\mathbb{E}{\parallel \psi \left(d\right)\parallel }^{2}d\varsigma \right)}^{\frac{1}{2}}<\infty }\hfill \end{array}\right\}.$$

Hence, $\mathrm{\Xi }$ is a fading memory space and $(\mathrm{\Xi },\parallel ·{\parallel }_{\mathrm{\Xi }})$ is a Banach space with the norm

$${\displaystyle {\parallel \psi \parallel }_{\mathrm{\Xi }}={\left({\int }_{-\infty }^0{e}^t\underset{t\le \varsigma \le 0}{sup}\mathsf{E}{\parallel \psi \left(\varsigma \right)\parallel }^{2}dt\right)}^{\frac{1}{2}},\psi \in \mathrm{\Xi }.}$$

By taking $q\left(t\right)={(y\left(t\right),u\left(t\right))}^T$, the family of linear operator $A\left(t\right)$ referred to as

$$A\left(t\right)\left(\begin{array}{c}y\\ u\end{array}\right)=\left(\begin{array}{cc}2+sin\left(\pi t\right)& 0\\ 0& 3+sin\left(\sqrt{2}t\right)\end{array}\right)\left(\begin{array}{c}y\\ u\end{array}\right) \mathrm{for} \left(\begin{array}{c}y\\ u\end{array}\right)\in \mathcal{D}\left(A\right)={\mathbb{R}}^2,$$

${\mathbb{A}}_0={(e^{-2}/4,e^{-2}/4)}^T$, $\mathrm{f}(t,q_t)={(a_1\left(t\right)f_1(t,y\left(t\right)),a_1\left(t\right)f_2(t,u\left(t\right)))}^T$, and $\mathrm{g}(t,q_t)={(a_2\left(t\right){\sigma }_1\left(y\left(t\right)\right),a_2\left(t\right){\sigma }_2\left(u\left(t\right)\right))}^T$; we transform the system (13) into the form (1). The evolution family $\left\{\mathcal{Q}\right(t,s),t\ge s\}$ generates by $\left\{A\right(t),t\in \mathbb{R}\}$ is expressed as

$$\mathcal{Q}(t,s)\left(\begin{array}{c}y\\ u\end{array}\right)=\left(\begin{array}{cc}exp(-2(t-s)-{\int }_s^{t}sin\left(\pi r\right))dr& 0\\ 0& exp(-3(t-s)+{\int }_s^{t}sin\left(\sqrt{2}r\right)dr)\end{array}\right)\left(\begin{array}{c}y\\ u\end{array}\right)$$

and satisfy $\parallel \mathcal{Q}(t,s)\parallel \le exp\left(-{\displaystyle \frac{(t-s)}{2}}\right)$ for $t\ge s$ (i.e $M=1$ and $\delta ={\displaystyle \frac{1}{2}}$). Now, we verify that the vector valued-processes $\mathrm{f},\mathrm{g}$ satisfies condition $\left(\mathbf{H}\mathbf{0}\right)$ and the global Lipschitz conditions. First, note that for $(t,y)\in (0,\infty )\times \mathrm{\Xi }$, we have

$$f_i(t,e^{ik\omega }y)=e^{ik\omega }{f}_i(t,y) \mathrm{and} {\sigma }_i(t,e^{ik\omega }y\left(t\right))=e^{ik\omega }{\sigma }_i(t,y),i=1,2.$$

For any $t\in (0,\infty ),q={(y,u)}^T\in {\mathrm{\Xi }}^2$,

$$\begin{array}{cc}\hfill & \mathsf{E}\parallel \mathrm{f}(t+1,q)-e^{ik\omega }\mathrm{f}(t,e^{-ik\omega }q){\parallel }^2\hfill \\ \hfill =& \mathsf{E}\left({∥a_1(t+1)f_1(t+1,y)-e^{ik\omega }{a}_1\left(t\right)f_1(t,e^{-ik\omega }y)∥}^2\right.\hfill \\ \hfill & +\left.{∥a_1(t+1)f_2(t+1,u)-e^{ik\omega }{a}_1\left(t\right)f_2(t,e^{-ik\omega }u)∥}^2\right)\hfill \\ \hfill =& \mathsf{E}\left({∥a_1\left(t\right)f_1(t+1,y)-a_1\left(t\right)f_1(t,y)∥}^2+{∥a_1\left(t\right)f_2(t+1,u)-a_1\left(t\right)f_2(t,u)∥}^2\right)\hfill \\ \hfill \le & \parallel a_1{\parallel }_{\infty }^2\mathsf{E}\left({∥f_1(t+1,y)-f_1(t,y)∥}^2+{∥f_2(t+1,u)-f_2(t,u)∥}^2\right)\hfill \\ \hfill \le & \parallel a_1{\parallel }_{\infty }^2{\left|{\displaystyle {\int }_{-\infty }^0{e}^{3s/2}\left({\displaystyle \frac{1}{\sqrt{2}+{(t+1)}^2}}-{\displaystyle \frac{1}{\sqrt{2}+{\left(t\right)}^2}}\right)ds}\right|}^2+{\left|{\int }_{-\infty }^0{e}^{3s/2}\left({\displaystyle \frac{1}{1+{(t+1)}^2}}-{\displaystyle \frac{1}{1+t^2}}\right)ds\right|}^2.\hfill \end{array}$$

Hence, we obtain $\mathsf{E}\parallel \mathrm{f}(t+1,q)-e^{ik\omega }\mathrm{f}(t,e^{-ik\omega }q){\parallel }^2\to 0 \mathrm{as} t\to \infty ,$ and similar arguments yields that

$$\mathsf{E}\parallel \mathrm{g}(t+1,q)-e^{ik\omega }\mathrm{g}(t,e^{-ik\omega }q){\parallel }^2\to 0 \mathrm{as} t\to \infty .$$

Therefore, $\left(\mathbf{H}\mathbf{0}\right)$ of Theorem 4 is also satisfied.

Now, for each $q_1=(y_1,u_1),q_2=(y_2,u_2)\in {\mathrm{\Xi }}^2$, we obtain

$$\begin{array}{cc}\hfill & \mathsf{E}\parallel \mathrm{f}(t,q_1)-\mathrm{f}(t,q_2){\parallel }^2\hfill \\ \hfill =& \mathsf{E}\parallel a_1\left(t\right)f_1(t,y_1\left(t\right))-a_1\left(t\right)f_1(t,y_2\left(t\right)){\parallel }^2+\mathsf{E}{\parallel a_2\left(t\right)f_2(t,u_1\left(t\right))-a_2\left(t\right)f_2(t,u_2\left(t\right))\parallel }^2\hfill \\ \hfill \le & \parallel a_1{\parallel }_{\infty }^2\mathsf{E}{\left({\int }_{-\infty }^0{e}^{3s/2}\left[\left({\displaystyle \frac{1}{2}}-\left(1+{\displaystyle \frac{1}{2t^2}}\right)\right)\left|y_2\left(s\right)-y_1\left(s\right)\right|\right]ds\right)}^2\hfill \\ \hfill & +\parallel a_1{\parallel }_{\infty }^2\mathsf{E}{\left({\int }_{-\infty }^0{e}^{3s/2}\left[\left({\displaystyle \frac{1}{2}}-\left(1+{\displaystyle \frac{1}{2{(1+t)}^2}}\right)\right)\left|u_2\left(s\right)-u_1\left(s\right)\right|\right]ds\right)}^2\hfill \\ \hfill \le & \parallel a_1{\parallel }_{\infty }^2{\displaystyle \frac{1}{4}}\mathsf{E}{\left({\int }_{-\infty }^0{e}^{3s/2}\left|y_2\left(s\right)-y_1\left(s\right)\right|ds\right)}^2+{\parallel a_1\parallel }_{\infty }^2{\displaystyle \frac{1}{4}}\mathsf{E}{\left({\int }_{-\infty }^0{e}^{3s/2}\left|u_2\left(s\right)-u_1\left(s\right)\right|ds\right)}^2\hfill \\ \hfill \le & \parallel a_1{\parallel }_{\infty }^2{\displaystyle \frac{1}{4}}\mathsf{E}\left({\int }_{-\infty }^0{e}^s\right)\left({\int }_{-\infty }^0{e}^s{\left|y_2\left(s\right)-y_1\left(s\right)\right|}^{2}ds\right)\hfill \\ \hfill & +\parallel a_1{\parallel }_{\infty }^2{\displaystyle \frac{1}{4}}\mathsf{E}\left({\int }_{-\infty }^0{e}^{s}ds\right)\left({\int }_{-\infty }^0{e}^s{\left|u_2\left(s\right)-u_1\left(s\right)\right|}^{2}ds\right)\hfill \\ \hfill \le & {\displaystyle \frac{1}{1600}}{∥q_2-q_1∥}_{\mathrm{\Xi }}^2.\hfill \end{array}$$

Because ${\sigma }_t^1\left(y\left(s\right)\right)={\displaystyle \frac{1}{2max(1,|y\left(s\right)\left|\right)}}-{\displaystyle \frac{1}{\sqrt{2}+t^2}}$, ${\sigma }_t^2\left(y\left(s\right)\right)={\displaystyle \frac{1}{2max(1,|y\left(s\right)\left|\right)}}-{\displaystyle \frac{1}{2{(1+t)}^2}}$ are $1/2$-Lipschitz and $G\left(y\right(s\left)\right)$ is 1-Lipschitz, similar calculations as above yield that

$$\mathsf{E}\parallel \mathrm{g}(t,q_1)-\mathrm{g}(t,q_2){\parallel }^2\le {\displaystyle \frac{1}{40}}{∥q_2-q_1∥}_{\mathrm{\Xi }}^2.$$

Moreover, we note that (with $L_{\mathrm{f}}={\displaystyle \frac{1}{1600}},L_{\mathrm{g}}={\displaystyle \frac{1}{40}}$, $\delta ={\displaystyle \frac{1}{2}}$, $L=M=1$ and $\parallel {\mathbb{A}}_0{\parallel }^2={\displaystyle \frac{e^{-4}}{8}}$)

$$\begin{array}{c}\hfill 3M^2\left({\displaystyle \frac{\parallel {\mathbb{A}}_0{\parallel }^2}{{\delta }^2}}+{\displaystyle \frac{L_{\mathrm{f}}L}{{\delta }^2}}+{\displaystyle \frac{L_{\mathrm{g}}L}{2\delta }}\right)=3\left({\displaystyle \frac{e^{-4}}{2}}+{\displaystyle \frac{1}{400}}+{\displaystyle \frac{1}{40}}\right)\approx 0,11<1.\end{array}$$

For the above considerations, we conclude that the system (12) has a unique square-mean S-asymptotically Bloch type ($(k,1),k\in \mathbb{R}$) periodic mild solution.

6. Conclusions and Perspectives

In this work, we investigate the class of stochastic processes, the square-mean S-asymptotically Bloch type periodic, which is more general than other classes of stochastic processes like square-mean S-asymptotically $\omega$-periodic processes or S-asymptotically $\omega$- anti periodic processes introduced in the literature. We establish results about this class of stochastic processes, such as invariance of the class by composition and convolution. These facts allow us to investigate a class of stochastic evolution equations with infinite delay and piecewise constant argument, under the situation that the operator $\mathbb{A}$ satisfy the so-called Acquistapace–Terreni conditions, and the drift and diffusion terms satisfying suitable Lipschitz type conditions in the spatial variable (local and global) and square-mean S-asymptotically Bloch type periodic conditions, based on the Banach fixed-point theorem. The outcomes of this article provide us results more general than those existing in the literature to date, allowing us to extend deterministic results to stochastic one and processes beyond the square-mean S-asymptotically Bloch type periodic class. We proposed as a future investigation the development of the discrete analogue, another class of periodic random sequence, mean S-asymptotically Bloch type periodic random sequence, and investigation of the S-asymptotically Bloch type periodic solution for stochastic singular difference equations. This investigation will allow us to obtain even more general outcomes, and to the best of our knowledge, this class has not been introduced in the literature yet.