Square-Mean S-Asymptotically(ω,c)-Periodic Solutions to Neutral Stochastic Impulsive Equations

Abstract

This paper investigates the existence of square-mean S-asymptotically $(\omega ,c)$-periodic solutions for a class of neutral impulsive stochastic differential equations driven by fractional Brownian motion, addressing the challenge of modeling long-range dependencies, delayed feedback, and abrupt changes in systems like biological networks or mechanical oscillators. By employing semigroup theory to derive mild solution representations and the Banach contraction principle, we establish sufficient conditions–such as Lipschitz continuity of nonlinear terms and growth bounds on the resolvent operator—that guarantee the uniqueness and existence of such solutions in the space ${\mathcal{S}AP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$. The important results demonstrate that under these assumptions, the mild solution exhibits square-mean S-asymptotic $(\omega ,c)$-periodicity, enabling robust asymptotic analysis beyond classical periodicity. We illustrate these findings with examples, such as a neutral stochastic heat equation with impulses, revealing stability thresholds and decay rates and highlighting the framework’s utility in predicting long-term dynamics. These outcomes advance stochastic analysis by unifying neutral, impulsive, and fractional noise effects, with potential applications in control theory and engineering.

1. Introduction and Preliminaries

The analysis of periodic and asymptotic phenomena in differential equations has been a central topic of research owing to its crucial role in describing the dynamics of systems across diverse areas of science. While classical concepts of periodicity have been extensively studied in deterministic frameworks, practical models frequently incorporate stochastic perturbations, which call for more flexible and generalized approaches. In this context, particular interest has been directed toward square-mean S-asymptotically $(\omega ,c)$-periodic solutions, which offer a powerful framework for characterizing the long-term dynamics of stochastic systems subject to structured periodic effects.

The notion of $(\omega ,c)$-periodicity, which has been introduced for the vector-valued functions in [1], generalizes the classical idea of periodicity by incorporating exponential modulations. This concept has since been extended to the framework of stochastic evolution equations, where significant attention has been devoted to the study of almost periodic and pseudo-almost periodic solutions.

Stochastic systems influenced by Lévy noise, fractional Brownian motion, or multiplicative perturbations require the development of refined periodicity frameworks. In this direction, Bezandry and Diagana [2] introduced the concept of $S^2$-almost periodic solutions through a measure-theoretic approach, and later Bezandry [3] extended these ideas to systems driven by fractional Brownian motion. Building on these foundations, Diop and collaborators [4,5,6] proposed notions of square-mean and Stepanov-like pseudo-S-asymptotically Bloch-type periodicity for stochastic equations, while Mbaye [7] established the theory of $\mu$-pseudo-almost periodicity within the framework of stochastic integro-differential systems.

Parallel advances have been made in the deterministic and fractional settings. Cuevas and de Souza [8,9] established the existence of S-asymptotically $\omega$-periodic (SAP) solutions for fractional integro-differential equations with infinite delay, and Henriquez et al. [10] subsequently generalized these results to abstract neutral equations. Brindle and N’Guérékata [11,12] further expanded SAP theory to encompass difference equations and integro-differential systems, while Dimbour [13] extended the framework by incorporating Stepanov-type SAP functions for equations with piecewise arguments. Additional progress was achieved by de Andrade and Cuevas [14], who considered SAP solutions in the case of non-dense domains. In related work, Cuesta [15] and Shu et al. [16] investigated the asymptotic behavior of fractional neutral equations, and Nicola and Pierri [17] refined several fundamental properties of the SAP framework.

Hybrid periodicity concepts that combine spatial modulation of the Bloch type with asymptotic behavior were advanced by Chang et al. [18,19,20], where pseudo-S-asymptotically Bloch-type periodicity was introduced and applied to fractional equations with Stepanov forcing. In a related direction, Kostić et al. [21] formalized Stepanov-like pseudo-S-asymptotic $(\omega ,c)$-periodicity for stochastic integro-differential systems, while weighted pseudo-S-aasymptotically $(\omega ,c)$-periodic solutions of fractional stochastic differential equations were analyzed in [22]. Similar problems to time-space fractional evolution equations were considered by Li et al. [23], and Mbaye et al. [24] studied square-mean S-asymptotically Bloch-type periodic solutions for stochastic evolution systems with piecewise-constant arguments.

In [25,26], the third named author has provided a systematic treatment of metrical almost periodicity and its applications to integro-differential equations. Further developments include Bloch-periodic solutions for fractional systems by Oueama-Guengai and N’Guérékata [27] and the unification of Stepanov-weighted pseudo-periodicity with fractional Brownian motion by Diop et al. [5]. Additional theoretical advances—ranging from measure-theoretic stability results [6] and abstract semigroup methods [14] to discretization techniques for fractional dynamics [15] and existence and uniqueness of solution of class of neutral stochastic integro-differential equation driven by fractional Brownian motion with impulses [28]—have broadened the methodological foundation of the field.

Taken together, these studies highlight the versatility of generalized periodicity in deterministic, stochastic, and fractional contexts. Major achievements include the integration of Bloch-type modulation with asymptotic periodicity [18,21,22], the establishment of measure-theoretic stability frameworks for stochastic systems [6], and semigroup-based formulations of Stepanov asymptotic periodicity [8,10]. Despite these advances, open challenges remain, particularly in the unification of multi-scale periodicity concepts and the computational identification of modulated behaviors. This synthesis underscores both the depth and interdisciplinary reach of the subject, with promising implications for stochastic control, fractional dynamics, and applied mathematical modeling.

Rizvi et al. [29] conducted an advanced analytical investigation of soliton solutions for the Dullin–Gottwald–Holm dynamical equation using elliptic functions and related mathematical techniques. Similarly, Seadawy et al. [30] derived solitary wave solutions for the conformable time-fractional coupled Konno–Oono model by applying several effective mathematical methods, some of which can also be employed for the numerical study of the proposed problem investigated in the present paper.

This is the first work that simultaneously addresses neutral structure, fractional Brownian motion, and impulsive perturbations within the square-mean S-aasymptotically $(\omega ,c)$-periodic framework. While [21] handles Stepanov-like pseudo-S-aasymptotically $(\omega ,c)$-periodic solutions on stochastic integro-differential equations with Brownian motion, it excludes neutrality, fractional Brownian motion and impulses; [22] includes neutrality but only for standard Brownian motion and almost periodicity; and [24] considers neutral stochastic evolution systems with piecewise-constant argument but without fBm or impulses. Similarly, Duan and Ren’s work on neutral equations with fBm and impulses addresses solvability and stability but not square-mean S-aasymptotically $(\omega ,c)$-periodic solutions in the functional setting considered here [28]. Our approach thus fills a significant gap in modeling real-world systems with memory, delay, and abrupt change such as neural networks with synaptic delays and sudden resets or mechanical systems with impacts under long-memory noise conditions. This concept provides a powerful tool to capture the long-term behavior of systems subject to structured oscillatory influences combined with stochastic perturbations, thereby offering a robust analytical framework that extends classical periodicity into a richer and more realistic setting. The results obtained not only broaden the theoretical foundations of stochastic impulsive equations but also pave the way for applications in various applied sciences where memory effects, random fluctuations, and abrupt changes naturally coexist.

The structure of the paper is as follows. In Section 2, we define and investigate square-mean S-aasymptotically $(\omega ,c)$-periodic processes. The results on their structure and composition under certain conditions are provided. Section 3 is devoted to the formulation and proof of our central result, devoted to the existence and uniqueness of square-mean S-aasymptotically $(\omega ,c)$-periodic of the investigated problem. Section 4 illustrates the applicability of the theoretical findings through two concrete examples. Finally, Section 5 contains concluding remarks and outlines potential directions for future research.

In this paper, we analyze the following fractional stochastic differential equation

$$\left\{\begin{array}{cc}\hfill & d\Psi \left(\tau \right)=A\Psi \left(\tau \right)d\tau +f(\tau ,{\Psi }_{\tau })d\tau +\sigma (\tau ,{\Psi }_{\tau })d{W}^H\left(\tau \right),\hfill \\ \hfill & \Delta \Psi \left({\tau }_k\right)=I_k\left(\Psi \left({\tau }_k\right)\right),k=1,2,\dots ,\hfill \\ \hfill & \Psi \left(\tau \right)=\phi \left(\tau \right),\tau \in (-\infty ,0]\mathrm{and}\phi \left(0\right)=0,\hfill \end{array}\right.$$

(1)

where A is the infinitesimal generator of a $C_0$-semigroup ${\left(R\left(\tau \right)\right)}_{\tau \ge 0}$ on a Hilbert space $\mathbb{H}$ with domain $D\left(A\right)$; $f,\sigma$ are appropriate functions; and $W^H$ is a fractional Brownian motion with Hurst parameter $H>1/2$ on the real, separable Hilbert space $\mathbb{H}$. For the impulsive moment ${\tau }_k$, we have $0<{\tau }_1<{\tau }_2<\dots$, ${lim}_{k\to \infty }{\tau }_k=\infty$, $I_k:\mathbb{H}\to \mathbb{H}$, and $\Delta \Psi \left(\tau \right)=\Psi \left({\tau }^{+}\right)-\Psi \left({\tau }^{-}\right)$, where $\Psi \left({\tau }^{+}\right)$ and $\Psi \left({\tau }^{-}\right)$ denote the right and the left limit of $\Psi (·)$ at $\tau$, respectively. The space $\mathcal{P}C=\mathcal{P}C([-s,0];L^2(\Omega ,\mathbb{H}))=\{\varphi :[-s,0]\to \mathbb{H},\varphi \left(\tau \right)$ is continuous everywhere except in a finite number of points ${\tau }_0 \mathrm{at} \mathrm{which}\varphi \left({\tau }_0^{-}\right),\varphi \left({\tau }_0^{+}\right)$ exist and $\varphi \left({\tau }_0^{-}\right)=\varphi \left({\tau }_0\right)\}$. For $\varphi \in \mathcal{P}C$, ${\parallel \varphi \parallel }_{\mathcal{P}C}={sup}_{r\in [-s,0]}\parallel \varphi \left(s\right)\parallel <+\infty$. For any function $\varphi$ and $\tau \in [0,b]$, ${\varphi }_{\tau }\left(\theta \right)=\varphi (\theta +\tau )$, $\theta \in [-s,0]$.

Preliminaries

We denote by $\mathbb{N}$ and $\mathbb{C}$ the sets of positive integers and complex numbers, respectively. Let $(\Omega ,\mathcal{F},p)$ be a probability space, and let $\mathbb{K}$ and $\mathbb{H}$ be complex separable Hilbert spaces with norm $\parallel ·\parallel$. The space of all bounded linear operators from $\mathbb{K}$ to $\mathbb{H}$ is denoted by $L(\mathbb{K},\mathbb{H})$ and is equipped with the topology induced by the operator norm. When $\mathbb{K}=\mathbb{H}$, we will use the notation $L\left(\mathbb{H}\right)$.

We define $L^2(\Omega ,\mathbb{H})$ as the space of all strongly measurable, square-integrable $\mathbb{H}$-valued random variables, which forms a complex Hilbert space for the norm

$${\parallel \Psi \parallel }_{L^2}={\left({E\parallel \Psi \parallel }^2\right)}^{{\displaystyle \frac{1}{2}}},\Psi \in L^2(\Omega ,\mathbb{H}),$$

where the expectation operator $E(·)$ is given by

$${E\parallel \Psi \parallel }^2={\int }_{\Omega }{\parallel \Psi \parallel }^{2}dp.$$

A stochastic process $\Psi :\mathbb{R}\to L^2(\Omega ,\mathbb{H})$ is said to be stochastically bounded if there exists a constant $C_1>0$ such that

$${E\parallel \Psi \left(\tau \right)\parallel }^2={\int }_{\Omega }{\parallel \Psi \left(\tau \right)\parallel }^{2}dp<C_1,\tau \in \mathbb{R}.$$

Furthermore, $\Psi :\mathbb{R}\to L^2(\Omega ,\mathbb{H})$ is said to be stochastically piecewise-continuous if $\Psi \in \mathcal{P}C(\mathbb{R},L^2(\Omega ,\mathbb{H}))$.

We denote by $\mathcal{BPC}([0,\infty ),L^2(\Omega ,\mathbb{H}))$ the space of all bounded, piecewise-continuous stochastic processes $\Psi :[0,\infty )\to L^2(\Omega ,\mathbb{H})$. This space, endowed with the norm

$$\parallel \Psi \parallel ={\left(\underset{\tau \in [0,\infty )}{sup}E{\parallel \Psi \left(\tau \right)\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}},$$

is a Banach space.

In the following part, we use the following definitions from [31]. By a phase space $(\mathcal{B},\parallel ·{\parallel }_{\mathcal{B}})$, we mean any vector space of functions $(-\infty ,0]\to L^2(\Omega ,\mathbb{H})$ endowed with the seminorm ${\parallel ·\parallel }_{\mathcal{B}}$ such that the following conditions hold:

(i)

If $u:(-\infty ,b+a]\to \mathbb{H}$, $a\ge 0$, $b\in \mathbb{R}$ is continuous on $[b,b+a)$ and $u_b\in \mathcal{B}$, then for every $\tau \in [b,b+a)$, the following holds:

(1)

$u_{\tau }\in \mathcal{B}$;

(2)

${E\parallel u\left(\tau \right)\parallel }^2\le C_1{\parallel u_{\tau }\parallel }_{\mathcal{B}}^2$;

(3)

$\parallel u_{\tau }{\parallel }_{\mathcal{B}}\le N_2{(\tau -b)sup\{\parallel u\left(s\right)\parallel }_{L^2}:0\le s\le \tau \}+N_3(\tau -b){\parallel u_b\parallel }_{\mathcal{B}}$, where $N_1>0$ is a constant, $N_2,N_3:[0,\infty )\to [0,\infty )$, $N_2$ is continuous, $N_3$ is locally bounded, and $N_1,N_2,N_3$ do not depend on $u(·)$.

(ii)

For the function $u(·)$ in (i), the function $\tau \mapsto u_{\tau }:[b,b+a]\to \mathcal{B}$ is continuous;

(iii)

The space $\mathcal{B}$ is complete;

(iv)

If ${\left({\Psi }_n\right)}_n$ is a uniformly bounded sequence of compactly supported continuous functions converging to $\Psi$ compactly on $(-\infty ,0]$, then $\Psi \in \mathcal{B}$ and $\parallel {\Psi }_n{-\Psi \parallel }_{\mathcal{B}}\to 0$, as $n\to \infty$.

For the phase space $\mathcal{B}$, it is said that it is a fading memory space if $\parallel R_0{\left(\tau \right)\left(\Psi \right)\parallel }_{\mathcal{B}}\to 0$ when $\tau \to \infty$ for every $\Psi \in \mathcal{B}$, where ${\mathcal{B}}_0=\{\Psi \in \mathcal{B}:\Psi \left(0\right)=0\}$, and for $\tau \ge 0$, the operator $R_0:{\mathcal{B}}_0\to {\mathcal{B}}_0$ is given by

$$R_0\left(\tau \right)\left(\Psi \right)\left(s\right)=\left\{\begin{array}{cc}\hfill & 0,-\tau \le s\le 0,\hfill \\ \hfill & \Psi (\tau +s),-\infty <s\le -\tau .\hfill \end{array}\right.$$

We note that if $\mathcal{B}$ is a fading memory space, then the functions $N_2$ and $N_3$ in (i) are bounded functions.

The standard fractional Brownian motion was defined in [32]. Two-sided fractional Brownian motion ${\beta }^H={\beta }^H\left(\tau \right)$ with Hurst parameter $H\in (0,1)$ is a centered Gaussian process with the covariance function

$$\begin{array}{c}\hfill C(\tau ,s)=E\left({\beta }^H\left(\tau \right){\beta }^H\left(s\right)\right)={\displaystyle \frac{1}{2}}\left({\left|\tau \right|}^{2H}+{\left|s\right|}^{2H}-{|\tau -s|}^{2H}\right).\end{array}$$

Here we consider case $H>1/2$. Then ${\beta }^H\left(\tau \right)$ has the following representation

$$\begin{array}{c}\hfill {\beta }^H\left(\tau \right)=\underset{0}{\overset{\tau }{\int }}K(\tau ,s)d\beta \left(s\right),\end{array}$$

where $\beta \left(s\right)$ is standard Brownian motion and the kernel is given by the following formula

$$\begin{array}{c}\hfill K(\tau ,s)=C_H{s}^{{\displaystyle \frac{1}{2}}-H}\underset{s}{\overset{\tau }{\int }}{(\mu -s)}^{H-{\displaystyle \frac{3}{2}}}{\mu }^{H-{\displaystyle \frac{1}{2}}}d\mu ,\tau \ge s,\end{array}$$

where $C_H\ge 0$ is a constant depending on H.

Let $\psi \in L^2\left([0,a]\right)$. The fractional Wiener integral of $\psi$, with respect of ${\beta }^H$ is defined by

$$\begin{array}{c}\hfill \underset{0}{\overset{a}{\int }}\psi \left(s\right)d{\beta }^H\left(s\right)=\underset{0}{\overset{a}{\int }}{K}_H^{*}\psi \left(s\right)d\beta \left(s\right),\end{array}$$

where $\left(K_H^{*}\psi \right)\left(s\right)={\int }_s^{\tau }\psi \left(\tau \right){\displaystyle \frac{\partial K(\tau ,s)}{\partial \tau }}d\tau$.

Let Q be a nonnegative self-adjoint operator in $L\left(\mathbb{H}\right)$ and $L_Q$ be the space consisting of all $\theta \in L\left(\mathbb{H}\right)$, such that $\theta Q^{{\displaystyle \frac{1}{2}}}$ is a Hilbert–Schmidt operator and its norm is given by ${\parallel \theta \parallel }_{L_Q}^2=tr\left(\theta Q{\theta }^{*}\right)$. Let $(e_n:n\in \mathbb{N})$ denote the complete orthonormal basis in $\mathbb{H},$ and let $Q\in L\left(\mathbb{H}\right)$ be an operator defined by $Q{e}_n={\lambda }_n{e}_n$ with finite trace given by $trQ={\sum }_{n=1}^{\infty }{\lambda }_n<\infty$, where ${\lambda }_n\ge 0$ are real numbers. We define the infinite dimensional fractional Brownian motion on $\mathbb{H}$ with covariance Q by the following formula

$$\begin{array}{c}\hfill W^H\left(\tau \right)=\sum _{n=1}^{\infty }{\beta }_n^H\left(\tau \right)Q^{{\displaystyle \frac{1}{2}}}{e}_n=\sum _{n=1}^{\infty }\sqrt{{\lambda }_n}{e}_n{\beta }_n^H\left(\tau \right),\end{array}$$

where ${\beta }_n^H\left(\tau \right)$, $n=1,2,\dots$ are real independent fractional Brownian motions. The process $W^H\left(\tau \right)$ is called $\mathbb{H}$-valued Q-fractional Brownian motion. For convenience, in the following we will use the term fractional Brownian motion.

The fractional Wiener integral of function $\phi :[0,a]\to L_Q$ with respect to fractional Brownian motion $W^H$ is given by

$$\begin{array}{c}\hfill \underset{0}{\overset{\tau }{\int }}\phi \left(s\right)d{W}^H\left(s\right)=\sum _{n=1}^{\infty }\underset{0}{\overset{\tau }{\int }}\phi \left(s\right)Q^{{\displaystyle \frac{1}{2}}}{e}_{n}d{\beta }_n^H\left(s\right)=\sum _{n=1}^{\infty }\underset{0}{\overset{\tau }{\int }}\left(K_H^{*}(\phi Q^{{\displaystyle \frac{1}{2}}}{e}_n)\right)\left(s\right)d{\beta }_n\left(s\right),\end{array}$$

where ${\beta }_n\left(s\right)$ is the standard Brownian motion with respect to ${\beta }_n^H$.

The following result is standard Itô-type isometry for fractional Brownian motion for $H>1/2$ (see [33]): For any $\mathbb{H}$-valued, square-integrable function $g:[0,\tau ]\to L^2(\Omega ,\mathbb{H})$, we have

$$\begin{array}{c}\hfill E{∥\underset{0}{\overset{\tau }{\int }}g\left(s\right)d{W}^H\left(s\right)∥}^2=H(2H-1)\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{\tau }{\int }}E⟨g\left(s\right),g\left(\tau \right)⟩{|s-\tau |}^{2H-2}dsd\tau .\end{array}$$

Let $c\in \mathbb{C}\setminus \left\{0\right\}$ and $\omega >0$. For a continuous function $\varphi :\mathbb{R}\to X$ is said to be $(\omega ,c)$-periodic if

$$\varphi (\tau +\omega )=c\varphi \left(\tau \right),\tau \in \mathbb{R}.$$

Here, $\omega$ is referred to as the c-period of $\varphi$. The notion of $(\omega ,c)$-periodic functions unifies several well-known types of recurrence. In particular, standard periodicity corresponds to the case $c=1$, and antiperiodicity arises when $c=-1$, while for $c=e^{ik\omega }$, where $k\in \mathbb{R}$, we obtain Bloch periodicity. A well-known example of such behavior appears in the solutions of Mathieu’s equation, ${\varphi }^{\prime \prime }\left(t\right)+\left(a-2qcos\left(2t\right)\right)\varphi \left(t\right)=0$, which describes the linearized motion of an inverted pendulum with a pivot oscillating periodically in the vertical direction. Similar types of solutions are also encountered in fluid dynamics and in seasonally forced population models, where periodic or quasi-periodic inputs generate long-term responses that are structured but scaled or shifted in time.

To formalize this, we use the principal branch of the complex logarithm and define

$$c^{{\displaystyle \frac{\tau }{\omega }}}:=exp\left({\displaystyle \frac{\tau }{\omega }}logc\right)=c^{^}\left(\tau \right).$$

A continuous function $\varphi \in \{f:\mathbb{R}\to X\mathrm{is} \mathrm{continuous}:{sup}_{t\in \mathbb{R}}\parallel c^{^}(-\tau )f\left(\tau \right)\parallel <+\infty \}$ is said to be an S-aasymptotically $(\omega ,c)$-periodic [20] if

$$\underset{\left|\tau \right|\to \infty }{lim}∥c^{^}(-\tau )\left(\varphi (\tau +\omega )-c\varphi \left(\tau \right)\right)∥=0,\mathrm{for} \mathrm{all}\tau \in \mathbb{R}.$$

For instance, the function $\varphi \left(\tau \right)=e^{-\delta \tau }sin\left(\omega \tau \right)$ is not periodic, but it satisfies

$$\underset{\left|\tau \right|\to \infty }{lim}\parallel c^{^}(-\tau )\left(\varphi (\tau +\omega )-c\varphi \left(\tau \right)\right)\parallel =0$$

with $c=e^{-\delta \omega }$ for all $\tau \in \mathbb{R}$, making it S-aasymptotically $(\omega ,c)$-periodic, which is a property typical to damped oscillatory systems.

In the following, we will always assume that $\left|c\right|=1$.

2. Square-Mean $\mathbf{S}$-Asymptotically $(\omega ,\mathbf{c})$-Periodic Processes

We begin this section with the definition of a square-mean S-aasymptotically $(\omega ,c)$-periodic process and some basic results about the space of square-mean S-aasymptotically $(\omega ,c)$-periodic processes.

Definition 1.

For a stochastic process $\Psi \in \mathcal{B}PC([0,\infty ),L^2(\Omega ,\mathbb{H}))$, it is said that it is mean-square S-aasymptotically $(\omega ,c)$-periodic if

$$\underset{\tau \to \infty }{lim}E{∥c^{^}\left(\tau \right)(\Psi (\tau +\omega )-c\Psi \left(\tau \right))∥}^2=0.$$

(2)

The space of all square-mean S-asymptotically $(\omega ,c)$-periodic stochastic processes $\Psi :[0,\infty )\to L(\Omega ,\mathbb{H})$ will be denoted by ${\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L(\Omega ,\mathbb{H}))$.

Following [4,21,24], we will state and prove the following theorem:

Theorem 1.

Let $\Psi ,{\Psi }_1,{\Psi }_2\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$. Then

 (i)

${\Psi }_1+a{\Psi }_2\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$ for every $a\in \mathbb{C}$;

 (ii)

Let $\mathcal{B}$ be a fading memory space and $\Psi \in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$ be a process with ${\Psi }_0\in \mathcal{B}$. Then ${\Psi }_{\tau }\in {\mathcal{S}SAP}_{\omega ,c}(\mathbb{R},\mathcal{B})$;

 (iii)

${\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$ endowed with the norm

$${\parallel \Psi \parallel }_{\infty }={\left(\underset{\tau \in [0,\infty )}{sup}E{\parallel \Psi \left(\tau \right)\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}\mathit{for}\Psi \in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H})),$$

is a Banach space.

Proof. 

(i) Let ${\Psi }_1,{\Psi }_2\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$. Then,

$$\begin{array}{cc}\hfill & E\parallel ({\Psi }_1+a{\Psi }_2)(\tau +\omega )-c({\Psi }_1+a{\Psi }_2)\left(\tau \right){\parallel }^2\hfill \\ \hfill & \le 2E\parallel {\Psi }_1(\tau +\omega )-c{\Psi }_1{\left(\tau \right)\parallel }^2+{2\left|a\right|}^{2}E{\parallel {\Psi }_2(\tau +\omega )-c{\Psi }_2\left(\tau \right)\parallel }^2\to 0,\hfill \end{array}$$

when $\tau \to \infty$. Hence, ${\Psi }_1+a{\Psi }_2\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$ for every $a\in \mathbb{C}$.

(ii)

It follows from the definition of the phase space $\mathcal{B}$ that $\tau \to {\Psi }_{\tau }$ is bounded on $[0,\infty )$. Next, we define for $\tau \in [0,\infty )$,

$$\begin{array}{c}\hfill \Phi \left(\tau \right)=\Psi (\tau +\omega )-c\Psi \left(\tau \right)={\Psi }_{\omega }\left(\tau \right)-c\Psi \left(\tau \right).\end{array}$$

So, ${\Phi }_0={\Psi }_{\omega }-{\Psi }_0\in \mathcal{B}$. Since $\Psi \in \mathcal{P}C([0,\infty ),L^2(\Omega ,\mathbb{H}))$, $\Phi \in \mathcal{B}PC([0,\infty ),L^2(\Omega ,\mathbb{H}))$ and $\Phi \left(\tau \right)\to 0$, in $L^2(\Omega ,\mathbb{H})$ when $\tau \to \infty$. Additionally, from [31], we have that

$$\begin{array}{c}\hfill \parallel {\Phi }_{\tau }{\parallel }_{\mathcal{B}}={\parallel {\Psi }_{\tau +\omega }-c{\Psi }_{\tau }\parallel }_{\mathcal{B}}\to 0\end{array}$$

as $\tau \to \infty$.

(iii)

Let ${\left({\Psi }_n\right)}_n$ be a sequence in ${\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$ such that ${\Psi }_n\to \Psi$. Hence for every $\epsilon >0$, there exists a constant $n_0>0$ and $K_1>0$ such that

(a)

For $n\ge n_0$, it holds that $\parallel {\Psi }_n{-\Psi \parallel }_{\infty }\le {\displaystyle \frac{\sqrt{\epsilon }}{3}}$;

(b)

For $\tau \ge K_1$, it holds that $E\parallel {\Psi }_n(\tau +\omega )-c{\Psi }_n{\left(\tau \right)\parallel }^2\le {\displaystyle \frac{\epsilon }{9}}$.

Now,

$$\begin{array}{cc}\hfill & {E\parallel \Psi (\tau +\omega )-c\Psi \left(\tau \right)\parallel }^2\hfill \\ \hfill =& E\parallel \Psi (\tau +\omega )-{\Psi }_n(\tau +\omega )+{\Psi }_n(\tau +\omega )-c{\Psi }_n\left(\tau \right)+c{\Psi }_n{\left(\tau \right)-c\Psi \left(\tau \right)\parallel }^2\hfill \\ \hfill \le & 3E\parallel \Psi (\tau +\omega )-{\Psi }_n{(\tau +\omega )\parallel }^2+3E\parallel {\Psi }_n(\tau +\omega )-c{\Psi }_n{\left(\tau \right)\parallel }^2+3E{\parallel c{\Psi }_n\left(\tau \right)-c\Psi \left(\tau \right)\parallel }^2\hfill \\ \hfill \le & 3\parallel {\Psi }_n{-\Psi \parallel }_{\infty }^2+3{\displaystyle \frac{\epsilon }{9}}+3{\parallel {\Psi }_n-\Psi \parallel }_{\infty }^2\le {\displaystyle \frac{\epsilon }{3}}+{\displaystyle \frac{\epsilon }{3}}+{\displaystyle \frac{\epsilon }{3}}=\epsilon ,\hfill \end{array}$$

so the space ${\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$ is a closed subspace of $\mathcal{B}PC([0,\infty ),L^2(\Omega ,\mathbb{H}))$; therefore, it is a Banach space equipped with the supremum norm.      □

Let $f,\sigma \in \mathcal{B}PC(\mathbb{R}\times \mathcal{B},L^2(\Omega ,\mathbb{H}))$, $I_k:\mathbb{H}\to \mathbb{H}$, and $\mathcal{B}$ be a fading memory space. We give a list of assumptions that will be imposed on certain places in the following:

(H0)

Let $u\in \mathcal{B}PC([0,\infty )\times \mathcal{B},L^2(\Omega ,\mathbb{H}))$ satisfy

$$\begin{array}{c}\hfill \underset{\tau \to \infty }{lim}E{\parallel u(\tau +\omega ,\Psi )-cu(\tau ,c^{-1}\Psi )\parallel }^2=0\end{array}$$

converging uniformly for any $\Psi$ in a bounded subset of $\mathcal{B}$;

(H1)

A is the infinitesimal generator of a strongly continuous semigroup ${\left(R\left(\tau \right)\right)}_{\tau \ge 0}$ on $\mathbb{H}$. There exists $p>0$ and $M\ge 0$ such that $\parallel R\left(\tau \right)\parallel \le M{e}^{-p\tau }$;

(H2)

There exists $m\in \mathbb{N}$ such that $\omega ={\tau }_m$, ${\tau }_{k+m}={\tau }_k+\omega$, for all $k\in \mathbb{N}$;

(H3)

Discrete S-asymptotic $(\omega ,c)$-periodicity condition:

$I_k\left({\tau }_k\right):=v_k,{lim}_{k\to \infty }E{\parallel v_{k+m}-c{v}_k\parallel }^2=0$ and ${sup}_{k\in \mathbb{N}}E{\parallel v_k\parallel }^2<+\infty$;

(H4)

There exists $L_f>0$ such that

$$\begin{array}{c}\hfill E\parallel f(\tau ,{\Phi }_{\tau })-f(\tau ,{\Psi }_{\tau }){\parallel }^2\le L_f{\parallel \Phi -\Psi \parallel }_{\infty }^2\end{array}$$

for all $\Phi ,\Psi \in \mathcal{P}C([0,\infty ),L^2(\Omega ,\mathbb{H}))$ and uniformly for all $\tau \in [0,\infty )$;

(H5)

There exists $L_{\sigma }>0$ such that

$$\begin{array}{c}\hfill E\parallel \sigma (\tau ,{\Phi }_{\tau })-\sigma (\tau ,{\Psi }_{\tau }){\parallel }^2\le L_{\sigma }{\parallel \Phi -\Psi \parallel }_{\infty }^2\end{array}$$

for all $\Phi ,\Psi \in \mathcal{P}C([0,\infty ),L^2(\Omega ,\mathbb{H}))$;

(H6)

There exists $L_I>0$ such that

$$\begin{array}{c}\hfill \parallel I_k\left(x\right)-I_k\left(y\right)\parallel \le L_I\parallel x-y\parallel \end{array}$$

for all $x,y\in H$ and $k\in \mathbb{N}$.

The composition results presented below play a crucial role in the following analysis. In particular, they provide essential tools for establishing the existence of mild solutions in a wide class of evolution systems.

Theorem 2.

Let $u\in \mathcal{B}C([0,\infty )\times \mathcal{B},L^2(\Omega ,\mathbb{H}))$ fulfill the assumptions (H0) and (H4). If $\Psi \in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),\mathcal{B}),$ then $u(·,\Psi (·))\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$.

Proof. 

Since $\Psi \in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),\mathcal{B}),$ we have ${lim}_{\tau \to \infty }{\parallel \Psi (\tau +\omega )-c\Psi \left(\tau \right)\parallel }_{\mathcal{B}}=0$. Hence,

$$\begin{array}{cc}\hfill & {E\parallel u(\tau +\omega ,\Psi (\tau +\omega ))-cu(\tau ,\Psi \left(\tau \right))\parallel }^2\hfill \\ \hfill & \le 2E\parallel u(\tau +\omega ,\Psi (\tau +\omega ))-cu(\tau ,c^{-1}\Psi (\tau +\omega )){\parallel }^2\hfill \\ \hfill & +2E\parallel cu(\tau ,c^{-1}\Psi (\tau +\omega ))-cu(\tau ,\Psi \left(\tau \right)){\parallel }^2\hfill \\ \hfill & \le 2E\parallel u(\tau +\omega ,\Psi (\tau +\omega ))-cu(\tau ,c^{-1}\Psi (\tau +\omega )){\parallel }^2\hfill \\ \hfill & +2L_u{\parallel c^{-1}\Psi (\tau +\omega )-\Psi \left(\tau \right)\parallel }_{\mathcal{B}}^2\to 0,\hfill \end{array}$$

as $\tau \to \infty$. Therefore, we have $u(·,\Psi (·))\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$. □

Now, we continue by starting the following result:

Theorem 3.

Let $u\in \mathcal{B}C([0,\infty )\times \mathcal{B},L^2(\Omega ,\mathbb{H}))$. (H0) holds, and for every $\epsilon >0$ and bounded subset $B\subset \mathcal{B}$, there are constants $C_{\epsilon ,B}$ and ${\delta }_{\epsilon ,B}>0$ such that

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

for all ${\Psi }_1,{\Psi }_2\in B$ with $E\parallel {\Psi }_1-{\Psi }_2{\parallel }_{\mathcal{B}}^2\le {\delta }_{\epsilon ,B}$ and $\tau \ge C_{\epsilon ,B}$. If $\Psi \in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),\mathcal{B}),$ then $u(·,\Psi (·))\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$.

Proof. 

From $\Psi \in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),\mathcal{B})$, we have that $B:=\left\{\Psi \right(\tau ):\tau \in [0,\infty \left)\right\}\subset \mathcal{B}$ is bounded. Let $\epsilon >0$ be arbitrary. By (H0) and $\Psi \in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),\mathcal{B})$, we obtain the existence of $C_{\epsilon }>0$ such that for $\tau \ge C_{\epsilon }$

$$\begin{array}{c}\hfill E\parallel u(\tau +\omega ,\Psi (\tau +\omega ))-cu(\tau ,c^{-1}\Psi \left(\tau \right)){\parallel }^2<{\displaystyle \frac{\epsilon }{4}}\end{array}$$

and

$$\begin{array}{c}\hfill {\parallel \Psi (\tau +\omega )-c\Psi \left(\tau \right)\parallel }_{\mathcal{B}}^2\le \epsilon .\end{array}$$

By the prescribed condition, we have the existence of ${\delta }_{\epsilon ,B}:=\epsilon$ and $C_{\epsilon ,B}:=C_{\epsilon }$ such that

$$\begin{array}{c}\hfill E\parallel u(\tau ,c^{-1}\Psi (\tau +\omega ))-u(\tau ,\Psi \left(\tau \right)){\parallel }^2\le {\displaystyle \frac{\epsilon }{4}},\end{array}$$

for ${\parallel \Psi (\tau +\omega )-c\Psi \left(\tau \right)\parallel }_{\mathcal{B}}^2\le \epsilon$ and $\tau \ge C_{\epsilon }$. So,

$$\begin{array}{cc}\hfill & {E\parallel u(\tau +\omega ,\Psi (\tau +\omega ))-cu(\tau ,\Psi \left(\tau \right))\parallel }^2\hfill \\ \hfill & \le 2E\parallel u(\tau +\omega ,\Psi (\tau +\omega )-cu(\tau ,c^{-1}\Psi (\tau +\omega )){\parallel }^2\hfill \\ \hfill & +2E\parallel cu(\tau ,c^{-1}\Psi (\tau +\omega ))-cu(\tau ,\Psi \left(\tau \right)){\parallel }^2\hfill \\ \hfill & \le 2E\parallel u(\tau +\omega ,\Psi (\tau +\omega ))-cu(\tau ,c^{-1}\Psi (\tau +\omega )){\parallel }^2\hfill \\ \hfill & +2E\parallel u(\tau ,c^{-1}\Psi (\tau +\omega ))-u(\tau ,\Psi \left(\tau \right)){\parallel }^2<\epsilon ,\hfill \end{array}$$

for $\tau \ge C_{\epsilon }$. Hence,

$$\begin{array}{c}\hfill \underset{\tau \to \infty }{lim}{E\parallel u(\tau +\omega ,\Psi (\tau +\omega )-cu(\tau ,\Psi \left(\tau \right))\parallel }^2=0,\end{array}$$

so $u(·,\Psi (·))\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$. □

The next three theorems are crucial for the proof of our central result in the next section.

Theorem 4.

Let ${\left(R\left(\tau \right)\right)}_{\tau \ge 0}$ be a strongly continuous family of operators, and (H1) holds. If $f\in \mathcal{S}SA{P}_{\omega ,c}([0,\infty )\times \mathcal{B},L^2(\Omega ,\mathbb{H}))\cap \mathcal{C}(\mathbb{R}\times \mathcal{B},L^2(\Omega ,\mathbb{H})),$ then

$$\begin{array}{c}\hfill \Phi \left(\tau \right)=\underset{0}{\overset{\tau }{\int }}R(\tau -s)f(s,{\Psi }_s)ds\in \mathcal{S}SA{P}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H})).\end{array}$$

Proof. 

It is easy to show that $\Phi \in \mathcal{B}PC([0,\infty ),L^2(\Omega ,\mathbb{H}))$.

• Since $f\in \mathcal{S}SA{P}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))\cap \mathcal{C}(\mathbb{R},L^2(\Omega ,\mathbb{H}))$, we have that

$$\begin{array}{c}\hfill \underset{\tau \to \infty }{lim}E{\parallel f(\tau +\omega ,{\Psi }_{\tau +\omega })-cf(\tau ,{\psi }_{\tau })\parallel }^2=0.\end{array}$$

Let $r\in (0,p/2)$. For every $\epsilon >0$, there exists ${\tau }_{\epsilon }>0$ such that

$$\begin{array}{cc}\hfill & e^{-{\displaystyle \frac{ps}{2}}}{\parallel R\left(s\right)-R(\tau -s)\parallel }^2<\left({\displaystyle \frac{p}{2}}-r\right)\epsilon ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & E\parallel f(\tau +\omega ,{\Psi }_{\tau +\omega })-f(\tau ,{\Psi }_{\tau }){\parallel }^2\le p\epsilon ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & E\parallel f(\tau -s,{\Psi }_{\tau -s})-f(s,{\Psi }_s){\parallel }^2<p\epsilon .\hfill \end{array}$$

Also, it holds that $E\parallel f(s,{\Psi }_s){\parallel }^2<M_1{\parallel f\parallel }_{\infty }^2$. Now, by applying the Cauchy–Schwartz inequality and change of variables, we get

$$\begin{array}{cc}\hfill & {E\parallel \Phi (\tau +\omega )-c\Phi \left(\tau \right)\parallel }^2\hfill \\ \hfill & =E{∥\underset{0}{\overset{\tau +\omega }{\int }}R(\tau +\omega -s)f(s,{\Psi }_s)ds-c\underset{0}{\overset{\tau }{\int }}R(\tau -s)f(s,{\Psi }_s)ds∥}^2\hfill \\ \hfill & \le 2E{∥\underset{\omega }{\overset{\tau +\omega }{\int }}R(\tau +\omega -s)f(s,{\Psi }_s)ds-c\underset{0}{\overset{\tau }{\int }}R(\tau -s)f(s,{\Psi }_s)ds∥}^2\hfill \\ \hfill & +2E{∥\underset{0}{\overset{\tau }{\int }}R(\tau +\omega -s)f(s,{\Psi }_s)ds∥}^2\hfill \\ \hfill & =2E{∥\underset{0}{\overset{\tau }{\int }}R\left(s\right)f(\tau +\omega -s,{\Psi }_{\tau +\omega -s})ds-c\underset{0}{\overset{\tau }{\int }}R(\tau -s)f(s,{\Psi }_s)ds∥}^2\hfill \\ \hfill & +2E{∥\underset{0}{\overset{\omega }{\int }}R(\tau +\omega -s)f(s,{\Psi }_s)ds∥}^2=2E\parallel \underset{0}{\overset{\tau }{\int }}R\left(s\right)f(\tau +\omega -s,{\Psi }_{\tau +\omega -s})ds\hfill \\ \hfill & -c\underset{0}{\overset{\tau }{\int }}R\left(s\right)f(\tau -s,{\Psi }_{\tau -s})ds+c\underset{0}{\overset{\tau }{\int }}R\left(s\right)f(\tau -s,{\Psi }_{\tau -s})ds\hfill \\ \hfill & -c\underset{0}{\overset{\tau }{\int }}R(\tau -s)f(\tau -s,{\Psi }_{\tau -s})ds+c\underset{0}{\overset{\tau }{\int }}R(\tau -s)f(\tau -s,{\Psi }_{\tau -s})ds\hfill \\ \hfill & -c\underset{0}{\overset{\tau }{\int }}R(\tau -s)f(s,{\Psi }_s)ds{\parallel }^2+2E{∥\underset{0}{\overset{\omega }{\int }}R(\tau +\omega -s)f(s,{\Psi }_s)ds∥}^2\hfill \\ \hfill & \le 6E{∥\underset{0}{\overset{\tau }{\int }}R\left(s\right)f(\tau +\omega -s,{\Psi }_{\tau +\omega -s})ds-c\underset{0}{\overset{\tau }{\int }}R\left(s\right)f(\tau -s,{\Psi }_{\tau -s})ds∥}^2\hfill \\ \hfill & +6E{∥c\left(\underset{0}{\overset{\tau }{\int }}R\left(s\right)f(\tau -s,{\Psi }_{\tau -s})ds-\underset{0}{\overset{\tau }{\int }}R(\tau -s)f(\tau -s,{\Psi }_{\tau -s})ds\right)∥}^2\hfill \\ \hfill & +6E{∥c\left(\underset{0}{\overset{\tau }{\int }}R(\tau -s)f(\tau -s,{\Psi }_{\tau -s})ds-\underset{0}{\overset{\tau }{\int }}R(\tau -s)f(s,{\Psi }_s)ds\right)∥}^2\hfill \\ \hfill & +2E{∥\underset{0}{\overset{\omega }{\int }}R(\tau +\omega -s)f(s,{\Psi }_s)ds∥}^2\hfill \\ \hfill & \le 6\underset{0}{\overset{\tau }{\int }}{\parallel R\left(s\right)\parallel }^2·E{\parallel f(\tau +\omega -s,{\Psi }_{\tau +\omega -s})-cf(\tau -s,{\Psi }_{\tau -s})\parallel }^{2}ds\hfill \\ \hfill & +6\underset{0}{\overset{\tau }{\int }}{\parallel R\left(s\right)-R(\tau -s)\parallel }^2·E{\parallel f(\tau -s,{\Psi }_{\tau -s})\parallel }^{2}ds\hfill \\ \hfill & +6\underset{0}{\overset{\tau }{\int }}{\parallel R(\tau -s)\parallel }^2·E{\parallel f(\tau -s,{\Psi }_{\tau -s})-f(s,{\Psi }_s)\parallel }^{2}ds\hfill \\ \hfill & +2E\underset{0}{\overset{\omega }{\int }}{\parallel R(\tau +\omega -s)\parallel }^2·E{\parallel f(s,{\Psi }_s)\parallel }^{2}ds\hfill \\ \hfill & =I_1\left(\tau \right)+I_2\left(\tau \right)+I_3\left(\tau \right)+I_4\left(\tau \right).\hfill \end{array}$$

We consider the integral $I_1\left(\tau \right)$. We have

$$\begin{array}{cc}\hfill & I_1\left(\tau \right)=6\underset{0}{\overset{\tau }{\int }}{\parallel R\left(s\right)\parallel }^2·E{\parallel f(\tau +\omega -s,{\Psi }_{\tau +\omega -s})-cf(\tau -s,{\Psi }_{\tau -s})\parallel }^{2}ds\hfill \\ \hfill & \le 6Mp\epsilon \underset{0}{\overset{\tau }{\int }}{e}^{-2ps}ds=3M\epsilon (1-e^{-p\tau }),\hfill \end{array}$$

meaning that $I_1\left(\tau \right)\to 0$ when $\tau \to \infty$. Using the Cauchy–Schwartz inequality again for $I_2\left(\tau \right),$ we obtain

$$\begin{array}{cc}\hfill & I_2\left(\tau \right)\hfill \\ \hfill & =6\underset{0}{\overset{\tau }{\int }}{\parallel R\left(s\right)-R(\tau -s)\parallel }^2·E{\parallel f(\tau -s,{\Psi }_{\tau -s})\parallel }^{2}ds\hfill \\ \hfill & \le \underset{0}{\overset{\tau }{\int }}{e}^{-rs}ds\underset{0}{\overset{\tau }{\int }}{e}^{rs}{\parallel R\left(s\right)-R(\tau -s)\parallel }^2·E{\parallel f(\tau -s,{\Psi }_{\tau -s})\parallel }^{2}ds\hfill \\ \hfill & \le {\displaystyle \frac{6M_1{\parallel f\parallel }_{\infty }^2}{r}}\left(4M^2\underset{0}{\overset{{\tau }_{\epsilon }}{\int }}{e}^{(r-2p)(\tau -s)}ds+\underset{{\tau }_{\epsilon }}{\overset{\tau }{\int }}{e}^{rs}{\parallel R\left(s\right)-R(\tau -s)\parallel }^{2}ds\right)\hfill \\ \hfill & \le {\displaystyle \frac{6M_1{\parallel f\parallel }_{\infty }^2}{r}}\left(4M^2\underset{0}{\overset{{\tau }_{\epsilon }}{\int }}{e}^{(r-2p)(\tau -s)}ds+\underset{{\tau }_{\epsilon }}{\overset{\tau }{\int }}{e}^{rs}·e^{-{\displaystyle \frac{ps}{2}}}{e}^{{\displaystyle \frac{ps}{2}}}{\parallel R\left(s\right)-R(\tau -s)\parallel }^{2}ds\right)\hfill \\ \hfill & \le {\displaystyle \frac{6M_1{\parallel f\parallel }_{\infty }^2}{r}}\left(4M^2\underset{0}{\overset{{\tau }_{\epsilon }}{\int }}{e}^{(r-2p)(\tau -s)}ds+\left({\displaystyle \frac{p}{2}}-r\right)\epsilon \underset{{\tau }_{\epsilon }}{\overset{\tau }{\int }}{e}^{\left(r-{\displaystyle \frac{p}{2}}\right)s}ds\right)\hfill \\ \hfill & \le {\displaystyle \frac{6M_1{\parallel f\parallel }_{\infty }^2}{r}}\left(4M^2\underset{0}{\overset{{\tau }_{\epsilon }}{\int }}{e}^{(r-2p)(\tau -s)}ds+\epsilon \right).\hfill \end{array}$$

Since $r-2p<0$, we obtain ${\in }_0^{{\tau }_{\epsilon }}{e}^{(r-2p)(\tau -s)}ds\to 0$ when $\tau \to \infty$.

Next, we show that $I_3\left(\tau \right)\to 0$, as $\tau \to \infty$. So,

$$\begin{array}{cc}\hfill & I_3\left(\tau \right)=6\underset{0}{\overset{\tau }{\int }}{\parallel R(\tau -s)\parallel }^2·E{\parallel f(\tau -s,{\Psi }_{\tau -s})-f(s,{\Psi }_s)\parallel }^{2}ds\hfill \\ \hfill & \le 6\underset{0}{\overset{\tau }{\int }}{M}^2{e}^{-2p(\tau -s)}·p\epsilon ds=6p\epsilon M^2·e^{-2p\tau }\underset{0}{\overset{\tau }{\int }}{e}^{2ps}ds\hfill \\ \hfill & =3\epsilon M^2(1-e^{-2p\tau }).\hfill \end{array}$$

Consequently, we have $I_3\left(\tau \right)\to 0$ when $\tau \to \infty$.

• Lastly, we consider $I_4\left(\tau \right)$. We obtain

$$\begin{array}{cc}\hfill & I_4\left(\tau \right)=2E\underset{0}{\overset{\omega }{\int }}{\parallel R(\tau +\omega -s)\parallel }^2·E{\parallel f(s,{\Psi }_s)\parallel }^{2}ds\hfill \\ \hfill & \le \underset{0}{\overset{\omega }{\int }}{M}^2{e}^{-2p(\tau +\omega -s)}{M}_1^2{\parallel f\parallel }_{\infty }^{2}ds\hfill \\ \hfill & =2M_1{M}^2{\parallel f\parallel }_{\infty }^2{e}^{-2p(\tau +\omega )}\underset{0}{\overset{\omega }{\int }}{e}^{2ps}ds\hfill \\ \hfill & ={\displaystyle \frac{M_1{M}^2{\parallel f\parallel }_{\infty }^2}{p}}\left(e^{-2p\tau }-e^{2p(\omega -\tau )}\right).\hfill \end{array}$$

Hence, we get $I_4\left(\tau \right)\to 0$ when $\tau \to \infty$.

• Finally, we conclude that $\Phi \in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$. □

Theorem 5.

Let ${\left(R\left(\tau \right)\right)}_{\tau \ge 0}$ be a strongly continuous family satisfying (H1). If

$$\sigma \in {\mathcal{S}SAP}_{\omega c,}([0,\infty )\times \mathcal{B},L^2(\Omega ,\mathbb{H}))\cap \mathcal{C}(\mathbb{R}\times \mathbb{B},L^2(\Omega ,\mathbb{H})),$$

then

$$\begin{array}{c}\hfill \Phi \left(\tau \right)=\underset{0}{\overset{\tau }{\int }}R(\tau -s)\sigma (s,{\Psi }_s)d{W}^H\left(s\right)\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H})),\end{array}$$

where $W^H$ is a fractional Brownian motion with Hurst parameter $H>1/2$.

Proof. 

We have

$$\begin{array}{cc}\hfill & \Phi (\tau +\omega )\hfill \\ \hfill =& \underset{0}{\overset{\tau +\omega }{\int }}R(\tau +\omega -s)\sigma (s,{\Psi }_s)d{W}^H\left(s\right)\hfill \\ \hfill =& \underset{0}{\overset{\omega }{\int }}R(\tau +\omega -s)\sigma (s,{\Psi }_s)d{W}^H\left(s\right)+\underset{\omega }{\overset{\tau +\omega }{\int }}R(\tau +\omega -s)\sigma (s,{\Psi }_s)d{W}^H\left(s\right)\hfill \\ \hfill =& \underset{0}{\overset{\omega }{\int }}R(\tau +\omega -s)\sigma (s,{\Psi }_s)d{W}^H\left(s\right)+\underset{0}{\overset{\tau }{\int }}R(\tau -r)\sigma (r+\omega ,{\Psi }_{r+\omega })d{W}^H(r+\omega ).\hfill \end{array}$$

The process ${\tilde{W}}^H\left(r\right)=W^H(r+\omega )-W^H\left(\omega \right)$ is again fractional Brownian motion with the same Hurst parameter and the same covariance kernel ${|r-s|}^{2H-2}$, so the second moment is identical whether we integrate against $d{W}^H\left(r\right)$ or $d{W}^H(r+\omega )$. So, we have

$$\begin{array}{cc}\hfill & {E\parallel \Phi (\tau +\omega )-c\Phi \left(\tau \right)\parallel }^2\hfill \\ \hfill & \le 2E{∥\underset{0}{\overset{\omega }{\int }}R(\tau +\omega -s)\sigma (s,{\Psi }_s)d{W}^H\left(s\right)∥}^2+2H(2H-1)\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{\tau }{\int }}{K}_{\tau }(r,s){|r-s|}^{2H-2}drds\hfill \\ \hfill & =I_1\left(\tau \right)+I_2\left(\tau \right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill K_{\tau }(r,s)=E⟨R(\tau -r)(\sigma (r+\omega ,{\Psi }_{r+\omega })-c\sigma (r,{\Psi }_r)),R(\tau -s)(\sigma (s+\omega ,{\Psi }_{s+\omega })-c\sigma (s,{\Psi }_s))⟩.\end{array}$$

Now, we show that $I_1\left(\tau \right)\to 0$ when $\tau \to \infty$. We use (H1)

$$\begin{array}{cc}\hfill & I_1\left(\tau \right)\hfill \\ \hfill =& 2E{∥\underset{0}{\overset{\omega }{\int }}R(\tau +\omega -s)\sigma (s,{\Psi }_s)d{W}^H\left(s\right)∥}^2\hfill \\ \hfill =& 2H(2H-1)\underset{0}{\overset{\omega }{\int }}\underset{0}{\overset{\omega }{\int }}E⟨R(\tau +\omega -s)\sigma (s,{\Psi }_s),R(\tau +\omega -r)\sigma (r,{\Psi }_r)⟩·{|s-r|}^{2H-2}drds\hfill \\ \hfill \le & 2(2H-1)M^2{e}^{-2p\tau }\underset{0}{\overset{\omega }{\int }}\underset{0}{\overset{\omega }{\int }}{\left(E\parallel \sigma (s,{\Psi }_s){\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}{\left(E\parallel \sigma (r,{\Psi }_r){\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}{|s-r|}^{2H-2}drds.\hfill \end{array}$$

The double integral is finite since $\sigma$ is bounded on $[0,\omega ]\times \mathcal{B}$ and $2H-2>-1$, so we obtain

$$\underset{\tau \to \infty }{lim}{I}_1\left(\tau \right)=0.$$

Let us consider the integral $I_2\left(\tau \right)$. Using the Cauchy–Schwartz inequality and (H1), we have

$$\begin{array}{cc}\hfill & |K_{\tau }(r,s)|\le M^2{e}^{-p(2\tau -r-s)}{\left(E\parallel \sigma (r+\omega ,{\Psi }_{r+\omega })-c\sigma (r,{\Psi }_r){\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & ·{\left(E\parallel \sigma (s+\omega ,{\Psi }_{s+\omega })-c\sigma (s,{\Psi }_s){\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

Letting $u=\tau -r$, $v=\tau -s$, so $0\le u\le \tau$, $0\le v\le \tau$, we obtain

$$\begin{array}{cc}\hfill & |K_{\tau }(\tau -u,\tau -v)|\le M^2{e}^{-p(u+v)}{\left(E\parallel \sigma (\tau -u+\omega ,{\Psi }_{\tau -u+\omega })-c\sigma (\tau -u,{\Psi }_{\tau -u}){\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & ·{\left(E\parallel \sigma (\tau -v+\omega ,{\Psi }_{\tau -v+\omega })-c\sigma (\tau -v,{\Psi }_{\tau -v}){\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

Using that $\sigma \in {\mathcal{S}SAP}_{\omega ,c}([0,\infty )\times \mathcal{B},L^2(\Omega ,\mathbb{H}))$, it follows that

$$\begin{array}{cc}\hfill & E\parallel \sigma (\tau -u+\omega ,{\Psi }_{\tau -u+\omega })-c\sigma (\tau -u,{\Psi }_{\tau -u}){\parallel }^2\to 0,\hfill \\ \hfill & E\parallel \sigma (\tau -v+\omega ,{\Psi }_{\tau -v+\omega })-c\sigma (\tau -v,{\Psi }_{\tau -v}){\parallel }^2\to 0,\hfill \end{array}$$

when $\tau \to \infty$, so $K_{\tau }(\tau -u,\tau -v)\to 0$, pointwise for each u and v when $\tau \to \infty$. Moreover,

$$\begin{array}{c}\hfill |K_{\tau }{(\tau -u,\tau -v)|·|u-v|}^{2H-2}\le M^2{e}^{-p(u+v)}·M_{\sigma }{|u-v|}^{2H-2},\end{array}$$

where

$$\begin{array}{c}\hfill M_{\sigma }=\underset{\tau \ge 0}{sup}E{\parallel \sigma (\tau +\omega ,{\Psi }_{\tau +\omega })-c\sigma (\tau ,{\Psi }_{\tau })\parallel }^2<\infty .\end{array}$$

The function $e^{-p(u+v)}{|u-v|}^{2H-2}$ is integrable over $[0,\infty )\times [0,\infty )$, so by the dominated convergence we get ${lim}_{\tau \to \infty }{I}_2\left(\tau \right)=0$. Finally, we conclude that

$$\Phi \in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H})).$$

□

Remark 1.

The proof appeals to the Wiener–Young fractional Brownian motion isometry with kernel ${|u-v|}^{2H-2}$, which is locally integrable only when $2H-2>-1$, i.e., $H>1/2$. For $H\le 1/2,$ one needs Skorohod–Malliavin or rough-path techniques and a different norm on the integrand space.

Theorem 6.

Let (H1)–(H3) hold. Then the impulsive convolution

$$J\left(\tau \right)=\sum _{0<{\tau }_k<\tau }R(\tau -{\tau }_k)v_k{v}_k:=I_k\left(\Psi \left({\tau }_k^{-}\right)\right)$$

is square-mean S-aasymptotically $(\omega ,c)$-periodic.

Proof. 

Given that $k=j+qm$, with $j\in \{1,2,\dots ,m\}$, $q\in {\mathbb{N}}_0$, we have ${\tau }_{j+qm}={\tau }_j+q\omega$. Let ${\Delta }_{j,k}=v_{j+(q+1)m}-c{v}_{j+qm}$. A direct reindexing gives, for $\tau >0$,

$$\begin{array}{cc}\hfill & J(\tau +\omega )-cJ\left(\tau \right)=\sum _{j=1}^m\sum _{q\ge 0:{\tau }_j+q\omega <\tau }R(\tau -({\tau }_j+q\omega )){\Delta }_{j,q}+\sum _{j=1}^{m}R(\tau +\omega -{\tau }_j)v_j\hfill \\ \hfill & =S\left(\tau \right)+B\left(\tau \right).\hfill \end{array}$$

First, we consider the finite term $B\left(\tau \right)$. By (H1), we have

$$\begin{array}{c}\hfill {E\parallel B\left(\tau \right)\parallel }^2\le m·\underset{1\le j\le m}{max}\parallel R(\tau +\omega -{\tau }_j){\parallel }^2·\underset{1\le j\le m}{max}E{\parallel v_j\parallel }^2\le C{e}^{-2p\tau }\to 0,\end{array}$$

when $\tau \to \infty$.

• Next, we consider the $S\left(\tau \right)$. By the Minkowski inequality and (H1), we have

$$\begin{array}{c}\hfill {\left({E\parallel S\left(\tau \right)\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}\le M\sum _{j=1}^m\sum _{q\ge 0:{\tau }_j+q\omega <\tau }{e}^{-p(\tau -{\tau }_j-q\omega )}{\left(E\parallel {\Delta }_{j,k}{\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}\le \epsilon .\end{array}$$

Fix $ϵ>0$. By (H3), there exists $\theta$ such that $(E\parallel {\Delta }_{j,k}{{\parallel }^2)}^{1/2}\le \epsilon$ for all $j,q\ge \theta$. We split the inner sum at $\theta$. The finitely many terms with $q<\theta$ are each multiplied by $e^{-p(\tau -{\tau }_j-q\omega )}$; hence, their sum is tending to zero when $\tau \to \infty$. For the tail, $q\ge \theta$,

$$\begin{array}{cc}\hfill & {\left({E\parallel S\left(\tau \right)\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}\le M\epsilon \sum _{j=1}^m\sum _{q\ge \theta :{\tau }_j+q\omega <\tau }{e}^{-p(\tau -{\tau }_j-q\omega )}\hfill \\ \hfill & \le M\epsilon \sum _{j=1}^m\sum _{r\ge 0}{e}^{-pr}\le C^{\prime }\epsilon .\hfill \end{array}$$

Since $\epsilon$ was arbitrary, ${E\parallel S\left(\tau \right)\parallel }^2\to 0$ when $\tau \to \infty$, so we can conclude that

$$J\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H})).$$

□

Remark 2. (i) Note that, if we let $c=1$, then ${lim}_{k\to \infty }E{\parallel v_{k+m}-v_k\parallel }^2=0$, which is the definition of square-mean S-asymptotically periodic processes.

 (ii)

Exponential stability is essential to eliminate the boundary and finitely many early terms; mere exponential boundedness with nonnegative growth does not suffice.

3. Existence and Uniqueness of Square-Mean $\mathbf{S}$-Aasymptotically $(\omega ,\mathbf{c})$-Periodic Solutions

In this section, we investigate the existence of square-mean S-aasymptotically $(\omega ,c)$-periodic solutions to Equation (1).

Definition 2.

For an $\mathbb{H}$-valued stochastic process $\Psi \left(\tau \right)$, $\tau \in (-\infty ,b]$, it is said that it is a mild solution of Equation (1) if the following holds:

 (i)

for $\tau \in (-\infty ,0]$, $\Psi \left(\tau \right)=\phi \left(\tau \right)$;

 (ii)

$\Psi (·)\in \mathcal{P}C((-\infty ,b],L^2(\Omega ,\mathbb{H}))$;

 (iii)

for each $\tau [0,b]$, Ψ satisfies the following equation p-a.s.

$$\begin{array}{cc}\hfill & \Psi \left(\tau \right)=\underset{0}{\overset{\tau }{\int }}R(\tau -s)f(s,{\Psi }_s)ds+\underset{0}{\overset{\tau }{\int }}R(\tau -s)\sigma (s,{\Psi }_s)d{W}^H\left(s\right)\hfill \\ \hfill & +\sum _{0<{\tau }_k<\tau }R(\tau -{\tau }_k)I_k\left(\Psi \left({\tau }_k^{-}\right)\right).\hfill \end{array}$$

In order to establish the existence and uniqueness of square-mean S-asymptotically $(\omega ,c)$-periodic solutions to Equation (1), we state the following theorem, which provides sufficient conditions under which the mild solution defined above belongs to the space ${\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$.

Theorem 7.

Let (H0)–(H6) hold. If $f(\tau ,{\Psi }_{\tau }),\sigma (\tau ,{\Psi }_{\tau })\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty )\times \mathcal{B},L^2(\Omega ,\mathbb{H})),$ then the neutral stochastic impulsive equation driven by fractional Brownian motion (1) has a unique solution in ${\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$, provided that $C_f{L}_f+C_{\sigma }{L}_{\sigma }+C_I{L}_I<1$, where

$$C_f={\displaystyle \frac{3M^2}{2p}},C_{\sigma }={\displaystyle \frac{3M^{2}H(2H-1)·\Gamma (2H-1)}{{\left(2p\right)}^{2H}}} and C_I={\displaystyle \frac{3M^{2}m}{1-e^{-2p\omega }}},$$

where by $\Gamma (·)$ we denote the Gamma function.

Proof. 

We define the operator

$$\begin{array}{c}\hfill \Theta :{\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))\to {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))\end{array}$$

by

$$\begin{array}{cc}\hfill & \left(\Theta \Psi \right)\left(\tau \right):=\underset{0}{\overset{\tau }{\int }}R(\tau -s)f(s,{\Psi }_s)ds+\underset{0}{\overset{\tau }{\int }}R(\tau -s)\sigma (s,{\Psi }_s)D{W}^H\left(s\right)\hfill \\ \hfill & +\sum _{0<{\tau }_k<\tau }R(\tau -{\tau }_k)I_k\left(\Psi \left({\tau }_k^{-}\right)\right).\hfill \end{array}$$

It is clear that the mapping $\Theta$ is well defined. By Theorems 4–6 we can conclude that the mapping $\Theta$ maps ${\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$ into itself.

Let $\Phi ,\Psi \in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$. Using the assumptions (H1) and (H4)–(H6), together with the isometry bounds for $H>1/2$, we have

$$\begin{array}{cc}\hfill & {E\parallel \left(\Theta \Phi \right)\left(\tau \right)-\left(\Theta \Psi \right)\left(\tau \right)\parallel }^2\hfill \\ \hfill & \le 3E{∥\underset{0}{\overset{\tau }{\int }}R(\tau -s)\left(f(s,{\Phi }_s)-f(s,{\Psi }_s)\right)ds∥}^2\hfill \\ \hfill & +3E{∥\underset{0}{\overset{\tau }{\int }}R(\tau -s)\left(\sigma (s,{\Phi }_s)-\sigma (s,{\Psi }_s)\right)d{W}^H\left(s\right)∥}^2\hfill \\ \hfill & +3{∥\sum _{0<{\tau }_k<\tau }R(\tau -{\tau }_k)\left(I_k\left(\Phi \left({\tau }_k^{-}\right)\right)-I_k\left(\Psi \left({\tau }_k^{-}\right)\right)\right)∥}^2.\hfill \end{array}$$

Now, for the first term, we have

$$\begin{array}{cc}\hfill & E\parallel \underset{0}{\overset{\tau }{\int }}R(\tau -s)\left(f(s,{\Phi }_s)-f(s,{\Psi }_s)\right)ds{\parallel }^2\hfill \\ \hfill & \le \underset{0}{\overset{\tau }{\int }}{\parallel R(\tau -s)\parallel }^2·{\parallel f(s,{\Phi }_s)-f(s,{\Psi }_s)\parallel }^{2}ds\hfill \\ \hfill & \le M^2{L}_f{\parallel \Phi -\Psi \parallel }_{\infty }^2\underset{0}{\overset{\tau }{\int }}{e}^{-2p(\tau -s)}ds\hfill \\ \hfill & ={\displaystyle \frac{M^2{L}_f}{2p}}{\parallel \Phi -\Psi \parallel }^2·e^{-2p\tau }(e^{2p\tau }-1)\hfill \\ \hfill & \le {\displaystyle \frac{M^2{L}_f}{2p}}{\parallel \Phi -\Psi \parallel }_{\infty }^2.\hfill \end{array}$$

Next, for the second term, using the fractional Brownian motion isometry, for fixed $\tau$, we have

$$\begin{array}{cc}\hfill & E{∥\underset{0}{\overset{\tau }{\int }}R(\tau -s)\left(\sigma (s,{\Phi }_s)-\sigma (s,{\Psi }_s)\right)d{W}^H\left(s\right)∥}^2\hfill \\ \hfill & =H(2H-1)\underset{{[0,\tau ]}^2}{\int \int }E⟨R(\tau -r)\left(\sigma (r,{\Phi }_r)-\sigma (r,{\Psi }_r)\right),R(\tau -s)\left(\sigma (s,{\Phi }_s)-\sigma (s,{\Psi }_s)\right)⟩\hfill \\ \hfill & {\times |s-r|}^{2H-2}dsdr.\hfill \end{array}$$

By (H1) and (H5), we obtain

$$\begin{array}{cc}\hfill & E\parallel \underset{0}{\overset{\tau }{\int }}R(\tau -s)\left(\sigma (s,{\Phi }_s)-\sigma (s,{\Psi }_s)\right)d{W}^H\left(s\right){\parallel }^2\hfill \\ \hfill & \le M^{2}H(2H-1)L_{\sigma }{·\parallel \Phi -\Psi \parallel }_{\infty }^2\underset{{[0,\infty )}^2}{\int \int }{e}^{-2p(u+v)}{|u-v|}^{2H-2}dudv.\hfill \end{array}$$

For the third term, we use the periodic structure of the impulses (H3) and (H6). So,

$$\begin{array}{cc}\hfill & E{∥\sum _{0<{\tau }_k<\tau }R(\tau -{\tau }_k)\left(I_k\left(\Phi \left({\tau }_k^{-}\right)\right)-I_k\left(\Psi \left({\tau }_k^{-}\right)\right)\right)∥}^2\hfill \\ \hfill & \le \sum _{0<{\tau }_k<\tau }\parallel R(\tau -{\tau }_k){\parallel }^2·E{\parallel I_k\left(\Phi \left({\tau }_k^{-}\right)\right)-I_k\left(\Psi \left({\tau }_k^{-}\right)\right)\parallel }^2\hfill \\ \hfill & \le M^2{L}_I{\parallel \Phi -\Psi \parallel }_{\infty }^2\sum _{0<{\tau }_k<\tau }{e}^{-2p(\tau -{\tau }_k)}.\hfill \end{array}$$

Write indices as $k=j+qm$, $j=1,2,\dots ,m$. The geometric series bound gives, uniformly in $\tau$,

$$\begin{array}{c}\hfill \sum _{0<{\tau }_k<\tau }{e}^{-2p(\tau -{\tau }_k)}\le \sum _{j=1}^m\sum _{q\ge 0}{e}^{-2pq\omega }={\displaystyle \frac{m}{1-e^{-2p\omega }}}.\end{array}$$

Hence,

$$\begin{array}{c}\hfill E{∥\sum _{0<{\tau }_k<\tau }R(\tau -{\tau }_k)\left(I_k\left(\Phi \left({\tau }_k^{-}\right)\right)-I_k\left(\Psi \left({\tau }_k^{-}\right)\right)\right)∥}^2\le {\displaystyle \frac{M^{2}m}{1-e^{-2p\omega }}}{L}_I{\parallel \Phi -\Psi \parallel }_{\infty }^2.\end{array}$$

Combining these parts, we get

$$\begin{array}{cc}\hfill & {E\parallel \left(\Theta \Phi \right)\left(\tau \right)-\left(\Theta \Psi \right)\left(\tau \right)\parallel }^2\hfill \\ \hfill & \le {\displaystyle \frac{3M^2}{2p}}·L_f·{\parallel \Phi -\Psi \parallel }_{\infty }^2\hfill \\ \hfill & +3M^{2}H(2H-1)·L_{\sigma }{·\parallel \Phi -\Psi \parallel }_{\infty }^2\underset{{[0,\infty )}^2}{\int \int }{e}^{-2p(u+v)}{|u-v|}^{2H-2}dudv\hfill \\ \hfill & +3{\displaystyle \frac{M^{2}m}{1-e^{-2p\omega }}}·L_I·{\parallel \Phi -\Psi \parallel }_{\infty }^2\hfill \\ \hfill & =(C_f·L_f+C_{\sigma }·L_{\sigma }+C_I·L_I)·{\parallel \Phi -\Psi \parallel }_{\infty }^2,\hfill \end{array}$$

where $C_f={\displaystyle \frac{3M^2}{2p}}$, $C_{\sigma }={\displaystyle \frac{3M^{2}H(2H-1)·\Gamma (2H-1)}{{\left(2p\right)}^{2H}}}$, $C_I={\displaystyle \frac{3M^{2}m}{1-e^{-2p\omega }}}$.

The assumption $C_f·L_f+C_{\sigma }·L_{\sigma }+C_I·L_I<1$ demonstrates that the mapping $\Theta$ is a contraction on the Banach space ${\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$. By the Banach fixed point theorem, we obtain the existence of a unique fixed point $\Psi \in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$, meaning that Equation (1) has a unique solution in ${\mathcal{S}SAP}_{\omega ,c}\left([0,\infty )and{L}^2(\Omega ,\mathbb{H})\right)$. □

4. Applications

To illustrate the scope and effectiveness of our theoretical results, we now present two applications. The theoretical results on square-mean S-aasymptotically $(\omega ,c)$-periodic solutions provide a powerful framework for modeling the long-term modulated dynamics of complex systems that combine delays, stochastic noise, and abrupt perturbations, with direct applications spanning neuroscience, engineering, and finance. In neuroscience, the neutral term elegantly captures synaptic transmission delays, fractional Brownian motion (fBm) faithfully represents long-memory neural noise with persistent correlations $(H>1/2)$, and impulsive effects model sudden neuronal firings. Applying Theorem 7 enables precise prediction of stability thresholds; for instance, $\left|c\right|=1$ sustains persistent oscillatory patterns, while $\left|c\right|<1$ induces exponential decay toward a modulated attractor. In engineering vibration control, these results empower the design of adaptive damping mechanisms that enforce asymptotic periodicity, thereby minimizing resonant vibrations in critical infrastructure such as suspension bridges subjected to turbulent, stochastic wind loads. Similarly, in financial modeling, the framework can be adapted to stochastic volatility processes incorporating impulse-driven market crashes and neutral delayed feedback from lagged investor sentiment.

The first example concerns the heat semigroup on a bounded domain, which serves as a classical model for diffusion phenomena and allows us to demonstrate the applicability of our approach to semigroups arising in partial differential equations. The second example focuses on a scalar linear model with explicitly computable constants, highlighting the tractability of our framework in concrete situations where explicit estimates are essential.

Example 1.

Let $\mathbb{H}=L^2\left(\Omega \right)$, where $\Omega \subset {\mathbb{R}}^n$ is a bounded ${\mathcal{C}}^2$ domain. Let $A=\Delta$ with the Dirichlet boundary condition on Ω. Then ${\left(R\left(\tau \right)\right)}_{\tau \ge 0}={\left(e^{\Delta \tau }\right)}_{\tau \ge 0}$ is a $C_0$-semigroup with $\parallel R\left(\tau \right)\parallel \le M{e}^{-p\tau }$, $M=1$, $p=p_1>0$, where $p_1$ is the first Dirichlet eigenvalue. Fix $\omega >0$ and $m\in \mathbb{N}$, with impulse times ${\tau }_k={\displaystyle \frac{k}{m}}\omega$, so that ${\tau }_{k+m}={\tau }_k+\omega$ and $\omega ={\tau }_m$. Let $W^H$ be a real fractional Brownian motion with $H>{\displaystyle \frac{1}{2}}$. Let F and G be functions such that

$$\begin{array}{c}\hfill \underset{\xi }{sup}\parallel F\left(\xi \right)\parallel =:M_F<\infty \mathit{and}\underset{\xi }{sup}\parallel G\left(\xi \right)\parallel =:M_G<\infty .\end{array}$$

Additionally, F and G are Lipschitz-continuous with Lipschitz constants $L_F$ and $L_G$, respectively. Define

$$f(\tau ,{\Psi }_{\tau })={\displaystyle \frac{1}{1+\tau }}F\left({\Psi }_{\tau }\right)\mathit{and}\sigma (\tau ,{\Psi }_{\tau })={(1+\tau )}^{-\gamma }G\left({\Psi }_{\tau }\right),$$

where $\gamma >0$. Let $a\left(\tau \right)={\displaystyle \frac{1}{1+\tau }}$ and $b\left(\tau \right)={(1+\tau )}^{-\gamma }$ for $\gamma >0$. Note that $a\left(\tau \right),b\left(\tau \right)\to 0$, as $\tau \to \infty$. Then for arbitrary process $\Psi \in \mathcal{B}PC([0,\infty ),L^2(\Omega ,\mathbb{H}))$, we have that $f(·,{\Psi }_{·})$, $\sigma (·,{\Psi }_{·})\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$. Indeed,

$$\begin{array}{cc}\hfill & E\parallel f(\tau +\omega ,{\Psi }_{\tau +\omega })-cf(\tau ,{\Psi }_{\tau }){\parallel }^2\hfill \\ \hfill & =E\parallel a(\tau +\omega )F\left({\Psi }_{\tau +\omega }\right)-ca\left(\tau \right)F\left({\Psi }_{\tau }\right){\parallel }^2\hfill \\ \hfill & \le {2\left|a(\tau +\omega )\right|}^{2}E\parallel F\left({\Psi }_{\tau +\omega }\right){\parallel }^2+{2\left|c\right|}^2|a\left(\tau \right){|}^{2}E{\parallel F\left({\Psi }_{\tau }\right)\parallel }^2\hfill \\ \hfill & \le 2M_F^2\left({\left|a(\tau +\omega )\right|}^2+{\left|a\left(\tau \right)\right|}^2\right)\to 0,\hfill \end{array}$$

when $\tau \to \infty$. The proof for $\sigma (·,{\Psi }_{·})\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$ is analogous. Let the impulses be linear: $I_k\left(x\right)=Kx$, for a fixed $K\subset L\left(H\right)$, with $\parallel K\parallel =\kappa$. Hence $I_{k+m}=I_k$.

It is obvious that (H0) and (H1) hold. So,

$$\begin{array}{c}\hfill {\left(E\parallel f(\tau ,{\Phi }_{\tau })-f(\tau ,{\Psi }_{\tau }){\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}\le M_F{\left({E\parallel \Phi \left(\tau \right)-\Psi \left(\tau \right)\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}},\end{array}$$

so $L_f=M_F$. Likewise,

$$\begin{array}{c}\hfill {\left(E\parallel \sigma (\tau ,{\Phi }_{\tau })-\sigma (\tau ,{\Psi }_{\tau }){\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}\le M_G{\left({E\parallel \Phi \left(\tau \right)-\Psi \left(\tau \right)\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}},\end{array}$$

so $L_{\sigma }=M_G$. For impulses,

$$\begin{array}{c}\hfill \parallel I_k\left(x\right)-I_k\left(y\right)\parallel \le \kappa \parallel x-y\parallel ,\end{array}$$

so $L_I=\kappa$. Also, $v_k=I_k\left(\Phi \left({\tau }_k^{-}\right)\right)=K\Phi \left({\tau }_k^{-}\right)$ satisfies the discrete ${\mathcal{S}SAP}_{\omega ,c}$ property by the Lipschitz continuity and the periodicity of $\left({\tau }_k\right)$; hence, the impulsive part is ${\mathcal{S}SAP}_{\omega ,c}$. Now, putting $M=1$, $p=p_1$,

$$C_f={\displaystyle \frac{3}{2p_1}},$$

$$C_{\sigma }={\displaystyle \frac{3H(2H-1)·\Gamma (2H-1)}{{\left(2p_1\right)}^{2H}}}$$

and

$$C_I={\displaystyle \frac{3m}{1-e^{-2p_1\omega }}}.$$

Therefore,

$$\begin{array}{c}\hfill K=C_f{L}_f+C_{\sigma }{L}_{\sigma }+C_I{L}_I={\displaystyle \frac{3M_F}{2p_1}}+{\displaystyle \frac{3H(2H-1)M_G·\Gamma (2H-1)}{{\left(2p_1\right)}^{2H}}}+{\displaystyle \frac{3m\kappa }{1-e^{-2p_1\omega }}}.\end{array}$$

By choosing small enough $M_F,M_G,\kappa$ or taking a domain with larger $p_1$, we get $K<1$, so by Theorem 7, the equation admits a unique solution in ${\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$.

Example 2.

Let $\mathbb{H}=\mathbb{R}$, $A=-a$, with $a>0$. Then $R\left(\tau \right)=e^{-a\tau }$, $M=1$, $p=a$. For a fixed period $\omega >0$ and impulse times ${\tau }_k={\displaystyle \frac{k}{m}}\omega$ for some $m\in \mathbb{N}$, so ${\tau }_{k+m}={\tau }_k+\omega$. Let $W^H$ be a real fractional Brownian motion with $H\in \left({\displaystyle \frac{1}{2}},1\right)$. Choose F and G as in the previous example. Now, define

$$\begin{array}{c}\hfill f(\tau ,{\Psi }_{\tau })=e^{-\lambda \tau }F\left({\Psi }_{\tau }\right),\lambda >0\mathit{and}\sigma (\tau ,{\Psi }_{\tau })={\displaystyle \frac{1}{ln(e+\tau )}}G\left({\Psi }_{\tau }\right).\end{array}$$

With the same arguments as before, we conclude that $f(·,{\Psi }_{·})$, $\sigma (·,{\Psi }_{·})\in {\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$ for arbitrary process $\Psi \in \mathcal{B}PC([0,\infty ),L^2(\Omega ,\mathbb{H}))$. Let the impulses be $I_k\left(x\right)=\gamma x$, with $\gamma \in \mathbb{R}$, so $I_{k+m}=I_k$. Because the model is linear, we can write the mild solution explicitly

$$\begin{array}{cc}\hfill & \Psi \left(\tau \right)=\underset{0}{\overset{\tau }{\int }}{e}^{-a(\tau -s)}{e}^{-\lambda s}F\left({\Psi }_s\right)ds+\underset{0}{\overset{\tau }{\int }}{e}^{-a(\tau -s)}{\displaystyle \frac{1}{ln(e+\tau )}}G\left({\Psi }_s\right),d{W}^H\left(s\right)\hfill \\ \hfill & +\sum _{0<{\tau }_k<\tau }{e}^{-a(\tau -{\tau }_k)}\gamma \Phi \left({\tau }_k^{-}\right).\hfill \end{array}$$

The Lipschitz constants are $L_f=M_F$, $L_{\sigma }=M_G$, and $L_I=\left|\gamma \right|$. Since $v_k=\gamma \Phi \left({\tau }_k^{-}\right)$ is a discrete ${\mathcal{S}SAP}_{\omega ,c}$ sequence, the impulsive part is in ${\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$. Let $R\left(\tau \right)=e^{-a\tau }$, $C_f={\displaystyle \frac{3}{2a}}$,

$$C_{\sigma }={\left(3H(2H-1)\underset{{[0,\infty )}^2}{\int \int }{e}^{-2a(u+v)}{|u-v|}^{2H-2}dudv\right)}^{{\displaystyle \frac{1}{2}}}$$

and

$$C_I={\displaystyle \frac{3m}{1-e^{-2a\omega }}}.$$

By evaluation of the double integral, we obtain that

$$\begin{array}{c}\hfill \underset{{[0,\infty )}^2}{\int \int }{e}^{-2a(u+v)}{|u-v|}^{2H-2}dudv={\displaystyle \frac{\Gamma (2H-1)}{{\left(2a\right)}^{2H}}},\end{array}$$

so

$$\begin{array}{c}\hfill C_{\sigma }={\displaystyle \frac{3H(2H-1)\Gamma (2H-1)}{{\left(2a\right)}^{2H}}}.\end{array}$$

Hence,

$$\begin{array}{c}\hfill K={\displaystyle \frac{M_F}{2a}}+{\displaystyle \frac{3H(2H-1)M_G\Gamma (2H-1)}{{\left(2a\right)}^{2H}}}+{\displaystyle \frac{3m\left|\gamma \right|}{1-e^{-2a\omega }}}.\end{array}$$

If $K<1$ (it can be achieved by choosing $M_F$ and $M_G$ (i.e., the functions F and G) and $\left|\gamma \right|$ to be sufficiently small, or using a larger decay a), by Theorem 7, the solution of the equation is unique and belongs to ${\mathcal{S}SAP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$.

Fractional calculus, via fractional derivatives or fractional noise like fBm $(H\ne 1/2)$, introduces memory effects that alter long-term dynamics: for $H>1/2$ (persistent noise), solutions exhibit long-range correlations, leading to slower decay toward equilibria and enhanced periodicity robustness against impulses. In impulsive SDEs, fractional terms smooth discontinuities, promoting quasi-periodic behaviors over chaotic ones. For periodicity, fractional orders can shift Bloch thresholds or stabilize S-asymptotic $(\omega ,c)$-periodic solutions by damping high-frequency modes, as seen in the paper’s semigroup convolutions where fBm isometry bounds ensure mean-square convergence.

5. Conclusions

In this paper we have established the existence of square-mean S-aasymptotically $(\omega ,c)$-periodic solutions for impulsive stochastic differential equations within the framework of fractional calculus. The main results in this paper provide sufficient conditions for the existence and uniqueness of square-mean S-aasymptotically $(\omega ,c)$-periodic mild solutions of the considered stochastic differential equation, namely, (i) the operator A generates a compact analytic $C_0$-semigroup with exponential decay; (ii) the functions $f,\sigma ,I_k$ are uniformly Lipschitz-continuous and square-mean S-aasymptotically $(\omega ,c)$-periodic functions; (iii) the sequence of impulsive times are quasi-periodic with period $\omega$; (iv) the technical conditions (H0) and $C_f{L}_f+C_{\sigma }{L}_{\sigma }+C_I{L}_I<1$ hold. The applications to the heat semigroup on a bounded domain and to a scalar linear model with explicit constants demonstrate that our abstract results are not only of theoretical interest but also extend naturally to concrete and computable models.

The approach proposed in this paper ensures global existence and uniqueness of solutions without introducing discretization errors, making it particularly suitable for infinite-dimensional spaces such as the Hilbert space $\mathbb{H}$. However, its main limitations lie in its non-constructive nature (absence of explicit solutions), sensitivity to the Lipschitz constants, and high computational cost when applied to simulations.

As the paper’s framework encompasses neutral impulsive SDEs driven by fBm, Theorem 7 can be applied to population models like Lotka-Volterra SDEs with impulses (predator introductions) and neutral delays (gestation periods), ensuring the existence of S-aasymptotically $(\omega ,c)$-periodic solutions under Lipschitz conditions.

Several directions for further investigation remain open. One natural extension is to consider nonlinear systems, where the interplay between impulsive effects and stochastic perturbations may produce richer dynamical behaviors. Another promising direction is the study of such problems in Banach spaces with weaker topological structures, which may broaden the applicability to more general classes of evolution equations. Furthermore, it would be of interest to analyze the stability and robustness of the obtained solutions with perturbations of the impulses or stochastic terms, as well as to investigate numerical approximation schemes that preserve the S-asymptotic periodicity properties. Finally, potential applications to models in physics, biology, and engineering—such as population dynamics with stochastic perturbations or diffusion processes with impulses—deserve detailed study.