Approximate Controllability of Higher-Order Hilfer Fractional Neutral Stochastic Systems Driven by Fractional Brownian Motion, Poisson Jumps, and Non-Instantaneous Impulses

Abstract

This paper addresses the existence of mild solutions and the approximate controllability of a class of higher-order Hilfer fractional semi-linear neutral stochastic differential equations with non-instantaneous impulses in Hilbert spaces. The system is driven by both fractional Brownian motion and Poisson jumps, thereby capturing long-range dependence as well as random discontinuities. By combining techniques from fractional calculus, stochastic analysis, and operator theory, we establish sufficient conditions for the existence of mild solutions. The analysis is carried out through the construction of suitable solution operator families and the application of Sadovskii’s fixed point theorem in an appropriate phase space framework. In addition, we investigate the controllability properties of the system and derive criteria ensuring approximate controllability of the underlying fractional neutral dynamics. The proposed approach relies on the structural properties of the higher-order Hilfer fractional derivative, estimates for stochastic integrals with respect to fractional Brownian motion, and compactness arguments adapted to non-instantaneous impulsive effects. The inclusion of Poisson jumps and neutral terms introduces significant analytical difficulties, which are overcome using refined resolvent operator techniques and fractional power estimates. An illustrative example is presented to demonstrate the applicability of the theoretical results. The results obtained generalize and unify several recent developments in the theory of fractional stochastic systems and provide a flexible framework for analyzing controlled dynamical models with memory, randomness, and impulsive behavior.

1. Introduction

Fractional calculus concerns the formulation of noninteger-order integral and differential operators and has evolved into a powerful analytical framework for modeling systems with memory and hereditary characteristics [1,2,3]. In recent decades, fractional differential equations have emerged as a versatile tool for describing complex phenomena such as anomalous transport, viscoelastic behavior, scale-invariant responses, and feedback mechanisms in control systems [4,5,6]. Classical developments have largely relied on the Riemann–Liouville and Caputo derivatives, which underpin much of the established theory of mild solutions ($\mathbb{MS}$s), whereas more generalized operators have received comparatively limited attention. Among these, the Hilfer fractional derivative (HFD), introduced by Hilfer [7], provides an interpolative structure that unifies and extends classical formulations. Foundational contributions by Furati and Kassim, Wang and Zhang, and, later, Gu and Trujillo [8,9,10] have advanced the analytical treatment of such operators, particularly through noncompactness techniques. Owing to its structural flexibility, the Hilfer framework has found applications across diverse scientific domains, including materials with memory, porous media, fluid dynamics, and engineering systems [11,12]. In addition to modeling aspects, qualitative properties such as stability and robustness have also attracted considerable attention. In particular, Ulam-type stability for integral equations with time-varying delays has been investigated in recent studies [13]. Recent investigations have further incorporated stochastic effects, higher-order dynamics, and abstract functional settings, with emphasis on well-posedness and controllability in Banach spaces via fixed-point methods and measures of non-compactness [14,15,16].

Stochastic differential equations provide a fundamental paradigm for modeling systems influenced by random perturbations. In their classical form, driven by Brownian motion, they represent Gaussian diffusion processes [17,18]. The integration of fractional operators into stochastic models enables the simultaneous representation of memory effects and randomness, leading to fractional stochastic differential inclusions with long-range dependence [19,20]. However, many real-world phenomena exhibit discontinuities that cannot be captured by continuous noise alone. Poisson random measures are therefore essential for modeling jump-type disturbances arising in applications such as finance, biology, and engineering. Consequently, fractional stochastic systems with Poisson jumps have attracted increasing attention, as evidenced by the works of Tamilalagan [21], Rihan et al. [22], Balasubramaniam et al. [23], and, more recently, Sayed Ahmed et al. [24].

In parallel, fractional Brownian motion (fBm), introduced by Kolmogorov and later developed by Mandelbrot and Van Ness [25], extends classical Brownian motion through the Hurst parameter $H\in (0,1)$, encoding self-similarity and long-range dependence. Depending on whether $H<\frac{1}{2}$ or $H>\frac{1}{2}$, the process exhibits antipersistent or persistent behavior, respectively. This feature renders fBm particularly suitable for modeling systems with temporal correlations. Significant progress in stochastic evolution equations driven by fBm has been achieved by Maslowski and Nualart, Caraballo et al., among others [26,27], who established existence and qualitative properties of $\mathbb{MS}$s in Hilbert spaces. The combined presence of fractional derivatives, fBm, and Poisson jumps thus yields a highly expressive framework for capturing the interplay between memory, diffusion, and abrupt perturbations.

Controllability constitutes a central concept in the qualitative analysis of dynamical systems, describing the ability to steer system trajectories toward desired configurations [28,29]. While exact controllability is often unattainable in infinite-dimensional or stochastic contexts, alternative notions have been developed. Among these, approximate controllability ($\mathbb{AC}$) provides a more realistic criterion by requiring that the target state be approached arbitrarily closely [30,31]. This notion is particularly suitable for fractional stochastic systems, where memory effects and random disturbances hinder exact steering. As a result, $\mathbb{AC}$ has become a key analytical tool for assessing the influence of control mechanisms in complex dynamical environments [32,33,34]. Recent developments in this direction include the works of Ahmed et al. [35] and Yadav et al. [36], which address approximate controllability of stochastic fractional systems with impulsive effects and complex noise structures.

Impulsive differential equations extend classical models by incorporating instantaneous or non-instantaneous state changes at specified moments [37,38]. Such mechanisms arise naturally in applications involving sudden transitions, including mechanical impacts, biological interventions, and engineering adjustments. When combined with fractional and stochastic dynamics, impulsive effects yield hybrid systems in which continuous evolution interacts with discrete perturbations, thereby enhancing modeling fidelity. The study of existence and controllability for such systems is therefore essential for understanding and regulating processes characterized by both gradual and abrupt dynamics.

Despite substantial progress in fractional stochastic analysis, the specific class of systems considered herein has not been comprehensively addressed. In particular, higher-order Hilfer fractional semi-linear neutral systems driven by fBm and Poisson jumps, together with non-instantaneous impulses and control inputs, remain largely unexplored. The coexistence of memory effects, long-range dependence, jump discontinuities, and neutral structures introduces significant analytical challenges.

Motivated by these considerations, this paper investigates a class of higher-order Hilfer fractional stochastic control systems with $1<\wp <2$ in a separable Hilbert space (sHS). The model incorporates non-instantaneous impulses, semilinear neutral terms, and stochastic perturbations driven by fBm and Poisson jumps. The main objective is to establish the existence of $\mathbb{MS}$s via Sadovskii’s fixed point theorem and analyze the $\mathbb{AC}$ of the associated system, thereby extending the current theoretical framework for complex fractional stochastic dynamics.

Let ${}^{\mathbb{H}}D_{0+}^{\wp ,\Re }$ denote the higher-order HFD with order $1<\wp <2$ and type parameter $\Re \in [0,1]$. Let $\mathcal{X}$ be a real sHS endowed with the inner product ${⟨·,·⟩}_{\mathcal{X}}$ and the associated norm ${\parallel ·\parallel }_{\mathcal{X}}$. We consider the state trajectory $z(·):[0,\mathfrak{a}]\to \mathcal{X}$ governed by the following higher-order Hilfer fractional stochastic differential equation (HFSDE) involving nonlocal initial conditions and non-instantaneous impulsive effects:

$$\left\{\begin{array}{cc}\hfill {}^{\mathbb{H}}D_{0+}^{\wp ,\Re }& \left[z(\ell )+\mathcal{F}\left(\ell ,z(\ell ),z\left({\zeta }_1(\ell )\right),\cdots ,z\left({\zeta }_m(\ell )\right)\right)\right]\hfill \\ \hfill =& \mathcal{A}\left[z(\ell )+\mathcal{F}\left(\ell ,z(\ell ),z\left({\zeta }_1(\ell )\right),\cdots ,z\left({\zeta }_m(\ell )\right)\right)\right]+\mathfrak{B}u(\ell )\hfill \\ \hfill & +{\int }_0^{\ell }\mathfrak{F}\left(s,z\left(s\right),z\left(\widehat{{\zeta }_1}\left(s\right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(s\right)\right)\right)d\mathcal{W}\left(s\right)\hfill \\ \hfill & +\mathfrak{O}\left(\ell ,z(\ell ),z\left({\xi }_1(\ell )\right),\cdots ,z\left({\xi }_m(\ell )\right)\right){\displaystyle \frac{d{\mathcal{W}}^H(\ell )}{d\ell }}\hfill \\ \hfill & +{\int }_{\mathsf{\Pi }}\hslash \left(\ell ,z(\ell ),z\left({\varrho }_1(\ell )\right),\cdots ,z\left({\varrho }_m(\ell )\right),\eta \right)\widehat{N}(d\ell ,d\eta ),\hfill \\ \hfill & \ell \in (s_j,{\ell }_{j+1}]\subset [0,\mathfrak{a}],j=0,1,2,\cdots ,\beta ,\hfill \\ \hfill z(\ell )& ={\mathbb{O}}_j^1(\ell ,z(\ell )),z^{\prime }(\ell )={\mathbb{O}}_j^2(\ell ,z(\ell )),\hfill \\ \hfill & \ell \in ({\ell }_j,s_j],j=1,2,\cdots ,\beta ,\hfill \\ \hfill \left({}^{\mathbb{RL}}\mathbf{I}_{0+}^{2-\mu }z\right)\left(0\right)& +{\mathfrak{g}}_1\left(z\right)=z_0,\hfill \\ \hfill {\left({}^{\mathbb{RL}}\mathbf{I}_{0+}^{2-\mu }z\right)}^{\prime }\left(0\right)& +{\mathfrak{g}}_2\left(z\right)=z_1.\hfill \end{array}\right.$$

(1)

Here, ${}^{\mathbb{RL}}\mathbf{I}_{0+}^{2-\mu }$ denotes the Riemann–Liouville fractional integral of order $2-\mu$, with $\mu =\wp +\Re (2-\wp )$. The linear operator $\mathcal{A}:\mathbb{D}\left(\mathcal{A}\right)\subset \mathcal{X}\to \mathcal{X}$ generates a strongly continuous (SC) cosine family ${\left\{\widehat{C}(\ell )\right\}}_{\ell \ge 0}$ on the Hilbert space $\mathcal{X}$, facilitating the representation of higher-order dynamics. The bounded linear control operator $\mathfrak{B}:\mathcal{U}\to \mathcal{X}$ mediates between the control space $\mathcal{U}$, a sHS of admissible inputs, and the state space $\mathcal{X}$. The control signal $u(·)$ belongs to ${\mathcal{L}}_{\widehat{Y}}^2([0,\mathfrak{a}];\mathcal{U})$, the space of square–integrable, $\left\{{\widehat{Y}}_{\ell }\right\}$–adapted, $\mathcal{U}$–valued stochastic processes. Let $\mathcal{Y}$ be an auxiliary sHS endowed with the norm ${\parallel ·\parallel }_{\mathcal{Y}}$ and inner product ${⟨·,·⟩}_{\mathcal{Y}}$. The stochastic forcing of the system is characterized by a $\mathcal{Y}$–valued Brownian motion ${\left\{\mathcal{W}(\ell )\right\}}_{\ell \ge 0}$ defined on a complete filtered probability space $(\mathrm{\Omega },\widehat{Y},{\left\{{\widehat{Y}}_{\ell }\right\}}_{\ell \ge 0},\mathbb{P})$, with covariance operator $\mathbb{Q}\ge 0$ possessing finite trace. Moreover, ${\left\{{\mathcal{W}}^H(\ell )\right\}}_{\ell \ge 0}$ represents a fBm with Hurst index $H\in (1/2,1)$, taking values in $\mathcal{Y}$ and adapted to the same filtered space. Let the sequences ${\left\{s_{\mu }\right\}}_{\mu =0}^{\beta }$ and ${\left\{{\ell }_{\mu }\right\}}_{\mu =1}^{\beta +1}$ satisfy $0=s_0<{\ell }_1\le s_1\le {\ell }_2<\cdots <s_{\beta -1}<{\ell }_{\beta }\le s_{\beta }\le {\ell }_{\beta +1}=\mathfrak{a}$ and let ${\mathbb{O}}_j^1,{\mathbb{O}}_j^2$ denote the non–instantaneous impulsive operators for all $j=1,2,\cdots ,\beta$. The mappings $\mathcal{F},\mathfrak{F},\mathfrak{O},{\mathbb{O}}_j^1,{\mathbb{O}}_j^2,\hslash ,{\mathfrak{g}}_1$, and ${\mathfrak{g}}_2$ are assumed to be appropriately defined nonlinear functions, whose precise properties will be specified subsequently.

This work develops a unified analytical framework for higher-order Hilfer-type semi-linear neutral stochastic differential equations (HFSDEs) incorporating nonlocal initial conditions, non-instantaneous impulses, and stochastic perturbations. The underlying dynamics are governed by a HFD of order $\wp \in (1,2)$ and type $\Re \in [0,1]$, which jointly encode hereditary memory effects and nonlocal temporal dependence. The system is further perturbed by fractional Brownian motion (fBm) and Poisson jump processes, thereby capturing the coexistence of long-range dependence and abrupt stochastic discontinuities within a hybrid stochastic environment.

In contrast to many existing studies that either treat fractional Brownian motion or jump-type noise separately, or adopt unified Lévy-driven formulations, the present work adopts a decomposed stochastic structure in which continuous long-memory effects (fBm) and discontinuous jump phenomena (Poisson measures) are modeled simultaneously but distinctly. This separation enables a more transparent analytical treatment of memory and jump mechanisms and allows their individual influence on controllability properties to be rigorously characterized.

The proposed formulation is established in a sHS and integrates neutral operators, impulsive effects, and mixed stochastic drivers within a single coherent structure. This setting significantly extends existing deterministic and single-noise models, yielding a more expressive framework for systems influenced simultaneously by memory, diffusion-type randomness, and discontinuous events. The considered model is particularly relevant in applications where hereditary effects, persistent correlations, and sudden stochastic shocks interact in a nontrivial manner.

The first main contribution concerns the existence and well-posedness of $\mathbb{MS}$s. By employing Sadovskii’s fixed point theorem in combination with measures of non-compactness, fractional integral estimates, and generalized Gronwall–Bihari-type inequalities, we derive sufficient conditions ensuring the solvability of the considered HFSDE. This approach establishes a rigorous connection between operator-theoretic techniques and stochastic fractional evolution systems of neutral type, thereby extending several recent results in the literature.

The second contribution addresses the $\mathbb{AC}$ property of the associated control system. Through refined estimates of stochastic resolvent operators and convolution-type integral representations, together with compactness properties of stochastic semigroups, we obtain sufficient conditions guaranteeing that controlled trajectories can be steered arbitrarily close to a prescribed terminal state. Notably, $\mathbb{AC}$ is achieved despite the simultaneous presence of fractional memory, impulsive dynamics, neutral effects, and stochastic perturbations, even in regimes where exact controllability fails.

The main objective of this work is to develop a unified analytical framework for higher-order, Hilfer-type, semi-linear, neutral stochastic differential equations with nonlocal initial conditions, non-instantaneous impulses, and mixed stochastic perturbations driven by fractional Brownian motion and Poisson jump processes. A constructive example is included to illustrate the applicability of the theoretical results and demonstrate how the proposed conditions can be explicitly verified in practice. The obtained theoretical results are applicable to a wide range of real-world systems where memory effects, stochastic fluctuations, and impulsive behaviors coexist. Typical examples include viscoelastic materials with hereditary stress–strain relations, financial systems subject to sudden market jumps, and biological or engineering control systems where external interventions occur at discrete moments. In such settings, the proposed framework provides a rigorous mathematical foundation for analyzing controllability and stability properties under the combined influence of memory, randomness, and discontinuous dynamics. It is worth emphasizing that the proposed model is not artificially constructed but instead emerges naturally from the interaction of three independently established mechanisms in stochastic dynamics: (i) fractional-order memory effects described by Hilfer operators, (ii) long-range dependent stochastic fluctuations modeled by fractional Brownian motion, and (iii) discontinuous jump phenomena represented by Poisson random measures. Their combination reflects realistic systems in which memory, persistence, and abrupt stochastic events coexist rather than act in isolation.

The specific structure of the proposed system is motivated by the necessity to unify several independently observed mechanisms in stochastic dynamics that cannot be adequately captured within a single classical framework. It is important to emphasize that the proposed model is not introduced as an abstract mathematical generalization without physical or applied motivation. Rather, the considered system naturally arises from the simultaneous interaction of three fundamental mechanisms observed in complex dynamical environments: fractional-order memory effects, long-range dependent stochastic fluctuations, and discontinuous jump phenomena. Fractional derivatives of the Hilfer type describe hereditary behavior in viscoelastic and anomalous diffusion systems, while fractional Brownian motion captures persistent correlations observed in financial and networked systems. In contrast, Poisson random measures model sudden and unpredictable shocks commonly encountered in insurance risk, finance, and engineering systems subject to abrupt external disturbances. The coupling of these mechanisms with neutral terms, non-instantaneous impulses, and nonlocal initial conditions reflects realistic situations in which system evolution depends not only on its current state but also on past history and delayed interventions. Therefore, the present framework provides a mathematically rigorous and unified description of systems in which memory, persistence, and abrupt stochastic events coexist in a nontrivial and inherently coupled manner.

The remainder of the paper is organized as follows. Section 2 presents the necessary functional setting, notation, and auxiliary results. Section 3 establishes the existence theory of $\mathbb{MS}$s for the considered HFSDE of order $1<\wp <2$ under non-instantaneous impulses and stochastic perturbations driven by fBm and Poisson jumps. Section 4 investigates the $\mathbb{AC}$ problem for the corresponding control system and analyzes the combined effects of memory, stochasticity, and impulsive behavior on system trajectories. Finally, Section 5 provides a constructive example illustrating the theoretical findings and their applicability.

2. Preliminaries

Throughout this section, we assemble the essential structural ingredients, operator–theoretic settings, and analytical machinery required to rigorously articulate and substantiate the main results of this work.

Fix a time interval $[0,\mathfrak{a}]$ and let $(\mathrm{\Omega },\widehat{Y},{\left\{{\widehat{Y}}_{\ell }\right\}}_{\ell \ge 0},\mathbb{P})$ be a complete probability space. Let $(\mathcal{V},\mathsf{\Phi },\kappa (d\nu \left)\right)$ be a $\sigma$–finite measurable space. Consider a stationary Poisson point process ${\left\{p_{\ell }\right\}}_{\ell \ge 0}$ defined on the probability space $(\mathrm{\Omega },\widehat{Y},{\left\{{\widehat{Y}}_{\ell }\right\}}_{\ell \ge 0},\mathbb{P})$ taking values in $\mathcal{V}$ and governed by the intensity measure $\kappa$. The associated counting measure is denoted by $\aleph (\ell ,d\nu )$ and satisfies $\widehat{\aleph }(\ell ,\mathfrak{b}):=\mathbb{E}\left[\aleph (\ell ,\mathfrak{b})\right]=\ell \kappa \left(\mathfrak{b}\right),\mathfrak{b}\in \mathsf{\Phi }.$ The compensated Poisson random measure is then introduced by $\widehat{\aleph }(\ell ,d\nu ):=\aleph (\ell ,d\nu )-\ell \kappa \left(d\nu \right),$ which constitutes a martingale measure corresponding to the jump structure of the Poisson point process ${\left\{p_{\ell }\right\}}_{\ell \ge 0}$.

A standard fBm $\{{\mathcal{W}}^H(\ell ),\ell \in \mathbb{R}\}$ with Hurst index $H\in (0,1)$ is understood as a mean–zero Gaussian process whose trajectories are almost surely continuous and whose covariance kernel is prescribed by

$$CO{V}_H(\ell ,s)=\mathbb{E}\left[{\mathcal{W}}^H(\ell ){\mathcal{W}}^H\left(s\right)\right]={\displaystyle \frac{1}{2}}\left({\ell }^{2H}+s^{2H}-{|\ell -s|}^{2H}\right),\ell ,s\in \mathbb{R}.$$

Remark 1.

For the long-memory regime $H>\frac{1}{2}$, it is well known (see, e.g., [39]) that the covariance function of fractional Brownian motion admits a mixed second-order derivative in the sense of distributions (or generalized functions), given by

$${\displaystyle \frac{{\partial }^{2}CO{V}_H}{\partial \ell \partial s}}(\ell ,s)=H(2H-1){|\ell -s|}^{2H-2},\ell ,s\ge 0,$$

which is locally integrable on ${\mathbb{R}}^2$ since $2H-2>-1$. Consequently, the covariance kernel can be represented in the integral form

$$CO{V}_H(\ell ,s)=H(2H-1){\int }_0^{\ell }{\int }_0^s{|u-v|}^{2H-2}dudv.$$

Let $\mathcal{X}$ and $\mathcal{Y}$ be real, separable Hilbert spaces, and denote by $\mathcal{L}(\mathcal{Y},\mathcal{X})$ the Banach space consisting of all bounded linear mappings from $\mathcal{Y}$ into $\mathcal{X}$. Consider a linear operator $\widehat{\mathrm{\Theta }}\in \mathcal{L}\left(\mathcal{Y}\right)$ which is positive, self–adjoint, and of trace class. Thus, there exists a complete orthonormal system ${\left\{{\mathfrak{e}}_n\right\}}_{n\ge 1}$ in $\mathcal{Y}$ together with a sequence of nonnegative eigenvalues ${\left\{{\mathfrak{v}}_n\right\}}_{n\ge 1}$ such that

$$\widehat{\mathrm{\Theta }}{\mathfrak{e}}_n={\mathfrak{v}}_n{\mathfrak{e}}_n,{\mathfrak{v}}_n\ge 0,{\displaystyle \sum _{n=1}^{\infty }}{\mathfrak{v}}_n=Tr\left(\widehat{\mathrm{\Theta }}\right)<\infty .$$

The $\mathcal{Y}$–valued $\widehat{\mathrm{\Theta }}$–fBm $\{{\mathcal{W}}^H(\ell ),\ell \ge 0\}$ is introduced through the spectral expansion

$${\mathcal{W}}^H(\ell )={\mathcal{W}}_{\widehat{\mathrm{\Theta }}}^H(\ell )={\displaystyle \sum _{n=1}^{\infty }}\sqrt{{\mathfrak{v}}_n}{\mathfrak{e}}_n{\mathcal{W}}_n^H(\ell ),$$

where ${\left\{{\mathcal{W}}_n^H\right\}}_{n\ge 1}$ denotes a family of mutually independent real-valued fBms sharing the same Hurst index H. This series converges in ${\mathcal{L}}^2(\mathrm{\Omega };\mathcal{Y})$ and defines a centered Gaussian process with values in $\mathcal{Y}$. Moreover, its covariance structure is given by

$$\mathbb{E}\left[{⟨{\mathcal{W}}^H(\ell ),x⟩}_{\mathcal{Y}}{⟨{\mathcal{W}}^H\left(s\right),y⟩}_{\mathcal{Y}}\right]=CO{V}_H(\ell ,s){⟨\widehat{\mathrm{\Theta }}x,y⟩}_{\mathcal{Y}},x,y\in \mathcal{Y},\ell ,s\in [0,\mathfrak{a}].$$

Let ${\mathcal{L}}_0^2(\mathcal{Y},\mathcal{X})$ represent the collection of all $\mathrm{\Theta }$–Hilbert–Schmidt operators mapping from $\mathcal{Y}$ into $\mathcal{X}$, explicitly defined by

$${\mathcal{L}}_0^2(\mathcal{Y},\mathcal{X}):=\left\{\varphi \in {\mathcal{L}(\mathcal{Y},\mathcal{X}):\parallel \varphi \parallel }_{{\mathcal{L}}_0^2}^2:={\displaystyle \sum _{n=1}^{\infty }}{\mathfrak{v}}_n{\parallel \varphi {\mathfrak{e}}_n\parallel }_{\mathcal{X}}^2<\infty \right\}.$$

Endowed with the canonical inner product

$${⟨\psi ,\varphi ⟩}_{{\mathcal{L}}_0^2}:={\displaystyle \sum _{n=1}^{\infty }}{\mathfrak{v}}_n{⟨\psi {\mathfrak{e}}_n,\varphi {\mathfrak{e}}_n⟩}_{\mathcal{X}},$$

this space constitutes a separable Hilbert structure suitable for stochastic integration with respect to $\widehat{\mathrm{\Theta }}$–fBm.

Let $\varphi :[0,\mathfrak{a}]\to {\mathcal{L}}_0^2(\mathcal{Y},\mathcal{X})$ be a measurable operator-valued function satisfying

$${\int }_0^{\mathfrak{a}}{\parallel \varphi \left(s\right)\parallel }_{{\mathcal{L}}_0^2}^{2}ds<\infty .$$

The stochastic convolution of $\varphi$ with respect to the $\widehat{\mathrm{\Theta }}$–fBm ${\mathcal{W}}^H$ is then articulated as

$${\int }_0^{\ell }\varphi \left(s\right)d{\mathcal{W}}^H\left(s\right):={\displaystyle \sum _{n=1}^{\infty }}{\int }_0^{\ell }\sqrt{{\mathfrak{v}}_n}\varphi \left(s\right){\mathfrak{e}}_{n}d{\mathcal{W}}_n^H\left(s\right),\ell \in [0,\mathfrak{a}],$$

which is well-posed under the prescribed Hilbert–Schmidt norm.

Lemma 1

([26]). Let $\varphi :[0,\mathfrak{a}]\to {\mathcal{L}}_0^2(\mathcal{Y},\mathcal{X})$ satisfy

$${\int }_0^{\mathfrak{a}}{\parallel \varphi \left(s\right)\parallel }_{{\mathcal{L}}_0^2}^{2}ds<\infty .$$

Then the stochastic integral ${\int }_0^{\ell }\varphi \left(s\right)d{\mathcal{W}}^H\left(s\right)$ is well-defined as an $\mathcal{X}$-valued random variable, and the following moment estimate holds:

$$\mathbb{E}\parallel {\int }_0^{\ell }\varphi \left(s\right)d{\mathcal{W}}^H\left(s\right){\parallel }_{\mathcal{X}}^2\le 2H{\ell }^{2H-1}{\int }_0^{\ell }{\parallel \varphi \left(s\right)\parallel }_{{\mathcal{L}}_0^2}^{2}ds,\ell \in [0,\mathfrak{a}].$$

Lemma 2

([17]). Let $\mathfrak{k}:[0,\mathfrak{a}]\times \mathrm{\Omega }\to {\mathcal{L}}_{\mathbb{Q}}^2(\mathcal{Y},\mathcal{X})$ be a strongly measurable mapping such that

$${\int }_0^{\mathfrak{a}}\mathbb{E}\parallel \mathfrak{k}(\ell ){\parallel }_{{\mathcal{L}}_{\mathbb{Q}}^2}^{2}d\ell <\infty .$$

Then, the stochastic integral with respect to the $\mathbb{Q}$-Wiener process $\mathcal{W}$ satisfies

$$\mathbb{E}\parallel {\int }_0^{\mathfrak{a}}\mathfrak{k}(\ell )d\mathcal{W}(\ell ){\parallel }_{\mathcal{X}}^2\le Tr\left(\mathbb{Q}\right){\int }_0^{\mathfrak{a}}\mathbb{E}\parallel \mathfrak{k}(\ell ){\parallel }_{{\mathcal{L}}_{\mathbb{Q}}^2}^{2}d\ell ,\forall \ell \in [0,\mathfrak{a}],$$

where ${\mathcal{L}}_{\mathbb{Q}}^2(\mathcal{Y},\mathcal{X})$ denotes the Banach space of all $\mathbb{Q}$–Hilbert–Schmidt operators mapping $\mathcal{Y}$ into $\mathcal{X}$.

Definition 1

([2]). Let $\wp >0$ and let $n:=\lceil \wp \rceil$ be the smallest integer such that $n-1<\wp <n$. Assume that $f:[0,\infty )\to \mathbb{R}$ satisfies $f\in {\mathcal{L}}_{\mathrm{loc}}^1\left([0,\infty )\right)$ and that its n-th order derivative exists in the classical sense. The Riemann–Liouville fractional derivative of order ℘, taken from the lower limit 0, is defined by

$${}^{RL}D_{0+}^{\wp }f(\ell ):={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\wp )}}{\displaystyle \frac{d^n}{d{\ell }^n}}{\int }_0^{\ell }{(\ell -s)}^{n-\wp -1}f\left(s\right)ds,\ell >0,$$

where $\mathrm{\Gamma }(·)$ denotes the Euler gamma function. This operator captures nonlocal memory effects through the weakly singular kernel ${(\ell -s)}^{n-\wp -1}$.

Definition 2

([2]). Let $\wp >0$ and let $f:[0,\infty )\to \mathbb{R}$ be a locally integrable function. The Riemann–Liouville fractional integral of order ℘, taken from the lower limit 0, is defined by

$${}^{RL}\mathbf{I}_{0+}^{\wp }f(\ell ):={\displaystyle \frac{1}{\mathrm{\Gamma }(\wp )}}{\int }_0^{\ell }{(\ell -s)}^{\wp -1}f\left(s\right)ds,\ell >0,$$

which represents a Volterra-type convolution operator encoding hereditary accumulation with a power-law kernel ${(\ell -s)}^{\wp -1}$.

Definition 3

([2]). Let $f:[0,\infty )\to \mathbb{R}$ be an absolutely continuous function whose derivatives up to order n are continuous. For a fractional order $\wp \in (n-1,n)$, the Caputo fractional derivative of f of order ℘ is introduced as

$${}^{C}D_{0+}^{\wp }f(\ell ):={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\wp )}}{\int }_0^{\ell }{(\ell -s)}^{n-\wp -1}{f}^{\left(n\right)}\left(s\right)ds,\ell >0,$$

which evaluates the nth classical derivative of f against a weakly singular memory kernel of order $n-\wp -1$.

Definition 4

([7]). For a fractional index ℘ satisfying $n-1<\wp <n$ and an interpolation parameter $\Re \in [0,1]$, the Hilfer fractional derivative of a function f is introduced through the composite operator

$${}^{\mathbb{H}}D_{0+}^{\wp ,\Re }f(\ell ):=\left({}^{RL}\mathbf{I}_{0+}^{\Re (n-\wp )}\right){\displaystyle \frac{d^n}{d{\ell }^n}}\left({}^{RL}\mathbf{I}_{0+}^{(1-\Re )(n-\wp )}f\right)(\ell ),$$

which provides a continuous interpolation between the Riemann–Liouville fractional derivative (corresponding to $\Re =0$) and the Caputo-type fractional derivative (corresponding to $\Re =1$), thereby unifying both classical fractional operators within a single framework.

Remark 2.

Let $1<\mu <2$ and assume that $f\in L_{\mathrm{loc}}^1\left([0,\infty )\right)$ is such that $\left({}^{RL}\mathbf{I}_{0^{+}}^{2-\mu }f\right)(\ell )$ is continuous on $[0,\mathfrak{a}]$ and ${\left({}^{RL}\mathbf{I}_{0^{+}}^{2-\mu }f\right)}^{\prime }(\ell )$ is absolutely continuous on $(0,\mathfrak{a}]$. Then the following identity holds:

$${}^{RL}\mathbf{I}_{0^{+}}^{\mu }\left({}^{RL}D_{0^{+}}^{\mu }f\right)(\ell )=f(\ell )-{\displaystyle \frac{\left({}^{\mathit{RL}}\mathbf{I}_{0+}^{2-\mu }f\right)\left(0\right)}{\mathrm{\Gamma }(\mu -1)}}{\ell }^{\mu -2}-{\displaystyle \frac{{\left({}^{\mathit{RL}}\mathbf{I}_{0+}^{2-\mu }f\right)}^{\prime }\left(0\right)}{\mathrm{\Gamma }\left(\mu \right)}}{\ell }^{\mu -1}.$$

This relation represents the classical inversion formula for Riemann–Liouville fractional operators under appropriate regularity assumptions.

Throughout the analysis, we suppose that there exists a constant $\mathfrak{d}\ge 1$ for which the cosine family ${\left\{\widehat{C}(\ell )\right\}}_{\ell \ge 0}$ satisfies the uniform boundedness condition

$$\parallel \widehat{C}{(\ell )\parallel }_{\mathcal{L}\left(\mathcal{X}\right)}\le \mathfrak{d},\mathrm{for}\mathrm{all}\ell \ge 0.$$

It then follows, by definition of the associated sine operator family

$$S(\ell )={\int }_0^{\ell }\widehat{C}\left(s\right)ds,$$

that a linear growth bound is inherited, namely,

$${\parallel S(\ell )\parallel }_{\mathcal{L}\left(\mathcal{X}\right)}\le \mathfrak{d}\ell ,\mathrm{for}\mathrm{every}\ell \ge 0.$$

Let $\mathcal{C}\left([0,\mathfrak{a}],{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})\right)$ denote the Banach space formed by all continuous mappings

$$z:[0,\mathfrak{a}]⟶{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})$$

for which the second–moment norm is uniformly bounded, that is,

$${\parallel z\parallel }_{\mathcal{C}}:=\underset{\ell \in [0,\mathfrak{a}]}{sup}\mathbb{E}{∥z(\ell )∥}_{\mathcal{X}}^2<\infty .$$

Define the weighted space

$${\mathcal{C}}_{2-\mu }\left([0,\mathfrak{a}],{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})\right)=\left\{z:[0,\mathfrak{a}]\to {\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X}):{\ell }^{2-\mu }z(\ell )\in \mathcal{C}\left([0,\mathfrak{a}],{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})\right)\right\},$$

endowed with the norm

$${\parallel z\parallel }_{{\mathcal{C}}_{2-\mu }}={\left(\underset{\ell \in [0,\mathfrak{a}]}{sup}\mathbb{E}{∥{\ell }^{2-\mu }z(\ell )∥}_{\mathcal{X}}^2\right)}^{{\displaystyle \frac{1}{2}}}.$$

Then $\left({\mathcal{C}}_{2-\mu }([0,\mathfrak{a}],{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})),{\parallel ·\parallel }_{{\mathcal{C}}_{2-\mu }}\right)$ is a Banach space.

Remark 3.

The elements of the space ${\mathcal{C}}_{2-\mu }([0,\mathfrak{a}],{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X}))$ are strongly measurable stochastic processes $z:[0,\mathfrak{a}]\to {\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})$ such that the weighted mapping $\ell \mapsto {\ell }^{2-\mu }z(\ell )$ is continuous with respect to the mean-square norm. Equivalently, this space can be characterized as

$${\mathcal{C}}_{2-\mu }=\left\{z:\underset{\ell \in [0,\mathfrak{a}]}{sup}\mathbb{E}{\parallel {\ell }^{2-\mu }z(\ell )\parallel }_{\mathcal{X}}^2<\infty ,{\ell }^{2-\mu }z(\ell )\mathit{is}\mathit{continuous}\mathit{on}[0,\mathfrak{a}]\right\}.$$

Moreover, the mapping $z\mapsto \left(\ell \mapsto {\ell }^{2-\mu }z(\ell )\right)$ defines an isometric embedding of ${\mathcal{C}}_{2-\mu }$ into the Banach space $\mathcal{C}([0,\mathfrak{a}],{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X}))$. Consequently, ${\mathcal{C}}_{2-\mu }$ is a Banach space, being isometrically isomorphic to its image in $\mathcal{C}([0,\mathfrak{a}],{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X}))$, which is closed due to the preservation of continuity under uniform convergence (see, e.g., [16]).

Remark 4.

The weighted phase space ${\mathcal{C}}_{2-\mu }([0,\mathfrak{a}],{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X}))$ is chosen to properly reflect the singular and memory-dependent behavior induced by fractional-order dynamics. Such spaces are standard in fractional stochastic systems and are physically interpreted as capturing nonlocal initial memory rather than classical pointwise initial states. It is also worth noting that this space is not a Hilbert space in general, since its norm is defined via a supremum-type structure and is not induced by an inner product satisfying the parallelogram law.

Let ${\left\{{\ell }_j\right\}}_{j=0}^{\beta +1}$ be a partition of $[0,\mathfrak{a}]$ with $0={\ell }_0<{\ell }_1<\cdots <{\ell }_{\beta }<{\ell }_{\beta +1}=\mathfrak{a}$. We introduce the piecewise weighted spaces

$$\begin{array}{ccc}\hfill \mathbb{S}& :=& P{C}_{2-\mu }\left([0,\mathfrak{a}],{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})\right)=\{z:[0,\mathfrak{a}]\to {\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X}):{(\ell -{\ell }_j)}^{2-\mu }z(\ell )\in \hfill \\ & & \mathcal{C}\left(({\ell }_j,{\ell }_{j+1}],{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})\right),z\left({\ell }_j^{+}\right)\mathrm{exists},z\left({\ell }_j^{-}\right)=z\left({\ell }_j\right),j=1,\cdots ,\beta ,\hfill \\ & & \underset{\ell \to {\ell }_j^{+}}{lim}{(\ell -{\ell }_j)}^{2-\mu }z(\ell )\mathrm{exists}\}.\hfill \end{array}$$

Remark 5.

The partition ${\left\{{\ell }_j\right\}}_{j=0}^{\beta +1}$ is fixed throughout the analysis and determines the set of admissible discontinuity points of the system. Accordingly, the phase space $\mathbb{S}={\mathcal{PC}}_{2-\mu }([0,\mathfrak{a}],{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X}))$ consists of stochastic processes that are piecewise continuous with respect to this partition, and any discontinuities are allowed only at the points ${\ell }_j$. Each non-instantaneous impulse is defined on a sub-interval of the form $({\ell }_j,s_j]$, where $s_j$ lies strictly inside $({\ell }_j,{\ell }_{j+1}]$. Hence, these points do not introduce additional discontinuities. Moreover, all operators involved in the system, including stochastic convolution terms, fractional resolvent operators, and impulsive mappings, preserve this structure. Therefore, the set of discontinuities remains invariant under the solution operator, and all admissible solutions belong to the same phase space $\mathbb{S}$.

$$\begin{array}{cc}\hfill \widehat{\mathbb{S}}& =\{z:[0,\mathfrak{a}]\to {\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X}):{(\ell -{\ell }_j)}^{2-\mu }z(\ell )\in {\mathcal{C}}^1\left(({\ell }_j,{\ell }_{j+1}],{\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})\right),\hfill \\ \hfill & z^{\prime }\left({\ell }_j^{+}\right)\mathrm{exists},z^{\prime }\left({\ell }_j^{-}\right)=z^{\prime }\left({\ell }_j\right),j=1,\cdots ,\beta ,\underset{\ell \to {\ell }_j^{+}}{lim}{(\ell -{\ell }_j)}^{2-\mu }{z}^{\prime }(\ell )\mathrm{exists}\}.\hfill \end{array}$$

Both $\mathbb{S}$ and $\widehat{\mathbb{S}}$ are Banach spaces when equipped with the norm

$${\parallel z\parallel }_{\mathbb{S}}={\left(\underset{\ell \in [0,\mathfrak{a}]}{sup}\mathbb{E}{∥{\ell }^{2-\mu }z(\ell )∥}_{\mathcal{X}}^2\right)}^{{\displaystyle \frac{1}{2}}}.$$

In order to formulate the $\mathbb{MS}$ corresponding to problem (1), we make use of the Wright-type kernel ${\mathbb{W}}_j$, which constitutes a central analytical tool in the theory of fractional resolvents. For parameters $0<j<1$ and $\varphi \in \mathbb{C}$, the function ${\mathbb{W}}_j\left(\varphi \right)$ is defined by

$${\mathbb{W}}_j\left(\varphi \right)={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{{(-\varphi )}^{k-1}}{(k-1)!\mathrm{\Gamma }(1-jk)}}.$$

An important property of ${\mathbb{W}}_j$ is the moment-type relation

$${\int }_0^{\infty }{\varphi }^{\xi }{\mathbb{W}}_j\left(\varphi \right)d\varphi ={\displaystyle \frac{\mathrm{\Gamma }(1+\xi )}{\mathrm{\Gamma }(1+j\xi )}},\xi \ge 0.$$

This kernel arises naturally in integral representations of fractional cosine and sine operator families generated by the Hilfer derivative ${}^{\mathbb{H}}D_{0+}^{\wp ,\Re }$. Adopting the approach developed in [40,41], we employ ${\mathbb{W}}_j$ to derive the $\mathbb{MS}$ for the system (1), thereby capturing hereditary effects together with stochastic disturbances within a Hilbert-space setting.

Definition 5

([42]). A one–parameter operator family ${\left\{\widehat{C}(\ell )\right\}}_{\ell \in \mathbb{R}}$ acting on $\mathcal{X}$ and consisting of bounded linear operators is termed an SC cosine family provided that the following axioms are fulfilled:

1. 

Normalization at the origin holds in the form

$$\widehat{C}\left(0\right)=\mathcal{I},$$

where $\mathcal{I}$ denotes the identity operator on $\mathcal{X}$.

2. 

The d’Alembert functional relation is satisfied, namely,

$$\widehat{C}(\ell +s)+\widehat{C}(\ell -s)=2\widehat{C}(\ell )\widehat{C}\left(s\right),\forall \ell ,s\in \mathbb{R}.$$

3. 

For every $z\in \mathcal{X}$, the orbit map

$$\ell ⟼\widehat{C}(\ell )z$$

is continuous on the real line.

The associated SC sine family ${\left\{S(\ell )\right\}}_{\ell \in \mathbb{R}}$ is defined in terms of ${\left\{\widehat{C}(\ell )\right\}}_{\ell \in \mathbb{R}}$ through the operator-valued integral

$$S(\ell )z={\int }_0^{\ell }\widehat{C}\left(s\right)zds,\ell \in \mathbb{R},z\in \mathcal{X}.$$

An operator $\mathcal{A}$ is called the infinitesimal generator of the cosine family ${\left\{\widehat{C}(\ell )\right\}}_{\ell \in \mathbb{R}}$ if it is given by the second–order strong derivative at the origin,

$$\mathcal{A}z={\left.{\displaystyle \frac{d^2}{d{\ell }^2}}\widehat{C}(\ell )z\right|}_{\ell =0},z\in \mathbb{D}\left(\mathcal{A}\right),$$

where the domain of $\mathcal{A}$ is specified by

$$\mathbb{D}\left(\mathcal{A}\right)=\left\{z\in \mathcal{X}:\ell \mapsto \widehat{C}(\ell )z\mathit{is}\mathit{twice}\mathit{continuously}\mathit{differentiable}\mathit{on}\mathbb{R}\right\}.$$

Lemma 3

([42]). Assume that ${\left\{\widehat{C}(\ell )\right\}}_{\ell \in \mathbb{R}}$ forms an SC cosine operator family on $\mathcal{X}$, for which there exist constants ${\mathfrak{d}}_0\ge 1$ and $\widehat{\omega }\ge 0$ such that

$$\parallel \widehat{C}{(\ell )\parallel }_{\mathcal{L}\left(\mathcal{X}\right)}\le {\mathfrak{d}}_0{e}^{\widehat{\omega }|\ell |},\forall \ell \in \mathbb{R}.$$

Let $\mathcal{A}$ denote the infinitesimal generator of ${\left\{\widehat{C}(\ell )\right\}}_{\ell \in \mathbb{R}}$. Then, for every $j\in \mathbb{C}$ with $Rej>\widehat{\omega }$, it holds that $j^2\in \rho \left(\mathcal{A}\right)$, and for each $z\in \mathcal{X}$, the Laplace transform representation of the resolvent is given by

$$jR(j^2;\mathcal{A})z={\int }_0^{\infty }{e}^{-j\ell }\widehat{C}(\ell )zd\ell ,R(j^2;\mathcal{A})z={\int }_0^{\infty }{e}^{-j\ell }S(\ell )zd\ell ,$$

where $R(\lambda ;\mathcal{A})={(\lambda I-\mathcal{A})}^{-1}$ denotes the resolvent operator associated with $\mathcal{A}$.

Remark 6.

It is important to emphasize that the $C^2$-regularity condition appearing in the definition of the infinitesimal generator $\mathcal{A}$ pertains exclusively to the domain $\mathbb{D}\left(\mathcal{A}\right)$ and does not impose differentiability requirements on the solution trajectories considered in this work. In the present framework, solutions are understood in the mild sense and are formulated via integral representations involving the associated cosine and sine operator families. Such mild solutions are well-defined for initial data in $\mathcal{X}$ and, in general, do not belong to $\mathbb{D}\left(\mathcal{A}\right)$, nor are they required to possess classical second-order differentiability. This distinction between classical (strong) solutions and mild solutions is standard in the theory of abstract second-order evolution equations and is particularly essential in the treatment of fractional and stochastic systems, where solution regularity is inherently limited.

We now establish a fundamental theorem by invoking a fixed-point framework to ensure the existence of solutions for the stochastic system under consideration.

Definition 6.

Let $\mathcal{X}$ be a Banach space and $B\subset \mathcal{X}$ be a bounded set. The Kuratowski measure of non-compactness $\alpha \left(B\right)$ is defined as

$$\alpha \left(B\right):=inf\left\{\delta >0:B\subset {\displaystyle \bigcup _{i=1}^n}{B}_i,diam\left(B_i\right)\le \delta ,n\in \mathbb{N}\right\}.$$

Theorem 1

(Sadovskii’s Fixed Point Theorem, [43]). Let Ψ be a condensing operator on a Banach space $\mathcal{X}$, i.e., Ψ is continuous, maps bounded sets into bounded sets, and for every bounded set $\widehat{Z}\subset \mathcal{X}$ with $\alpha \left(\widehat{Z}\right)>0$, we have

$$\alpha \left(\mathrm{\Psi }\left(\widehat{Z}\right)\right)\le \alpha \left(\widehat{Z}\right),$$

where $\alpha (·)$ denotes Kuratowski’s measure of noncompactness. If there exists a convex, closed, and bounded set $Z\subset \mathcal{X}$ such that

$$\mathrm{\Psi }\left(Z\right)\subset Z,$$

then Ψ possesses at least one fixed point in $\mathcal{X}$.

Remark 7.

The use of Sadovskii’s fixed point theorem is motivated by the fact that the solution operator associated with the considered system is not necessarily compact due to the presence of fractional operators, stochastic integrals, and non-instantaneous impulses. Consequently, classical fixed point results based on compactness (such as Krasnoselskii’s theorem) are not directly applicable. Instead, we work within the framework of condensing operators and employ the measure of non-compactness to establish the existence of fixed points under weaker compactness assumptions.

Remark 8.

It is worth emphasizing that classical fixed-point results such as Krasnosel’skii’s theorem are based on a decomposition of the underlying operator into a contraction and a completely continuous mapping. However, in the present framework, the solution operator associated with the stochastic fractional neutral system is neither compact nor admits a natural decomposition satisfying the classical Krasnosel’skii structure due to the simultaneous presence of fractional derivatives, stochastic integrals, neutral terms, and non-instantaneous impulses. Therefore, the direct application of Krasnosel’skii’s theorem is not feasible without imposing restrictive assumptions that would significantly reduce the generality of the model. In contrast, Sadovskii’s fixed point theorem, based on measures of non-compactness and condensing operators, provides an appropriate and less restrictive framework that is fully compatible with the analytical structure of the system.

Definition 7.

A stochastic process $z(\ell ):[0,\mathfrak{a}]\to \mathcal{X}$, adapted to the filtration ${\widehat{Y}}_{\ell }$, is termed a mild solution of the system (1) if

1. 

$z_0,z_1\in {\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})$;

2. 

$\left({}^{\mathbb{RL}}\mathbf{I}_{0+}^{2-\mu }z\right)\left(0\right)+{\mathfrak{g}}_1\left(z\right)=z_0,{\left({}^{\mathbb{RL}}\mathbf{I}_{0+}^{2-\mu }z\right)}^{\prime }\left(0\right)+{\mathfrak{g}}_2\left(z\right)=z_1$;

3. 

The stochastic integral equation that ensues is affirmed:

$$z(\ell )=\left\{\begin{array}{cc}{\mho }_1,\hfill & \ell \in [0,{\ell }_1],\hfill \\ ⋮\hfill & \\ {\mathbb{O}}_j^1(\ell ,z(\ell )),\hfill & \ell \in ({\ell }_j,s_j],j=1,2,\cdots ,\beta \hfill \\ ⋮\hfill & \\ {\mho }_2,\hfill & \ell \in (s_j,{\ell }_{j+1}],j=1,2,\cdots ,\beta ,\hfill \end{array}\right.$$

$$\begin{array}{ccc}\hfill {\mho }_1& =& {\mathcal{R}}_{\wp ,\Re }(\ell )\left[z_0-{\mathfrak{g}}_1\left(z\right)+\mathcal{F}(0,z\left(0\right),z\left({\zeta }_1\left(0\right)\right),\cdots ,z\left({\zeta }_m\left(0\right)\right))\right]\hfill \\ & & -\mathcal{F}(\ell ,z(\ell ),z\left({\zeta }_1(\ell )\right),\cdots ,z\left({\zeta }_m(\ell )\right))+{\mathcal{T}}_{\wp ,\Re }(\ell )\left[z_1-{\mathfrak{g}}_2\left(z\right)+\Im \right]\hfill \\ & & +{\int }_0^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{B}u\left(s\right)ds\hfill \\ & & +{\int }_0^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s){\int }_0^s\mathfrak{F}(\tau ,z\left(\tau \right),z\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(\tau \right)\right))d\mathcal{W}\left(\tau \right)ds\hfill \\ & & +{\int }_0^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{O}(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right))d{\mathcal{W}}^H\left(s\right)\hfill \\ & & +{\int }_0^{\ell }{\int }_{\mathsf{\Pi }}{\mathcal{V}}_{\varsigma }(\ell -s)\hslash (s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta )\widehat{N}(ds,d\eta ),\hfill \\ \hfill {\mho }_2& =& {\mathcal{R}}_{\wp ,\Re }(\ell -s_j)\left[{\mathbb{O}}_j^1(s_j,z\left(s_j\right))+\mathcal{F}(s_j,z\left(s_j\right),z\left({\zeta }_1\left(s_j\right)\right),\cdots ,z\left({\zeta }_m\left(s_j\right)\right))\right]\hfill \\ & & -\mathcal{F}(\ell ,z(\ell ),z\left({\zeta }_1(\ell )\right),\cdots ,z\left({\zeta }_m(\ell )\right))+{\mathcal{T}}_{\wp ,\Re }(\ell -s_j)\left[{\mathbb{O}}_j^2(s_j,z\left(s_j\right))+{\widehat{\Im }}_j\right]\hfill \\ & & +{\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{B}u\left(s\right)ds\hfill \\ & & +{\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s){\int }_0^s\mathfrak{F}(\tau ,z\left(\tau \right),z\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(\tau \right)\right))d\mathcal{W}\left(\tau \right)ds\hfill \\ & & +{\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{O}(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right))d{\mathcal{W}}^H\left(s\right)\hfill \\ & & +{\int }_{s_j}^{\ell }{\int }_{\mathsf{\Pi }}{\mathcal{V}}_{\varsigma }(\ell -s)\hslash (s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta )\widehat{N}(ds,d\eta ),\hfill \end{array}$$

where $\widehat{N}(d\ell ,d\eta )$ signifies the compensated Poisson random measure (alternatively, $N(d\ell ,d\eta )$ in the absence of compensation) and the associated stochastic integrals are interpreted in the classical Itô–martingale framework, taking values in the Hilbert space ${\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})$ and

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel {\displaystyle \frac{d}{d\ell }}\left[\mathcal{F}(\ell ,z(\ell ),z\left({\zeta }_1(\ell )\right),\cdots ,z\left({\zeta }_m(\ell )\right))\right]{\parallel }_{\ell =s_j}^2& =& {\widehat{\Im }}_j,\hfill \\ \hfill \mathbb{E}\parallel {\displaystyle \frac{d}{d\ell }}\left[\mathcal{F}(\ell ,z(\ell ),z\left({\zeta }_1(\ell )\right),\cdots ,z\left({\zeta }_m(\ell )\right))\right]{\parallel }_{\ell =0}^2& =& \Im .\hfill \end{array}$$

In this context,

$$\begin{array}{ccc}\hfill {\mathcal{R}}_{\wp ,\Re }(\ell )& =& {}^{RL}D_{0+}^{1-\tau }{\ell }^{\varsigma -1}{{\rm Y}}_{\varsigma }(\ell ),{\mathcal{T}}_{\wp ,\Re }(\ell )={\mathbf{I}}_{0+}^{\tau }{\ell }^{\varsigma -1}{{\rm Y}}_{\varsigma }(\ell ),{\mathcal{V}}_{\varsigma }={\ell }^{\varsigma -1}{{\rm Y}}_{\varsigma }(\ell ).\hfill \end{array}$$

Remark 9.

The above definition of mild solution is given directly in its integral form, which is standard in the study of stochastic fractional evolution equations with non-instantaneous impulses and neutral terms. In such settings, the classical differential formulation is not directly applicable due to the presence of fractional operators, stochastic integrals, and impulsive effects. Therefore, the notion of mild solution is naturally characterized through the corresponding stochastic integral equation, which provides a suitable framework for analysis in the Hilbert space setting.

Remark 10.

The adopted analytical framework is fully consistent with classical nonlinear functional analysis in infinite-dimensional spaces. In particular, the combination of mild solution theory, measures of non-compactness, and condensing operator techniques constitutes a well-established methodology for studying stochastic evolution equations in settings where compactness assumptions are not satisfied. This approach has been extensively used in the literature on fractional stochastic systems, especially in the presence of memory effects, impulsive dynamics, and neutral terms.

Lemma 4

([44]). Let $\tau =\Re (2-\wp )\in (0,1)$ and $\varsigma ={\displaystyle \frac{\wp }{2}}\in \left({\displaystyle \frac{1}{2}},1\right)$. We introduce the parametric family of operators ${\left\{{{\rm Y}}_{\varsigma }(\ell )\right\}}_{\ell >0}$ through the representation

$${{\rm Y}}_{\varsigma }(\ell )={\int }_0^{\infty }\varsigma \varphi {\mathbb{W}}_{\varsigma }\left(\varphi \right)s\left({\ell }^{\varsigma }\varphi \right)d\varphi ,\varphi \in (0,\infty ),$$

where ${\mathbb{W}}_{\varsigma }\left(\varphi \right)$ designates the Wright-type density specified by

$${\mathbb{W}}_{\varsigma }\left(\varphi \right)={\displaystyle \frac{1}{\varsigma }}{\varphi }^{-{\displaystyle \frac{1}{\varsigma }}-1}{\ell }_{\varsigma }\left({\varphi }^{-{\displaystyle \frac{1}{\varsigma }}}\right),{\ell }_{\varsigma }\left(\varphi \right)={\displaystyle \frac{1}{\pi }}{\displaystyle \sum _{n=1}^{\infty }}{(-1)}^{n-1}{\varphi }^{-\varsigma n-1}{\displaystyle \frac{\mathrm{\Gamma }(\varsigma n+1)}{n!}}sin\left(\varsigma n\pi \right).$$

Consequently, for every $\ell >0$ and each $z\in \mathcal{X}$, the following assertions are fulfilled:

1. 

The mapping $\ell \mapsto {{\rm Y}}_{\varsigma }(\ell )$ enjoys uniform continuity on $(0,\infty )$ in the operator norm topology; explicitly,

$$\parallel {{\rm Y}}_{\varsigma }\left({\ell }_2\right)-{{\rm Y}}_{\varsigma }\left({\ell }_1\right)\parallel ⟶0\mathit{as}{\ell }_2\to {\ell }_1.$$

2. 

The operator assemblages ${{\rm Y}}_{\varsigma }(\ell )$, ${\mathcal{R}}_{\wp ,\Re }(\ell )$, ${\mathcal{T}}_{\wp ,\Re }(\ell )$, and ${\mathcal{V}}_{\varsigma }(\ell )$ admit the subsequent growth estimates in $\mathcal{X}$: for every $z\in \mathcal{X}$,

$$\begin{array}{cc}\hfill \parallel {{\rm Y}}_{\varsigma }{(\ell )z\parallel }_{\mathcal{X}}& \le \mathfrak{d}{\ell }^{\varsigma }\mathrm{\Gamma }{\left(2\varsigma \right)}^{-1}{\parallel z\parallel }_{\mathcal{X}},\hfill \\ \hfill \parallel {\mathcal{R}}_{\wp ,\Re }{(\ell )z\parallel }_{\mathcal{X}}& \le \mathfrak{d}{\ell }^{2\varsigma +\tau -2}{\left[(2\varsigma -1)\mathrm{\Gamma }(\tau +2\varsigma -1)\right]}^{-1}{\parallel z\parallel }_{\mathcal{X}},\hfill \\ \hfill \parallel {\mathcal{T}}_{\wp ,\Re }{(\ell )z\parallel }_{\mathcal{X}}& \le \mathfrak{d}{\ell }^{2\varsigma +\tau -1}\mathrm{\Gamma }{(\tau +2\varsigma )}^{-1}{\parallel z\parallel }_{\mathcal{X}},\hfill \\ \hfill \parallel {\mathcal{V}}_{\varsigma }{(\ell )z\parallel }_{\mathcal{X}}& \le \mathfrak{d}{\ell }^{2\varsigma -1}\mathrm{\Gamma }{\left(2\varsigma \right)}^{-1}{\parallel z\parallel }_{\mathcal{X}}.\hfill \end{array}$$

3. 

Furthermore, the temporal derivative of the weighted operator ${\ell }^{\varsigma -1}{{\rm Y}}_{\varsigma }(\ell )z$ admits the decomposition

$${\displaystyle \frac{d}{d\ell }}\left({\ell }^{\varsigma -1}{{\rm Y}}_{\varsigma }(\ell )z\right)=(\varsigma -1){\ell }^{\varsigma -2}{{\rm Y}}_{\varsigma }(\ell )z+{\varsigma }^2{\ell }^{2(\varsigma -1)}{\int }_0^{\infty }{\varphi }^2{\mathbb{W}}_{\varsigma }\left(\varphi \right)C\left({\ell }^{\varsigma }\varphi \right)zd\varphi ,$$

from which one infers the norm bound

$$\parallel {\displaystyle \frac{d}{d\ell }}\left({\ell }^{\varsigma -1}{{\rm Y}}_{\varsigma }(\ell )z\right){\parallel }_{\mathcal{X}}\le \mathfrak{d}{\ell }^{2(\varsigma -1)}\mathrm{\Gamma }{\left(2\varsigma \right)}^{-1}{\parallel z\parallel }_{\mathcal{X}}.$$

To investigate the $\mathbb{AC}$ properties of the fractional control system (1), we first formulate the ensuing linear fractional dynamical system, which serves as an auxiliary framework for the subsequent analysis.

$$\begin{array}{ccc}& & {}^{\mathbb{H}}D_{0+}^{\wp ,\Re }z(\ell )=\mathcal{A}z(\ell )+\mathfrak{B}u(\ell ),\ell \in [0,\mathfrak{a}],\hfill \\ & & \left({}^{\mathbb{RL}}\mathbf{I}_{0+}^{2-\mu }z\right)\left(0\right)+{\mathfrak{g}}_1\left(z\right)=z_0,{\left({}^{\mathbb{RL}}\mathbf{I}_{0+}^{2-\mu }z\right)}^{\prime }\left(0\right)+{\mathfrak{g}}_2\left(z\right)=z_1.\hfill \end{array}$$

(2)

We introduce the controllability operator corresponding to (2) by

$${\mathfrak{Q}}_0^{\mathfrak{a}}={\int }_0^{\mathfrak{a}}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s)\mathfrak{B}{\mathfrak{B}}^{*}{\mathcal{V}}_{\varsigma }^{*}(\mathfrak{a}-s)ds,$$

and define the associated resolvent operator

$$\mathfrak{R}(\mathfrak{a},{\mathfrak{Q}}_0^{\mathfrak{a}})={(\mathfrak{a}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1},\mathfrak{a}>0,$$

where ${\mathfrak{B}}^{*}$ and ${\mathcal{V}}_{\varsigma }^{*}(\ell )$ stand for the Hilbert adjoints of $\mathfrak{B}$ and ${\mathcal{V}}_{\varsigma }(\ell )$, respectively. Under the standing hypotheses, ${\mathfrak{Q}}_0^{\mathfrak{a}}$ constitutes a bounded linear operator on $\mathcal{X}$.

Lemma 5

([45]). The linear fractional system (2) is approximately controllable on the interval $[0,\mathfrak{a}]$ if and only if

$$\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}⟶0\mathit{as}\mathfrak{x}\to 0^{+},$$

with convergence understood in the strong operator topology.

Remark 11.

Convergence in the strong operator topology (SOT) means pointwise convergence on the underlying Hilbert space. More precisely, a family of bounded linear operators $\left\{{\mathcal{M}}_{\mathfrak{x}}\right\}$ on $\mathcal{X}$ converges to the zero operator in the strong operator topology if and only if

$$\parallel {\mathcal{M}}_{\mathfrak{x}}{z\parallel }_{\mathcal{X}}⟶0\mathit{as}\mathfrak{x}\to 0^{+},\forall z\in \mathcal{X}.$$

Accordingly, the condition

$$\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}⟶0\mathit{in}\mathit{SOT}$$

means that for every $z\in \mathcal{X}$,

$${∥\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}z∥}_{\mathcal{X}}⟶0,\mathit{as}\mathfrak{x}\to 0^{+}.$$

This notion of convergence is weaker than norm convergence but is sufficient for characterizing $\mathbb{AC}$ in infinite-dimensional systems.

Let $z(\ell ;z_0,z_1,u)$ denote the trajectory of system (1) driven by a control input $u(·)$, and write $z_{\mathfrak{a}}(z_0,z_1,u)$ for the corresponding terminal configuration at time $\mathfrak{a}$. The attainable set associated with (1) is then specified by

$$\mathrm{\Xi }(\mathfrak{a};z_0,z_1)=\left\{z_{\mathfrak{a}}(z_0,z_1,u):u(·)\in {\mathcal{L}}_{\widehat{Y}}^2([0,\mathfrak{a}];\mathcal{U})\right\},$$

and we denote by $\overline{\mathrm{\Xi }(\mathfrak{a};z_0,z_1)}$ its closure in ${\mathcal{L}}^2(\mathrm{\Omega };\mathcal{X})$.

Definition 8.

The system (1) is said to be approximately controllable on $[0,\mathfrak{a}]$ whenever

$$\overline{\mathrm{\Xi }(\mathfrak{a};z_0,z_1)}={\mathcal{L}}^2(\mathrm{\Omega },{\widehat{Y}}_{\mathfrak{a}};\mathcal{X}).$$

Lemma 6

([46]). For an arbitrary terminal configuration $z_{\mathfrak{a}}\in {\mathcal{L}}^2(\mathrm{\Omega },{\widehat{Y}}_{\mathfrak{a}};\mathcal{X})$, there exist stochastic processes $\widehat{\varrho }(·)$ and $\widehat{\vartheta }\in {\mathcal{L}}^2\left(\mathrm{\Omega };{\mathcal{L}}^2([0,\mathfrak{a}];{\mathcal{L}}_0^2)\right)$ such that the terminal state admits the decomposition

$$z_{\mathfrak{a}}=\mathbb{E}\left[z_{\mathfrak{a}}\right]+{\int }_0^{\mathfrak{a}}\widehat{\varrho }\left(s\right)d\mathcal{W}\left(s\right)+{\int }_0^{\mathfrak{a}}\widehat{\vartheta }\left(s\right)d{\mathcal{W}}^H\left(s\right),$$

that is, it can be expressed as the sum of its mean component and stochastic integrals with respect to the driving martingale measures. Moreover, for each $\mathfrak{x}>0$ and $z_{\mathfrak{a}}\in {\mathcal{L}}^2(\mathrm{\Omega },{\widehat{Y}}_{\mathfrak{a}};\mathcal{X})$, the control function

$$u^{\mathfrak{x}}(\ell )={\mathfrak{B}}^{*}{\mathcal{V}}_{\varsigma }^{*}(\mathfrak{a}-\ell )\mathfrak{R}(\mathfrak{x},{\mathfrak{Q}}_0^{\mathfrak{a}})\mathrm{\Theta }\left(z(·)\right),$$

is admissible, where

$$\begin{array}{ccc}\hfill \mathrm{\Theta }\left(z\right(·\left)\right)& =& \mathbb{E}\left[z_{\mathfrak{a}}\right]+{\int }_0^{\mathfrak{a}}\widehat{\varrho }\left(s\right)d\mathcal{W}\left(s\right)+{\int }_0^{\mathfrak{a}}\widehat{\vartheta }\left(s\right)d{\mathcal{W}}^H\left(s\right)\hfill \\ & & -{\mathcal{R}}_{\wp ,\Re }\left(\mathfrak{a}\right)\left[z_0-{\mathfrak{g}}_1\left(z\right)+\mathcal{F}(0,z\left(0\right),z\left({\zeta }_1\left(0\right)\right),\cdots ,z\left({\zeta }_m\left(0\right)\right))\right]\hfill \\ & & +\mathcal{F}(\mathfrak{a},z\left(\mathfrak{a}\right),z\left({\zeta }_1\left(\mathfrak{a}\right)\right),\cdots ,z\left({\zeta }_m\left(\mathfrak{a}\right)\right))-{\mathcal{T}}_{\wp ,\Re }\left(\mathfrak{a}\right)\left[z_1-{\mathfrak{g}}_2\left(z\right)+\Im \right]\hfill \\ & & -{\int }_0^{\mathfrak{a}}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s){\int }_0^s\mathfrak{F}(\tau ,z\left(\tau \right),z\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(\tau \right)\right))d\mathcal{W}\left(\tau \right)ds\hfill \\ & & -{\int }_0^{\mathfrak{a}}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s)\mathfrak{O}(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right))d{\mathcal{W}}^H\left(s\right)\hfill \\ & & -{\int }_0^{\mathfrak{a}}{\int }_{\mathsf{\Pi }}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s)\hslash (s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta )\widehat{N}(ds,d\eta ),\ell \in [0,{\ell }_1],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \mathrm{\Theta }\left(z\right(·\left)\right)& =& \mathbb{E}\left[z_{\mathfrak{a}}\right]+{\int }_0^{\mathfrak{a}}\widehat{\varrho }\left(s\right)d\mathcal{W}\left(s\right)+{\int }_0^{\mathfrak{a}}\widehat{\vartheta }\left(s\right)d{\mathcal{W}}^H\left(s\right)\hfill \\ & & -{\mathcal{R}}_{\wp ,\Re }(\mathfrak{a}-s_j)\left[{\mathbb{O}}_j^1(s_j,z\left(s_j\right))+\mathcal{F}(s_j,z\left(s_j\right),z\left({\zeta }_1\left(s_j\right)\right),\cdots ,z\left({\zeta }_m\left(s_j\right)\right))\right]\hfill \\ & & +\mathcal{F}(\mathfrak{a},z\left(\mathfrak{a}\right),z\left({\zeta }_1\left(\mathfrak{a}\right)\right),\cdots ,z\left({\zeta }_m\left(\mathfrak{a}\right)\right))-{\mathcal{T}}_{\wp ,\Re }(\mathfrak{a}-s_j)\left[{\mathbb{O}}_j^2(s_j,z\left(s_j\right))+{\widehat{\Im }}_j\right]\hfill \\ & & -{\int }_{s_j}^{\mathfrak{a}}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s){\int }_0^s\mathfrak{F}(\tau ,z\left(\tau \right),z\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(\tau \right)\right))d\mathcal{W}\left(\tau \right)ds\hfill \\ & & -{\int }_{s_j}^{\mathfrak{a}}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s)\mathfrak{O}(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right))d{\mathcal{W}}^H\left(s\right)\hfill \\ & & -{\int }_{s_j}^{\mathfrak{a}}{\int }_{\mathsf{\Pi }}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s)\hslash (s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta )\widehat{N}(ds,d\eta ),\hfill \\ & & \ell \in (s_j,{\ell }_{j+1}],j=1,2,\cdots ,\beta .\hfill \end{array}$$

3. Existence

In this section, we establish a structured framework of analytical hypotheses and, upon this foundation, deduce a set of rigorously justifiable sufficient conditions that secure the existence of an $\mathbb{MS}$ for the higher-order HFSDE (1), encompassing nonlocal initial specifications together with non–instantaneous impulsive dynamics.

(H1) 1.

The mapping $\mathcal{F}:[0,\mathfrak{a}]\times {\mathcal{X}}^{m+1}\to \mathcal{X}$ is continuous. Moreover, there exist a constant $\gamma \in (0,1)$ and positive constants $M_{\mathcal{F}},{\widehat{M}}_{\mathcal{F}}$ such that the transformed function ${(-\mathcal{A})}^{\gamma }\mathcal{F}$ fulfils the Lipschitz-type estimate

$$\begin{array}{ccc}& & \mathbb{E}\parallel {\mathcal{A}}^{\gamma }\mathcal{F}({\ell }_1,z_0,z_1,\cdots ,z_m)-{\mathcal{A}}^{\gamma }\mathcal{F}({\ell }_2,{\widehat{z}}_0,{\widehat{z}}_1,\cdots ,{\widehat{z}}_m){\parallel }^2\hfill \\ & & \le M_{\mathcal{F}}\left(|{\ell }_1-{\ell }_2|+\underset{0\le i\le m}{max}\mathbb{E}{\parallel z_i-{\widehat{z}}_i\parallel }^2\right),\hfill \end{array}$$

for all $0\le {\ell }_1,{\ell }_2\le \mathfrak{a}$ and $z_i,{\widehat{z}}_i\in \mathcal{X}$, $0\le i\le m$.

         2.

In addition, the growth bound

$$\mathbb{E}\parallel {\mathcal{A}}^{\gamma }\mathcal{F}(\ell ,z_0,z_1,\cdots ,z_m){\parallel }^2\le {\widehat{M}}_{\mathcal{F}}\left(\underset{0\le i\le m}{max}\mathbb{E}{\parallel z_i\parallel }^2+1\right)$$

holds for every $(\ell ,z_0,\cdots ,z_m)\in [0,\mathfrak{a}]\times {\mathcal{X}}^{m+1}$.

(H2)

The operator-valued function $\mathfrak{F}:[0,\mathfrak{a}]\times {\mathcal{X}}^{k+1}\to \mathcal{L}(\mathcal{Y},\mathcal{X})$ satisfies the following:

(i)

For each $\ell \in [o,\mathfrak{a}]$, the mapping $\mathfrak{F}(\ell ,·):{\mathcal{X}}^{m+1}\to \mathcal{L}(\mathcal{Y},\mathcal{X})$ is continuous; for every $(z_0,\cdots ,z_m)\in {\mathcal{X}}^{m+1}$, the trajectory $\mathfrak{F}(·,z_0,\cdots ,z_m)$ is ${\widehat{Y}}_{\ell }$–measurable.

(ii)

For every $\iota \in \mathbb{N}$ there exists a nonnegative function ${\psi }_{\iota }:(0,\mathfrak{a}]\to {\mathbb{R}}_{+}$ such that

$$\underset{\parallel z_0{\parallel }^2,\cdots ,{\parallel z_m\parallel }^2\le \iota }{sup}{\int }_0^{\ell }\mathbb{E}\parallel \mathfrak{F}(s,z_0,\cdots ,z_m){\parallel }_{\mathbb{Q}}^{2}ds\le {\psi }_{\iota }(\ell ),$$

the mapping $s\mapsto {(\ell -s)}^{2\varsigma -1}{\psi }_{\iota }\left(s\right)$ belongs to ${\mathcal{L}}^1((0,\mathfrak{a}],{\mathbb{R}}_{+})$, and there exists ${\mathrm{\Lambda }}_1>0$ such that

$$\underset{\iota \to \infty }{lim}inf{\displaystyle \frac{{\displaystyle {\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}{\psi }_{\iota }\left(s\right)ds}}{\iota }}={\mathrm{\Lambda }}_1<\infty ,\ell \in (0,\mathfrak{a}].$$

(H3)

The diffusion coefficient $\mathfrak{O}:[0,\mathfrak{a}]\times {\mathcal{X}}^{m+1}\to {\mathcal{L}}_0^2(\mathcal{Y},\mathcal{X})$ satisfies the following:

(i)

For each $\ell \in [0,\mathfrak{a}]$, the mapping $\mathfrak{O}(\ell ,·):{\mathcal{X}}^{m+1}\to {\mathcal{L}}_0^2(\mathcal{Y},\mathcal{X})$ is continuous, and for fixed $(z_0,\cdots ,z_m)$, the function $\mathfrak{O}(·,z_0,\cdots ,z_m)$ is ${\widehat{Y}}_{\ell }$–measurable.

(ii)

For each $\iota \in \mathbb{N}$ there exists a nonnegative function ${\widehat{\psi }}_{\iota }:(0,\mathfrak{a}]\to {\mathbb{R}}_{+}$ such that

$$\underset{\parallel z_0{\parallel }^2,\cdots ,{\parallel z_m\parallel }^2\le \iota }{sup}\mathbb{E}\parallel \mathfrak{O}(\ell ,z_0,\cdots ,z_m){\parallel }_{{\mathcal{L}}_0^2}^2\le {\widehat{\psi }}_{\iota }(\ell ),$$

with $s\mapsto {(\ell -s)}^{2\varsigma -1}{\widehat{\psi }}_{\iota }\left(s\right)\in {\mathcal{L}}^1((0,\mathfrak{a}],{\mathbb{R}}_{+})$, and there exists ${\mathrm{\Lambda }}_2>0$ such that

$$\underset{\iota \to \infty }{lim}inf{\displaystyle \frac{{\displaystyle {\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}{\widehat{\psi }}_{\iota }\left(s\right)ds}}{\iota }}={\mathrm{\Lambda }}_2<\infty ,\ell \in (0,\mathfrak{a}].$$

(H4)

The non-instantaneous impulsive mappings ${\mathbb{O}}_j^l\in \mathcal{C}(({\ell }_j,s_j]\times \mathcal{X},\mathcal{X})$, $l=1,2$ satisfy the following structural conditions:

(i)

There exist positive constants $M_{{\mathbb{O}}_j}^l>0$ such that

$$\mathbb{E}\parallel {\mathbb{O}}_j^l(\ell ,z_1)-{\mathbb{O}}_j^l(\ell ,z_2){\parallel }^2\le M_{{\mathbb{O}}_j}^l\mathbb{E}{\parallel z_1-z_2\parallel }^2,$$

for all $\ell \in ({\ell }_j,s_j)$ and $z_1,z_2\in \mathcal{X}$.

(ii)

There exist constants ${\widehat{M}}_{{\mathbb{O}}_j}^l>0$ such that

$$\mathbb{E}\parallel {\mathbb{O}}_j^l(\ell ,z){\parallel }^2\le {\widehat{M}}_{{\mathbb{O}}_j}^l\mathbb{E}{\parallel z\parallel }^2,$$

for all $z\in \mathcal{X}$ and $\ell \in ({\ell }_j,s_j]$.

(H5)

The nonlocal operators ${\mathfrak{g}}_l:\mathcal{C}([0,\mathfrak{a}],\mathcal{X})\to \mathcal{X}$, $l=1,2$, are assumed to fulfil the following structural properties:

(i)

There exist strictly positive constants $M_{\mathfrak{g}}^l$ and ${\tilde{M}}_{\mathfrak{g}}^l$ such that the quadratic growth estimate

$$\mathbb{E}\parallel {\mathfrak{g}}_l\left(z\right){\parallel }^2\le M_{\mathfrak{g}}^l\mathbb{E}{\parallel z\parallel }^2+{\tilde{M}}_{\mathfrak{g}}^l,\forall z\in \mathcal{C}([0,\mathfrak{a}],\mathcal{X})$$

is satisfied.

(ii)

There exists a constant ${\widehat{M}}_{\mathfrak{g}}^l>0$ such that the Lipschitz-type inequality

$$\mathbb{E}\parallel {\mathfrak{g}}_l\left(z_1\right)-{\mathfrak{g}}_l\left(z_2\right){\parallel }^2\le {\widehat{M}}_{\mathfrak{g}}^l\mathbb{E}{\parallel z_1-z_2\parallel }^2,\forall z_1,z_2\in \mathcal{C}([0,\mathfrak{a}],\mathcal{X}),$$

holds true.

(H6)

The jump–diffusion kernel $\hslash :[0,\mathfrak{a}]\times {\mathcal{X}}^{m+1}\times \mathsf{\Pi }\to \mathcal{X}$ is presumed to comply with the following structural requirements:

• The mapping $\hslash (\ell ,z_0,\cdots ,z_m,\eta )$ is continuous with respect to its arguments.
• For every $\iota \in \mathbb{N}$ there exists a nonnegative function ${\widehat{\varphi }}_{\iota }:(0,\mathfrak{a}]\to {\mathbb{R}}_{+}$ such that

  $$\underset{\parallel z_0{\parallel }^2,\cdots ,{\parallel z_m\parallel }^2\le \iota }{sup}{\int }_{\mathsf{\Pi }}\mathbb{E}\parallel \hslash (\ell ,z_0,\cdots ,z_m,\eta )\parallel \kappa \left(d\eta \right)\le {\widehat{\varphi }}_{\iota }(\ell ),$$

  and the weighted mapping $s\mapsto {(\ell -s)}^{2\varsigma -1}{\widehat{\varphi }}_{\iota }\left(s\right)$ belongs to ${\mathcal{L}}^1((0,\mathfrak{a}],{\mathbb{R}}_{+})$. Furthermore, there exists a constant ${\mathrm{\Lambda }}_3>0$ satisfying

  $$\underset{\iota \to \infty }{lim}inf{\displaystyle \frac{{\displaystyle {\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}{\widehat{\varphi }}_{\iota }\left(s\right)ds}}{\iota }}={\mathrm{\Lambda }}_3<\infty ,\ell \in (0,\mathfrak{a}].$$

Remark 12.

Conditions (H2)–(H6), particularly their second parts, are imposed to control the growth of the nonlinear and stochastic terms in a weighted integral sense. These assumptions are standard in the study of fractional stochastic evolution equations with memory, impulses, and jump perturbations. The functions ${\psi }_{\iota }$, ${\widehat{\psi }}_{\iota }$, and ${\widehat{\varphi }}_{\iota }$ are not required to be explicitly known; rather, they serve as bounding functions depending on bounded subsets of the state space. Such estimates are satisfied in many practical situations, including systems with Lipschitz continuity, polynomial growth nonlinearities, or locally bounded stochastic coefficients. Therefore, the above conditions are not restrictive and can be verified for a wide class of fractional stochastic models after suitable estimation.

Theorem 2.

Whenever hypotheses (H1)–(H6) are fulfilled, the dynamical system (1) admits an $\mathbb{MS}$ on the interval $[0,\mathfrak{a}]$, contingent upon the validity of the following additional requirements:

$$\begin{array}{ccc}& & {\mathfrak{a}}^{2(2-\mu )}{\widehat{M}}_{{\mathbb{O}}_j}^1+7{\mathfrak{a}}^{2(2-\mu )}[{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -2}}{(2\varsigma -1)\mathrm{\Gamma }(\tau +2\varsigma -1)}}\right)}^2\left[M_{\mathfrak{g}}^1+{\widehat{M}}_{{\mathbb{O}}_j}^1+M_{*}^2{\widehat{M}}_{\mathcal{F}}\right]\hfill \\ & & +M_{*}^2{\widehat{M}}_{\mathcal{F}}+{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -1}}{\mathrm{\Gamma }(\tau +2\varsigma )}}\right)}^2\left\{M_{\mathfrak{g}}^2+{\widehat{M}}_{{\mathbb{O}}_j}^2\right\}+\mathrm{Tr}\left(\mathbb{Q}\right){\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\mathrm{\Lambda }}_1+{\displaystyle \frac{2H{\mathfrak{d}}^2{\mathfrak{a}}^{2H+4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\mathrm{\Lambda }}_2\hfill \\ & & +{\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\mathrm{\Lambda }}_3]\left[1+7{\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{6\varsigma -1}{M}_{\mathfrak{B}}^2}{2\varsigma {\mathrm{\Gamma }}^2(2\varsigma +1)}}{\left({\displaystyle \frac{M_{\mathfrak{B}}\mathfrak{d}{\mathfrak{a}}^{2\varsigma -1}}{\mathfrak{x}\mathrm{\Gamma }\left(2\varsigma \right)}}\right)}^2\right]<1\hfill \end{array}$$

(3)

and

$$\begin{array}{ccc}\hfill \mathfrak{I}& :=& 4[{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -\mu }}{(2\varsigma -1)\mathrm{\Gamma }(\tau +2\varsigma -1)}}\right)}^2\left({\widehat{M}}_{\mathfrak{g}}^1+M_{*}^2{M}_{\mathcal{F}}+M_{{\mathbb{O}}_j}^1\right)\hfill \\ & & +{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -\mu +1}}{\mathrm{\Gamma }(\tau +2\varsigma )}}\right)}^2\left({\widehat{M}}_{\mathfrak{g}}^2+M_{{\mathbb{O}}_j}^2\right)+{\mathfrak{a}}^{2(2-\mu )}\left(M_{{\mathbb{O}}_j}^1+M_{*}^2{M}_{\mathcal{F}}\right)]<1.\hfill \end{array}$$

(4)

Proof. 

Before proceeding to the proof, we emphasize that all operators and mappings involved are well-defined on the phase space $\mathbb{S}$ and take values in ${\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})$. More precisely:

• The deterministic and stochastic operators define mappings from $\mathbb{S}$ into ${\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})$ and are continuous with respect to the norm of $\mathbb{S}$.
• The fractional resolvent, cosine family, and associated sine family are bounded linear operators on $\mathcal{X}$, and their induced integral operators are well-defined on $\mathbb{S}$.
• The impulsive operators are well-defined on each sub-interval induced by the fixed partition and preserve the structure of $\mathbb{S}$.
• All stochastic integrals appearing in the formulation are interpreted in the Itô sense and are square-integrable in ${\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})$.

Let $\mathfrak{x}>0$ be fixed. We now define the operator ${\mathrm{\Xi }}^{\mathfrak{x}}:\mathbb{S}\to \mathbb{S}$ by

$$\left({\mathrm{\Xi }}^{\mathfrak{x}}z\right)(\ell )=\left\{\begin{array}{cc}{\mho }_1,\hfill & \ell \in [0,{\ell }_1],\hfill \\ ⋮\hfill & \\ {\mathbb{O}}_j^1(\ell ,z(\ell )),\hfill & \ell \in ({\ell }_j,s_j],j=1,2,\cdots ,\beta ,\hfill \\ ⋮\hfill & \\ {\mho }_2,\hfill & \ell \in (s_j,{\ell }_{j+1}],j=1,2,\cdots ,\beta ,\hfill \end{array}\right.$$

(5)

$$\begin{array}{ccc}\hfill {\widehat{\mho }}_1& =& {\mathcal{R}}_{\wp ,\Re }(\ell )\left[z_0-{\mathfrak{g}}_1\left(z\right)+\mathcal{F}(0,z\left(0\right),z\left({\zeta }_1\left(0\right)\right),\cdots ,z\left({\zeta }_m\left(0\right)\right))\right]\hfill \\ & & -\mathcal{F}(\ell ,z(\ell ),z\left({\zeta }_1(\ell )\right),\cdots ,z\left({\zeta }_m(\ell )\right))+{\mathcal{T}}_{\wp ,\Re }(\ell )\left[z_1-{\mathfrak{g}}_2\left(z\right)+\Im \right]\hfill \\ & & +{\int }_0^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{B}{u}^{\mathfrak{x}}\left(s\right)ds\hfill \\ & & +{\int }_0^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s){\int }_0^s\mathfrak{F}(\tau ,z\left(\tau \right),z\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(\tau \right)\right))d\mathcal{W}\left(\tau \right)ds\hfill \\ & & +{\int }_0^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{O}(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right))d{\mathcal{W}}^H\left(s\right)\hfill \\ & & +{\int }_0^{\ell }{\int }_{\mathsf{\Pi }}{\mathcal{V}}_{\varsigma }(\ell -s)\hslash (s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta )\widehat{N}(ds,d\eta ),\hfill \\ \hfill {\widehat{\mho }}_2& =& {\mathcal{R}}_{\wp ,\Re }(\ell -s_j)\left[{\mathbb{O}}_j^1(s_j,z\left(s_j\right))+\mathcal{F}(s_j,z\left(s_j\right),z\left({\zeta }_1\left(s_j\right)\right),\cdots ,z\left({\zeta }_m\left(s_j\right)\right))\right]\hfill \\ & & -\mathcal{F}(\ell ,z(\ell ),z\left({\zeta }_1(\ell )\right),\cdots ,z\left({\zeta }_m(\ell )\right))+{\mathcal{T}}_{\wp ,\Re }(\ell -s_j)\left[{\mathbb{O}}_j^2(s_j,z\left(s_j\right))+{\widehat{\Im }}_j\right]\hfill \\ & & +{\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{B}{u}^{\mathfrak{x}}\left(s\right)ds\hfill \\ & & +{\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s){\int }_0^s\mathfrak{F}(\tau ,z\left(\tau \right),z\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(\tau \right)\right))d\mathcal{W}\left(\tau \right)ds\hfill \\ & & +{\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{O}(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right))d{\mathcal{W}}^H\left(s\right)\hfill \\ & & +{\int }_{s_j}^{\ell }{\int }_{\mathsf{\Pi }}{\mathcal{V}}_{\varsigma }(\ell -s)\hslash (s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta )\widehat{N}(ds,d\eta ),\hfill \end{array}$$

By virtue of Theorem 1, the mapping ${\mathrm{\Xi }}^{\mathfrak{x}}$ is guaranteed to exhibit a fixed point; this invariant element constitutes an $\mathbb{MS}$ of (1). For given $\mathfrak{x}>0$ and $ı>0$, let ${\mathbb{B}}_{ı}=\left\{z\in {\mathbb{S}:\parallel z\parallel }^2\le ı\right\}.$ It is readily verified that ${\mathbb{B}}_{ı}$ forms a bounded, closed, and convex subset of $\mathbb{S}$. Invoking assumption (H2) in conjunction with the Burkholder–Davis–Gundy inequality, one derives

$$\begin{array}{ccc}& & \mathbb{E}\parallel {\int }_0^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s){\int }_0^s\mathfrak{F}(\tau ,z\left(\tau \right),z\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(\tau \right)\right))d\mathcal{W}\left(\tau \right)ds{\parallel }^2\hfill \\ & & \le \mathrm{Tr}\left(\mathbb{Q}\right){\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}\hfill \\ & & \times \underset{\parallel z_0{\parallel }^2,\cdots ,{\parallel z_m\parallel }^2\le \iota }{sup}{\int }_0^s\mathbb{E}\parallel \mathfrak{F}\left(\tau ,z\left(\tau \right),z\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(\tau \right)\right)\right){\parallel }_Q^{2}d\tau ds\hfill \\ & & \le \mathrm{Tr}\left(\mathbb{Q}\right){\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}{\psi }_{\iota }\left(s\right)ds.\hfill \end{array}$$

Similarly, invoking assumption (H3) in conjunction with the Burkholder–Davis–Gundy inequality for fBm, we infer

$$\begin{array}{ccc}& & \mathbb{E}\parallel {\int }_0^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{O}(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right))d{\mathcal{W}}^H\left(s\right){\parallel }^2\hfill \\ & & \le {\displaystyle \frac{2H{\mathfrak{d}}^2{\mathfrak{a}}^{2H+4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}\hfill \\ & & \times \underset{\parallel z_0{\parallel }^2,\cdots ,{\parallel z_m\parallel }^2\le \iota }{sup}\mathbb{E}\parallel \mathfrak{O}(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right)){\parallel }_{{\mathcal{L}}_0^2}^{2}ds\hfill \\ & & \le {\displaystyle \frac{2H{\mathfrak{d}}^2{\mathfrak{a}}^{2H+4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}{\widehat{\psi }}_{\iota }\left(s\right)ds.\hfill \end{array}$$

Moreover, by appealing to hypothesis (H6) in conjunction with an application of Hölder’s inequality, one arrives at the following bound.

$$\begin{array}{ccc}& & \mathbb{E}\parallel {\int }_0^{\ell }{\int }_{\mathsf{\Pi }}{\mathcal{V}}_{\varsigma }(\ell -s)\hslash (s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta )\widehat{N}(ds,d\eta ){\parallel }^2\hfill \\ & & \le {\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}\hfill \\ & & \times \underset{\parallel z_0{\parallel }^2,\cdots ,{\parallel z_m\parallel }^2\le \iota }{sup}{\int }_{\mathsf{\Pi }}\mathbb{E}\parallel \hslash (s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta ){\parallel }^2\kappa \left(d\eta \right)ds\hfill \\ & & \le {\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}{\widehat{\varphi }}_{\iota }\left(s\right)ds.\hfill \end{array}$$

Furthermore, by invoking hypotheses (H1)–(H6) in concert with Hölder’s inequality, one derives the ensuing estimate.

$$\begin{array}{c}\hfill \mathbb{E}\parallel {{\int }_0^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{B}{u}^{\mathfrak{x}}\left(s\right)ds\parallel }^2\le {\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}{M}_{\mathfrak{B}}^2}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}\mathbb{E}{\parallel u^{\mathfrak{x}}\left(s\right)\parallel }^{2}ds,\end{array}$$

where

$$\begin{array}{ccc}\hfill \mathbb{E}{∥u^{\mathfrak{x}}\left(s\right)∥}^2& \le & 7{\left({\displaystyle \frac{M_{\mathfrak{B}}\mathfrak{d}{\mathfrak{a}}^{2\varsigma -1}}{\mathfrak{x}\mathrm{\Gamma }\left(2\varsigma \right)}}\right)}^2\{\mathbb{E}{∥\mathbb{E}\left[z_{\mathfrak{a}}\right]+{\int }_0^{\mathfrak{a}}\widehat{\varrho }\left(s\right)d\mathcal{W}\left(s\right)+{\int }_0^{\mathfrak{a}}\widehat{\vartheta }\left(s\right)d{\mathcal{W}}^H\left(s\right)∥}^2\hfill \\ & & +\mathbb{E}\parallel {\mathcal{R}}_{\wp ,\Re }\left(\mathfrak{a}\right)\left[z_0-{\mathfrak{g}}_1\left(z\right)+\mathcal{F}(0,z\left(0\right),z\left({\zeta }_1\left(0\right)\right),\cdots ,z\left({\zeta }_m\left(0\right)\right))\right]{\parallel }^2\hfill \\ & & +\mathbb{E}\parallel \mathcal{F}(\mathfrak{a},z\left(\mathfrak{a}\right),z\left({\zeta }_1\left(\mathfrak{a}\right)\right),\cdots ,z\left({\zeta }_m\left(\mathfrak{a}\right)\right)){\parallel }^2+\mathbb{E}\parallel {\mathcal{T}}_{\wp ,\Re }\left(\mathfrak{a}\right)\left[z_1-{\mathfrak{g}}_2\left(z\right)+\Im \right]{\parallel }^2\hfill \\ & & +\mathbb{E}\parallel {\int }_0^{\mathfrak{a}}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s){\int }_0^s\mathfrak{F}(\tau ,z\left(\tau \right),z\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(\tau \right)\right))d\mathcal{W}\left(\tau \right)ds{\parallel }^2\hfill \\ & & +\mathbb{E}\parallel {\int }_0^{\mathfrak{a}}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s)\mathfrak{O}(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right))d{\mathcal{W}}^H\left(s\right){\parallel }^2\hfill \\ & & +\mathbb{E}\parallel {\int }_0^{\mathfrak{a}}{\int }_{\mathsf{\Pi }}\hslash (s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta )\widehat{N}(ds,d\eta ){\parallel }^2\}\hfill \\ & \le & 7{\left({\displaystyle \frac{M_{\mathfrak{B}}\mathfrak{d}{\mathfrak{a}}^{2\varsigma -1}}{\mathfrak{x}\mathrm{\Gamma }\left(2\varsigma \right)}}\right)}^2{\mathcal{K}}^{\mathfrak{x}}.\hfill \end{array}$$

Here,

$$\begin{array}{ccc}\hfill {\mathcal{K}}^{\mathfrak{x}}& =& \{3\mathbb{E}{∥z_{\mathfrak{a}}∥}^2+3\mathrm{Tr}\left(\mathbb{Q}\right){\int }_0^{\mathfrak{a}}\mathbb{E}{∥\widehat{\varrho }\left(s\right)∥}^{2}ds+6H{\mathfrak{a}}^{2H-1}{\int }_0^{\mathfrak{a}}\mathbb{E}{∥\widehat{\vartheta }\left(s\right)∥}^{2}ds\hfill \\ & & +3{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -2}}{(2\varsigma -1)\mathrm{\Gamma }(\tau +2\varsigma -1)}}\right)}^2\left[\mathbb{E}{∥z_0∥}^2+M_{\mathfrak{g}}^1ı+{\tilde{M}}_{\mathfrak{g}}^1+M_{*}^2{\widehat{M}}_{\mathcal{F}}(ı+1)\right]\hfill \\ & & +M_{*}^2{\widehat{M}}_{\mathcal{F}}(ı+1)+3{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -1}}{\mathrm{\Gamma }(\tau +2\varsigma )}}\right)}^2\left[\mathbb{E}{∥{\mathfrak{z}}_{\mathfrak{1}}∥}^2+M_{\mathfrak{g}}^2ı+{\tilde{M}}_{\mathfrak{g}}^2+\mathbb{E}{∥\Im ∥}^2\right]\hfill \\ & & +\mathrm{Tr}\left(\mathbb{Q}\right){\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\mathfrak{a}}{(\mathfrak{a}-s)}^{2\varsigma -1}{\psi }_{\iota }\left(s\right)ds+{\displaystyle \frac{2H{\mathfrak{d}}^2{\mathfrak{a}}^{2H+4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\mathfrak{a}}{(\mathfrak{a}-s)}^{2\varsigma -1}{\widehat{\psi }}_{\iota }\left(s\right)ds\hfill \\ & & +{\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\mathfrak{a}}{(\mathfrak{a}-s)}^{2\varsigma -1}{\widehat{\varphi }}_{\iota }\left(s\right)ds\}\hfill \end{array}$$

and for $\ell \in ({\mathfrak{s}}_{\mathfrak{j}},{\ell }_{\mathfrak{j}+\mathfrak{1}}]$,

$$\begin{array}{ccc}\hfill \mathbb{E}{∥u^{\mathfrak{x}}\left(s\right)∥}^2& \le & 7{\left({\displaystyle \frac{M_{\mathfrak{B}}\mathfrak{d}{\mathfrak{a}}^{2\varsigma -1}}{\mathfrak{x}\mathrm{\Gamma }\left(2\varsigma \right)}}\right)}^2{\widehat{\mathcal{K}}}^{\mathfrak{x}},\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\widehat{\mathcal{K}}}^{\mathfrak{x}}& =& \{3\mathbb{E}{∥z_{\mathfrak{a}}∥}^2+3\mathrm{Tr}\left(\mathbb{Q}\right){\int }_0^{\mathfrak{a}}\mathbb{E}{∥\widehat{\varrho }\left(s\right)∥}^{2}ds+6H{\mathfrak{a}}^{2H-1}{\int }_0^{\mathfrak{a}}\mathbb{E}{∥\widehat{\vartheta }\left(s\right)∥}^{2}ds\hfill \\ & & +2{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -2}}{(2\varsigma -1)\mathrm{\Gamma }(\tau +2\varsigma -1)}}\right)}^2\left[{\widehat{M}}_{{\mathbb{O}}_j}^1ı+M_{*}^2{\widehat{M}}_{\mathcal{F}}(ı+1)\right]+M_{*}^2{\widehat{M}}_{\mathcal{F}}(ı+1)\hfill \\ & & +2{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -1}}{\mathrm{\Gamma }(\tau +2\varsigma )}}\right)}^2\left[{\widehat{M}}_{{\mathbb{O}}_j}^2ı+\mathbb{E}{∥{\widehat{\Im }}_j∥}^2\right]+\mathrm{Tr}\left(\mathbb{Q}\right){\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_{s_j}^{\mathfrak{a}}{(\mathfrak{a}-s)}^{2\varsigma -1}{\psi }_{\iota }\left(s\right)ds\hfill \\ & & +{\displaystyle \frac{2H{\mathfrak{d}}^2{\mathfrak{a}}^{2H+4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_{s_j}^{\mathfrak{a}}{(\mathfrak{a}-s)}^{2\varsigma -1}{\widehat{\psi }}_{\iota }\left(s\right)ds+{\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_{s_j}^{\mathfrak{a}}{(\mathfrak{a}-s)}^{2\varsigma -1}{\widehat{\varphi }}_{\iota }\left(s\right)ds\}.\hfill \end{array}$$

We assert that there exists a constant $q>0$ such that ${\mathrm{\Xi }}^{\mathfrak{x}}\left({\mathbb{B}}_{ı}\right)\subseteq {\mathbb{B}}_{ı}$. Assume, to the contrary, that this assertion fails. Then for every $ı>0$, one can find a function $z_{ı}(·)\in {\mathbb{B}}_{ı}$ satisfying ${\mathrm{\Xi }}^{\mathfrak{x}}\left(z_{ı}\right)\notin {\mathbb{B}}_{ı}$. Equivalently, there exists $\ell =\ell (ı)\in [0,\mathfrak{a}]$ such that

$$\parallel \left({\mathrm{\Xi }}^{\mathfrak{x}}{z}_{ı}\right)(\ell ){\parallel }_{\mathbb{S}}^2>ı,$$

where the notation $\ell (ı)$ emphasizes the dependence of ℓ on ı.

Nevertheless, by virtue of hypotheses (H1)–(H6), in conjunction with the foregoing relations, we deduce that for every $\ell \in (0,{\ell }_1]$, the following estimate holds:

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel {\mathrm{\Xi }}^{\mathfrak{x}}{z}_{ı}{(\ell )\parallel }_{\mathbb{S}}^2& \le & 7\underset{\ell }{sup}{\ell }^{2(2-\mu )}\{\mathbb{E}\parallel {\mathcal{R}}_{\wp ,\Re }(\ell )\left[z_0-{\mathfrak{g}}_1\left(z\right)+\mathcal{F}(0,z\left(0\right),z\left({\zeta }_1\left(0\right)\right),\cdots ,z\left({\zeta }_m\left(0\right)\right))\right]{\parallel }^2\hfill \\ & & +\mathbb{E}\parallel \mathcal{F}(\ell ,z(\ell ),z\left({\zeta }_1(\ell )\right),\cdots ,z\left({\zeta }_m(\ell )\right)){\parallel }^2+\mathbb{E}\parallel {\mathcal{T}}_{\wp ,\Re }(\ell )\left[z_1-{\mathfrak{g}}_2\left(z\right)+\Im \right]{\parallel }^2\hfill \\ & & +\mathbb{E}\parallel {\int }_0^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{B}{u}^{\mathfrak{x}}\left(s\right)ds{\parallel }^2+\mathbb{E}{∥{\int }_0^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s){\int }_0^s\mathfrak{F}(\tau ,z\left(\tau \right),z\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(\tau \right)\right))d\mathcal{W}\left(\tau \right)ds∥}^2\hfill \\ & & +\mathbb{E}{∥{\int }_0^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{O}(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right))d{\mathcal{W}}^H\left(s\right)∥}^2\hfill \\ & & +\mathbb{E}{∥{\int }_0^{\ell }{\int }_{\mathsf{\Pi }}{\mathcal{V}}_{\varsigma }(\ell -s)\hslash (s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta )\widehat{N}(ds,d\eta )∥}^2\}\hfill \\ & \le & 7{\mathfrak{a}}^{2(2-\mu )}[3{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -2}}{(2\varsigma -1)\mathrm{\Gamma }(\tau +2\varsigma -1)}}\right)}^2\left[\mathbb{E}{∥z_0∥}^2+M_{\mathfrak{g}}^1ı+{\tilde{M}}_{\mathfrak{g}}^1+M_{*}^2{\widehat{M}}_{\mathcal{F}}(ı+1)\right]+M_{*}^2{\widehat{M}}_{\mathcal{F}}(ı+1)\hfill \\ & & +3{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -1}}{\mathrm{\Gamma }(\tau +2\varsigma )}}\right)}^2\left[\mathbb{E}{∥{\mathfrak{z}}_{\mathfrak{1}}∥}^2+M_{\mathfrak{g}}^2ı+{\tilde{M}}_{\mathfrak{g}}^2+\mathbb{E}{∥\Im ∥}^2\right]+\mathrm{Tr}\left(\mathbb{Q}\right){\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}{\psi }_{\iota }\left(s\right)ds\hfill \\ & & +{\displaystyle \frac{2H{\mathfrak{d}}^2{\mathfrak{a}}^{2H+4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}{\widehat{\psi }}_{\iota }\left(s\right)ds+{\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_0^{\ell }{(\ell -s)}^{2\varsigma -1}{\widehat{\varphi }}_{\iota }\left(s\right)ds\hfill \\ & & +7{\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{6\varsigma -1}{M}_{\mathfrak{B}}^2}{2\varsigma {\mathrm{\Gamma }}^2(2\varsigma +1)}}{\left({\displaystyle \frac{M_{\mathfrak{B}}\mathfrak{d}{\mathfrak{a}}^{2\varsigma -1}}{\mathfrak{x}\mathrm{\Gamma }\left(2\varsigma \right)}}\right)}^2{\mathcal{K}}^{\mathfrak{x}}].\hfill \end{array}$$

(6)

For $\ell \in ({\ell }_j,s_j]$,

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel {\mathrm{\Xi }}^{\mathfrak{x}}{z}_{ı}{(\ell )\parallel }_{\mathbb{S}}^2& \le & {\mathfrak{a}}^{2(2-\mu )}{\widehat{M}}_{{\mathbb{O}}_j}^1ı,\hfill \end{array}$$

(7)

and for $\ell \in (s_j,{\ell }_{j+1}]$,

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel {\mathrm{\Xi }}^{\mathfrak{x}}{z}_{ı}{(\ell )\parallel }_{\mathbb{S}}^2& \le & 7\underset{\ell }{sup}{\ell }^{2(2-\mu )}\{\mathbb{E}\parallel {\mathcal{R}}_{\wp ,\Re }(\ell -s_j)\left[{\mathbb{O}}_j^1(s_j,z\left(s_j\right))+\mathcal{F}(s_j,z\left(s_j\right),z\left({\zeta }_1\left(s_j\right)\right),\cdots ,z\left({\zeta }_m\left(s_j\right)\right))\right]{\parallel }^2\hfill \\ & & +\mathbb{E}\parallel \mathcal{F}(\ell ,z(\ell ),z\left({\zeta }_1(\ell )\right),\cdots ,z\left({\zeta }_m(\ell )\right)){\parallel }^2+\mathbb{E}\parallel {\mathcal{T}}_{\wp ,\Re }(\ell -s_j)\left[{\mathbb{O}}_j^2(s_j,z\left(s_j\right))+{\widehat{\Im }}_j\right]{\parallel }^2\hfill \\ & & +\mathbb{E}\parallel {\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{B}{u}^{\mathfrak{x}}\left(s\right)ds{\parallel }^2+\mathbb{E}{∥{\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s){\int }_0^s\mathfrak{F}(\tau ,z\left(\tau \right),z\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(\tau \right)\right))d\mathcal{W}\left(\tau \right)ds∥}^2\hfill \\ & & +\mathbb{E}{∥{\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{O}(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right))d{\mathcal{W}}^H\left(s\right)∥}^2\hfill \\ & & +\mathbb{E}{∥{\int }_{s_j}^{\ell }{\int }_{\mathsf{\Pi }}{\mathcal{V}}_{\varsigma }(\ell -s)\hslash (s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta )\widehat{N}(ds,d\eta )∥}^2\}\hfill \end{array}$$

$$\begin{array}{ccc}& \le & 7{\mathfrak{a}}^{2(2-\mu )}[2{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -2}}{(2\varsigma -1)\mathrm{\Gamma }(\tau +2\varsigma -1)}}\right)}^2\left[{\widehat{M}}_{{\mathbb{O}}_j}^1ı+M_{*}^2{\widehat{M}}_{\mathcal{F}}(ı+1)\right]+M_{*}^2{\widehat{M}}_{\mathcal{F}}(ı+1)\hfill \\ & & +2{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -1}}{\mathrm{\Gamma }(\tau +2\varsigma )}}\right)}^2\left[{\widehat{M}}_{{\mathbb{O}}_j}^2ı+\mathbb{E}{∥{\widehat{\Im }}_j∥}^2\right]+\mathrm{Tr}\left(\mathbb{Q}\right){\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_{s_j}^{\ell }{(\ell -s)}^{2\varsigma -1}{\psi }_{\iota }\left(s\right)ds\hfill \\ & & +{\displaystyle \frac{2H{\mathfrak{d}}^2{\mathfrak{a}}^{2H+4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_{s_j}^{\ell }{(\ell -s)}^{2\varsigma -1}{\widehat{\psi }}_{\iota }\left(s\right)ds+{\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\int }_{s_j}^{\ell }{(\ell -s)}^{2\varsigma -1}{\widehat{\varphi }}_{\iota }\left(s\right)ds\hfill \\ & & +7{\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{6\varsigma -1}{M}_{\mathfrak{B}}^2}{2\varsigma {\mathrm{\Gamma }}^2(2\varsigma +1)}}{\left({\displaystyle \frac{M_{\mathfrak{B}}\mathfrak{d}{\mathfrak{a}}^{2\varsigma -1}}{\mathfrak{x}\mathrm{\Gamma }\left(2\varsigma \right)}}\right)}^2{\widehat{\mathcal{K}}}^{\mathfrak{x}}].\hfill \end{array}$$

(8)

By amalgamating inequalities (6)–(8) with the relation

$$ı\le \parallel \left({\mathrm{\Xi }}^{\mathfrak{x}}{z}_{ı}\right)(\ell ){\parallel }_{\mathbb{S}}^2$$

and subsequently normalizing both sides by ı, we proceed to pass to the lower limit as $ı\to +\infty$. This yields

$$\begin{array}{ccc}& & {\mathfrak{a}}^{2(2-\mu )}{\widehat{M}}_{{\mathbb{O}}_j}^1+7{\mathfrak{a}}^{2(2-\mu )}[{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -2}}{(2\varsigma -1)\mathrm{\Gamma }(\tau +2\varsigma -1)}}\right)}^2\left[M_{\mathfrak{g}}^1+{\widehat{M}}_{{\mathbb{O}}_j}^1+M_{*}^2{\widehat{M}}_{\mathcal{F}}\right]+M_{*}^2{\widehat{M}}_{\mathcal{F}}+{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -1}}{\mathrm{\Gamma }(\tau +2\varsigma )}}\right)}^2\left\{M_{\mathfrak{g}}^2+{\widehat{M}}_{{\mathbb{O}}_j}^2\right\}\hfill \\ & & +\mathrm{Tr}\left(\mathbb{Q}\right){\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\mathrm{\Lambda }}_1+{\displaystyle \frac{2H{\mathfrak{d}}^2{\mathfrak{a}}^{2H+4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\mathrm{\Lambda }}_2+{\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{4\varsigma -1}}{2\varsigma {\mathrm{\Gamma }}^2\left(2\varsigma \right)}}{\mathrm{\Lambda }}_3]\left[1+7{\displaystyle \frac{{\mathfrak{d}}^2{\mathfrak{a}}^{6\varsigma -1}{M}_{\mathfrak{B}}^2}{2\varsigma {\mathrm{\Gamma }}^2(2\varsigma +1)}}{\left({\displaystyle \frac{M_{\mathfrak{B}}\mathfrak{d}{\mathfrak{a}}^{2\varsigma -1}}{\mathfrak{x}\mathrm{\Gamma }\left(2\varsigma \right)}}\right)}^2\right]\ge 1.\hfill \end{array}$$

This conclusion stands in direct opposition to (3). Consequently, there must exist a positive constant ı such that

$${\mathrm{\Xi }}^{\mathfrak{x}}\left({\mathbb{B}}_{ı}\right)\subseteq {\mathbb{B}}_{ı}.$$

We now proceed to establish that the operator ${\mathrm{\Xi }}^{\mathfrak{x}}$ admits a fixed point in ${\mathbb{B}}_{ı}$, a fact that consequently ensures that Equation (1) possesses an $\mathbb{MS}$.

To this end, we decompose ${\mathrm{\Xi }}^{\mathfrak{x}}$ as

$${\mathrm{\Xi }}^{\mathfrak{x}}={\mathrm{\Xi }}_1^{\mathfrak{x}}+{\mathrm{\Xi }}_2^{\mathfrak{x}},$$

where the mappings ${\mathrm{\Xi }}_1^{\mathfrak{x}}$ and ${\mathrm{\Xi }}_2^{\mathfrak{x}}$ are defined on ${\mathbb{B}}_{ı}$ as follows:

$$\begin{array}{c}\hfill \left({\mathrm{\Xi }}_1^{\mathfrak{x}}z\right)(\ell )=\left\{\begin{array}{c}{\displaystyle {\mathcal{R}}_{\wp ,\Re }(\ell )\left[z_0-{\mathfrak{g}}_1\left(z\right)+\mathcal{F}\left(0,z\left(0\right),z\left({\zeta }_1\left(0\right)\right),\cdots ,z\left({\zeta }_m\left(0\right)\right)\right)\right]}\hfill \\ {\displaystyle -\mathcal{F}\left(\ell ,z(\ell ),z\left({\zeta }_1(\ell )\right),\cdots ,z\left({\zeta }_m(\ell )\right)\right)}\hfill \\ {\displaystyle +{\mathcal{T}}_{\wp ,\Re }(\ell )\left[z_1-{\mathfrak{g}}_2\left(z\right)+\Im \right],\ell \in [0,{\ell }_1],}\hfill \\ {\mathbb{O}}_j^1\left(\ell ,z(\ell )\right),\ell \in ({\ell }_j,s_j],j=1,2,\cdots ,\beta ,\hfill \\ {\displaystyle {\mathcal{R}}_{\wp ,\Re }(\ell -s_j)\left[{\mathbb{O}}_j^1\left(s_j,z\left(s_j\right)\right)+\mathcal{F}\left(s_j,z\left(s_j\right),z\left({\zeta }_1\left(s_j\right)\right),\cdots ,z({\zeta }_m\left(s_j\right)\right)\right]}\hfill \\ {\displaystyle -\mathcal{F}\left(\ell ,z(\ell ),z\left({\zeta }_1(\ell )\right),\cdots ,z\left({\zeta }_m(\ell )\right)\right)}\hfill \\ {\displaystyle +{\mathcal{T}}_{\wp ,\Re }(\ell -s_j)\left[{\mathbb{O}}_j^2\left(s_j,z\left(s_j\right)\right)+{\widehat{\Im }}_j\right],\ell \in (s_j,{\ell }_{j+1}],j=1,2,\cdots ,\beta .}\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill \left({\mathrm{\Xi }}_2^{\mathfrak{x}}z\right)(\ell )=\left\{\begin{array}{c}{\displaystyle {\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{B}{u}^{\mathfrak{x}}\left(s\right)ds}\hfill \\ {\displaystyle +{\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\left({\int }_0^s\mathfrak{F}\left(\tau ,z\left(\tau \right),z\left({\widehat{\zeta }}_1\left(\tau \right)\right),\cdots ,z\left({\widehat{\zeta }}_m\left(\tau \right)\right)\right)d\mathcal{W}\left(\tau \right)\right)ds}\hfill \\ {\displaystyle +{\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{O}\left(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right)\right)d{\mathcal{W}}^H\left(s\right)}\hfill \\ {\displaystyle +{\int }_{s_j}^{\ell }{\int }_{\mathsf{\Pi }}{\mathcal{V}}_{\varsigma }(\ell -s)\hslash \left(s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta \right)\widehat{N}(ds,d\eta ),}\hfill \\ \ell \in (s_j,{\ell }_{j+1}],j=0,1,2,\cdots ,\beta ,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

We proceed to demonstrate that ${\mathrm{\Xi }}_1^{\mathfrak{x}}$ fulfills a contraction property, whereas ${\mathrm{\Xi }}_2^{\mathfrak{x}}$ constitutes a compact operator.

To establish the contractive nature of ${\mathrm{\Xi }}_1^{\mathfrak{x}}$, let $z^{*},z^{**}\in {\mathbb{B}}_{ı}$ be arbitrary. Then, for each $\ell \in [0,\mathfrak{a}]$, and under hypotheses (H1), (H4), and (H5), it follows that for $\ell \in (0,{\ell }_1]$,

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel {\mathrm{\Xi }}_1^{\mathfrak{x}}{z}^{*}(\ell )-{\mathrm{\Xi }}_1^{\mathfrak{x}}{z}^{**}(\ell ){\parallel }^2& \le & 4[\mathbb{E}\parallel {\mathcal{R}}_{\wp ,\Re }(\ell )\left[{\mathfrak{g}}_1\left(z^{*}\right)-{\mathfrak{g}}_1\left(z^{**}\right)\right]{\parallel }^2+\mathbb{E}\parallel {\mathcal{T}}_{\wp ,\Re }(\ell )\left[{\mathfrak{g}}_2\left(z^{*}\right)-{\mathfrak{g}}_2\left(z^{**}\right)\right]{\parallel }^2\hfill \\ & & +\mathbb{E}\parallel {\mathcal{R}}_{\wp ,\Re }(\ell )[\mathcal{F}\left(0,z^{*}\left(0\right),z^{*}\left({\zeta }_1\left(0\right)\right),\cdots ,z^{*}\left({\zeta }_m\left(0\right)\right)\right)\hfill \\ & & -\mathcal{F}\left(0,z^{**}\left(0\right),z^{**}\left({\zeta }_1\left(0\right)\right),\cdots ,z^{**}\left({\zeta }_m\left(0\right)\right)\right)]{\parallel }^2\hfill \\ & & +\mathbb{E}\parallel \mathcal{F}\left(\ell ,z^{*}(\ell ),z^{*}\left({\zeta }_1(\ell )\right),\cdots ,z^{*}\left({\zeta }_m(\ell )\right)\right)-\mathcal{F}\left(\ell ,z^{**}(\ell ),z^{**}\left({\zeta }_1(\ell )\right),\cdots ,z^{**}\left({\zeta }_m(\ell )\right)\right){\parallel }^2]\hfill \\ & \le & 4\left[{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -2}}{(2\varsigma -1)\mathrm{\Gamma }(\tau +2\varsigma -1)}}\right)}^2\left({\widehat{M}}_{\mathfrak{g}}^1+M_{*}^2{M}_{\mathcal{F}}\right)+{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -1}}{\mathrm{\Gamma }(\tau +2\varsigma )}}\right)}^2{\widehat{M}}_{\mathfrak{g}}^2+M_{*}^2{M}_{\mathcal{F}}\right]\mathbb{E}\parallel z^{*}-z^{**}{\parallel }^2;\hfill \end{array}$$

for $\ell \in ({\ell }_j,s_j]$,

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel {\mathrm{\Xi }}_1^{\mathfrak{x}}{z}^{*}(\ell )-{\mathrm{\Xi }}_1^{\mathfrak{x}}{z}^{**}(\ell ){\parallel }^2& \le & \mathbb{E}\parallel {\mathbb{O}}_j^1\left(\ell ,z^{*}(\ell )\right)-{\mathbb{O}}_j^1\left(\ell ,z^{**}(\ell )\right){\parallel }^2\hfill \\ & \le & M_{{\mathbb{O}}_j}^1\mathbb{E}\parallel z^{*}-z^{**}{\parallel }^2;\hfill \end{array}$$

and for $\ell \in (s_j,{\ell }_{j+1}]$,

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel {\mathrm{\Xi }}_1^{\mathfrak{x}}{z}^{*}(\ell )-{\mathrm{\Xi }}_1^{\mathfrak{x}}{z}^{**}(\ell ){\parallel }^2& \le & 4[\mathbb{E}\parallel {\mathcal{R}}_{\wp ,\Re }(\ell -s_j)\left[{\mathbb{O}}_j^1\left(s_j,z^{*}\left(s_j\right)\right)-{\mathbb{O}}_j^1\left(s_j,z^{**}\left(s_j\right)\right)\right]{\parallel }^2\hfill \\ & & +\mathbb{E}\parallel {\mathcal{T}}_{\wp ,\Re }(\ell -s_j)\left[{\mathbb{O}}_j^2\left(s_j,z^{*}\left(s_j\right)\right)-{\mathbb{O}}_j^2\left(s_j,z^{**}\left(s_j\right)\right)\right]{\parallel }^2\hfill \\ & & +\mathbb{E}\parallel {\mathcal{R}}_{\wp ,\Re }(\ell -s_j)[\mathcal{F}\left(s_j,z^{*}\left(s_j\right),z^{*}\left({\zeta }_1\left(s_j\right)\right),\cdots ,z^{*}\left({\zeta }_m\left(s_j\right)\right)\right)\hfill \\ & & -\mathcal{F}\left(s_j,z^{**}\left(s_j\right),z^{**}\left({\zeta }_1\left(s_j\right)\right),\cdots ,z^{**}\left({\zeta }_m\left(s_j\right)\right)\right)]{\parallel }^2\hfill \\ & & +\mathbb{E}\parallel \mathcal{F}\left(\ell ,z^{*}(\ell ),z^{*}\left({\zeta }_1(\ell )\right),\cdots ,z^{*}\left({\zeta }_m(\ell )\right)\right)-\mathcal{F}\left(\ell ,z^{**}(\ell ),z^{**}\left({\zeta }_1(\ell )\right),\cdots ,z^{**}\left({\zeta }_m(\ell )\right)\right){\parallel }^2]\hfill \\ & \le & 4\left[{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -2}}{(2\varsigma -1)\mathrm{\Gamma }(\tau +2\varsigma -1)}}\right)}^2\left(M_{{\mathbb{O}}_j}^1+M_{*}^2{M}_{\mathcal{F}}\right)+{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -1}}{\mathrm{\Gamma }(\tau +2\varsigma )}}\right)}^2{M}_{{\mathbb{O}}_j}^2+M_{*}^2{M}_{\mathcal{F}}\right]\mathbb{E}\parallel z^{*}-z^{**}{\parallel }^2.\hfill \end{array}$$

Taking the supremum with respect to $\ell \in [0,\mathfrak{a}]$ of ${\ell }^{2(2-\mu )}$ on both sides of the preceding inequality, we arrive at

$$\begin{array}{ccc}\hfill \underset{\ell \in [0,\mathfrak{a}]}{sup}{\ell }^{2(2-\mu )}\mathbb{E}\parallel {\mathrm{\Xi }}_1^{\mathfrak{x}}{z}^{*}(\ell )-{\mathrm{\Xi }}_1^{\mathfrak{x}}{z}^{**}(\ell ){\parallel }^2& \le & 4[{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -\mu }}{(2\varsigma -1)\mathrm{\Gamma }(\tau +2\varsigma -1)}}\right)}^2\left({\widehat{M}}_{\mathfrak{g}}^1+M_{*}^2{M}_{\mathcal{F}}+M_{{\mathbb{O}}_j}^1\right)\hfill \\ & & +{\left({\displaystyle \frac{\mathfrak{d}{\mathfrak{a}}^{2\varsigma +\tau -\mu +1}}{\mathrm{\Gamma }(\tau +2\varsigma )}}\right)}^2\left({\widehat{M}}_{\mathfrak{g}}^2+M_{{\mathbb{O}}_j}^2\right)+{\mathfrak{a}}^{2(2-\mu )}\left(M_{{\mathbb{O}}_j}^1+M_{*}^2{M}_{\mathcal{F}}\right)]\mathbb{E}\parallel z^{*}-z^{**}{\parallel }^2\hfill \end{array}$$

Consequently, invoking the characterization of $\mathbb{S}$ together with relation (4), we deduce that

$$\parallel {\mathrm{\Xi }}_1^{\mathfrak{x}}{z}^{*}-{\mathrm{\Xi }}_1^{\mathfrak{x}}{z}^{**}{\parallel }_{\mathbb{S}}^2\le \mathfrak{I}\parallel z^{*}-z^{**}{\parallel }_{\mathbb{S}}^2$$

Consequently, the operator ${\mathrm{\Xi }}_1^{\mathfrak{x}}$ satisfies a contraction property. To establish the compactness of ${\mathrm{\Xi }}_2^{\mathfrak{x}}$, we begin by verifying that ${\mathrm{\Xi }}_2^{\mathfrak{x}}$ is continuous on ${\mathbb{B}}_{ı}$.

Let $\left\{z_n\right\}\subset {\mathbb{B}}_{ı}$ be a sequence such that $z_n\to z$ in ${\mathbb{B}}_{ı}$, and denote the control function by $u^{\mathfrak{x}}(\ell )=u^{\mathfrak{x}}(\ell ,z)$. Then, for every $s\in [0,\mathfrak{a}]$, it follows that $z_n\left(s\right)\to z\left(s\right)$. By assumptions (H2), (H3) and (H6), we deduce that

$$\mathfrak{F}(s,z_n\left(s\right),z_n\left(\widehat{{\zeta }_1}\left(s\right)\right),\cdots ,z_n\left(\widehat{{\zeta }_m}\left(s\right)\right))⟶\mathfrak{F}(s,z\left(s\right),z\left(\widehat{{\zeta }_1}\left(s\right)\right),\cdots ,z\left(\widehat{{\zeta }_m}\left(s\right)\right))$$

as $n\to \infty$,

$$\hslash \left(s,z_n\left(s\right),z_n\left({\varrho }_1\left(s\right)\right),\cdots ,z_n\left({\varrho }_m\left(s\right)\right),\eta \right)⟶\hslash \left(s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta \right)$$

as $n\to \infty$, and, similarly,

$$\mathfrak{O}(s,z_n\left(s\right),z_n\left({\xi }_1\left(s\right)\right),\cdots ,z_n\left({\xi }_m\left(s\right)\right))⟶\mathfrak{O}(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right)),n\to \infty .$$

Invoking the dominated convergence theorem yields

$$\begin{array}{cc}\hfill \parallel {\mathrm{\Xi }}_2^{\mathfrak{x}}{z}^{*}-{\mathrm{\Xi }}_2^{\mathfrak{x}}{z}^{**}{\parallel }_{\mathbb{S}}^2& =\underset{\ell \in [0,\mathfrak{a}]}{sup}{\ell }^{2(2-\mu )}\mathbb{E}\parallel {\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)\mathfrak{B}\left(u^{\mathfrak{x}}(s,z_n)-u^{\mathfrak{x}}(s,z)\right)ds\hfill \\ \hfill & +{\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s){\int }_0^s(\mathfrak{F}\left(\tau ,z_n\left(\tau \right),z_n\left({\widehat{\zeta }}_1\left(\tau \right)\right),\cdots ,z_n\left({\widehat{\zeta }}_m\left(\tau \right)\right)\right)\hfill \\ \hfill & -\mathfrak{F}\left(\tau ,z\left(\tau \right),z\left({\widehat{\zeta }}_1\left(\tau \right)\right),\cdots ,z\left({\widehat{\zeta }}_m\left(\tau \right)\right)\right))d\mathcal{W}\left(\tau \right)ds\hfill \\ \hfill & +{\int }_{s_j}^{\ell }{\mathcal{V}}_{\varsigma }(\ell -s)(\mathfrak{O}\left(s,z_n\left(s\right),z_n\left({\xi }_1\left(s\right)\right),\cdots ,z_n\left({\xi }_m\left(s\right)\right)\right)\hfill \\ \hfill & -\mathfrak{O}\left(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right)\right))d{\mathcal{W}}^H\left(s\right)\hfill \\ \hfill & +{\int }_{s_j}^{\ell }{\int }_{\mathsf{\Pi }}{\mathcal{V}}_{\varsigma }(\ell -s)(\hslash \left(s,z_n\left(s\right),z_n\left({\varrho }_1\left(s\right)\right),\cdots ,z_n\left({\varrho }_m\left(s\right)\right),\eta \right)\hfill \\ \hfill & -\hslash \left(s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta \right))\widehat{N}(ds,d\eta ){\parallel }^2⟶0,\hfill \end{array}$$

as $n\to \infty$, which confirms the continuity of ${\mathrm{\Xi }}_2^{\mathfrak{x}}$.

Subsequently, we verify that the set

$$\left\{{\mathrm{\Xi }}_2^{\mathfrak{x}}z:z\in {\mathbb{B}}_{ı}\right\}$$

constitutes an equicontinuous family of functions. Let $\epsilon >0$ be arbitrarily small and select points satisfying $s_j<\widehat{\ell }<\overline{\ell }\le {\ell }_{j+1}$. Under these circumstances, one obtains the estimate

$$\begin{array}{cc}\hfill \mathbb{E}\parallel {\mathrm{\Xi }}_2^{\mathfrak{x}}z\left(\overline{\ell }\right)-{\mathrm{\Xi }}_2^{\mathfrak{x}}z\left(\widehat{\ell }\right){\parallel }^2& \le \mathbb{E}\parallel {\int }_{\widehat{\ell }}^{\overline{\ell }}{\mathcal{V}}_{\varsigma }(\overline{\ell }-s)\mathfrak{B}{u}^{\mathfrak{x}}\left(s\right)ds{\parallel }^2\hfill \\ \hfill & +\mathbb{E}\parallel {\int }_{s_j}^{\widehat{\ell }-\epsilon }\left({\mathcal{V}}_{\varsigma }(\overline{\ell }-s)-{\mathcal{V}}_{\varsigma }(\widehat{\ell }-s)\right)\mathfrak{B}{u}^{\mathfrak{x}}\left(s\right)ds{\parallel }^2\hfill \\ \hfill & +\mathbb{E}\parallel {\int }_{\widehat{\ell }-\epsilon }^{\widehat{\ell }}\left({\mathcal{V}}_{\varsigma }(\overline{\ell }-s)-{\mathcal{V}}_{\varsigma }(\widehat{\ell }-s)\right)\mathfrak{B}{u}^{\mathfrak{x}}\left(s\right)ds{\parallel }^2\hfill \\ \hfill & +\mathbb{E}\parallel {\int }_{s_j}^{\widehat{\ell }-\epsilon }\left({\mathcal{V}}_{\varsigma }(\overline{\ell }-s)-{\mathcal{V}}_{\varsigma }(\widehat{\ell }-s)\right){\int }_0^s\mathfrak{F}\left(\tau ,z\left(\tau \right),z\left({\widehat{\zeta }}_1\left(\tau \right)\right),\cdots ,z\left({\widehat{\zeta }}_m\left(\tau \right)\right)\right)d\mathcal{W}\left(\tau \right)ds{\parallel }^2\hfill \\ \hfill & +\mathbb{E}\parallel {\int }_{\widehat{\ell }-\epsilon }^{\widehat{\ell }}\left({\mathcal{V}}_{\varsigma }(\overline{\ell }-s)-{\mathcal{V}}_{\varsigma }(\widehat{\ell }-s)\right){\int }_0^s\mathfrak{F}\left(\tau ,z\left(\tau \right),z\left({\widehat{\zeta }}_1\left(\tau \right)\right),\cdots ,z\left({\widehat{\zeta }}_m\left(\tau \right)\right)\right)d\mathcal{W}\left(\tau \right)ds{\parallel }^2\hfill \\ \hfill & +\mathbb{E}\parallel {\int }_{\widehat{\ell }}^{\overline{\ell }}{\mathcal{V}}_{\varsigma }(\overline{\ell }-s){\int }_0^s\mathfrak{F}\left(\tau ,z\left(\tau \right),z\left({\widehat{\zeta }}_1\left(\tau \right)\right),\cdots ,z\left({\widehat{\zeta }}_m\left(\tau \right)\right)\right)d\mathcal{W}\left(\tau \right)ds{\parallel }^2\hfill \\ \hfill & +\mathbb{E}\parallel {\int }_{s_j}^{\widehat{\ell }-\epsilon }\left({\mathcal{V}}_{\varsigma }(\overline{\ell }-s)-{\mathcal{V}}_{\varsigma }(\widehat{\ell }-s)\right)\mathfrak{O}\left(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right)\right)d{\mathcal{W}}^H\left(s\right){\parallel }^2\hfill \\ \hfill & +\mathbb{E}\parallel {\int }_{\widehat{\ell }-\epsilon }^{\widehat{\ell }}\left({\mathcal{V}}_{\varsigma }(\overline{\ell }-s)-{\mathcal{V}}_{\varsigma }(\widehat{\ell }-s)\right)\mathfrak{O}\left(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right)\right)d{\mathcal{W}}^H\left(s\right){\parallel }^2\hfill \\ \hfill & +\mathbb{E}\parallel {\int }_{\widehat{\ell }}^{\overline{\ell }}{\mathcal{V}}_{\varsigma }(\overline{\ell }-s)\mathfrak{O}\left(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right)\right)d{\mathcal{W}}^H\left(s\right){\parallel }^2\hfill \\ \hfill & +\mathbb{E}\parallel {\int }_{s_j}^{\widehat{\ell }-\epsilon }\left({\mathcal{V}}_{\varsigma }(\overline{\ell }-s)-{\mathcal{V}}_{\varsigma }(\widehat{\ell }-s)\right){\int }_{\mathsf{\Pi }}\hslash \left(s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta \right)\widehat{N}(ds,d\eta ){\parallel }^2\hfill \\ \hfill & +\mathbb{E}\parallel {\int }_{\widehat{\ell }-\epsilon }^{\widehat{\ell }}\left({\mathcal{V}}_{\varsigma }(\overline{\ell }-s)-{\mathcal{V}}_{\varsigma }(\widehat{\ell }-s)\right){\int }_{\mathsf{\Pi }}\hslash \left(s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta \right)\widehat{N}(ds,d\eta ){\parallel }^2\hfill \\ \hfill & +\mathbb{E}\parallel {\int }_{\widehat{\ell }}^{\overline{\ell }}{\mathcal{V}}_{\varsigma }(\overline{\ell }-s){\int }_{\mathsf{\Pi }}\hslash \left(s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta \right)\widehat{N}(ds,d\eta ){\parallel }^2.\hfill \end{array}$$

It can be observed that

$$\mathbb{E}\parallel {\mathrm{\Xi }}_2^{\mathfrak{x}}z\left(\overline{\ell }\right)-{\mathrm{\Xi }}_2^{\mathfrak{x}}z\left(\widehat{\ell }\right){\parallel }^2⟶0\mathit{as}\overline{\ell }\to \widehat{\ell },$$

uniformly for all $z\in {\mathbb{B}}_{ı}$. This uniform convergence is ensured by the continuity of the resolvent family ${\mathcal{R}}_{\wp ,\Re }(\ell )$ in the uniform operator topology together with its compactness for $\ell >0$, which implies uniform continuity with respect to time variations.

More precisely, for sufficiently small $\epsilon >0$, the difference of the operator values can be made arbitrarily small uniformly on bounded sets of ${\mathbb{B}}_{ı}$, which yields equicontinuity on $(0,\mathfrak{a}]$. Furthermore, by similar arguments, one can verify that the mapping ${\mathrm{\Xi }}_2^{\mathfrak{x}}z$ is also equicontinuous at $\ell =0$. Consequently, the operator ${\mathrm{\Xi }}_2^{\mathfrak{x}}$ maps bounded subsets of ${\mathbb{B}}_{ı}$ into equicontinuous families in $\mathbb{S}$.

It remains to verify that

$$\mathbb{V}(\ell )=\left\{\left({\mathrm{\Xi }}_2^{\mathfrak{x}}z\right)(\ell ):z\in {\mathbb{B}}_{ı}\right\}$$

is relatively compact in ${\mathbb{B}}_{ı}$. Clearly, in view of hypothesis (H3), the set $\mathbb{V}\left(0\right)$ is relatively compact in ${\mathbb{B}}_{ı}$.

Let $\ell \in (s_j,{\ell }_{j+1}]$ be fixed and choose $\epsilon$ such that $s_j<\epsilon <\ell$. For an arbitrary $\theta >0$ and any $z\in {\mathbb{B}}_{ı}$, we introduce

$$\begin{array}{ccc}\hfill \left({\mathrm{\Xi }}_{2,\epsilon ,\theta }^{\mathfrak{x}}z\right)(\ell )& =& {\displaystyle \frac{S\left({\epsilon }^{\varsigma }\theta \right)}{{\epsilon }^{\varsigma }\theta }}{\int }_{s_j}^{\ell -\epsilon }{\int }_{\theta }^{\infty }\varsigma \phi {(\ell -s)}^{\varsigma -1}{\mathbb{W}}_{\varsigma }\left(\phi \right)S({(\ell -s)}^{\varsigma }\phi -{\epsilon }^{\varsigma }\theta )\mathfrak{B}{u}^{\mathfrak{x}}\left(s\right)d\phi ds\hfill \\ & & +{\displaystyle \frac{S\left({\epsilon }^{\varsigma }\theta \right)}{{\epsilon }^{\varsigma }\theta }}{\int }_{s_j}^{\ell -\epsilon }{\int }_{\theta }^{\infty }\varsigma \phi {(\ell -s)}^{\varsigma -1}{\mathbb{W}}_{\varsigma }\left(\phi \right)S({(\ell -s)}^{\varsigma }\phi -{\epsilon }^{\varsigma }\theta )\hfill \\ & & \times \left({\int }_0^s\mathfrak{F}\left(\tau ,z\left(\tau \right),z\left({\widehat{\zeta }}_1\left(\tau \right)\right),\cdots ,z\left({\widehat{\zeta }}_m\left(\tau \right)\right)\right)d\mathcal{W}\left(\tau \right)\right)d\phi ds\hfill \\ & & +{\displaystyle \frac{S\left({\epsilon }^{\varsigma }\theta \right)}{{\epsilon }^{\varsigma }\theta }}{\int }_{s_j}^{\ell -\epsilon }{\int }_{\theta }^{\infty }\varsigma \phi {(\ell -s)}^{\varsigma -1}{\mathbb{W}}_{\varsigma }\left(\phi \right)S({(\ell -s)}^{\varsigma }\phi -{\epsilon }^{\varsigma }\theta )\hfill \\ & & \times \mathfrak{O}\left(s,z\left(s\right),z\left({\xi }_1\left(s\right)\right),\cdots ,z\left({\xi }_m\left(s\right)\right)\right)d{\mathcal{W}}^H\left(s\right)d\phi ds\hfill \\ & & +{\displaystyle \frac{S\left({\epsilon }^{\varsigma }\theta \right)}{{\epsilon }^{\varsigma }\theta }}{\int }_{s_j}^{\ell -\epsilon }{\int }_{\theta }^{\infty }{\int }_{\mathsf{\Pi }}\varsigma \phi {(\ell -s)}^{\varsigma -1}{\mathbb{W}}_{\varsigma }\left(\phi \right)S({(\ell -s)}^{\varsigma }\phi -{\epsilon }^{\varsigma }\theta )\hfill \\ & & \times \left(\hslash \left(s,z\left(s\right),z\left({\varrho }_1\left(s\right)\right),\cdots ,z\left({\varrho }_m\left(s\right)\right),\eta \right)\widehat{N}(ds,d\eta )\right)d\phi .\hfill \end{array}$$

The operator $S\left({\epsilon }^{\varsigma }\theta \right)$ is compact, which implies that the set $\{{\mathrm{\Xi }}_{2,\epsilon ,\theta }^{\mathfrak{x}}z(\ell ):z\in {\mathbb{B}}_{ı}\}$ is relatively compact in ${\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})$. The operator ${\mathrm{\Xi }}_{2,\epsilon ,\theta }^{\mathfrak{x}}$ is introduced as a regularized approximation of ${\mathrm{\Xi }}_2^{\mathfrak{x}}$, designed to isolate compactness properties and enable the application of Arzelà–Ascoli-type arguments in the stochastic framework. Moreover, using Lemma 4 together with dominated convergence arguments, we obtain

$$\underset{\epsilon \to 0}{lim}\underset{\theta \to \infty }{lim}\mathbb{E}\parallel {\mathrm{\Xi }}_2^{\mathfrak{x}}z(\ell )-{\mathrm{\Xi }}_{2,\epsilon ,\theta }^{\mathfrak{x}}z(\ell ){\parallel }^2=0,$$

which shows that the original operator ${\mathrm{\Xi }}_2^{\mathfrak{x}}$ can be approximated arbitrarily closely by compact operators. Consequently, one can construct relatively compact subsets arbitrarily close to the image set

$$\mathbb{V}(\ell )=\{\left({\mathrm{\Xi }}_2^{\mathfrak{x}}z\right)(\ell ):z\in {\mathbb{B}}_{ı}\}.$$

This implies that $\mathbb{V}(\ell )$ is relatively compact in ${\mathbb{B}}_{ı}$, and, hence, by the Arzelà–Ascoli theorem, the operator ${\mathrm{\Xi }}_2^{\mathfrak{x}}$ is compact. Finally, since ${\mathrm{\Xi }}_1^{\mathfrak{x}}$ is shown to be a contraction, the decomposition

$${\mathrm{\Xi }}^{\mathfrak{x}}={\mathrm{\Xi }}_1^{\mathfrak{x}}+{\mathrm{\Xi }}_2^{\mathfrak{x}}$$

ensures that ${\mathrm{\Xi }}^{\mathfrak{x}}$ is a condensing operator on ${\mathbb{B}}_{ı}$. Therefore, by Theorem 1 (Sadovskii’s fixed point theorem), ${\mathrm{\Xi }}^{\mathfrak{x}}$ admits at least one fixed point, which corresponds to a mild solution of system (1). □

4. Approximate Controllability

To rigorously establish the $\mathbb{AC}$ of problem (1), we impose the following auxiliary conditions:

• (H7) The stochastic fractional linear system (2) is assumed to exhibit $\mathbb{AC}$.
• (H8) The operators $\mathcal{F}$, $\mathfrak{F}$, $\mathfrak{O}$, and ℏ are uniformly bounded over their respective domains.

Theorem 3.

Presuming the validity of all hypotheses delineated in Theorem 2, along with the fulfillment of (H7)–(H8), it follows that the system (1) achieves $\mathbb{AC}$ over the temporal domain $[0,\mathfrak{a}]$.

Proof. 

Let $z^{\mathfrak{x}}$ denote a fixed point of the operator ${\mathrm{\Xi }}^{\mathfrak{x}}$ in the Hilbert space $\mathbb{S}$. According to Theorem 2, this element corresponds to the $\mathbb{MS}$ of system (1) associated with the control function $u^{\mathfrak{x}}(·)$. Employing the stochastic Fubini theorem, the terminal state $z^{\mathfrak{x}}\left(\mathfrak{a}\right)$ admits the representation

$$\begin{array}{ccc}\hfill z^{\mathfrak{x}}\left(\mathfrak{a}\right)& =& z_{\mathfrak{a}}-\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}[\mathbb{E}\left[z_{\mathfrak{a}}\right]+{\int }_0^{\mathfrak{a}}\widehat{\varrho }\left(s\right)d\mathcal{W}\left(s\right)+{\int }_0^{\mathfrak{a}}\widehat{\vartheta }\left(s\right)d{\mathcal{W}}^H\left(s\right)-{\mathcal{R}}_{\wp ,\Re }(\mathfrak{a}-s_{\beta })({\mathbb{O}}_{\beta }^1(s_{\beta },z^{\mathfrak{x}}\left(s_{\beta }\right))\hfill \\ & & +\mathcal{F}(s_{\beta },z^{\mathfrak{x}}\left(s_{\beta }\right),z^{\mathfrak{x}}\left({\zeta }_1\left(s_{\beta }\right)\right),\cdots ,z^{\mathfrak{x}}\left({\zeta }_m\left(s_{\beta }\right)\right)))+\mathcal{F}(\mathfrak{a},z^{\mathfrak{x}}\left(\mathfrak{a}\right),z^{\mathfrak{x}}\left({\zeta }_1\left(\mathfrak{a}\right)\right),\cdots ,z^{\mathfrak{x}}\left({\zeta }_m\left(\mathfrak{a}\right)\right))\hfill \\ & & -{\mathcal{T}}_{\wp ,\Re }(\mathfrak{a}-s_{\beta })\left({\mathbb{O}}_{\beta }^2(s_{\beta },z^{\mathfrak{x}}\left(s_{\beta }\right))+{\widehat{\Im }}_{\beta }\right)-{\int }_{s_{\beta }}^{\mathfrak{a}}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s){\int }_0^s\mathfrak{F}(\tau ,z^{\mathfrak{x}}\left(\tau \right),z^{\mathfrak{x}}\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z^{\mathfrak{x}}\left(\widehat{{\zeta }_m}\left(\tau \right)\right))d\mathcal{W}\left(\tau \right)ds\hfill \\ & & -{\int }_{s_{\beta }}^{\mathfrak{a}}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s)\mathfrak{O}(s,z^{\mathfrak{x}}{z}^{\mathfrak{x}}\left(s\right),z^{\mathfrak{x}}{z}^{\mathfrak{x}}\left({\xi }_1\left(s\right)\right),\cdots ,z^{\mathfrak{x}}{z}^{\mathfrak{x}}\left({\xi }_m\left(s\right)\right))d{\mathcal{W}}^H\left(s\right)\hfill \\ & & -{\int }_{s_{\beta }}^{\mathfrak{a}}{\int }_{\mathsf{\Pi }}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s)\hslash (s,z^{\mathfrak{x}}\left(s\right),z^{\mathfrak{x}}\left({\varrho }_1\left(s\right)\right),\cdots ,z^{\mathfrak{x}}\left({\varrho }_m\left(s\right)\right),\eta )\widehat{N}(ds,d\eta )].\hfill \end{array}$$

(9)

In view of the boundedness assumptions imposed on $\mathcal{F}$, $\mathfrak{F}$, $\mathfrak{O}$, and ℏ, there exists a constant $\mathfrak{D}>0$ such that

$$\parallel \mathcal{F}(s,z^{\mathfrak{x}}\left(s\right),z^{\mathfrak{x}}\left({\zeta }_1\left(s\right)\right),\cdots ,z^{\mathfrak{x}}\left({\zeta }_m\left(s\right)\right)){\parallel }^2\le \mathfrak{D},$$

$$\parallel \mathfrak{F}(s,z^{\mathfrak{x}}\left(s\right),z^{\mathfrak{x}}\left(\widehat{{\zeta }_1}\left(s\right)\right),\cdots ,z^{\mathfrak{x}}\left(\widehat{{\zeta }_m}\left(s\right)\right)){\parallel }^2\le \mathfrak{D},$$

$$\parallel \mathfrak{O}(s,z^{\mathfrak{x}}\left(s\right),z^{\mathfrak{x}}\left({\xi }_1\left(s\right)\right),\cdots ,z^{\mathfrak{x}}\left({\xi }_m\left(s\right)\right)){\parallel }^2\le \mathfrak{D},$$

and

$$\parallel \hslash (s,z^{\mathfrak{x}}\left(s\right),z^{\mathfrak{x}}\left({\varrho }_1\left(s\right)\right),\cdots ,z^{\mathfrak{x}}\left({\varrho }_m\left(s\right)\right),\eta ){\parallel }^2\le \mathfrak{D},$$

As a consequence, the sequences

$$\left\{\mathcal{F}(s,z^{\mathfrak{x}}\left(s\right),z^{\mathfrak{x}}\left({\zeta }_1\left(s\right)\right),\cdots ,z^{\mathfrak{x}}\left({\zeta }_m\left(s\right)\right))\right\},\left\{\mathfrak{F}(s,z^{\mathfrak{x}}\left(s\right),z^{\mathfrak{x}}\left(\widehat{{\zeta }_1}\left(s\right)\right),\cdots ,z^{\mathfrak{x}}\left(\widehat{{\zeta }_m}\left(s\right)\right))\right\},$$

and

$$\left\{\mathfrak{O}(s,z^{\mathfrak{x}}\left(s\right),z^{\mathfrak{x}}\left({\xi }_1\left(s\right)\right),\cdots ,z^{\mathfrak{x}}\left({\xi }_m\left(s\right)\right))\right\},\left\{\hslash (s,z^{\mathfrak{x}}\left(s\right),z^{\mathfrak{x}}\left({\varrho }_1\left(s\right)\right),\cdots ,z^{\mathfrak{x}}\left({\varrho }_m\left(s\right)\right),\eta )\right\},$$

form bounded families in the spaces ${\mathcal{L}}^2([0,\mathfrak{a}],\mathcal{X})$, ${\mathcal{L}}^2\left({\mathcal{L}}_{\mathbb{Q}}^2(\mathcal{Y},\mathcal{X})\right)$, ${\mathcal{L}}^2\left({\mathcal{L}}_0^2(\mathcal{Y},\mathcal{X})\right)$, and $\mathcal{X}$, respectively.

Hence, by weak compactness in Hilbert spaces, there exist subsequences—still denoted by the same symbols for simplicity—that converge weakly to $\mathcal{F}\left(s\right)$, $\mathfrak{F}\left(s\right)$, $\mathfrak{O}\left(s\right)$, and $\hslash (s,\eta )$ in ${\mathcal{L}}^2([0,\mathfrak{a}],\mathcal{X})$, ${\mathcal{L}}^2\left({\mathcal{L}}_{\mathbb{Q}}^2(\mathcal{Y},\mathcal{X})\right)$, ${\mathcal{L}}^2\left({\mathcal{L}}_0^2(\mathcal{Y},\mathcal{X})\right)$, and $\mathcal{X}$, respectively.

Employing relation (9), we derive the following estimate.

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel z^{\mathfrak{x}}\left(\mathfrak{a}\right)-z_{\mathfrak{a}}{\parallel }^2& \le & 11\mathbb{E}\parallel \mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}[\mathbb{E}\left[z_{\mathfrak{a}}\right]-{\mathcal{R}}_{\wp ,\Re }(\mathfrak{a}-s_{\beta })({\mathbb{O}}_{\beta }^1(s_{\beta },z^{\mathfrak{x}}\left(s_{\beta }\right))\hfill \\ & & +\mathcal{F}(s_{\beta },z^{\mathfrak{x}}\left(s_{\beta }\right),z^{\mathfrak{x}}\left({\zeta }_1\left(s_{\beta }\right)\right),\cdots ,z^{\mathfrak{x}}\left({\zeta }_m\left(s_{\beta }\right)\right))\left)\right]{\parallel }^2\hfill \\ & & +11\mathbb{E}\parallel {\int }_0^{\mathfrak{a}}\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}\widehat{\varrho }\left(s\right)d\mathcal{W}\left(s\right){\parallel }^2+11\mathbb{E}\parallel {\int }_0^{\mathfrak{a}}\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}\widehat{\vartheta }\left(s\right)d{\mathcal{W}}^H\left(s\right){\parallel }^2\hfill \\ & & +11\mathbb{E}\parallel \mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}{\mathcal{T}}_{\wp ,\Re }(\mathfrak{a}-s_{\beta })\left({\mathbb{O}}_{\beta }^2(s_{\beta },z^{\mathfrak{x}}\left(s_{\beta }\right))+{\widehat{\Im }}_{\beta }\right){\parallel }^2\hfill \\ & & +11\mathbb{E}\parallel \mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}\mathcal{F}(\mathfrak{a},z^{\mathfrak{x}}\left(\mathfrak{a}\right),z^{\mathfrak{x}}\left({\zeta }_1\left(\mathfrak{a}\right)\right),\cdots ,z^{\mathfrak{x}}\left({\zeta }_m\left(\mathfrak{a}\right)\right)){\parallel }^2\hfill \\ & & +11\mathbb{E}\parallel {\int }_{s_{\beta }}^{\mathfrak{a}}\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s){\int }_0^s\left[\mathfrak{F}(\tau ,z^{\mathfrak{x}}\left(\tau \right),z^{\mathfrak{x}}\left(\widehat{{\zeta }_1}\left(\tau \right)\right),\cdots ,z^{\mathfrak{x}}\left(\widehat{{\zeta }_m}\left(\tau \right)\right))-\mathfrak{F}\left(\tau \right)\right]d\mathcal{W}\left(\tau \right)ds{\parallel }^2\hfill \\ & & +11\mathbb{E}\parallel {\int }_{s_{\beta }}^{\mathfrak{a}}\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s){\int }_0^s\mathfrak{F}\left(\tau \right)d\mathcal{W}\left(\tau \right)ds{\parallel }^2\hfill \\ & & +11\mathbb{E}\parallel {\int }_{s_{\beta }}^{\mathfrak{a}}\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s)\left[\mathfrak{O}(s,z^{\mathfrak{x}}{z}^{\mathfrak{x}}\left(s\right),z^{\mathfrak{x}}{z}^{\mathfrak{x}}\left({\xi }_1\left(s\right)\right),\cdots ,z^{\mathfrak{x}}{z}^{\mathfrak{x}}\left({\xi }_m\left(s\right)\right))-\mathfrak{O}\left(s\right)\right]d{\mathcal{W}}^H\left(s\right){\parallel }^2\hfill \\ & & +11\mathbb{E}\parallel {\int }_{s_{\beta }}^{\mathfrak{a}}\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s)\mathfrak{O}\left(s\right)d{\mathcal{W}}^H\left(s\right){\parallel }^2\hfill \\ & & +11\mathbb{E}\parallel {\int }_{s_{\beta }}^{\mathfrak{a}}{\int }_{\mathsf{\Pi }}\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s)\left[\hslash (s,z^{\mathfrak{x}}\left(s\right),z^{\mathfrak{x}}\left({\varrho }_1\left(s\right)\right),\cdots ,z^{\mathfrak{x}}\left({\varrho }_m\left(s\right)\right),\eta )-\hslash (s,\eta )\right]\widehat{N}(ds,d\eta ){\parallel }^2\hfill \\ & & +11\mathbb{E}\parallel {\int }_{s_{\beta }}^{\mathfrak{a}}\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}{\mathcal{V}}_{\varsigma }(\mathfrak{a}-s){\int }_{\mathsf{\Pi }}\hslash (s,\eta )\widehat{N}(ds,d\eta ){\parallel }^2\hfill \\ & & \to 0,\mathit{as}\mathfrak{x}\to 0^{+}.\hfill \end{array}$$

In light of Lemma 5, the family of operators

$$\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}$$

converges strongly to zero in $\mathcal{L}\left(\mathcal{X}\right)$ as $\mathfrak{x}\to 0^{+}$, and is uniformly bounded by the identity operator, i.e.,

$$∥\mathfrak{x}{(\mathfrak{x}I+{\mathfrak{Q}}_0^{\mathfrak{a}})}^{-1}∥\le 1.$$

We apply the classical Lebesgue dominated convergence theorem in the Hilbert space ${\mathcal{L}}^2(\mathrm{\Omega },\mathcal{X})$ to the sequence of stochastic variables

$${\left\{z^{\mathfrak{x}}\left(\mathfrak{a}\right)\right\}}_{\mathfrak{x}>0},$$

where domination is guaranteed by the uniform boundedness of the resolvent operator family and the square-integrability of the trajectories. Since the integrand converges pointwise (in mean square) due to the strong operator convergence stated in Lemma 5, we obtain

$$\mathbb{E}\parallel z^{\mathfrak{x}}\left(\mathfrak{a}\right)-z_{\mathfrak{a}}{\parallel }^2⟶0\mathrm{as}\mathfrak{x}\to 0^{+}.$$

This establishes the $\mathbb{AC}$ property of system (1). □

5. Example

This section is devoted to the presentation of an illustrative example demonstrating the applicability of the principal results. Specifically, we examine a stochastic control Hilfer fractional partial differential equation perturbed by fBm:

$${\displaystyle \left\{\begin{array}{c}{}^{\mathbb{H}}D_{0+}^{\frac{5}{4},\frac{1}{3}}\left[{\displaystyle z(\ell ,\mathfrak{w})+\frac{e^{-\ell }}{1+{\ell }^2}{\int }_0^{\pi }K(\mathfrak{w},s)\left({\displaystyle \sum _{i=0}^m}{z}_i\left(s\right)\right)ds}\right]={\displaystyle \frac{{\partial }^2}{\partial {\mathfrak{w}}^2}}\left[{\displaystyle z(\ell ,\mathfrak{w})+\frac{e^{-\ell }}{1+{\ell }^2}{\int }_0^{\pi }K(\mathfrak{w},s)\left({\displaystyle \sum _{i=0}^m}{z}_i\left(s\right)\right)ds}\right]\hfill \\ {\displaystyle +u(\ell ,\mathfrak{w})+{\int }_0^{\ell }{4}^{-s}z(s,\mathfrak{w})d\mathcal{W}\left(s\right)+{\displaystyle \frac{sin\ell }{1+sin\ell }}z(\ell ,\mathfrak{w})d{\mathcal{W}}^H(\ell )}\hfill \\ {\displaystyle +{\int }_{\mathsf{\Pi }}cos(\ell )e^{-4\ell }{\displaystyle \frac{z(\ell ,\mathfrak{w},\eta )}{9}}\widehat{N}(d\ell ,d\eta ),\ell \in (0,\frac{1}{3}]\cup (\frac{2}{3},1],0\le \mathfrak{w}\le \pi ,}\hfill \\ z(\ell ,0)=z(\ell ,\pi )=0,\ell \in (0,1],\hfill \\ z(\ell ,\mathfrak{w})={\displaystyle \frac{1}{5}}{e}^{-(\ell -\frac{1}{3})}\frac{∥z(\ell ,\mathfrak{w})∥}{1+∥z(\ell ,\mathfrak{w})∥},{\partial }_{\ell }z(\ell ,\mathfrak{w})={\displaystyle \frac{1}{10}}{e}^{-(\ell -\frac{2}{3})}\frac{∥z(\ell ,\mathfrak{w})∥}{1+∥z(\ell ,\mathfrak{w})∥},\ell \in (\frac{1}{3},\frac{2}{3}],0\le \mathfrak{w}\le \pi ,\hfill \\ {}^{\mathbb{RL}}\mathbf{I}_{0+}^{\frac{1}{2}}z(0,\mathfrak{w})+{\displaystyle \sum _{i=1}^2}{\mathfrak{b}}_{i}z({\ell }_i,\mathfrak{w})=z_0\left(\mathfrak{w}\right),{}^{\mathbb{RL}}\mathbf{I}_{0+}^{\frac{1}{2}}{z}^{\prime }(0,\mathfrak{w})+{\displaystyle \sum _{i=1}^2}{c}_{i}z({\ell }_i,\mathfrak{w})=z_1\left(\mathfrak{w}\right)0\le \mathfrak{w}\le \pi ,\hfill \end{array}\right.}$$

(10)

where ${}^{\mathbb{H}}D_{0+}^{{\displaystyle \frac{5}{4}},{\displaystyle \frac{1}{3}}}$ denotes the HFD with parameters $\wp ={\displaystyle \frac{5}{4}}$ and $\Re ={\displaystyle \frac{1}{3}}$. The process $\mathcal{W}$ represents a Wiener process, while ${\mathcal{W}}^H$ is an fBm with Hurst parameter $H\in (\frac{1}{2},1)$.

Let $\mathcal{X}=\mathcal{Y}=\mathcal{U}={\mathcal{L}}^2\left([0,\pi ]\right)$, and consider the linear operator $\mathcal{A}:\mathbb{D}\left(\mathcal{A}\right)\subset \mathcal{X}\to \mathcal{X}$ defined by

$$\mathcal{A}z=-{\displaystyle \frac{{\partial }^{2}z}{\partial {\mathfrak{w}}^2}},$$

where the domain is given by

$$\mathbb{D}\left(\mathcal{A}\right)=\{z\in \mathcal{X}:z,z^{\prime }\mathrm{are}\mathrm{absolutely}\mathrm{continuous},z^{\prime \prime }\in \mathcal{X},z\left(0\right)=z\left(\pi \right)=0\}.$$

The operator $\mathcal{A}$ admits the spectral decomposition

$$\mathcal{A}z=-{\displaystyle \sum _{n=1}^{\infty }}{n}^2⟨z,{\mathfrak{e}}_n⟩{\mathfrak{e}}_n,z\in \mathbb{D}\left(\mathcal{A}\right),$$

where the eigenfunctions

$${\mathfrak{e}}_n\left(s\right)=\sqrt{{\displaystyle \frac{2}{\pi }}}sin\left(ns\right),n=1,2,\cdots$$

constitute a complete orthonormal basis of $\mathcal{X}$.

It is well established that the operator $-\mathcal{A}$ generates a strongly continuous and uniformly bounded cosine family ${\left\{\widehat{C}(\ell )\right\}}_{\ell \ge 0}$ on $\mathcal{X}$. According to [42], this family can be expressed as

$$\widehat{C}(\ell )z={\displaystyle \sum _{n=1}^{\infty }}cos\left({\displaystyle \frac{n^2}{1+n^2}}\ell \right)⟨z,{\mathfrak{e}}_n⟩{\mathfrak{e}}_n,$$

while the associated sine operator is given by

$$S(\ell )z={\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1+n^2}{n^2}}sin\left({\displaystyle \frac{n^2}{1+n^2}}\ell \right)⟨z,{\mathfrak{e}}_n⟩{\mathfrak{e}}_n.$$

The operator $S(\ell )$ is compact for each $\ell >0$, and $\parallel \widehat{C}(\ell )\parallel \le 1$ holds for all $\ell \ge 0$. Furthermore, as reported in [42], the fractional operator family ${{\rm Y}}_{\varsigma }(\ell )$ admits the representation

$${{\rm Y}}_{\varsigma }(\ell )z={\displaystyle \sum _{n=1}^{\infty }}{\ell }^{\wp /2}{\mathcal{E}}_{\wp ,\wp }(-n^2{\ell }^{\wp })⟨z,{\mathfrak{e}}_n⟩{\mathfrak{e}}_n,\varsigma ={\displaystyle \frac{\wp }{2}}\in \left({\displaystyle \frac{1}{2}},1\right),$$

where

$${\mathcal{E}}_{\wp ,\wp }\left(z\right)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{z^n}{\mathrm{\Gamma }(\wp (n+1\left)\right)}}$$

denotes the Mittag–Leffler function.

The control operator $\mathfrak{B}$ is defined by $\mathfrak{B}u=u(\ell ,\mathfrak{w})$, $0\le \mathfrak{w}\le \pi$, for $u\in \mathcal{U}$. If $u\in {\mathcal{L}}_{\widehat{Y}}^2([0,\mathfrak{a}];\mathcal{U})$, then $\mathfrak{B}={\mathfrak{B}}^{*}=I$.

We define the nonlinear mapping

$$\mathcal{F}:[0,\mathfrak{a}]\times {\mathcal{X}}^{m+1}⟶\mathcal{X}$$

by

$$\mathcal{F}(\ell ,z_0,z_1,\cdots ,z_m)\left(\mathfrak{w}\right)={\displaystyle \frac{e^{-\ell }}{1+{\ell }^2}}{\int }_0^{\pi }K(\mathfrak{w},s)\left({\displaystyle \sum _{i=0}^m}{z}_i\left(s\right)\right)ds,\mathfrak{w}\in (0,\pi ),$$

where the kernel $K:(0,\pi )\times (0,\pi )\to \mathbb{R}$ is measurable and satisfies

$${\int }_0^{\pi }{\int }_0^{\pi }{|K(\mathfrak{w},s)|}^{2}dsd\mathfrak{w}<\infty .$$

Since K is square–integrable, the associated integral operator

$$\left(T\phi \right)\left(\mathfrak{w}\right)={\int }_0^{\pi }K(\mathfrak{w},s)\phi \left(s\right)ds,\phi \in \mathcal{X}=L^2(0,\pi ),$$

is a Hilbert–Schmidt operator on $\mathcal{X}$ and therefore bounded. Hence, there exists a constant $C_K>0$ such that

$$\parallel T\phi \parallel \le C_K\parallel \phi \parallel ,\phi \in \mathcal{X}.$$

Assume moreover that

$$T\left(\mathcal{X}\right)\subset \mathbb{D}\left({\mathcal{A}}^{\gamma }\right)\mathrm{and}\parallel {\mathcal{A}}^{\gamma }T\phi \parallel \le C_K\parallel \phi \parallel ,\phi \in \mathcal{X}.$$

Consequently,

$$\mathcal{F}(\ell ,z_0,\cdots ,z_m)={\displaystyle \frac{e^{-\ell }}{1+{\ell }^2}}T\left({\displaystyle \sum _{i=0}^m}{z}_i\right).$$

Since the coefficient ${\displaystyle \frac{e^{-\ell }}{1+{\ell }^2}}$ is continuous and bounded on $[0,\mathfrak{a}]$, the mapping $\mathcal{F}$ is continuous from $[0,\mathfrak{a}]\times {\mathcal{X}}^{m+1}$ into $\mathcal{X}$.

Let ${\ell }_1,{\ell }_2\in [0,\mathfrak{a}]$ and $z_i,{\widehat{z}}_i\in \mathcal{X}$ for $i=0,1,\cdots ,m$. Using the boundedness of T and the triangle inequality, we obtain

$$\begin{array}{cc}& \parallel \mathcal{F}({\ell }_1,z_0,\cdots ,z_m)-\mathcal{F}({\ell }_2,{\widehat{z}}_0,\cdots ,{\widehat{z}}_m)\parallel \hfill \\ & \le C\left(|{\ell }_1-{\ell }_2|+{\displaystyle \sum _{i=0}^m}\parallel z_i-{\widehat{z}}_i\parallel \right)\hfill \end{array}$$

for some constant $C>0$.

Since $T\left(\mathcal{X}\right)\subset D\left({\mathcal{A}}^{\gamma }\right)$, we may apply the operator ${\mathcal{A}}^{\gamma }$ to obtain

$$\begin{array}{cc}& \parallel {\mathcal{A}}^{\gamma }\mathcal{F}({\ell }_1,z_0,\cdots ,z_m)-{\mathcal{A}}^{\gamma }\mathcal{F}({\ell }_2,{\widehat{z}}_0,\cdots ,{\widehat{z}}_m)\parallel \hfill \\ & \le M_{\mathcal{F}}\left(|{\ell }_1-{\ell }_2|+\underset{0\le ii\le m}{max}\parallel z_i-{\widehat{z}}_i\parallel \right),\hfill \end{array}$$

where $M_{\mathcal{F}}>0$ is a suitable constant.

Therefore,

$$\mathbb{E}{∥{\mathcal{A}}^{\gamma }\mathcal{F}({\ell }_1,z_0,\cdots ,z_m)-{\mathcal{A}}^{\gamma }\mathcal{F}({\ell }_2,{\widehat{z}}_0,\cdots ,{\widehat{z}}_m)∥}^2\le M_{\mathcal{F}}\left(|{\ell }_1-{\ell }_2|+\underset{0\le i\le m}{max}\mathbb{E}{\parallel z_i-{\widehat{z}}_i\parallel }^2\right)$$

for some constant $M_{\mathcal{F}}>0$.

Moreover, using the boundedness of the coefficient and the operator T, we have

$$\parallel \mathcal{F}(\ell ,z_0,\cdots ,z_m)\parallel \le \widehat{C}{\displaystyle \sum _{i=0}^m}\parallel z_i\parallel .$$

Hence,

$$\mathbb{E}{∥{\mathcal{A}}^{\gamma }\mathcal{F}(\ell ,z_0,\cdots ,z_m)∥}^2\le {\widehat{M}}_{\mathcal{F}}\left(\underset{0\le i\le m}{max}\mathbb{E}{\parallel z_i\parallel }^2+1\right).$$

Therefore, the mapping $\mathcal{F}$ satisfies assumption (H1). For $\ell \in (0,\frac{1}{3}]\cup (\frac{2}{3},1]$ and $0\le \mathfrak{w}\le \pi$, we choose the stochastic coefficients as

$$\mathfrak{F}(\ell ,z)\left(\mathfrak{w}\right)=4^{-\ell }z(\ell ,\mathfrak{w}),$$

$$\mathfrak{O}(\ell ,z)\left(\mathfrak{w}\right)={\displaystyle \frac{sin\ell }{1+sin\ell }}z(\ell ,\mathfrak{w}),$$

and

$$\hslash (\ell ,z,\eta )\left(\mathfrak{w}\right)=cos(\ell )e^{-4\ell }{\displaystyle \frac{z(\ell ,\mathfrak{w},\eta )}{9}},\eta \in \mathsf{\Pi }.$$

Since the scalar coefficients $4^{-\ell }$, ${\displaystyle \frac{sin\ell }{1+sin\ell }}$, and $cos(\ell )e^{-4\ell }$ are continuous and bounded on $[0,\mathfrak{a}]$, the mappings $\mathfrak{F}$, $\mathfrak{O}$, and ℏ are continuous with respect to their arguments.

Moreover, for any $\iota \in \mathbb{N}$ and ${\parallel {\mathfrak{z}}_0\parallel }^2,\cdots ,{\parallel {\mathfrak{z}}_m\parallel }^2\le \iota$, there exist positive functions ${\psi }_{\iota }(\ell )$, ${\widehat{\psi }}_{\iota }(\ell )$, and ${\widehat{\varphi }}_{\iota }(\ell )$ such that

$$\underset{\parallel z_0{\parallel }^2,\cdots ,{\parallel z_m\parallel }^2\le \iota }{sup}\mathbb{E}\parallel \mathfrak{F}(\ell ,z_0,\cdots ,z_m){\parallel }_{\mathbb{Q}}^2\le {\psi }_{\iota }(\ell ),$$

$$\underset{\parallel z_0{\parallel }^2,\cdots ,{\parallel z_m\parallel }^2\le \iota }{sup}\mathbb{E}\parallel \mathfrak{O}(\ell ,z_0,\cdots ,z_m){\parallel }_{{\mathcal{L}}_0^2}^2\le {\widehat{\psi }}_{\iota }(\ell ),$$

and

$$\underset{\parallel z_0{\parallel }^2,\cdots ,{\parallel z_m\parallel }^2\le \iota }{sup}{\int }_{\mathsf{\Pi }}\mathbb{E}\parallel \hslash (\ell ,z_0,\cdots ,z_m,\eta )\parallel \kappa \left(d\eta \right)\le {\widehat{\varphi }}_{\iota }(\ell ).$$

Hence, the mappings $\mathfrak{F}$, $\mathfrak{O}$, and ℏ satisfy assumptions (H2), (H3), and (H6), respectively. Let the non–instantaneous impulsive operators be defined by

$${\mathbb{O}}_j^1(\ell ,z)\left(\mathfrak{w}\right)={\displaystyle \frac{1}{5}}{e}^{-(\ell -{\displaystyle \frac{1}{3}})}{\displaystyle \frac{\parallel z(\ell ,\mathfrak{w})\parallel }{1+\parallel z(\ell ,\mathfrak{w})\parallel }},{\mathbb{O}}_j^2(\ell ,z)\left(\mathfrak{w}\right)={\displaystyle \frac{1}{10}}{e}^{-(\ell -{\displaystyle \frac{2}{3}})}{\displaystyle \frac{\parallel z(\ell ,\mathfrak{w})\parallel }{1+\parallel z(\ell ,\mathfrak{w})\parallel }},\ell \in (\frac{1}{3},\frac{2}{3}],0\le \mathfrak{w}\le \pi .$$

Since the function

$$\varphi \left(r\right)={\displaystyle \frac{r}{1+r}},r\ge 0,$$

is globally Lipschitz on ${\mathbb{R}}_{+}$, there exists a constant $L_{\varphi }>0$ such that

$$|\varphi \left(r_1\right)-\varphi \left(r_2\right)|\le L_{\varphi }|r_1-r_2|,r_1,r_2\ge 0.$$

Let $z_1,z_2\in \mathcal{X}$. Then

$$\mathbb{E}\parallel {\mathbb{O}}_j^l(\ell ,z_1(\ell ))-{\mathbb{O}}_j^l(\ell ,z_2(\ell )){\parallel }^2\le M_{{\mathbb{O}}_j}^l\mathbb{E}{\parallel z_1-z_2\parallel }^2$$

for some constant $M_{{\mathbb{O}}_j}^l>0$.

Moreover, using the boundedness of $\varphi \left(r\right)$ and the exponential coefficients, we obtain

$${\parallel \mathbb{O}}_j^l(\ell ,z)\parallel \le C\parallel z\parallel ,$$

which yields

$$\mathbb{E}\parallel {\mathbb{O}}_j^l(\ell ,z){\parallel }^2\le {\widehat{M}}_{{\mathbb{O}}_j}^l\mathbb{E}{\parallel z\parallel }^2.$$

Hence, the non–instantaneous impulsive mappings ${\mathbb{O}}_j^l$ satisfy assumption (H4). In the present problem, the nonlocal operators are specified by

$${\mathfrak{g}}_1\left(z\right)\left(\mathfrak{w}\right)={\displaystyle \sum _{i=1}^2}{\mathfrak{b}}_{i}z({\ell }_i,\mathfrak{w}),{\mathfrak{g}}_2\left(z\right)\left(\mathfrak{w}\right)={\displaystyle \sum _{i=1}^2}{c}_{i}z({\ell }_i,\mathfrak{w}),\mathfrak{w}\in [0,\pi ],$$

where ${\mathfrak{b}}_i,c_i\in \mathbb{R}$ are fixed constants. We verify that these operators satisfy assumption (H5). For any $z\in \mathcal{C}\left(\right[0,\mathfrak{a}],\mathcal{X})$, by the triangle inequality and Cauchy–Schwarz inequality, we have

$$\mathbb{E}\parallel {\mathfrak{g}}_l{\left(z\right)\parallel }^2\le 2{\displaystyle \sum _{i=1}^2}{|{\alpha }_i^{\left(l\right)}|}^2\underset{\ell \in [0,\mathfrak{a}]}{sup}\mathbb{E}{\parallel z(\ell )\parallel }^2,$$

where ${\alpha }_i^{\left(1\right)}={\mathfrak{b}}_i$ and ${\alpha }_i^{\left(2\right)}=c_i$. This establishes the growth condition in (H5)(i). Moreover, for any $z_1,z_2\in \mathcal{C}([0,\mathfrak{a}],\mathcal{X})$, we obtain

$$\mathbb{E}\parallel {\mathfrak{g}}_l\left(z_1\right)-{\mathfrak{g}}_l\left(z_2\right){\parallel }^2\le 2{\displaystyle \sum _{i=1}^2}{|{\alpha }_i^{\left(l\right)}|}^2\underset{\ell \in [0,\mathfrak{a}]}{sup}\mathbb{E}{\parallel z_1(\ell )-z_2(\ell )\parallel }^2,$$

which verifies the Lipschitz condition in (H5)(ii). Therefore, the nonlocal operators ${\mathfrak{g}}_1$ and ${\mathfrak{g}}_2$ satisfy assumption (H5), provided that the coefficients $\left\{{\mathfrak{b}}_i\right\}$ and $\left\{c_i\right\}$ are fixed and bounded.

Accordingly, the operator mappings $\mathcal{F}$, $\mathfrak{F}$, $\mathfrak{O}$, ℏ, ${\mathbb{O}}_j^l$, and ${\mathfrak{g}}_l$ rigorously satisfy the structural postulates delineated in assumptions (H1)–(H6). Within this abstract framework, system (10) can be recast in the canonical form of Equation (1). Moreover, the associated linearized evolution system exhibits $\mathbb{AC}$. Hence, all preconditions mandated by Theorems 2 and 3 are verifiably fulfilled. Consequently, the stochastic Hilfer fractional partial differential equation driven by fBm with nonlocal initial conditions, as specified in (10), possesses $\mathbb{AC}$ on the temporal interval $(0,1]$.

6. Conclusions

In this work, we have investigated a class of higher-order Hilfer fractional semi-linear neutral stochastic differential equations with non-instantaneous impulses in Hilbert spaces. By incorporating both fractional Brownian motion and Poisson jumps, the model captures long-range memory effects as well as stochastic discontinuities, providing a highly realistic framework for complex dynamical systems. We established sufficient conditions for the existence of mild solutions using fractional calculus, stochastic analysis, and operator-theoretic methods, notably leveraging Sadovskii’s fixed point theorem. Moreover, criteria for approximate controllability of the system were derived, highlighting the interplay between the higher-order Hilfer derivative, stochastic integral estimates, and compactness properties induced by non-instantaneous impulses.

The analysis demonstrates that neutral terms and jump discontinuities introduce nontrivial challenges, which were overcome via refined resolvent operator techniques and fractional power estimates. An illustrative example verified the theoretical findings and illustrated the practical effectiveness of the proposed methodology.

Looking forward, several promising research avenues emerge from this study:

• Optimal Control and Feedback Synthesis: Extending the approximate controllability results to design optimal control laws and feedback mechanisms for Hilfer fractional stochastic systems with impulses.
• Applications to Complex Networks: Applying the framework to stochastic fractional dynamics in networked systems, including biological networks, financial systems, and cyber-physical infrastructures, where impulsive effects and memory are prevalent.
• Robustness and Uncertainty Quantification: Studying the sensitivity, robustness, and probabilistic guarantees of controlled systems under fractional noise, jump perturbations, and parameter uncertainty.

Overall, the present study provides a versatile mathematical framework for the analysis and control of fractional stochastic systems with memory and impulsive effects, paving the way for both theoretical advances and practical implementations in emerging fields of fractional dynamics and stochastic control.