Stability and Controllability Analysis of Stochastic Fractional Differential Equations Under Integral Boundary Conditions Driven by Rosenblatt Process with Impulses

Abstract

Differential equations are frequently used to mathematically describe many problems in real life, but they are always subject to intrinsic phenomena that are neglected and could influence how the model behaves. In some cases like ecosystems, electrical circuits, or even economic models, the model may suddenly change due to outside influences. Occasionally, such changes start off impulsively and continue to exist for specific amounts of time. Non-instantaneous impulses are used in the creation of the models for this kind of scenario. In this paper, a new class of non-instantaneous impulsive $\psi$-Caputo fractional stochastic differential equations under integral boundary conditions driven by the Rosenblatt process was examined. Semigroup theory, stochastic theory, the Banach fixed-point theorem, and fractional calculus were applied to investigating the existence of piecewise continuous mild solutions for the systems under consideration. The impulsive Gronwall’s inequality was employed to establish the unique stability conditions for the system under consideration. Furthermore, we examined the controllability results of the proposed system. Finally, some examples were provided to demonstrate the validity of the presented work.

1. Introduction

Fractional calculus has acquired significant popularity and importance due to its numerous applications in many mathematical, physical, and engineering disciplines, such as systems of diffusion phenomena [1], optimal control problems [2,3], chaotic synchronization systems [4], stability [5], controllability [6], quantum theory [7], thermoelasticity systems [8], solutions of differential systems [9,10,11], delay problems [12], and impulsive problems [13]. Compared with integral calculus, fractional calculus provides a more accurate and significant description of many real-world problems. New discoveries in several scientific and technological areas have shown that integer-order differential systems have been rapidly replaced by fractional-order differential systems. The primary advantage of fractional calculus is its ability to describe the memory effect and hereditary properties of different materials and processes through the use of fractional derivatives (FDs). Different types of FDs have been used in the analysis of fractional differential systems (DSs), such as Riemann–Liouville, Hadamard, Caputo, Caputo–Hadamard, Riesz, and Grünwald–Letnikov FDs. Recently, Almeida [14] introduced a new $\psi$-Caputo FD in relation to a different function. The $\psi$-Caputo fractional derivative is one such extension, which is particularly useful in the context of fractional stochastic DSs. This allows the model to account for phenomena where the system’s past states influence its future behavior, such as in viscoelastic materials, anomalous diffusion, or systems exhibiting long-range dependence. To find out more about these new $\psi$-Caputo fractional DSs, and fractional calculus, see [15,16,17,18,19]. For their applications, refer to [20,21,22] and the references therein.

Nonlinear analysis has been known to actively research the area of impulsive DSs. Differential equations, including impulse conditions, have been extensively studied in the literature. It has been noted in ecological models, population dynamics, engineering systems, stability analysis, controllability analysis, optimal control theory, electronics, medicine, biology, and biotechnology that impulsive differential systems are suitable formulations for systems facing short-term perturbations. Impulsive effects are often classified into two distinct categories based on their duration. First, an impulse is considered instantaneous if its impulsive effect occurs immediately; an impulse effect that occurs after a period of time is classified as non-instantaneous. Non-instantaneous impulses were introduced by Hernandez and O’Regan [23] while modeling evolution systems. Fixed-point theorems have been used in the literature to investigate the qualitative properties of instantaneous and non-instantaneous DSs. See [23,24,25] and the references therein for more studies.

Considering noise and uncontrollably perturbed systems as common occurrences in both natural and artificial systems, stochastic models require further investigation instead of deterministic ones. In including disturbance, differential system theory has been broadened to include stochastic DSs. Recently, many researchers have given significant attention to stochastic DSs for explaining a variety of occurrences in technological engineering, population motion, biological science, neuroscience, and many other areas of science and technology [26]. Existence, controllability, stability, uniqueness, and other quantitative and qualitative analyses of the solutions for stochastics DSs have attracted many researchers; see [27,28] and the references therein. Studies have indicated that when it comes to analyzing and explaining an event, stochastic frameworks work exceptionally effectively and more precisely, such as for population display, stock prices, and memory-containing materials.

Stochastic DSs describe systems that evolve randomly over time and are frequently used in fields such as finance, physics, and biology. When these systems are driven by the Rosenblatt process, a type of self-similar process with stationary increments that generalizes fractional Brownian motion (fBm), they can model more complex dependencies and long-range interactions. The Rosenblatt process captures memory effects and is widely used in modeling phenomena like turbulent fluid flows and signal processing, where correlations exist over long time scales.

In a presentation at Wisconsin University in 1940, Ulam [29] introduced the concept of stability in functional equations. He put out the following problem: When is it possible that there exists an additive mapping close to an approximation additive mapping? In 1941, Hyers was the first to answer Ulam’s question, particularly in the context of Banach spaces [30]. This form of stability eventually came to be known as Ulam–Hyers stability. By including variables, Rassias presented a significant extension of Ulam–Hyers stability in 1978 [31]. The idea of stability in functional equations appears when an inequality acts as a perturbation of the original equations. Hence, the difference between the solutions of the inequality and the supplied functional equation is the central problem concerning the stability of functional equations. Ulam–Hyers and Ulam–Hyers–Rassias stability in different forms of functional equations has received a significant amount of attention; these topics are discussed in the monographs by [32,33,34,35]. Further insights can be found in [36,37].

Controllability is the ability of a dynamical system to be transferred through a set of controls from its starting state to any desired state, which is a basic and essential concept in control theory. In modern mathematical control theory, controllability is closely related to structural decomposition, quadratic optimization, and other concepts and is crucial to the growth of engineering. In 1960, Kalman was the first mathematician to develop the idea of controllability. Since then, many researchers have extensively used various methodologies to work on the controllability of different DSs. For more details about controllability and its applications, see [38,39,40,41]. See [42,43] for related results.

Recently, many researchers have conducted stability and controllability analyses of various fractional differential equations and systems. In [18], Dhayal and Zhu studied the stability and controllability of impulsive $\psi$-Hilfer fractional DSs; Shen et al. [44] examined the stability and controllability of stochastic fractional differential systems driven by the Rosenblatt process; and Kumar and Djemai [45] established results on the existence, stability, and controllability of piecewise dynamic systems under the influence of impulses. Moreover, Dhayal et al. [46] derived results on the existence, stability, and controllability of non-instantaneous impulsive stochastic DSs, and Kumar and Malik [47] ascertained the existence, Hyers–Ulam stability, and controllability of an impulsive hybrid neutral switched system. In particular, Dhayal et al. [48] recently studied the stability and controllability of $\psi$-Caputo impulsive fractional stochastic DSs driven by the Rosenblatt process. But to the best of our knowledge, no research works concerning the stability and controllability of non-instantaneous $\psi$-Caputo fractional stochastic differential systems driven by Rosenblatt process under integral boundary conditions has been found in the literature as of now.

Motivated by these research works and to fill the gap mentioned above, this research aims to discuss the stability and controllability of non-instantaneous $\psi$-Caputo fractional stochastic differential systems driven by the Rosenblatt process with integral boundary conditions. In this study, first, we investigate the stability of the following problem:

$$\left\{\begin{array}{c}{C}_{D_{\psi }^{\mu }}\omega (\varrho )=\mathcal{T}\omega (\varrho )+{\Phi }_1(\varrho ,\omega (\varrho ))+{\Phi }_2(\varrho )\frac{d{\widehat{\mathcal{E}}}^{\mathcal{K}}(\varrho )}{d\varrho },\varrho \in \bigcup _{z=0}^{\delta }(m_z,{\varrho }_{z+1}],\hfill \\ \omega (\varrho )={\mathcal{F}}_z(\varrho ,\omega ({\varrho }_z^{-})),\varrho \in \bigcup _{z=0}^{\delta }(m_z,{\varrho }_{z+1}],\hfill \\ \omega (0)={\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma ,\hfill \end{array}\right.$$

(1)

Then, we determine the controllability of the following problem:

$$\left\{\begin{array}{c}{C}_{D_{\psi }^{\mu }}\omega (\varrho )=\mathcal{T}\omega (\varrho )+\aleph u(\varrho )+{\Phi }_1(\varrho ,\omega (\varrho ))+{\Phi }_2(\varrho )\frac{d{\widehat{\mathcal{E}}}^{\mathcal{K}}(\varrho )}{d\varrho },\varrho \in \bigcup _{z=0}^{\delta }(m_z,{\varrho }_{z+1}],\hfill \\ \omega (\varrho )={\mathcal{F}}_z(\varrho ,\omega ({\varrho }_z^{-})),\varrho \in \bigcup _{z=0}^{\delta }(m_z,{\varrho }_{z+1}],\hfill \\ \omega (0)={\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma ,\hfill \end{array}\right.$$

(2)

where $C_{D_{\psi }^{\mu }}$ is the $\psi$-Caputo FD of order $\frac{1}{2}<\mu <1$, and $\omega (·)$ takes points in the real separable Hilbert space $\mathcal{M}$. $0=m_0={\varrho }_0<{\varrho }_1<{\varrho }_2<\dots <{\varrho }_{\delta }<m_{\delta }<{\varrho }_{\delta +1}=d<\infty ,\mathcal{J}=[0,d]$, and $\mathcal{T}$ is the generator of a $C_0$-semigroup ${\{\mathcal{S}(\varrho )\}}_{\varrho \ge 0}$ on $\mathcal{M}$. ${\widehat{\mathcal{E}}}^{\mathcal{K}}=\{{\widehat{\mathcal{E}}}^{\mathcal{K}}(\varrho ):\varrho \ge 0\}$ is a $\mathcal{Y}$-valued Rosenblatt process with the parameter $\mathcal{K}\in (\frac{1}{2},1)$, where $\mathcal{Y}$ is a real separable Hilbert space. ${\mathcal{F}}_z(\varrho ,\omega ({\varrho }_z^{-}))$ denotes non-instantaneous impulses on $({\varrho }_z,m_z],z=1,2,\dots ,\delta .$$u(·)\in L^2(\mathcal{J},\mathcal{X})$ is the control function, and $\aleph :\mathcal{X}\to \mathcal{M}$ is a bounded and linear operator where $\mathcal{X}$ is a real separable Hilbert space. The functions ${\Phi }_1:\mathcal{J}\times \mathcal{M}\to \mathcal{M}$, ${\Phi }_2:\mathcal{J}\to {\mathcal{L}}_2^0({\mathcal{P}}^{\frac{1}{2}}\mathcal{Y},\mathcal{M})$, ${\Phi }_3:\mathcal{J}\times \mathcal{M}\to \mathcal{M}$, and ${\mathcal{F}}_z:({\varrho }_z,m_z]\times \mathcal{M}\to \mathcal{M},z=1,2,\dots ,\delta$ are satisfied by certain conditions, which will be explained later.

The significant outcomes of this study are presented as follows:

• The stability and controllability of non-instantaneous $\psi$-Caputo fractional stochastic differential systems driven by the Rosenblatt process under integral boundary conditions are rarely available in the literature which is the key inspiration to our research work in this paper.
• The existence of a mild solution for system (1) on $\mathcal{J}$ is suitably proved by using the Banach fixed-point theorem and Lipschitz criteria.
• Unlike conventional exponential stability results, a novel result has been determined on the stability of impulsive stochastic $\psi$-Caputo fractional DSs driven by the Rosenblatt process.
• We examined the controllability results for the proposed system (2) on $\mathcal{J}$ by using the Banach fixed-point theorem and a new piecewise control function. Finally, illustrative examples were provided to demonstrate the authenticity and validity of the presented work.

This article is organized in the following manner. A few important preliminary materials that will be utilized subsequently are provided in Section 2. In Section 3, we investigate the existence of solutions and new stability results for the proposed system (1). Section 4 investigates the findings on the controllability for the suggested system (2). In the last part, Section 5, two examples are presented to demonstrate the certainty of the obtained theory.

2. Preliminaries

Let $L(\mathcal{Y},\mathcal{M})$ denote the space of all bounded linear operators and $\parallel ·\parallel$ represent the norms of the vectors belonging to the spaces $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{M}$, and $L(\mathcal{Y},\mathcal{M})$. Assume $L^2({\mathcal{G}}_d,\mathcal{M})$ denotes the Banach space of all ${\mathcal{G}}_d$-measurable, square integrable, random variables on $\mathcal{M}$. Let $(\Theta ,\mathcal{G},Q)$ be a complete probability space, and for $\varrho \ge 0$, ${\mathcal{G}}_{\varrho }$ represents the $\sigma$-field generated by $\{{\widehat{\mathcal{E}}}^{\widehat{\mathcal{K}}}(p):p\in [0,\varrho ]\}$ and the Q-null sets.

Assume ${\mathcal{L}}_2^0={\mathcal{L}}_2^0({\mathcal{P}}^{\frac{1}{2}}\mathcal{Y},\mathcal{M})$ denotes the separable Hilbert space of all Hilbert–Schmidt operators from ${\mathcal{P}}^{\frac{1}{2}}\mathcal{Y}$ into $\mathcal{M}$, with

$${\parallel \varphi \parallel }_{{\mathcal{L}}_2^0}^2={\parallel \varphi {\mathcal{P}}^{\frac{1}{2}}\parallel }^2=Tr(\varphi \mathcal{P}{\varphi }^{*})$$

Let $\{{\widehat{\mathcal{E}}}^{\mathcal{K}}(\varrho ):\varrho \ge 0\}$ denote the one-dimensional Rosenblatt process with the parameter $\mathcal{K}\in (\frac{1}{2},1)$ and written as [49]

$${\widehat{\mathcal{E}}}^{\mathcal{K}}(\varrho )=q(\mathcal{K}){\int }_0^{\varrho }{\int }_0^{\varrho }\left[{\int }_{y_1\vee y_2}^{\varrho }\frac{\partial H^{\acute{\mathcal{K}}}}{\partial t}(t,y_1)\frac{\partial H^{\acute{\mathcal{K}}}}{\partial t}(t,y_2)\right]d\widehat{\mathcal{E}}(y_1)d\widehat{\mathcal{E}}(y_2),$$

where $\widehat{\mathcal{E}}$ is a Wiener process, and $H^{\mathcal{K}}(q,p)$ is defined as

$$H^{\mathcal{K}}(q,p)=a_{\mathcal{K}}{p}^{(\frac{1}{2}-\mathcal{K})}{\int }_p^{\varrho }{(t-p)}^{\mathcal{K}-(\frac{3}{2})}{t}^{\mathcal{K}-(\frac{1}{2})}dt,$$

$\varrho \ge p$ and $H^{\mathcal{K}}(\varrho ,p)=0$ when $\varrho \le p$. Here, $a_{\mathcal{K}}=\sqrt{\frac{\mathcal{K}(2\mathcal{K}-1)}{\Gamma (2-2\mathcal{K},\mathcal{K}-\frac{1}{2})}}$, $\acute{\mathcal{K}}=\frac{(1+\mathcal{K})}{2}$, $q(\mathcal{K})=(\frac{1}{(1+\mathcal{K})})\sqrt{\frac{\mathcal{K}}{(2\mathcal{K}-1)}}$ is a normalizing constant, and $\Gamma (·,·)$ represents the gamma function. The Rosenblatt process covariance satisfies

$$\mathbb{E}({\widehat{\mathcal{E}}}^{\mathcal{K}}(\varrho ),{\widehat{\mathcal{E}}}^{\mathcal{K}}(\varrho ))=\frac{1}{2}\left(p^{2\mathcal{K}}+{\varrho }^{2\mathcal{K}}-{|\varrho -p|}^{2\mathcal{K}}\right).$$

Let ${\widehat{\mathcal{E}}}^{\mathcal{P}}(\varrho )$ be a $\mathcal{Y}$-valued process given by

$${\widehat{\mathcal{E}}}^{\mathcal{P}}(\varrho )=\sum _{i=1}^{\infty }{q}_i(\varrho ){\mathcal{P}}^{\frac{1}{2}}{t}_i,\varrho \ge 0,$$

where $q_i(\varrho )$ represents a series of mutually independent, two-sided, one-dimensional Rosenblatt processes on $(\Theta ,\mathcal{G},Q)$. Assume $P\in L(\mathcal{Y},\mathcal{Y})$ is an operator defined by $P{t}_i={\lambda }_i{t}_i$ with the finite-trace $\mathrm{Tr}(P)={\sum }_{i=1}^{\infty }{\lambda }_i<\infty$, where $\{{\lambda }_i\ge 0:i=1,2,\dots \}$ are real numbers and $\{t_i:i=1,2,\dots \}$ is a complete orthonormal basis in $\mathcal{Y}$. Let ${\widehat{\mathcal{E}}}^{\mathcal{P}}(\varrho )$ be a $\mathcal{Y}$-valued Rosenblatt process with the covariance P as

$${\widehat{\mathcal{E}}}^{\mathcal{P}}(\varrho )=\sum _{i=1}^{\infty }\sqrt{{\lambda }_i}{q}_i(\varrho )t_i.$$

Definition 1

([49]). Assume $\varphi :\mathcal{J}\to {\mathcal{L}}_2^0({\mathcal{P}}^{\frac{1}{2}}\mathcal{Y},\mathcal{M})$ such that ${\sum }_{i=1}^{\infty }{\parallel H_{\mathcal{K}}^{*}(\varphi {\mathcal{P}}^{\frac{1}{2}}{t}_i)\parallel }_{L^2(\mathcal{J},\mathcal{M})}\le \infty .$ Then, for $\varrho \ge 0$, with respect to the Rosenblatt process, its stochastic integral is defined as

$$\begin{array}{ccc}\hfill {\int }_0^{\varrho }\varphi (p)d{\widehat{\mathcal{E}}}^{\mathcal{P}}(p)& =& \sum _{i=1}^{\infty }{\int }_0^{\varrho }\varphi (p){\mathcal{P}}^{\frac{1}{2}}{t}_{i}d{q}_i(p)\hfill \\ & =& \sum _{i=1}^{\infty }{\int }_0^{\varrho }{\int }_0^{\varrho }{H}_{\mathcal{K}}^{*}(\varphi {\mathcal{P}}^{\frac{1}{2}}{t}_i)(y_1,y_2)d\widehat{\mathcal{E}}(y_1)d\widehat{\mathcal{E}}(y_2).\hfill \end{array}$$

Lemma 1

([50]). For $\varphi :\mathcal{J}\to {\mathcal{L}}_2^0({\mathcal{P}}^{\frac{1}{2}}{t}_i\mathcal{Y},\mathcal{M})$ such that ${\sum }_{i=1}^{\infty }{\parallel \varphi {\mathcal{P}}^{\frac{1}{2}}{t}_i\parallel }_{L^{\frac{1}{\mathcal{K}}}}(\mathcal{J},\mathcal{M})<\infty$ holds, and for any $c,d\in \mathcal{J}$ with $d>c$, we have

$$\mathbb{E}\left[\parallel {\int }_c^d\varphi (p)d{\widehat{\mathcal{E}}}^{\mathcal{P}}(p){\parallel }^2\right]\le a_{\mathcal{K}}{(d-c)}^{2\mathcal{K}-1}\sum _{i=1}^{\infty }{\int }_c^d{\parallel \varphi (p){\mathcal{P}}^{\frac{1}{2}}{t}_i\parallel }^{2}dp.$$

If ${\sum }_{i=1}^{\infty }\parallel \varphi {\mathcal{P}}^{\frac{1}{2}}{t}_i\parallel$ is uniformly convergent for $\varrho \in \mathcal{J}$, then it satisfies

$$\mathbb{E}\left[\parallel {\int }_c^d\varphi (p)d{\widehat{\mathcal{E}}}^{\mathcal{P}}(p){\parallel }^2\right]\le a_{\mathcal{K}}{(d-c)}^{2\mathcal{K}-1}{\int }_c^d{\parallel \varphi (p)\parallel }_{{\mathcal{L}}_2^0}^{2}dp.$$

Let ${\mathcal{J}}_2=[c,d]$ and $\psi \in C^n({\mathcal{J}}_2,\mathbb{R})$ be an increasing function such that ${\psi }^{\prime }(q)\ne 0$ for all $q\in {\mathcal{J}}_2$.

Definition 2

([51]). The ψ-Caputo FD of order $\mu (h-1<\mu <h,h\in \mathbb{N})$ for the function $\mathcal{W}$ is defined as

$$\left(C_{D_{\psi }^{\mu }}\mathcal{W}\right)(\varrho )=(c{J}_{\psi }^{h-\mu }{\mathcal{W}}^{\left[h\right]})=\frac{1}{\Gamma (h-\mu )}{\int }_c^{\varrho }{(\psi (\varrho )-\psi (p))}^{h-\mu -1}{\mathcal{W}}^{\left[h\right]}(p){\psi }^{\prime }(p)dp,$$

where $h=\left[\mu \right]+1$ and ${\mathcal{W}}^{\left[h\right]}(\varrho )={\left(\frac{1}{{\psi }^{\prime }(p)}\frac{d}{d\varrho }\right)}^h\mathcal{W}(\varrho ).$

Lemma 2

([51]). Let $\mathcal{W}\in C^h([c,d])$ and $\mu >0$. Then, we have

$$c^{J_{\psi }^{\mu }}{C}_{D_{\psi }^{\mu }}\mathcal{W}(\varrho )=\mathcal{W}(\varrho )-\sum _{m=0}^{h-1}\frac{{\mathcal{W}}^{\left[m\right]}(c^{+})}{m!}{(\psi (\varrho )-\psi (p))}^m.$$

Particularly, for $\mu \in (0,1)$, we obtain

$$c^{J_{\psi }^{\mu }}{C}_{D_{\psi }^{\mu }}\mathcal{W}(\varrho )=\mathcal{W}(\varrho )-\mathcal{W}(c).$$

Lemma 3

([52]). Let $\mu >0$ and $\Pi >0$; then,

i 

$J_{\psi }^{\mu }{(\psi (\varrho )-\psi (c))}^{\Pi -1}(\varrho )=\frac{\Gamma (\Pi )}{\Gamma (\Pi +\mu )}{(\psi (\varrho )-\psi (p))}^{\Pi +\mu -1}$.

ii 

$C_{J_{\psi }^{\mu }}{(\psi (\varrho )-\psi (c))}^{\Pi -1}(\varrho )=\frac{\Gamma (\Pi )}{\Gamma (\Pi -\mu )}{(\psi (\varrho )-\psi (p))}^{\Pi -\mu -1}$.

Lemma 4.

The ψ-Caputo fractional-order Cauchy system has a solution:

$$\left\{\begin{array}{c}{C}_{D_{\psi }^{\mu }}\omega (\varrho )=\mathcal{T}\omega (\varrho )+\Re (\varrho ),\varrho \in (0,d],\hfill \\ \omega (0)={\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma .\hfill \end{array}\right.$$

(3)

given by

$$\left\{\begin{array}{cc}\omega (\varrho )=\hfill & {\mathcal{T}}_{\psi }^{\mu }(\varrho ,0){\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma \hfill \\ \hfill & +{\int }_c^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p)\Re (p){\psi }^{\prime }(p)dp,\hfill \end{array}\right.$$

(4)

where

$${\mathcal{T}}_{\psi }^{\mu }(\varrho ,p)\omega ={\int }_0^{\infty }{\phi }_{\mu }(\vartheta )\mathcal{S}({(\psi (\varrho )-\psi (p))}^{\mu }\vartheta )\omega d\vartheta ,$$

$${\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)\omega =\mu {\int }_0^{\infty }\vartheta {\phi }_{\mu }(\vartheta )\mathcal{S}({(\psi (\varrho )-\psi (p))}^{\mu }\vartheta )\omega d\vartheta ,0\le p\le \varrho \le d,$$

and ${\phi }_{\mu }(\vartheta )=\frac{1}{\mu }{\vartheta }^{(-1-(\frac{1}{\mu }))}{\varsigma }_{\mu }({\vartheta }^{-\frac{1}{\mu }})$ is the probability density function defined on $(0,-\infty )$, i.e., ${\phi }_{\mu }(\vartheta )\ge 0,$$\vartheta \in (0,\infty )$ and ${\int }_0^{\infty }{\phi }_{\mu }(\vartheta )d\vartheta =1$.

Proof. 

We can rewrite Equation (3) in integral equation form as

$$\begin{array}{cc}\hfill \omega (\varrho )=& {\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma \hfill \\ \hfill & +\frac{1}{\Gamma (\mu )}+{\int }_c^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}[\mathcal{T}\omega (p)+\Re (p)]{\psi }^{\prime }(p)dp,\hfill \end{array}$$

(5)

given that Equation (5) exists. Assume that $\Pi \ge 0$. With the help of a Laplace transform, one can obtain

$$\widehat{M}(\Pi )=\frac{{\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma }{\Pi }+\frac{1}{{\Pi }^{\mu }}\left(\mathcal{T}\widehat{M}(\Pi )+\widehat{\Re }(\Pi )\right),$$

where

$$\widehat{M}(\Pi )={\int }_0^{\infty }{e}^{-\Pi (\psi (\zeta )-\psi (0))}\omega (\zeta ){\psi }^{\prime }(\zeta )d\zeta ,$$

and

$$\widehat{\Re }(\Pi )={\int }_0^{\infty }{e}^{-\Pi (\psi (\zeta )-\psi (0))}\Re (\zeta ){\psi }^{\prime }(\zeta )d\zeta .$$

It follows that

$$\begin{array}{ccc}\hfill \widehat{M}(\Pi )& =& {\Pi }^{\mu -1}{({\Pi }^{\mu }J-\mathcal{T})}^{-1}{\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma +{({\Pi }^{\mu }J-\mathcal{T})}^{-1}\widehat{\Re }(\Pi )\hfill \\ & =& {\Pi }^{\mu -1}{\int }_0^{\infty }{e}^{-{\Pi }^{\mu }p}\mathcal{S}(p){\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma dp\hfill \\ & & +{\int }_0^{\infty }{e}^{-{\Pi }^{\mu }p}\mathcal{S}(p)\widehat{\Re }(\Pi )dp.\hfill \end{array}$$

Taking $p=\widehat{{\varrho }^{\mu }}$, we obtain

$$\begin{array}{ccc}\hfill \widehat{M}(\Pi )& =& \mu {\int }_0^{\infty }{(\Pi \widehat{\varrho })}^{\mu -1}{e^{-(\Pi \widehat{\varrho })}}^{\mu }\mathcal{S}({\widehat{\varrho }}^{\mu }){\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma d\widehat{\varrho }\hfill \\ & & +\mu {\int }_0^{\infty }{(\widehat{\varrho })}^{\mu -1}{e^{-(\Pi \widehat{\varrho })}}^{\mu }\mathcal{S}({\widehat{\varrho }}^{\mu })\widehat{\Re }(\Pi )d\widehat{\varrho }\hfill \\ & =& J_1+J_2,\hfill \end{array}$$

where

$$J_1=\mu {\int }_0^{\infty }{(\Pi \widehat{\varrho })}^{\mu -1}{e^{-(\Pi \widehat{\varrho })}}^{\mu }\mathcal{S}({\widehat{\varrho }}^{\mu }){\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma d\widehat{\varrho }$$

$$J_2=\mu {\int }_0^{\infty }{(\widehat{\varrho })}^{\mu -1}{e^{-(\Pi \widehat{\varrho })}}^{\mu }\mathcal{S}(\widehat{\varrho \mu })\widehat{\Re }(\Pi )d\widehat{\varrho }$$

Taking $\widehat{\varrho }=\psi (\varrho )-\psi (0)$, we obtain

$$\begin{array}{ccc}\hfill J_1& =& \mu {\int }_0^{\infty }{({\Pi }^{\mu -1}(\psi (\varrho )-\psi (0)))}^{\mu -1}{e^{-(\Pi (\psi (\varrho )-\psi (0)))}}^{\mu }\mathcal{S}{((\psi (\varrho )-\psi (0)))}^{\mu }\hfill \\ & & \times {\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma {\psi }^{\prime }(\varrho )d\varrho \hfill \\ & =& {\int }_0^{\infty }\frac{-1}{\Pi }\frac{d}{d\varrho }\left({e^{-(\Pi (\psi (\varrho )-\psi (0)))}}^{\mu }\right)\mathcal{S}{((\psi (\varrho )-\psi (0)))}^{\mu }\hfill \\ & & \times {\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma {\psi }^{\prime }(\varrho )d\varrho .\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill J_2& =& \mu {\int }_0^{\infty }{((\psi (\varrho )-\psi (0)))}^{\mu -1}{e^{-(\Pi (\psi (\varrho )-\psi (0)))}}^{\mu }\mathcal{S}{((\psi (\varrho )-\psi (0)))}^{\mu }\widehat{\Re }(\Pi ){\psi }^{\prime }(\varrho )d\varrho \hfill \\ & =& {\int }_0^{\infty }{\int }_0^{\infty }\mu {((\psi (\varrho )-\psi (0)))}^{\mu -1}{e^{-(\Pi (\psi (\varrho )-\psi (0)))}}^{\mu }\mathcal{S}{((\psi (\varrho )-\psi (0)))}^{\mu }\hfill \\ & & \times e^{-(\Pi (\psi (\varrho )-\psi (0)))}\Re {\psi }^{\prime }(p){\psi }^{\prime }(\varrho )dpd\varrho .\hfill \end{array}$$

Now, we take the below one-sided stable probability density:

$${\varsigma }_{\mu }(\vartheta )=\frac{1}{\pi }\sum _{j=1}^{\infty }{(-1)}^{j-1}{\vartheta }^{-\mu j-1}\frac{\Gamma (\mu j+1)}{j!}sin(j\pi \mu ),\vartheta \in (0,\infty ),$$

whose integration is defined as follows:

$${\int }_0^{\infty }{e}^{-\Pi \vartheta }{\varsigma }_{\mu }(\vartheta )d\vartheta =e^{-{\Pi }^{\mu }},\mu \in (0,1).$$

(6)

Using Equation (6), we obtain

$$\begin{array}{ccc}\hfill J_1& =& {\int }_0^{\infty }\frac{-1}{\Pi }\frac{d}{d\varrho }\left({\int }_0^{\infty }{e^{-(\Pi (\psi (\varrho )-\psi (0)))}}^{\vartheta }{\varsigma }_{\mu }(\vartheta )d\vartheta \right)\mathcal{S}{((\psi (\varrho )-\psi (0)))}^{\mu }\hfill \\ & & \times {\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma d\varrho \hfill \\ & =& {\int }_0^{\infty }{\int }_0^{\infty }\vartheta {\varsigma }_{\mu }(\vartheta ){e^{-(\Pi (\psi (\varrho )-\psi (0)))}}^{\vartheta }\mathcal{S}{((\psi (\varrho )-\psi (0)))}^{\mu }\hfill \\ & & \times {\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma {\psi }^{\prime }(\varrho )d\vartheta d\varrho \hfill \\ & =& {\int }_0^{\infty }{e}^{-(\Pi (\psi (\varrho )-\psi (0)))}\left({\int }_0^{\infty }{\varsigma }_{\mu }(\vartheta )\mathcal{S}\left(\frac{(\psi (\varrho )-\psi (0)}{{\vartheta }^{\mu }}\right)d\vartheta \right)\hfill \\ & & \times {\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma {\psi }^{\prime }(\varrho )d\varrho .\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill J_2& =& {\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }\mu {((\psi (\varrho )-\psi (0)))}^{\mu -1}{\varsigma }_{\mu }(\vartheta )e^{-(\Pi (\psi (\varrho )-\psi (0)))\vartheta }\mathcal{S}({((\psi (\varrho )-\psi (0)))}^{\mu })\hfill \\ & & \times e^{-(\Pi (\psi (\varrho )-\psi (0)))}\Re (p){\psi }^{\prime }(p){\psi }^{\prime }(\varrho )d\vartheta dpd\varrho \hfill \\ & =& {\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }\mu e^{-(\Pi (\psi (\varrho )+\psi (p)-2\psi (0)))}\frac{{((\psi (\varrho )-\psi (0)))}^{\mu -1}}{{\vartheta }^{\mu }}\hfill \\ & & \times \left(\frac{{((\psi (\varrho )-\psi (0)))}^{\mu -1}}{{\vartheta }^{\mu }}\right)\Re (p){\psi }^{\prime }(p){\psi }^{\prime }(\varrho )d\vartheta dpd\varrho \hfill \\ & =& {\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }\mu e^{-(\Pi (\psi (\varrho )-\psi (0)))}{\varsigma }_{\mu }(\vartheta )\frac{{((\psi (\varrho )-\psi (0)))}^{\mu -1}}{{\vartheta }^{\mu }}\hfill \\ & & \times \mathcal{S}\left(\frac{{((\psi (\varrho )-\psi (0)))}^{\mu }}{{\vartheta }^{\mu }}\right)\Re ({\psi }^{-1}(\psi (\zeta )-\psi (\varrho )+\psi (0))){\psi }^{\prime }(p){\psi }^{\prime }(\varrho )d\vartheta d\zeta d\varrho \hfill \\ & =& {\int }_0^{\infty }{e}^{-(\Pi (\psi (\varrho )-\psi (0)))}({\int }_o^{\zeta }{\int }_0^{\infty }\mu {\varsigma }_{\mu }(\vartheta )\frac{{((\psi (\varrho )-\psi (0)))}^{\mu -1}}{{\vartheta }^{\mu }}\hfill \\ & & \times \mathcal{S}\left(\frac{{((\psi (\zeta )-\psi (p)))}^{\mu }}{{\vartheta }^{\mu }}\right)\Re (p){\psi }^{\prime }(p)d\vartheta dp){\psi }^{\prime }(\zeta )d\zeta .\hfill \end{array}$$

Hence, we obtain

$$\begin{array}{ccc}\hfill \widehat{M}(\Pi )& =& {\int }_0^{\infty }{e}^{-(\Pi (\psi (\varrho )-\psi (0)))}({\int }_o^{\zeta }{\varsigma }_{\mu }(\vartheta )\mathcal{S}\left(\frac{{((\psi (\zeta )-\psi (p)))}^{\mu }}{{\vartheta }^{\mu }}\right)\hfill \\ & & \times {\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma d\vartheta ){\psi }^{\prime }(\varrho )d\varrho \hfill \\ & & +{\int }_0^{\infty }{e}^{-(\Pi (\psi (\varrho )-\psi (0)))}({\int }_o^{\zeta }\hfill \\ & & \times {\int }_0^{\infty }\mu {\varsigma }_{\mu }(\vartheta )\frac{{((\psi (\varrho )-\psi (0)))}^{\mu -1}}{{\vartheta }^{\mu }}\hfill \\ & & \times \mathcal{S}\left(\frac{{((\psi (\zeta )-\psi (p)))}^{\mu }}{{\vartheta }^{\mu }}\right)\Re (p){\psi }^{\prime }(p)d\vartheta dp){\psi }^{\prime }(\zeta )d\zeta .\hfill \end{array}$$

By using an inverse Laplace transform, we obtain

$$\begin{array}{ccc}\hfill \omega (\varrho )& =& {\int }_o^{\infty }{\varsigma }_{\mu }(\vartheta )\mathcal{S}\left(\frac{{((\psi (\zeta )-\psi (p)))}^{\mu }}{{\vartheta }^{\mu }}\right){\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma d\vartheta \hfill \\ & & +{\int }_0^{\varrho }{\int }_0^{\infty }\mu {\varsigma }_{\mu }(\vartheta )\frac{{((\psi (\varrho )-\psi (p)))}^{\mu -1}}{{\vartheta }^{\mu }}\mathcal{S}\left(\frac{{((\psi (\zeta )-\psi (p)))}^{\mu }}{{\vartheta }^{\mu }}\right)\Re (p){\psi }^{\prime }(p)d\vartheta dp.\hfill \end{array}$$

Thus, we obtain

$$\begin{array}{ccc}\hfill \omega (\varrho )& =& {\int }_o^{\infty }{\phi }_{\mu }(\vartheta )\mathcal{S}({((\psi (\zeta )-\psi (p)))}^{\mu }\vartheta )\hfill \\ & & +\mu {\int }_0^{\varrho }{\int }_0^{\infty }\vartheta {\phi }_{\mu }(\vartheta ){((\psi (\varrho )-\psi (p)))}^{\mu -1}\mathcal{S}{((\psi (\zeta )-\psi (p)))}^{\mu }\vartheta )\Re (p){\psi }^{\prime }(p)d\vartheta dp,\hfill \end{array}$$

where ${\phi }_{\mu }(\vartheta )=\frac{1}{\mu }{\vartheta }^{-1-\frac{1}{\mu }}{\varsigma }_{\mu }({\vartheta }^{-\frac{1}{\mu }})$. For any $\omega \in \mathcal{M}$, the operators ${\mathcal{T}}_{\psi }^{\mu }(\varrho ,p)\omega$ and ${\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)\omega$ are defined as

$${\mathcal{T}}_{\psi }^{\mu }(\varrho ,p)\omega ={\int }_0^{\infty }{\phi }_{\mu }(\vartheta )\mathcal{S}({(\psi (q)-\psi (p))}^{\mu }\vartheta )\omega d\vartheta ,$$

and

$${\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)\omega =\mu {\int }_0^{\infty }\vartheta {\phi }_{\mu }(\vartheta )\mathcal{S}({(\psi (q)-\psi (p))}^{\mu }\vartheta )\omega d\vartheta ,)\le p\le \varrho \le d.$$

Hence, we obtain

$$\begin{array}{ccc}\hfill \omega (\varrho )& =& {\mathcal{T}}_{\psi }^{\mu }(\varrho ,0){\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma \hfill \\ & & +{\int }_c^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p)\Re (p){\psi }^{\prime }(p)dp.\hfill \end{array}$$

□

Lemma 5

([51]). For any fixed $\varrho \ge p\ge o$, ${\mathcal{T}}_{\psi }^{\mu }(\varrho ,p)$ and ${\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)$ are linear bounded operators, and

$$\begin{array}{ccc}\hfill \parallel {\mathcal{T}}_{\psi }^{\mu }(\varrho ,p)(\omega )\parallel & \le & \widehat{\mathcal{W}}\parallel \omega \parallel ,\hfill \\ \hfill \parallel {\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)(\omega )\parallel & \le & \frac{\mu \widehat{\mathcal{W}}}{\Gamma (1+\mu )}\parallel \omega \parallel =\frac{\widehat{\mathcal{W}}}{\Gamma (\mu )}\parallel \omega \parallel .\hfill \end{array}$$

Definition 3.

A ${\mathcal{G}}_{\varrho }$-adapted stochastic process $\omega :\mathcal{J}\to \mathcal{M}$ is said to be a mild solution of the system (1) if for any $\varrho \in \mathcal{J},\omega (\varrho )$ satisfies $\omega (0)={\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma$, and $\omega (\varrho )={\mathcal{F}}_z(\varrho ,\omega ({\varrho }_z^{-}))$, $\varrho \in ({\varrho }_z,m_z],$$z=1,2,\dots ,\delta$, and

$$\begin{array}{ccc}\hfill \omega (\varrho )& =& {\mathcal{T}}_{\psi }^{\mu }(\varrho ,0){\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma \hfill \\ & & +{\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p){\Phi }_1(p,\omega (p)){\psi }^{\prime }(p)dp\hfill \\ & & +{\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{E}}}^{\mathcal{H}}(p),\hfill \end{array}$$

for all $\varrho \in [0,{\varrho }_1]$, $z=0$, and

$$\begin{array}{ccc}\hfill \omega (\varrho )& =& {\mathcal{T}}_{\psi }^{\mu }(\varrho ,m_z){\mathcal{F}}_z(m_z,\omega ({\varrho }_z^{-}))\hfill \\ & & +{\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p){\Phi }_1(p,\omega (p)){\psi }^{\prime }(p)dp\hfill \\ & & +{\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{E}}}^{\mathcal{H}}(p),\hfill \end{array}$$

for all $\varrho \in (m_z,{\varrho }_{z+1}]$, $z=1,2,\dots ,\delta$.

Definition 4

([53]). System (1) assures a stable mild solution ω if for any $ϵ>0$, $\mho >0$ exists such that

$\mathbb{E}\parallel \omega (\varrho )-\widehat{\omega }(\varrho )\parallel <ϵ$, whenever $\mathbb{E}\parallel \omega (0)-{\widehat{\omega }}_0\parallel <\mho$, where $\widehat{\omega }$ is the mild solution of (1) with initial conditions $\widehat{\omega }(0)={\widehat{\omega }}_0$, and the impulsive conditions $\omega (\varrho )={\mathcal{f}}_z(\varrho ,\widehat{\omega }({\varrho }_z^{-})),$$\varrho \in ({\varrho }_z,m_z]$, $z=1,2,\dots ,\delta$.

Let $\mathcal{Q}\mathcal{C}(\mathcal{M})$ be the space formed by all ${\mathcal{G}}_{\varrho }$-adapted measurable, $\mathcal{M}$-valued stochastic process $\{\omega (\varrho ):\varrho \in \mathcal{J}\}$ such that $\omega$ is continuous at $\varrho \ne {\varrho }_z$, $\omega ({\varrho }_z^{-})=\omega ({\varrho }_z^{+})$ and $\omega ({\varrho }_z^{+})$ exist for every $z=1,2,\dots ,\delta$ with the norm described as

$${\parallel \omega \parallel }_{\mathcal{Q}\mathcal{C}}={\left(\underset{0\le \varrho \le d}{sup}\mathbb{E}{\parallel \omega (\varrho )\parallel }^2\right)}^{\frac{1}{2}}.$$

Then $(\mathcal{Q}\mathcal{C}(\mathcal{M}),\parallel .{\parallel }_{\mathcal{Q}\mathcal{C}})$ represents a Banach Space.

Lemma 6

([54]). Assume $\mathcal{V}\in \mathcal{Q}\mathcal{C}(\mathcal{J},\mathbb{R})$ satisfies the inequality

$$\mathcal{V}(\varrho )\le \mathcal{U}(\varrho )+f(\varrho ){\int }_0^{\varrho }{(\psi (\varrho )-\psi (t))}^{\mu -1}\mathcal{V}(t){\psi }^{\prime }(t)dt+\sum _{0<{\varrho }_z<\varrho }{\beta }_z\mathcal{V}({\varrho }_z^{-}),\varrho >0,$$

where f is an increasing function, $\mathcal{U}\in \mathcal{Q}\mathcal{C}(\mathcal{J},\mathbb{R})$ and $\mathcal{U}\ge 0$, ${\beta }_z>0$ for $z=1,2,\dots ,\delta$, then

$$\begin{array}{ccc}\hfill \mathcal{V}(\varrho )& \le & \mathcal{U}(\varrho )\left[\prod _{j=1}^z\{1+{\beta }_j{Z}_{\mu }{(f(\varrho )\Gamma (\mu )(\psi ({\varrho }_z)-\psi (0)))}^{\mu })\}\right]\hfill \\ & & \times Z_{\mu }{(f(\varrho )\Gamma (\mu )(\psi ({\varrho }_z)-\psi (0)))}^{\mu }),\varrho \in (m_z,{\varrho }_{z+1}].\hfill \end{array}$$

3. Stability Results

In this section, we delve into the stability analysis of stochastic fractional differential equations under integral boundary conditions driven by the rosenblatt process with impulses. These equations are pivotal in modeling systems exhibiting memory and hereditary properties, where the future state depends not only on the current state but also on past states. The inclusion of stochastic elements, particularly the Rosenblatt process, introduces complexities due to its self-similarity and long-range dependence characteristics. Additionally, the presence of impulsive effects—sudden, significant changes at specific moments—adds further intricacy to the system’s behavior.

The following assumptions are made to ensure that the system described by the equations has well-defined solutions, specifically a unique mild solution. The assumptions provide the necessary conditions for the continuity, Lipschitz continuity, and bounded growth of the functions involved in the system, which are key to applying standard existence and uniqueness results from functional analysis, such as fixed-point theorems.

C1

${\Phi }_1:\mathcal{J}\times \mathcal{M}\to \mathcal{M}$ is a continuous function, and ${\widehat{\mathcal{H}}}_{{\Phi }_1},{\widehat{ϵ}}_{{\Phi }_1}>0$ exists such that

$$\mathbb{E}\parallel {\Phi }_1{(\varrho ,\omega )\parallel }^2\le {\widehat{\mathcal{H}}}_{{\Phi }_1}{(1+\mathbb{E}\parallel \omega \parallel }^2),\forall \omega \in \mathcal{M}$$

$$\mathbb{E}\parallel {\Phi }_1(\varrho ,{\omega }_1)-{\Phi }_1(\varrho ,{\omega }_2){\parallel }^2\le {\widehat{ϵ}}_{{\Phi }_1}\mathbb{E}{\parallel {\omega }_1-{\omega }_2\parallel }^2,\forall {\omega }_1,{\omega }_2\in \mathcal{M}.$$

C2

${\mathcal{F}}_z:({\varrho }_z,m_z]\times \mathcal{M}\to \mathcal{M}$ and $z=1,2,\dots ,\delta$ are continuous functions, and ${\widehat{\mathcal{H}}}_{{\mathcal{F}}_z},{\widehat{ϵ}}_{{\mathcal{F}}_z}>0$ exists such that

$$\mathbb{E}\parallel {\mathcal{F}}_z{(\varrho ,\omega )\parallel }^2\le {\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}{(1+\mathbb{E}\parallel \omega \parallel }^2),\forall \omega \in \mathcal{M}$$

$$\mathbb{E}\parallel {\mathcal{F}}_z(\varrho ,{\omega }_1)-{\mathcal{F}}_z(\varrho ,{\omega }_2){\parallel }^2\le {\widehat{ϵ}}_{{\mathcal{F}}_z}\mathbb{E}{\parallel {\omega }_1-{\omega }_2\parallel }^2,\forall {\omega }_1,{\omega }_2\in \mathcal{M}.$$

C3

${\Phi }_1:\mathcal{J}\to {\mathcal{L}}_2^0({\mathcal{P}}^{\frac{1}{2}}\mathcal{Y},\mathcal{M})$ satisfying ${\int }_0^{\varrho },{\parallel {\Phi }_2(p)\parallel }_{{\mathcal{L}}_2^0}^{2}dp<\infty$, and ${ϵ}_{{\Phi }_2}>0$ exists such that $\parallel {\Phi }_2{(\varrho )\parallel }_2^0\le {ϵ}_{{\Phi }_2}$.

C4

The following inequality holds:

$$\underset{1\le z\le \delta }{max}\left\{{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}+3{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}+\frac{3{\widehat{\mathcal{W}}}^{2}d{\psi }^{\prime }(d)}{(2\mu -1){(\Gamma (\mu ))}^2}{(\psi (d)-\psi (0))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Psi }_1}\right\}<1.$$

C5

${\mathcal{K}}_{{\Phi }_3},{\mathcal{N}}_{{\Phi }_3}>0$ exists such that $|{\Phi }_3(\varsigma ,\omega (\varsigma )\parallel \le {\mathcal{K}}_{{\Phi }_3}\parallel \omega (\varsigma )-\overline{\omega }(\varsigma )\parallel ,$ and ${\mathcal{N}}_{{\Phi }_3}=max\parallel {\Phi }_3(\varsigma ,0)\parallel$.

Theorem 1.

If assumptions [C1]–[C5] hold, then (1) has a unique mild solution on $\mathcal{J}$, if

$$\underset{1\le z\le \delta }{max}\left[{\widehat{O}}_0,{\widehat{ϵ}}_{{\mathcal{F}}_z},{\widehat{O}}_z\right]<1,$$

(7)

where

$${\widehat{O}}_0=\frac{\widehat{\mathcal{W}}{\varrho }_1{\psi }^{\prime }({\varrho }_1)}{(2\mu -1){(\Gamma (\mu ))}^2}{\widehat{ϵ}}_{{\Phi }_1}{(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1},$$

$${\widehat{O}}_z=2{\widehat{\mathcal{W}}}^2{\widehat{ϵ}}_{{\mathcal{F}}_z}+\frac{2{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1})}{(2\mu -1){(\Gamma (\mu ))}^{2\mu -1}}{\widehat{ϵ}}_{{\Phi }_1}{(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}.$$

Proof. 

Taking $\beta >0$, we define

$$H_{\beta }=\left\{\omega \in {\mathcal{Q}\mathcal{C}(\mathcal{M}):\parallel \omega \parallel }_{\mathcal{P}\mathcal{C}}^2\le \beta \right\}.$$

Obviously, $H_{\beta }$ is a closed and bounded subset of $\mathcal{Q}\mathcal{C}(\mathcal{M}).$ The operator $\mathfrak{G}$ on $H_{\beta }$ is defined as follows:

$$(\mathfrak{G}\omega )(\varrho )=\left\{\begin{array}{c}{\mathcal{T}}_{\psi }^{\mu }(\varrho ,0){\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma \hfill \\ +{\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p){\Phi }_1(p,\omega (p)){\psi }^{\prime }(p)dp\hfill \\ +{\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{E}}}^{\mathcal{H}}(p),\hfill \\ \varrho \in [0,{\varrho }_1],z=0\hfill \\ {\mathcal{F}}_z(\varrho ,\omega ({\varrho }_z^{-})),\varrho \in ({\varrho }_z,m_z],z\ge 1.\hfill \\ {\mathcal{T}}_{\psi }^{\mu }(\varrho ,m_z){\mathcal{F}}_z(m_z,\omega ({\varrho }_z^{-}))\hfill \\ +{\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p){\Phi }_1(p,\omega (p)){\psi }^{\prime }(p)dp\hfill \\ +{\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{E}}}^{\mathcal{H}}(p),\hfill \\ \varrho \in (m_z,{\varrho }_{z+1}],z\ge 1.\hfill \end{array}\right.$$

Step 1

There exists $\beta >0$ such that $\mathcal{G}(H_{\beta })\subset H_{\beta }$. On the contrary, let the assumption not hold; then, for $\beta >0$, let $\varrho \in \mathcal{J}$ and ${\omega }^{\beta }\in H_{\beta }$ such that $\mathbb{E}\parallel (\mathfrak{G}{\omega }^{\beta })(\varrho ){\parallel }^2>\beta$. For $\varrho \in [0,{\varrho }_1]$, we obtain

$$\begin{array}{ccc}\hfill \beta & <& \mathbb{E}\parallel (\mathfrak{G}{\omega }^{\beta })(\varrho ){\parallel }^2\hfill \\ & \le & 3\mathbb{E}\parallel {\mathcal{T}}_{\psi }^{\mu }(\varrho ,0){\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3{(\varsigma ,\omega (\varsigma ))d\varsigma \parallel }^2\hfill \\ & & +3\mathbb{E}\parallel {\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p){\Phi }_1(p,\omega (p)){\psi }^{\prime }{(p)dp\parallel }^2\hfill \\ & & +3\mathbb{E}\parallel {\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{E}}}^{\mathcal{H}}{(p)\parallel }^2\hfill \\ & \le & 3{\widehat{\mathcal{W}}}^2\mathbb{E}{\mathcal{K}}_{{\Phi }_3}\frac{{\hslash }^{\varrho }}{\Gamma (\varrho +1)}{\parallel \omega (\varsigma )-\overline{\omega }(\varsigma )\parallel }^2\hfill \\ & & +\frac{3\widehat{\mathcal{W}}{\varrho }_1{\psi }^{\prime }({\varrho }_1)}{{(\Gamma (\mu ))}^2}{\widehat{\mathcal{H}}}_{{\Phi }_1}(1+\beta ){\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{2\mu -2}{\psi }^{\prime }(p)dp\hfill \\ & & +\frac{3a_{\mathcal{K}}{\widehat{\mathcal{W}}}^2{\varrho }_1^{2\mathcal{K}-1}{\psi }^{\prime }({\varrho }_1)}{{(\Gamma (\mu ))}^2}{ϵ}_{{\Phi }_2}{\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{2\mu -2}{\psi }^{\prime }(p)dp\hfill \\ & \le & 3{\widehat{\mathcal{W}}}^2\mathbb{E}{\mathcal{K}}_{{\Phi }_3}\frac{{\hslash }^{\varrho }}{\Gamma (\varrho +1)}{\parallel \omega (\varsigma )-\overline{\omega }(\varsigma )\parallel }^2\hfill \\ & & +\frac{3\widehat{\mathcal{W}}{\varrho }_1{\psi }^{\prime }({\varrho }_1)}{(2\mu -1){(\Gamma (\mu ))}^2}{(\psi (\varrho )-\psi (p))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Phi }_1}(1+\beta )\hfill \\ & & +\frac{3a_{\mathcal{K}}{\widehat{\mathcal{W}}}^2{\varrho }_1^{2\mathcal{K}-1}{\psi }^{\prime }({\varrho }_1)}{(2\mu -1){(\Gamma (\mu ))}^2}{(\psi (\varrho )-\psi (p))}^{2\mu -1}{ϵ}_{{\Phi }_2}.\hfill \end{array}$$

If $\varrho \in ({\varrho }_z,m_z],$$z=1,2,\dots ,\delta$, then we obtain

$$\begin{array}{ccc}\hfill \beta & <& \mathbb{E}\parallel (\mathfrak{G}{\omega }^{\beta })(\varrho ){\parallel }^2\hfill \\ & \le & 3\mathbb{E}\parallel {\mathcal{F}}_z(z,{\omega }^{\beta }({\varrho }_z^{-})){\parallel }^2\hfill \\ & \le & {\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}(1+\beta ).\hfill \end{array}$$

Similarly, if $\varrho \in (m_z,{\varrho }_{z+1}],z=1,2,\dots ,\delta$, then we obtain

$$\begin{array}{ccc}\hfill \beta & <& \mathbb{E}\parallel (\mathfrak{G}{\omega }^{\beta })(\varrho ){\parallel }^2\hfill \\ & \le & 3\mathbb{E}\parallel {\mathcal{T}}_{\psi }^{\mu }(\varrho ,m_z){\mathcal{F}}_z(m_z,{\omega }^{\beta }({\varrho }_z^{-})){\parallel }^2\hfill \\ & & +3\mathbb{E}\parallel {\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p){\Phi }_1(\varrho ,{\omega }^{\beta }(p)){\parallel }^2\hfill \\ & & +3\mathbb{E}\parallel {\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{E}}}^{\mathcal{H}}{(p)\parallel }^2\hfill \\ & \le & 3{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}(1+\beta )+\frac{3{\widehat{\mathcal{W}}}^2{\varrho }_{Z+1}{\psi }^{\prime }({\varrho }_{z+1})}{{(\Gamma (\mu ))}^2}{\widehat{\mathcal{H}}}_{{\Phi }_1}(1+\beta )\hfill \\ & & (\times ){\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{2\mu -2}{\psi }^{\prime }(p)dp\hfill \\ & \le & 3{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{K}}}_{{\mathcal{F}}_z}(1+\beta )+\frac{3\widehat{\mathcal{W}}{\varrho }_1{\psi }^{\prime }({\varrho }_1)}{(2\mu -1){(\Gamma (\mu ))}^2}{(\psi (\varrho )-\psi (p))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Phi }_1}(1+\beta )\hfill \\ & & +\frac{3a_{\mathcal{K}}{\widehat{\mathcal{W}}}^2{\varrho }_1^{2\mathcal{K}-1}{\psi }^{\prime }({\varrho }_1)}{(2\mu -1){(\Gamma (\mu ))}^2}{(\psi (\varrho )-\psi (p))}^{2\mu -1}{ϵ}_{{\Phi }_2}{\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{2\mu -2}{\psi }^{\prime }(p)dp\hfill \end{array}$$

$$\begin{array}{ccc}& \le & 3{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}(1+\beta )+\frac{3{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{r+1})}{(2\mu -1){(\Gamma (\mu ))}^2}{(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Phi }_1}(1+\beta )\hfill \\ & & +\frac{3a_{\mathcal{K}}{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}^{2\mathcal{K}-1}{\psi }^{\prime }({\varrho }_{z+1})}{(2\mu -1){(\Gamma (\mu ))}^2}{(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}{ϵ}_{{\Phi }_2}.\hfill \end{array}$$

For any $\varrho \in \mathcal{J}$, we obtain

$$\begin{array}{cc}\hfill \beta & <\mathbb{E}\parallel (\mathfrak{G}{\omega }^{\beta })(\varrho ){\parallel }^2\hfill \\ \hfill & \le {\mathcal{M}}^{*}+{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}\beta +3{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}\beta +\frac{3{\widehat{\mathcal{W}}}^{2}d{\psi }^{\prime }(d)}{(2\mu -1){(\Gamma (\mu ))}^2}{(\psi (d)-\psi (0))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Phi }_1}\beta .\hfill \end{array}$$

(8)

where

$$\begin{array}{ccc}\hfill {\mathcal{M}}^{*}& =& \underset{1\le z\le \delta }{max}\{3{\widehat{\mathcal{W}}}^2\mathbb{E}{\mathcal{K}}_{{\Phi }_3}\frac{{\hslash }^{\varrho }}{\Gamma (\varrho +1)}{\parallel \omega (\varsigma )-\overline{\omega }(\varsigma )\parallel }^2+{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}+3{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}\hfill \\ & & +\frac{3{\widehat{\mathcal{W}}}^{2}d{\psi }^{\prime }(d)}{(2\mu -1){(\Gamma (\mu ))}^2}{(\psi (d)-\psi (0))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Phi }_1}\hfill \\ & & ++\frac{3a_{\mathcal{K}}{\widehat{\mathcal{W}}}^2{d}^{2\mathcal{K}-1}{\psi }^{\prime }(d)}{(2\mu -1){(\Gamma (\mu ))}^2}{(\psi (d)-\psi (0))}^{2\mu -1}{ϵ}_{{\Phi }_2}\}.\hfill \end{array}$$

So, ${\mathcal{M}}^{*}$ is independent of $\beta$. By dividing both sides of Equation (8) by $\beta$ and taking $\beta \to \infty$, we obtain

$$\begin{array}{ccc}\hfill 1& <& {\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}+3{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}+\frac{3{\widehat{\mathcal{W}}}^{2}d{\psi }^{\prime }(d)}{(2\mu -1){(\Gamma (\mu ))}^2}{(\psi (d)-\psi (0))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Phi }_1}\hfill \end{array}$$

which contradicts [C4]. Therefore, we have $\mathcal{G}(H_{\beta })\subset H_{\beta }$ for some $\beta >0$.

Step 2

In this step, we prove that $\mathfrak{G}$ is a contraction mapping on $H_{\beta }$.

$$\begin{array}{cc}\hfill \mathbb{E}\parallel (\mathfrak{G}{\omega }_1(\varrho )-\mathfrak{G}{\omega }_2(\varrho ){\parallel }^2\le & \mathbb{E}\parallel {\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)\hfill \\ \hfill & \times [{\Phi }_1(p,{\omega }_1(p))-{\Phi }_1(p,{\omega }_2(p))]{\psi }^{\prime }(p)dp{\parallel }^2\hfill \\ \hfill \le & \frac{{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1)}{{(\Gamma (\mu ))}^2}{\widehat{ϵ}}_{{\Phi }_1}{\parallel {\omega }_1-{\omega }_2\parallel }_{\mathcal{Q}\mathcal{C}}^2\hfill \\ \hfill \times & {\int }_0^{\varrho }{(\psi (\varrho )-\psi (\varrho ))}^{2\mu -2}{\psi }^{\prime }(p)dp\hfill \\ \hfill \le & \frac{{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1)}{{(\Gamma (\mu ))}^2}{\widehat{ϵ}}_{{\Phi }_1}{(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1}{\parallel {\omega }_1-{\omega }_2\parallel }_{\mathcal{Q}\mathcal{C}}^2.\hfill \end{array}$$

(9)

If $\varrho \in ({\varrho }_z,m_z],$$z=1,2,\dots ,\delta$, then we obtain

$$\begin{array}{cc}\hfill \mathbb{E}\parallel (\mathfrak{G}{\omega }_1(\varrho )-\mathfrak{G}{\omega }_2(\varrho ){\parallel }^2& \le \mathbb{E}\parallel {\mathcal{F}}_z(\varrho ,{\omega }_1({\varrho }_z^{-}-{\mathcal{F}}_z(\varrho ,{\omega }_2{({\varrho }_z^{-}\parallel }^2\hfill \\ \hfill & \le {\widehat{ϵ}}_{{\mathcal{F}}_z}{\parallel {\omega }_1-{\omega }_2\parallel }_{\mathcal{Q}\mathcal{C}}^2.\hfill \end{array}$$

(10)

Similarly, if $\varrho \in (m_z,{\varrho }_{z+1}],$$z=1,2,\dots ,\delta$, then we obtain

$$\begin{array}{cc}\hfill \mathbb{E}\parallel (\mathfrak{G}{\omega }_1)(\rho )-& (\mathfrak{G}{\omega }_2)(\rho ){\parallel }^2\le 2{\widehat{\mathcal{W}}}^2\mathbb{E}{\parallel {\mathcal{F}}_z(m_z,{\omega }_1({\varrho }_z^{-}))-{\mathcal{F}}_z(m_z,{\omega }_2({\varrho }_z^{-}))\parallel }^2\hfill \\ \hfill & +2\mathbb{E}\parallel {\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (\varrho ))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)\hfill \\ \hfill & \times [{\Phi }_1(p,{\omega }_1(p))-{\Phi }_2(p,{\omega }_2(p))]{\psi }^{\prime }(p)dp\parallel \hfill \\ \hfill \le & 2{\widehat{\mathcal{W}}}^2{\widehat{ϵ}}_{{\mathcal{F}}_z}{\parallel {\omega }_1-{\omega }_2\parallel }_{\mathcal{Q}\mathcal{C}}^2\hfill \\ \hfill & +\frac{2{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1})}{{(\Gamma (\mu ))}^2}{\widehat{ϵ}}_{{\mathcal{F}}_z}{\parallel {\omega }_1-{\omega }_2\parallel }_{\mathcal{Q}\mathcal{C}}^2\hfill \\ \hfill & \times {\int }_{m_z}^{\varrho }{(\psi (\varrho ))-\psi (p))}^{2\mu -2}{\psi }^{\prime }(p)dp\hfill \\ \hfill \le & \left(2{\widehat{\mathcal{W}}}^2{\widehat{ϵ}}_{{\mathcal{F}}_z}+\frac{2{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1})}{(2\mu -1){(\Gamma (\mu ))}^2}{\widehat{ϵ}}_{{\Phi }_1}(\psi {({\varrho }_{z+1}-\psi (m_z))}^{2\mu -1}\right)\hfill \\ \hfill & \times \parallel {\omega }_1-{\omega }_2{\parallel }_{\mathcal{Q}\mathcal{C}}^2\hfill \end{array}$$

(11)

From Equations (9)–(11), we obtain

$$\mathbb{E}\parallel (\mathfrak{G}{\omega }_1)(\rho )-(\mathfrak{G}{\omega }_2)(\rho ){\parallel }^2\le \widehat{O}{\parallel {\omega }_1-{\omega }_2\parallel }_{\mathcal{Q}\mathcal{C}}^2$$

where

$$\widehat{O}=\underset{1\le z\le delta}{max}[{\widehat{O}}_0,{\widehat{ϵ}}_{{\mathcal{F}}_z},{\widehat{O}}_z],$$

with

$${\widehat{O}}_0=\frac{\widehat{\mathcal{W}}{\varrho }_1{\psi }^{\prime }({\varrho }_1)}{(2\mu -1){(\Gamma (\mu ))}^2}{\widehat{ϵ}}_{{\Phi }_1}9\psi ({\varrho }_1)-\psi (0){)}^{2\mu -1},$$

$${\widehat{O}}_z=2{\widehat{\mathcal{W}}}^2{\widehat{ϵ}}_{{\mathcal{F}}_z}+\frac{2{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1})}{(2\mu -1){(\Gamma (\mu ))}^{2\mu -1}}{\widehat{ϵ}}_{{\Phi }_1}{(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}.$$

Hence,

$$\parallel \mathfrak{G}{\omega }_1-\mathfrak{G}{\omega }_2{\parallel }_{\mathcal{Q}\mathcal{C}}^2\le \widehat{O}{\parallel {\omega }_1-{\omega }_2\parallel }_{\mathcal{Q}\mathcal{C}}^2$$

From Equation (7), $\mathfrak{G}$ is shown to be a contraction mapping on $H_{\beta }$. Therefore, system (1) has a unique mild solution on $\mathcal{J}$.

□

Theorem 2.

If assumptions [C1]–[C5] hold, system (1) has a unique stable mild solution on $\mathcal{J}$.

Proof. 

From Theorem 1, we conclude that (1) possesses a unique mild solution $\omega (\varrho )$. Suppose that $\widehat{\omega }(\varrho )$ represents any mild solution of (1) with the conditions $\widehat{\omega }(0)={\widehat{\omega }}_0$, and $\widehat{\omega }(\varrho )={\mathcal{F}}_z(\varrho ,\widehat{\omega }({\varrho }_z^{-}))$, $\varrho \in ({\varrho }_z,m_z]$, $z=1,2,\dots ,\delta$. For $\varrho \in [0,{\varrho }_1]$, we have

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel \omega (\varrho )-\widehat{\omega }{(\varrho )\parallel }^2& \le & 2{\widehat{W}}^2\mathbb{E}{\parallel {\omega }_0-{\widehat{\omega }}_0\parallel }^2\hfill \\ & & +2\mathbb{E}\parallel {\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)\hfill \\ & & \times [{\Phi }_1(p,\omega (p))-{\Phi }_2(p,\widehat{\omega }(p))]{\psi }^{\prime }(p)dp{\parallel }^2\hfill \end{array}$$

$$\begin{array}{ccc}& \le & 2{\widehat{\mathcal{W}}}^2\mathbb{E}{\parallel {\omega }_0-{\widehat{\omega }}_0\parallel }^2+\frac{2{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1)}{{(\Gamma (\mu ))}^2}\hfill \\ & & \times {\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{2\mu -2}\mathbb{E}{\parallel {\Phi }_1(p,\omega (p))-{\Phi }_2(p,\widehat{\omega }(p))\parallel }^2{\psi }^{\prime }(p)dp\hfill \\ & \le & 2{\widehat{\mathcal{W}}}^2\mathbb{E}{\parallel {\omega }_0-{\widehat{\omega }}_0\parallel }^2\hfill \\ & & +\frac{2{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1){\widehat{ϵ}}_{{\Phi }_1}}{{(\Gamma (\mu ))}^2}{\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{2\mu -2}\mathbb{E}{\parallel \omega (q)-\widehat{\omega }(q)\parallel }^2{\psi }^{\prime }(p)dp\hfill \end{array}$$

For $\varrho \in ({\varrho }_z,m_z]$, $z=1,2,\dots ,\delta$, we have

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel \omega (\varrho )-\widehat{\omega }{(\varrho )\parallel }^2& \le & \mathbb{E}\parallel {\mathcal{F}}_z(\varrho ,\omega ({\varrho }_z^{-}))-{\mathcal{F}}_z(\varrho ,\widehat{\omega }({\varrho }_z^{-})){\parallel }^2\hfill \\ & =& {\widehat{ϵ}}_{{\mathcal{F}}_z}\mathbb{E}\parallel \omega ({\varrho }_z^{-}-\widehat{\omega }({\varrho }_z^{-}){\parallel }^2.\hfill \end{array}$$

For $\varrho \in (m_z,{\varrho }_{z+1}]$, $z=1,2,\dots ,\delta$, we have

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel \omega (\varrho )-\widehat{\omega }{(\varrho )\parallel }^2& \le & 2{\widehat{W}}^2\mathbb{E}{\parallel {\mathcal{F}}_z(m_z,\omega ({\varrho }_z^{-}))-{\mathcal{F}}_z(m_z,\widehat{\omega }({\varrho }_z^{-}))\parallel }^2\hfill \\ & & +2\mathbb{E}\parallel {\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)\hfill \\ & & \times [{\Phi }_1(p,\omega (p))-{\Phi }_2(p,\widehat{\omega }(p))]{\psi }^{\prime }(p)dp{\parallel }^2\hfill \\ & \le & 2{\widehat{\mathcal{W}}}^2{\widehat{ϵ}}_{{\mathcal{F}}_z}\mathbb{E}{\parallel \omega ({\varrho }_z^{-})-\widehat{\omega }({\varrho }_z^{-})\parallel }^2+\frac{2{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1})}{{(\Gamma (\mu ))}^2}\hfill \\ & & \times {\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{2\mu -2}\mathbb{E}{\parallel {\Phi }_1(p,\omega (p))-{\Phi }_2(p,\widehat{\omega }(p))\parallel }^2{\psi }^{\prime }(p)dp\hfill \\ & \le & 2{\widehat{\mathcal{W}}}^2{\widehat{ϵ}}_{{\mathcal{F}}_z}\mathbb{E}{\parallel \omega ({\varrho }_z^{-})-\widehat{\omega }({\varrho }_z^{-})\parallel }^2+\frac{2{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1}){\widehat{ϵ}}_{{\Phi }_1}}{{(\Gamma (\mu ))}^2}\hfill \\ & & \times {\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{2\mu -2}\mathbb{E}{\parallel \omega (q)-\widehat{\omega }(q)\parallel }^2{\psi }^{\prime }(p)dp.\hfill \end{array}$$

For $\varrho \in \mathcal{J}$, we have

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel \omega (\varrho )-\widehat{\omega }{(\varrho )\parallel }^2& \le & 2{\widehat{W}}^2\mho +\sum _{z=1}^{\delta }({\widehat{ϵ}}_{{\mathcal{F}}_z}+2{\widehat{\mathcal{W}}}^2{\widehat{ϵ}}_{{\mathcal{F}}_z})\mathbb{E}\parallel \omega ({\varrho }_z^{-})-\widehat{\omega }({\varrho }_z^{-})\parallel \hfill \\ & & +\frac{2{\widehat{\mathcal{W}}}^{2}d{\psi }^{\prime }(d){\widehat{ϵ}}_{{\Phi }_1}}{{(\Gamma (\mu ))}^2}{\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{2\mu -2}\mathbb{E}{\parallel \omega (q)-\widehat{\omega }(q)\parallel }^2{\psi }^{\prime }(p)dp.\hfill \end{array}$$

Using Lemma (6), we obtain

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel \omega (\varrho )-\widehat{\omega }{(\varrho )\parallel }^2& \le & 2{\widehat{W}}^2\mho \left[\prod _{i=1}^z\right(1+{\widehat{ϵ}}_{{\mathcal{F}}_i}+2\widehat{\mathcal{W}}{\widehat{\mathcal{W}}}^2{\widehat{ϵ}}_{{\mathcal{F}}_i}\hfill \\ & & \times E_{\mu }\left(\Re \Gamma (\mu )(\psi {({\varrho }_i-\psi (0))}^{2\mu -1}\right)\left)\right]{ϵ}_0\hfill \\ & \le & {ϵ}_{\mho },\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill ϵ& =& 2{\widehat{\mathcal{W}}}^2\mho \left[\prod _{i=1}^z(1+({\widehat{ϵ}}_{{\mathcal{F}}_i}+2{\widehat{\mathcal{W}}}^2{\widehat{ϵ}}_{{\mathcal{F}}_i}{E}_{\mu }(\Re \Gamma (\mu )(\psi {({\varrho }_i-\psi (0))}^{2\mu -1}))\right]{ϵ}_0,\hfill \end{array}$$

$\Re =\frac{2{\widehat{\mathcal{W}}}^{2}d{\psi }^{\prime }{\widehat{ϵ}}_{{\Phi }_1}}{{(\Gamma (\mu ))}^2}$, ${ϵ}_0=E_{\mu }(\Re \Gamma (\mu ){(\psi (\varrho )-\psi (0))}^{2\mu -1}).$

Next, we can select a $\mho >0$ such that $\mho <\frac{\epsilon }{ϵ}$; then,

$$\mathbb{E}\parallel \omega (\varrho )-\widehat{\omega }{(\varrho )\parallel }^2<\epsilon .$$

Therefore, system (1) assures a unique stable mild solution on $\mathcal{J}$. □

4. Controllability Results

This section focuses on determining the setting in which a controllability analysis of stochastic fractional DSs driven by the Rosenblatt process under integral boundary conditions with impulses can be achieved taking into account the combined effects of the boundary conditions, fractional derivatives, impulsive effects, and stochastic noise. Consider the following assumptions:

C6

The linear operator ${\Delta }_{m_z}^{{\varrho }_{z+1}}:L^2((m_z,{\varrho }_{z+1}],\mathcal{X})\to \mathcal{M}$, $z=0,1,2,\dots ,\delta$, defined by

$$\begin{array}{ccc}\hfill {\Delta }_{m_z}^{{\varrho }_{z+1}}\mu & =& {\int }_{m_z}^{{\varrho }_{z+1}}{(\psi (\varrho -z+1)-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }({\varrho }_{z+1},p)\aleph \mu (p){\psi }^{\prime }(p)dp,\hfill \end{array}$$

has bounded invertible operators ${({\Delta }_{m_z}^{{\varrho }_{z+1}})}^{-1}$ which transfer the values in $\frac{L^2((m_z,{\varrho }_{z+1}],\mathcal{X})}{Ker({\Delta }_{m_z}^{{\varrho }_{z+1}})}$, and ${\Delta }_z>0$, $z=1,2,\dots ,\delta$ exist such that $\aleph {({\Delta }_{m_z}^{{\varrho }_{z+1}})}^{-1}\parallel \le {\Delta }_z.$

C7

The following inequalities hold: ${max}_{1\le z\le \delta }\left[O_{{\mathfrak{H}}_0},{\widehat{ϵ}}_{{\mathcal{F}}_z},O_{{\mathfrak{H}}_z}\right]<1,$ where

$$O_{{\mathfrak{H}}_0}=\left(1+\frac{{\Delta }_0^2{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1){(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\right)\frac{{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1){\widehat{ϵ}}_{{\Phi }_1}{(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2},$$

$$\begin{array}{ccc}\hfill O_{{\mathfrak{H}}_z}& =& \left(1+\frac{2{\Delta }_z^2{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_1){(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\right)\hfill \\ & & \times \left(3\widehat{\mathcal{W}}{\widehat{ϵ}}_{{\mathcal{F}}_z}+\frac{3{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1}){\widehat{ϵ}}_{{\Phi }_1}{(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\right).\hfill \end{array}$$

For simplicity, we assume

$b_z=\frac{{\Delta }_z^2{\widehat{\mathcal{W}}}^{2}d{\psi }^{\prime }(d){(\psi (d)-\psi (0))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}$, and $e_d=\frac{{\widehat{\mathcal{W}}}^{2}d{\psi }^{\prime }}{(2\mu -1){(\Gamma (\mu ))}^2}.$

Definition 5.

System (2) is said to be controllable on $\mathcal{J}$ if for every ${\omega }_1\in \mathcal{M}$, there exists suitable control $\mu \in L^2(\mathcal{J},\mathcal{X})$ such that the mild solution of system (2) satisfies the following condition:

$$\omega (d)={\omega }_1,$$

where ${\omega }_1$ and d are the terminal state and preassigned time, respectively.

Definition 6.

A ${\mathcal{G}}_{\varrho }$-adapted stochastic process $\omega :\mathcal{J}\to \mathcal{M}$ is said to be a mild solution of system (2) if for any $\varrho \in \mathcal{J}$, $\omega (\varrho )$ satisfies $\omega (0)={\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma ,$ and $\omega (\varrho )={\mathcal{F}}_z(\varrho ,\omega ({\varrho }_z^{-}))$, $(\varrho \in 9{\varrho }_z,m_z]$, $z=1,2,\dots \delta ,$

$$\begin{array}{ccc}\hfill \omega (\varrho )& =& {\mathcal{T}}_{\psi }^{\mu }(\varrho ,0){\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma \hfill \\ & & +{\int }_0^{\varrho }{(\psi (\varrho )-\psi (\varrho ))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)[\aleph \mu (p)+{\Phi }_1(p,\omega (p))]{\psi }^{\prime }(p)dp\hfill \\ & & +{\int }_0^{\varrho }{(\psi (\varrho )-\psi (\varrho ))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{Z}}}^{\mathcal{K}}(p),\hfill \end{array}$$

for all $\varrho \in [0,{\varrho }_1]$, $z=0$, and

$$\begin{array}{cc}\hfill \omega (\varrho )& ={\mathcal{T}}_{\psi }^{\mu }(\varrho ,m_z){\mathcal{F}}_z(m_z,\omega ({\varrho }_z^{-}))\hfill \\ \hfill & +{\int }_0^{\varrho }{(\psi (\varrho )-\psi (\varrho ))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)[\aleph \mu (p)+{\Phi }_1(p,\omega (p))]{\psi }^{\prime }(p)dp\hfill \\ \hfill & +{\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (\varrho ))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{Z}}}^{\mathcal{K}}(p)\hfill \end{array}$$

(12)

for all $\varrho \in (m_z,{\varrho }_{z+1}]$, $z=1,2,\dots ,\delta$.

Let the control function ${\mu }_{\omega }(\varrho )$ be described by

$$\begin{array}{cc}\hfill {\mu }_{\omega }(\varrho )=& {\Delta }_{m_z}^{{\varrho }_{z+1}}[\omega ({\varrho }_{z+1})-{\mathcal{T}}_{\psi }^{\mu }({\varrho }_{z+1},m_z){\mathcal{F}}_z(m_z,\omega ({\varrho }_z^{-}))\hfill \\ \hfill & -{\int }_{m_z}^{{\varrho }_{z+1}}{(\psi ({\varrho }_{z+1})-\psi (q))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }({\varrho }_{z+1},p)[{\Phi }_1(p,\omega (p))]{\psi }^{\prime }(p)dp\hfill \\ \hfill & -{\int }_{m_z}^{{\varrho }_{z+1}}{(\psi ({\varrho }_{z+1})-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }({\varrho }_{r+1},p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{Z}}}^{\mathcal{K}}(p)],\hfill \end{array}$$

(13)

for all $\varrho \in (m_z,{\varrho }_{z+1}]$, $z=1,2,\dots ,\delta$, where ${\mathcal{F}}_0(0,·)={\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma$.

Theorem 3.

If assumptions [C1]–[C3] and [C5]–[C7] hold, then system (2) is controllable on $\mathcal{J}$ if

$$\underset{1\le z\le \delta }{max}\left[{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}+(1+4b_z)\left(4{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}+4e_d{(\psi (d)-\psi (0))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Phi }_1}\right)\right]<1.$$

(14)

Proof. 

For $\nu >0$, define

$${\Re }_{\nu }=\left\{\omega \in {\mathcal{Q}\mathcal{C}(\mathcal{M}):\parallel \omega \parallel }_{\mathcal{Q}\mathcal{C}}^2\le \nu \right\}.$$

Obviously, ${\Re }_{\nu }$ is a closed and bounded subset of $\mathcal{Q}\mathcal{C}(\mathcal{M})$. Define $\mathfrak{H}$ on ${\Re }_{\nu }$ as follows:

$$(\mathfrak{H}\omega )(\varrho )=\left\{\begin{array}{c}{\mathcal{T}}_{\psi }^{\mu }(\varrho ,0){\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma \hfill \\ +{\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p)[\aleph {\mu }_{\omega }(p)+{\Phi }_1(p,\omega (p))]{\psi }^{\prime }(p)dp\hfill \\ +{\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{E}}}^{\mathcal{H}}(p),\hfill \\ \varrho \in [0,{\varrho }_z],z=0\hfill \\ {\mathcal{F}}_z(\varrho ,\omega ({\varrho }_z^{-})),\varrho \in ({\varrho }_z,m_z],z\ge 1.\hfill \\ {\mathcal{T}}_{\psi }^{\mu }(\varrho ,m_z){\mathcal{F}}_z(m_z,\omega ({\varrho }_z^{-}))\hfill \\ +{\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)[\aleph {\mu }_{\omega }(p)+{\Phi }_1(p,\omega (p)){\psi }^{\prime }(p)dp\hfill \\ +{\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{E}}}^{\mathcal{H}}(p),\hfill \\ \varrho \in (m_z,{\varrho }_{z+1}],z\ge 1.\hfill \end{array}\right.$$

Step 1

$\nu >0$ exists such that $\mathfrak{H}({\Re }_{\nu })\subset {\Re }_{\nu }.$ If we assume this assumption is not true, then for $\nu >0$, we take $\varrho \in \mathcal{J}$ and ${\omega }^{\nu }\in {\Re }_{\nu }$ such that $\mathbb{E}\parallel (\mathfrak{H}{\omega }^{\mu })(\varrho ){\parallel }^2>\nu .$ For $\varrho \in [0,{\varrho }_1]$, we obtain

$$\begin{array}{ccc}\hfill \nu & <& \mathbb{E}\parallel (\mathfrak{H}{\omega }^{\mu })(\varrho ){\parallel }^2\hfill \\ & \le & 4\mathbb{E}\parallel {\mathcal{T}}_{\psi }^{\mu }(\varrho ,0){\int }_0^{\hslash }\frac{{(\hslash -\varsigma )}^{\varrho }}{\Gamma (\varrho )}{\Phi }_3(\varsigma ,\omega (\varsigma ))d\varsigma {\parallel }^2\hfill \\ & & +4\mathbb{E}\parallel {\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p)[{\Phi }_1(p,{\omega }^{\nu }(p))]{\psi }^{\prime }(p)dp{\parallel }^2\hfill \\ & & +4\mathbb{E}\parallel {\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p)\aleph {\mu }_{{\omega }^{\nu }}(p){\psi }^{\prime }(p)dp{\parallel }^2\hfill \\ & & +4\mathbb{E}\parallel {\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(q,p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{E}}}^{\mathcal{H}}(p){\parallel }^2\hfill \\ & \le & \frac{16{\Delta }_0^2{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1){(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\mathbb{E}{\parallel {\omega }_{{\varrho }_1}\parallel }^2\hfill \\ & & +\left(1+\frac{4{\Delta }_0^2{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1){(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\right)\hfill \\ & & \times (4{\widehat{\mathcal{W}}}^2\mathbb{E}{\mathcal{K}}_{{\Phi }_3}\frac{{\hslash }^{\varrho }}{\Gamma (\varrho +1)}{\parallel \omega (\varsigma )-\overline{\omega }(\varsigma )\parallel }^2\hfill \\ & & +\frac{4{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1){(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Phi }_1}(1+\nu )}{(2\mu -1){(\Gamma (\mu ))}^2}\hfill \\ & & +\frac{4a_{\mathcal{K}}{\widehat{\mathcal{W}}}^2{\varrho }_1^{2\mathcal{K}-1}{\psi }^{\prime }({\varrho }_1){(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}{ϵ}_{{\Phi }_2}).\hfill \end{array}$$

If $\varrho \in ({\varrho }_z,m_z]$, $z=1,2,\dots \delta$, then we obtain

$$\begin{array}{cc}\hfill \nu <& \mathbb{E}\parallel (\mathfrak{H}{\omega }^{\nu })(\varrho ){\parallel }^2\hfill \\ \hfill & \le \mathbb{E}\parallel {\mathcal{F}}_z(\varrho ,{\omega }^{\nu }({\varrho }_z^{-})){\parallel }^2\hfill \\ \hfill & \le {\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}(1+\nu ).\hfill \end{array}$$

(15)

Similarly, if $\mu \in (m_z,{\varrho }_{z+1}]$, $z=1,2,\dots ,\delta$, then we obtain

$$\begin{array}{cc}\hfill \nu <& \mathbb{E}\parallel (\mathfrak{H}{\omega }^{\mu })(\varrho ){\parallel }^2\hfill \\ \hfill \le & 4\mathbb{E}\parallel {\mathcal{T}}_{\psi }^{\mu }(\varrho ,m_z){\mathcal{F}}_z(m_z,\omega ({\varrho }_z^{-})){\parallel }^2\hfill \\ \hfill & +4\mathbb{E}\parallel {\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p){\Phi }_1(p,{\omega }^{\nu }(p)){\psi }^{\prime }(p)dp{\parallel }^2\hfill \\ \hfill & +4\mathbb{E}\parallel {\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)\aleph {\mu }_{{\omega }^{\nu }}(p){\psi }^{\prime }(p)dp{\parallel }^2\hfill \\ \hfill & +4\mathbb{E}\parallel {\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p){\Phi }_2(p){\psi }^{\prime }(p)d{\widehat{\mathcal{E}}}^{\mathcal{H}}(p){\parallel }^2\hfill \\ \hfill \le & \frac{16{\Delta }_0^2{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1}){(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\mathbb{E}{\parallel {\omega }_{{\varrho }_{z+1}}\parallel }^2\hfill \\ \hfill & +\left(1+\frac{4{\Delta }_0^2{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1}){(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\right)(4{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}(1+\nu )\hfill \\ \hfill & +\frac{4{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1}){(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Phi }_1}(1+\nu )}{(2\mu -1){(\Gamma (\mu ))}^2}\hfill \\ \hfill & +\frac{4a_{\mathcal{K}}{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}^{2\mathcal{K}-1}{\psi }^{\prime }({\varrho }_{z+1}){(\psi ({\varrho }_{z+1})-\psi (m_r))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}{ϵ}_{{\Phi }_2}).\hfill \end{array}$$

For each $\varrho \in \mathcal{J}$, we obtain

$$\begin{array}{cc}\hfill \nu =& \mathbb{E}\parallel (\mathfrak{H}{\omega }^{\mu })(\varrho ){\parallel }^2\hfill \\ \hfill \le & {\mathcal{Y}}^{*}+{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}\nu +(1+4b_z)\left(4{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}\nu +4e_d{(\psi (d)-\psi (0))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Phi }_1}\nu \right),\hfill \end{array}$$

(16)

where

$$\begin{array}{ccc}\hfill {\mathcal{Y}}^{*}& =& \underset{1\le z\le \delta }{max}\{16b_0\mathbb{E}\parallel {\omega }_{{\varrho }_1}{\parallel }^2+16b_z\mathbb{E}{\parallel {\omega }_{{\varrho }_{z+1}}\parallel }^2+{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}\hfill \\ & & +(1+4b_z)(4{\widehat{\mathcal{W}}}^2\mathbb{E}{\mathcal{K}}_{{\Phi }_3}\frac{{\hslash }^{\varrho }}{\Gamma (\varrho +1)}{\parallel \omega (\varsigma )-\overline{\omega }(\varsigma )\parallel }^2\hfill \\ & & +4{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}+4e_d{(\psi (d)-\psi (0))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Phi }_1}\hfill \\ & & +\frac{4a_{\mathcal{K}}{\widehat{\mathcal{W}}}^2{d}^{2\mathcal{K}-1}{\psi }^{\prime }(d){(\psi (d)-\psi (0))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}{ϵ}_{{\Phi }_2}\left)\right\}.\hfill \end{array}$$

Here, ${\mathcal{Y}}^{*}$ does not contain the term $\nu$; taking as the limit $\nu \to \infty$ and dividing both sides of Equation (16) by $\nu$, we obtain

$$1<{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}+(1+4b_z)\left(4{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{H}}}_{{\mathcal{F}}_z}+4e_d{(\psi (d)-\psi (0))}^{2\mu -1}{\widehat{\mathcal{H}}}_{{\Phi }_1}\right).$$

This contradicts Equation (14). Therefore, $\mathfrak{H}({\Re }_{\nu })\subset {\Re }_{\nu }$, for some $\nu >0$.

Step 2

In this step, we prove that $\mathfrak{H}$ is a contraction mapping on ${\Re }_{\nu }$. For all ${\omega }_1,{\omega }_2\in {\Re }_{\nu }$, if $\varrho \in [0,{\varrho }_1]$, then we obtain

$$\begin{array}{cc}\hfill \mathbb{E}\parallel (\mathfrak{H}{\omega }_1)(\varrho )-(\mathfrak{H}{\omega }_2)(\varrho ){\parallel }^2\le & 2\mathbb{E}\parallel {\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)[{\Phi }_1(p,{\omega }_1(p))\hfill \\ \hfill & -{\Phi }_2(p,{\omega }_2(p))]{\psi }^{\prime }(p)dP{\parallel }^2\hfill \\ \hfill & +2\mathbb{E}\parallel {\int }_0^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)\aleph ({\mu }_{{\omega }_1}(p)\hfill \\ \hfill & -{\mu }_{{\omega }_2}(p)){\psi }^{\prime }(p)dp{\parallel }^2\hfill \\ \hfill & +\left(1+\frac{{\Delta }_0^2{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1){(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\right)\hfill \\ \hfill & \times 2{\widehat{\mathcal{W}}}^2{\widehat{ϵ}}_{{\Phi }_1}{(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1}{\parallel {\omega }_1-{\omega }_2\parallel }_{\mathcal{Q}\mathcal{C}}^2.\hfill \end{array}$$

(17)

If $\varrho \in ({\varrho }_z,m_z]$, $z=1,2,\dots ,\delta$, then we obtain

$$\begin{array}{cc}\hfill \mathbb{E}\parallel (\mathfrak{H}{\omega }_1)(\varrho )-(\mathfrak{H}{\omega }_2)(\varrho ){\parallel }^2\le & \mathbb{E}\parallel {\mathcal{F}}_z(\varrho ,{\omega }_1({\varrho }_z^{-}))-{\mathcal{F}}_z(\varrho ,{\omega }_2)({\varrho }_z^{-}){)\parallel }^2\hfill \\ \hfill \le & {\widehat{ϵ}}_{{\mathcal{F}}_z}{\parallel {\omega }_1-{\omega }_2\parallel }_{\mathcal{Q}\mathcal{C}}^2\hfill \end{array}$$

(18)

Similarly, if $\varrho \in (m_z,{\varrho }_{z+1}]$, $z=1,2,\dots ,\delta$, then we obtain

$$\begin{array}{cc}\hfill \mathbb{E}\parallel (\mathfrak{H}{\omega }_1)(\varrho )-& (\mathfrak{H}{\omega }_2)(\varrho ){\parallel }^2\le 3{\widehat{\mathcal{W}}}^2\mathbb{E}{\parallel {\mathcal{F}}_z(m_z,{\omega }_1({\varrho }_z^{-}))-{\mathcal{F}}_z(\varrho ,{\omega }_2)({\varrho }_z^{-})\parallel }^2\hfill \\ \hfill & +3\mathbb{E}\parallel {\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)[{\Phi }_1(p,{\omega }^{\nu }(p))\hfill \\ \hfill & -{\Phi }_2(p,{\omega }^{\nu }(p))]{\psi }^{\prime }(p)dp{\parallel }^2\hfill \\ \hfill & +3\mathbb{E}\parallel {\int }_{m_z}^{\varrho }{(\psi (\varrho )-\psi (p))}^{\mu -1}{\mathcal{S}}_{\psi }^{\mu }(\varrho ,p)\aleph ({\nu }_{{\omega }_1}(p)\hfill \\ \hfill & -{\nu }_{{\omega }_2}(p)){\psi }^{\prime }(p)dp{\parallel }^2\hfill \\ \hfill \le & \left(1+\frac{2{\Delta }_z^2{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1}){(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\right)\hfill \\ \hfill & \times \left(3{\widehat{\mathcal{W}}}^2{\widehat{ϵ}}_{{\mathcal{F}}_z}+\frac{3{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1}){\widehat{ϵ}}_{{\Phi }_1}{(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\right)\hfill \\ \hfill & \times \parallel {\omega }_1-{\omega }_2{\parallel }_{\mathcal{Q}\mathcal{C}}^2.\hfill \end{array}$$

(19)

Using Equations (17)–(19), we find

$$\mathbb{E}\parallel (\mathfrak{H}{\omega }_1)(\varrho )-(\mathfrak{H}{\omega }_2)(\varrho ){\parallel }^2\le O_{\mathfrak{H}}{\parallel {\omega }_1-{\omega }_2\parallel }_{\mathcal{Q}\mathcal{C}}^2,$$

where $O_{\mathfrak{H}}={max}_{1\le z\le \delta }\left[O_{{\mathfrak{H}}_0},{\widehat{ϵ}}_{{\mathcal{F}}_z},O_{{\mathfrak{H}}_z}\right],$ with

$$O_{{\mathfrak{H}}_0}=\left(1+\frac{{\Delta }_0^2{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1){(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\right)\frac{{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1){\widehat{ϵ}}_{{\Phi }_1}{(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2},$$

$$\begin{array}{ccc}\hfill O_{{\mathfrak{H}}_z}& =& \left(1+\frac{2{\Delta }_z^2{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_1){(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\right)\hfill \\ & & \times \left(3\widehat{\mathcal{W}}{\widehat{ϵ}}_{{\mathcal{F}}_z}+\frac{3{\widehat{\mathcal{W}}}^2{\varrho }_{z+1}{\psi }^{\prime }({\varrho }_{z+1}){\widehat{ϵ}}_{{\Phi }_1}{(\psi ({\varrho }_{z+1})-\psi (m_z))}^{2\mu -1}}{(2\mu -1){(\Gamma (\mu ))}^2}\right).\hfill \end{array}$$

Therefore,

$$\parallel \mathfrak{H}{\omega }_1-\mathfrak{H}({\omega }_2){\parallel }_{\mathcal{Q}\mathcal{C}}^2\le O_{\mathfrak{H}}{\parallel {\omega }_1-{\omega }_2\parallel }_{\mathcal{Q}\mathcal{C}}^2$$

Using [C6], we find that $\mathfrak{H}$ is a contraction mapping on ${\Re }_{\nu }$. Therefore, system (2) has a unique mild solution. So, (2) is controllable on $\mathcal{J}$.

□

5. Examples

Example 1.

Assume the ψ-Caputo fractional-order stochastic differential system of the following form:

$$\begin{array}{cc}\hfill C_{D_{\psi }^{\mu }}\omega (\varrho ,\mho )=& {\omega }_{\mho \mho }(\varrho ,\mho )+\frac{\sqrt{\varrho }{e}^{-\varrho }\omega (\varrho ,\mho )}{5(1+|\omega (\varrho ,\mho )|)}+e^{-\varrho }\frac{d{\widehat{\mathcal{E}}}^{\mathcal{K}}(\varrho )}{d\varrho },\hfill \\ \hfill & \varrho \in (0,0.30]\bigcup (0.60,1],\mho \in [0,\pi ],\hfill \\ \hfill \omega (\varrho ,\mho )& =\frac{1}{3}(sin\varrho )\omega (0.30,\mho ),\varrho \in (0.30,0.60],\mho \in [0,\pi ],\hfill \\ \hfill \omega (\varrho ,0)& =0=\omega (\varrho ,\pi ),\hfill \\ \hfill \omega (\varrho ,\mho )=& {\int }_0^1\frac{{(\hslash -\varsigma )}^{\varrho -1}}{\Gamma (\varrho )}\frac{1}{9}{e}^{-\omega (\varsigma )}d\varsigma ,\hfill \end{array}$$

(20)

where $\mu =\frac{2}{3}$ and $0=m_0={\varrho }_0<{\varrho }_1<m_1<{\varrho }_2=d$, with ${\varrho }_1=0.30$, $m_1=0.60$, and ${\varrho }_2=1.$ Let $\psi (\varrho )=\varrho$ and $\mathcal{M}=\mathcal{X}=L^2([0,\pi ])$. Define an operator $\mathcal{T}:D(\mathcal{T})\subseteq \mathcal{M}\to \mathcal{M}$ with $\mathcal{T}\varphi ={\varphi }^{″}$ and $D(\mathcal{T})=\{\varphi \in \mathcal{M}:\varphi ,{\varphi }^{\prime }$ are absolutely continuous and ${\varphi }^{″}\in \mathcal{M},\varphi (0)=0=\varphi (\pi )\}$. $\mathcal{T}$ has a discrete spectrum; the normalized eigenvectors $e_m(\mho )=\sqrt{\frac{2}{\pi }}sin(m\mho )$ associated with the eigenvalue are $m^2,m\in \mathbb{N}$; and $\mathcal{T}$ generates an analytic semigroup ${\{\mathcal{S}(\varrho )\}}_{\varrho }$ in $\mathcal{M}$, which is uniformly bounded and defined as

$$\mathcal{S}(\varrho )\omega =\sum _{m=1}^{\infty }{e}^{-m^2\varrho }{\langle \mho ,e_m\rangle }_m,\mho \in \mathcal{M},$$

with $\parallel \mathcal{S}(\varrho )\parallel \le e^{-\varrho }\forall \varrho \ge 0.$ Thus, we choose $\widehat{\mathcal{W}}=1$, which implies that ${sup}_{\varrho \in [0,\infty )}\parallel \mathcal{S}(\varrho )\parallel =1.$

Assume $\omega (\varrho )(\mho )=\omega (\varrho ,\mho )$ and the functions ${\Phi }_1,{\Phi }_2{\Phi }_3$, and ${\mathcal{F}}_1$ are defined as

$${\Phi }_{1}9\varrho ,\omega )(\mho )=\frac{\sqrt{\varrho }{e}^{-\varrho }\omega (\varrho ,\mho )}{5(1+|\omega (\varrho ,\mho )|)},$$

$${\mathcal{F}}_1(\varrho ,\omega ({\varrho }_1^{-}))(\mho )=\frac{1}{3}(sin\varrho )\omega (0.{30}^{-},\mho ),$$

${\Phi }_2=e^{-\varrho },$ and ${\Phi }_3=\frac{1}{9}{e}^{-\omega (\varsigma )}.$ Clearly,

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel {\Phi }_1(\varrho ,{\omega }_1)-{\Phi }_1(\varrho ,{\omega }_2){\parallel }^2& \le & \frac{\varrho e^{-2\varrho }}{25}\mathbb{E}{\parallel {\omega }_1-{\omega }_2\parallel }^2\hfill \\ & \le & \frac{1}{25}\mathbb{E}{\parallel {\omega }_1-{\omega }_2\parallel }^2,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \mathbb{E}\parallel {\mathcal{F}}_1(\varrho ,{\omega }_1({\varrho }_1^{-}))-{\mathcal{F}}_1(\varrho ,{\omega }_2({\varrho }_1^{-})){\parallel }^2& \le & \frac{1}{25}\mathbb{E}{\parallel {\omega }_1(0.{30}^{-},\mho )-{\omega }_2(0.{30}^{-},\mho )\parallel }^2.\hfill \end{array}$$

and

$$\mathbb{E}\parallel {\Phi }_1{(\varrho ,\omega )\parallel }^2\le \frac{\varrho e^{-2\varrho }}{25}{\mathbb{E}\parallel \omega \parallel }^2\le \frac{1}{25}{\parallel \omega \parallel }^2,$$

$$\mathbb{E}\parallel {\mathcal{F}}_1(\varrho ,\omega ({\varrho }_1^{-})){\parallel }^2\le \frac{1}{25}\mathbb{E}{\parallel \omega (0.{30}^{-},\mho )\parallel }^2.$$

Next, we obtain ${\widehat{\mathcal{H}}}_{{\Phi }_1}={\widehat{ϵ}}_{{\Phi }_1}=\frac{1}{25}$, ${\widehat{\mathcal{H}}}_{{\mathcal{F}}_1}={\widehat{ϵ}}_{{\mathcal{F}}_1}=\frac{1}{9}$, and

$${\widehat{O}}_0=\frac{{\widehat{\mathcal{W}}}^2{\varrho }_1{\psi }^{\prime }({\varrho }_1)}{(2\mu -1){(\Gamma (\mu ))}^2}{\widehat{ϵ}}_{{\Phi }_1}{(\psi ({\varrho }_1)-\psi (0))}^{2\mu -1}=0.037,$$

$${\widehat{O}}_1=2{\widehat{\mathcal{W}}}^2{\widehat{ϵ}}_{{\mathcal{F}}_1}+\frac{{\widehat{\mathcal{W}}}^2{\varrho }_2{\psi }^{\prime }({\varrho }_2)}{(2\mu -1){(\Gamma (\mu ))}^2}{\widehat{ϵ}}_{{\Phi }_1}{(\psi ({\varrho }_2)-\psi (m_1))}^{2\mu -1}=0.3479,$$

and

$$max\left[{\widehat{O}}_0,{\widehat{ϵ}}_{{\mathcal{F}}_1},{\widehat{O}}_1\right]=max[0.037,\frac{1}{9},0.3479]=0.3479<1,$$

$${\widehat{\mathcal{H}}}_{{\mathcal{F}}_1}+5{\widehat{\mathcal{W}}}^2{\widehat{\mathcal{H}}}_{{\mathcal{F}}_1}+\frac{5{\widehat{\mathcal{W}}}^{2}d{\psi }^{\prime }(d)}{(2\mu -1){(\Gamma (\mu ))}^2}{\widehat{\mathcal{H}}}_{{\Phi }_1}{(\psi (d)-\psi (0))}^{2\mu -1}=0.7054<1.$$

Hence, all of the assumptions of Theorem (2) are fulfilled. Therefore, system (1) has a unique stable mild solution on $\mathcal{J}=[0,1].$

Example 2.

Consider a ψ-Caputo fractional-order stochastic differential system of the following form:

$$\begin{array}{cc}\hfill C_{D_{\psi }^{\mu }}\omega (\varrho ,\mho )=& {\omega }_{\mho \mho }(\varrho ,\mho )+\mu (\varrho ,\mho )+\frac{\sqrt{\varrho }{e}^{-\varrho }\omega (\varrho ,\mho )}{5(1+|\omega (\varrho ,\mho )|)}\hfill \\ \hfill & +e^{-\varrho }\frac{d{\widehat{\mathcal{E}}}^{\mathcal{K}}(\varrho )}{d\varrho }\varrho \in (0,0.30]\bigcup (0.60,1],\mho \in [0,\pi ],\hfill \\ \hfill \omega (\varrho ,\mho )& =\frac{1}{3}(sin\varrho )\omega (0.30,\mho ),\varrho \in (0.30,0.60],\mho \in [0,\pi ],\hfill \\ \hfill \omega (\varrho ,0)& =0=\omega (\varrho ,\pi ),\hfill \\ \hfill \omega (\varrho ,\mho )=& {\int }_0^1\frac{{(\hslash -\varsigma )}^{\varrho -1}}{\Gamma (\varrho )}\frac{1}{9}{e}^{-\omega (\varsigma )}d\varsigma ,\hfill \end{array}$$

(21)

By utilizing the control function of type (13), we can easily verify that all the assumptions of Theorem (3) hold when the unknown parameters are chosen suitably. Therefore, system (21) is controllable on $\mathcal{J}=[0,1]$.

6. Conclusions

In this paper, the stability and controllability of non-instantaneous impulsive $\psi$-Caputo fractional stochastic differential systems under integral boundary conditions have been examined and influenced by the Rosenblatt process. To derive existence and stability results for the proposed system, we employed the concept of a piecewise continuous mild solution, the Banach fixed-point theorem, stochastic theory, and fractional calculus. Further, to obtain the controllability results, we used the concept of a new piecewise control function. In the end, two examples were provided to support the authenticity of our main results. In the future, the findings of this paper could be generalized to the stability and controllability of noninstantaneous impulsive $\psi$-Caputo fractional stochastic differential systems under integral boundary conditions, driven by the Rosenblatt process, via a measure of non-compactness. It would be interesting for researchers to investigate the sensitivity to the noise range and develop computational and numerical methods for approximating the results as a further avenue of investigation and an extension of the present study.