Well-Posedness and Hyers–Ulam Stability of Fractional Stochastic Delay Systems Governed by the Rosenblatt Process

Abstract

Under the effect of the Rosenblatt process, the well-posedness and Hyers–Ulam stability of nonlinear fractional stochastic delay systems are considered. First, depending on fixed-point theory, the existence and uniqueness of solutions are proven. Next, utilizing the delayed Mittag–Leffler matrix functions and Grönwall’s inequality, sufficient criteria for Hyers–Ulam stability are established. Ultimately, an example is presented to demonstrate the effectiveness of the obtained findings.

1. Introduction

Fractional stochastic delay differential systems (FSDDSs) and their applications have attracted an abundance of research attention owing to their efficient modeling in many scientific and engineering areas, such as diffusion processes, fluid dynamics, biology, physics, control theory, viscoelastic systems, and many more (see, for instance, [1,2,3,4,5,6,7,8,9,10,11]). Specifically, many new results on how to represent the solutions to time-delay systems were obtained in the novel study [12], and were applied to stability analysis and control problems, for example, representation of solutions [13,14], controllability analysis [15,16,17,18], stability analysis [14] and the references therein.

The Wiener–Ito multiple integral of order p is defined as

$$Z_H^p\left(\mathsf{\ell }\right)=a\left(H,p\right){\int }_{R^p}\left({\int }_0^{\mathsf{\ell }}\prod _{j=1}^p{\left(\varsigma -{\Im }_j\right)}_{+}^{-\left(\frac{1}{2}+\frac{1-H}{p}\right)}\mathrm{d}\varsigma \right)\mathrm{d}\mathcal{G}\left({\Im }_1\right)\dots \mathrm{d}\mathcal{G}\left({\Im }_p\right),$$

(1)

in terms of the standard Wiener process ${\left(\mathcal{G}\left(\Im \right)\right)}_{\Im \in \mathbb{R}}$, where E${\left(Z_H^p\left(1\right)\right)}^2=1$ and ${\Im }_{+}=max\left(\Im ,0\right)$ are the conditions under which $a\left(H,p\right)$ is a normalizing constant. The process ${\left(Z_H^p\left(\mathsf{\ell }\right)\right)}_{\mathsf{\ell }\ge 0}$, defined by (1), is called the Hermite process. The Hermite process is the fractional Brownian motion (fBm) with Hurst parameter $H\in \left(\frac{1}{2},1\right)$ for $p=1$, while it is not Gaussian for $p=2$. Additionally, the Hermite process denoted by (1) for $p=2$ is referred to as the Rosenblatt process. Most studies [19,20,21] involve fBm because of its self-similarity, long-range dependence, and more straightforward Gaussian calculus. However, fBm fails in the concrete situation of non-Gaussian, smoothed models. In this situation, the Rosenblatt process is applicable. Non-Gaussian processes like the Rosenblatt process have numerous intriguing characteristics such as stationarity of the increments, long-range dependence, and self-similarity; for more details, see [22,23,24,25,26,27,28]. As a result, studying a novel class of FSDDSs driven by the Rosenblatt process seems interesting.

On the other hand, studying the stability of FSDDE solutions is essential, and Hyers–Ulam stability (HUS) is a crucial topic. In 1940, Ulam [29] made the first proposal that functional equations are stable during a lecture at Wisconsin University. In 1941, Hyers [30] provided a solution to this problem, after which HUS was established. In addition to providing a solid theoretical foundation for the well-posedness and HUS of FSDDEs, the study of the HUS of FSDDEs also provides a solid theoretical foundation for the approximate solution to FSDDEs. When it is rather difficult to acquire the precise solution for the system with HUS, we may substitute an approximate solution for an accurate one, and HUS can, to a certain extent, ensure the dependability of the estimated solution. Recently, many researchers have examined the HUS of diverse kinds of FSDDEs, see [31,32,33,34] and the references therein.

However, as far as we know, the standard literature has not dealt with the well-posedness and HUS of nonlinear FSDDEs driven by the Rosenblatt process. Therefore, in this study, we try for the first time to analyze such a topic.

Our study focuses on determining the well-posedness and HUS of nonlinear FSDDEs driven by the Rosenblatt process, taking into account previous research.

$$\begin{array}{c}\left({}^{C}D_{0^{+}}^{\alpha }\aleph \right)\left(\mathsf{\ell }\right)+\mathbb{X}\aleph \left(\mathsf{\ell }-\zeta \right)=\hslash \left(\mathsf{\ell },\aleph \left(\mathsf{\ell }\right)\right)+\Delta \left(\mathsf{\ell },\aleph \left(\mathsf{\ell }\right)\right)\frac{\mathrm{d}{Z}_H\left(\mathsf{\ell }\right)}{\mathrm{d}\mathsf{\ell }},\mathsf{\ell }\in \mp :=\left[0,\varpi \right],\\ \aleph \left(\mathsf{\ell }\right)\equiv \psi \left(\mathsf{\ell }\right),{\aleph }^{\prime }\left(\mathsf{\ell }\right)\equiv {\psi }^{\prime }\left(\mathsf{\ell }\right),\mathsf{\ell }\in {\mp }_1:=\left[-\zeta ,0\right],\end{array}$$

(2)

where the so-called Caputo fractional derivative of order $\alpha \in \left(1,2\right]$ is denoted by ${}^{C}D_{0^{+}}^{\alpha }$ with the lower index zero, $\aleph \left(\mathsf{\ell }\right)\in {\mathbb{R}}^n$ represents the state vector, $\zeta >0$ denotes a delay, $\varpi >\left(m-1\right)\zeta$, $m=1,2,\cdots$, $\psi \in C^1\left(\left[-\zeta ,0\right],{\mathbb{R}}^n\right)$, $\mathbb{X}\in {\mathbb{R}}^{n\times n}$ is any matrix, and $\hslash \in C\left(\mp \times {\mathbb{R}}^n,{\mathbb{R}}^n\right)$ is a given function. In the separable Hilbert space ${\mathbb{R}}^n$, let $\aleph \left(·\right)$ have value, and let the norm be $∥·∥$ and the inner product be $⟨·,·⟩$ with parameter $H\in \left(\frac{1}{2},1\right)$. $Z_H\left(\mathsf{\ell }\right)$ is a Rosenblatt process on another real separable Hilbert space $\left(K,{∥·∥}_K,{⟨·,·⟩}_K\right)$. Furthermore, consider $\Delta \in C\left(\mp \times {\mathbb{R}}^n,L_2^0\right)$, where $L_2^0=L_2\left(Q^{\frac{1}{2}}K,{\mathbb{R}}^n\right)$.

The remainder of this paper is structured as follows: In Section 2, we present some notations and necessary preliminaries. In Section 3, by utilizing Krasnoselskii’s fixed-point theorem, some sufficient conditions are established for the existence and uniqueness of solutions to system (2). In Section 4, we prove the Hyers–Ulam stability of (2) via Grönwall’s inequality lemma approach. Finally, we give a numerical example to illustrate the effectiveness of the derived results.

2. Preliminaries

During the entire paper, consider $\left(\mathit{\Sigma },ð,\mathbb{P}\right)$ to represent the complete probability space with probability measure $\mathbb{P}$ on $\mathit{\Sigma }$ and a filtration $\left\{{ð}_{\mathsf{\ell }}|\mathsf{\ell }\in \mp \right\}$ produced by $\left\{Z_H\left(s\right)|s\in \left[0,\mathsf{\ell }\right]\right\}$. For some $1<\mu <\infty$, consider $L^{\mu }\left(\mathit{\Sigma },{ð}_{\varpi },{\mathbb{R}}^n\right)$ to represent the Hilbert space of all ${ð}_{\varpi }$-measurable $\mu$th-integrable variables with values in ${\mathbb{R}}^n$ with norm ${∥\aleph ∥}_{L^{\mu }}^{\mu }=\mathbf{E}{∥\aleph \left(\mathsf{\ell }\right)∥}^{\mu }$, where the expectation $\mathbf{E}$ is defined by $\mathbf{E}\aleph ={\int }_{\mathit{\Sigma }}\aleph \mathrm{d}\mathbb{P}$. Assume that $\mathcal{A}$, $\mathcal{B}$ are two Banach spaces, $Q\in L_b\left(\mathcal{A},\mathcal{A}\right)$ denotes a non-negative self-adjoint trace class operator on $\mathcal{A}$, and $L_b\left(\mathcal{A},\mathcal{B}\right)$ is the space of bounded linear operators from $\mathcal{A}$ to $\mathcal{B}$. Let $L_2^0=L_2\left(Q^{\frac{1}{2}}\mathcal{A},\mathcal{B}\right)$ be the space of all Q-Hilbert–Schmidt operators from $Q^{\frac{1}{2}}\mathcal{A}$ to $\mathcal{B}$, equipped with the norm

$${∥\mathit{\Xi }∥}_{L_2^0}^2={∥\mathit{\Xi }{Q}^{\frac{1}{2}}∥}^2=\mathrm{Tr}\left(\mathit{\Xi }Q{\mathit{\Xi }}^T\right).$$

Given a norm ${∥\mathit{\Xi }∥}_{\mathcal{E}}={\left({sup}_{\mathsf{\ell }\in \mp }\mathbf{E}{∥\mathit{\Xi }\left(\mathsf{\ell }\right)∥}^{\mu }\right)}^{1/\mu }$, let $\mathcal{E}:=C$($\left[-\zeta ,\varpi \right]$, $L^{\mu }\left(\mathit{\Sigma },{ð}_{\varpi },\mathbb{P},{\mathbb{R}}^n\right)$) be the Banach space of all $\mu$th-integrable and ${ð}_{\varpi }$-adapted processes $\mathit{\Xi }$. A norm $∥·∥$ on ${\mathbb{R}}^n$ can be represented by the matrix norm (column sum):

$$∥\mathbb{X}∥=max\left\{\sum _{i=1}^n\left|x_{i1}\right|,\sum _{i=1}^n\left|x_{i2}\right|,\dots ,\sum _{i=1}^n\left|x_{in}\right|\right\},$$

where $\mathbb{X}:{\mathbb{R}}^n⟶{\mathbb{R}}^n$. Furthermore, consider

$$\begin{array}{cc}\hfill & C^1\left(\mp ,L^{\mu }\left(\mathit{\Sigma },{ð}_{\varpi },\mathbb{P},{\mathbb{R}}^n\right)\right)\hfill \\ \hfill & =\left\{\aleph \in C\left(\mp ,L^{\mu }\left(\mathit{\Sigma },{ð}_{\varpi },\mathbb{P},{\mathbb{R}}^n\right)\right):{\aleph }^{\prime }\in C\left(\mp ,L^{\mu }\left(\mathit{\Sigma },{ð}_{\varpi },\mathbb{P},{\mathbb{R}}^n\right)\right)\right\}.\hfill \end{array}$$

Finally, we assume the initial values

$${∥\psi ∥}_C={\left(\underset{s\in {\mp }_1}{sup}\mathbf{E}{∥\psi \left(s\right)∥}^{\mu }\right)}^{1/\mu }\mathrm{and}{∥{\psi }^{\prime }∥}_C={\left(\underset{s\in {\mp }_1}{sup}\mathbf{E}{∥{\psi }^{\prime }\left(s\right)∥}^{\mu }\right)}^{1/\mu }.$$

Some of the basic definitions and lemmas employed in this study are discussed.

Definition 1

([13]). The delayed Mittag–Leffler-type matrix functions ${\mathcal{H}}_{\zeta ,\alpha }\left(\mathbb{X}{\mathsf{\ell }}^{\alpha }\right)$, ${\mathcal{M}}_{\zeta ,\alpha }\left(\mathbb{X}{\mathsf{\ell }}^{\alpha }\right)$ and ${\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\mathsf{\ell }}^{\alpha }\right)$ are formulated, respectively, by

$${\mathcal{H}}_{\zeta ,\alpha }\left(\mathbb{X}{\mathsf{\ell }}^{\alpha }\right):=\left\{\begin{array}{c}\mathit{\Theta },-\infty <\mathsf{\ell }<-\zeta ,\hfill \\ \mathbb{I},-\zeta \le \mathsf{\ell }<0,\hfill \\ \mathbb{I}-\mathbb{X}\frac{{\mathsf{\ell }}^{\alpha }}{\mathit{\Gamma }\left(1+\alpha \right)},0\le \mathsf{\ell }<\zeta ,\hfill \\ ⋮⋮\hfill \\ \mathbb{I}-\mathbb{X}\frac{{\mathsf{\ell }}^{\alpha }}{\mathit{\Gamma }\left(1+\alpha \right)}+{\mathbb{X}}^2\frac{{\left(\mathsf{\ell }-\zeta \right)}^{2\alpha }}{\mathit{\Gamma }\left(1+2\alpha \right)}\hfill \\ +\cdots +{(-1)}^{\iota }{\mathbb{X}}^{\iota }\frac{{\left(\mathsf{\ell }-\left(\iota -1\right)\zeta \right)}^{\iota \alpha }}{\mathit{\Gamma }\left(1+\iota \alpha \right)},\left(\iota -1\right)\zeta \le \mathsf{\ell }<\iota \zeta ,\hfill \end{array}\right.$$

(3)

$${\mathcal{M}}_{\zeta ,\alpha }\left(\mathbb{X}{\mathsf{\ell }}^{\alpha }\right):=\left\{\begin{array}{c}\mathit{\Theta },-\infty <\mathsf{\ell }<-\zeta ,\hfill \\ \mathbb{I}\left(\mathsf{\ell }+\zeta \right),-\zeta \le \mathsf{\ell }<0,\hfill \\ \mathbb{I}\left(\mathsf{\ell }+\zeta \right)-\mathbb{X}\frac{{\mathsf{\ell }}^{\alpha +1}}{\mathit{\Gamma }\left(2+\alpha \right)},0\le \mathsf{\ell }<\zeta ,\hfill \\ ⋮⋮\hfill \\ \mathbb{I}\left(\mathsf{\ell }+\zeta \right)-\mathbb{X}\frac{{\mathsf{\ell }}^{\alpha +1}}{\mathit{\Gamma }\left(2+\alpha \right)}+{\mathbb{X}}^2\frac{{\left(\mathsf{\ell }-\zeta \right)}^{2\alpha +1}}{\mathit{\Gamma }\left(2+2\alpha \right)}\hfill \\ +\cdots +{(-1)}^{\iota }{\mathbb{X}}^{\iota }\frac{{\left(\mathsf{\ell }-\left(\iota -1\right)\zeta \right)}^{\iota \alpha +1}}{\mathit{\Gamma }\left(2+\iota \alpha \right)},\left(\iota -1\right)\zeta \le \mathsf{\ell }<\iota \zeta ,\hfill \end{array}\right.$$

(4)

and

$${\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\mathsf{\ell }}^{\alpha }\right):=\left\{\begin{array}{c}\mathit{\Theta },-\infty <\mathsf{\ell }<-\zeta ,\hfill \\ \mathbb{I}\frac{{\left(\mathsf{\ell }+\zeta \right)}^{\alpha -1}}{\mathit{\Gamma }\left(\alpha \right)},-\zeta \le \mathsf{\ell }<0,\hfill \\ \mathbb{I}\frac{{\left(\mathsf{\ell }+\zeta \right)}^{\alpha -1}}{\mathit{\Gamma }\left(\alpha \right)}-\mathbb{X}\frac{{\mathsf{\ell }}^{2\alpha -1}}{\mathit{\Gamma }\left(2\alpha \right)},0\le \mathsf{\ell }<\zeta ,\hfill \\ ⋮⋮\hfill \\ \mathbb{I}\frac{{\left(\mathsf{\ell }+\zeta \right)}^{\alpha -1}}{\mathit{\Gamma }\left(\alpha \right)}-\mathbb{X}\frac{{\mathsf{\ell }}^{2\alpha -1}}{\mathit{\Gamma }\left(2\alpha \right)}+{\mathbb{X}}^2\frac{{\left(\mathsf{\ell }-\zeta \right)}^{3\alpha -1}}{\mathit{\Gamma }\left(3\alpha \right)}\hfill \\ +\cdots +{(-1)}^{\iota }{\mathbb{X}}^{\iota }\frac{{\left(\mathsf{\ell }-\left(\iota -1\right)\zeta \right)}^{\alpha \left(\iota +1\right)-1}}{\mathit{\Gamma }\left(\alpha \left(\iota +1\right)\right)},\left(\iota -1\right)\zeta \le \mathsf{\ell }<\iota \zeta ,\hfill \end{array}\right.$$

(5)

where Γ is a gamma function, $\mathbb{I}$ denotes the identity matrix, Θ denotes the null matrix, and $\iota =0,1,2,\cdots$

Lemma 1

([13]). The solution of (2) can be expressed in the following form:

$$\begin{array}{cc}\hfill \aleph \left(\mathsf{\ell }\right)& ={\mathcal{H}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right)\psi \left(0\right)+{\mathcal{M}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{X}{\int }_{-\zeta }^0{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-2\zeta -\varsigma \right)}^{\alpha }\right)\psi \left(\varsigma \right)\mathrm{d}\varsigma \hfill \\ \hfill & +{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\hslash \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}\varsigma \hfill \\ \hfill & +{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right).\hfill \end{array}$$

Lemma 2

([28]). If $\sigma :\mp ⟶L_2^0$ satisfies

$${\int }_0^{\varpi }{∥\sigma \left(\varsigma \right)∥}_{L_2^0}^2\mathrm{d}\varsigma <\infty ,$$

then

$$\mathbf{E}{∥{\int }_0^{\mathsf{\ell }}\sigma \left(\varsigma \right)\mathrm{d}{Z}_H\left(\varsigma \right)∥}^2\le 2H{\mathsf{\ell }}^{2H-1}{\int }_0^{\mathsf{\ell }}{∥\sigma \left(\varsigma \right)∥}_{L_2^0}^2\mathrm{d}\varsigma .$$

Lemma 3

([35]). For $\sigma :\mp ⟶L_2^0$ such that

$${\int }_0^{\mathsf{\ell }}{∥\sigma \left(\varsigma \right)∥}_{L_2^0}^{\mu }\mathrm{d}\varsigma <\infty ,$$

after applying Hölder’s inequality and the Kahane–Khinchin inequality, there is a constant ${\tau }_{\mu }$ such that

$$\begin{array}{cc}\hfill \mathbf{E}{∥{\int }_0^{\mathsf{\ell }}\sigma \left(\varsigma \right)\mathrm{d}{Z}_H\left(\varsigma \right)∥}^{\mu }& \le {\tau }_{\mu }{\left\{\mathbf{E}{∥{\int }_0^{\mathsf{\ell }}\sigma \left(\varsigma \right)\mathrm{d}{Z}_H\left(\varsigma \right)∥}^2\right\}}^{\mu /2}\hfill \\ \hfill & \le {\tau }_{\mu }{\left\{2H{\mathsf{\ell }}^{2H-1}{\int }_0^{\mathsf{\ell }}{∥\sigma \left(\varsigma \right)∥}_{L_2^0}^2\mathrm{d}\varsigma \right\}}^{\mu /2}\hfill \\ \hfill & \le {\tau }_{\mu }{\left(2H{\mathsf{\ell }}^{2H-1}\right)}^{\mu /2}{\left({\int }_0^{\mathsf{\ell }}\mathrm{d}\varsigma \right)}^{\mu /2-1}{\int }_0^{\mathsf{\ell }}{\left({∥\sigma \left(\varsigma \right)∥}_{L_2^0}^2\right)}^{\mu /2}\mathrm{d}\varsigma \hfill \\ \hfill & ={\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\mathsf{\ell }}^{\mu H-1}{\int }_0^{\mathsf{\ell }}{∥\sigma \left(\varsigma \right)∥}_{L_2^0}^{\mu }\mathrm{d}\varsigma .\hfill \end{array}$$

Definition 2

([36]). The system (2) is Hyers–Ulam stable on $[0,\varpi ]$ if there exists a number $W>0$ such that for every $\kappa >0$ and every $\mathit{\Xi }\in C\left(\mp ,{\mathbb{R}}^n\right)$,

$$\mathbf{E}{∥\left({}^{C}D_{0^{+}}^{\alpha }\mathit{\Xi }\right)\left(\mathsf{\ell }\right)+\mathbb{X}\mathit{\Xi }\left(\mathsf{\ell }-\zeta \right)-\hslash \left(\mathsf{\ell },\mathit{\Xi }\left(\mathsf{\ell }\right)\right)-\Delta \left(\mathsf{\ell },\mathit{\Xi }\left(\mathsf{\ell }\right)\right)\mathrm{d}{Z}_H\left(\mathsf{\ell }\right)∥}^{\mu }\le \kappa ,\mathsf{\ell }\in \mp$$

(6)

which implies that there exists a solution $\aleph \in C\left(\mp ,{\mathbb{R}}^n\right)$ of (2) such that

$$\mathbf{E}{∥\mathit{\Xi }\left(\mathsf{\ell }\right)-\aleph \left(\mathsf{\ell }\right)∥}^{\mu }\le W\kappa ,\mathit{for}\mathit{all}\mathsf{\ell }\in \mp .$$

Remark 1

([36]). A function $\mathit{\Xi }\in C\left(\mp ,{\mathbb{R}}^n\right)$ is a solution of the inequality (6) if and only if there exists a function $\mathcal{Q}\in C\left(\mp ,{\mathbb{R}}^n\right)$ such that

(i) 

$\mathbf{E}{∥\mathcal{Q}\left(\mathsf{\ell }\right)∥}^{\mu }\le \kappa$, $\mathsf{\ell }\in \mp$.

(ii) 

$\left({}^{C}D_{0^{+}}^{\alpha }\mathit{\Xi }\right)\left(\mathsf{\ell }\right)=-\mathbb{X}\mathit{\Xi }\left(\mathsf{\ell }-\zeta \right)+\left(\mathsf{\ell },\mathit{\Xi }\left(\mathsf{\ell }\right)\right)\mathrm{d}{Z}_H\left(\mathsf{\ell }\right)+\hslash \left(\mathsf{\ell },\mathit{\Xi }\left(\mathsf{\ell }\right)\right)+\mathcal{Q}\left(\mathsf{\ell }\right)$, $\mathsf{\ell }\in \mp$.

Definition 3

([5]). The derivative of a function $\aleph :\left[-\zeta ,\infty \right)\to {\mathbb{R}}^n$ with lower index 0, known as the Caputo fractional of order $\alpha \in \left(1,2\right]$, is expressed by

$$\left({}^{C}D_{0^{+}}^{\alpha }\aleph \right)\left(\mathsf{\ell }\right)=\frac{1}{\mathit{\Gamma }\left(2-\alpha \right)}{\int }_0^{\mathsf{\ell }}\frac{{\aleph }^{″}\left(\upsilon \right)}{{\left(\mathsf{\ell }-\upsilon \right)}^{\alpha -1}}d\upsilon ,\mathsf{\ell }>0.$$

Definition 4

([5]). The Mittag–Leffler function containing two parameters is defined as

$${\mathbb{E}}_{\Delta ,\gamma }\left(\mathsf{\ell }\right)=\sum _{\iota =0}^{\infty }\frac{{\mathsf{\ell }}^{\iota }}{\mathit{\Gamma }\left(\iota +\gamma \right)},\Delta ,\gamma >0,\mathsf{\ell }\in \mathbb{C},$$

when $\gamma =1$. Then,

$${\mathbb{E}}_{\Delta ,1}\left(\mathsf{\ell }\right)={\mathbb{E}}_{\Delta }\left(\mathsf{\ell }\right)=\sum _{\iota =0}^{\infty }\frac{{\mathsf{\ell }}^{\iota }}{\mathit{\Gamma }\left(\iota +1\right)},\Delta >0.$$

Lemma 4

([14]). For any $\mathsf{\ell }\in \left[\left(\iota -1\right)\zeta ,\iota \zeta \right]$, $\iota =1,2,\cdots$, and $\alpha \in \left(1,2\right]$, we obtain

$$∥{\mathcal{H}}_{\zeta ,\alpha }\left(\mathbb{X}{\mathsf{\ell }}^{\alpha }\right)∥\le {\mathbb{E}}_{\alpha }\left(∥\mathbb{X}∥{\mathsf{\ell }}^{\alpha }\right),$$

$$∥{\mathcal{M}}_{\zeta ,\alpha }\left(\mathbb{X}{\mathsf{\ell }}^{\alpha }\right)∥\le \left(\mathsf{\ell }+\zeta \right){\mathbb{E}}_{\alpha ,2}\left(∥\mathbb{X}∥{\left(\mathsf{\ell }+\zeta \right)}^{\alpha }\right),$$

and

$$∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\mathsf{\ell }}^{\alpha }\right)∥\le {\left(\mathsf{\ell }+\zeta \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }+\zeta \right)}^{\alpha }\right).$$

Lemma 5

(Grönwall’s inequality [37]). Let $\hslash \left(\mathsf{\ell }\right)$ and ℘$\left(\mathsf{\ell }\right)$ be non-negative, continuous functions on $0\le \mathsf{\ell }\le T$, for which the inequality

$$\hslash \left(\mathsf{\ell }\right)\le \eta +{\int }_0^{\mathsf{\ell }}\wp \left(s\right)\hslash \left(s\right)\mathrm{d}s,\mathit{for}\mathsf{\ell }\in \left[0,T\right],$$

holds, where $\eta \ge 0$ is a constant. Then,

$$\hslash \left(\mathsf{\ell }\right)\le \eta exp\left({\int }_0^{\mathsf{\ell }}\wp \left(s\right)\mathrm{d}s\right),\mathit{for}\mathsf{\ell }\in \left[0,T\right].$$

Finally, we present Krasnoselskii’s fixed point theorem.

Lemma 6

([38]). Assume that $\mathcal{J}$ is a closed, bounded and non-empty convex subset of a Banach space $\mathcal{U}$. If $O_1$ and $O_2$ are mappings from $\mathcal{J}$ into $\mathcal{U}$ such that

(i) 

$O_1\mathsf{\ell }+O_2\aleph \in \mathcal{J}$ for every pair ℓ, $\aleph \in \mathcal{J}$,

(ii) 

$O_2$ is a contraction mapping.

(iii) 

$O_1$ is continuous and compact.

then there is $\Im \in \mathcal{J}$ such that $\Im =O_1\Im +O_2\Im$.

3. Main Results

In this section, we present and prove the well-posedness and Hyers–Ulam stability results of (2). To prove our main results, the following assumptions are assumed:

(G1) 

There exist a continuous function $\Delta :\mp \times {\mathbb{R}}^n⟶L_2^0$ and a constant $U_{\Delta }\in L^{r_2}\left(\mp ,{\mathbb{R}}^{+}\right)$, where $r_2>1$, such that

$$\mathbf{E}{∥\Delta \left(\mathsf{\ell },{\aleph }_1\right)-\Delta \left(\mathsf{\ell },{\aleph }_2\right)∥}_{L_2^0}^{\mu }\le U_{\Delta }\left(\mathsf{\ell }\right)\mathbf{E}{∥{\aleph }_1-{\aleph }_2∥}^{\mu },\mathrm{for}\mathrm{all}\mathsf{\ell }\in \mp ,{\aleph }_1,{\aleph }_2\in {\mathbb{R}}^n.$$

Let $\mu \in \left[2,\infty \right)$ and ${sup}_{\mathsf{\ell }\in \mp }\mathbf{E}{∥\Delta \left(\mathsf{\ell },0\right)∥}_{L_2^0}^{\mu }=W_{\Delta }<\infty$.

(G2) 

There exist a continuous function $\hslash :\mp \times {\mathbb{R}}^n⟶L_2^0$ and a constant $U_{\hslash }\in L^{r_2}\left(\mp ,{\mathbb{R}}^{+}\right)$, where $r_2>1$, such that

$$\mathbf{E}{∥\hslash \left(\mathsf{\ell },{\aleph }_1\right)-\hslash \left(\mathsf{\ell },{\aleph }_2\right)∥}^{\mu }\le U_{\hslash }\left(\mathsf{\ell }\right)\mathbf{E}{∥{\aleph }_1-{\aleph }_2∥}^{\mu },\mathbf{E}{∥\hslash \left(\mathsf{\ell },\aleph \right)∥}^{\mu }\le U_{\hslash }\left(\mathsf{\ell }\right)\left(1+\mathbf{E}{∥\aleph ∥}^{\mu }\right),$$

for all $\mathsf{\ell }\in \mp$, ${\aleph }_1,{\aleph }_2\in {\mathbb{R}}^n$.

Using Krasnoselskii’s fixed-point theorem, we now prove the existence and uniqueness results.

Theorem 1.

If $\left(\mathbf{G}\mathbf{1}\right)$–$\left(\mathbf{G}\mathbf{2}\right)$ hold, then there exists a unique mild solution of the nonlinear stochastic system (2) provided that

$$2^{\mu -1}{W}_2+W_3<1,$$

(7)

where

$$W_2:=\frac{{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu \left(H+\alpha -1\right)-\frac{1}{r_2}}}{{\left(\mu r_1\left(\alpha -1\right)+1\right)}^{\frac{1}{r_1}}}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }{∥U_{\Delta }∥}_{L^{r_2}\left(\mp ,{\mathbb{R}}^{+}\right)},$$

and

$$W_3:=\frac{{\varpi }^{\mu \left(\alpha -1\right)+\frac{1}{r_1}}}{{\left(\mu r_1\left(\alpha -1\right)+1\right)}^{\frac{1}{r_1}}}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }{∥U_{\hslash }∥}_{L^{r_2}\left(\mp ,{\mathbb{R}}^{+}\right)},$$

for $\frac{1}{r_1}+\frac{1}{r_2}=1$, $r_1$, $r_2>1$.

Proof. 

We deal with the set

$${\mathcal{T}}_{\varrho }=\left\{\aleph \in \mathcal{E}:{∥\aleph ∥}_{\mathcal{E}}^{\mu }=\underset{\mathsf{\ell }\in \mp }{sup}\mathbf{E}{∥\aleph \left(\mathsf{\ell }\right)∥}^{\mu }\le \varrho \right\},$$

for each positive number $\varrho$. Let $\mathsf{\ell }\in \mp$. Applying Lemma 1, we then transform problem (2) into a fixed-point problem and define an operator $F:\mathcal{E}⟶\mathcal{E}$ by

$$\begin{array}{cc}\hfill \left(F\aleph \right)\left(\mathsf{\ell }\right)& ={\mathcal{H}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right)\psi \left(0\right)+{\mathcal{M}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{X}{\int }_{-\zeta }^0{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-2\zeta -\varsigma \right)}^{\alpha }\right)\psi \left(\varsigma \right)\mathrm{d}\varsigma \hfill \\ \hfill & +{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\hslash \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}\varsigma \hfill \\ \hfill & +{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right).\hfill \end{array}$$

for $\mathsf{\ell }\in \mp$. Decomposing the operator F, we can define the operators ${\mathit{\Pi }}_1$, ${\mathit{\Pi }}_2$ on ${\mathcal{T}}_{\varrho }$ as follows:

$$\begin{array}{cc}\hfill \left({\mathit{\Pi }}_1\aleph \right)\left(\mathsf{\ell }\right)& ={\mathcal{H}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right)\psi \left(0\right)+{\mathcal{M}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{X}{\int }_{-\zeta }^0{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-2\zeta -\varsigma \right)}^{\alpha }\right)\psi \left(\varsigma \right)\mathrm{d}\varsigma \hfill \\ \hfill & +{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\hslash \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}\varsigma ,\hfill \end{array}$$

(8)

$$\left({\mathit{\Pi }}_2\aleph \right)\left(\mathsf{\ell }\right)={\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right).$$

(9)

At this point, we observe that ${\mathcal{T}}_{\varrho }$ is a convex set, closed and bounded of $\mathcal{E}$. Consequently, our proof consists of three essential steps:

Step 1. We show the existence of $\varrho >0$ such that ${\mathit{\Pi }}_1\aleph +{\mathit{\Pi }}_2\Im \in {\mathcal{T}}_{\varrho }$ for all ℵ, $\Im \in {\mathcal{T}}_{\varrho }$.

For each $\mathsf{\ell }\in \mp$ and ℵ, $\Im \in {\mathcal{T}}_{\varrho }$, and using (8) and (9), we get

$$\begin{array}{cc}\hfill & {∥{\mathit{\Pi }}_1\aleph +{\mathit{\Pi }}_2\Im ∥}_{\mathcal{E}}^{\mu }\hfill \\ \hfill & =\underset{\mathsf{\ell }\in \mp }{sup}\mathbf{E}{∥\left({\mathit{\Pi }}_1\aleph +{\mathit{\Pi }}_2\Im \right)\left(\mathsf{\ell }\right)∥}^{\mu }\hfill \\ \hfill & \le 5^{\mu -1}\left[{∥{\mathcal{H}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right)∥}^{\mu }\mathbf{E}{∥\psi \left(0\right)∥}^{\mu }+{∥{\mathcal{M}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right)∥}^{\mu }\mathbf{E}{∥{\psi }^{\prime }\left(0\right)∥}^{\mu }\right.\hfill \\ \hfill & +{∥\mathbb{X}∥}^{\mu }\mathbf{E}{∥{\int }_{-\zeta }^0{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-2\zeta -\varsigma \right)}^{\alpha }\right)\psi \left(\varsigma \right)\mathrm{d}\varsigma ∥}^{\mu }\hfill \\ \hfill & +\mathbf{E}{∥{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\hslash \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}\varsigma ∥}^{\mu }\hfill \\ \hfill & \left.+\mathbf{E}{∥{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\Im \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right)∥}^{\mu }\right]\hfill \\ \hfill & ={\sum }_{n=1}^5{\mathbf{I}}_n.\hfill \end{array}$$

(10)

From Lemma 4, we have

$$\begin{array}{cc}\hfill {\mathbf{I}}_1& =5^{\mu -1}{∥{\mathcal{H}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right)∥}^{\mu }\mathbf{E}{∥\psi \left(0\right)∥}^{\mu }\hfill \\ \hfill & \le 5^{\mu -1}{\left({\mathbb{E}}_{\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥\psi ∥}_C^{\mu },\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathbf{I}}_2& =5^{\mu -1}{∥{\mathcal{M}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right)∥}^{\mu }\mathbf{E}{∥{\psi }^{\prime }\left(0\right)∥}^{\mu }\hfill \\ \hfill & \le 5^{\mu -1}{\left(\mathsf{\ell }{\mathbb{E}}_{\alpha ,2}\left(∥\mathbb{X}∥{\mathsf{\ell }}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥{\psi }^{\prime }∥}_C^{\mu },\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathbf{I}}_3& =5^{\mu -1}{∥\mathbb{X}∥}^{\mu }\mathbf{E}{∥{\int }_{-\zeta }^0{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-2\zeta -\varsigma \right)}^{\alpha }\right)\psi \left(\varsigma \right)\mathrm{d}\varsigma ∥}^{\mu }\hfill \\ \hfill & \le 5^{\mu -1}{∥\mathbb{X}∥}^{\mu }{\zeta }^{\mu -1}\mathbf{E}{∥\psi ∥}_C^{\mu }{\int }_{-\zeta }^0{∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-2\zeta -\varsigma \right)}^{\alpha }\right)∥}^{\mu }\mathrm{d}\varsigma \hfill \\ \hfill & \le 5^{\mu -1}{∥\mathbb{X}∥}^{\mu }{\zeta }^{\mu }{\left({\mathsf{\ell }}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\mathsf{\ell }}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥\psi ∥}_C^{\mu },\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathbf{I}}_4& =5^{\mu -1}\mathbf{E}{∥{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\Im \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right)∥}^{\mu }\hfill \\ \hfill & =5^{\mu -1}\mathbf{E}{\left\{{∥{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\Im \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right)∥}^2\right\}}^{\mu /2}.\hfill \end{array}$$

Applying Lemmas 2 and 3, we obtain

$$\begin{array}{cc}\hfill {\mathbf{I}}_4& \le 5^{\mu -1}{\tau }_{\mu }{\left\{\mathbf{E}{∥{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\Im \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right)∥}^2\right\}}^{\mu /2}\hfill \\ \hfill & \le 5^{\mu -1}{\tau }_{\mu }{\left\{2H{\mathsf{\ell }}^{2H-1}{\int }_0^{\mathsf{\ell }}\mathbf{E}{∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\Im \left(\varsigma \right)\right)∥}_{L_2^0}^2\mathrm{d}\varsigma \right\}}^{\mu /2}\hfill \\ \hfill & \le 5^{\mu -1}{\tau }_{\mu }{\left(2H{\mathsf{\ell }}^{2H-1}\right)}^{\mu /2}{\left\{{\int }_0^{\mathsf{\ell }}\mathbf{E}{∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\Im \left(\varsigma \right)\right)∥}_{L_2^0}^2\mathrm{d}\varsigma \right\}}^{\mu /2}\hfill \\ \hfill & \le 5^{\mu -1}{\tau }_{\mu }{\left(2H{\mathsf{\ell }}^{2H-1}\right)}^{\mu /2}\hfill \\ \hfill & \times {\left\{{\left({\int }_0^{\mathsf{\ell }}{\left(\mathbf{E}{∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\Im \left(\varsigma \right)\right)∥}_{L_2^0}^2\right)}^{\mu /2}\mathrm{d}\varsigma \right)}^{2/\mu }{\left({\int }_0^{\mathsf{\ell }}\mathrm{d}\varsigma \right)}^{\frac{\mu -2}{\mu }}\right\}}^{\mu /2}\hfill \\ \hfill & \le 5^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu H-1}{\int }_0^{\mathsf{\ell }}\mathbf{E}{∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\Im \left(\varsigma \right)\right)∥}_{L_2^0}^{\mu }\mathrm{d}\varsigma .\hfill \end{array}$$

Using Lemma 4 and $\left(\mathbf{G}\mathbf{1}\right)$, we obtain

$$\begin{array}{cc}\hfill {\mathbf{I}}_4& \le 5^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu H-1}{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥\Delta \left(\varsigma ,\Im \left(\varsigma \right)\right)∥}_{L_2^0}^{\mu }\mathrm{d}\varsigma \hfill \\ \hfill & \le 5^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu H-1}\hfill \\ \hfill & \times 2^{\mu -1}\left\{{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥\Delta \left(\varsigma ,\Im \left(\varsigma \right)\right)-\Delta \left(\varsigma ,0\right)∥}_{L_2^0}^{\mu }\mathrm{d}\varsigma \right.\hfill \\ \hfill & \left.+{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥\Delta \left(\varsigma ,0\right)∥}_{L_2^0}^{\mu }\mathrm{d}\varsigma \right\}\hfill \\ \hfill & \le {\left(10\right)}^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu H-1}\hfill \\ \hfill & \times \left\{{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }{U}_{\Delta }\left(\varsigma \right)\mathbf{E}{∥\Im \left(\varsigma \right)∥}^{\mu }\mathrm{d}\varsigma \right.\hfill \\ \hfill & \left.+W_{\Delta }{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }\mathrm{d}\varsigma \right\}\hfill \\ \hfill & \le {\left(10\right)}^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu H-1}\left\{{∥\Im ∥}_{\mathcal{E}}^{\mu }{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }{U}_{\Delta }\left(\varsigma \right)\mathrm{d}\varsigma \right.\hfill \\ \hfill & \left.+\frac{{\varpi }^{\mu \left(\alpha -1\right)+1}{W}_{\Delta }}{\mu \left(\alpha -1\right)+1}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }\right\}.\hfill \end{array}$$

(11)

Additionally, using the Hölder inequality and $\left(\mathbf{G}\mathbf{1}\right)$, we obtain

$$\begin{array}{cc}\hfill & {\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }{U}_{\Delta }\left(\varsigma \right)\mathrm{d}\varsigma \hfill \\ \hfill & \le {\left({\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu r_1}\mathrm{d}\varsigma \right)}^{\frac{1}{r_1}}{\left({\int }_0^{\mathsf{\ell }}{U}_{\Delta }^{r_2}\left(\varsigma \right)\mathrm{d}\varsigma \right)}^{\frac{1}{r_2}}\hfill \\ \hfill & \le {\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }{\left({\int }_0^{\mathsf{\ell }}{\left(\mathsf{\ell }-\varsigma \right)}^{\mu r_1\left(\alpha -1\right)}\mathrm{d}\varsigma \right)}^{\frac{1}{r_1}}{\left({\int }_0^{\mathsf{\ell }}{U}_{\Delta }^{r_2}\left(\varsigma \right)\mathrm{d}\varsigma \right)}^{\frac{1}{r_2}}\hfill \\ \hfill & \le \frac{{\varpi }^{\mu \left(\alpha -1\right)+\frac{1}{r_1}}}{{\left(\mu r_1\left(\alpha -1\right)+1\right)}^{\frac{1}{r_1}}}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }{∥U_{\Delta }∥}_{L^{r_2}\left(\mp ,{\mathbb{R}}^{+}\right)}.\hfill \end{array}$$

(12)

Substituting (12) into (11), we get

$$\begin{array}{cc}\hfill {\mathbf{I}}_4& \le {\left(10\right)}^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu H-1}\hfill \\ \hfill & \times \left\{\frac{\varrho {\varpi }^{\mu \left(\alpha -1\right)+\frac{1}{r_1}}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }}{{\left(\mu r_1\left(\alpha -1\right)+1\right)}^{\frac{1}{r_1}}}{∥U_{\Delta }∥}_{L^{r_2}\left(\mp ,{\mathbb{R}}^{+}\right)}+\frac{{\varpi }^{\mu \left(\alpha -1\right)+1}{W}_{\Delta }}{\mu \left(\alpha -1\right)+1}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }\right\}\hfill \\ \hfill & ={\left(10\right)}^{\mu -1}{W}_2\varrho +\frac{{\left(10\right)}^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu \left(H+\alpha -1\right)}{W}_{\Delta }}{\mu \left(\alpha -1\right)+1}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }.\hfill \end{array}$$

Furthermore, using (12) and $\left(\mathbf{G}\mathbf{2}\right)$, we obtain

$$\begin{array}{cc}\hfill {\mathbf{I}}_5& =5^{\mu -1}\mathbf{E}{∥{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\hslash \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}\varsigma ∥}^{\mu }\hfill \\ \hfill & \le 5^{\mu -1}{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥\hslash \left(\varsigma ,\aleph \left(\varsigma \right)\right)∥}^{\mu }\mathrm{d}\varsigma \hfill \\ \hfill & \le 5^{\mu -1}{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }{U}_{\hslash }\left(\varsigma \right)\left(1+\mathbf{E}{∥\aleph ∥}^{\mu }\right)\mathrm{d}\varsigma \hfill \\ \hfill & \le \frac{5^{\mu -1}\left(1+\varrho \right){\varpi }^{\mu \left(\alpha -1\right)+\frac{1}{r_1}}}{{\left(\mu r_1\left(\alpha -1\right)+1\right)}^{\frac{1}{r_1}}}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }{∥U_{\hslash }∥}_{L^{r_2}\left(\mp ,{\mathbb{R}}^{+}\right)}\hfill \\ \hfill & =5^{\mu -1}\left(1+\varrho \right)W_3.\hfill \end{array}$$

From ${\mathbf{I}}_1$ to ${\mathbf{I}}_5$, (10) becomes

$$\begin{array}{cc}\hfill & {∥{\mathit{\Pi }}_1\aleph +{\mathit{\Pi }}_2\Im ∥}_{\mathcal{E}}^{\mu }\hfill \\ \hfill & \le 5^{\mu -1}\left\{{\left({\mathbb{E}}_{\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥\psi ∥}_C^{\mu }\right.\hfill \\ \hfill & +{\left(\mathsf{\ell }{\mathbb{E}}_{\alpha ,2}\left(∥\mathbb{X}∥{\mathsf{\ell }}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥{\psi }^{\prime }∥}_C^{\mu }\hfill \\ \hfill & +{∥\mathbb{X}∥}^{\mu }{\zeta }^{\mu }{\left({\mathsf{\ell }}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\mathsf{\ell }}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥\psi ∥}_C^{\mu }\hfill \\ \hfill & +2^{\mu -1}{W}_2\varrho +\frac{2^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu \left(H+\alpha -1\right)}{W}_{\Delta }}{\mu \left(\alpha -1\right)+1}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }\hfill \\ \hfill & \left.+\left(1+\varrho \right)W_3\right\}\hfill \\ \hfill & \le 5^{\mu -1}\left\{\theta \left(\varpi \right)+\varrho \left(2^{\mu -1}{W}_2+W_3\right)+W_3\right\},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \theta \left(\mathsf{\ell }\right)& ={\left({\mathbb{E}}_{\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥\psi ∥}_C^{\mu }+{\left(\mathsf{\ell }{\mathbb{E}}_{\alpha ,2}\left(∥\mathbb{X}∥{\mathsf{\ell }}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥{\psi }^{\prime }∥}_C^{\mu }\hfill \\ \hfill & +{∥\mathbb{X}∥}^{\mu }{\zeta }^{\mu }{\left({\mathsf{\ell }}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\mathsf{\ell }}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥\psi ∥}_C^{\mu }\hfill \\ \hfill & +\frac{2^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu \left(H+\alpha -1\right)}{W}_{\Delta }}{\mu \left(\alpha -1\right)+1}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\mathsf{\ell }}^{\alpha }\right)\right)}^{\mu }.\hfill \end{array}$$

As a result, from (7), we obtain ${\mathit{\Pi }}_1\aleph +{\mathit{\Pi }}_2\Im \in {\mathcal{T}}_{\varrho }$ for some sufficiently large $\varrho$.

Step 2. We show that ${\mathit{\Pi }}_1:{\mathcal{T}}_{\varrho }⟶\mathcal{E}$ is a contraction.

For each $\mathsf{\ell }\in \mp$ and ℵ, $\Im \in {\mathcal{T}}_{\varrho }$, using (8) and $\left(\mathbf{G}\mathbf{2}\right)$, we get

$$\begin{array}{cc}\hfill & \mathbf{E}{∥\left({\mathit{\Pi }}_1\aleph \right)\left(\mathsf{\ell }\right)-\left({\mathit{\Pi }}_1\Im \right)\left(\mathsf{\ell }\right)∥}^{\mu }\hfill \\ \hfill & =\mathbf{E}{∥{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\left[\hslash \left(\varsigma ,\aleph \left(\varsigma \right)\right)-\hslash \left(\varsigma ,\Im \left(\varsigma \right)\right)\right]\mathrm{d}\varsigma ∥}^{\mu }\hfill \\ \hfill & \le \mathbf{E}{∥\aleph -\Im ∥}_{\mathcal{E}}^{\mu }{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }{U}_{\hslash }\left(\varsigma \right)\mathrm{d}\varsigma \hfill \\ \hfill & \le W_3{∥\aleph -\Im ∥}_{\mathcal{E}}^{\mu }.\hfill \end{array}$$

As we can see from (7), noting $W_3<1$, ${\mathit{\Pi }}_1$ is a contraction mapping.

Step 3. We show that ${\mathit{\Pi }}_2:{\mathcal{T}}_{\varrho }⟶\mathcal{E}$ is a continuous compact operator.

First, we verify the continuity of ${\mathit{\Pi }}_2$. Consider ${\aleph }_n\in {\mathcal{T}}_{\varrho }$ with ${\aleph }_n⟶\aleph$ as $n⟶\infty$ in ${\mathcal{T}}_{\varrho }$. Thus, using Lebesgue’s dominated convergence theorem and (9), we get, for each $\mathsf{\ell }\in \mp$,

$$\begin{array}{cc}\hfill & \mathbf{E}{∥\left({\mathit{\Pi }}_2{\aleph }_n\right)\left(\mathsf{\ell }\right)-\left({\mathit{\Pi }}_2\aleph \right)\left(\mathsf{\ell }\right)∥}^{\mu }\hfill \\ \hfill & \le {\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu H-1}{\int }_0^{\mathsf{\ell }}{∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)∥}^{\mu }\mathbf{E}{∥\Delta \left(\varsigma ,{\aleph }_n\left(\varsigma \right)\right)-\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)∥}_{L_2^0}^{\mu }\mathrm{d}\varsigma \hfill \\ \hfill & \le {\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu H-1}{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }{U}_{\Delta }\left(\varsigma \right)\hfill \\ \hfill & \times {∥{\aleph }_n-\aleph ∥}_{\mathcal{E}}^{\mu }\mathrm{d}\varsigma ⟶0,\mathrm{as}n⟶\infty .\hfill \end{array}$$

This proves the continuity of ${\mathit{\Pi }}_2:{\mathcal{T}}_{\varrho }⟶\mathcal{E}$.

Thereafter, we show that ${\mathit{\Pi }}_2$ is uniformly bounded on ${\mathcal{T}}_{\varrho }$. For each $\mathsf{\ell }\in \mp$, $\aleph \in {\mathcal{T}}_{\varrho }$, we have

$$\begin{array}{cc}\hfill {∥{\mathit{\Pi }}_2\aleph ∥}_{\mathcal{E}}^{\mu }& =\underset{\mathsf{\ell }\in \mp }{sup}\mathbf{E}{∥\left({\mathit{\Pi }}_2\aleph \right)\left(\mathsf{\ell }\right)∥}^{\mu }\hfill \\ \hfill & \le \underset{\mathsf{\ell }\in \mp }{sup}\left\{\mathbf{E}{∥{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right)∥}^{\mu }\right\}\hfill \\ \hfill & \le 2^{\mu -1}{W}_2\varrho +\frac{2^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu \left(H+\alpha -1\right)}{W}_{\Delta }}{\mu \left(\alpha -1\right)+1}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }.\hfill \end{array}$$

This indicates that, on ${\mathcal{T}}_{\varrho }$, ${\mathit{\Pi }}_2$ is uniformly bounded.

Showing that ${\mathit{\Pi }}_2$ is equicontinuous is still necessary. For each ${\mathsf{\ell }}_2$, ${\mathsf{\ell }}_3\in \mp$, $0<{\mathsf{\ell }}_2<{\mathsf{\ell }}_3\le \varpi$ and $\aleph \in {\mathcal{T}}_{\varrho }$, using (9), we obtain

$$\begin{array}{cc}\hfill & \left({\mathit{\Pi }}_2\aleph \right)\left({\mathsf{\ell }}_3\right)-\left({\mathit{\Pi }}_2\aleph \right)\left({\mathsf{\ell }}_2\right)\hfill \\ \hfill & ={\int }_0^{{\mathsf{\ell }}_3}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_3-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right)\hfill \\ \hfill & -{\int }_0^{{\mathsf{\ell }}_2}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_2-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right)\hfill \\ \hfill & ={\mathit{\Psi }}_1+{\mathit{\Psi }}_2,\hfill \end{array}$$

where

$${\mathit{\Psi }}_1={\int }_{{\mathsf{\ell }}_2}^{{\mathsf{\ell }}_3}{\mathcal{M}}_{\zeta }\left(\mathbb{X}\left({\mathsf{\ell }}_3-\zeta -\varsigma \right)\right)\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right),$$

and

$${\mathit{\Psi }}_2={\int }_0^{{\mathsf{\ell }}_2}\left[{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_3-\zeta -\varsigma \right)}^{\alpha }\right)-{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_2-\zeta -\varsigma \right)}^{\alpha }\right)\right]\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right).$$

Thus,

$$\begin{array}{cc}\hfill \mathbf{E}{∥\left({\mathit{\Pi }}_2\aleph \right)\left({\mathsf{\ell }}_3\right)-\left({\mathit{\Pi }}_2\aleph \right)\left({\mathsf{\ell }}_2\right)∥}^{\mu }& =\mathbf{E}{∥{\mathit{\Psi }}_1+{\mathit{\Psi }}_2∥}^{\mu }\hfill \\ \hfill & \le 2^{\mu -1}\left\{\mathbf{E}{∥{\mathit{\Psi }}_1∥}^{\mu }+\mathbf{E}{∥{\mathit{\Psi }}_2∥}^{\mu }\right\}.\hfill \end{array}$$

(13)

Now, we can check $∥{\mathit{\Psi }}_r∥⟶0$ as ${\mathsf{\ell }}_2⟶{\mathsf{\ell }}_3$, when $r=1$, 2. For ${\mathit{\Psi }}_1$, we get

$$\begin{array}{cc}\hfill \mathbf{E}{∥{\mathit{\Psi }}_1∥}^{\mu }& =\mathbf{E}{∥{\int }_{{\mathsf{\ell }}_2}^{{\mathsf{\ell }}_3}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_3-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right)∥}^{\mu }\hfill \\ \hfill & \le {\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\left({\mathsf{\ell }}_3-{\mathsf{\ell }}_2\right)}^{\mu H-1}{\int }_{{\mathsf{\ell }}_2}^{{\mathsf{\ell }}_3}\mathbf{E}{∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)∥}_{L_2^0}^{\mu }\mathrm{d}\varsigma \hfill \\ \hfill & \le 2^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\left({\mathsf{\ell }}_3-{\mathsf{\ell }}_2\right)}^{\mu H-1}\hfill \\ \hfill & \times \left\{\varrho {\int }_{{\mathsf{\ell }}_2}^{{\mathsf{\ell }}_3}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }{U}_{\Delta }\left(\varsigma \right)\mathrm{d}\varsigma \right.\hfill \\ \hfill & \left.+\frac{{\left({\mathsf{\ell }}_3-{\mathsf{\ell }}_2\right)}^{\mu \left(\alpha -1\right)+1}{W}_{\Delta }}{\mu \left(\alpha -1\right)+1}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left({\mathsf{\ell }}_3-{\mathsf{\ell }}_2\right)}^{\alpha }\right)\right)}^{\mu }\right\}⟶0,\mathrm{as}{\mathsf{\ell }}_2⟶{\mathsf{\ell }}_3.\hfill \end{array}$$

For ${\mathit{\Psi }}_2$, we get

$$\begin{array}{cc}\hfill & \mathbf{E}{∥{\mathit{\Psi }}_2∥}^{\mu }\hfill \\ \hfill & =\mathbf{E}{∥{\int }_0^{{\mathsf{\ell }}_2}\left[{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_3-\zeta -\varsigma \right)}^{\alpha }\right)-{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_2-\zeta -\varsigma \right)}^{\alpha }\right)\right]\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right)∥}^{\mu }\hfill \\ \hfill & \le {\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\mathsf{\ell }}_2^{\mu H-1}\hfill \\ \hfill & \times {\int }_0^{{\mathsf{\ell }}_2}\mathbf{E}{∥\left[{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_3-\zeta -\varsigma \right)}^{\alpha }\right)-{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_2-\zeta -\varsigma \right)}^{\alpha }\right)\right]\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)∥}_{L_2^0}^{\mu }\mathrm{d}\varsigma \hfill \\ \hfill & \le 2^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\mathsf{\ell }}_2^{\mu H-1}\hfill \\ \hfill & \times \left\{\varrho {\int }_0^{{\mathsf{\ell }}_2}{∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_3-\zeta -\varsigma \right)}^{\alpha }\right)-{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_2-\zeta -\varsigma \right)}^{\alpha }\right)∥}^{\mu }{U}_{\Delta }\left(\varsigma \right)\mathrm{d}\varsigma \right.\hfill \\ \hfill & \left.+W_{\Delta }{\int }_0^{{\mathsf{\ell }}_2}{∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_3-\zeta -\varsigma \right)}^{\alpha }\right)-{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_2-\zeta -\varsigma \right)}^{\alpha }\right)∥}^{\mu }\mathrm{d}\varsigma \right\}\hfill \\ \hfill & \le 2^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\mathsf{\ell }}_2^{\mu H-1}\hfill \\ \hfill & \times \left\{\varrho {∥U_{\Delta }∥}_{L^{r_2}\left(\mp ,{\mathbb{R}}^{+}\right)}\right.\hfill \\ \hfill & \times {\left({\int }_0^{{\mathsf{\ell }}_2}{\left({∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_3-\zeta -\varsigma \right)}^{\alpha }\right)-{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_2-\zeta -\varsigma \right)}^{\alpha }\right)∥}^{\mu }\right)}^{r_1}\right)}^{1/r_1}\mathrm{d}\varsigma \hfill \\ \hfill & \left.+W_{\Delta }{\int }_0^{{\mathsf{\ell }}_2}{∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_3-\zeta -\varsigma \right)}^{\alpha }\right)-{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_2-\zeta -\varsigma \right)}^{\alpha }\right)∥}^{\mu }\mathrm{d}\varsigma \right\}.\hfill \end{array}$$

From (5), knowing that ${\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\mathsf{\ell }}^{\alpha }\right)$ is uniformly continuous for $\mathsf{\ell }\in \mp ,$ we get

$$∥{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_3-\zeta -\varsigma \right)}^{\alpha }\right)-{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left({\mathsf{\ell }}_2-\zeta -\varsigma \right)}^{\alpha }\right)∥⟶0,\mathrm{as}{\mathsf{\ell }}_2⟶{\mathsf{\ell }}_3.$$

Therefore, we have $∥{\mathit{\Psi }}_r∥⟶0$ as ${\mathsf{\ell }}_2⟶{\mathsf{\ell }}_3$ when $r=1$, 2, which leads, via (13), to

$$\mathbf{E}{∥\left({\mathit{\Pi }}_2\aleph \right)\left({\mathsf{\ell }}_3\right)-\left({\mathit{\Pi }}_2\aleph \right)\left({\mathsf{\ell }}_2\right)∥}^{\mu }⟶0,\mathrm{as}{\mathsf{\ell }}_2⟶{\mathsf{\ell }}_3,$$

for all $\aleph \in {\mathcal{T}}_{\varrho }$. Then, ${\mathit{\Pi }}_2$ is compact on ${\mathcal{T}}_{\varrho }$ via the Arzelà–Ascoli theorem (see [38]). As a result, $F\aleph ={\mathit{\Pi }}_1\aleph +{\mathit{\Pi }}_2\aleph$ has a fixed point ℵ in ${\mathcal{T}}_{\varrho }$, in accordance with Lemma 6. Furthermore, ℵ is also a solution of (2) and $\left({\mathit{\Pi }}_1\aleph +{\mathit{\Pi }}_2\aleph \right)\left(\varpi \right)={\aleph }_1$. Therefore, the system (2) has a mild solution. This completes the proof. □

Next, we verify the Hyers–Ulam stability results via Grönwall’s inequality lemma approach.

Theorem 2.

If the assumptions of Theorem 1 hold, then nonlinear stochastic system (2) has the Ulam–Hyers stability.

Proof. 

Assume that ℵ is the unique solution of (2) and $\mathit{\Xi }\in C\left(\mp ,{\mathbb{R}}^n\right)$ is a solution of the inequality (6) with the aid of Theorem 1. Then,

$$\begin{array}{cc}\hfill \aleph \left(\mathsf{\ell }\right)& ={\mathcal{H}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right)\psi \left(0\right)+{\mathcal{M}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{X}{\int }_{-\zeta }^0{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-2\zeta -\varsigma \right)}^{\alpha }\right)\psi \left(\varsigma \right)\mathrm{d}\varsigma \hfill \\ \hfill & +{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\hslash \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}\varsigma \hfill \\ \hfill & +{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right).\hfill \end{array}$$

Based on Remark 1, then

$$\left({}^{C}D_{0^{+}}^{\alpha }\mathit{\Xi }\right)\left(\mathsf{\ell }\right)=-\mathbb{X}\mathit{\Xi }\left(\mathsf{\ell }-\zeta \right)+\Delta \left(\mathsf{\ell },\mathit{\Xi }\left(\mathsf{\ell }\right)\right)\mathrm{d}{Z}_H\left(\mathsf{\ell }\right)+\hslash \left(\mathsf{\ell },\mathit{\Xi }\left(\mathsf{\ell }\right)\right)+\mathcal{Q}\left(\mathsf{\ell }\right),\mathsf{\ell }\in \mp ,$$

can be expressed as

$$\begin{array}{cc}\hfill \mathit{\Xi }\left(\mathsf{\ell }\right)& ={\mathcal{H}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right)\psi \left(0\right)+{\mathcal{M}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta \right)}^{\alpha }\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{X}{\int }_{-\zeta }^0{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-2\zeta -\varsigma \right)}^{\alpha }\right)\psi \left(\varsigma \right)\mathrm{d}\varsigma \hfill \\ \hfill & +{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\Delta \left(\varsigma ,\mathit{\Xi }\left(\varsigma \right)\right)\mathrm{d}{Z}_H\left(\varsigma \right)\hfill \\ \hfill & +{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\hslash \left(\varsigma ,\mathit{\Xi }\left(\varsigma \right)\right)\mathrm{d}\varsigma \hfill \\ \hfill & +{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\mathcal{Q}\left(\varsigma \right)\mathrm{d}\varsigma .\hfill \end{array}$$

In the same manner as in the proof of Theorem 1 and by consequence of (10), we have

$$\begin{array}{cc}\hfill & \mathbf{E}{∥\mathit{\Xi }\left(\mathsf{\ell }\right)-\aleph \left(\mathsf{\ell }\right)∥}^{\mu }\hfill \\ \hfill & \le 3^{\mu -1}\left\{\mathbf{E}{∥{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\left[\Delta \left(\varsigma ,\mathit{\Xi }\left(\varsigma \right)\right)-\Delta \left(\varsigma ,\aleph \left(\varsigma \right)\right)\right]\mathrm{d}{Z}_H\left(\varsigma \right)∥}^{\mu }\right.\hfill \\ \hfill & +\mathbf{E}{∥{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\left[\hslash \left(\mathsf{\ell },\mathit{\Xi }\left(\mathsf{\ell }\right)\right)-\hslash \left(\mathsf{\ell },\aleph \left(\mathsf{\ell }\right)\right)\right]\mathrm{d}\varsigma ∥}^{\mu }\hfill \\ \hfill & \left.+\mathbf{E}{∥{\int }_0^{\mathsf{\ell }}{\mathcal{S}}_{\zeta ,\alpha }\left(\mathbb{X}{\left(\mathsf{\ell }-\zeta -\varsigma \right)}^{\alpha }\right)\mathcal{Q}\left(\varsigma \right)\mathrm{d}\varsigma ∥}^{\mu }\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le 3^{\mu -1}\left\{{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu H-1}{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }{U}_{\Delta }\left(\varsigma \right)\right.\hfill \\ \hfill & \times \mathbf{E}{∥\mathit{\Xi }\left(\varsigma \right)-\aleph \left(\varsigma \right)∥}^{\mu }\mathrm{d}\varsigma \hfill \\ \hfill & +{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }{U}_{\hslash }\left(\varsigma \right)\mathbf{E}{∥\mathit{\Xi }\left(\varsigma \right)-\aleph \left(\varsigma \right)∥}^{\mu }\mathrm{d}\varsigma \hfill \\ \hfill & \left.+{\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }\mathbf{E}{∥\mathcal{Q}\left(\varsigma \right)∥}^{\mu }\mathrm{d}\varsigma \right\}\hfill \\ \hfill & \le {\int }_0^{\mathsf{\ell }}{\left({\left(\mathsf{\ell }-\varsigma \right)}^{\alpha -1}{\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\left(\mathsf{\ell }-\varsigma \right)}^{\alpha }\right)\right)}^{\mu }\left(3^{\mu -1}{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu H-1}{U}_{\Delta }\left(\varsigma \right)+3^{\mu -1}{U}_{\hslash }\left(\varsigma \right)\right)\hfill \\ \hfill & \times \mathbf{E}{∥\mathit{\Xi }\left(\varsigma \right)-\aleph \left(\varsigma \right)∥}^{\mu }\mathrm{d}\varsigma \hfill \\ \hfill & +\frac{3^{\mu -1}{\varpi }^{\mu \left(\alpha -1\right)+1}\kappa }{\mu \left(\alpha -1\right)+1}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }.\hfill \end{array}$$

Applying Grönwall’s inequality (Lemma 5), we get

$$\mathbf{E}{∥\mathit{\Xi }\left(\mathsf{\ell }\right)-\aleph \left(\mathsf{\ell }\right)∥}^{\mu }\le \frac{3^{\mu -1}{\varpi }^{\mu \left(\alpha -1\right)+1}\kappa }{\mu \left(\alpha -1\right)+1}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }exp\left(3^{\mu -1}\left(W_2+W_3\right)\right),$$

which implies that

$$\mathbf{E}{∥\mathit{\Xi }\left(\mathsf{\ell }\right)-\aleph \left(\mathsf{\ell }\right)∥}^{\mu }\le W\kappa ,$$

where

$$W:=\frac{3^{\mu -1}{\varpi }^{\mu \left(\alpha -1\right)+1}}{\mu \left(\alpha -1\right)+1}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }exp\left(3^{\mu -1}\left(W_2+W_3\right)\right).$$

Therefore, there exists W that satisfies Definition 2. This ends the proof. □

4. An Example

Consider the following nonlinear fractional stochastic delay system driven by the Rosenblatt process:

$$\begin{array}{c}\left({}^{C}D_{0^{+}}^{1.8}\aleph \right)\left(\mathsf{\ell }\right)+\mathbb{X}\aleph \left(\mathsf{\ell }-0.3\right)=\hslash \left(\mathsf{\ell },\aleph \left(\mathsf{\ell }\right)\right)+\Delta \left(\mathsf{\ell },\aleph \left(\mathsf{\ell }\right)\right)\frac{\mathrm{d}{Z}_H\left(\mathsf{\ell }\right)}{\mathrm{d}\mathsf{\ell }},\mathrm{for}\mathsf{\ell }\in \mp :=\left[0,1\right],\\ \aleph \left(\mathsf{\ell }\right)\equiv \psi \left(\mathsf{\ell }\right),{\aleph }^{\prime }\left(\mathsf{\ell }\right)\equiv {\psi }^{\prime }\left(\mathsf{\ell }\right)\mathrm{for}-0.3\le \mathsf{\ell }\le 0,\end{array}$$

(14)

where $\alpha =1.8$, $\zeta =0.3$,

$$\aleph \left(\mathsf{\ell }\right)=\left(\genfrac{}{}{0pt}{}{{\aleph }_1\left(\mathsf{\ell }\right)}{{\aleph }_2\left(\mathsf{\ell }\right)}\right),\mathbb{X}=\left(\begin{array}{cc}0.4& 0.2\\ 0.6& 0.3\end{array}\right),$$

and

$$\hslash \left(\mathsf{\ell },\aleph \left(\mathsf{\ell }\right)\right)=\left(\begin{array}{c}0.2{sin}^2\left({\aleph }_1\left(\mathsf{\ell }\right)\right)+1\\ 0.2{cos}^2\left({\aleph }_2\left(\mathsf{\ell }\right)\right)+\sqrt{\mathsf{\ell }}\end{array}\right),\Delta \left(\mathsf{\ell },\aleph \left(\mathsf{\ell }\right)\right)=\left(\begin{array}{c}arctan\left({\aleph }_1\left(\mathsf{\ell }\right)\right)\\ e^{-\mathsf{\ell }}sin\left({\aleph }_2\left(\mathsf{\ell }\right)\right)\end{array}\right).$$

Next, by choosing $\mu =r_1=r_2=2$, we get

$$\begin{array}{cc}\hfill \mathbf{E}{∥\Delta \left(\mathsf{\ell },\aleph \right)-\left(\mathsf{\ell },\Im \right)∥}_{L_2^0}^2& \le \left[{\left({\aleph }_1\left(\mathsf{\ell }\right)-{\Im }_1\left(\mathsf{\ell }\right)\right)}^2+{\left({\aleph }_2\left(\mathsf{\ell }\right)-{\Im }_2\left(\mathsf{\ell }\right)\right)}^2\right]\hfill \\ \hfill & =\mathbf{E}{∥\aleph -\Im ∥}^2,\hfill \end{array}$$

for all $\mathsf{\ell }\in \mp$, and $\aleph \left(\mathsf{\ell }\right)$, $\Im \left(\mathsf{\ell }\right)\in {\mathbb{R}}^2$. We set $U_{\Delta }\left(\mathsf{\ell }\right)=1$ such that $U_{\Delta }\in L^2\left(\mp ,{\mathbb{R}}^{+}\right)$ in $\left(\mathbf{G}\mathbf{1}\right)$, and then we have ${∥U_{\Delta }∥}_{L^2\left(\mp ,{\mathbb{R}}^{+}\right)}=1$. Thus, selecting $H=0.75$ and ${\tau }_{\mu }=0.15$, we get

$$W_2=\frac{{\tau }_{\mu }{\left(2H\right)}^{\mu /2}{\varpi }^{\mu \left(H+\alpha -1\right)-\frac{1}{r_2}}}{{\left(\mu r_1\left(\alpha -1\right)+1\right)}^{\frac{1}{r_1}}}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }{∥U_{\Delta }∥}_{L^{r_2}\left(\mp ,{\mathbb{R}}^{+}\right)}=0.205.$$

Furthermore, we have

$$\begin{array}{cc}\hfill \mathbf{E}{∥\hslash \left(\mathsf{\ell },\aleph \right)-\hslash \left(\mathsf{\ell },\Im \right)∥}^2& \le 0.16\left[{\left({\aleph }_1\left(\mathsf{\ell }\right)-{\Im }_1\left(\mathsf{\ell }\right)\right)}^2+{\left({\aleph }_2\left(\mathsf{\ell }\right)-{\Im }_2\left(\mathsf{\ell }\right)\right)}^2\right]\hfill \\ \hfill & =U_{\hslash }\left(\mathsf{\ell }\right)\mathbf{E}{∥\aleph -\Im ∥}^2.\hfill \end{array}$$

We set $U_{\hslash }\left(\mathsf{\ell }\right)=0.16$ such that $U_{\hslash }\in L^2\left(\mp ,{\mathbb{R}}^{+}\right)$ in $\left(\mathbf{G}\mathbf{2}\right)$, and then we have ${∥U_{\hslash }∥}_{L^2\left(\mp ,{\mathbb{R}}^{+}\right)}=0.16$. Hence,

$$W_3=\frac{{\varpi }^{\mu \left(\alpha -1\right)+\frac{1}{r_1}}}{{\left(\mu r_1\left(\alpha -1\right)+1\right)}^{\frac{1}{r_1}}}{\left({\mathbb{E}}_{\alpha ,\alpha }\left(∥\mathbb{X}∥{\varpi }^{\alpha }\right)\right)}^{\mu }{∥U_{\hslash }∥}_{L^{r_2}\left(\mp ,{\mathbb{R}}^{+}\right)}=0.146.$$

Finally, we calculate that

$$2^{\mu -1}{W}_2+W_3=0.556<1,$$

which implies that all the assumptions of Theorems 1 and 2 hold. Therefore, system (14) has a unique mild solution ℵ and is Hyers–Ulam stable.

5. Conclusions

In this work, based on fixed-point theory and under the effect of the Rosenblatt process, we proved the existence and uniqueness of solutions of (2). After that, we derived the Hyers–Ulam stability results using the delayed Mittag–Leffler matrix functions and Grönwall’s inequality. Finally, we verified the theoretical results by giving an example with numerical simulations, which showed the effectiveness of the derived results. This is a novel study that proves the well-posedness and Hyers–Ulam stability of (2) using delayed Mittag–Leffler matrix functions.

In future work, studies will focus on the obtained results to ascertain the well-posedness and Hyers–Ulam stability of different types of stochastic delay systems, such as impulsive fractional stochastic delay systems or conformable fractional stochastic delay systems.