Approximate Controllability of Hilfer Fractional Stochastic Evolution Inclusions of Order 1 < q < 2

Abstract

This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order $1<\mathsf{q}<2$. Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects of finding mild solutions for the Hilfer fractional stochastic differential equation. Subsequently, we determined that the specified system is approximately controllable. Finally, an example displays the theoretical application of the results.

1. Introduction

Controllability, a core topic in mathematical control theory, is closely related to other concepts like structural decomposition, engineering, observer design, and pole assignment. From a mathematical standpoint, it is important to differentiate between exact and approximate controllability. A system with approximate controllability can be directed to any arbitrarily small neighborhood of the final state, whereas a system with exact controllability can be steered to any specific final state. Controllability issues arise in various fields of physics and engineering, including heat flow in materials with memory, viscoelasticity, and other physical phenomena, often modeled as abstract differential systems in infinite-dimensional spaces. For additional information on control theory, we recommend consulting the article and the references listed [1,2,3,4,5,6,7,8,9,10].

The capability of fractional calculus to capture inherited characteristics has garnered substantial interest recently from diverse scientific and technical fields, such as chaotic behavior, fractional differential equations, thermal physics, and biology. To address various phenomena in practical applications, different fractional derivatives have been developed, including the Riemann–Liouville, Caputo, Hadamard, and Grunwald–Letnikov fractional derivatives. In addition, a generalized version of the Riemann–Liouville (R-L) fractional derivative (R-LFD), called the Hilfer fractional (HF) derivative (HFD), was introduced by Hilfer [11]. It encompasses the R-L and Caputo fractional (CF) derivatives (CFD). Theoretical studies of dielectric relaxation in glass-forming materials gave rise to this operator. Since, several scholars have investigated fractional differential equations with HFD (see [12,13]). The authors of [14,15] studied the existence results for HFD with order $\mu \in (1,2)$ in Hilbert and Banach spaces. The existence of HF differential inclusions with order $\mu \in (1,2)$ has been studied by [16]. We direct interested readers to the monographs and papers listed here [17,18,19,20,21,22] and the references therein for further information.

Moreover, both artificial and natural systems are inevitably subjected to noise or stochastic issues. As a result, stochastic differential systems have gained substantial interest due to their extensive applications in modeling complex dynamic systems within biological, physical, and medical fields [23,24]. In [25], the authors evaluated controllability results of CF stochastic differential inclusions with nonlocal conditions. Currently, the authors [26] studied the approximate controllability results for HF stochastic differential equations with order $\mu \in (1,2)$. For further references, see these articles [27,28,29,30,31,32,33].

The research based on the approximate controllability of Hilfer fractional differential inclusions when order $1<\mathsf{q}<2$ is an untreated topic and motivated by articles [14,26], our goal is to fill this gap. Examine the Hilfer fractional stochastic differential inclusions system, together with its control term:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}_{0^{+}}^{\mathsf{p},\mathsf{q}}\mathtt{x}\left(\chi \right)\in \mathcal{A}\mathtt{x}\left(\chi \right)+\mathcal{B}w\left(\chi \right)+f(\chi ,\mathtt{x}\left(\chi \right))+\mathcal{H}(\chi ,\mathtt{x}\left(\chi \right)){\displaystyle \frac{d\mathcal{W}\left(\chi \right)}{d\chi }},\chi \in {\mathcal{I}}^{\prime }=(0,\mathfrak{y}],\hfill \\ \left(I_{0^{+}}^{2-\mathtt{v}}\mathtt{x}\right)\left(0\right)={\mathtt{x}}_{\mathtt{0}},{\left(I_{0^{+}}^{2-\mathtt{v}}\mathtt{x}\right)}^{\prime }\left(0\right)={\mathtt{x}}_{\mathtt{1}}.\hfill \end{array}\right.\end{array}$$

(1)

where $D_{0^{+}}^{\mathsf{p},\mathsf{q}}$ is the HFD with an order and type such that $\mathsf{q}\in (1,2)$ and $\mathsf{p}\in [0,1]$. The strongly continuous cosine family ${\left\{\mathtt{Q}\left(\chi \right)\right\}}_{\chi \ge 0}$ is represented by the infinitesimal generator $\mathcal{A}:D\left(\mathcal{A}\right)\subset \mathcal{H}\to \mathcal{H}$, where $\mathtt{x}(·)$ takes values in separable Hilbert space $\mathcal{H}$. $\mathcal{B}$ is a bounded linear operator from $\mathcal{Z}$ into $\mathcal{H}$, where $\mathcal{Z}$ is a Hilbert space, and $w(·)\in L_{\mathfrak{F}}^2(\mathcal{I},\mathcal{Z})$ is the control function. Let $\mathcal{G}$ be another real separable Hilbert space with ${\langle ·,·\rangle }_{\mathcal{G}}$ and the norm ${\parallel ·\parallel }_{\mathcal{G}}$. Assume that $\left\{\mathcal{W}\right(\chi ),\chi \ge 0\}$ is a $\mathcal{G}$-valued Brownian motion or Wiener process with a finite-trace nuclear covariance operator $\mathcal{Q}\ge 0$. The R-L integral of order $(2-\mathtt{v})$ (where $\mathtt{v}=\mathsf{q}+\mathsf{p}(2-\mathsf{q})$) is denoted by the symbol $I_{0^{+}}^{2-\mathtt{v}}$. The function f and $\mathcal{H}$ will be defined later. The initial conditions include ${\mathtt{x}}_0,{\mathtt{x}}_1\in L_0^2(\Omega ,\mathcal{H}).$

We organized the paper as follows:

$\left(i\right)$

The fundamental definitions of fractional calculus, multi-valued maps, and fixed point theory are given in Section 2 and are used to determine the main conclusions of the work.

$\left(ii\right)$

In Section 3, we prove that system (1) exists and show that it is approximately controlled by employing a fixed point approach.

$\left(iii\right)$

Finally, Section 4 provides a theoretical example.

2. Preliminaries

Given a normal filtration ${\mathfrak{F}}_{\chi }$, where $\chi \in \mathcal{I}=[0,\mathfrak{y}]$, let $(\Omega ,\mathfrak{F},{\left\{{\mathfrak{F}}_{\chi }\right\}}_{\chi \ge 0},\mathtt{P})$ be a complete probability space. Throughout the paper, we make use of the following:

• Take the real separable Hilbert spaces $\mathcal{H}$, $\mathcal{G}$.
• A Wiener process with linear bounded covariance operator $\mathcal{Q}$ such that $Tr\left(\mathcal{Q}\right)<\infty$ is represented as $\left\{\mathcal{W}\right(\chi ):\chi \ge 0\}$.
• From $\mathcal{H}$ to $\mathcal{G}$, the set of bounded linear operators is represented by $L(\mathcal{H},\mathcal{G})$.
• Let us assume that there is a complete orthonormal system ${\left\{{\mathtt{e}}_m\right\}}_{m\ge 1}$ in $\mathcal{G}$, a bounded sequence series of non-negative real numbers ${\mathtt{h}}_m$∋$\mathcal{Q}{\mathtt{e}}_m={\mathtt{h}}_m$, $m\in \mathbb{N}$ and a sequence of independent Brownian motions ${\left\{{\mathtt{Y}}_m\right\}}_{m\ge 1}$ such that

  $$\begin{array}{c}\hfill \langle \mathcal{W}\left(\chi \right),\mathtt{e}\rangle =\sum _{m=1}^{\infty }\sqrt{{\mathtt{h}}_m}\langle {\mathtt{e}}_m,\mathtt{e}\rangle {\mathtt{Y}}_m\left(\chi \right),\mathtt{e}\in \mathcal{G},\chi \in \mathcal{I}.\end{array}$$
• Let $L_0^2=L^2({\mathcal{Q}}^{{\displaystyle \frac{1}{2}}}\mathcal{G},\mathcal{H})$ represent the space of all Hilbert–Schmidt operators with the inner product ${\langle \mathtt{k},\mathtt{l}\rangle }_{\mathcal{Q}}=Tr\left[\mathtt{k}\mathcal{Q}\mathtt{l}\right]$ from ${\mathcal{Q}}^{{\displaystyle \frac{1}{2}}}\mathcal{G}$ to $\mathcal{H}$.
• Letting $L^2(\Omega ,\mathcal{H})$ be a Banach space containing all strongly measurable, square-integrable, and $\mathcal{H}$-valued random variables, we may define the norm as follows: ${\parallel \mathtt{x}(·)\parallel }_{L^2(\Omega ,\mathcal{H})}={\left({E\parallel \mathtt{x}(·,\mathcal{W})\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}$, where $E\left(\mathtt{x}\right)={\int }_{\Omega }\mathtt{x}\left(\mathcal{W}\right)d\mathtt{P}$ defines the expectation.
• Let $L_0^2(\Omega ,\mathcal{H})=\{y\in L_0^2(\Omega ,\mathcal{H}),y\mathrm{is}{\mathfrak{F}}_0-\mathrm{measurable}\}.$
• $L_{\mathfrak{F}}^2(\mathcal{I},\mathcal{Z}):=\{\mathtt{x}:\mathcal{I}\times \Omega \to \mathcal{Z}\mathrm{is}\mathrm{a}\mathrm{square}\mathrm{integrable}\mathrm{and}{\mathfrak{F}}_{\chi }-\mathrm{adapted}\mathrm{process}\}$.
• $L_{\mathfrak{F}}^2(\mathcal{I},\mathcal{Z})$ is the space of all ${\mathfrak{F}}_{\chi }$-adapted, $\mathcal{Z}$-valued measurable square integrable processes on $\mathcal{I}\times \Omega$.
• For any continuous map from $\mathcal{I}$ into $L^2(\Omega ,\mathcal{H})$, let $\mathfrak{C}(\mathcal{I},L^2(\Omega ,\mathcal{H}))$ be the Banach space and its norm be ${\parallel \mathtt{x}(·)\parallel }_{\mathfrak{C}}=({sup}_{\chi \in \mathcal{I}}{E\parallel \mathtt{x}\left(\chi \right)\parallel }^2{)}^{{\displaystyle \frac{1}{2}}}<\infty .$
• ${\mathfrak{C}}_{2-\mathtt{v}}(\mathcal{I},L^2(\Omega ,\mathcal{H})):=\{\mathtt{x}\in \mathfrak{C}({\mathcal{I}}^{\prime },L^2(\Omega ,\mathcal{H})):{lim}_{\chi \to 0^{+}}{\chi }^{2-\mathtt{v}}\mathtt{x}\left(\chi \right)\mathrm{exists}\mathrm{and}\mathrm{finite}\}$; with norm ${\parallel \mathtt{x}(·)\parallel }_{\mathfrak{S}}=({sup}_{\chi \in {\mathcal{I}}^{\prime }}E\parallel {\chi }^{2-\mathtt{v}}\mathtt{x}\left(\chi \right){{\parallel }^2)}^{{\displaystyle \frac{1}{2}}}$. Here, ${\mathfrak{C}}_{2-\mathtt{v}}=\mathfrak{S}$ is a Banach space.

Definition 1 

([34]). The R-L fractional integral is defined as follows:

$$I_{0^{+}}^{\mathsf{q}}\omega \left(\chi \right)={\displaystyle \frac{1}{\Gamma \left(\mathsf{q}\right)}}{\int }_0^{\chi }{\displaystyle \frac{\omega \left(\theta \right)}{{(\chi -\theta )}^{1-\mathsf{q}}}}d\theta ,\chi >0,\mathsf{q}>0.$$

where $\Gamma (·)$ is the Gamma function.

Definition 2 

([34]). The R-LFD is defined as follows:

$$\begin{array}{c}{ }^{RL}{D}_{0^{+}}^{\mathsf{q}}\omega \left(\chi \right)={\displaystyle \frac{1}{\Gamma (j-\mathsf{q})}}{\displaystyle \frac{d^j}{d{\chi }^j}}{\int }_0^{\chi }{\displaystyle \frac{\omega \left(\theta \right)}{{(\chi -\theta )}^{\mathsf{q}+1-j}}}d\theta ,\chi >0,j-1<\mathsf{q}<j.\end{array}$$

Definition 3 

([34]). The CFD is defined as follows:

$$\begin{array}{c}{ }^C{D}_{0^{+}}^{\mathsf{q}}\omega \left(\chi \right)={\displaystyle \frac{1}{\Gamma (j-\mathsf{q})}}{\int }_0^{\chi }{\displaystyle \frac{{\omega }^{\left(j\right)}\left(\theta \right)}{{(\chi -\theta )}^{\mathsf{q}+1-j}}}d\theta ,\chi >0,j-1<\mathsf{q}<j.\end{array}$$

Definition 4 

([11]). The HFD is defined as follows:

$$D_{0^{+}}^{\mathsf{p},\mathsf{q}}\omega \left(\chi \right)=(I_{0^{+}}^{\mathsf{p}(j-\mathsf{q})}{\displaystyle \frac{d^j}{d{\chi }^j}}\left(I_{0^{+}}^{(1-\mathsf{p})(j-\mathsf{q})}\omega \right))\left(\chi \right),\chi >0.$$

where $j-1<\mathsf{q}<j,$$0\le \mathsf{p}\le 1$.

Remark 1. 

Based on Definition 4, it follows that

(i) 

If $\mathsf{p}=0$, $j-1<\mathsf{q}<j,$ we have

$$D_{0^{+}}^{0,\mathsf{q}}\omega \left(\chi \right)={\displaystyle \frac{d^j}{d{\chi }^j}}{I}_{0^{+}}^{(j-\mathsf{q})}\omega \left(\chi \right)={ }^{RL}{D}_{0^{+}}^{\mathsf{q}}\omega \left(\chi \right).$$

(ii) 

If $\mathsf{p}=1$, $j-1<\mathsf{q}<j,$ we have

$$D_{0^{+}}^{1,\mathsf{q}}\omega \left(\chi \right)=I_{0^{+}}^{(j-\mathsf{q})}{\displaystyle \frac{d^j}{d{\chi }^j}}\omega \left(\chi \right)={ }^C{D}_{0^{+}}^{\mathsf{q}}\omega \left(\chi \right).$$

Definition 5 

([35]). If $\left(I_{0^{+}}^{2-\mathtt{v}}\omega \right)\left(\chi \right)$ is continuous, and ${\left(I_{0^{+}}^{2-\mathtt{v}}\omega \right)}^{\prime }\left(\chi \right)$ is absolutely continuous, then

$$\begin{array}{c}\hfill I_{0^{+}}^{\mathtt{v}}{(}^{RL}{D}_{0^{+}}^{\mathtt{v}}\omega \left(\chi \right))=\omega \left(\chi \right)-{\displaystyle \frac{\left(I_{0^{+}}^{2-\mathtt{v}}\omega \right)\left(0\right)}{\Gamma (\mathtt{v}-1)}}{\chi }^{\mathtt{v}-2}-{\displaystyle \frac{{\left(I_{0^{+}}^{2-\mathtt{v}}\omega \right)}^{\prime }\left(0\right)}{\Gamma \left(\mathtt{v}\right)}}{\chi }^{\mathtt{v}-1},\end{array}$$

where $\mathtt{v}\in (1,2).$

We will provide an overview of multi-valued analytical facts in the following pages. More details can be found in [36,37].

Definition 6. 

(i) A multivalued map $\mathcal{H}:\mathcal{H}\to 2^{\mathcal{H}}$ is convex (closed) valued for a Hilbert space $\mathcal{H}$, if $\mathcal{H}\left(\mathtt{x}\right)$ is convex (closed) for any $\mathtt{x}\in \mathcal{H}$. If $\mathtt{x}\in \mathcal{V}$, then $\mathcal{H}\left(\mathcal{V}\right)$ is bounded on bounded sets. For every bounded set $\mathcal{V}$ of $\mathcal{H}$ (that is, ${sup}_{\mathtt{x}\in \mathcal{V}}\{sup\{\parallel y\parallel :y\in \mathcal{H}\left(\mathtt{x}\right)\left\}\right\}<\infty )$, $\mathcal{H}\left(\mathtt{x}\right)$ is bounded in $\mathcal{H}$.

(ii) 

If, for every ${\mathtt{x}}^{*}\in \mathcal{H}$, the set $\mathcal{H}\left({\mathtt{x}}^{*}\right)$ is a nonempty, closed subset of $\mathcal{H}$, and if, ∀ open set $\mathcal{V}$ of $\mathcal{H}$ containing $\mathcal{H}\left({\mathtt{x}}^{*}\right)$, ∃ an open neighbourhood $\mathtt{N}$ of ${\mathtt{x}}^{*}$ such that $\mathcal{H}\left(\mathtt{N}\right)\in \mathcal{V}$, then $\mathcal{H}$ is termed upper semicontinuous (u.s.c.) on $\mathcal{H}$.

(iii) 

When $\tilde{\mathcal{V}}$ is a relatively compact bounded subset of every bounded subset $\tilde{\mathcal{V}}\subseteq \mathcal{H}$, then $\mathcal{H}$ is considered to be completely continuous.

(iv) 

When a multivalued map $\mathcal{H}$ has nonempty compact values and is completely continuous, it can only be said to be u.s.c. if it has a closed graph, meaning that ${\mathtt{x}}_n\to {\mathtt{x}}_{*}$, $y_n\to y_{*}$, and $y_n\in \mathcal{H}\left({\mathtt{x}}_n\right)$ imply $y_{*}\in \mathcal{H}\left({\mathtt{x}}_{*}\right)$.

(v) 

If there is an $\mathtt{x}$ in $\mathcal{H}$∋$\mathtt{x}\in \mathcal{H}\left(\mathtt{x}\right)$, then $\mathcal{H}$ has a fixed point.

(vi) 

The multivalued map $\mathcal{H}:\mathcal{I}\to B\mathfrak{C}\mathfrak{C}\left(\mathcal{H}\right)$ is considered measurable if the function $\mathcal{H}:\mathcal{I}\to \mathbb{R}$, defined by

$$\tilde{v}\left(\chi \right):d(\mathtt{x},\mathcal{H}\left(\chi \right))=inf\{\parallel \mathtt{x}-y\parallel :y\in \mathcal{H}\left(\chi \right)\}\in L^1(\mathcal{I},\mathbb{R})$$

is measurable. Here, $B\mathfrak{C}\mathfrak{C}\left(\mathcal{H}\right)$ is the set of all nonempty bounded, closed and convex subsets of $\mathcal{H}$.

Definition 7 

([37]). Let $\mathcal{H}:\mathcal{I}\times \mathcal{H}\to B\mathfrak{CC}\left(\mathcal{H}\right)$ be a $L^2$-Caratheodory if

$\left(i\right)$

For each $\mathtt{x}\in \mathcal{H},$$\chi \to \mathcal{H}(\chi ,\mathtt{x})$ is measurable;

$\left(ii\right)$

For all $\chi \in \mathcal{I},$$\mathtt{x}\to \mathcal{H}(\chi ,\mathtt{x})$ is u.s.c.;

$\left(iii\right)$

For each $\mathfrak{z}>0,$ there exists ${\mathbb{M}}_{\mathfrak{z}}\in L^1(\mathcal{I},\mathbb{R})$, such that

$${\parallel \mathcal{H}(\chi ,\mathtt{x})\parallel }^2=\underset{\mathtt{g}\in \mathcal{H}(\chi ,\mathtt{x})}{sup}{E\parallel \mathtt{g}\parallel }^2\le {\mathbb{M}}_{\mathfrak{z}}\left(\chi \right),\forall {\parallel \mathtt{x}\parallel }^2\le \mathfrak{z}andfora.e.\chi \in \mathcal{I}.$$

Definition 8 

([38]). Suppose $\mathcal{H}$ is a Hilbert space and $\mathcal{I}$ is a compact interval. If $\mathbb{Y}$ is a linear continuous mapping from $L^2(\mathcal{I},\mathcal{H})$ to $\mathfrak{S}$, and $\mathcal{H}$ is a Carath$\acute{e}$odory multivalued map with ${\mathtt{S}}_{\mathcal{H},\mathtt{x}}\ne \{\varnothing \}$, then the operator

$$\begin{array}{c}\hfill \mathbb{Y}\circ {\mathtt{S}}_{\mathcal{H}}:\mathfrak{S}\to B\mathfrak{CC}\left(\mathcal{H}\right),\mathtt{x}\to (\mathbb{Y}\circ {\mathtt{S}}_{\mathcal{H}})\left(\mathtt{x}\right)=\mathbb{Y}\left({\mathtt{S}}_{\mathcal{H},\mathtt{x}}\right)\end{array}$$

is a closed graph operator in $\mathfrak{S}\times \mathfrak{S}$, where $B\mathfrak{C}\mathfrak{C}\left(\mathcal{H}\right)$ is the set of all nonempty bounded, closed and convex subsets of $\mathcal{H}$, and ${\mathtt{S}}_{\mathcal{H},\mathtt{x}}$ is known as the selector set from $\mathcal{H}$, which is given by

$$\begin{array}{c}\hfill \mathtt{g}\in {\mathtt{S}}_{\mathcal{H},\mathtt{x}}=\{\mathtt{g}\in L^2(\mathcal{I},L(\mathcal{G},\mathcal{H})):\mathtt{g}\left(\chi \right)\in \mathcal{H}(\chi ,\mathtt{x})\mathrm{for}a.e.\chi \in \mathcal{I}\}.\end{array}$$

Definition 9 

([39]). Consider the space $\mathbb{X}$, and a bounded linear operator ${\mathtt{Q}\left(\chi \right)}_{\chi \in \mathbb{R}}:\mathcal{H}\to \mathcal{H}$ is considered a strongly continuous cosine family if:

$\left(i\right)$

$\mathtt{Q}(\chi +\mathtt{j})+\mathtt{Q}(\chi -\mathtt{j})=2\mathtt{Q}\left(\chi \right)\mathtt{Q}\left(\mathtt{j}\right)$ for all $\chi ,\mathtt{j}\in \mathbb{R};$

$\left(ii\right)$

$\mathtt{Q}\left(0\right)=I$;

$\left(iii\right)$

$\mathtt{Q}\left(\chi \right)\mathtt{x}$ is continuous in χ on $\mathbb{R}$∀ fixed point $\mathtt{x}\in \mathcal{H}$.

One parameter family, the sine function ${\mathtt{K}\left(\chi \right)}_{\chi \in \mathbb{R}}$, is defined by

$$\begin{array}{c}\hfill \mathtt{K}\left(\chi \right)\mathtt{x}={\int }_0^{\chi }\mathtt{Q}\left(\mathtt{j}\right)\mathtt{x}d\mathtt{j},\chi \in \mathbb{R},\mathtt{x}\in \mathcal{H}.\end{array}$$

where ${\mathtt{Q}\left(\chi \right)}_{\chi \in \mathbb{R}}$ is a strongly continuous cosine family in $\mathcal{H}$.

The operator $\mathcal{A}:\mathcal{H}\to \mathcal{H}$ is the infinitesimal generator of a strongly continuous cosine family ${\mathtt{Q}\left(\chi \right)}_{\chi \in \mathbb{R}}$. It is defined by

$$\begin{array}{c}\hfill \mathcal{A}y={\left({\displaystyle \frac{d^2}{d{\chi }^2}}\mathtt{Q}\left(\chi \right)\mathtt{x}\right)}_{\chi =0},\mathtt{x}\in D\left(A\right).\end{array}$$

Here,

$$\begin{array}{c}\hfill D\left(A\right)=\{y\in \mathcal{H}:\mathtt{Q}\left(\chi \right)\mathtt{x}\in {\mathfrak{C}}^2(\mathbb{R},\mathcal{H})\}.\end{array}$$

Lemma 1 

([39]). Strongly continuous cosine family ${\left\{\mathtt{Q}\left(\chi \right)\right\}}_{\chi \in \mathbb{R}}$ on $\mathcal{H}$ satisfying ${\parallel \mathtt{Q}\left(\chi \right)\parallel }_{\mathcal{H}}\le \mathfrak{I}{e}^{\mathtt{m}\left|\chi \right|},$ in $\mathcal{H}$, for all $\chi \ge 0$ and some $\mathtt{m}\ge 0$, $\mathfrak{I}\ge 1$, and $\mathcal{A}$ is the infinitesimal generator of ${\left\{\mathtt{Q}\left(\chi \right)\right\}}_{\chi \in \mathbb{R}}$. Then, for $Re$$\kappa >\mathtt{m}$, ${\kappa }^2\in \rho \left(\mathcal{A}\right)$ and

$$\begin{array}{c}\hfill \kappa \mathtt{R}({\kappa }^2;\mathcal{A})\mathtt{x}={\int }_0^{\infty }{e}^{-\kappa \chi }\mathtt{Q}\left(\chi \right)\mathtt{x}d\chi ,\mathtt{R}({\kappa }^2;\mathcal{A})\mathtt{x}={\int }_0^{\infty }{e}^{-\kappa \chi }\mathtt{K}\left(\chi \right)\mathtt{x}d\chi ,for\mathtt{x}\in \mathcal{H}.\end{array}$$

In this paper, since $\mathcal{A}$ is the infinitesimal generator of a strongly continuous cosine family of uniformly bounded linear operators ${\left\{\mathtt{Q}\left(\chi \right)\right\}}_{\chi \ge 0}$ in $\mathcal{H}$, there exists a constant $\mathfrak{I}\ge 1$ such that ${\parallel \mathtt{Q}\left(\chi \right)\parallel }_{L\left(\mathcal{H}\right)}\le \mathfrak{I}$ and ${\parallel \mathtt{K}\left(\chi \right)\parallel }_{L\left(\mathcal{H}\right)}\le \mathfrak{I}\chi$ for $\chi \ge 0$.

Definition 10 

([14]). $\mathtt{x}:{\mathcal{I}}^{\prime }\to \mathcal{H}$ is an ${\mathfrak{F}}_{\chi }$-adapted stochastic process; it is said to be the mild solution of the Cauchy problem (1), if $\mathtt{x}\in \mathfrak{S}$, $w(·)\in L_{\mathfrak{F}}^2(\mathcal{I},\mathcal{Z})$ and $\mathtt{g}\in L_0^2(\Omega ,\mathcal{H})$ such that $\mathtt{g}\left(\chi \right)\in \mathcal{H}(\chi ,\mathtt{x}(\chi \left)\right)$, ${\mathtt{x}}_0,{\mathtt{x}}_1\in L_0^2(\Omega ,\mathcal{H})$ and

$$\begin{array}{cc}\hfill \mathtt{x}\left(\chi \right)=& \mathcal{Y}\left(\chi \right){\mathtt{x}}_0+\mathcal{S}\left(\chi \right){\mathtt{x}}_1+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathcal{B}w\left(\mathtt{j}\right)d\mathtt{j}+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathtt{j}\hfill \\ & +{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathtt{g}\left(\mathtt{j}\right)d\mathcal{W}(\mathtt{j}),\hfill \end{array}$$

(2)

for $\chi \in {\mathcal{I}}^{\prime },\mathfrak{w}=\mathsf{p}(2-\mathsf{q})\in (0,1),\mathsf{q}=2\omega ,$ where

$$\begin{array}{cc}\hfill \mathcal{Y}\left(\chi \right)& ={ }^{RL}{D}_{0^{+}}^{1-\mathfrak{w}}{\chi }^{\omega -1}{\mathcal{Q}}_{\omega }\left(\chi \right),\mathcal{S}\left(\chi \right)=I_{0^{+}}^{\mathfrak{w}}{\chi }^{\omega -1}{\mathcal{Q}}_{\omega }\left(\chi \right),\mathcal{O}\left(\chi \right)={\chi }^{\omega -1}{\mathcal{Q}}_{\omega }\left(\chi \right),\hfill \\ \hfill {\mathcal{Q}}_{\omega }\left(\chi \right)& ={\int }_0^{\infty }\omega \mathfrak{k}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({\chi }^{\omega }\mathfrak{k}\right)d\mathfrak{k}.\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\mathfrak{P}}_{ı}\left(\mathfrak{k}\right)=\sum _{\mathfrak{l}=1}^{\infty }{\displaystyle \frac{{(-\mathfrak{k})}^{\mathfrak{l}-1}}{(\mathfrak{l}-1)!\Gamma (1-ı\mathfrak{l})}},0<ı<1,\mathfrak{k}\in \mathbb{C},\end{array}$$

is a function of a Wright type and ${\int }_0^{\infty }{\mathfrak{k}}^{\mathfrak{b}}{\mathfrak{P}}_{ı}\left(\mathfrak{k}\right)d\mathfrak{k}={\displaystyle \frac{\Gamma (1+\mathfrak{b})}{\Gamma (1+ı\mathfrak{b})}},for\mathfrak{b}\ge 0.$

Lemma 2 

([14]). The operator ${\mathcal{Q}}_{\omega }$ has the following properties:

$\left(i\right)$

For all ${\chi }_2,{\chi }_1\ge 0,$ the operators ${\mathcal{Q}}_{\omega }\left(\chi \right)$ are uniformly continuous;

$$\parallel {\mathcal{Q}}_{\omega }\left({\chi }_2\right)-{\mathcal{Q}}_{\omega }\left({\chi }_1\right)\parallel \to 0,as{\chi }_2\to {\chi }_1.$$

$\left(ii\right)$

The inequality below is true for any $\mathtt{x}\in \mathcal{H}$ and $\chi >0$,

$$\begin{array}{cc}\hfill & \parallel {\mathcal{Q}}_{\omega }\left(\chi \right)\mathtt{x}\parallel \le {\displaystyle \frac{\mathfrak{I}{\chi }^{\omega }}{\Gamma \left(2\omega \right)}}\parallel \mathtt{x}\parallel ,\parallel { }^{RL}{D}_{0^{+}}^{1-\mathfrak{w}}\left({\chi }^{\omega -1}{\mathcal{Q}}_{\omega }\left(\chi \right)\mathtt{x}\right)\parallel \le {\displaystyle \frac{\mathfrak{I}{\chi }^{2\omega +\mathfrak{w}-2}}{(2\omega -1)\Gamma (\mathfrak{w}+2\omega -1)}}\parallel \mathtt{x}\parallel ,\hfill \\ \hfill & \parallel I_{0^{+}}^{\mathfrak{w}}\left({\chi }^{\omega -1}{\mathcal{Q}}_{\omega }\left(\chi \right)\mathtt{x}\right)\parallel \le {\displaystyle \frac{\mathfrak{I}{\chi }^{2\omega +\mathfrak{w}-1}}{\Gamma (\mathfrak{w}+2\omega )}}\parallel \mathtt{x}\parallel .\hfill \end{array}$$

Definition 11. 

System (1) is said to be approximately controllable on $\mathcal{I}$ if

$$\overline{\mathfrak{R}(\mathfrak{y},{\mathtt{x}}_0,{\mathtt{x}}_1)}=L^2(\Omega ,\mathcal{H}),$$

where $\mathfrak{R}(\mathfrak{y},{\mathtt{x}}_0,{\mathtt{x}}_1)=\{{\mathtt{x}}_{\mathfrak{y}}({\mathtt{x}}_0,{\mathtt{x}}_1,w):w(·)\in L_{\mathfrak{F}}^2(\mathcal{I},\mathcal{Z})\}.$

To examine the approximate controllability of system (1), we present the following operators.

$\left(i\right)$

The controllability Grammian ${⨿}_0^{\mathfrak{y}}$ is defined by

$$\begin{array}{c}\hfill {⨿}_0^{\mathfrak{y}}={\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathtt{j})\mathcal{B}{\mathcal{B}}^{*}{\mathcal{O}}^{*}(\mathfrak{y}-\mathtt{j})d\mathtt{j}:\mathcal{H}\to \mathcal{H},\end{array}$$

where ${\mathcal{B}}^{*}$ and ${\mathcal{O}}^{*}\left(\chi \right)$ denotes the adjoint operator of $\mathcal{B}$ and $\mathcal{O}\left(\chi \right)$.

$\left(ii\right)$

The resolvent operator $\mathtt{R}(\hslash ,{⨿}_0^{\mathfrak{y}})={(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}:\mathcal{H}\to \mathcal{H}.$

We now assume the following hypothesis:

$\left({\mathbf{H}}_{\mathbf{0}}\right)$:

$\hslash \mathtt{R}(\hslash ,{⨿}_0^{\mathfrak{y}})\to 0^{+}$ as $\hslash \to 0^{+}$ in the strong topology.

Consider the following deterministic linear system associated with (1)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}_{0^{+}}^{\mathsf{p},\mathsf{q}}\mathtt{x}\left(\chi \right)=\mathcal{A}\mathtt{x}\left(\chi \right)+\mathcal{B}w\left(\chi \right),\chi \in {\mathcal{I}}^{\prime }=(0,\mathfrak{y}],\hfill \\ \left(I_{0^{+}}^{2-\mathtt{v}}\mathtt{x}\right)\left(0\right)={\mathtt{x}}_0,{\left(I_{0^{+}}^{2-\mathtt{v}}\mathtt{x}\right)}^{\prime }\left(0\right)={\mathtt{x}}_1,\hfill \end{array}\right.\end{array}$$

(3)

is approximately controllable on $[0,\mathfrak{y}]$ if and only if condition $\left({\mathbf{H}}_{\mathbf{0}}\right)$ is satisfied; refer to [40,41].

Lemma 3 

([42]). Let $\mathfrak{D}$ be a nonempty subset of $\mathcal{H}$ that is bounded, closed, and convex. Assume $\mathcal{H}:\mathfrak{D}\to 2^{\mathcal{H}}\setminus \{\varnothing \}$ is u.s.c. with closed, convex values, and that $\mathcal{H}\left(\mathfrak{D}\right)\subseteq \mathfrak{D}$ and $\mathcal{H}\left(\mathfrak{D}\right)$ is compact. In this case, $\mathcal{H}$ has a fixed point.

Lemma 4 

([40]). For any ${\mathtt{x}}_{\mathfrak{y}}\in L^2({\mathfrak{F}}_{\mathfrak{y}},\mathcal{H}),$ there exists $\mathtt{z}\in L_{\mathfrak{F}}^2(\Omega ;L^2(\mathcal{I},L(\mathcal{G},\mathcal{H})))$ such that

$${\mathtt{x}}_{\mathfrak{y}}=E{\mathtt{x}}_{\mathfrak{y}}+{\int }_0^{\mathfrak{d}}\mathtt{z}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right).$$

3. Main Results

To explore the existence results for the system (1), we present the following hypotheses:

(H₁):

The continuous function $f:\mathcal{I}\times \mathcal{H}\to \mathcal{H}$, and there exists a positive constant ${\mathcal{L}}_{\mathfrak{z}}>0$ such that the function satisfies that

$$E\parallel f(\chi ,\mathtt{x})-f(\chi ,{\mathtt{x}}_1){\parallel }^2\le {\mathcal{L}}_{\mathfrak{z}}{\chi }^{2(2-\mathtt{v})}\parallel \mathtt{x}-{\mathtt{x}}_1{\parallel }^2,E\parallel f(\chi ,\mathtt{x}){\parallel }^2\le {\mathcal{L}}_{\mathfrak{z}}(1+{\chi }^{2(2-\mathtt{v})}{\parallel \mathtt{x}\parallel }^2).$$

(H₂):

The multi-valued map $\mathcal{H}:\mathcal{I}\times \mathcal{H}\to B\mathfrak{CC}\left(L\right(\mathcal{G},\mathcal{H}\left)\right)$ is an $L^2$-Caratheodory function that satisfies the below conditions:

$\left(i\right)$

For each $\chi \in \mathcal{I},$ the function $\mathcal{H}(·,\mathtt{x}):\mathcal{H}\to B\mathfrak{CC}\left(L\right(\mathcal{G},\mathcal{H}\left)\right)$ is u.s.c; and for each $\mathtt{x}\in \mathcal{H},$ the function $\mathcal{H}(·,\mathtt{x})$ is measurable. And, for each fixed $\mathtt{x}\in \mathfrak{C},$ the set

$${\mathtt{S}}_{\mathcal{H},\mathtt{x}}=\{\mathtt{g}\in L^2(\mathcal{I},L(\mathcal{G},\mathcal{H})):\mathtt{g}\left(\chi \right)\in \mathcal{H}(\chi ,\mathtt{x})\mathrm{for}\mathrm{a}.\mathrm{e}.\chi \in \mathcal{I}\},$$

is non-empty.

$\left(ii\right)$

There exists a positive function ${\mathbb{M}}_{\mathfrak{z}}:\mathcal{I}\to {\mathbb{R}}^{+}$ such that

$${sup\{E\parallel \mathtt{g}\parallel }^2:\mathtt{g}\left(\chi \right)\in \mathcal{H}(\chi ,\mathtt{x})\}\le {\mathbb{M}}_{\mathfrak{z}}\left(\chi \right),$$

for a.e. $\chi \in \mathcal{I}$ and the function $\mathtt{j}\to {(\chi -\mathtt{j})}^{2(2\omega -1)}{\mathbb{M}}_{\mathfrak{z}}\left(\mathtt{j}\right)\in L^1(\mathcal{I},{\mathbb{R}}^{+})$ such that

$$\underset{\mathfrak{z}\to \infty }{lim}inf{\displaystyle \frac{{\int }_0^{\chi }{(\chi -\mathtt{j})}^{2(2\omega -1)}{\mathbb{M}}_{\mathfrak{z}}\left(\mathtt{j}\right)d\mathtt{j}}{\mathfrak{z}}}=\Lambda <\infty .$$

For any $\hslash >0$, we define the control function for system (1) as:

$$w^{\hslash }(\chi ,\mathtt{x})={\mathcal{B}}^{*}{\mathcal{O}}_{\omega }^{*}(\mathfrak{y}-\chi )\mathtt{R}(\hslash ,{⨿}_0^{\mathfrak{y}})\mathfrak{M}\left(\mathtt{x}(·)\right),$$

where

$$\begin{array}{cc}\hfill \mathfrak{M}\left(\mathtt{x}\right(·\left)\right)=& E{\mathtt{x}}_{\mathfrak{y}}+{\int }_0^{\mathfrak{y}}\mathtt{z}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right)-\mathcal{Y}\left(\mathfrak{y}\right){\mathtt{x}}_0-\mathcal{S}\left(\mathfrak{y}\right){\mathtt{x}}_1-{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathtt{j})f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathtt{j}\hfill \\ & -{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathtt{j})\mathtt{g}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right).\hfill \end{array}$$

Theorem 1. 

Assume hypotheses $\left({\mathbf{H}}_{\mathbf{0}}\right)$–$\left({\mathbf{H}}_{\mathbf{2}}\right)$ are satisfied. Then, the system (1) has a mild solution on $[0,\mathfrak{y}]$, provided that

$$\begin{array}{c}\hfill \left(1+5{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{I}}_{\mathcal{B}}}{{\hslash }^{\frac{1}{2}}\Gamma \left(2\omega \right)}}\right)}^4\left({\displaystyle \frac{{\mathfrak{y}}^{8\omega -2}}{2\omega (6\omega -2)}}\right)\right)5\left[{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2\omega -\mathtt{v}+2}}{\Gamma \left(2\omega +1\right)}}\right)}^2{\mathcal{L}}_{\mathfrak{z}}+Tr\left(\mathcal{Q}\right)\Lambda {\left({\displaystyle \frac{{\mathfrak{y}}^{2-\mathtt{v}}}{\Gamma \left(2\omega \right)}}\right)}^2\right]<1,\end{array}$$

where $\parallel \mathcal{B}\parallel \le {\mathfrak{I}}_{\mathcal{B}}.$

Proof. 

To demonstrate that system (1) has mild solutions, convert it into a fixed point problem. For any $\hslash >0$, take the operator $\mathfrak{H}:\mathfrak{S}\to 2^{\mathfrak{S}}$ defined by

$$\begin{array}{cc}\hfill \mathfrak{H}\left(\mathtt{x}\right)=\{\tau \in \mathfrak{S}:\tau (\chi )=& \mathcal{Y}\left(\chi \right){\mathtt{x}}_0+\mathcal{S}\left(\chi \right){\mathtt{x}}_1+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathcal{B}w\left(\mathtt{j}\right)d\mathtt{j}+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathtt{j}\hfill \\ & +{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathtt{g}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right),\mathtt{g}\in {\mathtt{S}}_{\mathcal{H},\mathtt{x}}\}.\hfill \end{array}$$

Our goal is to demonstrate that $\mathfrak{H}$ has a fixed point. We separate the proof into many steps for the ease of application.

Step 1. $\mathfrak{H}$ is convex for each $\mathtt{x}\in \mathfrak{S}$. If ${\tau }_1,{\tau }_2\in \mathfrak{H}$, then there exists ${\mathtt{g}}_1,{\mathtt{g}}_2\in {\mathtt{S}}_{\mathcal{H},\mathtt{x}}$, such that

$$\begin{array}{cc}\hfill {\tau }_i\left(\chi \right)=& \mathcal{Y}\left(\chi \right){\mathtt{x}}_0+\mathcal{S}\left(\chi \right){\mathtt{x}}_1+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathcal{B}{\mathcal{B}}^{*}{\mathcal{O}}^{*}(\mathfrak{y}-\mathtt{j})\mathtt{R}(\hslash ,{⨿}_0^{\mathfrak{y}})[E{\mathtt{x}}_{\mathfrak{y}}\hfill \\ & +{\int }_0^{\mathfrak{y}}\mathtt{z}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)-\mathcal{Y}\left(\mathfrak{y}\right){\mathtt{x}}_0-\mathcal{S}\left(\mathfrak{y}\right){\mathtt{x}}_1-{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x})f(\mathfrak{x},\mathtt{x}\left(\mathfrak{x}\right))d\mathfrak{x}\hfill \\ & -{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x}){\mathtt{g}}_i\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)]d\mathtt{j}+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathtt{j}\hfill \\ & +{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j}){\mathtt{g}}_i\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right),i=1,2.\hfill \end{array}$$

Let us assume that $0\le \mathfrak{s}\le 1$. Then for each $\chi \in \mathcal{I}$, we have

$$\begin{array}{cc}\hfill \mathfrak{s}{\tau }_1\left(\chi \right)+(1-\mathfrak{s}){\tau }_2\left(\chi \right)=& \mathcal{Y}\left(\chi \right){\mathtt{x}}_0+\mathcal{S}\left(\chi \right){\mathtt{x}}_1+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathcal{B}{\mathcal{B}}^{*}{\mathcal{O}}^{*}(\mathfrak{y}-\mathtt{j})\mathtt{R}(\hslash ,{⨿}_0^{\mathfrak{y}})\hfill \\ & \times [E{\mathtt{x}}_{\mathfrak{y}}+{\int }_0^{\mathfrak{y}}\mathtt{z}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)-\mathcal{Y}\left(\mathfrak{y}\right){\mathtt{x}}_0-\mathcal{S}\left(\mathfrak{y}\right){\mathtt{x}}_1\hfill \\ & -{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x})f(\mathfrak{x},\mathtt{x}\left(\mathfrak{x}\right))d\mathfrak{x}\hfill \\ & -{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x})[\mathfrak{s}{\mathtt{g}}_1\left(\mathfrak{x}\right)+(1-\mathfrak{s}){\mathtt{g}}_2\left(\mathfrak{x}\right)]d\mathcal{W}\left(\mathfrak{x}\right)]d\mathtt{j}\hfill \\ & +{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathtt{j}\hfill \\ & +{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})[\mathfrak{s}{\mathtt{g}}_1\left(\mathtt{j}\right)+(1-\mathfrak{s}){\mathtt{g}}_2\left(\mathtt{j}\right)]d\mathcal{W}\left(\mathtt{j}\right).\hfill \end{array}$$

Since ${\mathtt{S}}_{\mathcal{H},\mathtt{x}}$ is convex, $\mathfrak{s}{\mathtt{g}}_1+(1-\mathfrak{s}){\mathtt{g}}_2\in {\mathtt{S}}_{\mathcal{H},\mathtt{x}}$. Hence, $\mathfrak{s}{\tau }_1+(1-\mathfrak{s}){\tau }_2\in \mathfrak{H}\left(y\right).$

Step 2. We show that ∃$\mathfrak{z}>0$ such that $\mathfrak{H}\left({\mathtt{B}}_{\mathfrak{z}}\right)\subseteq {\mathtt{B}}_{\mathfrak{z}}$. Here, ${\mathtt{B}}_{\mathfrak{z}}={\{\mathtt{x}\in \mathfrak{S};\parallel \mathtt{x}\parallel }^2\le \mathfrak{z}\}$ is a bounded, closed, convex set in $\mathfrak{S}$.

If it is not true, then there exists $\hslash >0$ such that for every positive number $\mathfrak{z}$ and $\chi \in \mathcal{I}$, there exists a function ${\mathtt{x}}_{\mathfrak{z}}(·)\in {\mathtt{B}}_{\mathfrak{z}}$, but $\mathfrak{H}\left({\mathtt{x}}_{\mathfrak{z}}\right)\notin {\mathtt{B}}_{\mathfrak{z}}$, that is, $E\parallel \mathfrak{H}\left({\mathtt{x}}_{\mathfrak{z}}\right)\left(\chi \right){\parallel }^2\ge \mathfrak{z}$. For such $\hslash >0$, we can show that

$$\begin{array}{cc}\hfill \mathfrak{z}\le & \underset{\chi \in {\mathcal{I}}^{\prime }}{sup}{\chi }^{2(2-\mathtt{v})}5\{E\parallel \mathcal{Y}\left(\chi \right){\mathtt{x}}_0{\parallel }^2+E{\parallel \mathcal{S}\left(\chi \right){\mathtt{x}}_1\parallel }^2+E\parallel {\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathcal{B}{w}^{\hslash }(\mathtt{j},{\mathtt{x}}_{\mathfrak{z}})d\mathtt{j}{\parallel }^2\hfill \\ & +E\parallel {\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathtt{j}{\parallel }^2+E\parallel {\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathtt{g}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right){\parallel }^2\},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill E\parallel w^{\hslash }(\mathtt{j},{\mathtt{x}}_{\mathfrak{z}}){\parallel }^2\le & 5{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{I}}_{\mathcal{B}}}{\Gamma \left(2\omega \right)}}\right)}^2{\left({\displaystyle \frac{{(\mathfrak{y}-\chi )}^{2\omega -1}}{\hslash }}\right)}^2\{{\parallel E{\mathtt{x}}_{\mathfrak{y}}+{\int }_0^{\mathfrak{y}}\mathtt{z}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)\parallel }^2\hfill \\ & +{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2\omega +\mathfrak{w}-2}}{(2\omega -1)\Gamma (\mathfrak{w}+2\omega -1)}}\right)}^{2}E\parallel {\mathtt{x}}_0{\parallel }^2+{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2\omega +\mathfrak{w}-1}}{\Gamma (\mathfrak{w}+2\omega )}}\right)}^{2}E{\parallel {\mathtt{x}}_1\parallel }^2\hfill \\ & +{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2\omega }}{\Gamma (2\omega +1)}}\right)}^2{\mathcal{L}}_{\mathfrak{z}}(1+\mathfrak{z})+{\left({\displaystyle \frac{\mathfrak{I}}{\Gamma \left(2\omega \right)}}\right)}^{2}Tr\left(\mathcal{Q}\right){\int }_0^{\mathfrak{y}}{(\mathfrak{y}-\mathfrak{x})}^{2(2\omega -1)}{\mathbb{M}}_{\mathfrak{z}}\left(\mathfrak{x}\right)d\mathfrak{x}\}.\hfill \end{array}$$

By applying H$\ddot{\mathrm{o}}$lder’s inequality and $\left({\mathbf{H}}_{\mathbf{0}}\right)$–$\left({\mathbf{H}}_{\mathbf{2}}\right)$, we obtain

$$\begin{array}{cc}\hfill \mathfrak{z}\le & 5{\left({\displaystyle \frac{\mathfrak{I}}{(2\omega -1)\Gamma (2\omega +\mathfrak{w}-1)}}\right)}^{2}E\parallel {\mathtt{x}}_0{\parallel }^2+5{\left({\displaystyle \frac{\mathfrak{I}\mathfrak{y}}{\Gamma (\mathfrak{w}+2\omega )}}\right)}^{2}E{\parallel {\mathtt{x}}_1\parallel }^2+5{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{I}}_{\mathcal{B}}}{{\hslash }^{\frac{1}{2}}\Gamma \left(2\omega \right)}}\right)}^4\left({\displaystyle \frac{{\mathfrak{y}}^{8\omega -2\mathtt{v}+2}}{2\omega (6\omega -2)}}\right)\hfill \\ & \times \left[5\right\{E\parallel E{\mathtt{x}}_{\mathfrak{y}}+{\int }_0^{\mathfrak{y}}\mathtt{z}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right){\parallel }^2+{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2\omega +\mathfrak{w}-2}}{(2\omega -1)\Gamma (\mathfrak{w}+2\omega -1)}}\right)}^{2}E\parallel {\mathtt{x}}_0{\parallel }^2+{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2\omega +\mathfrak{w}-1}}{\Gamma (\mathfrak{w}+2\omega )}}\right)}^{2}E{\parallel {\mathtt{x}}_1\parallel }^2\hfill \\ & +{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2\omega }}{\Gamma (2\omega +1)}}\right)}^2{\mathcal{L}}_{\mathfrak{z}}(1+\mathfrak{z})+{\left({\displaystyle \frac{\mathfrak{I}}{\Gamma \left(2\omega \right)}}\right)}^{2}Tr\left(\mathcal{Q}\right){\int }_0^{\mathfrak{y}}{(\mathfrak{y}-\mathfrak{x})}^{2(2\omega -1)}{\mathbb{M}}_{\mathfrak{z}}\left(\mathfrak{x}\right)d\mathfrak{x}\left\}\right]\hfill \\ & +5{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2\omega -\mathtt{v}+2}}{\Gamma (2\omega +1)}}\right)}^2{\mathcal{L}}_{\mathfrak{z}}(1+\mathfrak{z})+5Tr\left(\mathcal{Q}\right){\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2-\mathtt{v}}}{\Gamma 2\omega }}\right)}^2{\int }_0^{\chi }{(\chi -\mathtt{j})}^{2(2\omega -1)}{\mathbb{M}}_{\mathfrak{z}}\left(\mathtt{j}\right)d\mathtt{j}.\hfill \end{array}$$

If we divide the above inequality by on both sides $\mathfrak{z}$ and assuming $\mathfrak{z}\to \infty$, we get that

$$\begin{array}{c}\hfill \left(1+5{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{I}}_{\mathcal{B}}}{{\hslash }^{\frac{1}{2}}\Gamma \left(2\omega \right)}}\right)}^4\left({\displaystyle \frac{{\mathfrak{y}}^{8\omega -2}}{2\omega (6\omega -2)}}\right)\right)5\left[{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2\omega -\mathtt{v}+2}}{\Gamma \left(2\omega +1\right)}}\right)}^2{\mathcal{L}}_{\mathfrak{z}}+Tr\left(\mathcal{Q}\right)\Lambda {\left({\displaystyle \frac{{\mathfrak{y}}^{2-\mathtt{v}}}{\Gamma \left(2\omega \right)}}\right)}^2\right]\ge 1,\end{array}$$

which is a contradiction to our assumption. Thus, for $\hslash >0$, for some positive number $\mathfrak{z}$ and some $\mathtt{g}\in {\mathtt{S}}_{\mathcal{H},\mathtt{x}}$, $\mathfrak{H}\left({\mathtt{B}}_{\mathfrak{z}}\right)\subseteq {\mathtt{B}}_{\mathfrak{z}}$.

Step 3. $\mathfrak{H}$ is equicontinuous on ${\mathtt{B}}_{\mathfrak{z}}.$ Take $\mathtt{E}=\{\mathfrak{f}\in \mathfrak{C}(\Omega ,L^2(\mathcal{G},\mathcal{H}));\mathfrak{f}\in {\chi }^{2-\mathtt{v}}\left(\mathfrak{H}\mathtt{x}\right)\left(\chi \right);\mathfrak{f}\left(0\right)=\mathfrak{f}\left(0^{+}\right);\mathtt{x}\in {\mathtt{B}}_{\mathfrak{z}}\}$ for ${\chi }_1=0$, $0<{\chi }_2\le \mathfrak{y}$; we can easily get $E\parallel \mathfrak{f}\left({\chi }_2\right)-\mathfrak{f}\left(0\right){\parallel }^2\to 0^{+}$ as ${\chi }_2\to 0^{+}$. For $0<{\chi }_1<{\chi }_2\le \mathfrak{y}$, there exists $\mathtt{g}\in {\mathtt{S}}_{\mathcal{H},\mathtt{x}}$, and we get

$$\begin{array}{cc}\hfill \mathfrak{f}\left(\chi \right)=& {\chi }^{2-\mathtt{v}}\{\mathcal{Y}\left(\chi \right){\mathtt{x}}_0+\mathcal{S}\left(\chi \right){\mathtt{x}}_1+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathcal{B}w\left(\mathtt{j}\right)d\mathtt{j}+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathtt{j}\hfill \\ & +{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathtt{g}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right)\}.\hfill \end{array}$$

Note that if we apply $C_r$-inequality, then we get

$$\begin{array}{cc}\hfill E\parallel \mathfrak{f}\left({\chi }_2\right)-\mathfrak{f}\left({\chi }_1\right){\parallel }^2\le & 13{\left({\chi }_2^{2-\mathtt{v}}-{\chi }_1^{2-\mathtt{v}}\right)}^{2}E\parallel \mathcal{Y}\left({\chi }_2\right){\mathtt{x}}_0{\parallel }^2+13{\chi }_1^{2(2-\mathtt{v})}E{\parallel \mathcal{Y}\left({\chi }_2\right){\mathtt{x}}_0-\mathcal{Y}\left({\chi }_1\right){\mathtt{x}}_0\parallel }^2\hfill \\ & +13{\left({\chi }_2^{2-\mathtt{v}}-{\chi }_1^{2-\mathtt{v}}\right)}^{2}E\parallel \mathcal{S}\left({\chi }_2\right){\mathtt{x}}_1{\parallel }^2+13{\chi }_1^{2(2-\mathtt{v})}E{\parallel \mathcal{S}\left({\chi }_2\right){\mathtt{x}}_1-\mathcal{S}\left({\chi }_1\right){\mathtt{x}}_1\parallel }^2\hfill \\ & +13E\parallel {\chi }_1^{2-\mathtt{v}}{\int }_{{\chi }_1}^{{\chi }_2}\mathcal{O}({\chi }_2-\mathtt{j})\mathcal{B}{w}^{\hslash }(\mathtt{j},\mathtt{x})d\mathtt{j}{\parallel }^2\hfill \\ & +13E\parallel {\chi }_1^{2-\mathtt{v}}{\int }_0^{{\chi }_1}\left(\mathcal{O}({\chi }_2-\mathtt{j})-\mathcal{O}({\chi }_1-\mathtt{j})\right)\mathcal{B}{w}^{\hslash }(\mathtt{j},\mathtt{x})d\mathtt{j}{\parallel }^2\hfill \\ & +13{({\chi }_2^{2-\mathtt{v}}-{\chi }_1^{2-\mathtt{v}})}^{2}E\parallel {\int }_0^{{\chi }_2}\mathcal{O}({\chi }_2-\mathtt{j})\mathcal{B}{w}^{\hslash }(\mathtt{j},\mathtt{x})d\mathtt{j}{\parallel }^2\hfill \\ & +13E\parallel {\chi }_1^{2-\mathtt{v}}{\int }_{{\chi }_1}^{{\chi }_2}\mathcal{O}({\chi }_2-\mathtt{j})f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathtt{j}{\parallel }^2\hfill \\ & +13E\parallel {\chi }_1^{2-\mathtt{v}}{\int }_0^{{\chi }_1}\left(\mathcal{O}({\chi }_2-\mathtt{j})-\mathcal{O}({\chi }_1-\mathtt{j})\right)f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathtt{j}{\parallel }^2\hfill \\ & +13{({\chi }_2^{2-\mathtt{v}}-{\chi }_1^{2-\mathtt{v}})}^{2}E\parallel {\int }_0^{{\chi }_2}\mathcal{O}({\chi }_2-\mathtt{j})f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathtt{j}{\parallel }^2\hfill \\ & +13E\parallel {\chi }_1^{2-\mathtt{v}}{\int }_{{\chi }_1}^{{\chi }_2}\mathcal{O}({\chi }_2-\mathtt{j})\mathtt{g}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right){\parallel }^2\hfill \\ & +13E\parallel {\chi }_1^{2-\mathtt{v}}{\int }_0^{{\chi }_1}\left(\mathcal{O}({\chi }_2-\mathtt{j})-\mathcal{O}({\chi }_1-\mathtt{j})\right)\mathtt{g}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right){\parallel }^2\hfill \\ & +13{({\chi }_2^{2-\mathtt{v}}-{\chi }_1^{2-\mathtt{v}})}^{2}E\parallel {\int }_0^{{\chi }_2}\mathcal{O}({\chi }_2-\mathtt{j})\mathtt{g}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right){\parallel }^2\hfill \\ \hfill :=& \sum _{♭=1}^{13}{\mathcal{M}}_{♭}.\hfill \end{array}$$

By applying H$\ddot{\mathrm{o}}$lder’s inequality and $\left({\mathbf{H}}_{\mathbf{0}}\right)$–$\left({\mathbf{H}}_{\mathbf{2}}\right)$, we obtain

$$\begin{array}{cc}\hfill {\mathcal{M}}_1\le & 13{\left({\chi }_2^{2-\mathtt{v}}-{\chi }_1^{2-\mathtt{v}}\right)}^2{\left({\displaystyle \frac{\mathfrak{I}}{\Gamma \left(\mathfrak{w}\right)\Gamma \left(2\omega \right)}}\right)}^2{\left({\int }_0^{{\chi }_2}({\chi }_2-\mathtt{j}){\mathtt{j}}^{2\omega -2}d\mathtt{j}\right)}^{2}E{\parallel {\mathtt{x}}_0\parallel }^2\hfill \\ \hfill \le & 13{\left({\chi }_2^{2-\mathtt{v}}-{\chi }_1^{2-\mathtt{v}}\right)}^2{\left({\displaystyle \frac{\mathfrak{I}{\chi }_2^{\mathfrak{w}+2\omega -2}}{(2\omega -1)\Gamma (\mathfrak{w}+2\omega -1)}}\right)}^{2}E{\parallel {\mathtt{x}}_0\parallel }^2,\hfill \\ \hfill {\mathcal{M}}_2\le & 26{\left({\displaystyle \frac{\mathfrak{I}{\chi }_1^{2-\mathtt{v}}}{\Gamma \left(\mathfrak{w}\right)\Gamma \left(2\omega \right)}}\right)}^2\{{\left({\int }_{{\chi }_1}^{{\chi }_2}{({\chi }_2-\mathtt{j})}^{\mathfrak{w}-1}{\mathtt{j}}^{2\omega -2}d\mathtt{j}\right)}^2\hfill \\ & +{\left({\int }_0^{{\chi }_1}[{({\chi }_2-\mathtt{j})}^{\mathfrak{w}-1}-{({\chi }_1-\mathtt{j})}^{\mathfrak{w}-1}]{\mathtt{j}}^{2\omega -2}d\mathtt{j}\right)}^2\}E{\parallel {\mathtt{x}}_0\parallel }^2,\hfill \\ \hfill {\mathcal{M}}_3\le & 13{\left({\chi }_2^{2-\mathtt{v}}-{\chi }_1^{2-\mathtt{v}}\right)}^2{\left({\displaystyle \frac{\mathfrak{I}}{\Gamma \left(\mathfrak{w}\right)\Gamma \left(2\omega \right)}}\right)}^2{\left({\int }_0^{{\chi }_2}({\chi }_2-\mathtt{j}){\mathtt{j}}^{2\omega -1}d\mathtt{j}\right)}^{2}E{\parallel {\mathtt{x}}_1\parallel }^2\hfill \\ \hfill \le & 13{\left({\chi }_2^{2-\mathtt{v}}-{\chi }_1^{2-\mathtt{v}}\right)}^2{\left({\displaystyle \frac{\mathfrak{I}{\chi }_2^{\mathfrak{w}+2\omega -1}}{\Gamma (\mathfrak{w}+2\omega )}}\right)}^{2}E{\parallel {\mathtt{x}}_1\parallel }^2,\hfill \\ \hfill {\mathcal{M}}_4\le & 26{\left({\displaystyle \frac{\mathfrak{I}{\chi }_1^{2-\mathtt{v}}}{\Gamma \left(2\omega \right)\Gamma \left(\mathfrak{w}\right)}}\right)}^2\{{\left({\int }_{{\chi }_1}^{{\chi }_2}{({\chi }_2-\mathtt{j})}^{\mathfrak{w}-1}{\mathtt{j}}^{2\omega -1}d\mathtt{j}\right)}^2\hfill \\ & +{\left({\int }_0^{{\chi }_1}[{({\chi }_2-\mathtt{j})}^{\mathfrak{w}-1}-{({\chi }_1-\mathtt{j})}^{\mathfrak{w}-1}]{\mathtt{j}}^{2\omega -1}d\mathtt{j}\right)}^2\}E{\parallel {\mathtt{x}}_1\parallel }^2,\hfill \\ \hfill {\mathcal{M}}_5\le & 13{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{I}}_{\mathcal{B}}{\chi }_1^{2-\mathtt{v}}}{\Gamma \left(2\omega \right)}}\right)}^2\left({\displaystyle \frac{{({\chi }_2-{\chi }_1)}^{2\omega }}{2\omega }}\right){\int }_{{\chi }_2}^{{\chi }_1}{({\chi }_2-\mathtt{j})}^{2\omega -1}E{\parallel w^{\hslash }(\mathtt{j},\mathtt{x})\parallel }^{2}d\mathtt{j},\hfill \\ \hfill {\mathcal{M}}_6\le & 13{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{I}}_{\mathcal{B}}{\chi }_1^{2-\mathtt{v}}}{(2\omega -1)\Gamma \left(2\omega \right)}}\right)}^2{\left({\int }_0^{{\chi }_1}\left({({\chi }_2-\mathtt{j})}^{2\omega -1}-{({\chi }_1-\mathtt{j})}^{2\omega -1}\right)E\parallel w^{\hslash }(\mathtt{j},\mathtt{x})\parallel d\mathtt{j}\right)}^2,\hfill \\ \hfill {\mathcal{M}}_7\le & 13{\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{I}}_{\mathcal{B}}({\chi }_2^{2-\mathtt{v}}-{\chi }_1^{2-\mathtt{v}})}{\Gamma \left(2\omega \right)}}\right)}^2\left({\displaystyle \frac{{\chi }_2^{2\omega }}{2\omega }}\right){\int }_0^{{\chi }_2}{({\chi }_2-\mathtt{j})}^{2\omega -1}E{\parallel w^{\hslash }(\mathtt{j},\mathtt{x})\parallel }^{2}d\mathtt{j},\hfill \\ \hfill {\mathcal{M}}_8\le & 13{\left({\displaystyle \frac{\mathfrak{I}{\chi }_1^{2-\mathtt{v}}}{\Gamma \left(2\omega \right)}}\right)}^2{\left({\displaystyle \frac{{({\chi }_2-{\chi }_1)}^{2\omega }}{2\omega }}\right)}^2{\mathcal{L}}_{\mathfrak{z}}(1+\mathfrak{z}),\hfill \\ \hfill {\mathcal{M}}_9\le & 13{\left({\displaystyle \frac{\mathfrak{I}{\chi }_1^{2-\mathtt{v}}}{(2\omega -1)\Gamma \left(2\omega \right)}}\right)}^2{\left({\int }_0^{{\chi }_1}\left({({\chi }_2-\mathtt{j})}^{2\omega -1}-{({\chi }_1-\mathtt{j})}^{2\omega -1}\right)E\parallel f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))\parallel d\mathtt{j}\right)}^2,\hfill \\ \hfill {\mathcal{M}}_{10}\le & 13{\left({\displaystyle \frac{\mathfrak{I}({\chi }_2^{2-\mathtt{v}}-{\chi }_1^{2-\mathtt{v}})}{\Gamma \left(2\omega \right)}}\right)}^2{\left({\displaystyle \frac{{\chi }_2^{2\omega }}{2\omega }}\right)}^2{\mathcal{L}}_{\mathfrak{z}}(1+\mathfrak{z}),\hfill \\ \hfill {\mathcal{M}}_{11}\le & 13Tr\left(\mathcal{Q}\right){\left({\displaystyle \frac{\mathfrak{I}{\chi }_1^{2-\mathtt{v}}}{\Gamma \left(2\omega \right)}}\right)}^2{\int }_{{\chi }_1}^{{\chi }_2}{({\chi }_2-\mathtt{j})}^{2(2\omega -1)}{\mathbb{M}}_{\mathfrak{z}}\left(\mathtt{j}\right)d\mathtt{j},\hfill \\ \hfill {\mathcal{M}}_{12}\le & 13Tr\left(\mathcal{Q}\right){\left({\displaystyle \frac{\mathfrak{I}{\chi }_1^{2-\mathtt{v}}}{(2\omega -1)\Gamma \left(2\omega \right)}}\right)}^2{\int }_0^{{\chi }_1}{\left({({\chi }_2-\mathtt{j})}^{2\omega -1}-{({\chi }_1-\mathtt{j})}^{2\omega -1}\right)}^2{\mathbb{M}}_{\mathfrak{z}}\left(\mathtt{j}\right)d\mathtt{j},\hfill \\ \hfill {\mathcal{M}}_{13}\le & 13Tr\left(\mathcal{Q}\right){\left({\displaystyle \frac{\mathfrak{I}({\chi }_2^{2-\mathtt{v}}-{\chi }_1^{2-\mathtt{v}})}{\Gamma \left(2\omega \right)}}\right)}^2{\int }_0^{{\chi }_2}{({\chi }_2-\mathtt{j})}^{2(2\omega -1)}{\mathbb{M}}_{\mathfrak{z}}\left(\mathtt{j}\right)d\mathtt{j}.\hfill \end{array}$$

According to the above analysis, ${\mathcal{M}}_1$ to ${\mathcal{M}}_{13}$ tends to $0^{+}$, i.e., ${lim}_{{\chi }_2-{\chi }_1}E{\parallel \mathfrak{f}\left({\chi }_2\right)-\mathfrak{f}\left({\chi }_1\right)\parallel }^2\to 0$ for ${\chi }_1,{\chi }_2\in [0,\mathfrak{y}]$; therefore, recalling the relationship of $\mathtt{E}$ and $\mathfrak{H}\left({\mathtt{B}}_{\mathfrak{z}}\right)$, one can easily deduce that $\mathfrak{H}$ is equicontinuous on ${\mathtt{B}}_{\mathfrak{z}}$.

Step 4. Next, we show that the set $\mathcal{E}\left(\chi \right)=\{\tau \left(\chi \right)\in \left(\mathfrak{H}\mathtt{x}\right)\left(\chi \right):\mathtt{x}\in {\mathtt{B}}_{\mathfrak{z}}\}$ is relatively compact in $L^2(\Omega ,\mathcal{H})$.

For $\chi =0$, it is evident that $\mathcal{E}\left(\chi \right)$ is relatively compact in $L^2(\Omega ,\mathcal{H})$ for $\chi \in [0,\mathfrak{y}]$. Assume that $0<\chi \le \mathfrak{y}$ is fixed. Then, we define the operator ${\tau }_{ϵ,\mathtt{i}}$ for any $ϵ\in (0,\chi )$, arbitrary $\mathtt{i}>0$ and $\mathtt{x}\in {\mathtt{B}}_{\mathfrak{z}}$ by

$$\begin{array}{cc}\hfill {\tau }_{ϵ,\mathtt{i}}\left(\chi \right)=& \mathcal{Y}\left(\chi \right){\mathtt{x}}_0+\mathcal{S}\left(\chi \right){\mathtt{x}}_1\hfill \\ & +{\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}{\int }_0^{\chi -ϵ}{\int }_{\mathtt{i}}^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}-{ϵ}^{\omega }\mathtt{i}\right)\mathcal{B}{w}^{\hslash }(\mathtt{j},\mathtt{x})d\mathfrak{k}d\mathtt{j}\hfill \\ & +{\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}{\int }_0^{\chi -ϵ}{\int }_{\mathtt{i}}^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}-{ϵ}^{\omega }\mathtt{i}\right)f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathfrak{k}d\mathtt{j}\hfill \\ & +{\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}{\int }_0^{\chi -ϵ}{\int }_{\mathtt{i}}^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}-{ϵ}^{\omega }\mathtt{i}\right)g\left(\mathtt{j}\right)d\mathfrak{k}d\mathcal{W}\left(\mathtt{j}\right).\hfill \end{array}$$

If ${\left\{\mathtt{K}\left(\chi \right)\right\}}_{\chi >0}$ is compact, then $\left\{{\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}\right\}$ is also compact. Then, for any $ϵ\in (0,\chi )$ and any $\mathtt{i}>0$, we can deduce that ${\mathcal{E}}_{ϵ,\mathtt{i}}\left(\chi \right)=\{{\tau }_{ϵ,\mathtt{i}}\left(\chi \right)\in \left(\mathfrak{H}\mathtt{x}\right)\left(\chi \right):\mathtt{x}\in {\mathtt{B}}_{\mathfrak{z}}\}$ is relatively compact in $L^2(\Omega ,\mathcal{H})$. In addition, for any $\mathtt{x}\in {\mathtt{B}}_{\mathfrak{z}}$, we obtain

$$\begin{array}{cc}\hfill E\parallel \tau \left(\chi \right)-{\tau }_{ϵ,\mathtt{i}}{\left(\chi \right)\parallel }^2\le & \underset{\chi \in {\mathcal{I}}^{\prime }}{sup}{\chi }^{2(2-\mathtt{v})}3E\parallel {\int }_0^{\chi }{\int }_0^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)\mathcal{B}{w}^{\hslash }(\mathtt{j},\mathtt{x})d\mathfrak{k}d\mathtt{j}\hfill \\ & -{\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}{\int }_0^{\chi -ϵ}{\int }_{\mathtt{i}}^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}-{ϵ}^{\omega }\mathtt{i}\right)\mathcal{B}{w}^{\hslash }(\mathtt{j},\mathtt{x})d\mathfrak{k}d\mathtt{j}{\parallel }^2\hfill \\ & +\underset{\chi \in {\mathcal{I}}^{\prime }}{sup}{\chi }^{2(2-\mathtt{v})}3E\parallel {\int }_0^{\chi }{\int }_0^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathfrak{k}d\mathtt{j}\hfill \\ & -{\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}{\int }_0^{\chi -ϵ}{\int }_{\mathtt{i}}^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}-{ϵ}^{\omega }\mathtt{i}\right)f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathfrak{k}d\mathtt{j}{\parallel }^2\hfill \\ & +\underset{\chi \in {\mathcal{I}}^{\prime }}{sup}{\chi }^{2(2-\mathtt{v})}3E\parallel {\int }_0^{\chi }{\int }_0^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)\mathtt{g}\left(\mathtt{j}\right)d\mathfrak{k}d\mathcal{W}\left(\mathtt{j}\right)\hfill \\ & -{\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}{\int }_0^{\chi -ϵ}{\int }_{\mathtt{i}}^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}-{ϵ}^{\omega }\mathtt{i}\right)\mathtt{g}\left(\mathtt{j}\right)d\mathfrak{k}d\mathcal{W}\left(\mathtt{j}\right){\parallel }^2\hfill \\ \hfill :=& 3\sum _{i=1}^3{\mathfrak{A}}_i.\hfill \end{array}$$

For $i=1$, we have

$$\begin{array}{cc}\hfill {\mathfrak{A}}_1\le & \underset{\chi \in {\mathcal{I}}^{\prime }}{sup}{\chi }^{2(2-\mathtt{v})}3\{E\parallel {\int }_0^{\chi }{\int }_0^{\mathtt{i}}\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)\mathcal{B}{w}^{\hslash }(\mathtt{j},\mathtt{x})d\mathfrak{k}d\mathtt{j}{\parallel }^2\hfill \\ & +E\parallel {\int }_{\chi -ϵ}^{\chi }{\int }_{\mathtt{i}}^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)\mathcal{B}{w}^{\hslash }(\mathtt{j},\mathtt{x})d\mathfrak{k}d\mathtt{j}{\parallel }^2\hfill \\ & +E\parallel {\int }_0^{\chi -ϵ}{\int }_{\mathtt{i}}^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\hfill \\ & \times \left(\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)-{\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}-{ϵ}^{\omega }\mathtt{i}\right)\right)\mathcal{B}{w}^{\hslash }(\mathtt{j},\mathtt{x})d\mathfrak{k}d\mathtt{j}{\parallel }^2\}\hfill \\ \hfill :=& 3[{\mathfrak{A}}_{11}+{\mathfrak{A}}_{12}+{\mathfrak{A}}_{13}].\hfill \end{array}$$

To prove ${\mathfrak{A}}_1\to 0$, we obtain

$$\begin{array}{cc}\hfill {\mathfrak{A}}_{11}\le & \left({\displaystyle \frac{{\mathfrak{I}}^2{\mathfrak{I}}_{\mathcal{B}}^2{\mathfrak{y}}^{2(\omega -\mathtt{v}+2)}}{2\omega }}\right){\int }_0^{\chi }{(\chi -\mathtt{j})}^{2\omega -1}E{\parallel w^{\hslash }(\mathtt{j},\mathtt{x})\parallel }^{2}d\mathtt{j}{\left({\int }_0^{\mathtt{i}}\omega {\mathfrak{k}}^2{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)d\mathfrak{k}\right)}^2\to 0,\mathrm{as}\mathtt{i}\to 0.\hfill \\ \hfill {\mathfrak{A}}_{12}\le & {\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{I}}_{\mathcal{B}}{\mathfrak{y}}^{2-\mathtt{v}}}{\Gamma \left(2\omega \right)}}\right)}^2\left({\displaystyle \frac{{ϵ}^{2\omega }}{2\omega }}\right){\int }_{\chi -ϵ}^{\chi }{(\chi -\mathtt{j})}^{2\omega -1}E{\parallel w^{\hslash }(\mathtt{j},\mathtt{x})\parallel }^{2}d\mathtt{j}\to 0,\mathrm{as}ϵ\to 0.\hfill \end{array}$$

Before examining the ${\mathfrak{A}}_{13}$, utilizing $\parallel \mathtt{K}\left(\chi \right)-\mathtt{K}\left(p\right)\parallel \le \mathfrak{I}|\chi -p|$ and the ${lim}_{\chi \to 0}\parallel {\displaystyle \frac{\mathtt{K}\left(\chi \right)y}{\chi }}-y\parallel =0,$ for all $\mathtt{x}\in \mathcal{H},$ we deduce that

$$\begin{array}{cc}\hfill \parallel \mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)-& {\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}-{ϵ}^{\omega }\mathtt{i}\right)\mathtt{x}\parallel \hfill \\ \hfill \le & \parallel \left({\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}-I\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)\mathtt{x}\parallel \hfill \\ & +\parallel {\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}\left(\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}-{ϵ}^{\omega }\mathfrak{k}\right)-\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)\right)\mathtt{x}\parallel \hfill \\ \hfill \le & \mathfrak{I}{(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\parallel \left({\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}-I\right)y\parallel \hfill \\ & +\mathfrak{I}\parallel \left(\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}-{ϵ}^{\omega }\mathfrak{k}\right)-\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)\right)\mathtt{x}\parallel \to 0,\mathrm{as}ϵ,\mathrm{i}\to 0.\hfill \end{array}$$

(4)

Now, we need to prove ${\mathfrak{A}}_{13}\to 0$, then

$$\begin{array}{cc}\hfill {\mathfrak{A}}_{13}\le & {\left({\displaystyle \frac{{\mathfrak{I}}_{\mathcal{B}}\mathfrak{I}(\mathfrak{I}+1){\mathfrak{y}}^{2-\mathtt{v}}}{\Gamma \left(2\omega \right)}}\right)}^2{\left({\int }_0^{\chi -ϵ}{(\chi -\mathtt{j})}^{2\omega -1}E\parallel w^{\hslash }(\mathtt{j},\mathtt{x})\parallel d\mathtt{j}\right)}^2\to 0,\mathrm{as}ϵ,\mathtt{i}\to 0.\hfill \end{array}$$

For $i=2$, we get

$$\begin{array}{cc}\hfill {\mathfrak{A}}_2\le & \underset{\chi \in {\mathcal{I}}^{\prime }}{sup}{\chi }^{2(2-\mathtt{v})}3\{E\parallel {\int }_0^{\chi }{\int }_0^{\mathtt{i}}\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathfrak{k}d\mathtt{j}{\parallel }^2\hfill \\ & +E\parallel {\int }_{\chi -ϵ}^{\chi }{\int }_{\mathtt{i}}^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathfrak{k}d\mathtt{j}{\parallel }^2\hfill \\ & +E\parallel {\int }_0^{\chi -ϵ}{\int }_{\mathtt{i}}^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\hfill \\ & \times \left(\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)-{\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}-{ϵ}^{\omega }\mathtt{i}\right)\right)f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathfrak{k}d\mathtt{j}{\parallel }^2\}\hfill \\ \hfill :=& 3[{\mathfrak{A}}_{21}+{\mathfrak{A}}_{22}+{\mathfrak{A}}_{23}].\hfill \end{array}$$

Let us take (4), and we get

$$\begin{array}{cc}\hfill {\mathfrak{A}}_{21}\le & {\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2\omega -\mathtt{v}+2}}{2\omega }}\right)}^2{\mathcal{L}}_{\mathfrak{z}}(1+\mathfrak{z}){\left({\int }_0^{\mathtt{i}}\omega {\mathfrak{k}}^2{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)d\mathfrak{k}\right)}^2\to 0,\mathrm{as}\mathtt{i}\to 0.\hfill \\ \hfill {\mathfrak{A}}_{22}\le & {\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2-\mathtt{v}}{ϵ}^{2\omega }}{\Gamma (2\omega +1)}}\right)}^2{\mathcal{L}}_{\mathfrak{z}}(1+\mathfrak{z})\to 0,\mathrm{as}ϵ\to 0.\hfill \\ \hfill {\mathfrak{A}}_{23}\le & {\left({\displaystyle \frac{\mathfrak{I}(\mathfrak{I}+1){\mathfrak{y}}^{2-\mathtt{v}}}{\Gamma \left(2\omega \right)}}\right)}^2{\left({\int }_0^{\chi -ϵ}{(\chi -\mathtt{j})}^{2\omega -1}E\parallel f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))\parallel d\mathtt{j}\right)}^2\to 0,\mathrm{as}ϵ,\mathtt{i}\to 0.\hfill \end{array}$$

For $i=3$, we have

$$\begin{array}{cc}\hfill {\mathfrak{A}}_3\le & \underset{\chi \in {\mathcal{I}}^{\prime }}{sup}{\chi }^{2(2-\mathtt{v})}3\{E\parallel {\int }_0^{\chi }{\int }_0^{\mathtt{i}}\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)\mathtt{g}\left(\mathtt{j}\right)d\mathfrak{k}d\mathcal{W}\left(\mathtt{j}\right){\parallel }^2\hfill \\ & +E\parallel {\int }_{\chi -ϵ}^{\chi }{\int }_{\mathtt{i}}^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)\mathtt{g}\left(\mathtt{j}\right)d\mathfrak{k}d\mathcal{W}\left(\mathtt{j}\right){\parallel }^2\hfill \\ & +E\parallel {\int }_0^{\chi -ϵ}{\int }_{\mathtt{i}}^{\infty }\omega \mathfrak{k}{(\chi -\mathtt{j})}^{\omega -1}{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)\hfill \\ & \times \left(\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}\right)-{\displaystyle \frac{\mathtt{K}\left({ϵ}^{\omega }\mathtt{i}\right)}{{ϵ}^{\omega }\mathtt{i}}}\mathtt{K}\left({(\chi -\mathtt{j})}^{\omega }\mathfrak{k}-{ϵ}^{\omega }\mathtt{i}\right)\right)\mathtt{g}\left(\mathtt{j}\right)d\mathfrak{k}d\mathcal{W}\left(\mathtt{j}\right){\parallel }^2\}\hfill \\ \hfill :=& 3[{\mathfrak{A}}_{31}+{\mathfrak{A}}_{32}+{\mathfrak{A}}_{33}].\hfill \end{array}$$

By using the Equation (4), we get

$$\begin{array}{cc}\hfill {\mathfrak{A}}_{31}\le & Tr\left(\mathcal{Q}\right){\left(\mathfrak{I}{\mathfrak{y}}^{2-\mathtt{v}}\right)}^2{\int }_0^{\chi }{(\chi -\mathtt{j})}^{2(2\omega -1)}{\mathbb{M}}_{\mathfrak{z}}\left(\mathtt{j}\right)d\mathtt{j}{\left({\int }_0^{\mathtt{i}}\omega {\mathfrak{k}}^2{\mathfrak{P}}_{\omega }\left(\mathfrak{k}\right)d\mathfrak{k}\right)}^2\to 0,\mathrm{as}\mathtt{i}\to 0.\hfill \\ \hfill {\mathfrak{A}}_{32}\le & Tr\left(\mathcal{Q}\right){\left({\displaystyle \frac{\mathfrak{I}{\mathfrak{y}}^{2-\mathtt{v}}}{\Gamma \left(2\omega \right)}}\right)}^2{\int }_{\chi -ϵ}^{\chi }{(\chi -\mathtt{j})}^{2(2\omega -1)}{\mathbb{M}}_{\mathfrak{z}}\left(\mathtt{j}\right)d\mathtt{j}\to 0,\mathrm{as}ϵ\to 0.\hfill \\ \hfill {\mathfrak{A}}_{33}\le & Tr\left(\mathcal{Q}\right){\left({\displaystyle \frac{\mathfrak{I}(\mathfrak{I}+1){\mathfrak{y}}^{2-\mathtt{v}}}{\Gamma \left(2\omega \right)}}\right)}^2{\int }_0^{\chi -ϵ}{(\chi -\mathtt{j})}^{2(2\omega -1)}{\mathbb{M}}_{\mathfrak{z}}\left(\mathtt{j}\right)d\mathtt{j}\to 0,\mathrm{as}ϵ,\mathtt{i}\to 0.\hfill \end{array}$$

By using the Lebesgue dominated convergence theorem, we derive that ${\mathfrak{A}}_1$ to ${\mathfrak{A}}_3\to 0$. Thus, $E\parallel \tau \left(\chi \right)-{\tau }_{ϵ,\mathtt{i}}{\left(\chi \right)\parallel }^2\to 0$. Therefore, the set $\mathcal{E}\left(\chi \right)$ is relatively compact in $L^2(\Omega ,\mathcal{H})$.

Step 5. $\mathfrak{H}$ has a closed graph.

Let ${\mathtt{x}}_n\to {\mathtt{x}}_{*}$, as $n\to \infty$, ${\tau }_n\in \mathfrak{H}{\mathtt{x}}_n$ for each $\mathtt{x}\in {\mathtt{B}}_{\mathfrak{z}}$, and ${\tau }_n\to {\tau }_{*}$ as $n\to \infty$. We shall show that ${\tau }_n\in \mathfrak{H}{\mathtt{x}}_n$. Since ${\tau }_{*}\in \mathfrak{H}{\mathtt{x}}_{*}$, then there exists ${\mathtt{g}}_n\in {\mathtt{S}}_{\mathcal{H},{\mathtt{x}}_n}$ such that

$$\begin{array}{cc}\hfill {\tau }_n\left(\chi \right)=& \mathcal{Y}\left(\chi \right){\mathtt{x}}_0+\mathcal{S}\left(\chi \right){\mathtt{x}}_1+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathcal{B}{\mathcal{B}}^{*}{\mathcal{O}}^{*}(\mathfrak{y}-\mathtt{j})\mathtt{R}(\hslash ,{⨿}_0^{\mathfrak{y}})[E{\mathtt{x}}_{\mathfrak{y}}\hfill \\ & +{\int }_0^{\mathfrak{y}}\mathtt{z}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)-\mathcal{Y}\left(\mathfrak{y}\right){\mathtt{x}}_0-\mathcal{S}\left(\mathfrak{y}\right){\mathtt{x}}_1-{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x})f(\mathfrak{x},{\mathtt{x}}_n\left(\mathfrak{x}\right))d\mathfrak{x}\hfill \\ & -{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x}){\mathtt{g}}_n\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)]d\mathtt{j}+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})f(\mathtt{j},{\mathtt{x}}_n\left(\mathtt{j}\right))d\mathtt{j}\hfill \\ & +{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j}){\mathtt{g}}_n\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right),\chi \in \mathcal{I}.\hfill \end{array}$$

We have to demonstrate that ${\mathtt{g}}_{*}\in {\mathtt{S}}_{\mathcal{H},{\mathtt{x}}_{*}}$ exists such that

$$\begin{array}{cc}\hfill {\tau }_{*}\left(\chi \right)=& \mathcal{Y}\left(\chi \right){\mathtt{x}}_0+\mathcal{S}\left(\chi \right){\mathtt{x}}_1+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathcal{B}{\mathcal{B}}^{*}{\mathcal{O}}^{*}(\mathfrak{y}-\mathtt{j})\mathtt{R}(\hslash ,{⨿}_0^{\mathfrak{y}})[E{\mathtt{x}}_{\mathfrak{y}}\hfill \\ & +{\int }_0^{\mathfrak{y}}\mathtt{z}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)-\mathcal{Y}\left(\mathfrak{y}\right){\mathtt{x}}_0-\mathcal{S}\left(\mathfrak{y}\right){\mathtt{x}}_1-{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x})f(\mathfrak{x},{\mathtt{x}}_{*}\left(\mathfrak{x}\right))d\mathfrak{x}\hfill \\ & -{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x}){\mathtt{g}}_{*}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)]d\mathtt{j}+{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})f(\mathtt{j},{\mathtt{x}}_{*}\left(\mathtt{j}\right))d\mathtt{j}\hfill \\ & +{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j}){\mathtt{g}}_{*}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right),\chi \in \mathcal{I}.\hfill \end{array}$$

To illustrate that

$$\begin{array}{cc}\hfill E\parallel & ({\tau }_n\left(\chi \right)-\mathcal{Y}\left(\chi \right){\mathtt{x}}_0-\mathcal{S}\left(\chi \right){\mathtt{x}}_1-{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathcal{B}{\mathcal{B}}^{*}{\mathcal{O}}^{*}(\mathfrak{y}-\mathtt{j})\mathtt{R}(\hslash ,{⨿}_0^{\mathfrak{y}})\hfill \\ & \times [E{\mathtt{x}}_{\mathfrak{y}}+{\int }_0^{\mathfrak{y}}\mathtt{z}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)-\mathcal{Y}\left(\mathfrak{y}\right){\mathtt{x}}_0-\mathcal{S}\left(\mathfrak{y}\right){\mathtt{x}}_1-{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x})f(\mathfrak{x},{\mathtt{x}}_n\left(\mathfrak{x}\right))d\mathfrak{x}\hfill \\ & -{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x}){\mathtt{g}}_n\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)]d\mathtt{j}-{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})f(\mathtt{j},{\mathtt{x}}_n\left(\mathtt{j}\right))d\mathtt{j}-{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j}){\mathtt{g}}_n\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right))\hfill \\ & -({\tau }_{*}\left(\chi \right)-\mathcal{Y}\left(\chi \right){\mathtt{x}}_0-\mathcal{S}\left(\chi \right){\mathtt{x}}_1-{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathcal{B}{\mathcal{B}}^{*}{\mathcal{O}}^{*}(\mathfrak{y}-\mathtt{j})\mathtt{R}(\hslash ,{⨿}_0^{\mathfrak{y}})\hfill \\ & \times [E{\mathtt{x}}_{\mathfrak{y}}+{\int }_0^{\mathfrak{y}}\mathtt{z}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)-\mathcal{Y}\left(\mathfrak{y}\right){\mathtt{x}}_0-\mathcal{S}\left(\mathfrak{y}\right){\mathtt{x}}_1-{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x})f(\mathfrak{x},{\mathtt{x}}_{*}\left(\mathfrak{x}\right))d\mathfrak{x}\hfill \\ & -{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x}){\mathtt{g}}_{*}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)]d\mathtt{j}-{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})f(\mathtt{j},{\mathtt{x}}_{*}\left(\mathtt{j}\right))d\mathtt{j}-{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j}){\mathtt{g}}_{*}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right)){\parallel }^2\to 0.\hfill \end{array}$$

Consider the linear continuous operator ${\rm Y}:L^2(\mathcal{I},\mathcal{H})\to \mathfrak{S}$, so we have

$$\begin{array}{cc}\hfill \left({\rm Y}\mathtt{g}\right)\left(\chi \right)=& {\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathtt{g}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right)-{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})B{B}^{*}{\mathcal{Q}}_{\omega }^{*}(\chi -\mathtt{j})\hfill \\ & \times \mathtt{R}(\hslash ,{⨿}_0^{\mathfrak{y}})\left({\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x})\mathtt{g}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)\right)d\mathtt{j}.\hfill \end{array}$$

Moreover, we have

$$\begin{array}{cc}\hfill & ({\tau }_n\left(\chi \right)-\mathcal{Y}\left(\chi \right){\mathtt{x}}_0-\mathcal{S}\left(\chi \right){\mathtt{x}}_1-{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathcal{B}{\mathcal{B}}^{*}{\mathcal{O}}^{*}(\mathfrak{y}-\mathtt{j})\mathtt{R}(\hslash ,{⨿}_0^{\mathfrak{y}})\hfill \\ & \times [E{\mathtt{x}}_{\mathfrak{y}}+{\int }_0^{\mathfrak{y}}\mathtt{z}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)-\mathcal{Y}\left(\mathfrak{y}\right){\mathtt{x}}_0-\mathcal{S}\left(\mathfrak{y}\right){\mathtt{x}}_1\hfill \\ & -{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x})f(\mathfrak{x},{\mathtt{x}}_n\left(\mathfrak{x}\right))d\mathfrak{x}]d\mathtt{j}-{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})f(\mathtt{j},{\mathtt{x}}_n\left(\mathtt{j}\right))d\mathtt{j})\in {\rm Y}\left({\mathtt{S}}_{\mathcal{H},{\mathtt{x}}_n}\right).\hfill \end{array}$$

Since ${\mathtt{x}}_n\to {\mathtt{x}}_{*}$, it follows from Lemma 8 that

$$\begin{array}{cc}\hfill & ({\tau }_{*}\left(\chi \right)-\mathcal{Y}\left(\chi \right){\mathtt{x}}_0-\mathcal{S}\left(\chi \right){\mathtt{x}}_1-{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})\mathcal{B}{\mathcal{B}}^{*}{\mathcal{O}}^{*}(\mathfrak{y}-\mathtt{j})\mathtt{R}(\hslash ,{⨿}_0^{\mathfrak{y}})\hfill \\ & \times [E{\mathtt{x}}_{\mathfrak{y}}+{\int }_0^{\mathfrak{y}}\mathtt{z}\left(\mathfrak{x}\right)d\mathcal{W}\left(\mathfrak{x}\right)-\mathcal{Y}\left(\mathfrak{y}\right){\mathtt{x}}_0-\mathcal{S}\left(\mathfrak{y}\right){\mathtt{x}}_1\hfill \\ & -{\int }_0^{\mathfrak{y}}\mathcal{O}(\mathfrak{y}-\mathfrak{x})f(\mathfrak{x},{\mathtt{x}}_{*}\left(\mathfrak{x}\right))d\mathfrak{x}]d\mathtt{j}-{\int }_0^{\chi }\mathcal{O}(\chi -\mathtt{j})f(\mathtt{j},{\mathtt{x}}_{*}\left(\mathtt{j}\right))d\mathtt{j})\in {\rm Y}\left({\mathtt{S}}_{\mathcal{H},{\mathtt{x}}_{*}}\right),\hfill \end{array}$$

therefore, $\mathfrak{H}$ has a closed graph.

Using the Arzela-Ascoli theorem, we deduce that $\mathfrak{H}$ is a compact multivalued map, u.s.c., with convex closed values, as a result of Steps 1 through 5. From Lemma 3, we may infer that $\mathfrak{H}$ has a mild solution of system (1) with a fixed point $\mathtt{x}(·)$ on ${\mathtt{B}}_{\mathfrak{z}}$. □

Theorem 2. 

Suppose that the assumptions of Theorem 1 hold. Then, the system (1) is approximately controllable on $\mathcal{I}$.

Proof. 

Let ${\mathtt{x}}^{\hslash }(·)\in {\mathtt{B}}_{\mathfrak{z}}$ be an operator $\mathfrak{H}$ fixed point. Any fixed point of $\mathfrak{H}$ is a mild solution of (1) according to Theorem 1. According to the stochastic Fubini theorem, this indicates that there exists ${\mathtt{x}}^{\hslash }\in \mathfrak{H}\left({\mathtt{x}}^{\hslash }\right)$, and ${\mathtt{g}}^{\hslash }\in {\mathtt{S}}_{\mathcal{H},{\mathtt{x}}^{\hslash }}$ such that

$$\begin{array}{cc}\hfill {\mathtt{x}}^{\hslash }=& {\mathtt{z}}_{\mathfrak{y}}-\hslash {(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}\left[E{\mathtt{x}}_{\mathfrak{y}}-\mathcal{Y}\left(\mathfrak{y}\right){\mathtt{x}}_0-\mathcal{S}\left(\mathfrak{y}\right){\mathtt{x}}_1\right]\hfill \\ & -\hslash {\int }_0^{\mathfrak{y}}{(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}\mathtt{z}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right)-\hslash {\int }_0^{\mathfrak{y}}{(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}\mathcal{O}(\mathfrak{y}-\mathtt{j})f(\mathtt{j},\mathtt{x}\left(\mathtt{j}\right))d\mathtt{j}\hfill \\ & -\hslash {\int }_0^{\mathfrak{y}}{(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}\mathcal{O}(\mathfrak{y}-\mathtt{j})\mathtt{g}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right).\hfill \end{array}$$

Furthermore, according to the Dunford–Pettis Theorem and the boundedness of f and $\mathcal{H}$, we have that the sequences $\left\{f(\mathtt{j},{\mathtt{x}}^{\hslash }\left(\mathtt{j}\right))\right\}$, and $\left\{{\mathtt{g}}^{\hslash }\left(\mathtt{j}\right)\right\}$ are weakly compact in $L^2(\mathcal{I},\mathcal{H})$ and $L^2(\mathcal{I},L(\mathcal{G},\mathcal{H}))$. As a result, some subsequences are still denoted by $\left\{f(\mathtt{j},{\mathtt{x}}^{\hslash }\left(\mathtt{j}\right))\right\}$, and $\left\{{\mathtt{g}}^{\hslash }\left(\mathtt{j}\right)\right\}$ that weakly converge to say, f and $\mathcal{H}$, respectively, in $L^2(\mathcal{I},\mathcal{H})$ and $L^2(\mathcal{I},L(\mathcal{G},\mathcal{H}))$.

Conversely, based on the assumption of $\left({\mathbf{H}}_{\mathbf{0}}\right)$, the operator $\hslash {(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}\to 0$ strongly as $\hslash \to 0^{+}$ for every $0\le \mathtt{j}\le \mathfrak{y}$, and additionally, $\parallel \hslash {(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}\parallel \le 1$. Thus, the compactness of ${\mathcal{Q}}_{\omega }\left(\chi \right)$ and the Lebesgue-dominated convergence theorem provide

$$\begin{array}{cc}\hfill E\parallel {\mathtt{x}}^{\hslash }\left(\chi \right)-{\mathtt{x}}_{\mathfrak{y}}{\parallel }^2\le & 6\parallel \hslash {(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}{\parallel }^{2}E\parallel \left[E{\mathtt{x}}_{\mathfrak{y}}-\mathcal{Y}\left(\mathfrak{y}\right){\mathtt{x}}_0-\mathcal{S}\left(\mathfrak{y}\right){\mathtt{x}}_1\right]{\parallel }^2\hfill \\ & +6E\parallel \hslash {\int }_0^{\mathfrak{y}}{(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}\mathtt{z}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right){\parallel }^2\hfill \\ & +6E\parallel \hslash {\int }_0^{\mathfrak{y}}{(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}\mathcal{O}(\mathfrak{y}-\mathtt{j})\left[f(\mathtt{j},{\mathtt{x}}^{\hslash }\left(\mathtt{j}\right))-f\left(\mathtt{j}\right)\right]d\mathtt{j}{\parallel }^2\hfill \\ & +6E\parallel \hslash {\int }_0^{\mathfrak{y}}{(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}\mathcal{O}(\mathfrak{y}-\mathtt{j})f\left(\mathtt{j}\right)d\mathtt{j}{\parallel }^2\hfill \\ & +6E\parallel \hslash {\int }_0^{\mathfrak{y}}{(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}\mathcal{O}(\mathfrak{y}-\mathtt{j})\left[{\mathtt{g}}^{\hslash }\left(\mathtt{j}\right)-\mathtt{g}\left(\mathtt{j}\right)\right]d\mathcal{W}\left(\mathtt{j}\right){\parallel }^2\hfill \\ & +6E\parallel \hslash {\int }_0^{\mathfrak{y}}{(\hslash I+{⨿}_0^{\mathfrak{y}})}^{-1}\mathcal{O}(\mathfrak{y}-\mathtt{j})\mathtt{g}\left(\mathtt{j}\right)d\mathcal{W}\left(\mathtt{j}\right){\parallel }^2\to 0,\mathrm{as}\hslash \to 0^{+}.\hfill \end{array}$$

It follows that $E{\parallel {\mathtt{x}}^{\hslash }\left(\chi \right)-{\mathtt{x}}_{\mathfrak{y}}\parallel }^2\to 0^{+}$ holds, demonstrating that the proof is complete and the system (1) is approximately controllable. □

4. Example

Consider the upcoming Hilfer fractional stochastic control system of the form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{\chi }^{\mathsf{p},\mathsf{q}}\mathfrak{h}(\chi ,\mathfrak{j})\in {\partial }_{\mathfrak{j}}^2\mathfrak{h}(\chi ,\mathfrak{j})+\varrho (\chi ,\mathfrak{j})+\tilde{f}(\chi ,\mathfrak{h}(\chi ,\mathfrak{j}))+\tilde{\mathfrak{g}}(\chi ,\mathfrak{h}(\chi ,\mathfrak{j})){\displaystyle \frac{d\mathcal{W}\left(\chi \right)}{d\chi }},\chi \in (0,\mathfrak{y}],\hfill \\ \mathfrak{j}\in [0,\pi ],\hfill \\ \mathfrak{h}(\chi ,0)=\mathfrak{h}(\chi ,\pi )=0,\chi \in (0,\mathfrak{y}],\hfill \\ \left(I_{0^{+}}^{2-\mathtt{v}}\mathfrak{h}\right)(0,\mathfrak{j})={\mathfrak{h}}_0\left(\mathfrak{j}\right),{\left(I_{0^{+}}^{2-\mathtt{v}}\mathfrak{h}\right)}^{\prime }(0,\mathfrak{j})={\mathfrak{h}}_1\left(\mathfrak{j}\right),\mathfrak{j}\in [0,\pi ],\hfill \end{array}\right.\end{array}$$

(5)

where ${\partial }_{\chi }^{\mathsf{p},\mathsf{q}}\mathfrak{h}(\chi ,\mathfrak{m})$ is the HF partial derivative $\mathsf{q}\in (1,2)$, $\mathsf{p}\in [0,1]$, and $I_{0^{+}}^{2-\mathtt{v}}$ is an R-L integral with order $2-\mathtt{v}$ ($\mathtt{v}=\mathsf{q}+\mathsf{p}(2-\mathsf{q})$).

Take $\mathcal{H}=\mathcal{Z}=L^2\left([0,\pi ]\right)$ and defined $\mathcal{A}$ satisfies $A\mathfrak{h}={\displaystyle \frac{d^2}{d{\mathfrak{j}}^2}}\mathfrak{h}$, $D\left(A\right)=\{\mathfrak{h}\in \mathcal{H}:{\mathfrak{h}}^{\prime \prime }\in \mathcal{H},\mathfrak{h}\left(0\right)=\mathfrak{h}\left(\pi \right)=0;{\mathfrak{h}}^{\prime },{\mathfrak{h}}^{\prime \prime }\mathrm{are}\mathrm{absolutely}\mathrm{continuous}\}.$ Then, for any $\chi \ge 0$, the uniformly bounded strongly continuous cosine family $\left\{\mathtt{Q}\right(\chi \left)\right\}$ has $\mathcal{A}$ as its infinitesimal generator. Take ${\mathfrak{X}}_{\mathfrak{m}}\left(\chi \right)=\sqrt{{\displaystyle \frac{2}{\pi }}}sin\left(\mathfrak{m}\pi \chi \right),$ implying that $(-{\mathfrak{m}}^2,\mathfrak{m}\in \mathbb{N})$ are eigenvalues of $\mathcal{A}$, and that ${\left\{{\mathfrak{X}}_{\mathfrak{m}}\right\}}_{\mathfrak{m}=1}^{\infty }$ is an orthonormal basis of $\mathcal{H}$. Now,

$$A\mathfrak{h}=-\sum _{\mathfrak{m}}^{\infty }{\mathfrak{m}}^2\langle \mathfrak{h},{\mathfrak{X}}_{\mathfrak{m}}\rangle {\mathfrak{X}}_{\mathfrak{m}},\mathfrak{h}\in D\left(A\right),$$

where $\langle ·,·\rangle$ is the inner product $\mathcal{H}$. From [39], we have

$$\begin{array}{c}\hfill \mathtt{Q}\left(\chi \right)\mathfrak{h}=\sum _{\mathfrak{m}=1}^{\infty }cos\left(\mathfrak{m}\pi \chi \right)\langle \mathfrak{h},{\mathfrak{X}}_{\mathfrak{m}}\rangle {\mathfrak{X}}_{\mathfrak{m}},\mathtt{K}\left(\chi \right)\mathfrak{h}=\sum _{\mathfrak{m}=1}^{\infty }{\displaystyle \frac{1}{\mathfrak{m}}}sin\left(\mathfrak{m}\pi \chi \right)\langle \mathfrak{h},{\mathfrak{X}}_{\mathfrak{m}}\rangle {\mathfrak{X}}_{\mathfrak{m}},\mathfrak{h}\in \mathcal{H}.\end{array}$$

According to [39], we have

$$\begin{array}{c}\hfill {\mathcal{Q}}_{\omega }\left(\chi \right)\mathfrak{h}=\sum _{\mathfrak{m}=1}^{\infty }{\chi }^{{\displaystyle \frac{\mathsf{q}}{2}}}{E}_{\mathsf{q},\mathsf{q}}(-{\mathfrak{m}}^2{\chi }^{\mathsf{q}})\langle \mathfrak{h},{\mathfrak{X}}_{\mathfrak{m}}\rangle {\mathfrak{X}}_{\mathfrak{m}},\omega ={\displaystyle \frac{\mathsf{q}}{2}},\end{array}$$

where $E_{\mathsf{q},\mathsf{q}}\left(\mathtt{x}\right)={\sum }_{\mathfrak{m}=0}^{\infty }{\displaystyle \frac{{\mathtt{x}}^{\mathfrak{m}}}{\Gamma \left(\mathsf{q}\right(\mathfrak{m}+1\left)\right)}}$ is the Mittag-Leffler function. Let $\mathtt{x}\left(\chi \right)\mathfrak{m}=\mathfrak{h}(\chi ,\mathfrak{m})$, and $\mathcal{B}w(\chi ,\mathtt{x})\mathfrak{m}=\varrho (\chi ,\mathfrak{m})$. The functions $f(\chi ,\mathtt{x}\left(\chi \right))\mathfrak{m}=\tilde{f}(\chi ,\mathfrak{h}(\chi ,\mathfrak{m}))$ and $\mathcal{H}(\chi ,\mathtt{x}\left(\chi \right))\mathfrak{m}=\tilde{g}(\chi ,\mathfrak{h}(\chi ,\mathfrak{m}))$ are satisfies $\left({\mathbf{H}}_{\mathbf{1}}\right)$ and $\left({\mathbf{H}}_{\mathbf{2}}\right)$.

Since (5) can be formulated as problem (1) in $\mathcal{H}$, it follows that the Theorem 1 satisfies all assumptions, and then that system (5) has a mild solution and is approximately controllable on $\mathcal{I}$.

5. Conclusions

This study examined the approximate controllability of Hilfer fractional stochastic differential inclusions with order $1<\mathsf{q}<2$. Cosine families, Hilfer fractional derivatives, fixed point theory, fractional calculus, and stochastic analysis all form the foundation of our findings. First, we obtain the mild solution for the given system (1) and show that it is approximately controllable. Lastly, an application that demonstrates the theory is presented. Further work will investigate the existence and approximate controllability of Hilfer fractional stochastic integrodifferential systems with order $1<\mathsf{q}<2$ using the noncompactness measure.