Study on Controllability for Ψ-Hilfer Fractional Stochastic Differential Equations

Abstract

The goal of this paper is to study the existence of a mild solution and controllability for a class of neutral stochastic differential equations (SDEs) involving the $\mathrm{\Psi }$-Hilfer fractional derivatives, a generalization of the well-known Riemann–Liouville fractional derivative using almost sectorial operators. Sufficient conditions for controllability are established using the notion of measure of noncompactness (MNC) and the Mönch fixed-point theorem. An example is given to illustrate the abstract findings.

1. Introduction

Over the years, fractional differential equations (FDEs) have been extensively studied by researchers, significantly advancing the mathematical analysis of these equations. FDEs accurately represent certain natural phenomena more accurately than integer-order differential equations. This ability has increased interest and research in the field, as fractional derivatives allow for varying the order of the differential equation, which is closely linked to solving FDEs. Lately, Sousa et al. [1] introduced a fractional mathematical model based on the diffusion equation of time-fractions, which more effectively describes nutrient concentration in the blood compared with the integer-order model. Additional studies and analyses of mathematical models using FDEs can be found in [2,3,4].

Utilizing broader classes of fractional derivatives, which encompass a broad range of fractional derivatives, including the Riemann–Liouville and Caputo derivatives, offers significant advantages. Sousa et al. [5] presented the $\mathrm{\Psi }$-Hilfer derivatives, which include several well-known derivatives, such as the Caputo and Riemann–Liouville derivatives. The benefit of analyzing nonlinear FDEs involving the $\mathrm{\Psi }$-Hilfer derivative lies in its ability to preserve the properties investigated in FDEs with other fractional derivative operators, as detailed in [5]. Foundational studies on the existence, stability, and uniqueness properties of solutions to FDEs involving the $\mathrm{\Psi }$-Hilfer (FDEs) can be found in [6,7,8].

Control theory is a crucial discipline for engineers, mathematicians, and researchers, providing the framework for developing and managing processes and systems. The range of control problems varies from straightforward scenarios like heat conduction through a rod to complex ones such as landing an aircraft on a carrier, managing a nation’s economy, or controlling epidemics. Controllability is a well-known attribute of a control system that is significant in multiple control problems in finite and infinite dimensional space. A great deal of emphasis has been focused on controllability issues for diverse, dynamic systems that are deterministic, stochastic, linear, and nonlinear.

Controllability is typically sufficient in many applications in modern science and technology. Many researchers are concerned with studying the controllability results for various dynamical systems, for instance, [9,10,11,12]. Recently, the controllability result of FDEs involving Caputo derivatives has been extensively studied. However, exploring the controllability concerning the Hilfer FDEs is still in the initial stage. In papers [13,14], researchers examined the approximate controllability of FDEs employing a fixed-point approach. Additionally, in [15,16,17], authors investigated the existence of Hilfer FDEs utilizing almost sectorial operators.

In [18], Bose and Udhayakumar focused on the controllability of the following Hilfer fractional integrodifferential equation of order $0<a<1$ and type $0\le \delta \le 1$, shown as follows,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{D}_{0^{+}}^{a,\delta }\left[x\left(\nu \right)-\mathcal{G}(\nu ,x_{\nu })\right]=Ax\left(\nu \right)+\mathfrak{F}\left({\displaystyle \nu ,x_{\nu },{\int }_0^{\nu }\mathcal{H}(\nu ,s,x_s)ds}\right)+Bv\left(\nu \right),\nu \in (0,d],\hfill & \\ I_{0^{+}}^{(1-a)(1-\delta )}x\left(0\right)=\xi \left(0\right)\in {\mathfrak{B}}_w,\nu \in (-\infty ,0],\hfill \end{array}\right.\end{array}$$

and they proved the controllability results for the considered system via MNC.

Stochastic models should be explored instead of deterministic ones since noise and fluctuations can occur in manufactured and natural systems. Unpredictability exists in the theoretical representation of a particular occurrence in differential equations with stochastic components. For a broad introduction to stochastic differential equations and its uses, see [19,20]. In [21], Sivasankar et al. studied the existence result of the following Hilfer neutral stochastic FDEs,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{D}_{0^{+}}^{\eta ,\zeta }[u\left(\nu \right)-⅁(\nu ,u_{\nu })]=Au\left(\nu \right)+\tilde{\mathcal{F}}(\nu ,u_{\nu })+\tilde{\mathcal{K}}(\nu ,u_{\nu }){\displaystyle \frac{dw\left(\nu \right)}{d\nu }},\nu \in (0,d],\hfill & \\ I_{0^{+}}^{(1-\eta )(1-\zeta )}u\left(0\right)+\tilde{N}\left(u_{\nu }\right)=\alpha \in L^2(\Delta ,{\mathfrak{B}}_{\varrho }),\nu \in (-\infty ,0],\hfill \end{array}\right.\end{array}$$

where order $0<\eta <1$, and type $0\le \zeta \le 1$.

To the best of our knowledge, the controllability of neutral SDEs utilizing the MNC, particularly incorporating the $\mathrm{\Psi }$-Hilfer FDEs with infinite delay, has yet to be addressed in the existing literature. This is the motivation of our work.

In this paper, we consider the following neutral $\mathrm{\Psi }$-Hilfer stochastic FDEs in a separable Hilbert space $H:$

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{D}_{0^{+}}^{\delta ,\gamma ;\mathrm{\Psi }}[x\left(\nu \right)-\mathcal{F}(\nu ,x_{\nu },x\left(\nu \right))]=Ax\left(\nu \right)+\mathcal{G}(\nu ,x_{\nu },x\left(\nu \right))+B\upsilon \left(\nu \right)\hfill & \\ +\mathcal{H}(\nu ,x_{\nu },x\left(\nu \right)){\displaystyle \frac{dw\left(\nu \right)}{d\nu }},\nu \in J=(0,d],\hfill & \\ I_{0^{+}}^{(1-\delta )(1-\gamma );\mathrm{\Psi }}x\left(0\right)=\mathfrak{H}\left(0\right)\in L^2(\mathrm{\Omega },{\mathfrak{D}}_h)\mathrm{for}\mathrm{a}.\mathrm{e}.\nu \in (-\infty ,0],\hfill \end{array}\right.\end{array}$$

(1)

where $1/2<\delta <1,0\le \gamma \le 1$, $D_{0^{+}}^{\delta ,\gamma ;\mathrm{\Psi }}$ denotes the $\mathrm{\Psi }$-Hilfer fractional derivatives with order $\delta$ and type $\gamma$, $I_{0^{+}}^{(1-\delta )(1-\gamma );\mathrm{\Psi }}$ denotes the $\mathrm{\Psi }$-Riemann–Liouville integral of order $(1-\delta )(1-\gamma )$. $x(·)\in H$ is the state function, A represents the almost sectorial operator that produces an analytic semigroup $\left\{T\right(\nu ),\nu \ge 0\}$ on H. The history function $x_{\nu }:(-\infty ,0]\to {\mathfrak{D}}_h$ defined by $x_{\nu }\left(s\right)=x(\nu +s),s\le 0$ takes values in some phase space ${\mathfrak{D}}_h$. Let K and U be another separable Hilbert spaces. $w(·)$ denotes the K-valued Wiener process. The control function $\upsilon \in L^2(J,U)$ is a set of admissible control functions on a separable Hilbert space U, and the operator B is bounded. $\mathcal{F},\mathcal{G}$, and $\mathcal{H}$ are given functions to be specified later.

2. Preliminaries and Assumptions

Let $(\mathrm{\Omega },\mathrm{{\rm Y}},\mathbb{P})$ be a complete probability space, ${\mathrm{{\rm Y}}}_{\nu },\nu \in J$ denotes a normal filtration satisfying the usual condition (i.e., right continuity and ${\mathrm{{\rm Y}}}_0$ contains all $\mathbb{P}$-null sets of $\mathrm{{\rm Y}}$). A stochastic process is a set of random variables $S=\{x(\nu ,\omega ):\mathrm{\Omega }\to H{|}_{\nu \in J}\}$. An H-valued random variable is a $\mathrm{{\rm Y}}$-measurable function $x\left(\nu \right):\mathrm{\Omega }\to H$.

Let ${\left\{e_n\right\}}_{n=1}^{\infty }$ be an orthonormal basis of K and $\left\{w\right(\nu ),\nu \ge 0\}$ be a K-valued Wiener process with finite trace nuclear covariance operator $Q\ge 0$. Then there exists a bounded sequence of non-negative real numbers ${\left\{{\alpha }_n\right\}}_{n=1}^{\infty }$ such that $Q{e}_n={\alpha }_n{e}_n,n=1,2, \dots$ satisfying $Tr\left(Q\right)={\displaystyle \sum _{n=1}^{\infty }}{\alpha }_n<\infty$. We define $w\left(\nu \right)={\displaystyle \sum _{n=1}^{\infty }}\sqrt{{\alpha }_n}{w}_n\left(\nu \right)e_n,n=1,2, \dots ,$ where $w_n$ are a mutually independent one-dimensional standard Wiener process. For every $\mu \in L(K,H),$ we define

$$\begin{array}{c}\hfill \parallel \mu {\parallel }_Q^2=Tr\left(\mu Q{\mu }^{\ast }\right)=\sum _{n=1}^{\infty }{∥\sqrt{{\alpha }_n}\mu e_n∥}^2.\end{array}$$

The term $\mu$ refers to a Q-Hilbert–Schmidt operator if ${\parallel \mu \parallel }_Q^2<\infty .$ We denote the space containing all Q-Hilbert–Schmidt operators $\mu :K\to H$ by $L_Q(K,H)$, where $L_Q(K,H)$ is a completion of $L(K,H),$ which is a Hilbert space with respect to the norm defined by ${\parallel \mu \parallel }_Q^2=<\mu ,\mu >.$

The set of all H-valued, square integrable, strongly measurable random variables is represented by the $L^2(\mathrm{\Omega },\mathrm{{\rm Y}},P;H)\equiv L^2(\mathrm{\Omega },H)$ equipped with the norm

$$\begin{array}{c}\hfill \parallel x(·){\parallel }_{L^2}={\left(\mathbb{E}\parallel (·,\omega ){\parallel }_H^2\right)}^{1/2},\end{array}$$

where $\mathbb{E}$ represents the expectation given by ${\displaystyle \mathbb{E}\left(x\right)={\int }_{\mathrm{\Omega }}x\left(\omega \right)d\mathbb{P},w\in \mathrm{\Omega }.}$

Assume that $J_1=(-\infty ,d]$, and let $C(J_1,L^2(\mathrm{\Omega },H)),$ denotes the Banach space of all continuous functions from $J_1$ to $L^2(\mathrm{\Omega },H)$ with the condition ${\displaystyle \underset{\nu \in J_1}{sup}\mathbb{E}{\parallel x\left(\nu \right)\parallel }^2<\infty }$.

We denote $J=(0,d]$ and $\zeta =\delta +\gamma -\delta \gamma$ and define

$$\begin{array}{c}\hfill C_{\zeta ;\mathrm{\Psi }}(J,L^2(\mathrm{\Omega },H))=\left\{x\in C(J,L^2(\mathrm{\Omega },H)):{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(0\right))}^{1-\zeta }x\left(\nu \right)\in C(J,L^2(\mathrm{\Omega },H))\right\}\end{array}$$

with the norm

$$\begin{array}{c}\hfill {\parallel x\parallel }_{C_{\zeta ;\mathrm{\Psi }}}^2=\underset{\nu \in J}{sup}{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(0\right))}^{1-\zeta }\mathbb{E}{\parallel x\left(\nu \right)\parallel }^2.\end{array}$$

Clearly, $(C_{\zeta ;\mathrm{\Psi }}(J,L^2(\mathrm{\Omega },H)),\parallel ·{\parallel }_{C_{\zeta ;\mathrm{\Psi }}})$ is a Banach space.

We discuss the abstract phase space ${\mathfrak{D}}_h$ (defined in [22]). Let $h:(-\infty ,0]\to (0,\infty )$ be a continuous function with ${\displaystyle {\int }_{-\infty }^{0}h\left(\nu \right)d\nu <\infty }$. The Banach space $({\mathfrak{D}}_h,\parallel ·{\parallel }_{{\mathfrak{D}}_h})$ induced by the function h is described as follows:

$$\begin{array}{ccc}\hfill {\mathfrak{D}}_h& =& \{\xi :(-\infty ,0]\to H:\mathrm{for}\mathrm{any}\mathsf{\ell }>0,\xi (\theta )\mathrm{is}\mathrm{measurable}\mathrm{and}\mathrm{bounded}\mathrm{function}\hfill \\ & & {\displaystyle \mathrm{on}[-\mathsf{\ell },0]\mathrm{and}{\int }_{-\infty }^{0}h\left(\nu \right)\underset{-\mathsf{\ell }\le \theta \le 0}{sup}{\mathbb{E}(\parallel \xi \left(\theta \right)\parallel }^2{)}^{1/2}d\nu <\infty \}}\hfill \end{array}$$

with the norm

$$\begin{array}{c}\hfill {\displaystyle {\parallel \xi \parallel }_{{\mathfrak{D}}_h}={\int }_{-\infty }^{0}h\left(\nu \right)\underset{-\mathsf{\ell }\le \theta \le 0}{sup}{\mathbb{E}(\parallel \xi \left(\theta \right)\parallel }^2{)}^{1/2}d\nu \forall \nu \in {\mathfrak{D}}_h.}\end{array}$$

We define the space

$$\begin{array}{c}\hfill {\mathfrak{D}}_h^{\prime }=\left\{x:(-\infty ,d]\to H\mathrm{such}\mathrm{that}{x}_{{|}_J}\in C(J,H),x_0=\mathfrak{H}\in {\mathfrak{D}}_h\right\}\end{array}$$

with the norm

$$\begin{array}{c}\hfill {\parallel x\parallel }_{{\mathfrak{D}}_h^{\prime }}={\parallel \mathfrak{H}\parallel }_{{\mathfrak{D}}_h}+\underset{\nu \in [0,d]}{sup}{\left({\mathbb{E}\parallel x\left(\nu \right)\parallel }^2\right)}^{1/2},x\in {\mathfrak{D}}_h^{\prime }.\end{array}$$

Lemma 1 

([22]). Suppose $x\in {\mathfrak{D}}_h^{\prime },$ then for $\nu \in J$, and $x_{\nu }\in {\mathfrak{D}}_h$,

$$\begin{array}{c}\hfill {k(\mathbb{E}\parallel x\left(\nu \right)\parallel }^2{)}^{1/2}\le \parallel x_{\nu }{\parallel }_{{\mathfrak{D}}_h}\le {\parallel \mathfrak{H}\parallel }_{{\mathfrak{D}}_h}+k^{\prime }\underset{\nu \in [0,d]}{sup}{\left({\mathbb{E}\parallel x\left(\nu \right)\parallel }^2\right)}^{1/2},\end{array}$$

where ${\displaystyle k^{\prime }={\int }_{-\infty }^{0}h\left(\nu \right)d\nu <\infty }$.

Definition 1 

([5]). Let $\mathrm{\Psi }\in C^k[c,d]$ be a positive function on $[c,d]$ such that $\mathrm{\Psi }\left(\nu \right)$ is continuous and ${\mathrm{\Psi }}^{\prime }\left(\nu \right)>0$ for all $\nu \in (c,d)$. Then, the left Ψ-Hilfer fractional derivative of order δ and type γ, is defined by

$$\begin{array}{ccc}\hfill D_{{\alpha }^{+}}^{\delta ,\gamma ;\mathrm{\Psi }}w\left(\nu \right)& =& I_{{\alpha }^{+}}^{\gamma (k-\delta ),\mathrm{\Psi }}{\left({\displaystyle \frac{1}{{\mathrm{\Psi }}^{\prime }\left(\nu \right)}}{\displaystyle \frac{d}{d\nu }}\right)}^k{I}_{{\alpha }^{+}}^{(1-\gamma )(k-\delta ),\mathrm{\Psi }}w\left(\nu \right),\hfill \end{array}$$

where $k=\left[\delta \right]+1$.

Definition 2 

([23]). If $I_{0^{+}}^{(1-\delta )(1-\gamma );\mathrm{\Psi }}x\left(0\right)=\mathfrak{H}\left(0\right)\in L^2(\mathrm{\Omega },{\mathfrak{D}}_h)forall\nu \in J_0=(-\infty ,0],$ then a stochastic process $x:(-\infty ,d]\to H$ is called a mild solution of (1), if x satisfies the following integral equation

$$\begin{array}{ccc}\hfill x\left(\nu \right)& =& S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,0)[\mathfrak{H}\left(0\right)-\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right))]+\mathcal{F}(\nu ,x_{\nu },x\left(\nu \right))\hfill \\ & & +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )A\mathcal{F}(\varrho ,x_{\varrho },x\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{G}(\varrho ,x_{\varrho },x\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )B\upsilon \left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{H}(\varrho ,x_{\varrho },x\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right),\nu \in J,\hfill \end{array}$$

(2)

where

$$\begin{array}{ccc}\hfill S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )& =& I_{0^{+}}^{\delta (1-\gamma );\mathrm{\Psi }}{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho ),\hfill \\ \hfill Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )& =& \delta {\int }_0^{\infty }\theta M_{\delta }\left(\theta \right)T\left({(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta }\theta \right)d\theta ,\hfill \end{array}$$

and $M_{\delta }(·)$ is the Wright function defined by

$$\begin{array}{c}\hfill M_{\delta }\left(\tau \right)=\sum _{k\in \mathbb{N}}{\displaystyle \frac{{(-\tau )}^{k-1}}{(k-1)!\Gamma (1-\delta k)}},\tau \in \mathbb{C},0<\delta <1,\end{array}$$

which satisfies the following

$$\begin{array}{c}\hfill {\int }_0^{\infty }{\theta }^{\alpha }{M}_{\delta }\left(\theta \right)d\theta ={\displaystyle \frac{\Gamma (1+\alpha )}{\Gamma (1+\delta \alpha )}},\tau \ge 0.\end{array}$$

Definition 3 

([24]). Consider $0<\rho <1$ and $0<\xi <{\displaystyle \frac{\pi }{2}}$, A ∈${\dot{\mathrm{\Theta }}}_{\xi }^{-\rho },$ where $\dot{\mathrm{\Theta }}$ is the family of closed linear operators, and A is called an almost sectorial operator on H.

(i) 

There exists a constant $M>0$ such that ${\parallel T\left(\nu \right)\parallel }_c\le M{\nu }^{\rho -1}$ for any $\nu >0$

Lemma 2 

([25]). The operators $S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )$ and $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )$ have the following characteristics:

(i) 

For fixed $\nu \ge \varrho \ge 0,$ $S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )$ and $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )$ are linear and bounded operators with

$$\begin{array}{c}\hfill ∥S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )x∥\le {\displaystyle \frac{M\Gamma \left(\rho \right){(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{-1+\gamma -\delta \gamma +\delta \rho }}{\Gamma \left(\gamma \right(1-\delta )+\delta \rho )}}\parallel x\parallel ,\end{array}$$

and

$$\begin{array}{c}\hfill ∥Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )x∥\le {\displaystyle \frac{M\Gamma \left(\rho \right){(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{-\delta +\delta \rho }}{\Gamma \left(\delta \right)}}\parallel x\parallel ,\end{array}$$

for all $x\in H$, where $L^{\prime }={\displaystyle \frac{M\Gamma \left(\rho \right)}{\Gamma \left(\delta \right)}}$ and $L^{\prime \prime }={\displaystyle \frac{M\Gamma \left(\rho \right)}{\Gamma \left(\gamma \right(1-\delta )+\delta \rho )}}$.

(ii) 

The operators $S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )$ and $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )$ are strongly continuous for $\nu \ge 0$, that is, for all $x\in H$ and $0\le \varrho \le {\nu }_1\le {\nu }_2,{\nu }_1,{\nu }_2\in J$, we have

$$\begin{array}{ccc}\hfill \parallel S_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )x-S_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_1,\varrho )x\parallel & \to & 0,\hfill \\ \hfill \parallel Q_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )x-Q_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_1,\varrho )x\parallel & \to & 0\mathit{as}{\nu }_2-{\nu }_1\to 0.\hfill \end{array}$$

Definition 4. 

The system (1) is controllable on $[0,d]$ if there exists $\upsilon \in L^2(J,U)$ such that for each continuous initial state $\xi \in L^2(\mathrm{\Omega },{\mathfrak{D}}_h)$ and $x^1\in H,$ the mild solution x of (1) satisfies $x\left(d\right)=x^1$.

Here, we discuss some of the definitions and properties of the measure of noncompactness (MNC).

Definition 5. 

Suppose ℧ is the bounded set in H; then, the Hausdorff MNC χ is defined by

$$\begin{array}{c}\hfill \chi (\mho )=inf\{ϵ>0:\mho canbecoveredby\mathrm{a}finitenumber\mathit{of}\mathit{balls}\mathit{with}\mathit{radii}ϵ\}.\end{array}$$

Lemma 3 

([26]). Let H be a Hilbert space and ${\mho }_1,{\mho }_2$ are bounded subsets of H. Then, the following statements hold.

• ${\mho }_1$ is relatively compact $\iff \chi \left(G_1\right)=0$.
• $\chi \left({\mho }_1\right)\le \chi \left({\mho }_2\right)$ if ${\mho }_1\subseteq {\mho }_2$.
• $\chi \left({\mho }_1\cup {\mho }_2\right)\le max\left\{\chi \left({\mho }_1\right),\chi \left({\mho }_2\right)\right\}$.
• For every $\lambda \in \mathbb{R}$, $\chi \left(\lambda {\mho }_1\right)=\left|\lambda \right|\chi \left({\mho }_1\right)$.
• $\chi \left({\mho }_1+{\mho }_2\right)\le \chi \left({\mho }_1\right)+\chi \left({\mho }_2\right)$, where ${\mho }_1+{\mho }_2=\left\{x_1+x_2:x_1\in {\mho }_1,x_2\in {\mho }_2\right\}$.
• $\chi \left({\mho }_1\right)=\chi \left({\overline{\mho }}_1\right)=\chi \left(\mathit{co}\left({\mho }_1\right)\right)$, where ${\overline{\mho }}_1$ and $\mathit{co}\left({\mho }_1\right)$ denote the closure and convex hull of ${\mho }_1$, respectively.
• If the operator $\mathrm{\Psi }:D\left(\mathrm{\Psi }\right)\subseteq H\to H_1$ is Lipschitz continuous with constant k, then ${\chi }_1\left(\mathrm{\Psi }\left({\mho }_1\right)\right)\le \chi \left({\mho }_1\right)$ for any bounded subset ${\mho }_1\subset$ $D\left(\mathrm{\Psi }\right)$, where ${\chi }_1$ represents the Hausdorff MNC in the Banach space $H_1$.

Lemma 4 

([26]). If $\mho \subset C\left(\right[c,d],H)$ is equicontinuous and bounded, then $\chi (\mho (\nu \left)\right)$ is continuous for $c\le \nu \le d$, and

$$\begin{array}{c}\hfill \chi (\mho )=sup\left\{\chi \right(\mho \left(\nu \right)),c\le \nu \le d\},\end{array}$$

where $\mho \left(\nu \right)=\left\{x\right(\nu ),\nu \in \mho \}\subseteq H$.

Theorem 1 

([26,27]). If ${\left\{a_k\right\}}_{k=1}^{\infty }$ is a sequence of Bochner integrable functions from J→H such that $∥a_k\left(\nu \right)∥\le \xi \left(\nu \right)$ for almost every $\nu \in J$ and for all $k\ge 1$, where ξ$\in L^1(J,\mathbb{R})$, then the function $\phi \left(\nu \right)=\mu \left(\left\{a_k\left(\nu \right),k\ge 1\right\}\right)$ $\in L^1(J,\mathbb{R})$, and satisfies

$$\begin{array}{c}\hfill \mu \left(\left\{{\int }_0^{\nu }{a}_k\left(\varrho \right)d\varrho ,k\ge 1\right\}\right)\le 2{\int }_0^{\nu }\phi \left(\varrho \right)d\varrho .\end{array}$$

Lemma 5 

([28]). Let ℧ be a convex and closed subset of a Banach space X and 0 $\in \mho$. If a continuous mapping of $f:\mho \to X$ satisfies Mönch’s condition, (i.e., ${\mho }_1\subset \mho$ is countable, and ${\mho }_1\subset \overline{\mathit{co}}\left(\left\{0\right\}\cup f\left({\mho }_1\right)\right)$ implies that $\overline{{\mho }_1}$ is compact), then f has a fixed point in ℧.

To deal with the measure of stochastic integral terms, we need the following lemma:

Lemma 6 

([29]). If $\mho \subset L^p([0,T],L_0^2)$ for all $p\ge 2$, then the Hausdorff MNC χ satisfies

$$\begin{array}{c}\hfill \chi \left({\int }_0^{\nu }\mho \left(\varrho \right)dw\left(\varrho \right)\right)\le \sqrt{T{\displaystyle \frac{p}{2}}(p-1)}\chi (\mho \left(\nu \right)),\end{array}$$

where

$$\begin{array}{c}\hfill {\int }_0^{\nu }\mho \left(\varrho \right)dw\left(\varrho \right)=\left\{{\int }_0^{\nu }x\left(\varrho \right)dw\left(\varrho \right)\forall x\in \mho ,\forall \nu \in [0,T]\right\}.\end{array}$$

and $w\left(\nu \right)$ is a Q-Wiener process.

Remark 1. 

For $p=2$ in Lemma 6, we have

$$\begin{array}{c}\hfill \chi \left({\int }_0^{\nu }\mho \left(\varrho \right)dw\left(\varrho \right)\right)\le \sqrt{TTr\left(Q\right)}\chi (\mho \left(\nu \right)).\end{array}$$

3. Results

To establish the main result, we need the following assumptions:

Hypothesis 1 

(H1). For all bounded subsets $\mathcal{C}\subseteq H$ and x$\in \mathcal{C}$,

$$\begin{array}{c}\hfill ∥T\left({\left(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(0\right)\right)}^{\delta }w\right)x-T\left({\left(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(0\right)\right)}^{\delta }w\right)x∥\to 0,{\nu }_2\to {\nu }_1,\end{array}$$

for each fixed w$\in (0,\infty )$.

Hypothesis 2 

(H2). The function $\mathcal{F}:J\times {\mathfrak{D}}_h\times H\to H$ satisfies the following properties:

(i) 

For all x$\in H$ and for any ν$\in J$, the function $\mathcal{F}$ is continuous and there exists $\alpha \in (0,1)$ s.t. $\mathcal{F}$ $\in D\left(A^{\alpha }\right)$ and ∀$x,y\in H,x_{\nu },y_{\nu }\in {\mathfrak{D}}_h,\nu \in J,A^{\alpha }\mathcal{F}(\nu ,·)$ satisfies

$$\begin{array}{ccc}\hfill \parallel A^{\alpha }\mathcal{F}(\nu ,x_{\nu },x\left(\nu \right))-A^{\alpha }\mathcal{F}(\nu ,y_{\nu },y\left(\nu \right)){\parallel }^2& \le & M_{\mathcal{F}}^{\prime }{\nu }^{1-\gamma +\delta \gamma -\delta \rho }[{∥x_{\nu }-y_{\nu }∥}_{{\mathfrak{D}}_h}^2\hfill \\ \hfill & & {+\parallel x\left(\nu \right)-y\left(\nu \right)\parallel }^2],\hfill \\ \hfill \parallel A^{\alpha }\mathcal{F}(\nu ,x_{\nu },x\left(\nu \right)){\parallel }^2& \le & M_{\mathcal{F}}^{\prime }\left(1+{\nu }^{1-\gamma +\delta \gamma -\delta \rho }\left[\parallel x_{\nu }{\parallel }_{{\mathfrak{D}}_h}^2+{\parallel x\parallel }^2\right]\right).\hfill \end{array}$$

(ii) 

$\mathcal{F}$ is completely continuous and for any bounded set $\mho \subset C(J,L^2(\mathrm{\Omega },H))$, the set $\{\nu \to \mathcal{F}(\nu ,x_{\nu },x\left(\nu \right)),x$ $\in \mho \}$ is equicontinuous in H.

Hypothesis 3 

(H3). The function $\mathcal{G}:J\times {\mathfrak{D}}_h\times H\to H$ satisfies the following conditions:

(i) 

$\mathcal{G}(·,y,x)$ is strongly measurable for every y$\in {\mathfrak{D}}_h$ and x$\in H$. $\mathcal{G}(\nu ,·,·)$ is continuous for a.e. $\nu \in J$, and $\mathcal{G}(\nu ,·,·):J\to H$ is strongly measurable function.

(ii) 

There exist integrable function $m_1$∈$L^{1/p_1}\left(J,{\mathbb{R}}^{+}\right),p_1$∈$(0,p)$ and non-decreasing real-valued continuous function $\tilde{g}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ s.t.

$$\begin{array}{c}\hfill {\parallel \mathcal{G}(\nu ,y,x)\parallel }^2\le m_1\left(\nu \right)\tilde{g}{(\parallel y\parallel }^2+{\parallel x\parallel }^2),y\in {\mathfrak{D}}_h,x\in H,\nu \in J,\end{array}$$

where $\tilde{g}$ satisfies

$$\begin{array}{c}\hfill \underset{\varrho \to \infty }{lim\; inf}{\displaystyle \frac{\tilde{g}\left(\varrho \right)}{\varrho }}=0.\end{array}$$

(iii) 

There exists a function $m_2$∈$L^{1/p_2}\left(J,{\mathbb{R}}^{+}\right),p_2$∈$(0,p)$ such that for each bounded subset $\tilde{\mho }\subset {\mathfrak{D}}_h$, $\mho \subset H$

$$\begin{array}{c}\hfill \chi \left(\mathcal{G}(\nu ,\tilde{\mho },\mho )\right)\le m_2\left(\nu \right)(\chi \left(\tilde{\mho }\right)+\chi (\mho ))\end{array}$$

for a.e. $\nu \in J$.

Hypothesis 4 

(H4). The function $\mathcal{H}:J\times {\mathfrak{D}}_h\times H\to L_2^0(\mathrm{\Omega },H)$ satisfies the following conditions:

(i) 

$\mathcal{H}(·,\tilde{\vartheta },\vartheta )$ is strongly measurable for every $y\in {\mathfrak{D}}_h,x\in H,\mathcal{H}(\nu ,·,·)$ is continuous for a.e. ν belong to J, and $\mathcal{H}(\nu ,·,·):J\to L_2^0(\mathrm{\Omega },H)$ is strongly measurable function.

(ii) 

There exists integrable function $m_3$∈$L^{1/p_3}\left(J,{\mathbb{R}}^{+}\right),p_3$∈$(0,p)$, and non-decreasing real-valued continuous function $\tilde{h}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that

$$\begin{array}{c}\hfill {\parallel \mathcal{H}(\nu ,y,x)\parallel }^2\le m_3\left(\nu \right)\tilde{h}{(\parallel y\parallel }^2+{\parallel x\parallel }^2),y\in {\mathfrak{D}}_h,x\in H,\nu \in J,\end{array}$$

where $\tilde{h}$ satisfies

$$\begin{array}{c}\hfill \underset{\varrho \to \infty }{lim\; inf}{\displaystyle \frac{\tilde{h}\left(\varrho \right)}{\varrho }}=0.\end{array}$$

(iii) 

There exists a function $m_4$ $\in L^{1/p_4}\left(J,{\mathbb{R}}^{+}\right),p_4$ $\in (0,p)$ such that for each bounded subset $\tilde{\mho }\subset {\mathfrak{D}}_h$, $\mho \subset H$

$$\begin{array}{c}\hfill \chi \left(\mathcal{H}(\nu ,\tilde{\mho },\mho )\right)\le m_4\left(\nu \right)(\chi \left(\tilde{\mho }\right)+\chi (\mho ))\end{array}$$

for a.e. ν$\in J$.

Hypothesis 5 

(H5). The operator B is bounded, i.e., $\parallel B\parallel \le M^{\prime }$, where $M^{\prime }$ is a positive constant.

Hypothesis 6 

(H6). The operator $\mathcal{K}$: $L^2(J,U)\to L^2(\mathrm{\Omega },H)$, defined by

$$\begin{array}{c}\hfill \mathcal{K}v={\int }_0^d{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )Bv\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \end{array}$$

satisfies the following:

(i) 

It has an inverse operator ${\mathcal{K}}^{-1}$, which take the values in $L^2(J,U)/$Ker $\mathcal{K}$, and there exists a positive constant $M^{″}$ such that $∥{\mathcal{K}}^{-1}∥\le M^{\prime \prime }$.

(ii) 

For ${\delta }_5$ $\in (0,\delta )$ and for any bounded subset $L\subseteq {\mathfrak{D}}_h,F\subseteq H$, there exists $m_5$ $\in L^{1/{\delta }_5}\left(J,{\mathbb{R}}^{+}\right)$ s.t.

$$\begin{array}{c}\hfill \chi \left(\left({\mathcal{K}}^{-1}\mho \right)\left(\nu \right)\right)\le m_5\left(\nu \right)\chi (\mho ).\end{array}$$

Take $\parallel A^{-\alpha }\parallel =M_0$ and $\parallel A^{1-\alpha }\parallel =M_1$.

Theorem 2. 

Suppose assumptions (H1)–(H6) are satisfied, then the system (1) is controllable on J if

$$\begin{array}{ccc}\hfill M^{\ast }& =& {\displaystyle \frac{2L^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta +\delta \rho )}}{\delta \rho }}\left[\parallel m_2{\parallel }_{L^{1/{\delta }_2}\left(J,{\mathbb{R}}^{+}\right)}+Tr\left(Q\right){\parallel m_4\parallel }_{L^{1/{\delta }_4}\left(J,{\mathbb{R}}^{+}\right)}\right]\times \hfill \\ & & \left\{1+{\displaystyle \frac{2L^{\prime }{M}^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta +\delta \rho )}}{\delta \rho }}{\parallel m_5\parallel }_{L^{1/{\delta }_5}\left(J,{\mathbb{R}}^{+}\right)}\right\}<1\hfill \\ \hfill and\theta >1+\rho .\end{array}$$

Proof. 

Using the assumption (H6), we define the control function $v_x$ as

$$\begin{array}{ccc}\hfill {\upsilon }_x\left(\nu \right)& =& {\mathcal{K}}^{-1}[x^1-S_{\delta ,\gamma ;\mathrm{\Psi }}(d,0)[\mathfrak{H}\left(0\right)-\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right))]-\mathcal{F}(d,x_d,x\left(d\right))\hfill \\ & & -{\int }_0^d{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(d,\varrho )A\mathcal{F}(\varrho ,x_{\varrho },x\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & -{\int }_0^d{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(d,\varrho )\mathcal{G}(\varrho ,x_{\varrho },x\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & -{\int }_0^d{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(d,\varrho )\mathcal{H}(\varrho ,x_{\varrho },x\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right)]\left(\nu \right).\hfill \end{array}$$

(3)

By using this control function, we show that the operator ${\mathcal{L}}^{\prime }:{\mathfrak{D}}_h^{\prime }\to {\mathfrak{D}}_h^{\prime }$, defined by

$$\begin{array}{c}\hfill {\mathcal{L}}^{\prime }\left(x\left(\nu \right)\right)=\left\{\begin{array}{cc}\mathfrak{H}\left(\nu \right),\nu \in (-\infty ,0],\hfill & \\ S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,0)[\mathfrak{H}\left(0\right)-\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right))]+\mathcal{F}(\nu ,x_{\nu },x\left(\nu \right))\hfill & \\ {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )A\mathcal{F}(\varrho ,x_{\varrho },x\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill & \\ {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{G}(\varrho ,x_{\varrho },x\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill & \\ {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )B{\upsilon }_x\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill & \\ {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{H}(\varrho ,x_{\varrho },x\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right),\nu \in J,}\hfill \end{array}\right.\end{array}$$

(4)

possesses a fixed point, which is a mild solution of (1). It is trivial that ${\mathcal{L}}^{\prime }\left(x\left(d\right)\right)=x^1$. So, the system (1) is controllable on $[0,d]$.

For $\mathfrak{H}\in {\mathfrak{D}}_h^{\prime }$, we define $\widehat{\mathfrak{H}}$ by

$$\begin{array}{c}\hfill \widehat{\mathfrak{H}}\left(\nu \right)=\left\{\begin{array}{cc}\mathfrak{H}\left(\nu \right),\hfill & \nu \in (-\infty ,0],\hfill \\ S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,0)\mathfrak{H}\left(0\right),\hfill & \nu \in J,\hfill \end{array}\right.\end{array}$$

(5)

then $\widehat{\mathfrak{H}}\in {\mathfrak{D}}_h^{\prime }$. Set $x\left(\nu \right)=\widehat{z}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right),\nu \in (-\infty ,d]$. It is trivial that x satisfies (2) if, and only if, $\widehat{z}$ satisfies ${\widehat{z}}_0=0$ and

$$\begin{array}{ccc}\hfill \widehat{z}\left(\nu \right)& =& -S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,0)\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right))+\mathcal{F}(\nu ,{\widehat{z}}_{\nu }+{\widehat{\mathfrak{H}}}_{\nu },\widehat{z}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right))\hfill \\ & & +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )B{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right),\hfill \end{array}$$

(6)

where ${\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}$ is obtained from (3) by replacing $x=\widehat{z}+\widehat{\mathfrak{H}}$.

Let ${\mathfrak{D}}_h^{″}=\left\{\widehat{z}\in {\mathfrak{D}}_h^{\prime }:{\widehat{z}}_0=0\in {\mathfrak{D}}_h\right\}$. For any $\widehat{z}\in {\mathfrak{D}}_h^{″}$, we have

$$\begin{array}{c}\hfill \parallel \widehat{z}{\parallel }_{{\mathfrak{D}}_h^{″}}={\parallel {\widehat{z}}_0\parallel }_{{\mathfrak{D}}_h}+\underset{\nu \in [0,d]}{sup}{\left(\mathbb{E}\parallel \widehat{z}{\left(\nu \right)\parallel }^2\right)}^{1/2}=\underset{\nu \in [0,d]}{sup}{\left(\mathbb{E}\parallel \widehat{z}{\left(\nu \right)\parallel }^2\right)}^{1/2},\end{array}$$

and $({\mathfrak{D}}_h^{″},\parallel ·{\parallel }_{{\mathfrak{D}}_h^{\prime \prime }})$ forms a Banach space.

For $q_0>0$, define the set

$$\begin{array}{c}\hfill B_{q_0}=\left\{\widehat{z}\in {\mathfrak{D}}_h^{″}:{\parallel \widehat{z}\parallel }_{{\mathfrak{D}}_h^{″}}^2\le q_0\right\}.\end{array}$$

The set $B_{q_0}$ is a convex, closed, and bounded set. For all $q_0>0$ and for all $\widehat{z}\in B_{q_0}$, by Lemma 1, we obtain

$$\begin{array}{ccc}\hfill \parallel {\widehat{z}}_{\nu }+{\widehat{\mathfrak{H}}}_{\nu }{\parallel }_{{\mathfrak{D}}_h}^2& \le & 2\left({∥{\widehat{z}}_{\nu }∥}_{{\mathfrak{D}}_h}^2+{\parallel {\widehat{\mathfrak{H}}}_{\nu }\parallel }_{{\mathfrak{D}}_h}^2\right)\hfill \\ & \le & 4\left(\parallel {\widehat{z}}_0{\parallel }_{{\mathfrak{D}}_h}^2+k^2\underset{\nu \in [0,d]}{sup}\mathbb{E}{\parallel \widehat{z}\left(\nu \right)\parallel }^2\right)+4\left(\parallel {\mathfrak{H}}_0{\parallel }_{{\mathfrak{D}}_h}^2+k^2\underset{\nu \in [0,d]}{sup}\mathbb{E}{\parallel \mathfrak{H}\left(\nu \right)\parallel }^2\right)\hfill \\ & \le & 4\left(k^2{q}_0+{\parallel \mathfrak{H}\parallel }_{{\mathfrak{D}}_h}^2\right)\hfill \\ & =& q_1^{\prime }.\hfill \end{array}$$

Similarly, $\parallel \widehat{z}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right){\parallel }^2\le q_2^{\prime }.$ Let us take $max\left\{q_1^{\prime },q_2^{\prime }\right\}=q_0^{\prime }$.

Consider the operator $\tilde{\mathcal{L}}:{\mathfrak{D}}_h^{″}\to {\mathfrak{D}}_h^{″}$ defined by

$$\begin{array}{c}\hfill \tilde{\mathcal{L}}\widehat{z}\left(\nu \right)=\left\{\begin{array}{cc}0,\nu \in (-\infty ,0],\hfill & \\ -S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,0)\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right))+\mathcal{F}(\nu ,{\widehat{z}}_{\nu }+{\widehat{\mathfrak{H}}}_{\nu },\widehat{z}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right))\hfill & \\ {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )A\mathcal{F}(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill & \\ {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{G}(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill & \\ {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )B{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill & \\ {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{H}(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right),\nu \in J.}\hfill \end{array}\right.\end{array}$$

(7)

Clearly, the existence of a fixed point of the operator ${\mathcal{L}}^{\prime }$ is equivalent to the existence of a fixed point of the operator $\tilde{\mathcal{L}}$. We prove that $\tilde{\mathcal{L}}$ has a fixed point. We divide the proof into several steps:

Step 1. There exists $q_0>0$ such that $\tilde{\mathcal{L}}\left(B_{q_0}\right)\subseteq B_{q_0}$. Suppose that the condition is not true; then, for all $q_0>0$, there exists ${\widehat{z}}^{q_0}$ $\in B_{q_0}$ such that $\tilde{\mathcal{L}}\left({\widehat{z}}^{q_0}\right)\notin B_{q_0}$, i.e., $\parallel \tilde{\mathcal{L}}\widehat{z}\left(\nu \right)\parallel >q_0$. From (7), we have

$$\begin{array}{ccc}\hfill q_0& \le & \mathbb{E}\parallel {(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta )}\tilde{\mathcal{L}}\left({\widehat{z}}^{q_0}\left(\nu \right)\right){\parallel }^2\hfill \\ \hfill & \le & sup{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}\mathbb{E}\parallel [-S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,0)\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right))\hfill \\ & & +\mathcal{F}(\nu ,{\widehat{z}}_{\nu }^{q_0}+{\widehat{\mathfrak{H}}}_{\nu },{\widehat{z}}^{q_0}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right))\hfill \\ & & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }^{q_0}+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^{q_0}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }^{q_0}+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^{q_0}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )B{\mathcal{K}}^{-1}\{x^1-S_{\delta ,\gamma ;\mathrm{\Psi }}(d,0)[\mathfrak{H}\left(0\right)-\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right))]}\hfill \\ & & -\mathcal{F}\left(d,{\widehat{z}}_d^{q_0}+{\widehat{\mathfrak{H}}}_d,{\widehat{z}}^{q_0}\left(d\right)+\widehat{\mathfrak{H}}\left(d\right)\right)\hfill \\ & & -{\int }_0^d{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(d,\varrho )A\mathcal{F}(\varrho ,{\widehat{z}}_{\varrho }^{q_0}+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^{q_0}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & -{\int }_0^d{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(d,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }^{q_0}+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^{q_0}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & -{\int }_0^d{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(d,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }^{q_{)}}+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^{q_0}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right)\}\left(\varrho \right)d\varrho \hfill \\ \hfill \\ & & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }^{q_0}+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^{q_0}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right)]{\parallel }^2}\hfill \\ & \le & 11{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}[\mathbb{E}{\parallel S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,0)\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right))\parallel }^2\hfill \\ & & +\mathbb{E}{∥\mathcal{F}\left(\nu ,{\widehat{z}}_{\nu }^{q_0}+{\widehat{\mathfrak{H}}}_{\nu },{\widehat{z}}^{q_0}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right)\right)∥}^2\hfill \\ & & +\mathbb{E}{∥{\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }^{q_0}+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^{q_0}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }∥}^2\hfill \\ & & +\mathbb{E}{∥{\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }^{q_0}+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^{q_0}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }∥}^2\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )B{\mathcal{K}}^{-1}\{\mathfrak{H}\left(0\right)}\hfill \\ & & -\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right))-\mathcal{F}\left(d,{\widehat{z}}_d^{q_0}+{\widehat{\mathfrak{H}}}_d,{\widehat{z}}^{q_0}\left(d\right)+\widehat{\mathfrak{H}}\left(d\right)\right)\hfill \\ & & -{\int }_0^d{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(d,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }^{q_0}+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^{q_0}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & -{\int }_0^d{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(d,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }^{q_0}+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^{q_0}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & -{\int }_0^d{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(d,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }^{q_0}+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^{q_0}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right){\parallel }^2\hfill \\ & & +\mathbb{E}{∥{\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }^{q_0}+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^{q_0}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right)}∥}^2]\hfill \\ & \le & 11{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}[{\left(L^{″}\right)}^2{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}{\left(M_{\mathcal{F}}^{\prime }\right)}^2{\left(M_0\right)}^2+{\left(M_0\right)}^2{\left(M_{\mathcal{F}}^{\prime }\right)}^2\hfill \\ & & \times (1+{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}{\left(q_0^{\prime }\right)}^2)+{\left({\displaystyle \frac{M_{\mathcal{F}}^{\prime }{L}^{\prime }{M}_1}{\delta \rho }}\right)}^2(1+{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}{\left(q_0^{\prime }\right)}^2)\hfill \\ & & \times {(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2\delta \rho }+{\left({\displaystyle \frac{{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{\delta \rho }}{\delta \rho }}\right)}^2{L}^{\prime 2}{m}_1\left(d\right)\tilde{g}\left(q_0^{\prime }\right)\hfill \\ & & +Tr\left(Q\right){\left({\displaystyle \frac{{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{\delta \rho }}{\delta \rho }}\right)}^2{L}^{\prime 2}{m}_3\left(d\right)\tilde{h}\left(q_0^{\prime }\right)\hfill \\ & & +L^{\prime 2}{\left({\displaystyle \frac{{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{\delta \rho }}{\delta \rho }}\right)}^2{M}^{\prime }{M}^{\prime \prime }\left)\right[{\parallel x^1\parallel }^2-{\left(L^{″}\right)}^2{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}[\mathfrak{H}\left(0\right)+{M_{\mathcal{F}}^{\prime }}^2{M_0}^2]\hfill \\ & & -\{{\left(M_0\right)}^2{\left(M_{\mathcal{F}}^{\prime }\right)}^2(1+{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}{\left(q_0^{\prime }\right)}^2)\hfill \\ & & +{\left({\displaystyle \frac{M_{\mathcal{F}}^{\prime }{L}^{\prime }{M}_1}{\delta \rho }}\right)}^2(1+{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}{\left(q_0^{\prime }\right)}^2){(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2\delta \rho }\hfill \\ & & +{\left({\displaystyle \frac{{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{\delta \rho }}{\delta \rho }}\right)}^2{L}^{\prime 2}{m}_1\left(d\right)\tilde{g}\left(q_0^{\prime }\right)\hfill \\ & & +Tr\left(Q\right){\left({\displaystyle \frac{{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{\delta \rho }}{\delta \rho }}\right)}^2{L}^{\prime 2}{m}_3\left(d\right)\tilde{h}\left(q_0^{\prime }\right)\left\}\right]]\hfill \\ & \le & 11{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}{M}^{\ast \ast },\hfill \end{array}$$

(8)

where

$$\begin{array}{ccc}& & M^{\ast \ast }\hfill \\ & & =[{\left(L^{″}\right)}^2{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}{\left(M_{\mathcal{F}}^{\prime }\right)}^2\left(M_0^2\right)+{\left(M_0\right)}^2{\left(M_{\mathcal{F}}^{\prime }\right)}^2(1+{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}{\left(q_0^{\prime }\right)}^2)\hfill \\ & & +{\left({\displaystyle \frac{M_{\mathcal{F}}^{\prime }{L}^{\prime }{M}_1}{\delta \rho }}\right)}^2(1+{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}{\left(q_0^{\prime }\right)}^2){(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2\delta \rho }\hfill \\ & & +{\left({\displaystyle \frac{{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{\delta \rho }}{\delta \rho }}\right)}^2{L}^{\prime 2}{m}_1\left(d\right)\tilde{g}\left(q_0^{\prime }\right)+Tr\left(Q\right){\left({\displaystyle \frac{{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{\delta \rho }}{\delta \rho }}\right)}^2{L}^{\prime 2}{m}_3\left(d\right)\tilde{h}\left(q_0^{\prime }\right)\hfill \\ & & +L^{\prime 2}\left({\displaystyle \frac{{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{\delta \rho }}{\delta \rho }}{)}^2{M}^{\prime }{M}^{″}\right)[{\parallel x^1\parallel }^2-{\left(L^{″}\right)}^2{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}[\mathfrak{H}\left(0\right)+{M_{\mathcal{F}}^{\prime }}^2{M_0}^2]\hfill \\ & & -\{{\left(M_0\right)}^2{\left(M_{\mathcal{F}}^{\prime }\right)}^2(1+{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}{\left(q_0^{\prime }\right)}^2)\hfill \\ & & +{\left({\displaystyle \frac{M_{\mathcal{F}}^{\prime }{L}^{\prime }{M}_1}{\delta \rho }}\right)}^2(1+{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}{\left(q_0^{\prime }\right)}^2){(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2\delta \rho }\hfill \\ & & +{\left({\displaystyle \frac{{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{\delta \rho }}{\delta \rho }})\right)}^2{L}^{\prime 2}{m}_1\left(d\right)\tilde{g}\left(q_0^{\prime }\right)+Tr\left(Q\right){\left({\displaystyle \frac{{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{\delta \rho }}{\delta \rho }}\right)}^2{L}^{\prime 2}{m}_3\left(d\right)\tilde{h}\left(q_0^{\prime }\right)\left\}\right]].\hfill \end{array}$$

By dividing Equation (8) from both sides with $q_0$ and applying the limit $q_0\to \infty$, we obtain $1\le 0$, which is a contradiction. Therefore, $\tilde{\mathcal{L}}\left(B_{q_0}\right)\subseteq B_{q_0}$.

Step 2. The operator $\tilde{\mathcal{L}}$ is continuous on $B_{q_0}$.

For any ${\widehat{z}}^k,\widehat{z}\in B_{q_0},k=0,1,2,3,\cdots$ with ${\displaystyle \underset{k\to \infty }{lim}{\widehat{z}}^k\to \widehat{z}}$, we have

$${\displaystyle \underset{k\to \infty }{lim}{\left(\mathrm{\Psi }\right(\nu )-\mathrm{\Psi }(0\left)\right)}^{1-\zeta }{\widehat{z}}^k\left(\nu \right)={\left(\mathrm{\Psi }\right(\nu )-\mathrm{\Psi }(0\left)\right)}^{1-\zeta }\widehat{z}\left(\nu \right).}$$

Using assumption (H2), we obtain

$$\begin{array}{ccc}\hfill \mathcal{G}(\nu ,x_{\nu }^k,x^k\left(\nu \right))& =& \mathcal{G}(\nu ,{\widehat{z}}_{\nu }^k+\widehat{\mathfrak{H}{}_{\nu }},{\widehat{z}}^k\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right))\to \mathcal{G}(\nu ,{\widehat{z}}_{\nu }+{\widehat{\mathfrak{H}}}_{\nu },\widehat{z}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right))\hfill \\ & =& \mathcal{G}(\nu ,x_{\nu }^k,x^k\left(\nu \right)),k\to \infty .\hfill \end{array}$$

Take

$$\begin{array}{c}\hfill {\mathcal{G}}^k\left(\nu \right)=\mathcal{G}(\nu ,{\widehat{z}}_{\nu }^k+\widehat{\mathfrak{H}{}_{\nu }},{\widehat{z}}^k\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right))\mathrm{and}\mathcal{G}\left(\nu \right)=\mathcal{G}(\nu ,{\widehat{z}}_{\nu }+{\widehat{\mathfrak{H}}}_{\nu },\widehat{z}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right)).\end{array}$$

Then, by the dominated convergence theorem with assumption (H2), we get

$$\begin{array}{c}\hfill {\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{2(\delta -1)})\parallel Q_{\delta ,\gamma ;\mathrm{\Psi }}{(\nu ,\varrho )\parallel }^2\mathbb{E}\left\{\parallel {\mathcal{G}}^k{\left(\varrho \right)-\mathcal{G}\left(\varrho \right)\parallel }^2{\mathrm{\Psi }}^{\prime }\left(\varrho \right)\right\}d\rho \to 0\mathrm{as}k\to \infty .}\end{array}$$

(9)

Using (H3), we obtain

$$\begin{array}{ccc}\hfill {\mathcal{H}}^K(\nu ,x_{\nu }^k,x^k\left(\nu \right))& =& {\mathcal{H}}^k(\nu ,{\widehat{z}}_{\nu }^k+\widehat{\mathfrak{H}{}_{\nu }},{\widehat{z}}^k\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right))\hfill \\ & \to & \mathcal{H}(\nu ,{\widehat{z}}_{\nu }+{\widehat{\mathfrak{H}}}_{\nu },\widehat{z}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right))=\mathcal{H}(\nu ,x_{\nu }^k,x^k\left(\nu \right)),k\to \infty .\hfill \end{array}$$

We assume that

$$\begin{array}{c}\hfill {\mathcal{H}}^k\left(\nu \right)=\mathcal{H}(\nu ,{\widehat{z}}_{\nu }^k+\widehat{\mathfrak{H}{}_{\nu }},{\widehat{z}}^k\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right)),\mathrm{and}\mathcal{H}\left(\nu \right)=\mathcal{H}(\nu ,{\widehat{z}}_{\nu }+{\widehat{\mathfrak{H}}}_{\nu },\widehat{z}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right)).\end{array}$$

Using the dominated convergence theorem together with (H3), we obtain

$$\begin{array}{c}\hfill {\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{2(\delta -1)})\parallel Q_{\delta ,\gamma ;\mathrm{\Psi }}{(\nu ,\varrho )\parallel }^2\mathbb{E}\left\{\parallel {\mathcal{H}}^k{\left(\varrho \right)-\mathcal{H}\left(\varrho \right)\parallel }^2{\mathrm{\Psi }}^{\prime }\left(\varrho \right)\right\}dw\left(\varrho \right)\to 0\mathrm{as}k\to \infty .}\end{array}$$

(10)

Using (H5), we obtain

$$\begin{array}{ccc}\hfill \mathcal{F}(\nu ,x_{\nu }^k,x^k\left(\nu \right))& =& \mathcal{F}(\nu ,{\widehat{z}}_{\nu }^k+\widehat{\mathfrak{H}{}_{\nu }},{\widehat{z}}^k\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right))\hfill \\ & \to & \mathcal{F}(\nu ,{\widehat{z}}_{\nu }+{\widehat{\mathfrak{H}}}_{\nu },\widehat{z}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right))=\mathcal{F}(\nu ,x_{\nu }^k,x^k\left(\nu \right))\mathrm{as}k\to \infty .\hfill \end{array}$$

Define

$$\begin{array}{c}\hfill {\mathcal{F}}^k\left(\nu \right)=\mathcal{F}(\nu ,{\widehat{z}}_{\nu }^k+\widehat{\mathfrak{H}{}_{\nu }},{\widehat{z}}^k\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right))\mathrm{and}\mathcal{F}\left(\nu \right)=\mathcal{F}(\nu ,{\widehat{z}}_{\nu }+{\widehat{\mathfrak{H}}}_{\nu },\widehat{z}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right)).\end{array}$$

Using the dominated convergence theorem together with (H3), we obtain

$$\begin{array}{c}\hfill {\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{2(\delta -1)})A\parallel Q_{\delta ,\gamma ;\mathrm{\Psi }}{(\nu ,\varrho )\parallel }^2\mathbb{E}\left\{\parallel {\mathcal{F}}^k{\left(\varrho \right)-\mathcal{F}\left(\varrho \right)\parallel }^2{\mathrm{\Psi }}^{\prime }\left(\varrho \right)\right\}d\rho \to 0\mathrm{as}k\to \infty ,}\end{array}$$

(11)

and

$$\begin{array}{c}\hfill \mathbb{E}\parallel {\mathcal{F}}^k{\left(\varrho \right)-\mathcal{F}\left(\varrho \right)\parallel }^2\to 0\mathrm{as}k\to \infty .\end{array}$$

From (3), we obtain

$$\begin{array}{ccc}\hfill & & \mathbb{E}{∥{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}^k\left(\nu \right)-{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}\left(\nu \right)∥}^2\hfill \\ & & \le 4\parallel {\mathcal{K}}^{-1}{\parallel }^2[\mathbb{E}{\parallel {\mathcal{F}}^k\left(\varrho \right)-\mathcal{F}\left(\varrho \right)\parallel }^2\hfill \\ & & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(\varrho \right))}^{2(\delta -1)}{\parallel A\parallel }^2\parallel Q_{\delta ,\gamma ;\mathrm{\Psi }}{(\nu ,\varrho )\parallel }^2\mathbb{E}{\parallel {\mathcal{F}}^k\left(\varrho \right)-\mathcal{F}\left(\varrho \right)\parallel }^2{\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(\varrho \right))}^{2(\delta -1)}{\parallel Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\parallel }^2\mathbb{E}\left\{\parallel {\mathcal{G}}^k{\left(\varrho \right)-\mathcal{G}\left(\varrho \right)\parallel }^2\right\}{\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{2(\delta -1)}{\parallel Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\parallel }^2\mathbb{E}\left\{\parallel {\mathcal{H}}^k{\left(\varrho \right)-\mathcal{H}\left(\varrho \right)\parallel }^2\right\}{\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right)].}\hfill \end{array}$$

(12)

Using (10)–(12), the above inequality converges to zero as $k\to \infty$.

From (7), we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\parallel \tilde{\mathcal{L}}{\widehat{z}}^k\left(\nu \right)-\tilde{\mathcal{L}}\widehat{z}{\left(\nu \right)\parallel }^2\hfill \\ \hfill & \le 5{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}\times \{\mathbb{E}{\parallel {\mathcal{F}}^k\left(\varrho \right)-\mathcal{F}\left(\varrho \right)\parallel }^2\hfill \\ \hfill & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{2(\delta -1)}A{\parallel Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\parallel }^2\mathbb{E}\left\{\parallel {\mathcal{F}}^k{\left(\varrho \right)-\mathcal{F}\left(\varrho \right)\parallel }^2{\mathrm{\Psi }}^{\prime }\left(\varrho \right)\right\}d\rho }\hfill \\ \hfill & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{2(\delta -1)}{\parallel Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\parallel }^2\mathbb{E}\left\{\parallel {\mathcal{G}}^k{\left(\varrho \right)-\mathcal{G}\left(\varrho \right)\parallel }^2\right\}{\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ \hfill & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{2(\delta -1)}{\parallel Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\parallel }^{2}B\mathbb{E}\left\{{∥{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}^k\left(\varrho \right)-{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}\left(\varrho \right)∥}^2\right\}{\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ \hfill & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{2(\delta -1)}{\parallel Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\parallel }^2\mathbb{E}\left\{\parallel {\mathcal{H}}^k{\left(\varrho \right)-\mathcal{H}\left(\varrho \right)\parallel }^2{\mathrm{\Psi }}^{\prime }\left(\varrho \right)\right\}dw\left(\varrho \right)\}.}\hfill \end{array}$$

Using (10)–(12), we obtain

$$\begin{array}{c}\hfill \mathbb{E}\parallel \tilde{\mathcal{L}}{\widehat{z}}^k\left(\nu \right)-\tilde{\mathcal{L}}\widehat{z}{\left(\nu \right)\parallel }^2\to 0\mathrm{as}k\to \infty ,\end{array}$$

which shows that $\tilde{\mathcal{L}}$ is continuous on $B_{q_0}$.

Step 3. To prove $\tilde{\mathcal{L}}$ is equicontinuous, we take $\widehat{z}\in B_{q_0}$ and $0\le {\nu }_1<{\nu }_2\le d$, then we have

$$\begin{array}{ccc}& & \mathbb{E}\parallel \tilde{\mathcal{L}}\widehat{z}\left({\nu }_2\right)-\tilde{\mathcal{L}}\widehat{z}\left({\nu }_1\right){\parallel }^2\hfill \\ & & \le 6{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta )}\{\mathbb{E}\parallel \left[S_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,0)-S_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_1,0)\right]\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right)){\parallel }^2\hfill \\ & & +\mathbb{E}\parallel \mathcal{F}({\nu }_2,{\widehat{z}}_{{\nu }_2}+{\widehat{\mathfrak{H}}}_{{\nu }_2},\widehat{z}\left({\nu }_2\right)+\widehat{\mathfrak{H}}\left({\nu }_2\right))-\mathcal{F}({\nu }_1,{\widehat{z}}_{{\nu }_1}+{\widehat{\mathfrak{H}}}_{{\nu }_1},\widehat{z}\left({\nu }_1\right)+\widehat{\mathfrak{H}}\left({\nu }_1\right)){\parallel }^2\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{{\nu }_2}{(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle -{\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_1,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho {\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{{\nu }_2}{(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle -{\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_1,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho {\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{{\nu }_2}{(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )B{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle -{\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_1,\varrho )B{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho {\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{{\nu }_2}{(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right)}\hfill \\ & & {\displaystyle -{\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_1,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right){\parallel }^2\}.}\hfill \\ & \le & 14(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }{\left(0\right)}^{2(1-\zeta )}\{\mathbb{E}\parallel \left[S_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,0)-S_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_1,0)\right]\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right)){\parallel }^2\hfill \\ & & +\mathbb{E}\parallel \mathcal{F}({\nu }_2,{\widehat{z}}_{{\nu }_2}+{\widehat{\mathfrak{H}}}_{{\nu }_2},\widehat{z}\left({\nu }_2\right)+\widehat{\mathfrak{H}}\left({\nu }_2\right))-\mathcal{F}({\nu }_1,{\widehat{z}}_{{\nu }_1}+{\widehat{\mathfrak{H}}}_{{\nu }_1},\widehat{z}\left({\nu }_1\right)+\widehat{\mathfrak{H}}\left({\nu }_1\right)){\parallel }^2\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle -{\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho {\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle -{\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_1,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho {\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_{{\nu }_1}^{{\nu }_2}{(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho {\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle -{\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho {\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle -{\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_1,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho {\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_{{\nu }_1}^{{\nu }_2}{(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho {\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )B{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle -{\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )B{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho {\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )B{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle -{\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_1,\varrho )B{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho {\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_{{\nu }_1}^{{\nu }_2}{(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )B{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho {\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right)}\hfill \\ & & {\displaystyle -{\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right){\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right)}\hfill \\ & & {\displaystyle -{\int }_0^{{\nu }_1}{(\mathrm{\Psi }\left({\nu }_1\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_1,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right){\parallel }^2}\hfill \\ & & {\displaystyle +\mathbb{E}\parallel {\int }_{{\nu }_1}^{{\nu }_2}{(\mathrm{\Psi }\left({\nu }_2\right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}({\nu }_2,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right){\parallel }^2\}.}\hfill \end{array}$$

By the strong continuity of the operators $S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )$ and $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )$ and the dominated convergence theorem, we find that

$$\begin{array}{c}\hfill \mathbb{E}\parallel \tilde{\mathcal{L}}\widehat{z}\left({\nu }_2\right)-\tilde{\mathcal{L}}\widehat{z}\left({\nu }_1\right){\parallel }^2\to 0\mathrm{as}{\nu }_1\to {\nu }_2.\end{array}$$

Hence, $\tilde{\mathcal{L}}$ is equicontinuous on $B_{q_0}$.

Step 4. The Mönch’s statement is true.

Let $\tilde{\mathcal{L}}={\tilde{\mathcal{L}}}_1+{\tilde{\mathcal{L}}}_2+{\tilde{\mathcal{L}}}_3+{\tilde{\mathcal{L}}}_4+{\tilde{\mathcal{L}}}_5+{\tilde{\mathcal{L}}}_6$, where

$$\begin{array}{ccc}\hfill {\tilde{\mathcal{L}}}_1& =& -S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,0)\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right)),\hfill \\ \hfill {\tilde{\mathcal{L}}}_2& =& \mathcal{F}(\nu ,{\widehat{z}}_{\nu }+{\widehat{\mathfrak{H}}}_{\nu },\widehat{z}\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right)),\hfill \\ \hfill {\tilde{\mathcal{L}}}_3& =& {\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho ,}\hfill \\ \hfill {\tilde{\mathcal{L}}}_4& =& {\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho ,}\hfill \\ \hfill {\tilde{\mathcal{L}}}_5& =& {\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )B{\upsilon }_{\widehat{z}+\widehat{\mathfrak{H}}}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho ,}\hfill \\ \hfill {\tilde{\mathcal{L}}}_6& =& {\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }+{\widehat{\mathfrak{H}}}_{\varrho },\widehat{z}\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right).}\hfill \end{array}$$

Let $\mathcal{P}\subseteq B_{q_0}$ be countable and $\mathcal{P}\subseteq \mathrm{conv}(\left\{0\right\}\cup \tilde{\mathcal{L}}\left(\mathcal{P}\right)).$ We show that $\chi \left(\mathcal{P}\right)=0$.

Suppose $\mathcal{P}={\{{\widehat{z}}^n+\widehat{\mathfrak{H}}\}}_{n=1}^{\infty }$. We have to show that $\tilde{\mathcal{L}}\left(\mathcal{P}\right)\left(\nu \right)$ is relatively compact in H for all $\nu \in J$. Using Theorem 1, we obtain

$$\begin{array}{ccc}\hfill \chi \left(\mathcal{P}\right(\nu \left)\right)& =& \chi \left({\{{\widehat{z}}^n+\widehat{\mathfrak{H}}\}}_{n=0}^{\infty }\right)\hfill \\ & =& \chi \left(\{{\widehat{z}}^0+\widehat{\mathfrak{H}}\}\right)\cup \chi \left({\{{\widehat{z}}^n+\widehat{\mathfrak{H}}\}}_{n=1}^{\infty }\right)\hfill \\ & =& \chi \left({\{{\widehat{z}}^n+\widehat{\mathfrak{H}}\}}_{n=1}^{\infty }\right)\hfill \\ \hfill \chi \left({\left\{\tilde{\mathcal{L}}{\widehat{z}}^n\left(\nu \right)\right\}}_{n=1}^{\infty }\right)& =& \chi \left[{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta )}\right\{-S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,0)\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right))\hfill \\ & & +\mathcal{F}(\nu ,{\widehat{z}}_{\nu }^n+{\widehat{\mathfrak{H}}}_{\nu },{\widehat{z}}^n\left(\nu \right)+\widehat{\mathfrak{H}}\left(\nu \right))\hfill \\ & & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )A\mathcal{F}\left(\varrho ,{\widehat{z}}_{\varrho }^n+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^n\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }^n+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^n\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \\ & & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )B{\upsilon }_{{\widehat{z}}^n+\widehat{\mathfrak{H}}}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\hfill \end{array}$$

$$\begin{array}{ccc}& & {\displaystyle +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }^n+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^n\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right)}\hfill \\ & & {\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right){\}}_{n=1}^{\infty }]\hfill \\ & & \le {\mathcal{J}}_1+{\mathcal{J}}_2+{\mathcal{J}}_3,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\mathcal{J}}_1& =& \chi ({(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta )}\hfill \\ & & {\displaystyle \times {\left\{{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }^n+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^n\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \right\}}_{n=1}^{\infty }),}\hfill \\ \hfill {\mathcal{J}}_2& =& \chi ({(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta )}\hfill \\ & & {\displaystyle \times {\left\{{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }^n+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^n\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right)\right\}}_{n=1}^{\infty }),}\hfill \\ \hfill {\mathcal{J}}_3& =& {\displaystyle \chi \left({(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta )}\times {\left\{{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )B{\upsilon }_{{\widehat{z}}^n}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \right\}}_{n=1}^{\infty }\right).}\hfill \end{array}$$

Using Theorem 1 and the assumptions (H2)–(H4), we obtain

$$\begin{array}{ccc}\hfill {\mathcal{J}}_1& =& \chi ({(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta )}\hfill \\ & & \times {\left\{{\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }^n+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^n\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\right\}}_{n=1}^{\infty })\hfill \\ & \le & 2L^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta )}\hfill \\ & & {\displaystyle \times \left\{{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta \rho -1}{m}_2\left(\varrho \right)\chi {\left\{\mathcal{G}\left(\varrho ,{\widehat{z}}_{\varrho }^n+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^n\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right)\right\}}_{n=1}^{\infty }{\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \right\}}\hfill \\ & \le & {\displaystyle \frac{2L^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta +\delta \rho )}}{\delta \rho }}{\parallel m_2\parallel }_{L^{1/{\delta }_2}\left(J,{\mathbb{R}}^{+}\right)}\underset{-\infty <\theta \le 0}{sup}\chi \left({\left\{{\widehat{z}}_{\nu }^n\left(\theta \right)\right\}}_{n=1}^{\infty }\right).\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\mathcal{J}}_2& =& \chi ({(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta )}\hfill \\ & & \times {\left\{{\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }^n+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^n\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right)}\right\}}_{n=1}^{\infty })\hfill \\ & \le & 2L^{\prime }Tr\left(Q\right){(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta )}\hfill \\ & & \times \left\{{\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta \rho -1}{m}_4\left(\varrho \right)\chi {\left\{\mathcal{H}\left(\varrho ,{\widehat{z}}_{\varrho }^n+{\widehat{\mathfrak{H}}}_{\varrho },{\widehat{z}}^n\left(\varrho \right)+\widehat{\mathfrak{H}}\left(\varrho \right)\right)\right\}}_{n=1}^{\infty }{\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\right\}\hfill \\ & \le & {\displaystyle \frac{2L^{\prime }Tr\left(Q\right){(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta +\delta \rho )}}{\delta \rho }}{\parallel m_4\parallel }_{L^{1/{\delta }_4}\left(J,{\mathbb{R}}^{+}\right)}\underset{-\infty <\theta \le 0}{sup}\chi \left({\left\{{\widehat{z}}_{\nu }^n\left(\theta \right)\right\}}_{n=1}^{\infty }\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\mathcal{J}}_3& =& \chi \left({(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta )}{\left\{{\displaystyle {\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )B{\upsilon }_{{\widehat{z}}^n+\widehat{\mathfrak{H}}}\left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho }\right\}}_{n=1}^{\infty }\right)\hfill \\ & \le & {\displaystyle 2L^{\prime }{M}^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta )}\left\{{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta \rho -1}{m}_5\left(\varrho \right)\right(\frac{2L^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta +\delta \rho )}}{\delta \rho }}\hfill \\ & & \parallel m_2{\parallel }_{L^{1/{\delta }_2}\left(J,{\mathbb{R}}^{+}\right)}\underset{-\infty <\theta \le 0}{sup}\chi \left({\left\{{\widehat{z}}_{\nu }^n\left(\theta \right)\right\}}_{n=1}^{\infty }\right)\hfill \\ & & +{\displaystyle \frac{2L^{\prime }Tr\left(Q\right){(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta +\delta \rho )}}{\delta \rho }}{\parallel m_4\parallel }_{L^{1/{\delta }_4}\left(J,{\mathbb{R}}^{+}\right)}\underset{-\infty <\theta \le 0}{sup}\chi \left({\left\{{\widehat{z}}_{\nu }^n\left(\theta \right)\right\}}_{n=1}^{\infty }\right)\left){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \right\}\hfill \\ & \le & {\displaystyle \frac{4L^{\prime 2}{M}^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta +\delta \rho )}}{{\left(\delta \rho \right)}^2}}\parallel m_5{\parallel }_{L^{1/{\delta }_5}\left(J,{\mathbb{R}}^{+}\right)}[{\parallel m_2\parallel }_{L^{1/{\delta }_2}\left(J,{\mathbb{R}}^{+}\right)}\hfill \\ & & +Tr\left(Q\right)\parallel m_4{\parallel }_{L^{1/{\delta }_4}\left(J,{\mathbb{R}}^{+}\right)}]\underset{-\infty <\theta \le 0}{sup}\chi \left({\left\{{\widehat{z}}_{\nu }^n\left(\theta \right)\right\}}_{n=1}^{\infty }\right).\hfill \end{array}$$

Also, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \chi \left({\left\{\tilde{\mathcal{L}}{\widehat{z}}^n\left(\nu \right)\right\}}_{n=1}^{\infty }\right)& \le {\displaystyle \frac{4L^{\prime 2}{M}^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{2(1-\zeta +\delta \rho )}}{{\left(\delta \rho \right)}^2}}{\parallel m_5\parallel }_{L^{1/{\delta }_5}\left(J,{\mathbb{R}}^{+}\right)}\hfill \\ & \times \left[\parallel m_2{\parallel }_{L^{1/{\delta }_2}\left(J,{\mathbb{R}}^{+}\right)}+Tr\left(Q\right){\parallel m_4\parallel }_{L^{1/{\delta }_4}\left(J,{\mathbb{R}}^{+}\right)}\right]\hfill \\ & +{\displaystyle \frac{2L^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta +\delta \rho )}}{\delta \rho }}[{\parallel m_2\parallel }_{L^{1/{\delta }_2}\left(J,{\mathbb{R}}^{+}\right)}\hfill \\ & +Tr\left(Q\right)\parallel m_4{\parallel }_{L^{1/{\delta }_4}\left(J,{\mathbb{R}}^{+}\right)}]\underset{-\infty <\theta \le 0}{sup}\chi \left({\left\{{\widehat{z}}_{\nu }^n\left(\theta \right)\right\}}_{n=1}^{\infty }\right).\hfill \\ & \le {\displaystyle \frac{2L^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta +\delta \rho )}}{\delta \rho }}\left[\parallel m_2{\parallel }_{L^{1/{\delta }_2}\left(J,{\mathbb{R}}^{+}\right)}+Tr\left(Q\right){\parallel m_4\parallel }_{L^{1/{\delta }_4}\left(J,{\mathbb{R}}^{+}\right)}\right]\hfill \\ & \times \left\{1+{\displaystyle \frac{2L^{\prime }{M}^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta +\delta \rho )}}{\delta \rho }}{\parallel m_5\parallel }_{L^{1/{\delta }_5}\left(J,{\mathbb{R}}^{+}\right)}\right\}\hfill \\ & \underset{-\infty <\theta \le 0}{sup}\chi \left({\left\{{\widehat{z}}_{\nu }^n\left(\theta \right)\right\}}_{n=1}^{\infty }\right),\hfill \end{array}\end{array}$$

where

$$\begin{array}{ccc}\hfill M^{\ast }& =& {\displaystyle \frac{2L^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta +\delta \rho )}}{\delta \rho }}\left[\parallel m_2{\parallel }_{L^{1/{\delta }_2}\left(J,{\mathbb{R}}^{+}\right)}+Tr\left(Q\right){\parallel m_4\parallel }_{L^{1/{\delta }_4}\left(J,{\mathbb{R}}^{+}\right)}\right]\hfill \\ & & \times \left\{1+{\displaystyle \frac{2L^{\prime }{M}^{\prime }{(\mathrm{\Psi }\left(d\right)-\mathrm{\Psi }\left(0\right))}^{(1-\zeta +\delta \rho )}}{\delta \rho }}{\parallel m_5\parallel }_{L^{1/{\delta }_5}\left(J,{\mathbb{R}}^{+}\right)}\right\}.\hfill \end{array}$$

Using Lemma 4, we have

$$\begin{array}{c}\hfill \chi \left(\tilde{\mathcal{L}}\left(\mathcal{P}\right)\right)\le M^{\ast }\chi \left(\mathcal{P}\right).\end{array}$$

Therefore, from Mönch’s condition we have

$$\begin{array}{c}\hfill \chi \left(\mathcal{P}\right)\le \chi \left(\mathrm{conv}\left(\left\{0\right\}\cup \tilde{\mathcal{L}}\left(\mathcal{P}\right)\right)\right)=\chi \left(\tilde{\mathcal{L}}\left(\mathcal{P}\right)\right)\le M^{\ast }\chi \left(\mathcal{P}\right),\end{array}$$

which implies that $\chi \left(\mathcal{P}\right)=0$. Hence, $\mathcal{P}$ is relatively compact.

Therefore, using Lemma 5, $\tilde{\mathcal{L}}$ has a fixed point $\widehat{z}$ in $B_{q_0}$. Therefore, $x=\widehat{z}+\widehat{\mathfrak{H}}$ is a fixed point of operator $\tilde{\mathcal{L}}$, which is a mild solution of (1) satisfying $x\left(d\right)=x^1$. Thus, the system (1) is controllable on J. This completes the proof. □

4. Applications

4.1. Application 1

Consider the following $\mathrm{\Psi }$-Hilfer fractional differential equation given by

$$\begin{array}{ccc}\hfill & & D_{0^{+}}^{3/4,\gamma ;\mathrm{\Psi }}\left[\vartheta (\nu ,z)-{\int }_{-\infty }^{\nu }{\zeta }_1(\varrho ,z)\vartheta (\varrho ,z)d\varrho \right]={\displaystyle \frac{{\partial }^2}{\partial z^2}}\vartheta (\nu ,z)+{\gamma }_1\left(\nu ,{\int }_{-\infty }^{\nu }{\delta }_1(\varrho -\nu )\vartheta (\varrho ,z)d\varrho \right)\hfill \\ & & +B\upsilon (\nu ,z)+{\gamma }_2\left(\nu ,{\int }_{-\infty }^{\nu }{\delta }_2(\varrho -\nu )\vartheta (\varrho ,z)dw\left(\varrho \right)\right),\hfill \\ & & I_{0^{+}}^{(1-3/4)(1-\gamma );\mathrm{\Psi }}\vartheta (0,z)=\xi \left(z\right),z\in [0,\pi ],\hfill \\ & & \vartheta (\nu ,0)=\vartheta (\nu ,\pi )=0,\nu \in (0,1],\hfill \\ & & \vartheta (\nu ,z)=\varphi (\nu ,z),0\le z\le \pi ,\nu \in (-\infty ,0),\hfill \end{array}$$

(13)

where $D_{0^{+}}^{3/4,\gamma ;\mathrm{\Psi }}$ denotes the $\mathrm{\Psi }$-Hilfer fractional derivative with order $0<3/4<1$ and type $0\le \gamma \le 1$. $I_{0^{+}}^{(1-3/4)(1-\gamma );\mathrm{\Psi }}$ denotes the $\mathrm{\Psi }$-Riemann–Liouville integral of order $(1-3/4)(1-\gamma )$. ${\gamma }_1$ and ${\gamma }_2$ are continuous functions. In addition, $\varphi$ is continuous and satisfies specific smoothness criteria, and ${\zeta }_1,{\delta }_1$, and ${\delta }_2$ are the appropriate functions. The filtered probability space $(\mathrm{\Omega },\mathrm{{\rm Y}},\mathbb{P})$ represents the one-dimensional Brownian motion in K, which is denoted as $w(·)$. $\upsilon$ is the control function in Hilbert space U.

To change the system into an abstract form, let $H=K=L^2[0,\pi ]$ be endowed with the norm, ${\parallel ·\parallel }_{L^2}$ and $A:D\left(A\right)\subset H\to H$ is defined as $A\vartheta ={\vartheta }^{″}$ with

$$\begin{array}{c}\hfill D\left(A\right)=\left\{\vartheta :H:\vartheta ,{\vartheta }^{\prime }\mathrm{are}\mathrm{absolutely}\mathrm{continuous},{\vartheta }^{″}\in H,\vartheta \left(0\right)=\vartheta \left(\pi \right)=0\right\},\end{array}$$

and

$$\begin{array}{c}\hfill A\vartheta =\sum _{m\in \mathbb{N}}{m}^2\langle \vartheta ,e_m\rangle e_m,\vartheta \in D\left(A\right),\end{array}$$

where $e_m\left(z\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{imz},m\in \mathbb{N}$ is the orthogonal set of eigen vectors of A.

Here, A is the infinitesimal generator of analytic semigroup $\left\{T\right(\nu ):\nu \ge 0\}$ on H. $T\left(\nu \right)$ is not a compact semigroup on H with $\chi \left(T\right(\nu \left)D\right)\le \chi \left(D\right)$, where $\chi$ denotes the Hausdorff measure of noncompactness and ∃${\mathcal{K}}_1>1$ s.t. $\underset{\nu \in (0,1]}{sup}\parallel T\left(\nu \right)\parallel \le {\mathcal{K}}_1{\nu }^{\rho -1}$. Also, $\nu \to \varpi ({\nu }^{3/4}\beta +u)\vartheta$ is equicontinuous, $u>0$, and $\beta \in (0,1)$.

Define $\vartheta \left(\nu \right)\left(z\right)=\vartheta (\nu ,z)$ and $\upsilon \left(\nu \right)\left(z\right)=\upsilon (\nu ,z)$, then

$$\begin{array}{ccc}\hfill \mathcal{F}(\nu ,{\vartheta }_{\nu },\vartheta \left(\nu \right))\left(z\right)& =& {\int }_{-\infty }^{\nu }{\zeta }_1(\varrho ,z)\vartheta (\varrho ,z)d\varrho ,\hfill \\ \hfill \mathcal{G}(\nu ,{\vartheta }_{\nu },\vartheta \left(\nu \right))\left(z\right)& =& {\gamma }_1\left(\nu ,{\int }_{-\infty }^{\nu }{\delta }_1(\varrho -\nu )\vartheta (\varrho ,z)d\varrho \right),\hfill \\ \hfill \mathcal{H}(\nu ,{\vartheta }_{\nu },\vartheta \left(\nu \right))\left(z\right)& =& {\gamma }_2\left(\nu ,{\int }_{-\infty }^{\nu }{\delta }_2(\varrho -\nu )\vartheta (\varrho ,z)dw\left(\varrho \right)\right).\hfill \end{array}$$

Then, the system (13) can be written in an abstract form of system (1). We also assume that all the assumptions of the Theorem 2 are satisfied. Therefore, using Theorem 2, we conclude that the system (13) is controllable.

4.2. Application 2

The $\mathrm{\Psi }$-Hilfer stochastic fractional differential equation is a useful tool for systems affected by stochastic noise since it expands the traditional $\mathrm{\Psi }$-Hilfer fractional differential equations to incorporate random effects. In addition to modeling uncertainties and random perturbations, which are inherent in digital signal processing, these equations incorporate the memory and heredity features of fractional calculus.

Stochastic fractional differential equations are used in digital filter systems to create sophisticated filtering mechanisms for signals impacted by stochastic noise. This is especially helpful in communication systems, control systems, and biomedical signal processing, where noise and unpredictability have a big impact on signal quality. When a signal is distorted or corrupted by noise, signal restoration techniques can be employed to recover the original signal. For instance, sound recordings produced by certain equipment may contain noise and one common approach to restoring the signal is to apply filters, often using convolution. When it comes to stochastic systems, the distortion can be explained by the existence of a random noise term, which is usually modeled as white noise or a Wiener process. Thus, a stochastic differential equation can be used to model the signal restoration process.

Motivated by the filter system presented in [30,31,32], we describe the digital filter system corresponding to the mild solution in (1).

The $\mathrm{\Psi }$-Hilfer stochastic fractional differential equation governing the filter can be expressed as defined in (1), where x is the output signal (filtered signal), the function $\mathcal{H}$ represents the stochastic effect of noise, and $\upsilon$ is the controller. The mild solution of differential Equation (1) is given as

$$\begin{array}{ccc}\hfill x\left(\nu \right)& =& S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,0)[\mathfrak{H}\left(0\right)-\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right))]+\mathcal{F}(\nu ,x_{\nu },x\left(\nu \right))\hfill \\ & & +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )A\mathcal{F}(\varrho ,x_{\varrho },x\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{G}(\varrho ,x_{\varrho },x\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )B\upsilon \left(\varrho \right){\mathrm{\Psi }}^{\prime }\left(\varrho \right)d\varrho \hfill \\ & & +{\int }_0^{\nu }{(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\mathcal{H}(\varrho ,x_{\varrho },x\left(\varrho \right)){\mathrm{\Psi }}^{\prime }\left(\varrho \right)dw\left(\varrho \right),\nu \in J.\hfill \end{array}$$

(14)

In this solution (14), the deterministic integral component captures the low-frequency signal, while the stochastic integral term represents the effect of random noise, allowing for adaptive filtering.

Figure 1 describes the following:

• Product modulator 1 accepts the input $x\left(\nu \right)$ and $\mathcal{F}$ produces the output $\mathcal{F}(\nu ,x_{\nu },x\left(\nu \right))$.
• Product modulator 2 accepts the input $\mathcal{F}(\nu ,x_{\nu },x\left(\nu \right))$ and A, and produces the output $A\mathcal{F}(\nu ,x_{\nu },x\left(\nu \right))$.
• Product modulator 3 accepts the input $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)$ and $A\mathcal{F}(·,·,·)$ and produces the output $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)A\mathcal{F}(·,·,·)$.
• Product modulator 4 accepts the input $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)A\mathcal{F}(·,·,·)$ and $\mathrm{\Psi }$-function and obtains the output ${(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }(·))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)A\mathcal{F}(·,·,·){\mathrm{\Psi }}^{\prime }(·)$.
• Product modulator 5 accepts the input $x\left(\nu \right)$ and $\mathcal{G}$ produces the output $\mathcal{G}(\nu ,x_{\nu },x\left(\nu \right))$.
• Product modulator 6 accepts the input $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)$ and $\mathcal{G}(·,·,·)$ and produces the output $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)\mathcal{G}(·,·,·)$.
• Product modulator 7 accepts the input $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)\mathcal{G}(·,·,·)$ and $\mathrm{\Psi }$-function, and obtains the output ${(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }(·))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)\mathcal{G}(·,·,·){\mathrm{\Psi }}^{\prime }(·)$.
• Product modulator 8 accepts the input $\upsilon \left(\nu \right)$ and B, and produces the output $B\upsilon \left(\nu \right)$.
• Product modulator 9 accepts the input $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)$ and $B\upsilon \left(\nu \right)$ and produces the output $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)B\upsilon \left(\nu \right)$.
• Product modulator 10 accepts the input $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)B\upsilon (·)$ and $\mathrm{\Psi }$-function and obtains the output ${(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }(·))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)B\upsilon (·){\mathrm{\Psi }}^{\prime }(·)$.
• Product modulator 11 accepts the input $x\left(\nu \right)$ and $\mathcal{H}$ produces the output $\mathcal{H}(\nu ,x_{\nu },x\left(\nu \right)){\displaystyle \frac{d\omega \left(\nu \right)}{d\nu }}$.
• Product modulator 12 accepts the input $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)$ and $\mathcal{H}(\nu ,x_{\nu },x\left(\nu \right)){\displaystyle \frac{d\omega \left(\nu \right)}{d\nu }}$ and produces the output $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)\mathcal{H}(\nu ,x_{\nu },x\left(\nu \right)){\displaystyle \frac{d\omega \left(\nu \right)}{d\nu }}$.
• Product modulator 13 accepts the input $Q_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)\mathcal{H}(·,·,·){\displaystyle \frac{d\omega (·)}{d\nu }}$ and $\mathrm{\Psi }$-function and obtains the output ${(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }(·))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,·)\mathcal{H}(·,·,·){\mathrm{\Psi }}^{\prime }(·){\displaystyle \frac{d\omega (·)}{d\nu }}$.
• Product modulator 14 accepts $\mathfrak{H}\left(0\right)-\mathcal{F}(0,\mathfrak{H}(0),\mathfrak{H}(0\left)\right)$ and $S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,0)$ at time $\nu =0$ and produces $S_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,0)[\mathfrak{H}\left(0\right)-\mathcal{F}(0,\mathfrak{H}\left(0\right),\mathfrak{H}\left(0\right))]$.
• The integrators execute the following value,

  $$\begin{array}{ccc}& & {(\mathrm{\Psi }\left(\nu \right)-\mathrm{\Psi }\left(\varrho \right))}^{\delta -1}{Q}_{\delta ,\gamma ;\mathrm{\Psi }}(\nu ,\varrho )\times \hfill \\ & & \left[A\mathcal{F}(\varrho ,x_{\varrho },x\left(\varrho \right))+\mathcal{G}(\varrho ,x_{\varrho },x\left(\varrho \right))+B\upsilon \left(\varrho \right)+\mathcal{H}(\varrho ,x_{\varrho },x\left(\varrho \right)){\displaystyle \frac{dw\left(\varrho \right)}{d\varrho }}\right]{\mathrm{\Psi }}^{\prime }\left(\varrho \right),\hfill \end{array}$$

  and produce the integral value over the period $\nu$.

Finally, all outputs from the integrators are directed to the summer network, resulting in the output $x\left(\nu \right)$, which is bounded and approximately controllable.

5. Conclusions

In this article, we studied the controllability results of fractional stochastic neutral differential equations involving $\mathrm{\Psi }$-Hilfer FDEs with infinite delay using the measure of noncompactness. Sufficient conditions for the controllability of $\mathrm{\Psi }$-Hilfer fractional stochastic differential equations were obtained by using weak compactness criteria, appropriate assumptions, the theory of semigroups of bounded linear operators, and Mönch’s fixed-point theorem via a measure of noncompactness.

Further, these results can be extended to study the approximate controllability of the Hilfer fractional differential equation in an infinite dimensional state space. We plan to examine the controllability of $(k,\mathrm{\Psi })$-Hilfer FDEs with infinite delay using the concept of the measure of noncompactness.