New Results Concerning Approximate Controllability of Conformable Fractional Noninstantaneous Impulsive Stochastic Evolution Equations via Poisson Jumps

Abstract

We introduce the conformable fractional (CF) noninstantaneous impulsive stochastic evolution equations with fractional Brownian motion (fBm) and Poisson jumps. The approximate controllability for the considered problem was investigated. Principles and concepts from fractional calculus, stochastic analysis, and the fixed-point theorem were used to support the main results. An example is applied to show the established results.

1. Introduction

The field of fractional calculus is constantly expanding, and applications range from engineering and natural phenomena to financial perspectives (see [1,2,3,4,5,6,7]). Recently, there seems to be much enthusiasm for the use of stochastic differential equations to describe a variety of phenomena in population dynamics, physics, electrical engineering, geography, psychology, biochemistry, and other areas of physics and technology (see [8,9,10,11,12,13,14]). Stochastic impulsive differential equations arise in a very natural way as mathematical models (see [15,16,17,18,19,20]). The introduction of drugs into the bloodstream and the consequent absorption into the body are gradual and continuous processes that can be described by noninstantaneous impulsive differential equations (see [21,22]). Now, several authors have discussed different types of controllability for fractional stochastic systems (see [23,24,25,26,27,28,29]). To the best of our knowledge, the approximate controllability of CF noninstantaneous impulsive stochastic evolution equations via fBm and Poisson jump mentioned in this study is an area of research that appears to give extra incentive for completing this research.

Assume that the CF noninstantaneous impulsive stochastic evolution equation via fBm, Poisson jump, and the control function has the following form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}^{\pounds }N(\wp )+\Delta N(\wp )=M(\wp ,N(\wp ),N\left({\ell }_1(\wp )\right),\dots ,N\left({\ell }_m(\wp )\right))+Bu(\wp )\hfill \\ +{\int }_0^{\wp }V(\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚥}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚥}_k\left(\mathbb{k}\right)\right))d\omega \left(\mathbb{k}\right)\hfill \\ +\mathsf{\Omega }(\wp ,N(\wp ),N\left(c_1(\wp )\right),\dots ,N\left(c_p(\wp )\right))\frac{d{\mathfrak{B}}_H}{d\wp }\hfill \\ +{\int }_F\mho (\wp ,N(\wp ),N\left({𝚤}_1(\wp )\right),\dots ,N\left({𝚤}_u(\wp )\right),f)W(d\wp ,df),\wp \in ({\mathbb{k}}_i,{\wp }_{i+1}],i\in [0,m]\hfill \\ N(\wp )={\hslash }_i(\wp ,N(\wp )),\wp \in ({\wp }_i,{\mathbb{k}}_i],i\in [1,m]\hfill \\ N\left(0\right)=N_0,\hfill \end{array}\right.\end{array}$$

(1)

where $D^{\pounds }$ is the conformable fractional derivative (CFD) of order $\frac{1}{2}<\pounds <1$ and $-\Delta$ generates semigroup $\mathsf{\Theta }(\wp ),\wp \ge 0,$ on ℵ. Here, ℵ and G are separable Hilbert spaces with $\Vert ·\Vert$ and $u(·)\in L_2(\mathsf{{\rm Y}},\mathfrak{U})$ is the control function, where $L_2(\mathsf{{\rm Y}},\mathfrak{U})$ is the Hilbert space of control functions with $\mathfrak{U}$ a Hilbert space. B is a bounded linear operator from $\mathfrak{U}$ into ℵ, and ${\hslash }_i$ is a noninstantaneous impulsive function for all $i=1,2,\dots ,m$. Suppose that the time interval is $\mathsf{{\rm Y}}=(0,b]$, where ${\wp }_i,{\mathbb{k}}_i$ are fixed numbers verifying $0={\mathbb{k}}_0<{\wp }_1\le {\mathbb{k}}_1\le {\wp }_2<\dots <{\mathbb{k}}_{m-1}<{\wp }_m\le {\mathbb{k}}_m\le {\wp }_{m+1}=b$. Assume ${\left\{\omega (\wp )\right\}}_{\wp \ge 0}$ is a K-Wiener process on $(\mathfrak{F},\mathfrak{S},{\left\{{\mathfrak{S}}_{\wp }\right\}}_{\wp \ge 0},\mathfrak{P})$ with values in G and ${\left\{{\mathfrak{B}}_H(\wp )\right\}}_{\wp \ge 0}$ is fBm with Hurst parameter $H\in (\frac{1}{2},1)$ defined on $(\mathfrak{F},\mathfrak{S}p,{\left\{{\mathfrak{S}}_{\wp }\right\}}_{\wp \ge 0},\mathfrak{P})$ with values in $Q;$ Q is a Hilbert space with $\Vert .\Vert .$ In this paper, $L(G,\aleph )$ and $L(Q,\aleph )$ are the space of all bounded linear operators from G into ℵ and from Q into $\aleph ,$ respectively, with $\Vert .\Vert .$ $M,V$, $\mathsf{\Omega }$, ℧, and ${\hslash }_i$ are defined in Section 2.

The contributions of the present work are as follows:

• The conformable fractional noninstantaneous impulsive stochastic evolution equation with fractional Brownian motion and Poisson jump is presented.
• To the best of the author’s knowledge, there has not been any research that has studied the approximate controllability of (1).
• An example is applied to show the established results.

2. Preliminaries

Here, we collect the basic concepts, definitions, theorems, and lemmas that are used in the paper.

Definition 1

(See [7]). The CFD of order $0<\pounds <1$ of $z(\wp )$ for $\wp >0$ is defined as

$$\frac{d^{\pounds }z(\wp )}{d{\wp }^{\pounds }}=\underset{\nu \to 0}{lim}\frac{z(\wp +\nu {\wp }^{1-\pounds })-z(\wp )}{\nu }.$$

Furthermore, the conformable integral is defined as

$$I^{\pounds }\left(z\right)(\wp )={\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}z\left(\mathbb{k}\right)d\mathbb{k}.$$

Suppose $(\mathfrak{F},\mathfrak{S},\mathfrak{P})$ is a full probability area connected with a normal filtration $\mathfrak{S}{p}_{\wp },\wp \in [0,b]$, where $\mathfrak{S}{n}_{\wp }$ is the $\sigma$-algebra generated by random variables $\{\omega \left(\mathbb{k}\right),{\mathfrak{B}}_H\left(\mathbb{k}\right),s\in [0,b]\}$ and all $\mathfrak{P}$-null sets. Let $(F,\xi ,\varsigma (df\left)\right)$ be a $\sigma$-finite measurable space. The stationary Poisson point process ${\left(p_{\wp }\right)}_{\wp \ge 0}$ is defined on $(\mathfrak{F},\mathfrak{S},\mathfrak{P})$ with values in F and characteristic measure $\varsigma .$ The counting measure of $p_{\wp }$ is denoted by $\zeta (\wp ,df)$ such that $W(\wp ,\vartheta ):=E\left(\zeta \right(\wp ,\vartheta \left)\right)=\wp \varsigma \left(\vartheta \right)$ for $\vartheta \in \xi$. Define $W(\wp ,df):=\zeta (\wp ,df)-\wp \lambda \left(df\right)$, the Poisson martingale generated by $p_{\wp }$.

Let $\mathfrak{Z}\in L(Q,Q)$ be an operator defined by $\mathfrak{Z}{\mathfrak{x}}_n={\lambda }_n{\mathfrak{x}}_n$ with $Tr\left(\mathfrak{Z}\right)={\sum }_{n=1}^{\infty }{\lambda }_n<\infty$, where ${\lambda }_n\ge 0(n=1,2,\dots )$ are non-negative real numbers and $\left\{{\mathfrak{x}}_n\right\}(n=1,2,\dots )$ is a complete orthonormal basis in Q.

We introduce the space $L_2^0:=L_2^0(Q,\aleph )$ of all A-Hilbert–Schmidt operators $\mu :Q\to \aleph .$

$\mu \in L(Q,\aleph )$ is called a A-Hilbert–Schmidt operator, if

$${\parallel \mu \parallel }_{L_2^0}^2:=\sum _{n=1}^{\infty }{\parallel {\sqrt{\lambda }}_n\mu N_n\parallel }^2<\infty .$$

Lemma 1

(see [30]). If $\mu :[0,b]\to L_2^0(Q,\aleph )$ satisfies ${\int }_0^b{\parallel \mu \left(\mathbb{k}\right)\parallel }_{L_2^0}^2<\infty$ then

$$E{∥{\int }_0^{\wp }\mu \left(\mathbb{k}\right)d{\mathfrak{B}}_H\left(\mathbb{k}\right)∥}^2\le 2H{\wp }^{2H-1}{\int }_0^{\wp }{\parallel \mu \left(\mathbb{k}\right)\parallel }_{L_2^0}^{2}d\mathbb{k}.$$

Theorem 1

(see [31]). Assume $(\yen ;\mathbb{A})$ is a compact metric space. For a family of functions $\mathbb{Z}\in C(\yen )$, then the following statements are equivalent:

(i) 

$\mathbb{Z}$is relatively compact;

(ii) 

$\mathbb{Z}$is equicontinuous and uniformly bounded.

Through this paper, let $L_2(\mathfrak{F},\aleph )$ be a Banach space with

$${\Vert N(·)\Vert }_{L_2(\mathfrak{F},\aleph )}^2=E{\Vert N(.,\omega )\Vert }^2,$$

where $E\left(N\right)={\int }_{\mathfrak{F}}N\left(\omega \right)d\mathfrak{P}.$ Assume $C(\mathsf{{\rm Y}},L_2(\mathfrak{F},\aleph ))$, from $\mathsf{{\rm Y}}$ into $L_2(\mathfrak{F},\aleph ),$ is the Banach space of all continuous functions and satisfies ${sup}_{\wp \in \mathsf{{\rm Y}}}E{\Vert N(\wp )\Vert }^2<\infty .$

Define $ϝ=\{·:N(\wp )\in C(\mathsf{{\rm Y}},L_2(\mathfrak{F},\aleph ))\},$ with

$${\parallel ·\parallel }_{ϝ}^2=\underset{\wp \in \mathsf{{\rm Y}}}{sup}E{\parallel N(\wp )\parallel }^2.$$

Obviously, $ϝ$ is a Banach space.

We require the following hypotheses:

$\left(A1\right)$$M:\mathsf{{\rm Y}}\times {\aleph }^{m+1}\to \aleph$ verifies the following:

(i)

$M:\mathsf{{\rm Y}}\times {\aleph }^{m+1}\to \aleph$ is continuous;

(ii)

$\forall \epsilon \in N;\epsilon >0\exists$$e_{\epsilon }(·):\mathsf{{\rm Y}}\to R^{+}$ such that

$$\underset{\Vert N_0{\Vert }^2,\dots ,{\Vert N_n\Vert }^2\le \epsilon }{sup}E{\parallel M(\wp ,N_0,N_1,\dots ,N_m)\parallel }^2\le e_{\epsilon }(\wp ),$$

the function $\mathbb{k}\to e_{\epsilon }\left(\mathbb{k}\right)\in L^1((0,b],R^{+})$ and ∃ a ${\chi }_1>0$ such that

$$\underset{\epsilon \to \infty }{lim}inf\frac{{\int }_0^{\wp }{e}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}}{\epsilon }={\chi }_1<\infty ,\wp \in (0,b].$$

$\left(A2\right)$$V:\mathsf{{\rm Y}}\times {\aleph }^{k+1}\to L(G,\aleph )$ verifies the following:

(i)

$\forall \wp \in \mathsf{{\rm Y}},$ the function $V(\wp ,.):{\aleph }^{k+1}\to L(G,\aleph )$ is continuous and $\forall (N_0,N_1,\dots ,N_n)\in {\aleph }^{n+1};$ the function $V(.,N_0,N_1,\dots ,N_k):\mathfrak{J}\to L(G,\aleph )$ is ${\mathfrak{S}}_{\wp }$-measurable;

(ii)

$\forall \epsilon \in N;\epsilon >0\exists$$r_{\epsilon }(·):(0,b]\to R^{+}$ such that

$$\underset{\Vert N_0{\Vert }^2,\dots ,{\Vert N_n\Vert }^2\le \epsilon }{sup}{\int }_0^{\wp }E{\Vert V(\mathbb{k},N_0,N_1,\dots ,N_k)\Vert }_A^{2}d\mathbb{k}\le r_{\epsilon }(\wp ),$$

the function $\mathbb{k}\to r_{\epsilon }\left(\mathbb{k}\right)\in L^1((0,b],R^{+})$ and ∃ a ${\chi }_2>0$ such that

$$\underset{\epsilon \to \infty }{lim}inf\frac{{\int }_0^{\wp }{r}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}}{\epsilon }={\chi }_2<\infty ,\wp \in (0,b].$$

$\left(A3\right)$$\mathsf{\Omega }:\mathsf{{\rm Y}}\times {\aleph }^{p+1}\to L_2^0(Q,\aleph )$ satisfies the following:

(i)

$\forall \wp \in \mathsf{{\rm Y}},$ the function $\mathsf{\Omega }(\wp ,.):{\aleph }^{p+1}\to L_2^0(Q,\aleph )$ is continuous and ∀$(N_0,N_1,\dots ,N_p)\in {\aleph }^{p+1};$ the function $\mathsf{\Omega }(.,N_0,N_1,\dots ,N_p):J\to L_2^0(Q,\aleph )$ is ${\mathfrak{S}}_{\wp }$-measurable;

(ii)

$\forall \epsilon \in N;\epsilon >0\exists$${\overline{r}}_{\epsilon }(·):(0,b]\to R^{+}$ such that

$$\underset{\Vert N_0{\Vert }^2,\dots ,{\Vert N_n\Vert }^2\le \epsilon }{sup}E{\Vert \mathsf{\Omega }(\wp ,N_0,N_1,\dots ,N_p)\Vert }_{L_2^0}^2\le {\overline{r}}_{\epsilon }(\wp ),$$

$\mathbb{k}\to {\overline{r}}_{\epsilon }\left(\mathbb{k}\right)\in L^1((0,b],R^{+})$ and ∃ a ${\chi }_3>0$ such that

$$\underset{\epsilon \to \infty }{lim}inf\frac{{\int }_0^{\wp }{\overline{r}}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}}{\epsilon }={\chi }_3<\infty ,\wp \in (0,b],$$

$\left(A4\right)$$\mho :\mathsf{{\rm Y}}\times {\aleph }^{u+1}\times F\to \aleph$ satisfies the following:

(i)

$\mho :\mathsf{{\rm Y}}\times {\aleph }^{u+1}\times F\to \aleph$ is continuous;

(ii)

$\forall \epsilon \in N;\epsilon >0\exists$$q_{\epsilon }(·):\mathsf{{\rm Y}}\to R^{+}$ such that

$$\underset{\Vert N_0{\Vert }^2,\dots ,{\Vert N_n\Vert }^2\le \epsilon }{sup}{\int }_{F}E{\parallel \mho (\wp ,N_0,N_1,\dots ,N_u,f)\parallel }^2\lambda \left(df\right)\le q_{\epsilon }(\wp ),$$

$\mathbb{k}\to q_{\epsilon }\left(\mathbb{k}\right)\in L^1((0,\wp ],R^{+})$, and ∃ a ${\chi }_4>0$ such that

$$\underset{r\to \infty }{lim}inf\frac{{\int }_0^{\wp }{q}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}}{\epsilon }={\chi }_4<\infty ,\wp \in (0,b].$$

$\left(A5\right)$${\hslash }_i:({\wp }_i,{\mathbb{k}}_i]\times \aleph \to \aleph$ is continuous and verifies the following:

(i)

$\exists {\varrho }_3>0$, such that

$$E{\parallel {\hslash }_i(\wp ,N)\parallel }^2\le {\varrho }_{3}E{\parallel N\parallel }^2,\forall N\in \aleph ;\wp \in ({\wp }_i,{\mathbb{k}}_i],i=1,2,\dots ,m;$$

(ii)

$\exists {\varrho }_4>0$, such that

$$E{\parallel {\hslash }_i(\wp ,N_1)-{\hslash }_i(\wp ,N_2)\parallel }^2\le {\varrho }_{4}E{\parallel N_1-N_2\parallel }^2,\forall N_1,N_2\in \aleph ;\wp \in ({\wp }_i,{\mathbb{k}}_i],i=1,2,\dots ,m.$$

$\left(A6\right)$$\Delta$ generates a compact semigroup $\left\{\mathsf{\Theta }\right(\wp ),\wp \ge 0\}$ in $\aleph .$

Definition 2

(see [32]). $N(\wp ):\mathsf{{\rm Y}}\to \aleph$ is a mild solution of $\left(1\right)$ if $N_0\in \aleph \forall s\in [0,b)$ the function ${\mathbb{k}}^{\pounds -1}M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))$ is integrable and

$$\begin{array}{c}\hfill N(\wp )=\left\{\begin{array}{c}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }}{\pounds }\right)N_0+{\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))d\mathbb{k}\hfill \\ +{\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right))d\omega \left(\tau \right)d\mathbb{k}\hfill \\ +{\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H\left(\mathbb{k}\right)\hfill \\ +{\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)W(d\mathbb{k},df)\hfill \\ +{\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)Bu\left(\mathbb{k}\right)d\mathbb{k},\wp \in (0,{\wp }_1]\hfill \\ {\hslash }_i(\wp ,N(\wp )),\wp \in ({\wp }_i,{\mathbb{k}}_i],i=1,2,\dots ,m\hfill \\ \mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}_i^{\pounds }}{\pounds }\right){\hslash }_i({\mathbb{k}}_i,N\left({\mathbb{k}}_i\right))\hfill \\ +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))d\mathbb{k}\hfill \\ +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right))d\omega \left(\tau \right)d\mathbb{k}\hfill \\ +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H\left(\mathbb{k}\right)\hfill \\ +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_V\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)W(d\mathbb{k},df)\hfill \\ +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)Bu\left(\mathbb{k}\right)d\mathbb{k},\wp \in ({\mathbb{k}}_i,{\wp }_{i+1}],i=1,2,\dots ,m,\hfill \end{array}\right.\end{array}$$

(2)

is verified.

3. Approximate Controllability

Here, we investigate the approximate controllability of (1).

Consider the linear conformable fractional evolution equation in the following form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}^{\pounds }N(\wp )+\Delta N(\wp )=Bu(\wp ),\wp \in (0,b]\hfill \\ N\left(0\right)=N_0.\hfill \end{array}\right.\end{array}$$

(3)

We present the operators associated with (3) as

$${\mathsf{\Xi }}_0^b={\int }_0^b{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)B{B}^{*}{\mathsf{\Theta }}^{*}\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)d\mathbb{k},$$

and $\mathsf{\Phi }(b,{\mathsf{\Xi }}_0^b)={(bI+{\mathsf{\Xi }}_0^b)}^{-1},b>0,$ where the adjoint of B and $\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)$ are denoted by $B^{*}$ and ${\mathsf{\Theta }}^{*}\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right),$ respectively.

The state value of $\left(1\right)$ at terminal state $b,$ corresponding to the control u and the initial value $N_0$, is denoted by $N(b;N_0,u).$ Furthermore, the reachable set of $\left(1\right)$ at terminal time b is denoted by $\mathsf{\Phi }(b,N_0)=\{N(b;N_0,u):u\in L_2(\mathsf{{\rm Y}},U)\}$, and its closure in ℵ is $\overline{\mathsf{\Phi }(b,N_0)}.$

Definition 3

([33]). Let $\left(1\right)$ be approximately controllable on Υ if $\overline{\mathsf{\Phi }(b,N_0)}=L_2(\mathfrak{F},\aleph ).$

Lemma 2

([33]). The linear system $\left(3\right)$ is approximately controllable on Υ if and only if

$x{(xI+{\mathsf{\Phi }}_0^b)}^{-1}\to 0$ as $x\to 0^{+}.$

Lemma 3.

$\forall {\overline{N}}_b\in L_2(\mathfrak{F},\aleph )\exists \overline{\psi }$ and $\overline{\phi }\in L_2(\mathfrak{F};L_2(\mathsf{{\rm Y}};L_2^0))$ such that

$${\overline{N}}_b=E{\overline{N}}_b+{\int }_0^b\overline{\psi }(\wp )d\omega (\wp )+{\int }_0^b\overline{\phi }(\wp )d{\mathfrak{B}}^H(\wp ).$$

We define the control function, for any $\delta >0$ and ${\overline{N}}_b\in L_2(\mathfrak{F},\aleph ),$ in the following form:

$$\begin{array}{c}\hfill u^{\rho }(\wp )=\left\{\begin{array}{c}{B}^{*}{\mathsf{\Theta }}^{*}\left(\frac{b^{\pounds }-{\wp }^{\pounds }}{\pounds }\right){(xI+{\mathsf{\Phi }}_0^b)}^{-1}[E{\overline{N}}_b+{\int }_0^b\overline{\psi }\left(\mathbb{k}\right)d\omega \left(\mathbb{k}\right)+{\int }_0^b\overline{\phi }\left(\mathbb{k}\right)d{B}^H\left(\mathbb{k}\right)\hfill \\ -\mathsf{\Theta }\left(\frac{b^{\pounds }}{\pounds }\right)N_0-{\int }_0^b{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))d\mathbb{k}\hfill \\ -{\int }_0^b{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right))d\omega \left(\tau \right)d\mathbb{k}\hfill \\ -{\int }_0^b{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)W(d\mathbb{k},df)\hfill \\ -{\int }_0^b{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H\left(\mathbb{k}\right)],\wp \in (0,{\wp }_1]\hfill \\ B^{*}{\mathsf{\Theta }}^{*}\left(\frac{b^{\pounds }-{\wp }^{\pounds }}{\pounds }\right){(xI+{\mathsf{\Phi }}_0^b)}^{-1}[E{\overline{N}}_b+{\int }_0^b\overline{\psi }\left(\mathbb{k}\right)d\omega \left(\mathbb{k}\right)+{\int }_0^b\overline{\phi }\left(\mathbb{k}\right)d{\mathfrak{B}}^H\left(\mathbb{k}\right)\hfill \\ -\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}_i^{\pounds }}{\pounds }\right){\hslash }_i({\mathbb{k}}_i,N\left({\mathbb{k}}_i\right))\hfill \\ -{\int }_{{\mathbb{k}}_i}^b{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))d\mathbb{k}\hfill \\ -{\int }_{{\mathbb{k}}_i}^b{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right))d\omega \left(\tau \right)d\mathbb{k}\hfill \\ -{\int }_{{\mathbb{k}}_i}^b{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)W(d\mathbb{k},df)\hfill \\ -{\int }_{{\mathbb{k}}_i}^b{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H\left(\mathbb{k}\right)],\hfill \\ \wp \in ({\mathbb{k}}_i,{\wp }_{i+1}].\hfill \end{array}\right.\end{array}$$

(4)

In this paper, we set $\varrho ={sup}_{·\in \mathsf{{\rm Y}}}\parallel \mathsf{\Theta }(·)\parallel ,{\varrho }_B=\parallel B\parallel$ and ${\varrho }_{B^{*}}=\parallel B^{*}\parallel .$

Theorem 2.

Suppose $\left(A1\right)$–$\left(A6\right)$ holds, then $\left(1\right)$ has a mild solution on $\mathsf{{\rm Y}},$ such that

$$\begin{array}{c}\hfill 36{\varrho }^2\left\{1+\frac{{\varrho }^2{\varrho }_{B^{*}}^2{\varrho }_B^2{b}^{2\pounds }}{(2\pounds -1)x^2}\right\}\left\{{\varrho }_3+\frac{b^{2\pounds -1}}{2\pounds -1}{\chi }_1+Tr\left(\mathfrak{Z}\right)\frac{b^{2\pounds -1}}{2\pounds -1}{\chi }_2+\frac{2H{b}^{2(H+\pounds -1)}}{2\pounds -1}{\chi }_3+\frac{b^{2\pounds -1}}{2\pounds -1}{\chi }_4\right\}+{\varrho }_3<1,\end{array}$$

(5)

and ${\gamma }_1=\left[\frac{8{\varrho }^2{b}^{2\pounds }}{2\pounds -1}+{\varrho }_4+4{\varrho }^2{\varrho }_4\right]<1.$

Proof. 

Consider the map $\mathsf{\Lambda }$ on $\overline{\mathfrak{C}}$ defined by to be verified:

$$\begin{array}{c}\hfill \left(\mathsf{\Lambda }N\right)(\wp )=\left\{\begin{array}{c}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }}{\pounds }\right)N_0+{\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))d\mathbb{k}\hfill \\ +{\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)Bu\left(\mathbb{k}\right)d\mathbb{k}\hfill \\ +{\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right))d\omega \left(\tau \right)d\mathbb{k}\hfill \\ +{\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H\left(\mathbb{k}\right)\hfill \\ +{\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)W(d\mathbb{k},df),\wp \in (0,{\wp }_1]\hfill \\ {\hslash }_i(\wp ,N(\wp )),\wp \in ({\wp }_i,{\mathbb{k}}_i],i=1,2,\dots ,m\hfill \\ \mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}_i^{\pounds }}{\pounds }\right){\hslash }_i({\mathbb{k}}_i,N\left({\mathbb{k}}_i\right))+{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))d\mathbb{k}\hfill \\ +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)Bu\left(\mathbb{k}\right)d\mathbb{k}\hfill \\ +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right))d\omega \left(\tau \right)d\mathbb{k}\hfill \\ +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H\left(\mathbb{k}\right)\hfill \\ +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_V\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)W(d\mathbb{k},df),\hfill \\ \wp \in ({\mathbb{k}}_i,{\wp }_{i+1}],i=1,2,\dots ,m.\hfill \end{array}\right.\end{array}$$

□

Next, show that $\mathsf{\Lambda }$ from $\overline{\mathfrak{C}}$ into itself has a fixed point. Set ${\mathfrak{B}}_{\epsilon }=\{N\in \overline{\mathfrak{C}}{,\Vert N\Vert }_{\overline{\mathfrak{C}}}^2\le \epsilon \},$ $\epsilon >0$, integer. Therefore, ${\mathfrak{B}}_{\epsilon }\subset \overline{\mathfrak{C}}$ is a bounded closed convex set in $\overline{\mathfrak{C}},\forall \epsilon$.

From $\left(A1\right)$ and Hölder’s inequality, we obtain

$$\begin{array}{ccc}\hfill & & E\Vert {\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right)){d\mathbb{k}\Vert }^2\hfill \\ & & \le \frac{{\varrho }^2{b}^{2\pounds -1}}{2\pounds -1}{\int }_0^{\wp }\underset{\Vert N_0{\Vert }^2,\dots ,{\Vert N_n\Vert }^2\le \epsilon }{sup}E{\parallel M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))\parallel }^{2}d\mathbb{k}\hfill \\ & & \le \frac{{\varrho }^2{b}^{2\pounds -1}}{2\pounds -1}{\int }_0^{\wp }{e}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}.\hfill \end{array}$$

(6)

It follows that ${\mathbb{k}}^{\pounds -1}M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))$ is integrable on $\mathsf{{\rm Y}},$ and by Bochner’s theorem, $\mathsf{\Lambda }$ is defined on ${\mathfrak{B}}_{\epsilon }.$

From $\left(A2\right)\left(ii\right)$ with Burkholder–Gundy’s inequality, we obtain

$$\begin{array}{ccc}\hfill & & E\Vert {\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left(a_n\left(\tau \right)\right)){d\omega \left(\tau \right)d\mathbb{k}\Vert }^2\hfill \\ & & \le Tr\left(\mathfrak{Z}\right)\frac{{\varrho }^2{b}^{2\pounds -1}}{2\pounds -1}{\int }_0^{\wp }\left(\underset{\Vert N_0{\Vert }^2,\dots ,{\Vert N_n\Vert }^2\le \epsilon }{sup}{\int }_0^{\mathbb{k}}E{\Vert V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right))\Vert }_A^{2}d\tau \right)d\mathbb{k}\hfill \\ & & \le Tr\left(\mathfrak{Z}\right)\frac{{\varrho }^2{b}^{2\pounds -1}}{2\pounds -1}{\int }_0^{\wp }{r}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}.\hfill \end{array}$$

(7)

From $\left(A3\right)\left(ii\right)$ with Burkholder-Gundy’s inequality, this yields

$$\begin{array}{ccc}\hfill & & E\parallel {\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H{\left(\mathbb{k}\right)\parallel }^2\hfill \\ & & \le \frac{2H{\varrho }^2{b}^{2(H+\pounds -1)}}{2\pounds -1}{\int }_0^{\wp }\underset{\Vert N_0{\Vert }^2,\dots ,{\Vert N_n\Vert }^2\le \epsilon }{sup}E{\parallel \mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))\parallel }_{L_2^0}^{2}d\mathbb{k}\hfill \\ & & \le \frac{2H{\varrho }^2{b}^{2(H+\pounds -1)}}{2\pounds -1}{\int }_0^{\wp }{\overline{r}}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}.\hfill \end{array}$$

(8)

From Hölder inequality’s and $\left(A4\right)\left(ii\right),$ we obtain

$$\begin{array}{ccc}\hfill & & E\parallel {\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f){W(d\mathbb{k},df)\parallel }^2\hfill \\ & & \le \frac{{\varrho }^2{b}^{2\pounds -1}}{2\pounds -1}{\int }_0^{\wp }\left(\underset{\Vert N_0{\Vert }^2,\dots ,{\Vert N_n\Vert }^2\le \epsilon }{sup}{\int }_{F}E{\Vert \mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)\Vert }^2\lambda df\right)d\mathbb{k}\hfill \\ & & \le \frac{{\varrho }^2{b}^{2\pounds -1}}{2\pounds -1}{\int }_0^{\wp }{q}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}.\hfill \end{array}$$

(9)

From Hölder’s inequality and Burkholder–Gundy’s inequality with $\left(A1\right)$–$\left(A4\right),$ we obtain, for $\wp \in (0,{\wp }_1]$,

$$\begin{array}{ccc}\hfill E\parallel {\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){Bu\left(\mathbb{k}\right)d\mathbb{k}\parallel }^2& \le & \frac{{\varrho }^4{\varrho }_{B^{*}}^2{\varrho }_B^2{b}^{2\pounds }}{(2\pounds -1)x^2}[E\parallel {\overline{N}}_b{\parallel }^2+Tr\left(\mathfrak{Z}\right){\int }_0^{b}E{\parallel \overline{\psi }\left(\mathbb{k}\right)\parallel }_{\mathfrak{Z}}^{2}d\mathbb{k}\hfill \\ & & +2H{b}^{2H-1}{\int }_0^{b}E\parallel \overline{\phi }{\left(\mathbb{k}\right)\parallel }_{L_2^0}^{2}d\mathbb{k}+E{\parallel N\left(0\right)\parallel }^2+\frac{b^{2\pounds -1}}{2\pounds -1}{\int }_0^{\wp }{e}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}\hfill \\ & & +Tr\left(\mathfrak{Z}\right)\frac{b^{2\pounds -1}}{2\pounds -1}{\int }_0^{\wp }{r}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}\hfill \\ & & +\frac{2H{b}^{2(H+\pounds -1)}}{2\pounds -1}{\int }_0^{\wp }{\overline{r}}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}+\frac{b^{2\pounds -1}}{2\pounds -1}{\int }_0^{\wp }{q}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}],\hfill \end{array}$$

and for $\wp \in ({\mathbb{k}}_i,{\wp }_{i+1}]$, we obtain

$$\begin{array}{ccc}\hfill E\parallel {\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){Bu\left(\mathbb{k}\right)d\mathbb{k}\parallel }^2& \le & \frac{{\varrho }^4{\varrho }_{B^{*}}^2{\varrho }_B^2{b}^{2\pounds }}{(2\pounds -1)x^2}[E\parallel {\overline{N}}_b{\parallel }^2+Tr\left(\mathfrak{Z}\right){\int }_0^{b}E{\parallel \overline{\psi }\left(\mathbb{k}\right)\parallel }_{\mathfrak{Z}}^{2}d\mathbb{k}\hfill \\ & & +2H{b}^{2H-1}{\int }_0^{b}E{\parallel \overline{\phi }\left(\mathbb{k}\right)\parallel }_{L_2^0}^{2}d\mathbb{k}+\epsilon {\varrho }_3+\frac{b^{2\pounds -1}}{2\pounds -1}{\int }_{{\mathbb{k}}_i}^{\wp }{e}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}\hfill \\ & & +Tr\left(\mathfrak{Z}\right)\frac{b^{2\pounds -1}}{2\pounds -1}{\int }_{{\mathbb{k}}_i}^{\wp }{r}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}\hfill \\ & & +\frac{2H{b}^{2(H+\pounds -1)}}{2\pounds -1}{\int }_{{\mathbb{k}}_i}^{\wp }{\overline{r}}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}+\frac{b^{2\pounds -1}}{2\pounds -1}{\int }_{{\mathbb{k}}_i}^{\wp }{q}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}].\hfill \end{array}$$

We claim that $\exists \epsilon >0$ such that $\mathsf{\Lambda }\left({\mathfrak{B}}_{\epsilon }\right)\subseteq {\mathfrak{B}}_{\epsilon }.$ If it false, then $\forall \epsilon >0$; there is a function $N_{\epsilon }(·)\in {\mathfrak{B}}_{\epsilon },$ but $\mathsf{\Lambda }\left(N_{\epsilon }\right)\notin {\mathfrak{B}}_{\epsilon },$ that is $\Vert \left(\mathsf{\Lambda }{N}_{\epsilon }\right)(\wp ){\Vert }_{\overline{\mathfrak{C}}}^2>\epsilon$ for some $\wp =\wp \left(\epsilon \right)\in \mathsf{{\rm Y}},$ where $\wp \left(\epsilon \right)$ means that ℘ is dependent on $\epsilon .$

From $\left(A5\right)$ and Equations (6)–(9), we have, for $\wp \in (0,{\wp }_1]$,

$$\begin{array}{ccc}\hfill \Vert \mathsf{\Lambda }{N}_{\epsilon }{\Vert }_{ϝ}^2& \le & 36\underset{\wp \in \mathsf{{\rm Y}}}{sup}\{E{\parallel \mathsf{\Theta }\left(\frac{{\wp }^{\pounds }}{\pounds }\right)N_0\parallel }^2\hfill \\ & & +E\Vert {\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M(\wp ,N(\wp ),N\left({\ell }_1(\wp )\right),\dots ,N\left({\ell }_m(\wp )\right)){d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){Bu\left(\mathbb{k}\right)d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right)){d\omega \left(\tau \right)d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H{\left(\mathbb{k}\right)\Vert }^2\hfill \\ & & +{\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f){W(d\mathbb{k},df)\Vert }^2\}\hfill \\ & \le & 36{\varrho }^2\left\{1+\frac{{\varrho }^2{\varrho }_{B^{*}}^2{\varrho }_B^2{b}^{2\pounds }}{(2\pounds -1)x^2}\right\}\{E{\parallel N\left(0\right)\parallel }^2+\frac{b^{2\pounds -1}}{2\pounds -1}{\int }_0^b{e}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}\hfill \\ & & +Tr\left(\mathfrak{Z}\right)\frac{b^{2\pounds -1}}{2\pounds -1}{\int }_0^b{r}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}+\frac{2H{b}^{2(H+\pounds -1)}}{2\pounds -1}{\int }_0^b{\overline{r}}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}\hfill \\ & & +\frac{b^{2\pounds -1}}{2\pounds -1}{\int }_0^b{q}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}\}+\frac{{\varrho }^4{\varrho }_{B^{*}}^2{\varrho }_B^2{b}^{2\pounds }}{(2\pounds -1)x^2}[E\parallel {\overline{N}}_b{\parallel }^2+Tr\left(\mathfrak{Z}\right){\int }_0^{b}E{\parallel \overline{\psi }\left(\mathbb{k}\right)\parallel }_{\mathfrak{Z}}^{2}d\mathbb{k}\hfill \\ & & +2H{b}^{2H-1}{\int }_0^{b}E{\parallel \overline{\phi }\left(\mathbb{k}\right)\parallel }_{L_2^0}^{2}d\mathbb{k}],\hfill \end{array}$$

(10)

for $\wp \in ({\wp }_i,{\mathbb{k}}_i]$

$$\begin{array}{c}\hfill \Vert \mathsf{\Lambda }{N}_{\epsilon }{\Vert }_{ϝ}^2\le \underset{\wp \in J}{sup}E{\parallel {\hslash }_i(\wp ,N(\wp ))\parallel }^2\le {\varrho }_3\epsilon .\end{array}$$

(11)

and for $\wp \in ({\mathbb{k}}_i,{\wp }_{i+1}]$

$$\begin{array}{ccc}\hfill \Vert \mathsf{\Lambda }{N}_{\epsilon }{\Vert }_{ϝ}^2& \le & 36\underset{\wp \in J}{sup}\{E{\parallel \mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}_i^{\pounds }}{\pounds }\right){\hslash }_i({\mathbb{k}}_i,N\left({\mathbb{k}}_i\right))\parallel }^2\hfill \\ & & +E\Vert {\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M(\wp ,N(\wp ),N\left({\ell }_1(\wp )\right),\dots ,N\left({\ell }_m(\wp )\right)){d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){Bu\left(\mathbb{k}\right)d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_{{\mathbb{k}}_i}^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right)){d\omega \left(\tau \right)d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H{\left(\mathbb{k}\right)\Vert }^2\hfill \\ & & +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f){W(d\mathbb{k},df)\Vert }^2\}\hfill \\ \hfill & \le & 36{\varrho }^2\left\{1+\frac{{\varrho }^2{\varrho }_{B^{*}}^2{\varrho }_B^2{b}^{2\pounds }}{(2\pounds -1)x^2}\right\}\{\epsilon {\varrho }_3+\frac{b^{2\pounds -1}}{2\pounds -1}{\int }_{{\mathbb{k}}_i}^b{e}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}+Tr\left(\mathfrak{Z}\right)\frac{b^{2\pounds -1}}{2\pounds -1}{\int }_{{\mathbb{k}}_i}^b{r}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}\hfill \\ & & +\frac{2H{\varrho }^2{b}^{2(H+\pounds -1)}}{2\pounds -1}{\int }_{{\mathbb{k}}_i}^b{\overline{r}}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}+\frac{b^{2\pounds -1}}{2\pounds -1}{\int }_{{\mathbb{k}}_i}^b{q}_{\epsilon }\left(\mathbb{k}\right)d\mathbb{k}\}\hfill \\ & & +\frac{{\varrho }^4{\varrho }_{B^{*}}^2{\varrho }_B^2{b}^{2\pounds }}{(2\pounds -1)x^2}[E\parallel {\overline{N}}_b{\parallel }^2+Tr\left(\mathfrak{Z}\right){\int }_0^{b}E{\parallel \overline{\psi }\left(\mathbb{k}\right)\parallel }_{\mathfrak{Z}}^{2}d\mathbb{k}\hfill \\ & & +2H{b}^{2H-1}{\int }_0^{b}E{\parallel \overline{\phi }\left(\mathbb{k}\right)\parallel }_{L_2^0}^{2}d\mathbb{k}].\hfill \end{array}$$

(12)

Adding (10), (11) and (12) in the inequality $\epsilon \le \Vert \left(\mathsf{\Lambda }{N}_{\epsilon }\right)(\wp ){\Vert }_{ϝ}^2$, dividing both sides of the inequality by $\epsilon$, and applying the limit $\epsilon \to +\infty$, then

$$\begin{array}{ccc}& & 36{\varrho }^2\left\{1+\frac{{\varrho }^2{\varrho }_{B^{*}}^2{\varrho }_B^2{b}^{2\pounds }}{(2\pounds -1)x^2}\right\}\{{\varrho }_3+\frac{b^{2\pounds -1}}{2\pounds -1}{\chi }_1+Tr\left(\mathfrak{Z}\right)\frac{b^{2\pounds -1}}{2\pounds -1}{\chi }_2\hfill \\ & & +\frac{2H{b}^{2(H+\pounds -1)}}{2\pounds -1}{\chi }_3+\frac{b^{2\pounds -1}}{2\pounds -1}{\chi }_4\}+{\varrho }_3\ge 1.\hfill \end{array}$$

This contradicts (5) Hence, for $\epsilon >0$, $\mathsf{\Lambda }\left({\mathfrak{B}}_{\epsilon }\right)\subseteq {\mathfrak{B}}_{\epsilon }.$ Next, we have to demonstrate that $\mathsf{\Lambda }$ has a fixed point on ${\mathfrak{B}}_{\epsilon }.$ We decompose $\mathsf{\Lambda }$ as $\mathsf{\Lambda }={\mathsf{\Lambda }}_1+{\mathsf{\Lambda }}_2,$ where ${\mathsf{\Lambda }}_1$ and ${\mathsf{\Lambda }}_2$ are defined on ${\mathfrak{B}}_{\epsilon }$ by

$$\left({\mathsf{\Lambda }}_{1}N\right)(\wp )=\left\{\begin{array}{c}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }}{\pounds }\right)N_0+{\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))d\mathbb{k},\wp \in (0,{\wp }_1]\hfill \\ {\hslash }_i(\wp ,N(\wp )),\wp \in ({\wp }_i,{\mathbb{k}}_i],i=1,2,\dots ,m\hfill \\ \mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}_i^{\pounds }}{\pounds }\right){\hslash }_i({\mathbb{k}}_i,N\left({\mathbb{k}}_i\right))+{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))d\mathbb{k},\hfill \\ \wp \in ({\mathbb{k}}_i,{\wp }_{i+1}],i=1,2,\dots ,m\hfill \end{array}\right.$$

$$\left({\mathsf{\Lambda }}_{2}N\right)(\wp )=\left\{\begin{array}{c}{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)Bu\left(\mathbb{k}\right)d\mathbb{k}\hfill \\ +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right))d\omega \left(\tau \right)d\mathbb{k}\hfill \\ +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H\left(\mathbb{k}\right)\hfill \\ +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)W(d\mathbb{k},df),\hfill \\ \wp \in ({\mathbb{k}}_i,{\wp }_{i+1}],i=0,1,\dots ,m\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

for $\wp \in \mathsf{{\rm Y}}.$ Next, we show that ${\mathsf{\Lambda }}_1$ is a contraction and ${\mathsf{\Lambda }}_2$ is a compact operator. To show that ${\mathsf{\Lambda }}_1$ is a contraction, let $N_1,N_2\in {\mathfrak{B}}_{\epsilon },$ then for each $\wp \in \mathsf{{\rm Y}}$ and by $\left(A1\right)$ and $\left(A5\right)$, we obtain

$$\begin{array}{ccc}\hfill E\parallel \left({\mathsf{\Lambda }}_1{N}_1\right)(\wp )-\left({\mathsf{\Lambda }}_1{N}_2\right)(\wp ){\parallel }^2& \le & 4E\parallel {\int }_0^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)[M(\mathbb{k},N_1\left(\mathbb{k}\right),N_1\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N_1\left({\ell }_m\left(\mathbb{k}\right)\right))\hfill \\ & & -M(\mathbb{k},N_2\left(\mathbb{k}\right),N_2\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N_2\left({\ell }_m\left(\mathbb{k}\right)\right))]d\mathbb{k}{\parallel }^2\hfill \\ & \le & \frac{4{\varrho }^2{b}^{2\pounds }}{2\pounds -1}E{\parallel N_1(\wp )-N_2(\wp )\parallel }^2,\wp \in (0,{\wp }_1]\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill E\Vert \left({\mathsf{\Lambda }}_1{N}_1\right)(\wp )-\left({\mathsf{\Lambda }}_1{N}_2\right)(\wp ){\Vert }^2& \le & E\Vert {\hslash }_i(\wp ,N_1(\wp ))-{\hslash }_i(\wp ,N_2(\wp )){\Vert }^2\hfill \\ & \le & {\varrho }_{4}E{\Vert N_1(\wp )-N_2(\wp )\Vert }^2,\wp \in (0,{\wp }_1]\wp \in ({\wp }_i,{\mathbb{k}}_i]\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill & & E\Vert \left({\mathsf{\Lambda }}_1{N}_1\right)(\wp )-\left({\mathsf{\Lambda }}_1{N}_2\right)(\wp ){\Vert }^2\hfill \\ & & \le 4E\Vert \mathsf{\Theta }\left(\frac{{(\wp -{\mathbb{k}}_i)}^{\pounds }}{\pounds }\right)({\hslash }_i({\mathbb{k}}_i,N_1\left({\mathbb{k}}_i\right))-{\hslash }_i({\mathbb{k}}_i,N_2\left({\mathbb{k}}_i\right))){\Vert }^2\hfill \\ & & +4E\Vert {\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)[M(\wp ,N_1(\wp ),N_1\left({\ell }_1(\wp )\right),\dots ,N_1\left({\ell }_m(\wp )\right))\hfill \\ & & -M(\wp ,N_2(\wp ),N_2\left({\ell }_1(\wp )\right),\dots ,N_2\left({\ell }_m(\wp )\right)){]d\mathbb{k}\Vert }^2\hfill \\ & & \le 4\left[{\varrho }^2{\varrho }_4+\frac{{\varrho }^2{b}^{2\pounds }}{2\pounds -1}\right]E{\Vert N_1(\wp )-N_2(\wp )\Vert }^2,\wp \in ({\mathbb{k}}_i,{\wp }_{i+1}].\hfill \end{array}$$

(15)

Combining (13), (14) and (15), we obtain

$$\begin{array}{ccc}\hfill E\Vert \left({\mathsf{\Lambda }}_1{N}_1\right)(\wp )-\left({\mathsf{\Lambda }}_1{N}_2\right)(\wp ){\Vert }^2& & \le \left[\frac{8{\varrho }^2{b}^{2\pounds }}{2\pounds -1}+{\varrho }_4+4{\varrho }^2{\varrho }_4\right]E{\Vert N_1(\wp )-N_2(\wp )\Vert }^2\hfill \\ & & \le {\gamma }_{1}E{\Vert N_1(\wp )-N_2(\wp )\Vert }^2.\hfill \end{array}$$

Taking ${sup}_{\wp \in \mathsf{{\rm Y}}}$ for both sides of the inequality, we obtain

$$\begin{array}{c}\hfill \underset{\wp \in J}{sup}E\Vert \left({\mathsf{\Lambda }}_1{N}_1\right)(\wp )-\left({\mathsf{\Lambda }}_1{N}_2\right)(\wp ){\Vert }^2\le {\gamma }_1\underset{\wp \in J}{sup}E{\Vert N_1(\wp )-N_2(\wp )\Vert }^2\end{array}$$

Hence,

$$\begin{array}{c}\hfill \Vert {\mathsf{\Lambda }}_1{N}_1-{\mathsf{\Lambda }}_1{N}_2{\Vert }_{ϝ}^2\le {\gamma }_1{\Vert N_1-N_2\Vert }_{ϝ}^2.\end{array}$$

Thus, ${\mathsf{\Lambda }}_1$ is a contraction.

We show that ${\mathsf{\Lambda }}_2$ is compact.

First, we prove the continuity of ${\mathsf{\Lambda }}_2$ on ${\mathfrak{B}}_{\epsilon }$.

Let $\left\{N_n\right\}\subseteq {\mathfrak{B}}_{\epsilon }$ with $N_n\to y$ in ${\mathfrak{B}}_{\epsilon }$ and the control function $u(\wp )=u(\wp ,N)$. Therefore, for each

$\mathbb{k}\in \mathsf{{\rm Y}},N_n\left(\mathbb{k}\right)\to N\left(\mathbb{k}\right)$ with $A2\left(i\right)$, $A3\left(i\right)$, and $A4\left(i\right)$, we obtain

$V(\mathbb{k},N_n\left(\mathbb{k}\right),N_n\left({𝚥}_1\left(\mathbb{k}\right)\right),\dots ,N_n\left({𝚥}_k\left(\mathbb{k}\right)\right))\to V(\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚥}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚥}_k\left(\mathbb{k}\right)\right)),$ as $n\to \infty$,

$\mathsf{\Omega }(\mathbb{k},N_n\left(\mathbb{k}\right),N_n\left(c_1\left(\mathbb{k}\right)\right),\dots ,N_n\left(c_p\left(\mathbb{k}\right)\right))\to \mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right)),$ as $n\to \infty$, and

$\mho (\mathbb{k},N_n\left(\mathbb{k}\right),N_n\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N_n\left({𝚤}_u\left(\mathbb{k}\right)\right),f)\to \mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f),$ as $n\to \infty .$

From Lebesgue’s dominated convergence theorem, we have

$$\begin{array}{ccc}& & \Vert {\mathsf{\Lambda }}_2{N}_n-{\mathsf{\Lambda }}_2{y\Vert }_{ϝ}^2=\underset{\wp \in J}{sup}\{E\parallel {\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\left(Bu(\mathbb{k},N_n)-Bu(\mathbb{k},N)\right)d\mathbb{k}\hfill \\ & & +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}(V(\tau ,N_n\left(\tau \right),N_n\left({𝚥}_1\left(\tau \right)\right),\dots ,N_n\left({𝚥}_k\left(\tau \right)\right))\hfill \\ & & -V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right)))d\omega \left(\tau \right)d\mathbb{k}\hfill \\ & & +{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)(\mathsf{\Omega }(\mathbb{k},N_n\left(\mathbb{k}\right),N_n\left(c_1\left(\mathbb{k}\right)\right),\dots ,N_n\left(c_p\left(\mathbb{k}\right)\right))\hfill \\ & & -\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right)))d{\mathfrak{B}}_H\left(\mathbb{k}\right)\hfill \\ \hfill \end{array}$$

$$\begin{array}{ccc}& & \hspace{1em}+{\int }_{{\mathbb{k}}_i}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F(\mho (\mathbb{k},N_n\left(\mathbb{k}\right),N_n\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N_n\left({𝚤}_u\left(\mathbb{k}\right)\right),f)\hfill \\ & & \hspace{1em}-\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)\left)W(d\mathbb{k},df){\parallel }^2\right\}\to 0,\hfill \end{array}$$

as $n\to \infty ,$ which is continuous.

Next, we show that $\{{\mathsf{\Lambda }}_{2}y:N\in {\mathfrak{B}}_{\epsilon }\}$ is an equicontinuous family of functions. Assume $ϵ>0,$ small, ${\mathbb{k}}_i<{\wp }_{\alpha }<{\wp }_{\beta }\le {\wp }_{i+1}$, then

$$\begin{array}{ccc}& & E\Vert \left({\mathsf{\Lambda }}_{2}N\right)\left({\wp }_{\beta }\right)-\left({\mathsf{\Lambda }}_{2}N\right)\left({\wp }_{\alpha }\right){\Vert }^2\hfill \\ & & \le E\Vert {\int }_{{\mathbb{k}}_i}^{{\wp }_{\alpha }-ϵ}{\mathbb{k}}^{\pounds -1}\left(\mathsf{\Theta }\left(\frac{{\wp }_{\beta }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)-\mathsf{\Theta }\left(\frac{{\wp }_{\alpha }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\right){Bu\left(\mathbb{k}\right)d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\wp }_{\alpha }-ϵ}^{{\wp }_{\alpha }}{\mathbb{k}}^{\pounds -1}\left(\mathsf{\Theta }\left(\frac{{\wp }_{\beta }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)-\mathsf{\Theta }\left(\frac{{\wp }_{\alpha }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\right){Bu\left(\mathbb{k}\right)d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\wp }_{\alpha }}^{{\wp }_{\beta }}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }_{\beta }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){Bu\left(\mathbb{k}\right)d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\mathbb{k}}_i}^{{\wp }_{\alpha }-ϵ}{\mathbb{k}}^{\pounds -1}\left(\mathsf{\Theta }\left(\frac{{\wp }_{\beta }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)-\mathsf{\Theta }\left(\frac{{\wp }_{\alpha }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\right)\hfill \\ & & \times {\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right)){d\omega \left(\tau \right)d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\wp }_{\alpha }-ϵ}^{{\wp }_{\alpha }}{\mathbb{k}}^{\pounds -1}\left(\mathsf{\Theta }\left(\frac{{\wp }_{\beta }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)-\mathsf{\Theta }\left(\frac{{\wp }_{\alpha }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\right)\hfill \\ & & \times {\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right)){d\omega \left(\tau \right)d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\wp }_{\alpha }}^{{\wp }_{\beta }}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }_{\beta }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right)){d\omega \left(\tau \right)d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\mathbb{k}}_i}^{{\wp }_{\alpha }-ϵ}{\mathbb{k}}^{\pounds -1}\left(\mathsf{\Theta }\left(\frac{{\wp }_{\beta }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)-\mathsf{\Theta }\left(\frac{{\wp }_{\alpha }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right)){)d{\mathfrak{B}}_H\left(\mathbb{k}\right)\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\wp }_{\alpha }-ϵ}^{{\wp }_{\alpha }}{\mathbb{k}}^{\pounds -1}\left(\mathsf{\Theta }\left(\frac{{\wp }_{\beta }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)-\mathsf{\Theta }\left(\frac{{\wp }_{\alpha }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right)){)d{\mathfrak{B}}_H\left(\mathbb{k}\right)\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\wp }_{\alpha }}^{{\wp }_{\beta }}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }_{\beta }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right)){)d{\mathfrak{B}}_H\left(\mathbb{k}\right)\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\mathbb{k}}_i}^{{\wp }_{\alpha }-ϵ}{\mathbb{k}}^{\pounds -1}\left(\mathsf{\Theta }\left(\frac{{\wp }_{\beta }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)-\mathsf{\Theta }\left(\frac{{\wp }_{\alpha }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\right)\hfill \\ & & \times {\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f){W(d\mathbb{k},df)\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\wp }_{\alpha }-ϵ}^{{\wp }_{\alpha }}{\mathbb{k}}^{\pounds -1}\left(\mathsf{\Theta }\left(\frac{{\wp }_{\beta }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)-\mathsf{\Theta }\left(\frac{{\wp }_{\alpha }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\right)\hfill \\ & & \times {\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f){W(d\mathbb{k},df)\Vert }^2\hfill \\ & & +E\Vert {\int }_{{\wp }_{\alpha }}^{{\wp }_{\beta }}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }_{\beta }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f){W(d\mathbb{k},df)\Vert }^2.\hfill \end{array}$$

As ${\wp }_{\beta }\to {\wp }_{\alpha },$ we see that $E\Vert \left({\mathsf{\Lambda }}_{2}N\right)\left({\wp }_{\beta }\right)-\left({\mathsf{\Lambda }}_{2}N\right)\left({\wp }_{\alpha }\right){\Vert }^2\to 0$ independently of $N\in {\mathfrak{B}}_{\epsilon }$, with $ϵ$ sufficiently small, because the compactness of $\mathsf{\Theta }(\wp )$ for $\wp >0$ tends to the continuity in the uniform operator topology. Furthermore, we can show that ${\mathsf{\Lambda }}_{2}N,N\in {\mathfrak{B}}_{\epsilon }$ are equicontinuous at $\wp =0.$ Then, ${\mathsf{\Lambda }}_2$ maps ${\mathfrak{B}}_{\epsilon }$ into a family of equicontinuous functions. Next, we show that $T(\wp )=\{\left({\mathsf{\Lambda }}_{2}N\right)(\wp ):N\in {\mathfrak{B}}_{\epsilon }\}$ is relatively compact in ${\mathfrak{B}}_{\epsilon }$. Clearly, $T\left(0\right)$ is relatively compact in ${\mathfrak{B}}_{\epsilon }.$

Assume ${\mathbb{k}}_i<\wp \le {\wp }_{i+1}$ to be fixed; ${\mathbb{k}}_i<ϵ<\wp$ for $N\in {\mathfrak{B}}_{\epsilon },$ we define

$$\begin{array}{ccc}\hfill \left({\mathsf{\Lambda }}_2^{ϵ}N\right)(\wp )& & ={\int }_{{\mathbb{k}}_i}^{\wp -ϵ}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)Bu\left(\mathbb{k}\right)d\mathbb{k}\hfill \\ & & +{\int }_{{\mathbb{k}}_i}^{\wp -ϵ}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right)d\omega \left(\tau \right)d\mathbb{k}\hfill \\ & & +{\int }_{{\mathbb{k}}_i}^{\wp -ϵ}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H\left(\mathbb{k}\right)\hfill \\ & & +{\int }_{{\mathbb{k}}_i}^{\wp -ϵ}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)W(d\mathbb{k},df)\hfill \\ & & =\mathsf{\Theta }\left(\frac{{ϵ}^{\pounds }}{\pounds }\right){\int }_{{\mathbb{k}}_i}^{\wp -ϵ}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }-{ϵ}^{\pounds }}{\pounds }\right)Bu\left(\mathbb{k}\right)d\mathbb{k}\hfill \\ & & +\mathsf{\Theta }\left(\frac{{ϵ}^{\pounds }}{\pounds }\right){\int }_{{\mathbb{k}}_i}^{\wp -ϵ}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }-{ϵ}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}\mho (\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right)d\omega \left(\tau \right)d\mathbb{k}\hfill \\ & & +\mathsf{\Theta }\left(\frac{{ϵ}^{\pounds }}{\pounds }\right){\int }_{{\mathbb{k}}_i}^{\wp -ϵ}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }-{ϵ}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H\left(\mathbb{k}\right)\hfill \\ & & +\mathsf{\Theta }\left(\frac{{ϵ}^{\pounds }}{\pounds }\right){\int }_{{\mathbb{k}}_i}^{\wp -ϵ}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }-{ϵ}^{\pounds }}{\pounds }\right)\hfill \\ & & \times {\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)W(d\mathbb{k},df).\hfill \end{array}$$

Since $\mathsf{\Theta }\left(\frac{{ϵ}^{\pounds }}{\pounds }\right),\frac{{ϵ}^{\pounds }}{\pounds }>0$ is a compact operator, hence $T^{ϵ}(\wp )=\{\left({\mathsf{\Lambda }}_2^{ϵ}N\right)(\wp ):N\in {\mathfrak{B}}_{\epsilon }\}$ is relatively compact in ℵ for every $ϵ,$ ${\mathbb{k}}_i<ϵ<\wp$.

Moreover, $\forall N\in {\mathfrak{B}}_{\epsilon },$ we have

$$\begin{array}{ccc}& & \Vert {\mathsf{\Lambda }}_{2}y-{\mathsf{\Lambda }}_2^{ϵ}{y\Vert }_{ϝ}^2\le 16\underset{\wp \in J}{sup}\{E{\Vert {\int }_{\wp -ϵ}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)Bu\left(\mathbb{k}\right)d\mathbb{k}\parallel }^2\hfill \\ & & +E\Vert {\int }_{\wp -ϵ}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V{(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right)d\omega \left(\tau \right)d\mathbb{k}\Vert }^2\hfill \\ & & +E\Vert {\int }_{\wp -ϵ}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H{\left(\mathbb{k}\right)\Vert }^2\hfill \\ & & +E\Vert {\int }_{\wp -ϵ}^{\wp }{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{{\wp }^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f){W(d\mathbb{k},df)\Vert }^2\}\hfill \\ & \le & 16{\varrho }^2\{\frac{(b^{2\pounds -1}-{(b-ϵ)}^{2\pounds -1}){\varrho }_B^2}{2\pounds -1}{\int }_{b-ϵ}^{b}E{\Vert u\left(\mathbb{k}\right)\Vert }^{2}d\mathbb{k}\hfill \\ & & +Tr\left(\mathfrak{Z}\right)\frac{(b^{2\pounds -1}-{(b-ϵ)}^{2\pounds -1})}{2\pounds -1}{\int }_{b-ϵ}^b{\int }_0^{\mathbb{k}}E\Vert V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right){\Vert }_A^{2}d\tau d\mathbb{k}\hfill \\ & & +2H{ϵ}^{2H-1}\frac{(b^{2\pounds -1}-{(b-ϵ)}^{2\pounds -1})}{2\pounds -1}{\int }_{b-ϵ}^{b}E{\Vert \mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))\Vert }_{L_2^0}^{2}d\mathbb{k}\hfill \\ & & +\frac{(b^{2\pounds -1}-{(b-ϵ)}^{2\pounds -1})}{2\pounds -1}{\int }_{b-ϵ}^b{\int }_{F}E{\Vert \mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)\Vert }^2\lambda dfd\mathbb{k}\}.\hfill \end{array}$$

We see that, for each $N\in {\mathfrak{B}}_{\epsilon },$ $\Vert {\mathsf{\Lambda }}_{2}y-{\mathsf{\Lambda }}_2^{ϵ}{\Vert }_{ϝ}^2\to 0$ as $ϵ\to 0^{+}.$ Therefore, there are relative compact sets arbitrarily close to $T(\wp )=\{\left({\mathsf{\Lambda }}_{2}N\right)(\wp ):N\in {\mathfrak{B}}_{\epsilon }\};$ hence, $T(\wp )$ is also relatively compact in ${\mathfrak{B}}_{\epsilon }.$

Thus, by the Arzela–Ascoli theorem ${\mathsf{\Lambda }}_2$ is a compact operator. Hence, $\mathsf{\Lambda }={\mathsf{\Lambda }}_1+{\mathsf{\Lambda }}_2$ is a condensing map on ${\mathfrak{B}}_{\epsilon }$, and by the fixed-point theorem of Sadovskii, there exists a fixed point $N(·)$ for $\mathsf{\Lambda }$ on ${\mathfrak{B}}_{\epsilon }$. Thus, the stochastic system $\left(1\right)$ has a mild solution on $\mathsf{{\rm Y}}$.

Theorem 3.

Suppose that Assumptions $\left(A1\right)$–$\left(A6\right)$ are satisfied. Moreover, if $M,V,\mho$ and Ω are uniformly bounded, then $\left(1\right)$ be approximately controllable on Υ.

Proof. 

Assume N is a fixed point of $\mathsf{\Lambda }.$ By the stochastic Fubini theorem, we obtain

$$\begin{array}{ccc}\hfill N\left(b\right)& =& {\overline{N}}_b-x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}\{E{\overline{N}}_b+{\int }_0^T\overline{\psi }\left(s\right)d\omega \left(s\right)+{\int }_0^T\overline{\phi }\left(s\right)d{B}^H\left(s\right)\hfill \\ & -& x{\int }_{{\mathbb{k}}_m}^b{(xI+{\mathsf{\Xi }}_0^b)}^{-1}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))d\mathbb{k}\hfill \\ & -& x{\int }_{{\mathbb{k}}_m}^b{(xI+{\mathsf{\Xi }}_0^b)}^{-1}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right))d\omega \left(\tau \right)d\mathbb{k}\hfill \\ & -& x{\int }_{{\mathbb{k}}_m}^b{(xI+{\mathsf{\Xi }}_0^b)}^{-1}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)W(d\mathbb{k},df)\hfill \\ & -& x{\int }_{{\mathbb{k}}_m}^b{(xI+{\mathsf{\Xi }}_0^b)}^{-1}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))d{\mathfrak{B}}_H\left(\mathbb{k}\right)\}.\hfill \end{array}$$

□

From the condition on $M,V,\mho$ and $\mathsf{\Omega },$ there exists $\mathfrak{D}>0$ such that

$$\parallel M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right){\parallel }^2\le \mathfrak{D},\parallel V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right)){\parallel }_{\mathfrak{Z}}^2\le \mathfrak{D},$$

$$\parallel \mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f){\parallel }^2\le \mathfrak{D},$$

$$\parallel \mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right)){\parallel }_{L_2^0}^2\le \mathfrak{D},$$

Consequently, the sequences

$\{M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right)\},\{\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)\},$

$\left\{V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right))\right\},\left\{\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))\right\}$ are weakly compact in

$L_2(J,\aleph ),L_2\left(L_{\mathfrak{Z}}(G,\aleph )\right)$ and $L_2\left(L_2^0(Q,\aleph )\right),$

so there are subsequences

$\{M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right)\},\{\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)\},$

$\left\{V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right))\right\},$$\left\{\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))\right\}$ that weakly converge to $\left\{M\right(\mathbb{k}\left)\right\},\{\mho (\mathbb{k},f\left)\right\},\left\{V\right(\mathbb{k}\left)\right\},\left\{\mathsf{\Omega }\right(\mathbb{k}\left)\right\}$ in $L_2(\mathsf{{\rm Y}},\aleph ),L_2\left(L_{\mathfrak{Z}}(G,\aleph )\right)$, and $L_2\left(L_2^0(Q,\aleph )\right).$

From the above, we have

$$\begin{array}{ccc}& & E\parallel N\left(b\right)-{\overline{N}}_b{\parallel }^2\hfill \\ & \le & 36E\parallel x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}E{\overline{N}}_b{\parallel }^2+36E\parallel {\int }_0^b{\parallel x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}\overline{\psi }\left(\mathbb{k}\right)d\omega \left(\mathbb{k}\right)\parallel }^2\hfill \\ & +& 36E\parallel {\int }_0^{b}x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}\overline{\phi }\left(\mathbb{k}\right)d{B}^H\left(\mathbb{k}\right){\parallel }^2\hfill \\ & +& 36E\parallel {\int }_{{\mathbb{k}}_m}^{b}x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\left[M(\mathbb{k},N\left(\mathbb{k}\right),N\left({\ell }_1\left(\mathbb{k}\right)\right),\dots ,N\left({\ell }_m\left(\mathbb{k}\right)\right))-M\left(\mathbb{k}\right)\right]{d\mathbb{k}\parallel }^2\hfill \\ & +& 36E\parallel {\int }_{{\mathbb{k}}_m}^{b}x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)M\left(\mathbb{k}\right){d\mathbb{k}\parallel }^2\hfill \\ & +& 36E\parallel {\int }_{{\mathbb{k}}_m}^{b}x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\hfill \\ & & \times {\int }_0^{\mathbb{k}}\left[V(\tau ,N\left(\tau \right),N\left({𝚥}_1\left(\tau \right)\right),\dots ,N\left({𝚥}_k\left(\tau \right)\right))-V\left(\tau \right)\right]{d\omega \left(\tau \right)d\mathbb{k}\parallel }^2\hfill \\ & +& 36E\parallel {\int }_{{\mathbb{k}}_m}^{b}x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_0^{\mathbb{k}}V\left(\tau \right)d\omega \left(\tau \right){d\mathbb{k}\parallel }^2\hfill \\ & +& 36E\parallel {\int }_{{\mathbb{k}}_m}^{b}x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\hfill \\ & & \times {\int }_F\left[\mho (\mathbb{k},N\left(\mathbb{k}\right),N\left({𝚤}_1\left(\mathbb{k}\right)\right),\dots ,N\left({𝚤}_u\left(\mathbb{k}\right)\right),f)-\mho \left(\mathbb{k}\right)\right]{W(d\mathbb{k},df)\parallel }^2\hfill \\ & +& 36E\parallel {\int }_{{\mathbb{k}}_m}^{b}x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right){\int }_F\mho \left(\mathbb{k}\right)W(d\mathbb{k},df){\parallel }^2\hfill \\ & +& 36E\parallel {\int }_{{\mathbb{k}}_m}^{b}x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\left[\mathsf{\Omega }(\mathbb{k},N\left(\mathbb{k}\right),N\left(c_1\left(\mathbb{k}\right)\right),\dots ,N\left(c_p\left(\mathbb{k}\right)\right))-\mathsf{\Omega }\left(\mathbb{k}\right)\right]d{\mathfrak{B}}_H\left(\mathbb{k}\right){\parallel }^2\hfill \\ & +& 36E\parallel {\int }_{{\mathbb{k}}_m}^{b}x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}{\mathbb{k}}^{\pounds -1}\mathsf{\Theta }\left(\frac{b^{\pounds }-{\mathbb{k}}^{\pounds }}{\pounds }\right)\mathsf{\Omega }\left(\mathbb{k}\right)d{\mathfrak{B}}_H\left(\mathbb{k}\right){\parallel }^2.\hfill \end{array}$$

By Lemma 2, $x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}\to 0$ strongly as $x\to 0^{+}$ for all ${\mathbb{k}}_m<\mathbb{k}\le b$, and furthermore, $\parallel x{(xI+{\mathsf{\Xi }}_0^b)}^{-1}\parallel \le 1$. Thus, $E\parallel N\left(b\right)-{\overline{N}}_b{\parallel }^2\to 0$ as $x\to 0^{+}$ by the Lebesgue-dominated convergence theorem and the compactness of $\mathsf{\Theta }\left(\mathbb{k}\right)$. Hence, the system (1) is approximate controllable.

4. Example

Consider the CF noninstantaneous impulsive stochastic partial differential equation with fBm and Poisson jump of the form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}_{0+}^{0.6}N(\wp ,z)+\frac{{\partial }^2}{\partial z^2}N(\wp ,z)=\frac{cos\wp }{1+cos\wp }N(\wp ,z)+\mathfrak{w}(\wp ,z)+{\int }_0^{\wp }{3}^{-\mathbb{k}}N(\mathbb{k},z)d\omega \left(\mathbb{k}\right)\hfill \\ +\frac{e^{\wp }}{1+e^{\wp }}N(\wp ,z)\frac{d{\mathfrak{B}}_H(\wp )}{d\wp }+{\int }_{F}h(\wp ,N(\wp ,z),f)W(d\wp ,df),\wp \in (0,\frac{2}{3}]\cup (\frac{5}{4},3],0\le z\le \pi ,\hfill \\ N(\wp ,0)=N(\wp ,\pi )=0,\wp \in (0,3],\hfill \\ N(\wp ,z)=\frac{2}{7}{e}^{-(\wp -\frac{2}{3})}\frac{\left|N\right(\wp ,z\left)\right|}{1+\left|N\right(\wp ,z\left)\right|},\wp \in (\frac{2}{3},\frac{5}{4}],0\le z\le \pi ,\hfill \\ \left(N(0,z)\right)=N_0\left(z\right),0\le z\le \pi ,\hfill \end{array}\right.\end{array}$$

(16)

where $D^{0.6}$ is the CFD of order $\pounds =0.6,$ $\omega$ is a Wiener process, and ${\mathfrak{B}}_H$ is an fBm with $H\in (\frac{1}{2},1)$. Assume $\aleph =Q=G=\mathfrak{U}=L_2\left([0,\pi ]\right)$ and $\Delta$, where $\Delta y=-\left(\frac{{\partial }^2}{\partial z^2}\right)N$ with domain $D(\Delta )=\{N\in \aleph :N,\frac{dN}{dz}$ are absolutely continuous and $\left(\frac{d^2}{d{z}^2}\right)N\in \aleph ,N\left(0\right)=N\left(\pi \right)=0\}.$

Then, $-\Delta$ generates a strongly continuous semigroup $\mathsf{\Theta }(·)$, which is compact, analytic, and self-adjoint. Moreover, $-\Delta$ has a discrete spectrum with eigenvalues $n^2,n\in N$ and the corresponding normalized eigenfunctions given by

$${\mathfrak{x}}_n=\sqrt{\frac{2}{\pi }}sinnN,n=1,2,...$$

In addition, ${\left({\mathfrak{x}}_n\right)}_{n\in N}$ is a complete orthonormal basis in $\aleph .$ Then,

$$-\Delta N=\sum _{n=1}^{\infty }{n}^2⟨N,{\mathfrak{x}}_n⟩{\mathfrak{x}}_n,N\in D(\Delta ).$$

Moreover, $-\Delta$ generates an analytic semigroup of the bounded linear operator, ${\left\{\mathsf{\Theta }(\wp )\right\}}_{\wp \ge 0}$ on ℵ, and is defined by

$$\mathsf{\Theta }(\wp )N=\sum _{n=1}^{\infty }{x}^{-n^2\wp }{⟨N,x_n⟩}_{x_n},N\in \aleph ,\wp \ge 0.$$

with $\parallel \mathsf{\Theta }(\wp )\parallel \le x^{-\wp }\le 1.$ We define $B:\mathfrak{U}\to \aleph$ by $Bu(\wp )\left(z\right)=\mathfrak{w}(\wp ,z),0\le z\le \pi ,u\in \mathfrak{U}.$ Furthermore, $M:\mathsf{{\rm Y}}\times \aleph \to \aleph ,$ $V:\mathsf{{\rm Y}}\times \aleph \to L(G,\aleph ),$ $\mathsf{\Omega }:\mathsf{{\rm Y}}\times \aleph \to L_2^0(Q,\aleph ),$ $\mho :J\times \aleph \times F\to \aleph ,$ and ${\hslash }_i:({\wp }_i,{\mathbb{k}}_i]\times \aleph \to \aleph$ are defined by $M(\wp ,N)\left(z\right)=\frac{cos\wp }{1+cos\wp }N(\wp ,z),V(\mathbb{k},N)\left(z\right)=3^{-\mathbb{k}}N(\mathbb{k},z),$ $\mathsf{\Omega }(\wp ,N)\left(z\right)=\frac{e^{\wp }}{1+e^{\wp }}N(\wp ,z),$ $\mho (\wp ,N)\left(z\right)=\overline{h}(\wp ,N(\wp ,z),f),$ and ${\hslash }_1(\wp ,N(\wp ))=\frac{2}{7}{e}^{-(\wp -\frac{2}{3})}\frac{\left|N\right(\wp ,·\left)\right|}{1+\left|N\right(\wp ,·\left)\right|}$, respectively. Then $M,V,\mathsf{\Omega },\mho ,$ and ${\hslash }_1$ verify $\left(A1\right)$–$\left(A6\right)$.

Let $B=B^{*}=I$. Therefore, all conditions of Theorems 2 and 3 are verified and

$$\begin{array}{c}\hfill 36{\varrho }^2\left\{1+\frac{{\varrho }^2{\varrho }_{B^{*}}^2{\varrho }_B^2{b}^{2\pounds }}{(2\pounds -1)x^2}\right\}\left\{{\varrho }_3+\frac{b^{2\pounds -1}}{2\pounds -1}{\chi }_1+Tr\left(\mathfrak{Z}\right)\frac{b^{2\pounds -1}}{2\pounds -1}{\chi }_2+\frac{2H{b}^{2(H+\pounds -1)}}{2\pounds -1}{\chi }_3+\frac{b^{2\pounds -1}}{2\pounds -1}{\chi }_4\right\}+{\varrho }_3<1,\end{array}$$

and ${\gamma }_1=\left[\frac{8{\varrho }^2{b}^{2\pounds }}{2\pounds -1}+{\varrho }_4+4{\varrho }^2{\varrho }_4\right]<1.$

Thus, (16) is approximately controllable on $\mathsf{{\rm Y}}.$

5. Conclusions

By using fractional calculus, a compact semigroup, Sadovskii’s fixed-point theorem, and stochastic analysis, we investigated the approximate controllability of the given system (1). The obtained theoretical conclusions were illustrated in the later portion with an example. The results can be extended to a fractional stochastic inclusion system.