Existence and Approximate Controllability of ψ-Caputo Fractional Stochastic Evolution Equations of Sobolev Type with Poisson Jumps and Nonlocal Conditions

Abstract

The existence of mild solutions and the approximate controllability for a class of Sobolev-type $\psi$-Caputo fractional stochastic evolution equations (SCFSEEs) subject to nonlocal conditions and Poisson jumps are investigated in this paper. First, the existence of mild solutions for the SCFSEEs is established by employing tools from fractional calculus, stochastic analysis, the theory of two characteristic solution operators, and Schauder’s fixed point theorem. Then, the approximate controllability of the system is proven. Finally, an example is provided to illustrate the applicability of the theoretical results.

1. Introduction

In this paper, we consider the following Sobolev-type $\psi$-Caputo fractional stochastic evolution equations with Poisson jumps:

$${}^{C}D_0^{\alpha ,\psi }Ex\left(t\right)=Ax\left(t\right)+Bu\left(t\right)+f(t,x\left(t\right))+\sigma (t,x\left(t\right)){\displaystyle \frac{dW}{dt}}+{\int }_{V}g(t,x\left(t\right),v)\overline{N}(dt,dv),t\in J,$$

(1)

and nonlocal conditions

$$x\left(0\right)=\sum _{i=1}^m{c}_{k}x\left(t_k\right),$$

(2)

where $0<\alpha <1$, $J=[0,b]$ ($b>0$) and ${}^{C}D_0^{\alpha ,\psi }x\left(t\right)$ is the Caputo fractional derivative of a function x with respect to another function $\psi$, where $\psi \in C^1\left(J\right)$ is an increasing function with ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in J$. The state $x(·)$ takes a value in a separable Hilbert space X, the operators $A:D\left(A\right)\subset X\to X$ are the infinitesimal generator of a strongly continuous semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ in X, $E:D\left(E\right)\subset X\to X$, the control function $u(·)$ is given in $L^2(J,U)$, the Hilbert space of admissible control functions with U as a separable Hilbert space, B is a bounded linear operator from U into X, $0<t_1<t_2<\cdots <t_m<b$, $m\in \mathbb{N}$, $c_k$ are real numbers, $c_k\ne 0$, $k=1,2,\dots ,m$, $f:J\times X\to X$, $\sigma :J\times X\to {\mathcal{L}}_2^0$, and $g:J\times X\times V\to X$.

In 1963, Kalman [1] first introduced the concept of controllability. Due to its significant applications in the field of physics, extensive research has since been conducted on the complete and approximate controllability of systems described by integer-order and fractional-order differential equations in Banach spaces. For reference, see [2,3,4,5,6] and the literature cited therein. Approximate controllability, a fundamental concept in the mathematical control theory of infinite-dimensional differential systems, plays a crucial role in both deterministic and stochastic control theories (see [7]). Roughly speaking, a system is approximately controllable if it can be driven to an arbitrarily small neighborhood of any target state. This concept proves sufficient for many practical applications. Consequently, research on the approximate controllability of fractional-order evolution equations has advanced significantly in recent years. In [8], Liu and Li studied control systems governed by fractional evolution differential equations involving Riemann–Liouville fractional derivatives in Banach spaces. Later, Lian et al. [9] investigated the approximate controllability for a class of semilinear fractional differential systems of order $1<q<2$ by using the resolvent operators. In [10], Yadav et al. examined the approximate controllability of a class of Sobolev-type non-instantaneous impulsive Hilfer fractional stochastic differential systems driven by the Rosenblatt process and Poisson jumps. Shukla et al. [11] established results on the approximate controllability of Hilfer fractional stochastic differential inclusions of order $1<q<2$ using stochastic analysis, cosine families, fixed point theory, and fractional calculus. In [12], Li and Luo studied the approximate controllability of Hilfer fractional stochastic evolution equations by employing the Tikhonov-type regularization method and Schauder’s fixed point theorem.

Due to environmental noise, deterministic models often exhibit random behavior. Stochastic models have consequently been widely applied over the past few decades in various engineering fields [13,14,15,16]. In the field of stochastic processes, Poisson jumps provide a powerful model for describing random discontinuities and abrupt changes in dynamical systems. Stochastic evolution equations with Poisson jumps have attracted considerable attention in recent years. In [17], Taniguchi and Luo investigated the existence of mild solutions for stochastic evolution equations with infinite delay driven by Poisson jump processes. Long et al. [18] employed the Krasnoselski–Schaefer fixed point theorem to derive sufficient conditions for the approximate controllability of stochastic partial differential equations with infinite delay driven by Poisson jumps. In [19], Muthukumar and Thiagu established the approximate controllability of fractional-order $1<q<2$ nonlocal neutral impulsive stochastic differential equations with Poisson jumps using differentiable resolvent operators.

The concept of “nonlocal conditions” was first introduced by Byszewski [20,21]. In contrast to classical initial conditions, nonlocal conditions can describe natural phenomena more accurately as they incorporate more information. Nonlocal conditions often prove more practical than the standard initial condition $u\left(0\right)=u_0$ for certain physical phenomena. In 2017, Wang et al. [22] investigated the approximate controllability of a class of Sobolev-type fractional evolution systems with nonlocal conditions in Hilbert spaces. Their approach involved constructing control functions containing Gramian controllability operators and applying Schauder’s fixed point theorem. More recently, Jia [23] studied the existence and approximate controllability of mild solutions for a class of nonlinear evolution hemivariational inequalities with nonlocal conditions. This work utilized fixed point theory and nonsmooth analysis in Hilbert spaces. In a related line of research, Chen et al. [24] established the existence of mild solutions and the approximate controllability for fractional evolution equations with nonlocal conditions. They considered the following system:

$${\displaystyle \left\{\begin{array}{c}{\displaystyle {}^{C}D_0^{\alpha }u\left(t\right)=Au\left(t\right)+Bv\left(t\right)+f(t,u\left(t\right)),t\in J,}\hfill \\ {\displaystyle x\left(0\right)=\sum _{i=1}^m{c}_{k}u\left(t_k\right).}\hfill \end{array}\right.}$$

(3)

As a powerful tool for describing the memory and hereditary properties of various processes, fractional calculus has significantly contributed to both the theory and application of fractional differential equations in physics, mathematics, and engineering. Extensive research has explored its applications in physics, such as in [25,26,27,28,29]. In recent years, fractional calculus operators have been widely generalized [30,31,32], as these broader frameworks enable more comprehensive analysis of various special cases. A key development was Almeida’s generalization of the Caputo fractional derivative, which defines the derivative of a function with respect to another function $\psi$ [33]. A major advantage of this $\psi$-Caputo derivative is that selecting an appropriate function $\psi$ can significantly improve model accuracy [34]. For instance, in physical applications, this concept has been used to generalize the Scott–Blair model for fluids with time-varying viscosity [35]. Furthermore, as demonstrated in [36], introducing fractional derivatives with respect to another function offers a superior framework for modeling the growth rate of U.S. GDP. The theoretical foundations of these operators are also being advanced. Suechori and Ngiamsunthorn [37] presented local and global existence and uniqueness theorems for mild solutions to a class of semilinear $\psi$-Caputo fractional evolution equations. In [38], Yang et al. investigated the controllability of a class of Sobolev-type impulsive $\psi$-Caputo fractional evolution equations in Banach spaces.

This paper addresses a research gap by investigating the approximate controllability of Sobolev-type $\psi$-Caputo fractional stochastic evolution equations with nonlocal conditions and a Poisson jump process (1)–(2), a problem that, to our knowledge, remains unexplored in the literature. We extend the work of [24] to this more general framework. Our methodology involves first recasting the problem into an equivalent integral equation by employing the solution operators $S_E^{\alpha ,\psi }$ and $T_E^{\alpha ,\psi }$ and a novel Green’s function. Subsequently, the existence and approximate controllability are established by an application of Schauder’s fixed point theorem, combined with tools from fractional and stochastic calculus.

The main contributions of this work are summarized as follows:

(i)

For the first time, the existence and approximate controllability of $\psi$-fractional stochastic evolution equations with Poisson jumps and nonlocal conditions are established.

(ii)

Our analysis effectively utilizes fractional calculus, stochastic inequalities, and two newly introduced characteristic solution operators.

(iii)

The present study offers a novel and more technical approach, extending the main results of [24].

The remainder of this paper is organized as follows. Section 2 provides the necessary definitions and preliminaries. In particular, the mild solution for Systems (1)–(2) is defined by introducing a new Green’s function. In Section 3, we prove the existence of mild solutions and the approximate controllability under weaker sufficient conditions. An example is presented in Section 4 to illustrate the theoretical results. Finally, conclusions are drawn in Section 5.

2. Preliminaries

In this section, we provided some basic definitions and lemmas that are used in the sequel.

Let X and Y be the separable Hilbert spaces and $\mathcal{L}(Y,X)$ be the space of bounded linear operators from Y into X. Let $\left\{W\right(t):t\ge 0\}$ be a Y-valued Wiener process, defined on the complete filtered probability space $(\mathrm{\Omega },\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathbb{P})$ with a finite trace nuclear covariance operator $Q\ge 0$. Here, $Q\in \mathcal{L}(Y,Y)$ is an operator defined by $Q{e}_n={\lambda }_n{e}_n$ with finite trace $Tr\left(Q\right)={\displaystyle \sum _{n=1}^{\infty }}{\lambda }_n<\infty$, where ${\lambda }_n\ge 0$ ($n=1,2,\cdots$) are non-negative real numbers and $\left\{e_n\right\}$ ($n=1,2,\cdots$) is a complete orthonormal basis in Y, and there exists a sequence ${\beta }_n$ of independent Brownian motions such that

$$W\left(t\right)=\sum _{n=1}^{\infty }\sqrt{{\lambda }_n}{\beta }_n\left(t\right)e_n,t\ge 0.$$

The above Y-valued stochastic process $W\left(t\right)$ is called a Q-Wiener process. ${\mathcal{L}}_2^0$, a space with inner product ${\langle x,y\rangle }_{{\mathcal{L}}_2^0}={\sum }_{n=1}^{\infty }\langle x{e}_n,y{e}_n\rangle$, is a separable Hilbert space if $x=y$, ${\parallel x\parallel }_{{\mathcal{L}}_2^0}^2={\parallel x{Q}^{{\displaystyle \frac{1}{2}}}\parallel }^2=Tr\left(xQ{x}^{\ast }\right)$.

Let $L^2(\Omega ,X)$ denote the Banach space of all ${\mathcal{F}}_b$-measurable, square-integrable random variables with values in the Hilbert space X, endowed with the norm ${\parallel x(·)\parallel }_{L^2}={\left({\mathbb{E}\parallel x(·)\parallel }^2\right)}^{{\displaystyle \frac{1}{2}}}$, where $\mathbb{E}(·)$ denotes the expectation defined by $\mathbb{E}x={\int }_{\mathrm{\Omega }}x(·)d\mathbb{P}$. Let $(V,\Phi ,\mathrm{\lambda }(dv\left)\right)$ be a $\sigma$-finite measurable space. Given the stationary Poisson point process ${\left(p_t\right)}_{t\ge 0}$, which is defined on $(\mathrm{\Omega },\mathcal{F},P)$ with values in V and with characteristic measure $\lambda$, we denote with $N(t,dv)$ the counting measure of $p_t$ such that $\overline{N}(t,\mathrm{\Theta }):=\mathbb{E}\left(N(t,\mathrm{\Theta })\right)=t\lambda \left(\mathrm{\Theta }\right)$ for $\mathrm{\Theta }\in \mathrm{\Phi }$. Define $\overline{N}(dt,dv):=N(dt,dv)-\lambda \left(dv\right)dt$ and the Poisson martingale measure generated by $p_t$.

Definition 1 

(see [39]). Let $\alpha >0$, f be an integrable function defined on $[a,b]$ and $\psi \in C^1\left([a,b]\right)$ be an increasing function with ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$. The left ψ-Riemann–Liouville fractional integral operator of order α of a function f is defined by

$$I_a^{\alpha ,\psi }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}f\left(s\right)ds.$$

(4)

Definition 2 

(see [33,39]). Let $n-1<\alpha <n$, $f\in C^n\left([a,b]\right)$, and $\psi \in C^1\left([a,b]\right)$ be an increasing function with ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$. The left ψ-Caputo fractional derivative of order α of a function f is defined by

$$\begin{array}{cc}\hfill & {\displaystyle {}^{C}D_0^{\alpha ,\psi }f\left(t\right)=\left(I_a^{n-\alpha ,\psi }{f}^{\left[n\right]}\right)\left(t\right)}\hfill \\ \hfill & {\displaystyle =\frac{1}{\Gamma (n-\alpha )}{\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{n-\alpha -1}{f}^{\left[n\right]}\left(s\right){\psi }^{\prime }\left(s\right)ds,}\hfill \end{array}$$

(5)

where $n=\left[\alpha \right]+1$ and $f^{\left[n\right]}\left(t\right):={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{n}f\left(t\right)$ on $[a,b]$.

In the following, we will present the operational formulas for fractional integrals and fractional derivatives of a function with respect to another function.

Lemma 1 

(see [33]). Let $f\in C^n\left([a,b]\right)$ and $n-1<\alpha <n$. Then, we have

(1) 

${\displaystyle {}^{C}D_0^{\alpha ,\psi }{I}_a^{\alpha ,\psi }f\left(t\right)=f\left(t\right)}$;

(2) 

${\displaystyle I_a^{\alpha ,\psi }{}^{C}D_0^{\alpha ,\psi }f\left(t\right)=f\left(t\right)-\sum _{k=0}^{n-1}\frac{f^{\left[k\right]}\left(a^{+}\right)}{k!}{(\psi \left(t\right)-\psi \left(a\right))}^k}$.

In particular, given $\alpha \in (0,1)$, one has

$${\displaystyle I_a^{\alpha ,\psi }{}^{C}D_0^{\alpha ,\psi }=f\left(t\right)-f\left(a\right).}$$

Definition 3 

(see [39]). Let $u,\psi :[a,\infty )\to \mathbb{R}$ be real-valued functions such that $\psi \left(t\right)$ is continuous and ${\psi }^{\prime }\left(t\right)>0$ on $[0,\infty )$. The generalized Laplace transform of u is denoted by

$${\mathcal{L}}_{\psi }\left\{u\left(t\right)\right\}\left(s\right)={\int }_a^{\infty }{e}^{-s\left(\psi \right(\tau )-\psi (a\left)\right)}u\left(\tau \right){\psi }^{\prime }\left(\tau \right)d\tau$$

(6)

for all s.

For (1), we introduce the following assumptions on the operators A and E.

$\left(H_1\right)$ A and E are linear operators, and A is closed.

$\left(H_2\right)$ $D\left(E\right)\subset D\left(A\right)$, and E is bijective.

$\left(H_3\right)$ The linear operator $E^{-1}:X\to D\left(E\right)\subset X$ is compact (which implies that $E^{-1}$ is bounded).

From $\left(H_3\right)$, we know that E is closed because $E^{-1}$ is closed and injective; thus, the inverse is also closed. Note from $\left(H_1\right)$–$\left(H_3\right)$ and the closed graph theorem the boundedness of the linear operator $-A{E}^{-1}:X\to X$. Consequently, $-A{E}^{-1}$ generates a semigroup $\left\{T\right(t),t\ge 0\}$, $T\left(t\right):=e^{-A{E}^{-1}t}$.

According to Definitions 1 and 2 and Lemma 1, we can rewrite Equation (1) in the equivalent fractional integral equation

$$\begin{array}{cc}\hfill & {\displaystyle Ex\left(t\right)=Ex\left(0\right)+\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}(Ax\left(s\right)+f(s,x\left(s\right))+Bu\left(s\right)){\psi }^{\prime }\left(s\right)ds}\hfill \\ \hfill & {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{\int }_V{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv),t\in J,}\hfill \end{array}$$

(7)

provided that the integral in (7) exists.

From the generalized Laplace transform (6), similar to the proof of Lemma 4 in [38], we can obtain the mild solution of $\psi$-fractional evolution Equation (1) with initial value $x\left(0\right)$ as follows.

Lemma 2. 

Assume that $\left(H_1\right)$–$\left(H_3\right)$ hold. If (7) holds, then we have

$$\begin{array}{cc}\hfill & {\displaystyle x\left(t\right)=S_E^{\alpha ,\psi }(t,0)Ex\left(0\right)+{\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t,s)(f(s,x\left(s\right))+Bu\left(s\right)){\psi }^{\prime }\left(s\right)ds}\hfill \\ \hfill & {\displaystyle +{\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t,s)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +{\int }_0^t{\int }_V{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t,s)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(dt,dv),t\in J.}\hfill \end{array}$$

(8)

Here, $S_E^{\alpha ,\psi }(t,s)$ and $T_E^{\alpha ,\psi }(t,s)$ are called characteristic solutions and are given by

$$S_E^{\alpha ,\psi }(t,s)u:={\int }_0^{\infty }{E}^{-1}{\xi }_{\alpha }\left(\theta \right)T\left({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta \right)ud\theta$$

(9)

and

$$T_E^{\alpha ,\psi }(t,s)u:=\alpha {\int }_0^{\infty }{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right)T\left({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta \right)ud\theta$$

(10)

for $0\le s\le t\le b$, where

$${\xi }_{\alpha }\left(\theta \right)={\displaystyle \frac{1}{\alpha }}{\theta }^{-1-{\displaystyle \frac{1}{\alpha }}}{\rho }_{\alpha }({\theta }^{-{\displaystyle \frac{1}{\alpha }}}),$$

$${\rho }_{\alpha }\left(\theta \right)={\displaystyle \frac{1}{\pi }}\sum _{k=1}^{\infty }{(-1)}^{k-1}{\theta }^{-\alpha k-1}{\displaystyle \frac{\Gamma (\alpha k+1)}{k!}}sin\left(k\pi \alpha \right),$$

and ${\xi }_{\alpha }$ is the probability density function defined on $(0,\infty )$.

From [37], we easily obtain the following properties of $S_E^{\alpha ,\psi }(t,s)$ and $T_E^{\alpha ,\psi }(t,s)$.

Lemma 3. 

Suppose that conditions $\left(H_1\right)$–$\left(H_3\right)$ hold. Then, the operators $S_E^{\alpha ,\psi }$ and $T_E^{\alpha ,\psi }$ have the following properties:

(i)

For any fixed $t\ge s\ge 0$, $S_E^{\alpha ,\psi }(t,s)$ and $T_E^{\alpha ,\psi }(t,s)$ are bounded linear operators with

$${\displaystyle \parallel S_E^{\alpha ,\psi }(t,s)\left(x\right)\parallel \le M\parallel E^{-1}\parallel \parallel x\parallel and\parallel T_E^{\alpha ,\psi }(t,s)\left(x\right)\parallel \le \frac{M\parallel E^{-1}\parallel }{\Gamma \left(\alpha \right)}\parallel x\parallel ,}$$

for each $x\in X$, where constant $M>0$.

(ii)

$S_E^{\alpha ,\psi }(t,s)$ and $T_E^{\alpha ,\psi }(t,s)$ are strongly continuous for all $t\ge s\ge 0$; that is, for $0\le s\le t_1<t_2\le b$, we have

$${\displaystyle \parallel S_E^{\alpha ,\psi }(t_2,s)-S_E^{\alpha ,\psi }(t_1,s)\parallel \to 0and\parallel T_E^{\alpha ,\psi }(t_2,s)-T_E^{\alpha ,\psi }(t_1,s)\parallel \to 0}$$

as $t_2\to t_1$.

(iii)

If $T\left(t\right)$ is compact operator for every $t>0$, then $S_E^{\alpha ,\psi }(t,s)$ and $T_E^{\alpha ,\psi }(t,s)$ are compact for all $t,s>0$.

(iv)

If $S_E^{\alpha ,\psi }(t,s)$ and $T_E^{\alpha ,\psi }(t,s)$ are a compact, strongly continuous semigroup of the bounded linear operator for $t,s>0$, then $S_E^{\alpha ,\psi }(t,s)$ and $T_E^{\alpha ,\psi }(t,s)$ are continuous in the uniform operator topology.

Throughout this paper, we assume that

$\left(H_4\right)$${\displaystyle \sum _{k=1}^m\left|c_k\right|<\frac{1}{M\parallel E\parallel \parallel E^{-1}\parallel }}$.

From the assumption $\left(H_4\right)$ and Lemma 3, we obtain

$$∥\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E∥\le M\parallel E^{-1}\parallel \sum _{k=1}^m|c_k|\parallel E\parallel <1.$$

(11)

From (11) and the operator spectrum theorem, it can be easily determined that

$$\mathrm{\Lambda }:={\left(I-\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E\right)}^{-1}$$

(12)

exists and is bounded, and $D\left(\mathrm{\Lambda }\right)=X$. Moreover, via Neumann expression, $\mathrm{\Lambda }$ can be expressed by

$$\mathrm{\Lambda }=\sum _{n=0}^{\infty }{\left(\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E\right)}^n.$$

(13)

Thus

$$\parallel \mathrm{\Lambda }\parallel \le \sum _{n=0}^{\infty }{∥\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E∥}^n\le {\displaystyle \frac{1}{1-M\parallel E\parallel \parallel E^{-1}\parallel {\displaystyle \sum _{k=1}^m}\left|c_k\right|}}.$$

(14)

From (8), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle x\left(t_k\right)=S_E^{\alpha ,\psi }(t_k,0)Ex\left(0\right)+{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)(f(s,x\left(s\right))+Bu\left(s\right)){\psi }^{\prime }\left(s\right)ds}\hfill \\ \hfill & {\displaystyle +{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +{\int }_0^{t_k}{\int }_V{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv),k=1,2,\cdots ,m.}\hfill \end{array}$$

(15)

From (2) and (15), we have

$$\begin{array}{cc}\hfill & {\displaystyle x\left(0\right)=\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)Ex\left(0\right)}\hfill \\ \hfill & {\displaystyle +\sum _{k=1}^m{c}_k{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)(f(s,x\left(s\right))+Bu\left(s\right)){\psi }^{\prime }\left(s\right)ds}\hfill \\ \hfill & {\displaystyle +\sum _{k=1}^m{c}_k{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +\sum _{k=1}^m{c}_k{\int }_0^{t_k}{\int }_V{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv).}\hfill \end{array}$$

From the definition of operator $\mathrm{\Lambda }$, we deduce that

$$\begin{array}{cc}\hfill & {\displaystyle x\left(0\right)=\sum _{k=1}^m{c}_k\mathrm{\Lambda }{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)(f(s,x\left(s\right))+Bu\left(s\right)){\psi }^{\prime }\left(s\right)ds}\hfill \\ \hfill & {\displaystyle +\sum _{k=1}^m{c}_k\mathrm{\Lambda }{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +\sum _{k=1}^m{c}_k\mathrm{\Lambda }{\int }_0^{t_k}{\int }_V{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv).}\hfill \end{array}$$

(16)

Substituting (16) into (8), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle x\left(t\right)=\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E\mathrm{\Lambda }{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)(f(s,x\left(s\right))+Bu\left(s\right)){\psi }^{\prime }\left(s\right)ds}\hfill \\ \hfill & {\displaystyle +\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E\mathrm{\Lambda }{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E\mathrm{\Lambda }{\int }_0^{t_k}{\int }_V{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)}\hfill \\ \hfill & {\displaystyle +{\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t,s)(f(s,x\left(s\right))+Bu\left(s\right)){\psi }^{\prime }\left(s\right)ds}\hfill \\ \hfill & {\displaystyle +{\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t,s)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +{\int }_0^t{\int }_V{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t,s)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv),t\in J.}\hfill \end{array}$$

(17)

For convenience, let Green’s function $G(t,s)$ be the following:

$$\begin{array}{cc}\hfill & {\displaystyle G(t,s)=\sum _{k=1}^m{\chi }_{t_k}\left(s\right){(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{S}_E^{\alpha ,\psi }(t_k,0)E\mathrm{\Lambda }{T}_E^{\alpha ,\psi }(t_k,s)}\hfill \\ \hfill & {\displaystyle +{\chi }_t\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t,s),}\hfill \end{array}$$

(18)

where

$${\chi }_{t_k}\left(s\right)=\left\{\begin{array}{cc}{c}_k,\hfill & s\in [0,t_k),\hfill \\ 0,\hfill & s\in [t_k,b],\hfill \end{array}\right.{\chi }_t\left(s\right)=\left\{\begin{array}{cc}1,\hfill & s\in [0,t),\hfill \\ 0,\hfill & s\in [t,b].\hfill \end{array}\right.$$

(19)

Then, from (17)–(19), we know that the mild solution $x\in C(J,L^2(\mathrm{\Omega },X))$ of (1)–(2) can be expressed by

$$\begin{array}{cc}\hfill & {\displaystyle x\left(t\right)={\int }_0^{b}G(t,s)(f(s,x\left(s\right))+Bu\left(s\right)){\psi }^{\prime }\left(s\right)ds+{\int }_0^{b}G(t,s)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +{\int }_0^b{\int }_{V}G(t,s)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv),t\in J.}\hfill \end{array}$$

(20)

Thus, we have the following definition.

Definition 4. 

A stochastic process $x(·)$ defined on $[0,b]$ is said to be a mild solution of SCFSEEs (1)–(2) if the following conditions are satisfied:

(i) 

$x\left(t\right)\in X$ has càdlàg paths on $t\in [0,b]$ to X, and $x\left(t\right)$ is ${\mathcal{F}}_t$-adapted;

(ii) 

${\mathbb{E}\left|x\left(t\right)\right|}^2<\infty$ for each $t\in [0,b]$;

(iii) 

For $u\in L_2^{\mathcal{F}}([0,b];U)$ (the set of all square integrable processes $u(·)$ with value in U adapted to ${\mathcal{F}}_t$), the process $x(·)$ satisfies the integral Equation (20).

3. Main Results

In this section, we will present and prove the existence of mild solutions and the approximate controllability of SCFSEEs (1)–(2).

From Lemma 7.2 in [13], we have the following.

Lemma 4. 

Let $\varphi \in [0,b]\times \Omega \to L_2^0$ be a strongly measurable mapping such that ${\int }_0^b\mathbb{E}{\parallel \varphi \left(s\right)\parallel }_{L_2^0}^{2}ds<\infty$. Then, for $t\in [0,b]$,

$$\mathbb{E}{∥{\int }_0^t\varphi \left(s\right)dW\left(s\right)∥}^2\le \overline{c}\mathbb{E}\left({\int }_0^t{\parallel \varphi \left(s\right)\parallel }_{L_2^0}^{2}ds\right),$$

where $\overline{c}$ is a constant.

Lemma 5 

([40]). Let $h:J\times V\to X$ and assume that

$$\mathbb{E}\left({\int }_0^t{\int }_V{\parallel h(s,v)\parallel }^p\lambda \left(dv\right)ds\right)<\infty ,p=2,4,$$

It holds that, for $t\in [0,b]$,

$$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}\left(\underset{0\le \tau \le t}{sup}{∥{\int }_0^{\tau }{\int }_{V}h(s,v)\overline{N}(ds,dv)∥}^2\right)}\hfill \\ \hfill & {\displaystyle \le D\left\{\mathbb{E}\left({\int }_0^t{\int }_V{\parallel h(s,v)\parallel }^2\lambda \left(dv\right)ds\right)+\mathbb{E}{\left({\int }_0^t{\int }_V{\parallel h(s,v)\parallel }^4\lambda \left(dv\right)ds\right)}^{\frac{1}{2}}\right\}.}\hfill \end{array}$$

for a constant $D>0$.

Lemma 6 

([41,42]). The Wright function ${\xi }_{\alpha }$ is an entire function and has the following properties:

(i) 

${\displaystyle {\xi }_{\alpha }\left(\theta \right)\ge 0for\theta \ge 0and{\int }_0^{\infty }{\xi }_{\alpha }\left(\theta \right)d\theta =1}$;

(ii) 

${\displaystyle {\int }_0^{\infty }{\xi }_{\alpha }\left(\theta \right){\theta }^{r}d\theta =\frac{\Gamma (1+r)}{\Gamma (1+\alpha r)}forr>-1}$;

(iii) 

${\displaystyle {\int }_0^{\infty }{\xi }_{\alpha }\left(\theta \right)e^{-z\theta }d\theta =E_{\alpha }(-z),z\in \mathbb{C}}$;

(iv) 

${\displaystyle \alpha {\int }_0^{\infty }\theta {\xi }_{\alpha }\left(\theta \right)e^{-z\theta }d\theta =E_{\alpha ,\alpha }(-z),z\in \mathbb{C}}$.

Besides $\left(H_1\right)$–$\left(H_4\right)$, we state the following hypotheses.

$\left(H_5\right)$ The function $f:J\times X\to X$ satisfies the following conditions:

(i)

For a.e. $t\in J$, $t\to f(t,x)$ is continuous;

(ii)

For each $x\in X$, $t\to f(t,x)$ is strongly measurable;

(iii)

There exists a continuous non-decreasing function ${\eta }_1:[0,\infty )\to [0,\infty )$ and ${\rho }_1\in L^{{\displaystyle \frac{1}{{\gamma }_1}}}(J,{\mathbb{R}}^{+})$ with ${\gamma }_1\in (0,2\alpha -1)$ such that

$${\mathbb{E}\parallel f(t,x\left(t\right))\parallel }^2\le {\rho }_1\left(t\right){\eta }_1{(\mathbb{E}\parallel x\left(t\right)\parallel }^2),\forall (t,x)\in J\times X,$$

and

$$\underset{r\to \infty }{lim}inf{\displaystyle \frac{{\eta }_1\left(r\right)}{r}}=0.$$

$\left(H_6\right)$ The function $\sigma :J\times X\to {\mathcal{L}}_2^0$ satisfies the following conditions:

(i)

For a.e. $t\in J$, $t\to \sigma (t,x)$ is continuous;

(ii)

For each $x\in X$, $t\to \sigma (t,x)$ is strongly measurable;

(iii)

There exists a continuous non-decreasing function ${\eta }_2:[0,\infty )\to [0,\infty )$ and ${\rho }_2\in L^{{\displaystyle \frac{1}{{\gamma }_2}}}(J,{\mathbb{R}}^{+})$ with ${\gamma }_2\in (0,2\alpha -1)$ such that

$${\mathbb{E}\parallel \sigma (t,x\left(t\right))\parallel }_{{\mathcal{L}}_2^0}^2\le {\rho }_2\left(t\right){\eta }_2{(\mathbb{E}\parallel x\left(t\right)\parallel }^2),\forall (t,x)\in J\times X,$$

and

$$\underset{r\to \infty }{lim}inf{\displaystyle \frac{{\eta }_2\left(r\right)}{r}}=0.$$

$\left(H_7\right)$ The function $g:J\times X\times V\to X$ satisfies the following conditions:

(i)

For a.e. $t\in J$, $t\to g(t,x,v)$ is continuous;

(ii)

For each $x\in X$, $t\to g(t,x,v)$ is strongly measurable;

(iii)

There exist continuous functions ${\rho }_3\left(t\right),{\rho }_4\left(t\right)$ and continuous non-decreasing functions ${\eta }_3,{\eta }_4:[0,\infty )\to [0,\infty )$ such that

$${\int }_V{\mathbb{E}\parallel g(t,x,v)\parallel }^2\lambda \left(dv\right)\le {\rho }_3\left(t\right){\eta }_3{(\mathbb{E}\parallel x\left(t\right)\parallel }^2),\forall (t,x,v)\in J\times X\times V,$$

$${\int }_V{\mathbb{E}\parallel g(t,x,v)\parallel }^4\lambda \left(dv\right)\le {\rho }_4\left(t\right){\eta }_4{(\mathbb{E}\parallel x\left(t\right)\parallel }^2),\forall (t,x,v)\in J\times X\times V,$$

and

$$\underset{r\to \infty }{lim}inf{\displaystyle \frac{{\eta }_3\left(r\right)}{r}}=0,\underset{r\to \infty }{lim}inf{\displaystyle \frac{{\eta }_4\left(r\right)}{r^2}}=0.$$

In the following, we present the concept of approximate controllability of SCFSEEs (1)–(2).

Definition 5. 

Let $x(b;x(0),u)$ be the state value of SCFSEEs (1)–(2) at terminal time b corresponding to the control $u\in L^2(J,U)$ and nonlocal initial condition $x\left(0\right)$. SCFSEEs (1)–(2) are said to be approximately controllable on the interval J if $\overline{{\mathfrak{R}}_b}=L^2(\Omega ;X)$, where the set

$${\mathfrak{R}}_b=\{x(b;x\left(0\right),u):u\in L^2(J,U)\}$$

is called the reachable set of SCFSEEs (1)–(2).

Let

$${\Gamma }_0^b={\int }_0^{b}G(b,s)B{B}^{\ast }{G}^{\ast }(b,s){\psi }^{\prime }\left(s\right)ds,$$

here

$$\begin{array}{cc}\hfill & {\displaystyle G^{\ast }(b,s)=\sum _{k=1}^m{\chi }_{t_k}\left(s\right){(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T_E^{\alpha ,\psi }}^{\ast }(t_k,0){\mathrm{\Lambda }}^{\ast }{E}^{\ast }{S_E^{\alpha ,\psi }}^{\ast }(t_k,s)}\hfill \\ \hfill & {\displaystyle +{\chi }_t\left(s\right){(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}{T_E^{\alpha ,\psi }}^{\ast }(b,s),}\hfill \end{array}$$

(21)

where $s\in [0,b]$ and $B^{\ast },E^{\ast },{\mathrm{\Lambda }}^{\ast },{S_E^{\alpha ,\psi }}^{\ast }$, and ${T_E^{\alpha ,\psi }}^{\ast }$ denote the adjoint operators of $B,E,\mathrm{\Lambda },S_E^{\alpha ,\psi }$, and $T_E^{\alpha ,\psi }$, respectively.

The resolvent operator $R(\lambda ,{\Gamma }_0^b):X\to X$ for $\lambda >0$ is defined as

$$R(\lambda ,{\Gamma }_0^b):={(\lambda I+{\Gamma }_0^b)}^{-1}.$$

Since the operator ${\Gamma }_0^b$ is clearly positive, $R(\lambda ,{\Gamma }_0^b)$ is well defined. We will always assume that

$\left(H_8\right)$ $\lambda R(\lambda ,{\Gamma }_0^b)\to 0$ as $\lambda \to 0^{+}$ in the strong operator topology.

For every $\lambda >0$ and $x_b\in L^2(\Omega ,X)$, we will prove that there exists a continuous function $x\in C(J,L^2(\Omega ,X))$ such that

$$\begin{array}{cc}\hfill & {\displaystyle x\left(t\right)={\int }_0^{b}G(t,s)(f(s,x\left(s\right))+B{u}_{\lambda }(s,x\left(s\right))){\psi }^{\prime }\left(s\right)ds+{\int }_0^{b}G(t,s)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +{\int }_0^b{\int }_{V}G(t,s)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv),}\hfill \end{array}$$

(22)

where the function $u_{\lambda }$ is the control function defined by

$$u_{\lambda }(t,x)=B^{\ast }{G}^{\ast }(b,t)R(\lambda ,{\Gamma }_0^b)q\left(x(·)\right),$$

(23)

with

$$\begin{array}{cc}\hfill & {\displaystyle q\left(x(·)\right)=x_b-{\int }_0^{b}G(b,s)f(s,x\left(s\right)){\psi }^{\prime }\left(s\right)ds-{\int }_0^{b}G(b,s)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle -{\int }_0^b{\int }_{V}G(b,s)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv).}\hfill \end{array}$$

(24)

From Lemma 3, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \parallel G^{\ast }(b,s)\parallel \le ∥\sum _{k=1}^m{\chi }_{t_k}\left(s\right){(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T_E^{\alpha ,\psi }}^{\ast }(t_k,0){\mathrm{\Lambda }}^{\ast }{E}^{\ast }{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{S_E^{\alpha ,\psi }}^{\ast }(t_k,s)∥}\hfill \\ \hfill & {\displaystyle +∥{\chi }_t\left(s\right){(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}{T_E^{\alpha ,\psi }}^{\ast }(b,s)∥}\hfill \\ \hfill & {\displaystyle \le \frac{M^2\parallel E^{-1}{\parallel }^2\parallel E\parallel \parallel \mathrm{\Lambda }\parallel }{\Gamma \left(\alpha \right)}\sum _{k=1}^m{c}_k{(\psi \left(t_k\right)-\psi \left(0\right))}^{\alpha -1}+\frac{M\parallel E^{-1}\parallel }{\Gamma \left(\alpha \right)}{(\psi \left(b\right)-\psi \left(0\right))}^{\alpha -1}}\hfill \\ \hfill & {\displaystyle :=N_0.}\hfill \end{array}$$

(25)

If $x\in D_r:=\{x\in C(J,L^2(\Omega ,X))$ is an ${\mathcal{F}}_t$-adapted càdlàg process $|{sup}_{t\in J}{\mathbb{E}\parallel x\left(t\right)\parallel }^2\le r\}$ ($r>0$), from $\left(H_5\right)$, $\left(H_6\right)$, and H$\ddot{\mathrm{o}}$lder’s inequality, one can obtain the following:

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_0^t{\mathbb{E}\parallel f(s,x\left(s\right))\parallel }^{2}ds\le {\int }_0^t{\rho }_1\left(s\right){\eta }_1{(\mathbb{E}\parallel x\left(s\right)\parallel }^2)ds\le {\int }_0^t{\rho }_1\left(s\right){\eta }_1\left(r\right)ds}\hfill \\ \hfill & {\displaystyle \le {\eta }_1\left(r\right){\left({\int }_0^t{1}^{\frac{1}{1-{\gamma }_1}}ds\right)}^{1-{\gamma }_1}{\left({\int }_0^t{\rho }_1^{\frac{1}{{\gamma }_1}}\left(s\right)ds\right)}^{{\gamma }_1}}\hfill \\ \hfill & {\displaystyle \le {\eta }_1\left(r\right)b^{1-{\gamma }_1}{\parallel {\rho }_1\parallel }_{\frac{1}{{\gamma }_1}}<\infty ,}\hfill \end{array}$$

(26)

where ${\displaystyle \parallel {\rho }_1{\parallel }_{\frac{1}{{\gamma }_1}}={\left({\int }_0^b{\rho }_1^{\frac{1}{{\gamma }_1}}\left(s\right)ds\right)}^{{\gamma }_1}}$ and

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }\left(s\right)\mathbb{E}{\parallel \sigma (s,x\left(s\right))\parallel }_{{\mathcal{L}}_2^0}^{2}ds}\hfill \\ \hfill & {\displaystyle \le {\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }\left(s\right){\rho }_2\left(s\right){\eta }_2{(\mathbb{E}\parallel x\left(s\right)\parallel }^2)ds}\hfill \\ \hfill & {\displaystyle \le {\eta }_2\left(r\right){\left({\int }_0^t{\left[{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }\left(s\right)\right]}^{\frac{1}{1-{\gamma }_2}}ds\right)}^{1-{\gamma }_2}{\left({\int }_0^t{\rho }_2^{\frac{1}{{\gamma }_2}}\left(s\right)ds\right)}^{{\gamma }_2}}\hfill \\ \hfill & {\displaystyle ={\eta }_2\left(r\right)K^{{\gamma }_2}{\left(\frac{1-{\gamma }_2}{2\alpha -1-{\gamma }_2}\right)}^{1-{\gamma }_2}{(\psi \left(t\right)-\psi \left(0\right))}^{2\alpha -1-{\gamma }_2}{\parallel {\rho }_2\parallel }_{\frac{1}{{\gamma }_2}}<\infty ,}\hfill \end{array}$$

(27)

where $K={max}_{t\in J}{\psi }^{\prime }\left(t\right)$. From $\left(H_7\right)$, we get

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_0^t{\int }_V{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }\left(s\right)\mathbb{E}{\parallel g(s,x\left(s\right),v)\parallel }^2\lambda \left(dv\right)ds}\hfill \\ \hfill & {\displaystyle \le {\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }\left(s\right){\rho }_3\left(s\right){\eta }_3{(\mathbb{E}\parallel x\left(s\right)\parallel }^2)ds}\hfill \\ \hfill & {\displaystyle \le {\overline{\rho }}_3{\eta }_3\left(r\right){\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }\left(s\right)ds}\hfill \\ \hfill & {\displaystyle \le \frac{{\overline{\rho }}_3{\eta }_3\left(r\right)}{2\alpha -1}{(\psi \left(b\right)-\psi \left(0\right))}^{2\alpha -1}<\infty ,}\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_0^t{\int }_V{(\psi \left(t\right)-\psi \left(s\right))}^{4(\alpha -1)}{\psi }^{\prime }\left(s\right)\mathbb{E}{\parallel g(s,x\left(s\right),v)\parallel }^4\lambda \left(dv\right)ds}\hfill \\ \hfill & {\displaystyle \le {\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{4(\alpha -1)}{\psi }^{\prime }\left(s\right){\rho }_4\left(s\right){\eta }_4{(\mathbb{E}\parallel x\left(s\right)\parallel }^2)ds}\hfill \\ \hfill & {\displaystyle \le \frac{{\overline{\rho }}_4{\eta }_4\left(r\right)}{4\alpha -3}{(\psi \left(b\right)-\psi \left(0\right))}^{4\alpha -3}<\infty ,}\hfill \end{array}$$

(29)

where ${\overline{\rho }}_i={max}_{t\in [0,b]}{\rho }_i\left(t\right)$, $i=3,4$.

For sake of convenience, we set

$$N_1:={\displaystyle \frac{2M^2{\parallel E^{-1}\parallel }^{2}K}{(2\alpha -1){\Gamma }^2\left(\alpha \right)}}\left(m{M}^2{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2{\parallel E^{-1}\parallel }^2\sum _{k=1}^m{c}_k^2+1\right){(\psi \left(b\right)-\psi \left(0\right))}^{2\alpha -1}{b}^{1-{\gamma }_1}{\parallel {\rho }_1\parallel }_{{\displaystyle \frac{1}{{\gamma }_1}}},$$

$$\begin{array}{cc}\hfill & {\displaystyle N_2:=\frac{2M^2{\parallel E^{-1}\parallel }^2\overline{c}{K}^{1+{\gamma }_2}}{{\Gamma }^2\left(\alpha \right)}{\left(\frac{1-{\gamma }_2}{2\alpha -1-{\gamma }_2}\right)}^{1-{\gamma }_2}}\hfill \\ \hfill & {\displaystyle \times \left(m{M}^2{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2{\parallel E^{-1}\parallel }^2\sum _{k=1}^m{c}_k^2+1\right){(\psi \left(b\right)-\psi \left(0\right))}^{2\alpha -1-{\gamma }_2}{\parallel {\rho }_2\parallel }_{\frac{1}{{\gamma }_2}},}\hfill \end{array}$$

$$N_3:={\displaystyle \frac{2M^2{\parallel E^{-1}\parallel }^{2}DK{\overline{\rho }}_3}{(2\alpha -1){\Gamma }^2\left(\alpha \right)}}\left(m{M}^2{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2{\parallel E^{-1}\parallel }^2\sum _{k=1}^m{c}_k^2+1\right){(\psi \left(b\right)-\psi \left(0\right))}^{2\alpha -1},$$

$$N_4:={\displaystyle \frac{2\sqrt{2{\overline{\rho }}_4}{M}^2{\parallel E^{-1}\parallel }^{2}D{K}^{\frac{3}{2}}}{\sqrt{4\alpha -3}{\Gamma }^2\left(\alpha \right)}}\left(\sqrt{m^3}{M}^2{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2{\parallel E^{-1}\parallel }^2{\left(\sum _{k=1}^m{c}_k^4\right)}^{{\displaystyle \frac{1}{2}}}+1\right){(\psi \left(b\right)-\psi \left(0\right))}^{{\displaystyle \frac{4\alpha -3}{2}}},$$

and

$$\begin{array}{cc}\hfill & {\displaystyle N_5:=\frac{{8b\parallel B\parallel }^4{N}_0^{2}K{M}^2{\parallel E^{-1}\parallel }^2}{{\lambda }^2(2\alpha -1){\Gamma }^2\left(\alpha \right)}\left(m{M}^2{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2{\parallel E^{-1}\parallel }^2\sum _{k=1}^m{c}_k^2+1\right){(\psi \left(b\right)-\psi \left(0\right))}^{2\alpha -1},}\hfill \end{array}$$

with $\overline{c}$ is as in Lemma 4 and D as in Lemma 5.

Theorem 1. 

Let ${\displaystyle \frac{3}{4}}<\alpha <1$. The Sobolev-type ψ-Caputo fractional stochastic evolution Equations (1) and (2) admit a mild solution on J if the hypotheses $\left(H_1\right)$–$\left(H_8\right)$ hold.

Proof. 

We define an operator $\mathcal{T}:C(J,L^2(\Omega ,X))\to C(J,L^2(\Omega ,X))$ as follows:

$$\begin{array}{cc}\hfill & {\displaystyle \left(\mathcal{T}x\right)\left(t\right)={\int }_0^{b}G(t,s)(f(s,x\left(s\right))+B{u}_{\lambda }(s,x\left(s\right))){\psi }^{\prime }\left(s\right)ds+{\int }_0^{b}G(t,s)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +{\int }_0^b{\int }_{V}G(t,s)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv),t\in J.}\hfill \end{array}$$

(30)

We know from direct calculation that the operator $\mathcal{T}$ is well defined on $C(J,L^2(\Omega ,X))$. From Definition 4, it is easy to see that the mild solution of (1)–(2) on J is equivalent to the fixed point of operator $\mathcal{T}$ defined by (30). In the following, we will applying Schauder’s fixed point theorem to prove that the operator $\mathcal{T}$ has at least one fixed point. For convenience, the proof is divided into four steps.

• Step 1. We claim the existence of a positive number r such that $\mathcal{T}\left(D_r\right)\subset D_r$. In fact, if this is not true, then for each positive number r there exists a function $x^r\in D_r$, $u_{\lambda }\in L^2(J,U)$, but $\mathcal{T}\left(x^r\right)\notin D_r$. This implies that for some $t\in J$, $\mathbb{E}\parallel \mathcal{T}\left(x^r\right)\left(t\right){\parallel }^2>r$. From (30), we have

  $$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}\parallel \mathcal{T}\left(x^r\right)\left(t\right){\parallel }^2 \le 4\left\{\mathbb{E}{∥{\int }_0^{b}G(t,s)f(s,x^r\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2+\mathbb{E}{∥{\int }_0^{b}G(t,s)B{u}_{\lambda }(s,x^r\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2\right.}\hfill \\ \hfill & {\displaystyle \left.+\mathbb{E}{∥{\int }_0^{b}G(t,s)\sigma (s,x^r\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)∥}^2+\mathbb{E}{∥{\int }_0^b{\int }_{V}G(t,s)g(s,x^r\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)∥}^2\right\}}\hfill \\ \hfill & {\displaystyle =4(W_1+W_2+W_3+W_4).}\hfill \end{array}$$

  (31)

  Using Lemma 3, $\left(H_5\right)$, (26), H$\ddot{\mathrm{o}}$lder’s inequality, and the following elementary inequality

  $${\left(\sum _{i=1}^n{a}_i\right)}^2\le n\sum _{i=1}^n{a}_i^2,a_i>0,$$

  we deduce that

  $$\begin{array}{cc}\hfill & {\displaystyle W_1=\mathbb{E}{∥{\int }_0^{b}G(t,s)f(s,x^r\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle \le \mathbb{E}\left[{\int }_0^b\sum _{k=1}^m|{\chi }_{t_k}\left(s\right)|{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}\parallel S_E^{\alpha ,\psi }(t_k,0)\parallel \parallel E\parallel \parallel \mathrm{\Lambda }\parallel \parallel T_E^{\alpha ,\psi }(t_k,s)\parallel \parallel f(s,x^r\left(s\right)\parallel {\psi }^{\prime }\left(s\right)ds\right.}\hfill \\ \hfill & {\displaystyle {\left.+{\int }_0^b|{\chi }_t{\left(s\right)|(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}\parallel T_E^{\alpha ,\psi }(t,s)\parallel \parallel f(s,x^r\left(s\right)\parallel {\psi }^{\prime }\left(s\right)ds\right]}^2}\hfill \\ \hfill & {\displaystyle \le 2(M\parallel E^{-1}{\parallel )}^2{\parallel E\parallel }^2{\parallel \mathrm{\Lambda }\parallel }^2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}\mathbb{E}{\left(\sum _{k=1}^m{c}_k{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}\parallel f(s,x^r\left(s\right)\parallel {\psi }^{\prime }\left(s\right)ds\right)}^2}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}\mathbb{E}{\left({\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}\parallel f(s,x^r\left(s\right)\parallel {\psi }^{\prime }\left(s\right)ds\right)}^2}\hfill \\ \hfill & {\displaystyle \le 2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}\sum _{k=1}^m{c}_k^2{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^{2}ds{\int }_0^{t_k}\mathbb{E}\parallel f(s,x^r\left(s\right){\parallel }^{2}ds}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}{\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^{2}ds{\int }_0^t\mathbb{E}\parallel f(s,x^r{\left(s\right)\parallel }^{2}ds}\hfill \\ \hfill & {\displaystyle \le 2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}\frac{K}{2\alpha -1}\sum _{k=1}^m{c}_k^2{(\psi \left(t_k\right)-\psi \left(0\right))}^{2\alpha -1}{\int }_0^{t_k}{\rho }_1\left(s\right){\eta }_1(\mathbb{E}\parallel x^r\left(s\right){\parallel }^2)ds}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}\frac{K}{2\alpha -1}{(\psi \left(t\right)-\psi \left(0\right))}^{2\alpha -1}{\int }_0^t{\rho }_1\left(s\right){\eta }_1(\mathbb{E}\parallel x^r\left(s\right){\parallel }^2)ds}\hfill \\ \hfill & {\displaystyle \le \left[2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}\frac{K}{2\alpha -1}\sum _{k=1}^m{c}_k^2{(\psi \left(t_k\right)-\psi \left(0\right))}^{2\alpha -1}{b}^{1-{\gamma }_1}{\parallel {\rho }_1\parallel }_{\frac{1}{{\gamma }_1}}\right.}\hfill \\ \hfill & {\displaystyle \left.+2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}\frac{K}{2\alpha -1}{(\psi \left(b\right)-\psi \left(0\right))}^{2\alpha -1}{b}^{1-{\gamma }_1}{\parallel {\rho }_1\parallel }_{\frac{1}{{\gamma }_1}}\right]{\eta }_1\left(r\right)}\hfill \\ \hfill & {\displaystyle \le N_1{\eta }_1\left(r\right).}\hfill \end{array}$$

  (32)

  On the same scales using (32), Lemmas 3 and 4, $\left(H_6\right)$, and (27), we obtain

  $$\begin{array}{cc}\hfill & {\displaystyle W_3=\mathbb{E}{∥{\int }_0^{b}G(t,s)\sigma (s,x^r\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)∥}^2}\hfill \\ \hfill & {\displaystyle \le 2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}\overline{c}\sum _{k=1}^m{c}_k^2{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{2(\alpha -1)}\mathbb{E}{\parallel \sigma (s,x^r\left(s\right))\parallel }_{{\mathcal{L}}_2^0}^2{\psi }^{\prime }{\left(s\right)}^{2}ds}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}\overline{c}{\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}\mathbb{E}{\parallel \sigma (s,x^r\left(s\right))\parallel }_{{\mathcal{L}}_2^0}^2{\psi }^{\prime }{\left(s\right)}^{2}ds}\hfill \\ \hfill & {\displaystyle \le 2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}\overline{c}\sum _{k=1}^m{c}_k^2{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^2{\rho }_2\left(s\right){\eta }_2(\mathbb{E}\parallel x^r\left(s\right){\parallel }^2)ds}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}\overline{c}{\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^2{\rho }_2\left(s\right){\eta }_2(\mathbb{E}\parallel x^r\left(s\right){\parallel }^2)ds}\hfill \end{array}$$

  $$\begin{array}{cc}\hfill & {\displaystyle \le \left[2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}\overline{c}{K}^{1+{\gamma }_2}{\left(\frac{1-{\gamma }_2}{2\alpha -1-{\gamma }_2}\right)}^{1-{\gamma }_2}\sum _{k=1}^m{c}_k^2{(\psi \left(t_k\right)-\psi \left(0\right))}^{2\alpha -1-{\gamma }_2}{\parallel {\rho }_2\parallel }_{\frac{1}{{\gamma }_2}}\right.}\hfill \\ \hfill & {\displaystyle \left.+2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}\overline{c}{K}^{1+{\gamma }_2}{\left(\frac{1-{\gamma }_2}{2\alpha -1-{\gamma }_2}\right)}^{1-{\gamma }_2}{(\psi \left(b\right)-\psi \left(0\right))}^{2\alpha -1-{\gamma }_2}{\parallel {\rho }_2\parallel }_{\frac{1}{{\gamma }_2}}\right]{\eta }_2\left(r\right)}\hfill \\ \hfill & {\displaystyle \le N_2{\eta }_2\left(r\right).}\hfill \end{array}$$

  (33)

  From Lemmas 3 and 5, $\left(H_7\right)$, (28), (29), and the following three elementary inequalities

  $${\left(\sum _{i=1}^n{a}_i\right)}^2\le n\sum _{i=1}^n{a}_i^2,a_i>0,$$

  $${\left(\sum _{i=1}^n{a}_i\right)}^4\le n^3\sum _{i=1}^n{a}_i^4,a_i>0,$$

  $$\sqrt{a_1+a_2}\le \sqrt{a_1}+\sqrt{a_2},a_1,a_2>0,$$

  we have

  $$\begin{array}{cc}\hfill & {\displaystyle W_4=\mathbb{E}{∥{\int }_0^b{\int }_{V}G(t,s)g(s,x^r\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)∥}^2}\hfill \\ \hfill & {\displaystyle \le 2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}D\sum _{k=1}^m{c}_k^2{\int }_0^{t_k}{\int }_V{(\psi \left(t_k\right)-\psi \left(s\right))}^{2(\alpha -1)}\mathbb{E}{\parallel g(s,x^r\left(s\right),v)\parallel }^2{\psi }^{\prime }{\left(s\right)}^2\lambda \left(dv\right)ds}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}D{\int }_0^t{\int }_V{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}\mathbb{E}{\parallel g(s,x^r\left(s\right),v)\parallel }^2{\psi }^{\prime }{\left(s\right)}^2\lambda \left(dv\right)ds}\hfill \\ \hfill & {\displaystyle +\sqrt{8m^3}{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}D{\left(\sum _{k=1}^m{c}_k^4{\int }_0^{t_k}{\int }_V{(\psi \left(t_k\right)-\psi \left(s\right))}^{4(\alpha -1)}\mathbb{E}{\parallel g(s,x^r\left(s\right),v)\parallel }^4{\psi }^{\prime }{\left(s\right)}^4\lambda \left(dv\right)ds\right)}^{\frac{1}{2}}}\hfill \\ \hfill & {\displaystyle +\sqrt{8}\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}D{\left({\int }_0^t{\int }_V{(\psi \left(t\right)-\psi \left(s\right))}^{4(\alpha -1)}\mathbb{E}{\parallel g(s,x^r\left(s\right),v)\parallel }^4{\psi }^{\prime }{\left(s\right)}^4\lambda \left(dv\right)ds\right)}^{\frac{1}{2}}}\hfill \\ \hfill & {\displaystyle \le 2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}DK\frac{{\overline{\rho }}_3{\eta }_3\left(r\right)}{2\alpha -1}\sum _{k=1}^m{c}_k^2{(\psi \left(t_k\right)-\psi \left(0\right))}^{2\alpha -1}}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}DK\frac{{\overline{\rho }}_3{\eta }_3\left(r\right)}{2\alpha -1}{(\psi \left(b\right)-\psi \left(0\right))}^{2\alpha -1}}\hfill \\ \hfill & {\displaystyle +\sqrt{8m^3}{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}D{K}^{\frac{3}{2}}{\left(\frac{{\overline{\rho }}_4{\eta }_4\left(r\right)}{4\alpha -3}\sum _{k=1}^m{c}_k^4{(\psi \left(t_k\right)-\psi \left(0\right))}^{4\alpha -3}\right)}^{\frac{1}{2}}}\hfill \\ \hfill & {\displaystyle +\sqrt{8}\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}D{K}^{\frac{3}{2}}{\left(\frac{{\overline{\rho }}_4{\eta }_4\left(r\right)}{4\alpha -3}{(\psi \left(b\right)-\psi \left(0\right))}^{4\alpha -3}\right)}^{\frac{1}{2}}}\hfill \\ \hfill & {\displaystyle \le \frac{2M^2{\parallel E^{-1}\parallel }^{2}DK{\overline{\rho }}_3}{(2\alpha -1){\Gamma }^2\left(\alpha \right)}\left(m{M}^2{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2{\parallel E^{-1}\parallel }^2\sum _{k=1}^m{c}_k^2+1\right){(\psi \left(b\right)-\psi \left(0\right))}^{2\alpha -1}{\eta }_3\left(r\right)}\hfill \\ \hfill & {\displaystyle +\frac{\sqrt{8{\overline{\rho }}_4}{M}^2{\parallel E^{-1}\parallel }^{2}D{K}^{\frac{3}{2}}}{\sqrt{4\alpha -3}{\Gamma }^2\left(\alpha \right)}\left(\sqrt{m^3}{M}^2{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2{\parallel E^{-1}\parallel }^2{\left(\sum _{k=1}^m{c}_k^4\right)}^{\frac{1}{2}}+1\right){(\psi \left(b\right)-\psi \left(0\right))}^{\frac{4\alpha -3}{2}}\sqrt{{\eta }_4\left(r\right)}}\hfill \\ \hfill & {\displaystyle =N_3{\eta }_3\left(r\right)+N_4\sqrt{{\eta }_4\left(r\right)}.}\hfill \end{array}$$

  (34)

  Due to $\left(H_8\right)$, we may assume w.l.o.g. that $\parallel R(\lambda ,{\Gamma }_0^b)\parallel \le {\displaystyle \frac{1}{\lambda }}$ for all $\lambda \in (0,1)$. From (23)–(25) and utilizing the estimates of $W_1,W_3$, and $W_4$, we get

  $$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}\parallel u_{\lambda }(t,x^r){\parallel }^2}\hfill \\ \hfill & {\displaystyle \le 4\mathbb{E}\parallel B^{\ast }{G}^{\ast }(b,t)R(\lambda ,{\Gamma }_0^b)x_b{\parallel }^2+4\mathbb{E}{∥B^{\ast }{G}^{\ast }(b,t)R(\lambda ,{\Gamma }_0^b){\int }_0^{b}G(b,s)f(s,x^r\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle +4\mathbb{E}{∥B^{\ast }{G}^{\ast }(b,t)R(\lambda ,{\Gamma }_0^b){\int }_0^{b}G(b,s)\sigma (s,x^r\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)∥}^2}\hfill \\ \hfill & {\displaystyle +4\mathbb{E}{∥B^{\ast }{G}^{\ast }(b,t)R(\lambda ,{\Gamma }_0^b){\int }_0^b{\int }_{V}G(b,s)g(s,x^r\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)∥}^2}\hfill \\ \hfill & {\displaystyle \le \frac{4}{{\lambda }^2}{\parallel B\parallel }^2{N}_0^2[\parallel x_b{\parallel }^2+N_1{\eta }_1\left(r\right)+N_2{\eta }_2\left(r\right)+N_3{\eta }_3\left(r\right)+N_4\sqrt{{\eta }_4\left(r\right)}].}\hfill \end{array}$$

  (35)
  From (35), one can easily estimate $W_2$ as follows:

  $$\begin{array}{cc}\hfill & {\displaystyle W_2=\mathbb{E}{∥{\int }_0^{b}G(t,s)B{u}_{\lambda }(s,x^r\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle \le 2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}{\parallel B\parallel }^2\sum _{k=1}^m{c}_k^2{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^{2}ds{\int }_0^{t_k}\mathbb{E}{\parallel u_{\lambda }^r(s,x^r\left(s\right))\parallel }^{2}ds}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}{\parallel B\parallel }^2{\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^{2}ds{\int }_0^t\mathbb{E}{\parallel u_{\lambda }(s,x^r\left(s\right))\parallel }^{2}ds}\hfill \\ \hfill & {\displaystyle \le \left(2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}{\parallel B\parallel }^{2}K\sum _{k=1}^m\frac{c_k^2}{2\alpha -1}{(\psi \left(t_k\right)-\psi \left(0\right))}^{2\alpha -1}\right.}\hfill \\ \hfill & {\displaystyle \left.+2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}{\parallel B\parallel }^{2}K\frac{1}{2\alpha -1}{(\psi \left(b\right)-\psi \left(0\right))}^{2\alpha -1}\right)}\hfill \\ \hfill & {\displaystyle \times \frac{4b}{{\lambda }^2}{\parallel B\parallel }^2{N}_0^2\left(\parallel x_b{\parallel }^2+N_1{\eta }_1\left(r\right)+N_2{\eta }_2\left(r\right)+N_3{\eta }_3\left(r\right)+N_4\sqrt{{\eta }_4\left(r\right)}\right)}\hfill \\ \hfill & {\displaystyle \le N_5\left[\parallel x_b{\parallel }^2+N_1{\eta }_1\left(r\right)+N_2{\eta }_2\left(r\right)+N_3{\eta }_3\left(r\right)+N_4\sqrt{{\eta }_4\left(r\right)}\right].}\hfill \end{array}$$

  Substituting $W_1$, $W_2$, $W_3$, and $W_4$ into (31), we get

  $$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}\parallel \mathcal{T}\left(x^r\right)\left(t\right){\parallel }^2\le 4N_1{\eta }_1\left(r\right)+4N_5\left[\parallel x_b{\parallel }^2+N_1{\eta }_1\left(r\right)+N_2{\eta }_2\left(r\right)+N_3{\eta }_3\left(r\right)+N_4\sqrt{{\eta }_4\left(r\right)}\right]}\hfill \\ \hfill & {\displaystyle +4N_2{\eta }_2\left(r\right)+4N_3{\eta }_3\left(r\right)+4N_4\sqrt{{\eta }_4\left(r\right)}.}\hfill \end{array}$$

  Dividing both sides by r and taking the lower limit as $r\to \infty$, we obtain $1\le 0$, which is a contradiction. Hence, $\mathcal{T}\left(D_r\right)\subset D_r$.

• Step 2. Now, we will show that $\mathcal{T}:D_r\to D_r$ is continuous. Let $\left\{x_n\right\}\subset D_r$ with $x_n\to x\in D_r$ as $n\to \infty$. From $\left(H_5\right)-\left(H_7\right)$, we deduce that

  $$f(t,x_n\left(t\right))\to f(t,x\left(t\right)),\forall t\in J,$$

  $$\sigma (t,x_n\left(t\right))\to \sigma (t,x\left(t\right)),\forall t\in J,$$

  $${\int }_{V}g(t,x_n\left(t\right),v)\lambda \left(dv\right)\to {\int }_{V}g(t,x\left(t\right),v)\lambda \left(dv\right),\forall t\in J,$$

  and $u_{\lambda }(t,x_n\left(t\right))\to u_{\lambda }(t,x\left(t\right))$. Using the above, (26)–(29), and (35), according to the Lebesgue dominated convergence theorem, for each $t\in J$, we have

  $$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}{∥{\int }_0^{b}G(t,s)(f(s,x_n\left(s\right))-f(s,x\left(s\right))){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle \le 2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{(2\alpha -1){\Gamma }^2\left(\alpha \right)}K\sum _{k=1}^m{c}_k^2{(\psi \left(t_k\right)-\psi \left(0\right))}^{2\alpha -1}{\int }_0^{t_k}\mathbb{E}{\parallel f(s,x_n\left(s\right))-f(s,x\left(s\right))\parallel }^{2}ds}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{(2\alpha -1){\Gamma }^2\left(\alpha \right)}K{(\psi \left(t\right)-\psi \left(0\right))}^{2\alpha -1}{\int }_0^t\mathbb{E}{\parallel f(s,x_n\left(s\right))-f(s,x\left(s\right))\parallel }^{2}ds}\hfill \\ \hfill & {\displaystyle \to 0\mathrm{as}n\to \infty ,}\hfill \end{array}$$

  (36)

  $$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}{∥{\int }_0^{b}G(t,s)(\sigma (s,x_n\left(s\right))-\sigma (s,x\left(s\right))){\psi }^{\prime }\left(s\right)dW\left(s\right)∥}^2}\hfill \\ \hfill & {\displaystyle \le 2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}\overline{c}\sum _{k=1}^m{c}_k^2{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{2\alpha -1}{\psi }^{\prime }{\left(s\right)}^2\mathbb{E}{\parallel \sigma (s,x_n\left(s\right))-\sigma (s,x\left(s\right))\parallel }_{{\mathcal{L}}_2^0}^{2}ds}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}\overline{c}{\int }_0^t{(\psi \left(t\right)-\psi \left(s\right))}^{2\alpha -1}{\psi }^{\prime }{\left(s\right)}^2\mathbb{E}{\parallel \sigma (s,x_n\left(s\right))-\sigma (s,x\left(s\right))\parallel }_{{\mathcal{L}}_2^0}^{2}ds}\hfill \\ \hfill & {\displaystyle \to 0\mathrm{as}n\to \infty ,}\hfill \end{array}$$

  (37)

  $$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}{∥{\int }_0^{b}G(t,s)B(u_{\lambda }(s,x_n\left(s\right))-u_{\lambda }(s,x\left(s\right))){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle \le 2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{(2\alpha -1){\Gamma }^2\left(\alpha \right)}{K\parallel B\parallel }^2\sum _{k=1}^m{c}_k^2{(\psi \left(t_k\right)-\psi \left(0\right))}^{2\alpha -1}{\int }_0^{t_k}\mathbb{E}{\parallel u_{\lambda }(s,x_n\left(s\right))-u_{\lambda }(s,x\left(s\right))\parallel }^{2}ds}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{(2\alpha -1){\Gamma }^2\left(\alpha \right)}{K\parallel B\parallel }^2{(\psi \left(t\right)-\psi \left(0\right))}^{2\alpha -1}{\int }_0^t\mathbb{E}{\parallel u_{\lambda }(s,x_n\left(s\right))-u_{\lambda }(s,x\left(s\right))\parallel }^{2}ds}\hfill \\ \hfill & {\displaystyle \to 0\mathrm{as}n\to \infty ,}\hfill \end{array}$$

  (38)

  and

  $$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}{∥{\int }_0^b{\int }_{V}G(t,s)(g(s,x_n\left(s\right),v)-g(s,x\left(s\right),v)){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)∥}^2}\hfill \\ \hfill & {\displaystyle \le 2m{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}D\sum _{k=1}^m{c}_k^2{\int }_0^{t_k}{\int }_V{(\psi \left(t_k\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^2\mathbb{E}{\parallel g(s,x_n\left(s\right),v)-g(s,x\left(s\right),v)\parallel }^2\lambda \left(dv\right)ds}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}D{\int }_0^t{\int }_V{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^2\mathbb{E}{\parallel g(s,x_n\left(s\right),v)-g(s,x\left(s\right),v)\parallel }^2\lambda \left(dv\right)ds}\hfill \\ \hfill & {\displaystyle +\sqrt{8m^3}{M}^4{\parallel \mathrm{\Lambda }\parallel }^2{\parallel E\parallel }^2\frac{\parallel E^{-1}{\parallel }^4}{{\Gamma }^2\left(\alpha \right)}D{\left(\sum _{k=1}^m{c}_k^4{\int }_0^{t_k}{\int }_V{(\psi \left(t_k\right)-\psi \left(s\right))}^{4(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^4\mathbb{E}{\parallel g(s,x_n\left(s\right),v)-g(s,x\left(s\right),v)\parallel }^4\lambda \left(dv\right)ds\right)}^{\frac{1}{2}}}\hfill \\ \hfill & {\displaystyle +\sqrt{8}\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}D{\left({\int }_0^t{\int }_V{(\psi \left(t\right)-\psi \left(s\right))}^{4(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^4\mathbb{E}{\parallel g(s,x_n\left(s\right),v)-g(s,x\left(s\right),v)\parallel }^4\lambda \left(dv\right)ds\right)}^{\frac{1}{2}}}\hfill \\ \hfill & {\displaystyle \to 0\mathrm{as}n\to \infty .}\hfill \end{array}$$

  (39)

  Thus, from (36)–(39), we have

  $$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}\parallel \left(\mathcal{T}{x}_n\right)\left(t\right)-\left(\mathcal{T}x\right)\left(t\right){\parallel }^2}\hfill \\ \hfill & {\displaystyle \le 4\mathbb{E}{∥{\int }_0^{b}G(t,s)\mathbb{E}(f(s,x_n\left(s\right))-f(s,x\left(s\right))){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle +4\mathbb{E}{∥{\int }_0^{b}G(t,s)B(u_{\lambda }(s,x_n\left(s\right))-u_{\lambda }(s,x\left(s\right))){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle +4\mathbb{E}{∥{\int }_0^{b}G(t,s)(\sigma (s,x_n\left(s\right))-\sigma (s,x\left(s\right))){\psi }^{\prime }\left(s\right)dW\left(s\right)∥}^2}\hfill \\ \hfill & {\displaystyle +4\mathbb{E}{∥{\int }_0^b{\int }_{V}G(t,s)(g(s,x_n\left(s\right),v)-g(s,x\left(s\right),v)){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)∥}^2}\hfill \\ \hfill & {\displaystyle \to 0\mathrm{as}n\to \infty .}\hfill \end{array}$$

  which implies that $\mathcal{T}:D_r\to D_r$ is a continuous operator.

• Step 3. Next, we show that $\mathcal{T}$ is equicontinuous on $D_r$. For any $x\in D_r$ and $0\le {\tau }_1<{\tau }_2\le b$, by using (30), we evaluate

  $$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}\parallel \left(\mathcal{T}x\right)\left({\tau }_2\right)-\left(\mathcal{T}x\right)\left({\tau }_1\right){\parallel }^2}\hfill \\ \hfill & {\displaystyle \le 4\mathbb{E}{∥{\int }_0^b(G({\tau }_2,s)-G({\tau }_1,s))f(s,x\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle +4\mathbb{E}{∥{\int }_0^b(G({\tau }_2,s)-G({\tau }_1,s))B{u}_{\lambda }(s,x\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle +4\mathbb{E}{∥{\int }_0^b(G({\tau }_2,s)-G({\tau }_1,s))\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)∥}^2}\hfill \\ \hfill & {\displaystyle +4\mathbb{E}{∥{\int }_0^b{\int }_V(G({\tau }_2,s)-G({\tau }_1,s))g(s,x\left(s\right),v)){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)∥}^2.}\hfill \end{array}$$

  (40)

  From (18), we obtain

  $$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}{∥{\int }_0^b(G({\tau }_2,s)-G({\tau }_1,s))f(s,x\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle \le 2\mathbb{E}{∥{\int }_0^b[{\chi }_{{\tau }_2}\left(s\right)-{\chi }_{{\tau }_1}\left(s\right)]{(\psi \left({\tau }_2\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }({\tau }_2,s)f(s,x\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle +2\mathbb{E}{∥{\int }_0^b{\chi }_{{\tau }_1}\left(s\right)[{(\psi \left({\tau }_2\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }({\tau }_2,s)-{(\psi \left({\tau }_1\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }({\tau }_1,s)]f(s,x\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle :=2(I_1+I_2).}\hfill \end{array}$$

  For $I_1$, from $\left(H_5\right)$ and (26), we get

  $$\begin{array}{cc}\hfill & {\displaystyle I_1\le \mathbb{E}{∥{\int }_{{\tau }_1}^{{\tau }_2}{(\psi \left({\tau }_2\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }({\tau }_2,s)f(s,x\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle \le \frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}{\int }_{{\tau }_1}^{{\tau }_2}{(\psi \left({\tau }_2\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^{2}ds{\int }_{{\tau }_1}^{{\tau }_2}\mathbb{E}{\parallel f(s,x\left(s\right))\parallel }^{2}ds}\hfill \\ \hfill & {\displaystyle \le \frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}\frac{K}{2\alpha -1}{(\psi \left({\tau }_2\right)-\psi \left({\tau }_1\right))}^{2\alpha -1}{\parallel \rho \parallel }_{\frac{1}{{\gamma }_1}}{b}^{1-{\gamma }_1}{\eta }_1\left(r\right)}\hfill \\ \hfill & {\displaystyle \to 0,\mathrm{as}{\tau }_2\to {\tau }_1.}\hfill \end{array}$$

  For $I_2$, from Lemma 3 (ii), $\left(H_5\right)$, and (26), we deduce that

  $$\begin{array}{cc}\hfill & {\displaystyle I_2\le \mathbb{E}{∥{\int }_0^{{\tau }_1}[{(\psi \left({\tau }_2\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }({\tau }_2,s)-{(\psi \left({\tau }_1\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }({\tau }_1,s)]f(s,x\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle \le 2\mathbb{E}{∥{\int }_0^{{\tau }_1}{(\psi \left({\tau }_2\right)-\psi \left(s\right))}^{\alpha -1}[T_E^{\alpha ,\psi }({\tau }_2,s)-T_E^{\alpha ,\psi }({\tau }_1,s)]f(s,x\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle +2\mathbb{E}{∥{\int }_0^{{\tau }_1}[{(\psi \left({\tau }_2\right)-\psi \left(s\right))}^{\alpha -1}-{(\psi \left({\tau }_1\right)-\psi \left(s\right))}^{\alpha -1}]T_E^{\alpha ,\psi }({\tau }_1,s)f(s,x\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle \le 2\parallel T_E^{\alpha ,\psi }({\tau }_1,s)-T_E^{\alpha ,\psi }({\tau }_1,s){\parallel }^2{\int }_0^{{\tau }_1}{(\psi \left({\tau }_2\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^{2}ds{\int }_0^{{\tau }_1}\mathbb{E}{\parallel f(s,x\left(s\right))\parallel }^{2}ds}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2}{{\Gamma }^2\left(\alpha \right)}{\int }_0^{{\tau }_1}|{(\psi \left({\tau }_2\right)-\psi \left(s\right))}^{\alpha -1}-{(\psi \left({\tau }_1\right)-\psi \left(s\right))}^{\alpha -1}{|}^2{\psi }^{\prime }{\left(s\right)}^{2}ds{\int }_0^{{\tau }_1}\mathbb{E}{\parallel f(s,x\left(s\right))\parallel }^{2}ds}\hfill \\ \hfill & {\displaystyle \le 2\parallel T_E^{\alpha ,\psi }({\tau }_1,s)-T_E^{\alpha ,\psi }({\tau }_1,s){\parallel }^2\frac{K{b}^{1-\gamma }}{2\alpha -1}\left[{(\psi \left({\tau }_2\right)-\psi \left(0\right))}^{2\alpha -1}-{(\psi \left({\tau }_2\right)-\psi \left({\tau }_1\right))}^{2\alpha -1}\right]{\parallel \rho \parallel }_{\frac{1}{{\gamma }_1}}{\eta }_1\left(r\right)}\hfill \\ \hfill & {\displaystyle +2\frac{M^2{\parallel E^{-1}\parallel }^2{b}^{1-\gamma }}{{\Gamma }^2\left(\alpha \right)}{\parallel \rho \parallel }_{\frac{1}{{\gamma }_1}}{\eta }_1\left(r\right){\int }_0^{{\tau }_1}{|{(\psi \left({\tau }_2\right)-\psi \left(s\right))}^{\alpha -1}-{(\psi \left({\tau }_1\right)-\psi \left(s\right))}^{\alpha -1}|}^2{\psi }^{\prime }{\left(s\right)}^{2}ds}\hfill \\ \hfill & {\displaystyle :=I_{21}+2\frac{M^2{\parallel E^{-1}\parallel }^2{b}^{1-\gamma }}{{\Gamma }^2\left(\alpha \right)}{\parallel \rho \parallel }_{\frac{1}{{\gamma }_1}}{\eta }_1\left(r\right)I_{22}.}\hfill \end{array}$$

  Obviously, $I_{21}\to 0$ as ${\tau }_2\to {\tau }_1$. Notice that ${(a_2-a_1)}^2\le a_2^2-a_1^2$ for $a_2>a_1>0$; we have

  $$\begin{array}{cc}\hfill & {\displaystyle I_{22}={\int }_0^{{\tau }_1}{|{(\psi \left({\tau }_2\right)-\psi \left(s\right))}^{\alpha -1}-{(\psi \left({\tau }_1\right)-\psi \left(s\right))}^{\alpha -1}|}^2{\psi }^{\prime }{\left(s\right)}^{2}ds}\hfill \\ \hfill & {\displaystyle \le {\int }_0^{{\tau }_1}\left[{(\psi \left({\tau }_1\right)-\psi \left(s\right))}^{2(\alpha -1)}-{(\psi \left({\tau }_2\right)-\psi \left(s\right))}^{2(\alpha -1)}\right]{\psi }^{\prime }{\left(s\right)}^{2}ds}\hfill \\ \hfill & {\displaystyle \le \frac{K}{2\alpha -1}\left[{(\psi \left({\tau }_1\right)-\psi \left(0\right))}^{2\alpha -1}+{(\psi \left({\tau }_2\right)-\psi \left({\tau }_1\right))}^{2\alpha -1}-{(\psi \left({\tau }_2\right)-\psi \left(0\right))}^{2\alpha -1}\right]}\hfill \\ \hfill & {\displaystyle \to 0,\mathrm{as}{\tau }_2\to {\tau }_1.}\hfill \end{array}$$

  Therefore, $I_2\to 0$ as ${\tau }_2\to {\tau }_1$. Thus,

  $${\displaystyle \mathbb{E}{∥{\int }_0^b(G({\tau }_2,s)-G({\tau }_1,s))f(s,x\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2\to 0,\mathrm{as}{\tau }_2\to {\tau }_1.}$$

  (41)

  Applying (27)–(29) and Lemma 3, similar to the proof of (41), we can obtain that

  $${\displaystyle \mathbb{E}{∥{\int }_0^b(G({\tau }_2,s)-G({\tau }_1,s))B{u}_{\lambda }(s,x\left(s\right)){\psi }^{\prime }\left(s\right)ds∥}^2\to 0,\mathrm{as}{\tau }_2\to {\tau }_1,}$$

  (42)

  $${\displaystyle \mathbb{E}{∥{\int }_0^b(G({\tau }_2,s)-G({\tau }_1,s))\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)∥}^2\to 0,\mathrm{as}{\tau }_2\to {\tau }_1,}$$

  (43)

  $${\displaystyle \mathbb{E}{∥{\int }_0^b{\int }_V(G({\tau }_2,s)-G({\tau }_1,s))g(s,x\left(s\right),v)){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)∥}^2\to 0,\mathrm{as}{\tau }_2\to {\tau }_1.}$$

  (44)

  Substituting (41)–(44) into (40), we get

  $${\displaystyle \mathbb{E}\parallel \left(\mathcal{T}x\right)\left({\tau }_2\right)-\left(\mathcal{T}x\right)\left({\tau }_1\right){\parallel }^2\to 0,\mathrm{as}{\tau }_2\to {\tau }_1,}$$

  which means that the operator $\mathcal{T}:D_r\to D_r$ is equicontinuous.

• Step 4. We will prove that the operator $\mathcal{T}:D_r\to D_r$ is compact. To prove this, we first show that $\{\left(\mathcal{T}x\right)\left(t\right):x\in D_r\}$ is relatively compact for each $t\in J$. For $x\in D_r$, for any $\epsilon >0$ and $\delta >0$, similar to [37], we define

  $$\begin{array}{cc}\hfill & {\displaystyle \left({\mathcal{T}}_{\epsilon ,\delta }x\right)\left(t\right)=\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E\mathrm{\Lambda }{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)(f(s,x\left(s\right))+B{u}_{\lambda }(s,x\left(s\right))){\psi }^{\prime }\left(s\right)ds}\hfill \\ \hfill & {\displaystyle +\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E\mathrm{\Lambda }{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E\mathrm{\Lambda }{\int }_0^{t_k}{\int }_V{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)}\hfill \\ \hfill & {\displaystyle +\alpha {\int }_0^{t-\epsilon }{\int }_{\delta }^{\infty }{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}T\left({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta \right)(f(s,x\left(s\right))+B{u}_{\lambda }(s,x\left(s\right))){\psi }^{\prime }\left(s\right)d\theta ds}\hfill \\ \hfill & {\displaystyle +\alpha {\int }_0^{t-\epsilon }{\int }_{\delta }^{\infty }{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}T\left({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta \right)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)d\theta dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +\alpha {\int }_0^{t-\epsilon }{\int }_{\delta }^{\infty }{\int }_V{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}T\left({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta \right)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)d\theta }\hfill \\ \hfill & {\displaystyle =\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E\mathrm{\Lambda }{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)(f(s,x\left(s\right))+B{u}_{\lambda }(s,x\left(s\right))){\psi }^{\prime }\left(s\right)ds}\hfill \\ \hfill & {\displaystyle +\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E\mathrm{\Lambda }{\int }_0^{t_k}{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E\mathrm{\Lambda }{\int }_0^{t_k}{\int }_V{(\psi \left(t_k\right)-\psi \left(s\right))}^{\alpha -1}{T}_E^{\alpha ,\psi }(t_k,s)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)}\hfill \end{array}$$

  $$\begin{array}{cc}\hfill & {\displaystyle +\alpha T\left({\epsilon }^{\alpha }\delta \right){\int }_0^{t-\epsilon }{\int }_{\delta }^{\infty }{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}T({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta -{\epsilon }^{\alpha }\delta )(f(s,x\left(s\right))+B{u}_{\lambda }(s,x\left(s\right))){\psi }^{\prime }\left(s\right)d\theta ds}\hfill \\ \hfill & {\displaystyle +\alpha T\left({\epsilon }^{\alpha }\delta \right){\int }_0^{t-\epsilon }{\int }_{\delta }^{\infty }{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}T({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta -{\epsilon }^{\alpha }\delta )\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)d\theta dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +\alpha T\left({\epsilon }^{\alpha }\delta \right){\int }_0^{t-\epsilon }{\int }_{\delta }^{\infty }{\int }_V{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}T({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta -{\epsilon }^{\alpha }\delta )g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)d\theta \overline{N}(ds,dv).}\hfill \end{array}$$

  Then, by the compactness of $T\left({\epsilon }^{\alpha }\delta \right)$ for ${\epsilon }^{\alpha }\delta >0$, we see that the set ${\mathcal{T}}_{\epsilon ,\delta }\left(t\right)=\{\left({\mathcal{T}}_{\epsilon ,\delta }x\right)\left(t\right):x\in D_r\}$ is relatively compact for all $\epsilon >0$ and $\delta >0$. Moreover, for any $x\in D_r$, we have

  $$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}\parallel \left(\mathcal{T}x\right)\left(t\right)-\left({\mathcal{T}}_{\epsilon ,\delta }x\right)\left(t\right){\parallel }^2}\hfill \\ \hfill & {\displaystyle \le 6{\alpha }^2\mathbb{E}{∥{\int }_0^{t-\epsilon }{\int }_0^{\delta }{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}T\left({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta \right)(f(s,x\left(s\right))+B{u}_{\lambda }(s,x\left(s\right))){\psi }^{\prime }\left(s\right)d\theta ds∥}^2}\hfill \\ \hfill & {\displaystyle +6{\alpha }^2\mathbb{E}{∥{\int }_0^{t-\epsilon }{\int }_0^{\delta }{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}T\left({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta \right)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)d\theta dW\left(s\right)∥}^2}\hfill \\ \hfill & {\displaystyle +6{\alpha }^2\mathbb{E}{∥{\int }_0^{t-\epsilon }{\int }_0^{\delta }{\int }_V{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}T\left({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta \right)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)d\theta \overline{N}(ds,dv)∥}^2}\hfill \\ \hfill & {\displaystyle +6{\alpha }^2\mathbb{E}{∥{\int }_{t-\epsilon }^t{\int }_0^{\infty }{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}T\left({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta \right)(f(s,x\left(s\right))+B{u}_{\lambda }(s,x\left(s\right))){\psi }^{\prime }\left(s\right)d\theta ds∥}^2}\hfill \\ \hfill & {\displaystyle +6{\alpha }^2\mathbb{E}{∥{\int }_{t-\epsilon }^t{\int }_0^{\infty }{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}T\left({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta \right)\sigma (s,x\left(s\right)){\psi }^{\prime }\left(s\right)d\theta dW\left(s\right)∥}^2}\hfill \\ \hfill & {\displaystyle +6{\alpha }^2\mathbb{E}{∥{\int }_{t-\epsilon }^t{\int }_0^{\infty }{\int }_V{E}^{-1}\theta {\xi }_{\alpha }\left(\theta \right){(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}T\left({(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }\theta \right)g(s,x\left(s\right),v){\psi }^{\prime }\left(s\right)d\theta \overline{N}(ds,dv)∥}^2}\hfill \\ \hfill & {\displaystyle =6{\alpha }^2(J_1+J_2+\cdots +J_6).}\hfill \end{array}$$

  Here, we only consider the estimation of $J_1$, $J_5$, and $J_6$, while other cases are handled similarly. From Lemma 6, we know that

  $${\int }_0^{\infty }\theta {\xi }_{\alpha }\left(\theta \right)d\theta ={\displaystyle \frac{1}{\Gamma (1+\alpha )}}.$$

  (45)

  From H$\ddot{\mathrm{o}}$lder’s inequality, (26), (35), and (45), one has

  $$\begin{array}{cc}\hfill & {\displaystyle J_1\le {\parallel E^{-1}\parallel }^2{M}^2{\left({\int }_0^{\delta }\theta {\xi }_{\alpha }\left(\theta \right)d\theta \right)}^2{\int }_0^{t-\epsilon }{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^{2}ds}\hfill \\ \hfill & {\displaystyle \times 2\left({\int }_0^{t-\epsilon }{\mathbb{E}\parallel f(s,x\left(s\right))\parallel }^{2}ds+{\parallel B\parallel }^2{\int }_0^{t-\epsilon }\mathbb{E}{\parallel u_{\lambda }(s,x\left(s\right))\parallel }^{2}ds\right)}\hfill \\ \hfill & {\displaystyle \le 2\parallel E^{-1}{\parallel }^2{M}^2\frac{K}{2\alpha -1}{(\psi \left(t\right)-\psi \left(0\right))}^{2\alpha -1}\left[{\eta }_1\left(r\right)b^{1-{\gamma }_1}{\parallel {\rho }_1\parallel }_{\frac{1}{{\gamma }_1}}\right.}\hfill \\ \hfill & {\displaystyle \left.+\frac{4b}{{\lambda }^2}{\parallel B\parallel }^4{N}_0^2\left(\parallel x_b{\parallel }^2+N_1{\eta }_1\left(r\right)+N_2{\eta }_2\left(r\right)+N_3{\eta }_3\left(r\right)+N_4\sqrt{{\eta }_4\left(r\right)}\right)\right]{\left({\int }_0^{\delta }\theta {\xi }_{\alpha }\left(\theta \right)d\theta \right)}^2}\hfill \\ \hfill & {\displaystyle \to 0,\mathrm{as}\delta \to 0^{+}.}\hfill \end{array}$$
  From (27) and (45), we deduce that

  $$\begin{array}{cc}\hfill & {\displaystyle J_5\le \overline{c}\parallel E^{-1}{\parallel }^2{M}^2{\left({\int }_0^{\infty }\theta {\xi }_{\alpha }\left(\theta \right)d\theta \right)}^2{\int }_{t-\epsilon }^t{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}{\psi }^{\prime }{\left(s\right)}^2\mathbb{E}{\parallel \sigma (s,x\left(s\right))\parallel }_{{\mathcal{L}}_2^0}^{2}ds}\hfill \\ \hfill & {\displaystyle \le \frac{\overline{c}{\parallel E^{-1}\parallel }^2{M}^2}{{\Gamma }^2(1+\alpha )}{K}^{1+{\gamma }_2}{\left(\frac{1-{\gamma }_2}{2\alpha -1-{\gamma }_2}\right)}^{1-{\gamma }_2}{\eta }_2\left(r\right){\parallel {\rho }_2\parallel }_{\frac{1}{{\gamma }_2}}{(\psi \left(t\right)-\psi (t-\epsilon ))}^{2\alpha -1-{\gamma }_2}}\hfill \\ \hfill & {\displaystyle \to 0,\mathrm{as}\epsilon \to 0^{+}.}\hfill \end{array}$$

  Moreover, from Lemma 5, (28), (29), and (45), we have

  $$\begin{array}{cc}\hfill & {\displaystyle J_6\le D\left\{\parallel E^{-1}{\parallel }^2{M}^2{\left({\int }_0^{\infty }\theta {\xi }_{\alpha }\left(\theta \right)d\theta \right)}^2{\int }_{t-\epsilon }^t{\int }_V{(\psi \left(t\right)-\psi \left(s\right))}^{2(\alpha -1)}\mathbb{E}{\parallel g(s,x\left(s\right),v)\parallel }^2{\psi }^{\prime }{\left(s\right)}^2\lambda \left(dv\right)ds\right.}\hfill \\ \hfill & {\displaystyle \left.+\parallel E^{-1}{\parallel }^2{M}^2{\left({\int }_0^{\infty }\theta {\xi }_{\alpha }\left(\theta \right)d\theta \right)}^2{\left({\int }_{t-\epsilon }^t{\int }_V{(\psi \left(t\right)-\psi \left(s\right))}^{4(\alpha -1)}\mathbb{E}{\parallel g(s,x\left(s\right),v)\parallel }^4{\psi }^{\prime }{\left(s\right)}^4\lambda \left(dv\right)ds\right)}^{\frac{1}{2}}\right\}}\hfill \\ \hfill & {\displaystyle \le \frac{\parallel E^{-1}{\parallel }^{2}D{M}^2}{{\Gamma }^2(1+\alpha )}\left(\frac{{\overline{\rho }}_3{\eta }_3\left(r\right)K}{2\alpha -1}{(\psi \left(t\right)-\psi (t-\epsilon ))}^{2\alpha -1}+\sqrt{\frac{{\overline{\rho }}_4{\eta }_4\left(r\right)K^3}{4\alpha -3}}{(\psi \left(t\right)-\psi (t-\epsilon ))}^{\frac{4\alpha -3}{2}}\right)}\hfill \\ \hfill & {\displaystyle \to 0,\mathrm{as}\epsilon \to 0^{+}.}\hfill \end{array}$$

  Similarly, we can obtain that $J_i\to 0$ ($i=2,3,4$). Thus, $\mathbb{E}\parallel \left(\mathcal{T}x\right)\left(t\right)-\left({\mathcal{T}}_{\epsilon ,\delta }x\right)\left(t\right){\parallel }^2\to 0$ as $\epsilon ,\delta \to 0^{+}$. As a result, for each $t\in J$, there exists a relatively compact set arbitrarily close to the set $\{\left(\mathcal{T}x\right)\left(t\right):x\in D_r\}$ in X. Thus, $\{\left(\mathcal{T}x\right)\left(t\right):x\in D_r\}$ is also relatively compact in X for $t\in J$.

Hence, from the Arzela–Ascoli theorem, it is established that $\mathcal{T}:D_r\to D_r$ is a compact operator. Thus, from the Schauder fixed point theorem, we obtain that $\mathcal{T}$ has at least one fixed point $x\in D_r$, which is in turn a mild solution of SCFSEEs (1)–(2) on J. □

Let x be the mild solution of SCFSEEs (1)–(2) corresponding to the control $u\in L^2(J,U)$. SCFSEEs (1)–(2) are said to be approximately controllable on interval $[0,b]$ if for every desired final state $x_b\in L^2(\Omega ,X)$ and $ϵ>0$, there exists a control $u\in L^2(J,U)$ such that x satisfies $\parallel x\left(b\right)-x_b\parallel <ϵ$.

In the following, we will present and prove the approximate controllability of SCFSEEs (1)–(2).

Theorem 2. 

Let ${\displaystyle \frac{3}{4}}<\alpha <1$. Assume that $\left(H_1\right)$–$\left(H_8\right)$ hold. Then, the Sobolev-type ψ-Caputo fractional stochastic evolution Equations (1) and (2) are approximately controllable on J.

Proof. 

From Theorem 1, we know that SCFSEEs (1)–(2) have at least one mild solution $x_{\lambda }\in D_r$, which means that

$$\begin{array}{cc}\hfill & {\displaystyle x_{\lambda }\left(t\right)={\int }_0^{b}G(t,s)(f(s,x_{\lambda }\left(s\right))+B{u}_{\lambda }(s,x_{\lambda }\left(s\right))){\psi }^{\prime }\left(s\right)ds+{\int }_0^{b}G(t,s)\sigma (s,x_{\lambda }\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +{\int }_0^b{\int }_{V}G(t,s)g(s,x_{\lambda }\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv),}\hfill \end{array}$$

(46)

with

$$u_{\lambda }(t,x_{\lambda })=B^{\ast }{G}^{\ast }(b,t)R(\lambda ,{\Gamma }_0^b)q\left(x_{\lambda }\right),$$

(47)

and

$$\begin{array}{cc}\hfill & {\displaystyle q\left(x_{\lambda }\right)=x_b-{\int }_0^{b}G(b,s)f(s,x_{\lambda }\left(s\right)){\psi }^{\prime }\left(s\right)ds-{\int }_0^{b}G(b,s)\sigma (s,x_{\lambda }\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle -{\int }_0^b{\int }_{V}G(b,s)g(s,x_{\lambda }\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv).}\hfill \end{array}$$

(48)

Thus, (46)–(48) combined with an easy computation shows that

$$\begin{array}{cc}\hfill & {\displaystyle x_{\lambda }\left(b\right)={\int }_0^{b}G(b,s)(f(s,x_{\lambda }\left(s\right))+B{u}_{\lambda }(s,x_{\lambda }\left(s\right))){\psi }^{\prime }\left(s\right)ds+{\int }_0^{b}G(b,s)\sigma (s,x_{\lambda }\left(s\right)){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle +{\int }_0^b{\int }_{V}G(b,s)g(s,x_{\lambda }\left(s\right),v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)}\hfill \\ \hfill & {\displaystyle =x_b-q\left(x_{\lambda }\right)+{\int }_0^{b}G(b,s)B{B}^{\ast }{G}^{\ast }(b,s)R(\lambda ,{\Gamma }_0^b)q\left(x_{\lambda }\right){\psi }^{\prime }\left(s\right)ds}\hfill \\ \hfill & {\displaystyle =x_b-q\left(x_{\lambda }\right)+{\Gamma }_0^{b}R(\lambda ,{\Gamma }_0^b)q\left(x_{\lambda }\right)}\hfill \\ \hfill & {\displaystyle =x_b-(\lambda I+{\Gamma }_0^b)R(\lambda ,{\Gamma }_0^b)q\left(x_{\lambda }\right)+{\Gamma }_0^{b}R(\lambda ,{\Gamma }_0^b)q\left(x_{\lambda }\right)}\hfill \\ \hfill & {\displaystyle =x_b-\lambda R(\lambda ,{\Gamma }_0^b)q\left(x_{\lambda }\right).}\hfill \end{array}$$

(49)

The assumptions on f, $\sigma$, and g imply that $\left\{f(t,x_{\lambda }\left(t\right))\right|\lambda \in (0,1)\}$, $\left\{\sigma (t,x_{\lambda }\left(t\right))\right|\lambda \in (0,1)\}$ and $\left\{g(t,x_{\lambda }\left(t\right),v)\right|\lambda \in (0,1)\}$ are bounded uniformly in $\lambda \in (0,1)$ in X, ${\mathcal{L}}_2^0$, and X, respectively, for all $t\in [0,b]$ and the stochastic variable $\omega \in \Omega$. Thus, there are subsequences written as $f(t,x_{\lambda }\left(t\right))$, $\sigma (t,x_{\lambda }\left(t\right))$, and $g(t,x_{\lambda }\left(t\right),v)$ that weakly converge to $f\left(t\right)$, $\sigma \left(t\right)$, and $g(t,v)$ in X, ${\mathcal{L}}_2^0$, and X, respectively, for each $t\in [0,b]$. Now, we write

$$\begin{array}{cc}\hfill & {\displaystyle \pi :=x_b-{\int }_0^{b}G(b,s)f\left(s\right){\psi }^{\prime }\left(s\right)ds-{\int }_0^{b}G(b,s)\sigma \left(s\right){\psi }^{\prime }\left(s\right)dW\left(s\right)}\hfill \\ \hfill & {\displaystyle -{\int }_0^b{\int }_{V}G(b,s)g(s,v){\psi }^{\prime }\left(s\right)\overline{N}(ds,dv).}\hfill \end{array}$$

(50)

Hence, from (48) and (50), we know that

$$\begin{array}{cc}\mathbb{E}\parallel q\left(x_{\lambda }\right){-\pi \parallel }^2\hfill & {\displaystyle \le 3\mathbb{E}{∥{\int }_0^{b}G(b,s)[f(s,x_{\lambda }\left(s\right))-f\left(s\right)]{\psi }^{\prime }\left(s\right)ds∥}^2}\hfill \\ \hfill & {\displaystyle +3\mathbb{E}{∥{\int }_0^{b}G(b,s)[\sigma (s,x_{\lambda }\left(s\right))-\sigma \left(s\right)]{\psi }^{\prime }\left(s\right)dW\left(s\right)∥}^2}\hfill \\ \hfill & {\displaystyle +3\mathbb{E}{∥{\int }_0^b{\int }_{V}G(b,s)[g(s,x_{\lambda }\left(s\right),v)-g(s,v)]{\psi }^{\prime }\left(s\right)\overline{N}(ds,dv)∥}^2.}\hfill \end{array}$$

(51)

From the fact operators $T_E^{\alpha ,\psi }(t,s)$ and $S_E^{\alpha ,\psi }(t,s)$ are compact for $t,s>0$, combined with (18), we obtain that Green’s function $G(t,s)$ is also compact for $t,s>0$. This means that (via the passing of subsequences if necessary)

$$G(b,s)f(s,x_{\lambda }\left(s\right))\to G(b,s)f\left(s\right),$$

$$G(b,s)\sigma (s,x_{\lambda }\left(s\right))\to G(b,s)\sigma \left(s\right),$$

$$G(b,s)g(s,x_{\lambda }\left(s\right),v)\to G(b,s)g(s,v),$$

which implies that

$${\displaystyle \mathbb{E}\parallel q\left(x_{\lambda }\right){-\pi \parallel }^2\to 0\mathrm{as}\lambda \to 0^{+}.}$$

(52)

Thus, (49), (52), and the assumption $\left(H_8\right)$ imply that

$$\begin{array}{cc}\hfill & {\displaystyle \mathbb{E}\parallel x_{\lambda }\left(b\right)-x_b{\parallel }^2=\mathbb{E}{\parallel \lambda R(\lambda ,{\Gamma }_0^b)q\left(x_{\lambda }\right)\parallel }^2}\hfill \\ \hfill & {\displaystyle \le 2\mathbb{E}\parallel \lambda R(\lambda ,{\Gamma }_0^b){\pi \parallel }^2+2\parallel \lambda R(\lambda ,{\Gamma }_0^b){\parallel }^2\mathbb{E}{\parallel q\left(x_{\lambda }\right)-\pi \parallel }^2}\hfill \\ \hfill & {\displaystyle \to 0\mathrm{as}\lambda \to 0^{+}.}\hfill \end{array}$$

Hence, SCFSEEs (1)–(2) are approximately controllable. This completes the proof of Theorem 2. □

4. An Example

In this section, we give an example to illustrate the applicability of our abstract results.

Example 1. 

Consider the following Sobolev-type ψ-Caputo fractional stochastic differential equation with Poisson jumps:

$$\left\{\begin{array}{cc}\hfill & {\displaystyle {}^{C}D_0^{\alpha ,\psi }[x(t,z)-x_{zz}(t,z)]=Ax\left(t\right)+Bu\left(t\right)+f(t,x(t,z))+\sigma (t,x(t,z))\frac{dW}{dt},}\hfill \\ \hfill & {\displaystyle +{\int }_{V}g(t,x(t,z),v)\overline{N}(dt,dv),t\in J=[0,1],z\in [0,\pi ],}\hfill \\ \hfill & {\displaystyle x(t,0)=x(t,\pi )=0,t\in J,}\hfill \\ \hfill & {\displaystyle x(0,z)=c_{1}x(t_1,z),z\in [0,\pi ],}\hfill \end{array}\right.$$

(53)

where $\alpha ={\displaystyle \frac{4}{5}}$, $\psi \left(t\right)=\sqrt{t+1}$, $|c_1|<1$, $b=1$, and $t_1\in (0,1)$. Let $X=L^2[0,\pi ]$. The operators $E:D\left(E\right)\subset X\to X$ and $A:D\left(A\right)\subset X\to X$ are defined as

$$Ex=x-x_{zz},Ax=x_{zz},$$

where

$$D\left(A\right)=D\left(E\right)=\{x\in X:x,x_z\mathrm{are}\mathrm{absolutely}\mathrm{continuous},x_{zz}\in X,x(t,0)=x(t,\pi )=0\}.$$

Moreover, we define

$$x\left(t\right)\left(z\right)=x(t,z),t\in J,z\in [0,\pi ],$$

$$f(t,x\left(t\right))\left(z\right)=f(t,x(t,z))={\displaystyle \frac{t^2{x}^{\frac{1}{3}}(t,z)}{2(1+x^2(t,z))}},$$

$$\sigma (t,x\left(t\right))\left(z\right)=\sigma (t,x(t,z))={\displaystyle \frac{t}{1+e^t}}{\displaystyle \frac{x^{\frac{2}{3}}(t,z)}{1+x^2(t,z)}},$$

$$g(t,x\left(t\right),v)\left(z\right)=g(t,x(t,z),v)=t^{3}sin{x}^{{\displaystyle \frac{1}{3}}}(t,z)v,$$

and taking ${\int }_V{v}^2\lambda \left(dv\right)<\infty$, ${\int }_V{v}^4\lambda \left(dv\right)<\infty$.

It follows that A has eigenvalues $-n^2$, $n\in \mathbb{N}$ with corresponding orthogonal eigenvectors $e_n\left(z\right)=\sqrt{{\displaystyle \frac{2}{\pi }}}sin\left(nz\right)$. From [43], A and E can be written as

$${\displaystyle Ax:=-\sum _{n=1}^{\infty }{n}^2\langle x,e_n\rangle e_n,x\in D\left(A\right),}$$

$${\displaystyle Ex:=\sum _{n=1}^{\infty }(1+n^2)\langle x,e_n\rangle e_n,x\in D\left(A\right),}$$

respectively. Furthermore, for each $x\in X$, one has

$${\displaystyle E^{-1}x:=\sum _{n=1}^{\infty }\frac{1}{1+n^2}\langle x,e_n\rangle e_n,-A{E}^{-1}x:=\sum _{n=1}^{\infty }\frac{-n^2}{1+n^2}\langle x,e_n\rangle e_n,}$$

and

$${\displaystyle T\left(t\right)x=\sum _{n=1}^{\infty }{e}^{\frac{-n^2}{1+n^2}t}\langle x,e_n\rangle e_n.}$$

We know that $E^{-1}$ is compact and bounded with $\parallel E^{-1}\parallel \le 1$, and $-A{E}^{-1}$ generates the above strongly continuous semigroup $T\left(t\right)$ on X with $\parallel T\left(t\right)\parallel \le e^{-t}\le 1$. Obviously, the assumptions $\left(H_1\right)$–$\left(H_4\right)$ hold. The two characteristic operators $S_E^{\alpha ,\psi }(·,·)$ and $T_E^{\alpha ,\psi }(·,·)$ can be written as

$${\displaystyle S_E^{\alpha ,\psi }(t,s):={\int }_0^{\infty }{E}^{-1}{\xi }_{\frac{4}{5}}\left(\theta \right)T\left({(\sqrt{t+1}-\sqrt{s+1})}^{\frac{4}{5}}\theta \right)d\theta ,}$$

and

$${\displaystyle T_E^{\alpha ,\psi }(t,s):=\frac{4}{5}{\int }_0^{\infty }{E}^{-1}\theta {\xi }_{\frac{4}{5}}\left(\theta \right)T\left({(\sqrt{t+1}-\sqrt{s+1})}^{\frac{4}{5}}\theta \right)d\theta .}$$

Clearly,

$${\displaystyle \parallel S_E^{\alpha ,\psi }(t,s)\parallel \le 1,\parallel T_E^{\alpha ,\psi }(t,s)\parallel \le \frac{1}{\Gamma \left(\frac{4}{5}\right)},0\le s\le t\le 1.}$$

For $f(t,x(t\left)\right)$, from Lyapunov’s inequality for moments, we have

$${\mathbb{E}\parallel f(t,x\left(t\right))\parallel }^2\le {\displaystyle \frac{t^4}{4}}{\mathbb{E}\parallel x\left(t\right)\parallel }^{{\displaystyle \frac{2}{3}}}\le {\displaystyle \frac{t^4}{4}}{(\mathbb{E}\parallel x\left(t\right)\parallel }^2{)}^{{\displaystyle \frac{1}{3}}}.$$

Therefore, for all $t\in J$, $x\in X$,

$${\mathbb{E}\parallel f(t,x)\parallel }^2\le {\rho }_1\left(t\right){\eta }_1{(\mathbb{E}\parallel x\parallel }^2),$$

where ${\rho }_1\left(t\right)={\displaystyle \frac{t^4}{4}}$, ${\eta }_1\left(s\right)=s^{{\displaystyle \frac{1}{3}}}$. Obviously, ${\rho }_1\in L^2\left(J\right)$ (${\gamma }_1={\displaystyle \frac{1}{2}}<{\displaystyle \frac{3}{5}}=2\alpha -1$) and $\underset{r\to \infty }{lim}{\displaystyle \frac{{\eta }_1\left(r\right)}{r}}=0$, which implies that $\left(H_5\right)$ holds. Similarly, we can check that $\left(H_6\right)$ and $\left(H_7\right)$ hold.

The justification of the assumption $\left(H_8\right)$ remains. To this end, we take

$$U=\left\{v=\sum _{n=1}^{\infty }{v}_n{e}_n:\sum _{n=1}^{\infty }{v}_n^2<\infty \right\},$$

with the norm

$$\parallel v\parallel ={\left(\sum _{n=1}^{\infty }{v}_n^2\right)}^2.$$

Then, U is a Hilbert space. Now, we define the linear continuous operator $B:U\to X$ as

$$Bv=2v_2{e}_1+\sum _{n=2}^{\infty }{v}_n{e}_n,vs.=\sum _{n=1}^{\infty }{v}_n{e}_n\in U.$$

It is easy to compute that

$$B^{\ast }v=(2v_1+v_2)e_2+\sum _{n=3}^{\infty }{v}_n{e}_n.$$

We will now demonstrate that $B^{\ast }{G}^{\ast }(b,s)$ is a positive operator. Since

$$(I-c_1{S}_E^{\alpha ,\psi }(t_1,0)E)v=\sum _{n=1}^{\infty }\left[1-c_1{\int }_0^{\infty }{\xi }_{{\displaystyle \frac{4}{5}}}\left(\theta \right)exp\left(-{\displaystyle \frac{n^2}{1+n^2}}{(\sqrt{1+t_1}-1)}^{{\displaystyle \frac{4}{5}}}\theta \right)d\theta \right]v_n{e}_n,$$

one has

$$\begin{array}{cc}\hfill & {\displaystyle \mathrm{\Lambda }v={\left(I-\sum _{k=1}^m{c}_k{S}_E^{\alpha ,\psi }(t_k,0)E\right)}^{-1}v}\hfill \\ \hfill & {\displaystyle =\sum _{n=1}^{\infty }{\left[1-c_1{\int }_0^{\infty }{\xi }_{\frac{4}{5}}\left(\theta \right)exp\left(-\frac{n^2}{1+n^2}{(\sqrt{1+t_1}-1)}^{\frac{4}{5}}\theta \right)d\theta \right]}^{-1}{v}_n{e}_n.}\hfill \end{array}$$

(54)

Furthermore, we have

$${\displaystyle G^{\ast }(b,s)v={\chi }_{t_1}\left(s\right){(\psi \left(t_1\right)-\psi \left(s\right))}^{\alpha -1}{T_E^{\alpha ,\psi }}^{\ast }(b,s){\mathrm{\Lambda }}^{\ast }{E}^{\ast }{S_E^{\alpha ,\psi }}^{\ast }(t_1,s)v+{\chi }_b\left(s\right){(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}{T_E^{\alpha ,\psi }}^{\ast }(b,s)v.}$$

(55)

and

$$\begin{array}{cc}\hfill & {\displaystyle {(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}{T_E^{\alpha ,\psi }}^{\ast }(b,s)v}\hfill \\ \hfill & {\displaystyle =\frac{4}{5}\sum _{n=1}^{\infty }\frac{1}{1+n^2}{(\sqrt{1+b}-\sqrt{1+s})}^{-\frac{1}{5}}{\int }_0^{\infty }\theta {\xi }_{\frac{4}{5}}\left(\theta \right)exp\left(-\frac{n^2}{1+n^2}{(\sqrt{1+b}-\sqrt{1+s})}^{\frac{4}{5}}\theta \right)d\theta v_n{e}_n}\hfill \\ \hfill & {\displaystyle =\sum _{n=1}^{\infty }{A}_n{v}_n{e}_n,}\hfill \end{array}$$

(56)

and

$$\begin{array}{cc}\hfill & {\displaystyle {(\psi \left(t_1\right)-\psi \left(s\right))}^{\alpha -1}{T_E^{\alpha ,\psi }}^{\ast }(b,s){\mathrm{\Lambda }}^{\ast }{E}^{\ast }{S_E^{\alpha ,\psi }}^{\ast }(t_1,s)v}\hfill \\ \hfill & {\displaystyle =\frac{4}{5}\sum _{n=1}^{\infty }\frac{1}{1+n^2}{(\sqrt{1+b}-\sqrt{1+s})}^{-\frac{1}{5}}{\int }_0^{\infty }\theta {\xi }_{\frac{4}{5}}\left(\theta \right)exp\left(-\frac{n^2}{1+n^2}{(\sqrt{1+b}-\sqrt{1+s})}^{\frac{4}{5}}\theta \right)d\theta }\hfill \\ \hfill & {\displaystyle \times {\int }_0^{\infty }\frac{{\xi }_{\frac{4}{5}}\left(\theta \right)exp\left(-\frac{n^2}{1+n^2}{(\sqrt{1+t_1}-\sqrt{1+s})}^{\frac{4}{5}}\theta \right)}{1-c_1{\int }_0^{\infty }{\xi }_{\frac{4}{5}}\left(\theta \right)exp\left(-\frac{n^2}{1+n^2}{(\sqrt{1+t_1}-1)}^{\frac{4}{5}}\theta \right)d\theta }d\theta v_n{e}_n}\hfill \\ \hfill & {\displaystyle =\sum _{n=1}^{\infty }{B}_n{v}_n{e}_n,s\in [0,t_1).}\hfill \end{array}$$

(57)

Thus, we get

$${\displaystyle B^{\ast }{G}^{\ast }(b,t)vs.=\left\{\begin{array}{c}{\displaystyle [2(c_1{B}_1+A_1)v_1+(c_1{B}_2+A_2)v_2]e_2+\sum _{n=3}^{\infty }(A_n+c_1{B}_n)v_n{e}_n,t\in [0,t_1),}\hfill \\ {\displaystyle [2A_1{v}_1+A_2{v}_2]e_2+\sum _{n=3}^{\infty }{A}_n{v}_n{e}_n,t\in [t_1,b),}\hfill \\ {\displaystyle 0,t=b,}\hfill \end{array}\right.}$$

(58)

where $v={\sum }_{n=1}^{\infty }{v}_n{e}_n\in X$. If $B^{\ast }{G}^{\ast }(b,t)vs.=0$, then $v_n=0$ for all $n\in \mathbb{N}$. That is, $v=0$. From [7,44], we conclude that assumption $\left(H_8\right)$ holds true. All the conditions of Theorem 2 are therefore fulfilled. Consequently, SCFSEE (53) is approximately controllable on the interval $[0,1]$.

5. Conclusions

This paper establishes sufficient conditions for the existence of mild solutions and the approximate controllability of a class of Sobolev-type $\psi$-Caputo fractional stochastic evolution equations incorporating Poisson jumps and nonlocal conditions. The analysis relies on tools from fractional calculus, stochastic analysis, and Schauder’s fixed point theorem. The flexibility of the $\psi$-Caputo derivative, achieved through the choice of kernel functions, underscores the broader applicability of our theoretical results. A numerical example illustrates the main findings.