The Averaging Principle for Hilfer Fractional Stochastic Evolution Equations with Lévy Noise

Abstract

This article focuses on deriving the averaging principle for Hilfer fractional stochastic evolution equations (HFSEEs) driven by Lévy noise. We show that the solutions of the averaged equations converge to the corresponding solutions of the original equations, both in the sense of mean square and of probability. Our results enable us to focus on the averaged system rather than the original, more complex one. Given that the existing literature on the averaging principle for Hilfer fractional stochastic differential equations has been established in finite-dimensional spaces, the novelty here is the derivation of the averaging principle for a class of HFSEEs in Hilbert space. Furthermore, an example is allotted to illustrate the feasibility and utility of our results.

1. Introduction

Stochastic differential equations (SDEs) have garnered significant attention due to their distinctive properties and successful application in resolving scientific and engineering problems. However, in many cases, stochastic differential systems depend not only on the current states but also on the past states, occasionally involving derivatives with delays (see Kolmanovskii and Myshkis [1]). This characteristic renders fractional-order stochastic differential equations (FSDEs) highly valuable and has fueled extensive research into FSDE theory. The utilization of these equations extends across various fields such as mechanics, electricity, economics, and physics, among others. Recently, there has been a growing interest among researchers in studying the properties of solutions, including well-posedness and the controllability of FSDEs (see [2,3]).

In addition, stochastic dynamics primarily involve Gaussian noise. However, it is acknowledged that pure Gaussian noise is unsuitable for simulating certain practical phenomena due to the inevitability of internal or external interference. Non-Gaussian Lévy noise already encompasses these types of perturbations as it possesses the advantage of exhibiting a long-tailed distribution, thereby introducing time discontinuity in the sample path. Hence, its significance and necessity cannot be overstated. Numerous reports on SDEs with Lévy noise (see [4,5,6]) have been published. For instance, Balasubramaniam [7] discussed the existence of solutions for FSDEs of Hilfer-type with non-instantaneous impulses excited by mixed Brownian motion and Lévy noise. Using the successive approximation method, Xu et al. [8] addressed the existence and uniqueness of solutions for SDEs driven by Lévy noise.

The averaging principle is commonly employed to approximate dynamical systems with random fluctuations, serving as a powerful tool for simplifying nonlinear dynamical systems. By studying relevant averaged systems instead, we are able to investigate the underlying complex dynamics of scientific and engineering problems. The initial literature on the average principle of SDEs was introduced by Khasminskii [9], which subsequently garnered significant attention due to its ability to reduce computational complexity in the original system. For further details regarding the averaging principle for SDEs, refer to [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30].

Furthermore, the Hilfer fractional derivative is a generalized fractional derivative that combines both the classical R-L fractional derivative and the Caputo fractional derivative as special cases. This highlights its significance in studying the averaging principle for Hilfer fractional stochastic differential systems. It is worth mentioning that Ahmed and Zhu [29] were the first to investigate the theory of averaging principle for Hilfer fractional stochastic differential equations (HFSDEs) with delay and Lévy noise in ${\mathbb{R}}^n$; however, their study did not consider evolution equations. Luo et al. [30] examined the averaging principle for HFSDEs involving non-Lipschitz coefficients without considering Lévy noise. Shen et al. [12] established the stochastic averaging method for FSDEs with variable delays driven by Lévy noise, but they did not involve Hilfer fractional derivatives.

The gap in existing literature lies in the fact that the averaging method for HFSDEs has only been considered in finite dimensional spaces. It is both natural and significantly essential to pose a significant question: Does some averaging principle for HFSEEs still hold in infinite-dimensional spaces? If it does hold, how can we establish the averaging principle? This serves as the motivation behind this paper.

Inspired by the aforementioned discussions, our objective in this work is to derive an averaging principle for a class of HFSEEs driven by Lévy noise. The theory of stochastic analysis, fractional calculus, semigroup properties, and inequality techniques are employed to obtain the desired results. The main contributions of this paper are as follows:

• Diverging from the existing literature on the average principle for HFSDEs established in ${\mathbb{R}}^n$, we consider here the average principle of SEEs of Hilfer-type driven by Lévy noise in Hilbert spaces.
• The existence, uniqueness, and average principle of the concerned system are established.
• The feasibility and effectiveness of the proposed results are verified through a numerical example.

We organized the structure of this paper as follows: Section 2 provides a review of some fundamental preliminary facts. In Section 3, we establish the existence and uniqueness theorem of mild solutions for a class of HFSEEs; based on this foundation, we derive the average principle for the concerned system. An illustrative example is presented to demonstrate the obtained theory in Section 4. Finally, we conclude by providing a comprehensive overview of the findings in Section 5.

2. Preliminary

Throughout this paper, $\mathbb{H},\mathbb{V}$ represent separable, real Hilbert spaces. ${\left\{W(\ell )\right\}}_{\ell \ge 0}$ denotes a $\mathbb{V}$-valued Q-Wiener process defined on $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$, which is a complete probability space; space $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$ is equipped with a filtration ${\left\{{\mathcal{F}}_{\ell }\right\}}_{\ell \ge 0}$ that satisfies the usual conditions. We assume that Q is a self-adjoint, positive, and trace class operator on $\mathbb{V}$. Let ${\mathbb{V}}_0=Q^{\frac{1}{2}}\mathbb{V}$, $L_2^0=L_2({\mathbb{V}}_0,\mathbb{H})$ be a separable Hilbert space consisting of all Hilbert–Schmidt operators from $Q^{\frac{1}{2}}\mathbb{V}$ to $\mathbb{H}$. The space $\mathcal{C}=C([-\varsigma ,0];\mathbb{H})$ is composed of all càdlàg functions $\varphi$ from $[-\varsigma ,0]$ to $\mathbb{H},$ for $\varphi \in \mathcal{C},|\varphi |=\underset{-\varsigma \le t\le 0}{sup}|\varphi (t)|$. For the sake of convenience, and to avoid confusion, the norms in $\mathbb{H},\mathbb{V},L_2^0$, and $L(\mathbb{V},\mathbb{H})$ are expressed by the same notation, $|·|$.

Moreover, $L_2(\mathsf{\Omega },\mathbb{H})$ denotes the space composed of all strongly measurable, square-integrable, $\mathbb{H}$-valued random variables; $L_2(\mathsf{\Omega },\mathbb{H})$ is a Banach space furnished with the norm ${|x(·)|}_{L_2(\mathsf{\Omega },\mathbb{H})}={\left({E|x(·)|}^2\right)}^{\frac{1}{2}}$. $C(J,L_2(\mathsf{\Omega },\mathbb{H}))$ denotes the Banach spaces of all $\mathbb{H}$-valued continuous functions from J to $L_2(\mathsf{\Omega },\mathbb{H})$, satisfying $\underset{\ell \in J}{sup}E{|x(\ell )|}^2<\infty$.

In this paper, we will investigate the following HFSEEs driven by the Lévy process

$$\left\{\begin{array}{cc}\hfill & D_{0^{+}}^{\gamma ,\beta }x(\ell )=Ax(\ell )+\kappa (\ell ,x(\ell ),x_{\ell })+\sigma (\ell ,x(\ell ),x_{\ell })\frac{dL(\ell )}{d\ell },\ell \in J=(0,T],\hfill \\ \hfill & x(\ell )=\phi (\ell ),-\varsigma \le \ell \le 0,\hfill \\ \hfill & I_{0^{+}}^{(1-\gamma )(1-\beta )}x(0)={\phi }_0,\hfill \end{array}\right.$$

(1)

where A is the infinitesimal generator of a strongly continuous semigroup $S(\ell )(\ell \ge 0)$ on $\mathbb{H}$. $D_{0^{+}}^{\gamma ,\beta }$ is the Hilfer fractional derivative, $\frac{1}{2}<\beta <1,0\le \gamma \le 1$, $\kappa$ is $\mathbb{H}$-valued, L is a $\mathbb{V}$-valued Lévy process, and $\sigma$ is $L(\mathbb{V},\mathbb{H})$-valued. The history process is $x_{\ell }=x(\ell +\tau ),-\varsigma \le \tau \le 0.$ The initial function is $\phi =\{\phi (\ell ):-\varsigma \le \ell \le 0\}$, satisfying $E\left(\underset{-\varsigma \le \ell \le 0}{sup}{|\phi (\ell )|}^2\right)<\infty .$

Next, some fundamental definitions of the fractional calculus are introduced.

Definition 1 

([31]). The left-sided Riemann–Liouville fractional integral of order β for a function $f:[a,+\infty )\to \mathbb{R}$ is presented by

$$I_{a^{+}}^{\beta }f(\ell )=\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_a^t\frac{f(s)}{{(\ell -s)}^{1-\beta }}ds,\ell >a,\beta >0,$$

where $\mathrm{\Gamma }(·)$ is the Gamma function.

Definition 2 

([32]). (Hilfer fractional derivative) The left-sided Hilfer fractional derivative of order $0<\beta <1$ and type $0\le \gamma \le 1$ of function f is presented by

$$D_{a^{+}}^{\gamma ,\beta }f(\ell )=(I_{a^{+}}^{\gamma (1-\beta )}D(I_{a^{+}}^{(1-\gamma )(1-\beta )}f))(\ell ),$$

where $D:=\frac{d}{d\ell }$.

Based on the following assumption, we will introduce some necessary lemmas.

Hypothesis 0 (H0). 

For $\ell \ge 0$, semigroup $S(\ell )$ is continuous in the uniform operator topology; furthermore, it is uniformly bounded, i.e., there exists $M>1$  such that  $\underset{\ell \in [0,+\infty )}{sup}|S(\ell )|<M.$

Lemma 1 

([32]). If hypothesis $(H0)$ holds for arbitrary fixed $\ell >0$ and arbitrary $y\in \mathbb{H}$, the operators ${\left\{{\mathcal{T}}_{\beta }(\ell )\right\}}_{\ell >0}$ and ${\left\{{\mathcal{S}}_{\gamma ,\beta }(\ell )\right\}}_{\ell >0}$ are linear and bounded, i.e.,

$$|{\mathcal{T}}_{\beta }(\ell )y|\le \frac{M}{\mathrm{\Gamma }(\beta )}|y|,|{\mathcal{S}}_{\gamma ,\beta }(\ell )y|\le \frac{M{\ell }^{(\gamma -1)(1-\beta )}}{\mathrm{\Gamma }(\gamma (1-\beta )+\beta )}|y|.$$

where ${\displaystyle {\mathcal{T}}_{\beta }(\ell )={\int }_0^{\infty }\beta \theta M_{\beta }(\theta )S({\ell }^{\beta }\theta )d\theta ,{\mathcal{P}}_{\beta }(\ell )={\ell }^{\beta -1}{\mathcal{T}}_{\beta }(\ell )}$ and ${\mathcal{S}}_{\gamma ,\beta }(\ell )=I_{0^{+}}^{\gamma (1-\beta )}{\mathcal{P}}_{\beta }(\ell ).$ The Wright function $M_{\beta }(\theta )$ is defined by

$$M_{\beta }(\theta )=\sum _{n=1}^{\infty }\frac{{(-\theta )}^{n-1}}{(n-1)!\mathrm{\Gamma }(1-\beta n)},0<\beta <1,\theta \in \mathbb{C}.$$

Proposition 1 

([4,5]). If L is a $\mathbb{V}$-valued Lévy process, then there exists an element $a\in \mathbb{V}$, a $\mathbb{V}$-valued Wiener process W with covariance operator Q, and an independent Poisson random measure on ${\mathbb{R}}^{+}\times (\mathbb{V}\setminus \left\{0\right\})$ such that for each $\ell \ge 0$,

$$L(\ell )=a\ell +W(\ell )+{\int }_{|v|<c}v\tilde{N}(\ell ,dv)+{\int }_{|v|\ge c}vN(\ell ,dv),$$

where $c>0$ is a constant, the Poisson random measure N has the intensity measure λ, and λ satisfies

$$\int (|v{|}_{\mathbb{V}}^2\wedge 1)\lambda (dv)<\infty .$$

Let $L_1(\ell )$ and $L_2(\ell ),\ell \ge 0,$ be two independent, identically distributed Lévy processes. We set

$$L(\ell ):=\left\{\begin{array}{c}\begin{array}{cc}& L_1(\ell ),\ell \ge 0,\hfill \\ & -L_2((-\ell )-),\ell <0.\hfill \end{array}\hfill \end{array}\right.$$

The process L is then a two-sided Lévy process defined on the filtered probability space $(\mathsf{\Omega },\mathcal{F},\mathbb{P},$${({\mathcal{F}}_{\ell })}_{\ell \in \mathbb{R}})$.

By the above proposition, Equation (1) can be rewritten into a more general representation:

$$\left\{\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle D_{0^{+}}^{\gamma ,\beta }x(\ell )=Ax(\ell )+f(\ell ,x(\ell ),x_{\ell })+h(\ell ,x(\ell ),x_{\ell })\frac{dW(\ell )}{d\ell }}\hfill \\ & {\displaystyle +\frac{1}{d\ell }{\int }_{|v|<c}H(\ell ,x(\ell ),x_{\ell },v)\tilde{N}(d\ell ,dv)}\hfill \\ & {\displaystyle +\frac{1}{d\ell }{\int }_{|v|\ge c}Q(\ell ,x(\ell ),x_{\ell },v)N(d\ell ,dv),\ell \in J=(0,T],}\hfill \\ \hfill & x(\ell )=\phi (\ell ),-\varsigma \le \ell \le 0,\hfill \\ \hfill & I_{0^{+}}^{(1-\gamma )(1-\beta )}x(0)={\phi }_0,\hfill \end{array}\end{array}\right.$$

(2)

where functions $f:J\times \mathbb{H}\times \mathcal{C}\to \mathbb{H},h:J\times \mathbb{H}\times \mathcal{C}\to L(\mathbb{V},\mathbb{H}),H:J\times \mathbb{H}\times \mathcal{C}\times \mathbb{V}\to \mathbb{H}$ and $Q:J\times \mathbb{H}\times \mathcal{C}\times \mathbb{V}\to \mathbb{H}$ are measurable. Applying the technique demonstrated in the literature [4], we just need to focus on the stochastic differential system without large jumps:

$$\left\{\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle D_{0^{+}}^{\gamma ,\beta }x(\ell )=Ax(\ell )+f(\ell ,x(\ell ),x_{\ell })+h(\ell ,x(\ell ),x_{\ell })\frac{dW(\ell )}{d\ell }}\hfill \\ \hfill & {\displaystyle +\frac{1}{d\ell }{\int }_{|v|<c}H(\ell ,x(\ell ),x_{\ell },v)\tilde{N}(d\ell ,dv),\ell \in J=(0,T],}\hfill \\ \hfill & x(\ell )=\phi (\ell ),-\varsigma \le \ell \le 0,\hfill \\ \hfill & I_{0^{+}}^{(1-\gamma )(1-\beta )}x(0)={\phi }_0.\hfill \end{array}\end{array}\right.$$

(3)

Lemma 2. 

The Equation (3) is equivalent to the integral equation

$$\left\{\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle x(\ell )=\frac{{\ell }^{(\gamma -1)(1-\beta )}{\phi }_0}{\mathrm{\Gamma }(\gamma (1-\beta )+\beta )}+\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}[Ax(\varrho )+f(\varrho ,x(\varrho ),x_{\varrho })]d\varrho }\hfill \\ & {\displaystyle +\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}h(\varrho ,x(\varrho ),x_{\varrho })dW(\varrho )}\hfill \\ & {\displaystyle +\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\int }_{|v|<c}H(\varrho ,x(\varrho ),x_{\varrho },v)\tilde{N}(d\varrho ,dv),\ell \in (0,T],}\hfill \\ & x(\ell )=\phi (\ell ),-\varsigma \le \ell \le 0.\hfill \end{array}\end{array}\right.$$

Proof. 

We recommend that readers refer to references [32,33]; we omit the proof here. □

Lemma 3 

([34]). (The generalized Gronwall inequality) Suppose $\alpha >0$, for $0\le \ell <T$ (some $T\le +\infty$), $a(\ell )$ is a locally integrable and nonnegative function, $g(\ell )$ is a nonnegative, nondecreasing continuous function, and there exists a constant G, such that $g(\ell )\le G$; suppose $u(\ell )$ is nonnegative and locally integrable on $0\le \ell <T$ with

$$u(\ell )\le a(\ell )+g(\ell ){\int }_0^{\ell }{(\ell -\varrho )}^{\alpha -1}u(\varrho )d\varrho ,$$

on this interval. Then

$$u(\ell )\le a(\ell )+{\int }_0^{\ell }\left[\sum _{n=1}^{\infty }\frac{{(g(\ell )\mathrm{\Gamma }(\alpha ))}^n}{\mathrm{\Gamma }(n\alpha )}{(\ell -\varrho )}^{n\alpha -1}a(\varrho )\right]d\varrho ,$$

furthermore, if $a(\ell )$ is nondecreasing on $0\le \ell <T$, then

$$u(\ell )\le a(\ell )\sum _{n=0}^{\infty }\frac{{(g(\ell )\mathrm{\Gamma }(\alpha ){\ell }^{\alpha })}^n}{\mathrm{\Gamma }(n\alpha +1)}=a(\ell )E_{\alpha }(g(\ell )\mathrm{\Gamma }(\alpha ){\ell }^{\alpha }),$$

where $E_{\alpha }$ is the Mittag–Leffler function for all $z>0$, ${\displaystyle E_{\alpha }(z)=\sum _{n=0}^{\infty }\frac{z^n}{\mathrm{\Gamma }(n\alpha +1)}}$.

Definition 3. 

If an $\mathbb{H}$-value stochastic variable $\{x(\ell ),-\varsigma \le \ell \le T\}$ satisfies

(i) 

$x(\ell )$ is ${\mathcal{F}}_{\ell }$-adapted; it has càdlàg paths a.s on $\ell \in J$ and $\underset{\ell \in J}{sup}E{|x(\ell )|}^2<\infty$,

(ii) 

$x(\ell )=\phi (\ell ),\ell \in [-\varsigma ,0],$

(iii) 

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x(\ell )& {\displaystyle ={\mathcal{S}}_{\gamma ,\beta }(\ell ){\phi }_0+{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho )f(\varrho ,x(\varrho ),x_{\varrho })d\varrho }\hfill \\ \hfill & {\displaystyle +{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho )h(\varrho ,x(\varrho ),x_{\varrho })dW(\varrho )}\hfill \\ \hfill & {\displaystyle +{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho ){\int }_{|v|<c}H(\varrho ,x(\varrho ),x_{\varrho },v)\tilde{N}(d\varrho ,dv),\ell \in J,}\hfill \end{array}\end{array}$$

(4)

then $x(\ell )$ is a mild solution of Equation (3).

3. Main Results

In this section, we first establish the existence and uniqueness theorem of a mild solution to Equation (3). Now, let $u=\beta +\gamma -\beta \gamma$, we characterize $C_u(J,L_2(\mathsf{\Omega },\mathbb{H}))=\left\{x:{\ell }^{1-u}x(\ell )\in C(J,L_2(\mathsf{\Omega },\mathbb{H}))\right\}$ with the norm

$${|x|}_u={\left(\underset{\ell \in J}{sup}E{|{\ell }^{1-u}x(\ell )|}^2\right)}^{\frac{1}{2}}.$$

Clearly, $C_u(J,L_2(\mathsf{\Omega },\mathbb{H}))$ is a Banach space.

Now we need to make assumptions on the functions $f,h$, and H that will enable us to establish the required result.

Hypothesis 1 (H1). 

For any $x_1,x_2\in \mathbb{H},y_1,y_2\in \mathcal{C}$ and $\ell \in J$, there exist positive constants $K,K_1$ such that

(i) 

$E|f(\ell ,x_1,y_1){|}^2\le K(1+|x_1{|}_u^2+|y_1{|}_u^2),$

(ii) 

${\displaystyle E|f(\ell ,x_1,y_1)-f(\ell ,x_2,y_2){|}^2\le K_1(|x_1-x_2{|}_u^2+|y_1-y_2{|}_u^2).}$

Hypothesis 2 (H2). 

There exist functions $N_1(\ell )\in L^{\frac{1}{2{\beta }_1-1}}(J),N_2(\ell )\in L^{\frac{1}{2{\beta }_1-1}}(J),{\beta }_1\in (\frac{1}{2},\beta )$, such that for any $x_1,x_2\in \mathbb{H},y_1,y_2\in \mathcal{C}$ and $\ell \in J$, we have

(i) 

$E|h(\ell ,x_1,y_1){|}^2\vee E\left({\int }_{|v|<c}{|H(\ell ,x_1,y_1,v)|}^2\lambda (dv)\right)\le N_1(\ell )(1+|x_1{|}_u^2+|y_1{|}_u^2),$

(ii) 

$$\begin{array}{cc}& E|h(\ell ,x_1,y_1)-h(\ell ,x_2,y_2){|}^2\vee E\left({\int }_{|v|<c}{|H(\ell ,x_1,y_1,v)-H(\ell ,x_2,y_2,v)|}^2\lambda (dv)\right)\hfill \\ & \le N_2(\ell )(|x_1-x_2{|}_u^2+|y_1-y_2{|}_u^2).\hfill \end{array}$$

For further convenience, we set

$$\begin{array}{c}\hfill {\displaystyle C_0=\frac{6K_1{M}^2{T}^{2(1-\gamma +\beta \gamma )}}{(2\beta -1){(\mathrm{\Gamma }(\beta ))}^2}+\frac{48M^2{T}^{2(1-u+\beta -{\beta }_1)}}{{(\mathrm{\Gamma }(\beta ))}^2}{\left(\frac{1-{\beta }_1}{\beta -{\beta }_1}\right)}^{2-2{\beta }_1}{|N_2|}_{L^{\frac{1}{2{\beta }_1-1}}},}\end{array}$$

$$\begin{array}{cc}& {\displaystyle C_1=\frac{4M^2{|{\phi }_0|}^2}{{(\mathrm{\Gamma }(\gamma (1-\beta )+\beta ))}^2}+\frac{4K{M}^2{T}^{2(1-\gamma +\beta \gamma )}}{(2\beta -1){(\mathrm{\Gamma }(\beta ))}^2}+\frac{32M^2{T}^{2(1-u+\beta -{\beta }_1)}}{{(\mathrm{\Gamma }(\beta ))}^2}{\left(\frac{1-{\beta }_1}{\beta -{\beta }_1}\right)}^{2-2{\beta }_1}{|N_1|}_{L^{\frac{1}{2{\beta }_1-1}}},}\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle C_2=\frac{8K{M}^2{T}^{2(1-\gamma +\beta \gamma )}}{(2\beta -1){(\mathrm{\Gamma }(\beta ))}^2}+\frac{64M^2{T}^{2(1-u+\beta -{\beta }_1)}}{{(\mathrm{\Gamma }(\beta ))}^2}{\left(\frac{1-{\beta }_1}{\beta -{\beta }_1}\right)}^{2-2{\beta }_1}{|N_1|}_{L^{\frac{1}{2{\beta }_1-1}}}.}\end{array}$$

Moreover, we define an operator $\mathrm{\Psi }$ as follows:

$$\begin{array}{cc}& {\displaystyle (\mathrm{\Psi }x)(\ell )={\mathcal{S}}_{\gamma ,\beta }(\ell ){\phi }_0+{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho )f(\varrho ,x(\varrho ),x_{\varrho })d\varrho }\hfill \\ & {\displaystyle +{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho )h(\varrho ,x(\varrho ),x_{\varrho })dW(\varrho )}\hfill \\ & {\displaystyle +{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho ){\int }_{|v|<c}H(\varrho ,x(\varrho ),x_{\varrho },v)\tilde{N}(d\varrho ,dv),\ell \in (0,T].}\hfill \end{array}$$

Obviously, x is a mild solution of Equation (3) if the operator equation $x=\mathrm{\Psi }x$ possesses a fixed point on $\mathbb{H}.$

Presently, we let ${\tilde{B}}_r=\{x\in C_u(J,L_2(\mathsf{\Omega },\mathbb{H})),{|x|}_u^2\le r,$ where $r\ge \frac{C_1}{1-C_2}\}$, the set ${\tilde{B}}_r$ is a bounded, closed, and convex subset of $C_u(J,L_2(\mathsf{\Omega },\mathbb{H}))$.

In the following, we present and prove the existence and uniqueness of a mild solution for Equation (3).

Theorem 1. 

Assume that Hypotheses $(H0)$–$(H2)$ fulfill and $C_0<1$, then Equation (3) possesses a unique mild solution on ${\tilde{B}}_r$.

Proof. 

We demonstrate the proof through the following three steps.

Step 1.

$\mathrm{\Psi }$ is continuous on ${\tilde{B}}_r.$

By simple arguments from (H1) and (H2), we can prove this assertion is true.

Step 2.

$\mathrm{\Psi }$ maps ${\tilde{B}}_r$ into itself.

In fact, for $\forall x\in {\tilde{B}}_r,t\in J$, by the Cauchy–Schwarz inequality, the B-D-G inequality, Lemma 1, and the Hölder inequality, a standard calculation yields that

$$\begin{array}{cc}& {|(\mathrm{\Psi }x)|}_u^z\hfill \\ & {\displaystyle \le 4\underset{\ell \in J}{sup}{\ell }^{2(1-u)}E{|{\mathcal{S}}_{\gamma ,\beta }(\ell ){\phi }_0|}^2+4\underset{\ell \in J}{sup}{\ell }^{2(1-u)}E|{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho )f(\varrho ,x(\varrho ),x_{\varrho })d\varrho {|}^2}\hfill \\ & {\displaystyle +4\underset{\ell \in J}{sup}{\ell }^{2(1-u)}E|{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho )h(\varrho ,x(\varrho ),x_{\varrho })dW(\varrho ){|}^2}\hfill \\ & {\displaystyle +4\underset{\ell \in J}{sup}{\ell }^{2(1-u)}E|{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho ){\int }_{|v|<c}H(\varrho ,x(\varrho ),x_{\varrho },v)\tilde{N}(d\varrho ,dv){|}^2}\hfill \\ & {\displaystyle \le \frac{4M^2{|{\phi }_0|}^2}{{(\mathrm{\Gamma }(\gamma (1-\beta )+\beta ))}^2}+\frac{4M^2{T}^{1-2\gamma +2\beta \gamma }}{(2\beta -1){(\mathrm{\Gamma }(\beta ))}^2}{\int }_0^{\ell }E{|f(\varrho ,x(\varrho ),x_{\varrho })|}^{2}d\varrho }\hfill \\ & {\displaystyle +\frac{16M^2{T}^{2(1-u)}}{{(\mathrm{\Gamma }(\beta ))}^2}{\int }_0^{\ell }{(\ell -\varrho )}^{2(\beta -1)}E{|h(\varrho ,x(\varrho ),x_{\varrho })|}^{2}d\varrho }\hfill \\ & {\displaystyle +\frac{16M^2{T}^{2(1-u)}}{{(\mathrm{\Gamma }(\beta ))}^2}{\int }_0^{\ell }{(\ell -\varrho )}^{2(\beta -1)}E{\int }_{|v|<c}{|H(\varrho ,x(\varrho ),x_{\varrho },v)|}^2\tilde{N}(d\varrho ,dv)}\hfill \\ & {\displaystyle \le \frac{4M^2{|{\phi }_0|}^2}{{(\mathrm{\Gamma }(\gamma (1-\beta )+\beta ))}^2}+\frac{4M^2{T}^{2(1-\gamma +\beta \gamma )}}{(2\beta -1){(\mathrm{\Gamma }(\beta ))}^2}{K(1+2|x|}_u^2)}\hfill \\ & {\displaystyle +\frac{32M^2{T}^{2(1-u)}}{{(\mathrm{\Gamma }(\beta ))}^2}{\left({\int }_0^{\ell }{(\ell -\varrho )}^{\frac{2(\beta -1)}{2-2{\beta }_1}}d\varrho \right)}^{2-2{\beta }_1}{\left({\int }_0^{\ell }{N}_1{(\varrho )}^{\frac{1}{2{\beta }_1-1}}d\varrho \right)}^{2{\beta }_1-1}{(1+2|x|}_u^2)}\hfill \\ & {\displaystyle \le \frac{4M^2{|{\phi }_0|}^2}{{(\mathrm{\Gamma }(\gamma (1-\beta )+\beta ))}^2}+\frac{4M^2{T}^{2(1-\gamma +\beta \gamma )}}{(2\beta -1){(\mathrm{\Gamma }(\beta ))}^2}K(1+2r)}\hfill \end{array}$$

$$\begin{array}{cc}& {\displaystyle +\frac{32M^2{T}^{2(1-u+\beta -{\beta }_1)}}{{(\mathrm{\Gamma }(\beta ))}^2}{\left(\frac{1-{\beta }_1}{\beta -{\beta }_1}\right)}^{2-2{\beta }_1}{|N_1|}_{L^{\frac{1}{2{\beta }_1-1}}}(1+2r)}\hfill \\ & =C_1+C_{2}r\le r.\hfill \end{array}$$

Step 3.

$\mathrm{\Psi }$ is a contraction on ${\tilde{B}}_r.$

In fact, for $\forall x,y\in {\tilde{B}}_r,\ell \in J$, by a standard calculation, one can obtain

$$\begin{array}{cc}& {|(\mathrm{\Psi }x)-(\mathrm{\Psi }y)|}_u^2\hfill \\ & {\displaystyle \le \frac{3M^2{T}^{1-2\gamma +2\beta \gamma }}{(2\beta -1){(\mathrm{\Gamma }(\beta ))}^2}{\int }_0^{\ell }E{|f(\varrho ,x(\varrho ),x_{\varrho })-f(\varrho ,y(\varrho ),y_{\varrho })|}^{2}d\varrho }\hfill \\ & {\displaystyle +\frac{12M^2{T}^{2(1-u)}}{{(\mathrm{\Gamma }(\beta ))}^2}{\int }_0^{\ell }{(\ell -\varrho )}^{2(\beta -1)}E{|h(\varrho ,x(\varrho ),x_{\varrho })-h(\varrho ,y(\varrho ),y_{\varrho })|}^{2}d\varrho }\hfill \\ & {\displaystyle +\frac{12M^2{T}^{2(1-u)}}{{(\mathrm{\Gamma }(\beta ))}^2}{\int }_0^{\ell }{(\ell -\varrho )}^{2(\beta -1)}E{\int }_{|v|<c}{|H(\varrho ,x(\varrho ),x_{\varrho },v)-H(\varrho ,y(\varrho ),y_{\varrho },v)|}^2\tilde{N}(d\varrho ,dv)}\hfill \\ & {\displaystyle \le \frac{6K_1{M}^2{T}^{2(1-\gamma +\beta \gamma )}}{(2\beta -1){(\mathrm{\Gamma }(\beta ))}^2}{|x-y|}_u^2+\frac{48M^2{T}^{2(1-u+\beta -{\beta }_1)}}{{(\mathrm{\Gamma }(\beta ))}^2}{\left(\frac{1-{\beta }_1}{\beta -{\beta }_1}\right)}^{2-2{\beta }_1}|N_2{|}_{L^{\frac{1}{2{\beta }_1-1}}}{|x-y|}_u^2}\hfill \\ & =C_0{|x-y|}_u^2<{|x-y|}_u^2,\hfill \end{array}$$

which shows that $\mathrm{\Psi }$ is a contraction mapping. Then $\mathrm{\Psi }$ possesses only one fixed point on the set ${\tilde{B}}_r$, which coincides with the mild solution of Equation (3) on ${\tilde{B}}_r.$ □

In what follows, we shall study the averaging principle to Equation (3).

The perturbed form of Equation (3) is defined as

$$\left\{\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle D_{0^{+}}^{\gamma ,\beta }{x}_{ϵ}(\ell )=A{x}_{ϵ}(\ell )+ϵf(\ell ,x_{ϵ}(\ell ),x_{ϵ,\ell })+\sqrt{ϵ}h(\ell ,x_{ϵ}(\ell ),x_{ϵ,\ell })\frac{dW(\ell )}{d\ell }}\hfill \\ & {\displaystyle +\frac{\sqrt{ϵ}}{d\ell }{\int }_{|v|<c}H(\ell ,x_{ϵ}(\ell ),x_{ϵ,\ell },v)\tilde{N}(d\ell ,dv),\ell \in J=(0,T],}\hfill \\ \hfill & x(\ell )=\phi (\ell ),-\varsigma \le \ell \le 0,\hfill \\ \hfill & I_{0^{+}}^{(1-\gamma )(1-\beta )}x(0)={\phi }_0,\hfill \end{array}\end{array}\right.$$

(5)

where the functions $f,h,H$ satisfy the same assumptions as in Equation (3), $ϵ\in (0,{ϵ}_0]$ is a small parameter, along with ${ϵ}_0(0<{ϵ}_0\ll 1)$ being a fixed number.

From Definition 3, the mild solution $x_{ϵ}(\ell )$ of Equation (5) can be given as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle x_{ϵ}(\ell ):={\mathcal{S}}_{\gamma ,\beta }(\ell ){\phi }_0+ϵ{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho )f(\varrho ,x_{ϵ}(\varrho ),x_{ϵ,\varrho })d\varrho }\hfill \\ \hfill & {\displaystyle +\sqrt{ϵ}{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho )h(\varrho ,x_{ϵ}(\varrho ),x_{ϵ,\varrho })dW(\varrho )}\hfill \\ \hfill & {\displaystyle +\sqrt{ϵ}{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho ){\int }_{|v|<c}H(\varrho ,x_{ϵ}(\varrho ),x_{ϵ,\varrho },v)\tilde{N}(d\varrho ,dv),\ell \in J,}\hfill \end{array}\end{array}$$

(6)

To establish the averaging principle for Equation (3), we first make an assumption on functions $f,h$, and H.

Hypothesis 3 (H3). 

The function $\mathcal{M}(\ell ,x):[0,\infty )\times [0,\infty )\to [0,\infty )$ exists, such that $\mathcal{M}(\ell ,x)$ is locally integrable with respect to ℓ for any fixed $x\ge 0$ and is nondecreasing, continuous, and concave with respect to x for each fixed $\ell \in [0,T]$, for any $\ell \in [0,T],{\int }_{0^{+}}\frac{1}{\mathcal{M}(\ell ,x)}dx=\infty$. For any $x_1,x_2\in \mathbb{H},y_1,y_2\in \mathcal{C}$ and $\ell \in J$, this inequality holds:

$$\begin{array}{cc}& |f(\ell ,x_1,y_1)-f(\ell ,x_2,y_2){|}^2\vee {|h(\ell ,x_1,y_1)-h(\ell ,x_2,y_2)|}^2\hfill \\ & {\displaystyle \vee {\int }_{|v|<c}{|H(\ell ,x_1,y_1,v)-H(\ell ,x_2,y_2,v)|}^2\lambda (dv)}\hfill \\ & \le \mathcal{M}(\ell ,|x_1-x_2{|}^2+|y_1-y_2{|}^2).\hfill \end{array}$$

For the objective to ensure the approximation of $x_{ϵ}(t)$ by a simpler stochastic variable, we present three measurable coefficient functions $\overline{f}:\mathbb{H}\times \mathcal{C}\to \mathbb{H}$, $\overline{h}:\mathbb{H}\times \mathcal{C}\to L(\mathbb{V},\mathbb{H}),\overline{H}:\mathbb{H}\times \mathcal{C}\times \mathbb{V}\to \mathbb{H}$ also satisfying (H3), and meeting the following hypothesis:

Hypothesis 4 (H4). 

For any $\tilde{T}\in J,x\in \mathbb{H},y\in \mathcal{C}$, there exist bounded functions ${\zeta }_i(\tilde{T})>0,i=1,2,3$, such that

$$\frac{1}{\tilde{T}}{\int }_0^{\tilde{T}}|f(\varrho ,x,y)-\overline{f}{(x,y)|}^{2}d\varrho \le {\zeta }_1(\tilde{T}){(1+|x|}^2+{|y|}^2),$$

$$\frac{1}{\tilde{T}}{\int }_0^{\tilde{T}}|h(\varrho ,x,y)-\overline{h}{(x,y)|}^{4}d\varrho \le {\zeta }_2(\tilde{T}){(1+|x|}^4+{|y|}^4),$$

$$\frac{1}{\tilde{T}}{\int }_0^{\tilde{T}}{\left({\int }_{|v|<c}{|H(\varrho ,x,y,v)-\overline{H}(x,y,v)|}^2\lambda (dv)\right)}^{2}d\varrho \le {\zeta }_3(\tilde{T}){(1+|x|}^4+{|y|}^4),$$

where ${\displaystyle \underset{\tilde{T}\to \infty }{lim}{\zeta }_i(\tilde{T})=0,i=1,2,3.}$

Next, we confirm that as $ϵ$ tends to 0, the solution $x_{ϵ}(\ell )$ of the original Equation (3) converges to the solution $z_{ϵ}(\ell )$, where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill z_{ϵ}(\ell ):& {\displaystyle ={\mathcal{S}}_{\gamma ,\beta }(\ell ){\phi }_0+ϵ{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho )\overline{f}(z_{ϵ}(\varrho ),z_{ϵ,\varrho })d\varrho }\hfill \\ \hfill & {\displaystyle +\sqrt{ϵ}{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho )\overline{h}(z_{ϵ}(\varrho ),z_{ϵ,\varrho })dW(\varrho )}\hfill \\ \hfill & {\displaystyle +\sqrt{ϵ}{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho ){\int }_{|v|<c}\overline{H}(z_{ϵ}(\varrho ),z_{ϵ,\varrho },v)\tilde{N}(d\varrho ,dv),\ell \in J,}\hfill \end{array}\end{array}$$

(7)

is the solution of the averaged equation

$$\left\{\begin{array}{c}\hfill \begin{array}{cc}\hfill & D_{0^{+}}^{\gamma ,\beta }{z}_{ϵ}(\ell )=A{z}_{ϵ}(\ell )+ϵ\overline{f}(z_{ϵ}(\ell ),z_{ϵ,\ell })+\sqrt{ϵ}\overline{h}(z_{ϵ}(\ell ),z_{ϵ,\ell })\frac{dW(\ell )}{d\ell }\hfill \\ & {\displaystyle +\sqrt{ϵ}\frac{1}{d\ell }{\int }_{|v|<c}\overline{H}(z_{ϵ}(\ell ),z_{ϵ,\ell },v)\tilde{N}(d\ell ,dv),\ell \in J,}\hfill \\ \hfill & z_{ϵ}(\ell )=\phi (\ell ),-\varsigma \le \ell \le 0,\hfill \\ \hfill & I_{0^{+}}^{(1-\gamma )(1-\beta )}{z}_{ϵ}(0)={\phi }_0.\hfill \end{array}\end{array}\right.$$

(8)

Now comes the main result of this paper.

Theorem 2. 

Assume $(H0)$–$(H4)$ hold, then for a given arbitrary small number $\rho >0$, there exist constants $M_0>0,{ϵ}_1\in (0,{ϵ}_0]$ and $\delta \in (0,1)$, such that for all $ϵ\in (0,{ϵ}_1],\beta \in (\frac{3}{4},1)$,

$$E(\underset{\ell \in [-\varsigma ,M_0{ϵ}^{-\delta }]}{sup}|x_{ϵ}(\ell )-z_{ϵ}(\ell ){|}^2)\le \rho .$$

Proof. 

From Equations (6) and (7), for $\ell \in J$, we obtain

$$\begin{array}{cc}\hfill & x_{ϵ}(\ell )-z_{ϵ}(\ell )\hfill \\ \hfill & {\displaystyle =ϵ{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho )[f(\varrho ,x_{ϵ}(\varrho ),x_{ϵ,\varrho })-\overline{f}(z_{ϵ}(\varrho ),z_{ϵ,\varrho })]d\varrho }\hfill \\ \hfill & {\displaystyle +\sqrt{ϵ}{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho )[h(\varrho ,x_{ϵ}(\varrho ),x_{ϵ,\varrho })-\overline{h}(z_{ϵ}(\varrho ),z_{ϵ,\varrho })]dW(\varrho )}\hfill \\ \hfill & {\displaystyle +\sqrt{ϵ}{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}{\mathcal{T}}_{\beta }(\ell -\varrho ){\int }_{|v|<c}[H(\varrho ,x_{ϵ}(\varrho ),x_{ϵ,\varrho },v)-\overline{H}(z_{ϵ}(\varrho ),z_{ϵ,\varrho },v)]\tilde{N}(d\varrho ,dv).}\hfill \end{array}$$

(9)

Taking the mathematical expectation of Equation (9), for any $\ell \in (0,b]\subset (0,T]$, by Lemma 1 and elementary inequality, we have

$$\begin{array}{cc}\hfill & E(\underset{0<\ell \le b}{sup}|x_{ϵ}(\ell )-z_{ϵ}(\ell ){|}^2)\hfill \\ \hfill & {\displaystyle \le 3{(\frac{M}{\mathrm{\Gamma }(\beta )})}^2{ϵ}^{2}E\left(\underset{0<\ell \le b}{sup}|{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}[f(\varrho ,x_{ϵ}(\varrho ),x_{ϵ,\varrho })-\overline{f}(z_{ϵ}(\varrho ),z_{ϵ,\varrho })]d\varrho {|}^2\right)}\hfill \\ \hfill & {\displaystyle +3{(\frac{M}{\mathrm{\Gamma }(\beta )})}^2ϵE\left(\underset{0<\ell \le b}{sup}|{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}[h(\varrho ,x_{ϵ}(\varrho ),x_{ϵ,\varrho })-\overline{h}(z_{ϵ}(\varrho ),z_{ϵ,\varrho })]dW(\varrho ){|}^2\right)}\hfill \\ \hfill & {\displaystyle +3{(\frac{M}{\mathrm{\Gamma }(\beta )})}^2ϵE(\underset{0<\ell \le b}{sup}|{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}}\hfill \\ \hfill & {\displaystyle \times {\int }_{|v|<c}[H(\varrho ,x_{ϵ}(\varrho ),x_{ϵ,\varrho },v)-\overline{H}(z_{ϵ}(\varrho ),z_{ϵ,\varrho },v)]\tilde{N}(d\varrho ,dv){|}^2)}\hfill \\ \hfill & :=I_1+I_2+I_3.\hfill \end{array}$$

(10)

We now calculate each term of Equation (10) separately. For the term $I_1$, we obtain the following estimation:

$$\begin{array}{cc}\hfill I_1& {\displaystyle \le 6{(\frac{M}{\mathrm{\Gamma }(\beta )})}^2{ϵ}^{2}E\left(\underset{0<\ell \le b}{sup}|{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}[f(\varrho ,x_{ϵ}(\varrho ),x_{ϵ,\varrho })-f(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho })]d\varrho {|}^2\right)}\hfill \\ \hfill & {\displaystyle +6{(\frac{M}{\mathrm{\Gamma }(\beta )})}^2{ϵ}^{2}E\left(\underset{0<\ell \le b}{sup}|{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}[f(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho })-\overline{f}(z_{ϵ}(\varrho ),z_{ϵ,\varrho })]d\varrho {|}^2\right)}\hfill \\ \hfill & :=I_{11}+I_{12}.\hfill \end{array}$$

(11)

Adopting the Cauchy–Schwarz inequality and (H3)–(H4), one can obtain

$$\begin{array}{cc}\hfill I_{11}& {\displaystyle \le {\wp }_{11}{ϵ}^{2}b{\int }_0^b{(b-\varrho )}^{2(\beta -1)}}\hfill \\ \hfill & {\displaystyle \times \mathcal{M}\left(\varrho ,E(\underset{0<{\varrho }_1\le \varrho }{sup}|x_{ϵ}({\varrho }_1)-z_{ϵ}({\varrho }_1){|}^2)+E(\underset{0<{\varrho }_1\le \varrho }{sup}|x_{ϵ,{\varrho }_1}-z_{ϵ,{\varrho }_1}{|}^2)\right)d\varrho ,}\hfill \end{array}$$

(12)

where ${\displaystyle {\wp }_{11}=\frac{6M^2}{{(\mathrm{\Gamma }(\beta ))}^2}}$.

And

$$\begin{array}{cc}\hfill I_{12}& {\displaystyle \le \frac{6M^2{ϵ}^2{b}^{2\beta -1}}{(2\beta -1){(\mathrm{\Gamma }(\beta ))}^2}E\left(\underset{0<\ell \le b}{sup}\ell \frac{1}{\ell }{\int }_0^{\ell }{|f(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho })-\overline{f}(z_{ϵ}(\varrho ),z_{ϵ,\varrho })|}^{2}d\varrho \right)}\hfill \\ & {\displaystyle \le {\wp }_{12}{ϵ}^2{b}^{2\beta },}\hfill \end{array}$$

(13)

where ${\displaystyle {\wp }_{12}=\frac{6M^2}{(2\beta -1){(\mathrm{\Gamma }(\beta ))}^2}\underset{0<\ell \le b}{sup}{\zeta }_1(\ell )\left(1+E(\underset{0<\ell \le b}{sup}|z_{ϵ}{(\ell )|}^2)+E(\underset{0<\ell \le b}{sup}|z_{ϵ,\ell }{|}^2)\right).}$

For the second term $I_2$, we have

$$\begin{array}{cc}\hfill & {\displaystyle I_2\le 6{(\frac{M}{\mathrm{\Gamma }(\beta )})}^2ϵE\left(\underset{0<\ell \le b}{sup}|{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}[h(\varrho ,x_{ϵ}(\varrho ),x_{ϵ,\varrho })-h(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho })]dW(\varrho ){|}^2\right)}\hfill \\ \hfill & {\displaystyle +6{(\frac{M}{\mathrm{\Gamma }(\beta )})}^2ϵE\left(\underset{0<\ell \le b}{sup}|{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}[h(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho })-\overline{h}(z_{ϵ}(\varrho ),z_{ϵ,\varrho })]dW(\varrho ){|}^2\right)}\hfill \\ \hfill & :=I_{21}+I_{22}.\hfill \end{array}$$

(14)

In view of the B-D-G inequality and Hypothesis (H3), one can obtain

$$\begin{array}{cc}\hfill I_{21}& {\displaystyle \le {\wp }_{21}ϵ{\int }_0^b{(b-\varrho )}^{2(\beta -1)}}\hfill \\ \hfill & {\displaystyle \times \mathcal{M}(\varrho ,E(\underset{0<{\varrho }_1\le \varrho }{sup}|x_{ϵ}({\varrho }_1)-z_{ϵ}({\varrho }_1){|}^2)+E(\underset{0<{\varrho }_1\le \varrho }{sup}|x_{ϵ,{\varrho }_1}-z_{ϵ,{\varrho }_1}{|}^2))d\varrho ,}\hfill \end{array}$$

(15)

where ${\displaystyle {\wp }_{21}=\frac{24M^2}{{(\mathrm{\Gamma }(\beta ))}^2}}$.

By Hypothesis (H4) and the Hölder inequality, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle I_{22}\le \frac{24M^2ϵ}{{(\mathrm{\Gamma }(\beta ))}^2}E\left(\underset{0<\ell \le b}{sup}{\int }_0^{\ell }{(\ell -\varrho )}^{2(\beta -1)}{|h(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho })-\overline{h}(z_{ϵ}(\varrho ),z_{ϵ,\varrho })|}^{2}d\varrho \right)}\hfill \\ \hfill & {\displaystyle \le \frac{24M^2ϵ}{{(\mathrm{\Gamma }(\beta ))}^2}({\int }_0^b{(b-\varrho )}^{4(\beta -1)}d\varrho {)}^{\frac{1}{2}}}\hfill \\ & \times E{\left(\underset{0<\ell \le b}{sup}{\int }_0^{\ell }{|h(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho })-\overline{h}(z_{ϵ}(\varrho ),z_{ϵ,\varrho })|}^{4}d\varrho \right)}^{\frac{1}{2}}\hfill \\ \hfill & {\displaystyle \le \frac{24M^2\sqrt{4\beta -3}ϵb^{2\beta -\frac{3}{2}}}{(4\beta -3){(\mathrm{\Gamma }(\beta ))}^2}E{\left(\underset{0<\ell \le b}{sup}\ell \frac{1}{\ell }{\int }_0^{\ell }{|h(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho })-\overline{h}(z_{ϵ}(\varrho ),z_{ϵ,\varrho })|}^{4}d\varrho \right)}^{\frac{1}{2}}}\hfill \\ & {\displaystyle \le \frac{24M^2\sqrt{4\beta -3}ϵb^{2\beta -1}}{(4\beta -3){(\mathrm{\Gamma }(\beta ))}^2}{\left[\underset{0<\ell \le b}{sup}{\zeta }_2(\ell )(1+E(\underset{0<\ell \le b}{sup}|z_{ϵ}{(\ell )|}^4)+E(\underset{0<\ell \le b}{sup}|z_{ϵ,\ell }{|}^4))\right]}^{\frac{1}{2}}}\hfill \\ \hfill & {\displaystyle \le {\wp }_{22}ϵb^{2\beta -1},}\hfill \end{array}$$

(16)

where ${\displaystyle {\wp }_{22}=\frac{24M^2\sqrt{4\beta -3}}{(4\beta -3){(\mathrm{\Gamma }(\beta ))}^2}{\left[\underset{0<\ell \le b}{sup}{\zeta }_2(\ell )(1+E(\underset{0<\ell \le b}{sup}|z_{ϵ}{(\ell )|}^4)+E(\underset{0<\ell \le b}{sup}|z_{ϵ,\ell }{|}^4))\right]}^{\frac{1}{2}}.}$

For the last term, we have

$$\begin{array}{cc}\hfill & {\displaystyle I_3\le 6{(\frac{M}{\mathrm{\Gamma }(\beta )})}^2ϵE(\underset{0<\ell \le b}{sup}|{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}}\hfill \\ \hfill & {\displaystyle \times {\int }_{|v|<c}[H(\varrho ,x_{ϵ}(\varrho ),x_{ϵ,\varrho },v)-H(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho },v)]\tilde{N}(d\varrho ,dv){|}^2)}\hfill \\ \hfill & {\displaystyle +6{(\frac{M}{\mathrm{\Gamma }(\beta )})}^2ϵE(\underset{0<\ell \le b}{sup}|{\int }_0^{\ell }{(\ell -\varrho )}^{\beta -1}}\hfill \\ \hfill & {\displaystyle \times {\int }_{|v|<c}[H(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho },v)-\overline{H}(z_{ϵ}(\varrho ),z_{ϵ,\varrho },v)]\tilde{N}(d\varrho ,dv){|}^2)}\hfill \\ \hfill & {\displaystyle :=I_{31}+I_{32}.}\hfill \end{array}$$

(17)

From the B-D-G inequality and Hypothesis (H3), we have

$$\begin{array}{cc}\hfill I_{31}& {\displaystyle \le {\wp }_{31}ϵ{\int }_0^b{(b-\varrho )}^{2(\beta -1)}}\hfill \\ \hfill & {\displaystyle \times \mathcal{M}(\varrho ,E(\underset{0<{\varrho }_1\le \varrho }{sup}|x_{ϵ}({\varrho }_1)-z_{ϵ}({\varrho }_1){|}^2)+E(\underset{0<{\varrho }_1\le \varrho }{sup}|x_{ϵ,{\varrho }_1}-z_{ϵ,{\varrho }_1}{|}^2))d\varrho ,}\hfill \end{array}$$

(18)

where ${\displaystyle {\wp }_{31}=\frac{24M^2}{{(\mathrm{\Gamma }(\beta ))}^2}}$.

By a similar argument as $I_{22}$, from Hypothesis (H4), we have

$$\begin{array}{cc}\hfill & {\displaystyle I_{32}\le \frac{24M^2ϵ}{{(\mathrm{\Gamma }(\beta ))}^2}E(\underset{0<\ell \le b}{sup}{\int }_0^{\ell }{(\ell -\varrho )}^{2(\beta -1)}}\hfill \\ & \times {\int }_{|v|<c}{|H(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho },v)-\overline{H}(z_{ϵ}(\varrho ),z_{ϵ,\varrho },v)|}^2\lambda (dv)d\varrho )\hfill \\ \hfill & {\displaystyle \le \frac{24M^2ϵ}{{(\mathrm{\Gamma }(\beta ))}^2}({\int }_0^b{(b-\varrho )}^{4(\beta -1)}d\varrho {)}^{\frac{1}{2}}}\hfill \\ & \times E{\left(\underset{0<\ell \le b}{sup}{\int }_0^{\ell }{\left({\int }_{|v|<c}{|H(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho },v)-\overline{H}(z_{ϵ}(\varrho ),z_{ϵ,\varrho },v)|}^2\lambda (dv)\right)}^{2}d\varrho \right)}^{\frac{1}{2}}\hfill \\ & {\displaystyle \le \frac{24M^2\sqrt{4\beta -3}ϵb^{2\beta -\frac{3}{2}}}{(4\beta -3){(\mathrm{\Gamma }(\beta ))}^2}}\hfill \\ & \times E{\left(\underset{0<\ell \le b}{sup}\ell \frac{1}{\ell }{\int }_0^{\ell }{\left({\int }_{|v|<c}{|H(\varrho ,z_{ϵ}(\varrho ),z_{ϵ,\varrho },v)-\overline{H}(z_{ϵ}(\varrho ),z_{ϵ,\varrho },v)|}^2\lambda (dv)\right)}^{2}d\varrho \right)}^{\frac{1}{2}}\hfill \\ & {\displaystyle \le \frac{24M^2\sqrt{4\beta -3}ϵb^{2\beta -1}}{(4\beta -3){(\mathrm{\Gamma }(\beta ))}^2}{\left[\underset{0<\ell \le b}{sup}{\zeta }_3(\ell )(1+E(\underset{0<\ell \le b}{sup}|z_{ϵ}{(\ell )|}^4)+E(\underset{0<\ell \le b}{sup}|z_{ϵ,\ell }{|}^4))\right]}^{\frac{1}{2}}}\hfill \\ & {\displaystyle \le {\wp }_{32}ϵb^{2\beta -1},}\hfill \end{array}$$

(19)

where ${\displaystyle {\wp }_{32}=\frac{24M^2\sqrt{4\beta -3}}{(4\beta -3){(\mathrm{\Gamma }(\beta ))}^2}{\left[\underset{0<\ell \le b}{sup}{\zeta }_3(\ell )(1+E(\underset{0<\ell \le b}{sup}|z_{ϵ}{(\ell )|}^4)+E(\underset{0<\ell \le b}{sup}|z_{ϵ,\ell }{|}^4))\right]}^{\frac{1}{2}}.}$

From the concavity of $\mathcal{M}(\ell ,x)$, there are two functions: $p(\ell )>0$ and $q(\ell )>0$, such that

$$\mathcal{M}(\ell ,x)\le p(\ell )+q(\ell )x,{\int }_0^{b}p(\ell )d\ell <\infty ,{\int }_0^{b}q(\ell )d\ell <\infty .$$

Plugging Equations (11)–(19) into Equation (10), one has

$$\begin{array}{cc}\hfill & E(\underset{0<\ell \le b}{sup}|x_{ϵ}(\ell )-z_{ϵ}(\ell ){|}^2)\hfill \\ \hfill & {\displaystyle \le \frac{{\wp }_{11}{ϵ}^2{b}^{2\beta }+({\wp }_{21}+{\wp }_{31})ϵb^{2\beta -1}}{2\beta -1}\left(\underset{0<\ell \le b}{sup}p(\ell )\right)}\hfill \\ \hfill & {\displaystyle +({\wp }_{11}{ϵ}^{2}b+({\wp }_{21}+{\wp }_{31})ϵ)\left(\underset{0<\ell \le b}{sup}q(\ell )\right){\int }_0^b{(b-\varrho )}^{2(\beta -1)}}\hfill \\ & {\displaystyle \times \left(E(\underset{0<{\varrho }_1\le \varrho }{sup}|x_{ϵ}({\varrho }_1)-z_{ϵ}({\varrho }_1){|}^2)+E(\underset{0<{\varrho }_1\le \varrho }{sup}|x_{ϵ,{\varrho }_1}-z_{ϵ,{\varrho }_1}{|}^2)\right)d\varrho }\hfill \\ \hfill & {\displaystyle +({\wp }_{12}{ϵ}^2{b}^{2\beta }+({\wp }_{22}+{\wp }_{32})ϵb^{2\beta -1}).}\hfill \end{array}$$

(20)

Let $\mathrm{\Theta }(b)=E(\underset{0<\ell \le b}{sup}|x_{ϵ}(\ell )-z_{ϵ}(\ell ){|}^2)$, from the fact that $E(\underset{-\varsigma \le \ell <0}{sup}|x_{ϵ}(\ell )-z_{ϵ}(\ell ){|}^2)=0,$ then we have

$$\begin{array}{c}\hfill E(\underset{0<{\varrho }_1\le \varrho }{sup}|x_{ϵ,{\varrho }_1}-z_{ϵ,{\varrho }_1}{|}^2)=\mathrm{\Theta }(\varrho -\varphi ),0<\varphi \le \varsigma .\end{array}$$

(21)

Therefore,

$$\begin{array}{cc}\hfill \mathrm{\Theta }(b)& {\displaystyle \le \frac{{\wp }_{11}{ϵ}^2{b}^{2\beta }+({\wp }_{21}+{\wp }_{31})ϵb^{2\beta -1}}{2\beta -1}\left(\underset{0<\ell \le b}{sup}p(\ell )\right)}\hfill \\ & {\displaystyle +({\wp }_{11}{ϵ}^{2}b+({\wp }_{21}+{\wp }_{31})ϵ)\left(\underset{0<\ell \le b}{sup}q(\ell )\right){\int }_0^b{(b-\varrho )}^{2(\beta -1)}\left(\mathrm{\Theta }(\varrho )+\mathrm{\Theta }(\varrho -\varphi )\right)d\varrho }\hfill \\ & {\displaystyle +({\wp }_{12}{ϵ}^2{b}^{2\beta }+({\wp }_{22}+{\wp }_{32})ϵb^{2\beta -1}).}\hfill \end{array}$$

(22)

For all $b\in (0,T]$, setting $\Delta (b)=\underset{-\varsigma \le \ell \le b}{sup}\mathrm{\Theta }(\ell )$, then $\mathrm{\Theta }(\varrho )\le \Delta (\varrho )$ and $\mathrm{\Theta }(\varrho -\varphi )\le \Delta (\varrho ),0<\varphi \le \varsigma$. Hence, we have

$$\begin{array}{cc}\hfill & {\displaystyle \mathrm{\Theta }(b)\le \frac{{\wp }_{11}{ϵ}^2{b}^{2\beta }+({\wp }_{21}+{\wp }_{31})ϵb^{2\beta -1}}{2\beta -1}\left(\underset{0<\ell \le b}{sup}p(\ell )\right)}\hfill \\ \hfill & {\displaystyle +({\wp }_{11}{ϵ}^{2}b+({\wp }_{21}+{\wp }_{31})ϵ)\left(\underset{0<\ell \le b}{sup}q(\ell )\right){\int }_0^b{(b-\varrho )}^{2(\beta -1)}2\Delta (\varrho )d\varrho }\hfill \\ \hfill & {\displaystyle +({\wp }_{12}{ϵ}^2{b}^{2\beta }+({\wp }_{22}+{\wp }_{32})ϵb^{2\beta -1}).}\hfill \end{array}$$

(23)

Thus, the calculations above lead to

$$\begin{array}{cc}\hfill & {\displaystyle \Delta (b)=\underset{-\varsigma \le \ell \le b}{sup}\mathrm{\Theta }(\ell )}\hfill \\ & {\displaystyle \le max\left\{\underset{-\varsigma \le \ell \le 0}{sup}\mathrm{\Theta }(\ell )+\underset{0\le \ell \le b}{sup}\mathrm{\Theta }(\ell )\right\}}\hfill \\ \hfill & {\displaystyle \le [({\Re }_1+{\wp }_{12}){ϵ}^2{b}^{2\beta }+({\Re }_2+{\wp }_{22}+{\wp }_{32})ϵb^{2\beta -1}]}\hfill \\ & {\displaystyle +2({\Re }_3{ϵ}^{2}b+{\Re }_4ϵ){\int }_0^b{(b-\varrho )}^{(2\beta -1)-1}\Delta (\varrho )d\varrho ,}\hfill \end{array}$$

(24)

where ${\Re }_1=\frac{{\wp }_{11}}{2\beta -1}\left(\underset{0<\ell \le b}{sup}p(\ell )\right),{\Re }_2=\frac{{\wp }_{21}+{\wp }_{31}}{2\beta -1}\left(\underset{0<\ell \le b}{sup}p(\ell )\right),{\Re }_3={\wp }_{11}\left(\underset{0<\ell \le b}{sup}q(\ell )\right),{\Re }_4=({\wp }_{21}+{\wp }_{31})\left(\underset{0<\ell \le b}{sup}q(\ell )\right).$

Moreover, $({\Re }_1+{\wp }_{12}){ϵ}^2{b}^{2\beta }+({\Re }_2+{\wp }_{22}+{\wp }_{32})ϵb^{2\beta -1}$ is nondecreasing on J, by Lemma 3, it yields that

$$\begin{array}{cc}\hfill & {\displaystyle \Delta (b)\le [({\Re }_1+{\wp }_{12}){ϵ}^2{b}^{2\beta }+({\Re }_2+{\wp }_{22}+{\wp }_{32})ϵb^{2\beta -1}]}\hfill \\ & {\displaystyle \times \sum _{n=0}^{\infty }\frac{{\left[2({\Re }_3{ϵ}^2{b}^{2\beta }+{\Re }_4ϵb^{2\beta -1})\mathrm{\Gamma }(2\beta -1)\right]}^n}{\mathrm{\Gamma }(n(2\beta -1)+1)}.}\hfill \end{array}$$

(25)

Thus,

$$\begin{array}{cc}\hfill & {\displaystyle E(\underset{-\varsigma \le \ell \le b}{sup}|x_{ϵ}(\ell )-z_{ϵ}(\ell ){|}^2)\le [({\Re }_1+{\wp }_{12}){ϵ}^2{b}^{2\beta }+({\Re }_2+{\wp }_{22}+{\wp }_{32})ϵb^{2\beta -1}]}\hfill \\ & {\displaystyle \times \sum _{n=0}^{\infty }\frac{{\left[2({\Re }_3{ϵ}^2{b}^{2\beta }+{\Re }_4ϵb^{2\beta -1})\mathrm{\Gamma }(2\beta -1)\right]}^n}{\mathrm{\Gamma }(n(2\beta -1)+1)}.}\hfill \end{array}$$

(26)

The estimation (26) enables us to claim that there exist $M_0>0$ and $\delta \in (0,1)$, such that

$$\begin{array}{cc}\hfill & E(\underset{-\varsigma \le \ell \le M_0{ϵ}^{-\delta }}{sup}|x_{ϵ}(\ell )-z_{ϵ}(\ell ){|}^2)\le \mu {ϵ}^{1-\delta }\hfill \end{array}$$

(27)

holds for all $\ell \in (0,M_0{ϵ}^{-\delta }]\subset (0,T]$, where

$$\begin{array}{cc}\hfill & {\displaystyle \mu =[({\Re }_1+{\wp }_{12})M_0^{2\beta }{ϵ}^{1+\delta -2\delta \beta }+({\Re }_2+{\wp }_{22}+{\wp }_{32})M_0^{2\beta -1}{ϵ}^{2\delta (1-\beta )}]}\hfill \\ & {\displaystyle \times \sum _{n=0}^{\infty }\frac{{\left[2({\Re }_3{M}_0^{2\beta }{ϵ}^{2(1-\delta \beta )}+{\Re }_4{M}_0^{2\beta -1}{ϵ}^{1+\delta -2\delta \beta })\mathrm{\Gamma }(2\beta -1)\right]}^n}{\mathrm{\Gamma }(n(2\beta -1)+1)}}\hfill \end{array}$$

(28)

is a constant.

Based on the above analysis, for an arbitrarily given number $\rho$, there exists ${ϵ}_1\in (0,{ϵ}_0]$, such that

$$E(\underset{\ell \in [-\varsigma ,M_0{ϵ}^{-\delta }]}{sup}|x_{ϵ}(\ell )-z_{ϵ}(\ell ){|}^2)\le \rho ,$$

for $\beta \in (\frac{3}{4},1),\forall ϵ\in (0,{ϵ}_1]$ and $\forall \ell \in [-\varsigma ,M_0{ϵ}^{-\delta }]$. This completes the proof. □

Corollary 1. 

Assume that $(H0)$–$(H4)$ hold, then for any number $\rho >0$, there exists ${ϵ}_1\in (0,{ϵ}_0)$ such that for all $ϵ\in (0,{ϵ}_1),\beta \in (\frac{3}{4},1)$,

$$\underset{ϵ\to 0}{lim}P(\underset{\ell \in [-\varsigma ,M_0{ϵ}^{-\delta }]}{sup}|x_{ϵ}(\ell )-z_{ϵ}(\ell )|>\rho )=0.$$

Proof. 

On the basis of the Chebyshev–Markov inequality, for an arbitrary given number $\rho >0$, we have

$$P(\underset{\ell \in [-\varsigma ,M_0{ϵ}^{-\delta }]}{sup}|x_{ϵ}(\ell )-z_{ϵ}(\ell )|>\rho )\le \frac{1}{{\rho }^2}E(\underset{\ell \in [-\varsigma ,M_0{ϵ}^{-\delta }]}{sup}|x_{ϵ}(\ell )-z_{ϵ}(\ell ){|}^2)\le \frac{\mu {ϵ}^{1-\delta }}{{\rho }^2},$$

when $ϵ$ tends to 0, the required assertion is true. □

4. Example

Consider the Hilfer fractional stochastic evolution equation below:

$$\left\{\begin{array}{cc}\hfill & D^{\frac{3}{4},\frac{7}{8}}{x}_{ϵ}(\ell ,\hslash )=A{x}_{ϵ}(\ell ,\hslash )+ϵ\vartheta x_{ϵ}(\ell ,\hslash ){cos}^2(\ell )+\sqrt{ϵ}\varpi {sin}^2(\ell )x_{ϵ}(\ell ,\hslash )\frac{d\omega (\ell )}{d\ell }\hfill \\ & +\sqrt{ϵ}\frac{1}{d\ell }{\int }_{|v|<c}2v^4{sin}^2(\ell )x_{ϵ}(\ell ,\hslash )\tilde{N}(d\ell ,dv),\hslash \in [0,\pi ],\ell \in (0,\pi ],\hfill \\ \hfill & x_{ϵ}(\ell ,0)=x_{ϵ}(\ell ,\pi )=0,\hfill \\ \hfill & x_{ϵ}(\theta ,\hslash )=\phi (\theta ,\hslash ),\theta \in [-\varsigma ,0],\hfill \\ \hfill & I_{0^{+}}^{\frac{1}{32}}{x}_{ϵ}(0)={\phi }_0.\hfill \end{array}\right.$$

(29)

In the above, $T>0$, $\vartheta ,\varpi$ are constants, we opt for the space $\mathbb{H}=L^2([0,\pi ])$. We define an operator A by $Ax=x^{\prime \prime }$ with the domain $D(A)=\{x\in \mathbb{H}:x,x^{\prime }$ being absolutely continuous, $x^{\prime \prime }\in \mathbb{H},x(0)=x(\pi )=0\}$. Then A generates a strongly continuous semigroup ${\left\{S(\ell )\right\}}_{\ell \ge 0}$ which is compact, analytic, and self-adjoint. Additionally, A has a discrete spectrum, the eigenvalues are $-n^2,n\in \mathbb{N}$, with corresponding orthogonal eigenvectors $e_n(z)=\sqrt{\frac{2}{\pi }}sin(nz)$. Then for each $x\in \mathbb{H}$, $Ax={\displaystyle \sum _{n=1}^{\infty }}{n}^2\langle x,e_n\rangle e_n$ and $S(\ell )x={\displaystyle \sum _{n=1}^{\infty }}{e}^{-n^2\ell }\langle x,e_n\rangle e_n.$ In particular, $S(·)$ is a uniformly stable semigroup and $\parallel S(\ell )\parallel \le e^{-\ell }.$ For each $x\in \mathbb{H},A^{-\frac{1}{2}}x={\displaystyle \sum _{n=1}^{\infty }}\frac{1}{n}\langle x,e_n\rangle e_n$ and $\parallel A^{-\frac{1}{2}}\parallel =1$. The operator $A^{\frac{1}{2}}$ is presented by $A^{\frac{1}{2}}x={\displaystyle \sum _{n=1}^{\infty }}n\langle x,e_n\rangle e_n$ on the space $D(A^{\frac{1}{2}})=\{x(·)\in \mathbb{H},\sum _{n=1}^{\infty }n\langle x,e_n\rangle e_n\in \mathbb{H}\}$. $\omega (\ell )$ is a standard Brownian motion defined on the filtered probability space $(\mathsf{\Omega },\mathcal{F},P)$.

Take $T_1=\pi$ and let

$$\overline{f}(x_{ϵ}(\ell ),x_{ϵ,\ell })=\frac{1}{\pi }{\int }_0^{\pi }f(s,x_{ϵ}(s),x_{ϵ,s})ds=\frac{\vartheta }{2}{x}_{ϵ},$$

$$\overline{h}(x_{ϵ}(\ell ),x_{ϵ,\ell })=\frac{1}{\pi }{\int }_0^{\pi }h(s,x_{ϵ}(s),x_{ϵ,s})ds=\frac{\varpi }{2}{x}_{ϵ},$$

$$\overline{H}(x_{ϵ}(\ell ),x_{ϵ,\ell },v)=\frac{1}{\pi }{\int }_0^{\pi }H(s,x_{ϵ}(s),x_{ϵ,s},v)ds=v^4{x}_{ϵ}.$$

Then, one can easily verify that all conditions (H0)–(H4) in Theorem 2 are fulfilled. Hence, the averaged equation for Equation (29) can be expressed as

$$\left\{\begin{array}{cc}\hfill & D^{\frac{3}{4},\frac{7}{8}}{z}_{ϵ}(\ell ,\hslash )=A{z}_{ϵ}(\ell ,\hslash )+\frac{\vartheta }{2}ϵz_{ϵ}(\ell ,\hslash )+\frac{\varpi }{2}\sqrt{ϵ}{z}_{ϵ}(\ell ,\hslash )\frac{d\omega (\ell )}{d\ell }\hfill \\ & +\sqrt{ϵ}\frac{1}{d\ell }{\int }_{|v|<c}{v}^4{z}_{ϵ}(\ell ,\hslash )\tilde{N}(d\ell ,dv),\hslash \in [0,\pi ],\ell \in (0,\pi ],\hfill \\ \hfill & z_{ϵ}(\ell ,0)=z_{ϵ}(\ell ,\pi )=0,\hfill \\ \hfill & z_{ϵ}(\theta ,\hslash )=\phi (\theta ,\hslash ),\theta \in [-\varsigma ,0],\hfill \\ \hfill & I_{0^{+}}^{\frac{1}{32}}{z}_{ϵ}(0)={\phi }_0.\hfill \end{array}\right.$$

(30)

Clearly, compared with the original Equation (29), the time-averaged Equation (30) is a much simpler equation. Furthermore, Theorem 2 ensures that their solutions possess a very small error.

5. Conclusions

This work derived an averaging principle for a class of Hilfer fractional stochastic evolution equations with Lévy noise. Compared with previous work, we take the Hilfer fractional derivative, evolution equations, and Lévy noise into account simultaneously. Thus, our proposed results extend the stochastic averaging method to Hilfer fractional differential equations. Moreover, since the Wiener process and fractional Brownian motion are very common in reality, it is significantly important for future research to investigate the averaging principle for HFSEEs driven by both the Wiener process and fractional Brownian motion (fBm). Moreover, it is possible for us to investigate the averaging principle of Hilfer fractional impulsive stochastic evolution equations driven by time-changed Lévy noise in forthcoming research.