Effects of Lévy noise and impulsive action on the averaging principle of Atangana–Baleanu fractional stochastic delay differential equations

Abstract

As delays are common, persistent, and ingrained in daily life, it is imperative to take them into account. In this work, we explore the averaging principle for impulsive Atangana–Baleanu fractional stochastic delay differential equations driven by Lévy noise. The link between the averaged equation solutions and the equivalent solutions of the original equations is shown in the sense of mean square. To achieve the intended outcomes, fractional calculus, semigroup properties, and stochastic analysis theory are used. We also provide an example to demonstrate the practicality and relevance of our research.

1 Introduction

Fractional calculus offers a robust framework for studying systems that display memory and hereditary characteristics beyond what integer-order calculus can handle. Its applications extend across various fields, providing enhanced understanding and more precise models for complex phenomena in both natural and engineered systems. As research progresses, fractional calculus is anticipated to become increasingly vital in tackling challenges across diverse fields [1–4].

Fractional stochastic differential equations (SDEs) combine the principles of fractional calculus with stochastic processes, representing an advanced field of study. They are versatile for modeling systems where memory and randomness are significant factors, with applications across diverse fields including physics, biology, and economics. These equations provide deeper insights into complex system behaviors and improve our ability to predict and manage uncertainties in different domains. As research advances, stochastic fractional differential equations are expected to increasingly contribute to solving modern challenges and advancing scientific knowledge [5–13]). Furthermore, Gaussian noise (GN) is largely present in stochastic dynamics. However, pure GN is inadequate for mimicking some actual occurrences. These kinds of perturbations are already covered by non-Gaussian Lévy noise, which has the benefit of having an elongated tail distribution that causes spatial discontinuity in the sample route. Its importance and necessity cannot thus be overemphasized. There have been numerous studies on SDEs with Lévy noise, such as: Balasubramaniam in [14] discussed the Hilfer fractional stochastic system driven by mixed Brownian motion and Lévy noise suffered by noninstantaneous impulses. Xu et al. in [15] investigated the existence and uniqueness of solutions for SDEs driven by Lévy noise using the successive approximation approach.

Atangana and Baleanu proposed a fractional derivative known as the Atangana–Baleanu (AB) fractional derivative, for more details (see [16]). Many models were studied in the case of the AB-fractional derivative; for example, Khan et al. [17] proved the existence, uniqueness, and data dependence of solutions for an Atangana–Baleanu–Caputo (ABC)-fractional-order differential impulsive system. Mallika et al. [18] discussed AB-fractional Volterra–Fredholm integrodifferential inclusions with noninstantaneous impulses. Balasubramaniam in [19] derived the necessary and sufficient conditions for the controllability of Atangana–Baleanu–Caputo neutral fractional differential equations with noninstantaneous impulses. Balasubramaniam in [20] investigated the existence result for the first time in the literature for Atangana–Baleanu Riemann–Liouville fractional stochastic systems.

In many cases, stochastic fractional differential equations exhibit complex behaviors that are difficult to analyze directly. The averaging principle helps in simplifying the analysis by focusing on the averaged behavior over long time scales or large ensembles of stochastic processes. The averaging principle provides a systematic approach to analyze the long-term behavior of stochastic systems with fractional dynamics. It allows researchers to derive approximate solutions and understand the asymptotic behavior of such systems. Some authors studied the averaging principle for fractional stochastic differential equations; for example, Ahmed and Zhu in [21] were the first to examine the theory of the averaging principle for Hilfer fractional stochastic differential equations with delay and Poisson jumps. Luo et al. in [22] investigated the averaging principle of a stochastic Hilfer-type fractional system involving non-Lipschitz coefficients. Shen et al. in [23] discussed the averaging principle for neutral stochastic fractional-order differential equations with variable delays driven by Lévy noise.

Our goal in this study, which is motivated by the aforementioned debates, is to study an averaging principle for a class of impulsive Atangana–Baleanu fractional stochastic delay differential equations (SDDEs) driven by Lévy noise.

In the light of the above, let us take into consideration the impulsive Atangana–Baleanu–Caputo (ABC) fractional SDDE with Lévy noise of the form:

$$\begin{aligned} \textstyle\begin{cases} {}^{\mathfrak{ABC}}D_{0+}^{\iota} \zeta (\mu ) = \mathfrak{E} \zeta ( \mu )+\varrho (\mu ,\zeta (\mu ),\zeta (\mu +\gamma ))+\vartheta ( \mu ,\zeta (\mu ),\zeta (\mu +\gamma )) \frac{d\mathfrak{V}(\mu )}{d\mu}, \\ \quad \mu \in \mathfrak{J}:=(0, \rho ], \mu \neq \mu _{\kappa} \\ \Delta \zeta | _{\mu =\mu _{\kappa}}=\mathfrak{g}_{\kappa}(\zeta ( \mu _{\kappa}^{-})), \quad \mu =\mu _{\kappa}, \kappa =1,2, \ldots ,\mathcal{T} \\ \zeta (\mu )=\Psi (\mu ), \quad -\xi \leq \mu \leq 0, \\ \zeta (0)=\Psi _{0}, \end{cases}\displaystyle \end{aligned}$$

(1)

where ${}^{\mathfrak{ABC}}D_{0+}^{\iota}$ is an ABC-fractional derivative of order $\frac{1}{2} < \iota < 1$. $\zeta (\cdot )$ takes its values in $\mathcal{A}$, the separable Hilbert space endowed with $\langle \cdot ,\cdot \rangle$ and $\Vert \cdot \Vert$. Let $\mathcal{B}$ be another separable Hilbert space with inner product $\langle \cdot ,\cdot \rangle _{\mathcal{B}}$ and norm $\Vert \cdot \Vert _{\mathcal{B}}$. Let $\mathfrak{V}$ be a $\mathcal{B}$-valued Lévy process. $\Delta \zeta | _{\mu =\mu _{\kappa}}= \zeta (\mu _{\kappa}^{+})- \zeta (\mu _{\kappa}^{-})$, where $\zeta (\mu _{\kappa}^{+})$ and $\zeta (\mu _{\kappa}^{-})$ represent the right and left limits of ζ at $\mu =\mu _{\kappa}$. The history process is $\zeta (\mu +\gamma )$, $-\xi \leq \mu \leq 0$. $\mathfrak{E}$ is the linear operator on $\mathcal{A}$. The complete probability space is $(\Omega ,\mathcal{K}, \mathcal{C})$ with probability measure $\mathcal{C}$ on Ω and filtration $\{\mathcal{K}_{\mu}\}_{\mu \geq 0}$. Let $(\bigvee , V, \lambda (d\mathfrak{X}))$ be a σ-finite measurable space. The stationary Poisson point process $(P_{\mu} )_{\mu \geq 0}$ is defined on $(\Omega ,\mathcal{K}, \mathcal{C})$ with values in ⋁ and characteristic measure λ. $\tilde{M}(\mu ,d\mathfrak{X})$ is the counting measure of $P_{\mu}$ such that (s.t.) $\aleph (\mu ,l)= \mathbb{E}(\tilde{M}(\mu ,l))=\mu \lambda (l)$ for $l\in V$. $\aleph (\mu ,d\mathfrak{X}):=\tilde{M}(\mu ,d\mathfrak{X})-\mu \lambda (d\mathfrak{X})$ is the Poisson martingale measure generated by $P_{\mu}$. ϱ is $\mathcal{A}$-valued and ϑ is $\mathcal{L}(\mathcal{B},\mathcal{A})$-valued. The initial function is $\Psi =\{\Psi (\mu ), -\xi \leq \mu \leq 0\}$, satisfying $\mathbb{E}\{\sup_{-\xi \leq \mu \leq 0} \Vert \Psi (\mu ) \Vert ^{2}\}<\infty$.

The main contributions of this paper are summarized as follows:

• For the first time in the literature, impulsive Atangana–Baleanu fractional stochastic delay differential equations with a Lévy process have been considered.

• By using fractional calculus, stochastic analysis, and semigroup theory, the average principle for the considered system is studied.

• Finally, an example is given to illustrate the theoretical results.

2 Preliminaries

The following definitions and lemmas are necessary in order to analyze the suggested problem.

Definition 2.1

([16]) The Atangana–Baleanu fractional derivative in the Caputo sense of order $0< \iota <1$ is defined as:

$$ {}^{\mathfrak{ABC}}D_{\tau +}^{\iota}\jmath (\mu )= \frac{\varpi (\iota )}{1-\iota} \int _{\tau}^{\mu}\jmath ^{\prime }( \mathfrak{S}) \mathbb{W}_{\iota}\bigl(-\eta (\mu -\mathfrak{S})^{\iota}\bigr)\,d\mathfrak{S}, $$

(2)

where the function $\eta =\frac{\iota}{1-\iota}$, and

$$ \mathbb{W}_{\iota}(z)=\sum_{n=0}^{\infty} \frac{z^{n}}{\Gamma (n\iota +1)} $$

is the Mittag-Leffler function, and the normalization function $\varpi (\iota )=(1-\iota )+\frac{\iota}{\Gamma (\iota )}$ is any function with $\varpi (0)=\varpi (1)=1$. The fractional integral related to the Atangana–Baleanu derivative is provided by

$$ {}^{\mathfrak{AB}}I_{\tau +}^{\iota} \jmath (\mu )= \frac{1-\mu}{\varpi (\iota )}\jmath (\mu )+ \frac{\iota}{\varpi (\iota )\Gamma (\iota )} \int _{\tau}^{\mu}(\mu - \mathfrak{S})^{\iota -1} \jmath (\mathfrak{S})\,d\mathfrak{S,} $$

(3)

where $\{\mathfrak{W}(\mu )\}_{\mu \geq 0}$ stands for a $\mathcal{B}$-valued Θ-Brownian motion process specified on $(\Omega ,\mathcal{K}, \mathcal{C})$. Let $\breve{\mathcal{B}}=\Theta ^{1/2}\mathcal{B}$, $\mathcal{L}_{\Theta}^{2}(\breve{\mathcal{B}}, \mathcal{A})$ be a separable Hilbert space consisting of all Hilbert–Schmidt operators from $\breve{\mathcal{B}}$ to $\mathcal{A}$. However, The space $\mathbb{S}=PC([-\xi ,0]; \mathcal{L}^{2}(\Omega , \mathcal{A}))$ denotes the family of all right-continuous functions Ψ with left-hand limits from $[-\xi ,0]$ to $\mathcal{L}^{2}(\Omega , \mathcal{A})$, equipped with the norm $\Vert \Psi \Vert ^{2}=\sup_{-\xi <\mu \leq 0} \mathbb{E} \Vert \Psi (\mu ) \Vert ^{2}$, and the initial function $\Psi (\mu )$ is a $\mathcal{K}_{0}$-measurable PC-valued random variable satisfying $\mathbb{E} \Vert \Psi \Vert ^{2}<\infty$. Moreover, $\mathcal{L}^{2}(\Omega ,\mathcal{A})$ denotes the space of all strongly measurable, square integrable, $\mathcal{A}$-valued random variables; $\mathcal{L}^{2}(\Omega ,\mathcal{A})$ is a Banach space equipped with the norm $\Vert \zeta (\mu ) \Vert _{\mathcal{L}^{2}(\Omega , \mathcal{A})}= (\mathbb{E} \Vert \zeta (\mu ) \Vert ^{2} )^{1/2}$. The space $\tilde{\mathbb{S}}=PC([-\xi ,\rho ], \mathcal{L}^{2}(\Omega , \mathcal{A}))$ denotes the Banach spaces of all continuous functions from $[-\xi ,\rho ]$ into $\mathcal{L}^{2}(\Omega ,\mathcal{A})$, with norm $\Vert \zeta \Vert =\sup_{-\xi \leq \mu \leq \rho} ( \mathbb{E} \Vert \zeta (\mu ) \Vert ^{2} )^{1/2}$.

Lemma 2.1

[24] If Lévy motion is in $\mathcal{B}$, then the expression

$$ \mathfrak{V}(\mu )= \mathfrak{Z}\mu +\mathfrak{W}(\mu )+ \int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{X}\tilde{\aleph}(d\mu ,d \mathfrak{X})+ \int _{ \Vert \mathfrak{X} \Vert \geq s} \mathfrak{X}\aleph (d\mu ,d \mathfrak{X}) $$

is called the Lévy–Itô decomposition, where $\mathfrak{Z}\in \mathcal{B}$, $s>0$ is a constant, a $\mathcal{B}$-valued Brownian motion process $\mathfrak{W}$ has covariance operator Θ, ℵ̃, controlling small jumps, is a Poisson random measure, ℵ, controlling large jumps, is a compensated Poisson random measure on $\mathbb{R}^{+}\times (\mathcal{B}\setminus \{0\})$, λ is the jump measure, and λ satisfies

$$ \int \bigl( \Vert \mathfrak{X} \Vert ^{2}_{ \mathcal{B}}\wedge 1 \bigr)\lambda (d\mathfrak{X})< \infty . $$

Based on the previous discourse, system (1) may be reformulated into a more comprehensive format:

$$\begin{aligned} \textstyle\begin{cases} {}^{\mathfrak{ABC}}D_{0+}^{\iota} \zeta (\mu ) \\ \quad = \mathfrak{E} \zeta ( \mu )+\tilde{\varrho}(\mu ,\zeta (\mu ),\zeta (\mu +\gamma ))+ \tilde{\vartheta}(\mu ,\zeta (\mu ),\zeta (\mu +\gamma )) \frac{d\mathfrak{W}(\mu )}{d\mu} \\ \qquad {} +\frac{1}{d\mu}\int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{R}(\mu , \zeta (\mu ), \zeta (\mu +\gamma ), \mathfrak{X}) \tilde{\aleph}(d\mu ,d\mathfrak{X}) \\ \qquad {} +\frac{1}{d\mu}\int _{ \Vert \mathfrak{X} \Vert \geq s} \mathfrak{T}(\mu , \zeta (\mu ), \zeta (\mu +\gamma ), \mathfrak{X})\aleph (d\mu ,d\mathfrak{X}) , \quad \mu \in \mathfrak{J}:=(0,\rho ], \mu \neq \mu _{\kappa} \\ \Delta \zeta | _{\mu =\mu _{\kappa}}=\mathfrak{g}_{\kappa}(\zeta ( \mu _{\kappa}^{-})), \quad \mu =\mu _{\kappa}, \kappa =1,2, \ldots ,\mathcal{T} \\ \zeta (\mu )=\Psi (\mu ),\quad -\xi \leq \mu \leq 0, \\ \zeta (0)=\Psi _{0}, \end{cases}\displaystyle \end{aligned}$$

(4)

where $\tilde{\varrho}: \mathfrak{J}\times \mathcal{A}\times \mathbb{S} \rightarrow \mathcal{A}$, $\tilde{\vartheta}: \mathfrak{J}\times \mathcal{A}\times \mathbb{S} \rightarrow \mathcal{L}(\mathcal{B}, \mathcal{A})$, $\mathfrak{R}: \mathfrak{J}\times \mathcal{A}\times \mathbb{S}\times \mathcal{B}\rightarrow \mathcal{A}$, and $\mathfrak{T}:\mathfrak{J}\times \mathcal{A}\times \mathbb{S}\times \mathcal{B}\rightarrow \mathcal{A}$ are measurable. Given that we must concentrate on the stochastic differential system with small jumps, we obtain

$$\begin{aligned} \textstyle\begin{cases} {}^{\mathfrak{ABC}}D_{0+}^{\iota} \zeta (\mu ) \\ \quad = \mathfrak{E} \zeta ( \mu )+\tilde{\varrho}(\mu ,\zeta (\mu ),\zeta (\mu +\gamma ))+ \tilde{\vartheta}(\mu ,\zeta (\mu ),\zeta (\mu +\gamma )) \frac{d\mathfrak{W}(\mu )}{d\mu} \\ \qquad {} +\frac{1}{d\mu}\int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{R}(\mu , \zeta (\mu ), \zeta (\mu +\gamma ), \mathfrak{X}) \tilde{\aleph}(d\mu ,d\mathfrak{X}) ,\quad \mu \in \mathfrak{J}:=(0, \rho ], \mu \neq \mu _{\kappa} \\ \Delta \zeta | _{\mu =\mu _{\kappa}}=\mathfrak{g}_{\kappa}(\zeta ( \mu _{\kappa}^{-})), \quad \mu =\mu _{\kappa}, \kappa =1,2, \ldots ,\mathcal{T} \\ \zeta (\mu )=\Psi (\mu ),\quad -\xi \leq \mu \leq 0, \\ \zeta (0)=\Psi _{0}. \end{cases}\displaystyle \end{aligned}$$

(5)

Definition 2.2

If an $\mathcal{A}$-valued stochastic process $\{\zeta (\mu ), \mu \in [-\xi , \rho ]\}$ satisfies

• $\mu \in [-\xi , 0]$, $\zeta (\mu )=\Psi (\mu )$;

• $\zeta (\mu ) \in \mathcal{L}^{2}(\Omega , \mathcal{A})$ is $\mathcal{K}_{\mu}$-adapted, and has $c\grave{a}\,dl\grave{a}g$ path on $\mathfrak{J}$ a.s.;

• $$\begin{aligned} \zeta (\mu ) =& \Re \mathfrak{N}_{\iota}(\mu )\Psi _{0} + \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \tilde{\varrho}\bigl(\mathfrak{S}, \zeta (\mathfrak{S}),\zeta ( \mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{S} \\ &{}+\frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \tilde{\vartheta}\bigl( \mathfrak{S},\zeta (\mathfrak{S}),\zeta ( \mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}(\mathfrak{S}) \\ &{}+\frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \biggl\{ \int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{R}\bigl(\mathfrak{S}, \zeta ( \mathfrak{S}), \zeta (\mathfrak{S}+\gamma ), \mathfrak{X}\bigr) \tilde{ \aleph}(d\mathfrak{S},d\mathfrak{X}) \biggr\} \\ &{}+\frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu}\mathfrak{Q}_{ \iota}(\mu - \mathfrak{S}) \bigl\{ \tilde{\varrho}\bigl(\mathfrak{S},\zeta ( \mathfrak{S}),\zeta (\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{S} \\ &{}+\frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu}\mathfrak{Q}_{ \iota}(\mu - \mathfrak{S}) \bigl\{ \tilde{\vartheta}\bigl(\mathfrak{S},\zeta ( \mathfrak{S}), \zeta (\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}( \mathfrak{S}) \\ &{}+\frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu}\mathfrak{Q}_{ \iota}(\mu - \mathfrak{S}) \biggl\{ \int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{R}\bigl(\mathfrak{S},\zeta ( \mathfrak{S}), \zeta (\mathfrak{S}+\gamma ), \mathfrak{X}\bigr)\tilde{\aleph}(d \mathfrak{S},d\mathfrak{X}) \biggr\} \\ &{}+\sum_{0< \mu _{\kappa}< \mu} \Re \mathfrak{N}_{\iota}( \mu -\mu _{ \kappa})\mathfrak{g}_{\kappa}\bigl(\zeta \bigl(\mu _{\kappa}^{-}\bigr)\bigr), \mu \in \mathfrak{J}, \end{aligned}$$

  where $\Re = \vartheta ^{*}(\vartheta ^{*}I-\mathfrak{E})^{-1}$, $\wp = -\delta ^{*}\mathfrak{E}(\vartheta ^{*}I-\mathfrak{E})^{-1}$; with $\vartheta ^{*}=\frac{\varpi (\iota )}{1-\iota}$, $\delta ^{*}= \frac{\iota}{1-\iota}$,

  $$\begin{aligned}& \mathfrak{N}_{\iota}(\mu ) = \mathbb{W}_{\iota}\bigl(-\wp \mu ^{\iota}\bigr)= \frac{1}{2\pi i} \int _{\Pi} e^{\mathfrak{S}\mu}\mathfrak{S}^{\iota -1} \bigl( \mathfrak{S}^{\iota}I-\wp \bigr)^{-1}\,d\mathfrak{S}, \\& \mathfrak{Q}_{\iota}(\mu ) = \mu ^{\iota -1}\mathbb{W}_{\iota ,\iota} \bigl(- \wp \mu ^{\iota}\bigr)=\frac{1}{2\pi i} \int _{\Pi} e^{\mathfrak{S}\mu}\bigl( \mathfrak{S}^{\iota}I-\wp \bigr)^{-1}\,d\mathfrak{S}, \end{aligned}$$

then $\zeta (\mu )$ is a mild solution of problem (5).

Remark 2.1

As we know, $\mathfrak{N}_{\iota}(\cdot )$ and $\mathfrak{Q}_{\iota}(\cdot )$ are called characteristic solution operators. These operators can be expressed as follows:

$$\begin{aligned} \mathfrak{N}_{\iota}(\mu ) :=& \int _{0}^{\infty}\mathbf{M}_{\iota}( \mathfrak{S}) \mathbb{A}\bigl(\mathfrak{S}\mu ^{\iota}\bigr)\,d\mathfrak{S} \\ \mathfrak{Q}_{\iota}(\mu ) :=&\mu ^{\iota -1} \int _{0}^{\infty}\iota \mathfrak{S}\mathbf{M}_{\iota}( \mathfrak{S})\mathbb{A}\bigl(\mathfrak{S} \mu ^{\iota}\bigr)\,d\mathfrak{S}, \end{aligned}$$

where $\{\mathbb{A}(\mu ):\mu \geq 0\}$ is the analytic semigroup generated by $\mathfrak{E}$ and

$$ \mathbf{M}_{\iota}(\mu )=\sum_{\kappa =0}^{\infty} \frac{(-\mu )^{\kappa}}{\kappa !\Gamma (-\iota \kappa +(1-\iota ))}. $$

For more details and for the proof (see [25], Lemma 9).

Remark 2.2

• Assume that $\mathfrak{E}\in \ell ^{\iota}(\alpha _{0},l_{0})$, then $\Vert \mathfrak{N}_{\iota }(\mu ) \Vert \leq \hat{\mathfrak{J}} e^{l\mu}$ and $\Vert \mathfrak{Q}_{\iota }(\mu ) \Vert \leq \Re e^{l\mu}(1+ \mu ^{\mu -1})$, for every $\mu >0$, $l>l_{0}$. Thus, $\mathfrak{J}^{*}=\sup_{\mu \geq 0}{ \Vert \mathfrak{N}_{\iota }( \mu ) \Vert }$, $\mathfrak{J}_{1}^{*}=\sup_{\mu \geq 0}{\Re e^{l\mu}(1+\mu ^{\iota -1})}$. Hence, we obtain $\Vert \mathfrak{N}_{\iota }(\mu ) \Vert \leq \mathfrak{J}^{*}$ and $\Vert \mathfrak{Q}_{\iota }(\mu ) \Vert \leq \mathfrak{J}_{1}^{*}\mu ^{\iota -1}$. For more information, see [26].

• Consider the bounded, closed, and convex set $Q_{\sigma}= \{\zeta \in \tilde{\mathbb{S}}: \Vert \zeta \Vert _{\tilde{\mathbb{S}}}^{2}\leq \sigma , \sigma >0, \sigma \geq \frac{h_{1}}{1-h_{2}} \}$.

3 Main results

To begin with, we prove in this section that there is a mild solution to (5) and that it is unique. The functions ϱ̃, ϑ̃, and $\mathfrak{R}$ require assumptions in order for us to be able to prove the necessary result:

• (A1) $\forall \zeta _{1}, \zeta _{2}\in \mathcal{A}$, $\eta _{1}, \eta _{2}\in \mathbb{S}$ and $\mu \in \mathfrak{J}$, ∃ positive constants $C_{1}$, $C_{2}$ s.t.

  • 1. $\mathbb{E} \Vert \tilde{\varrho }(\mu , \zeta _{1}, \eta _{1}) \Vert ^{2}\leq C_{1} \{1+ \Vert \zeta _{1} \Vert ^{2}+ \Vert \eta _{1} \Vert ^{2} \}$,

  • 2. $\mathbb{E} \Vert \tilde{\varrho }(\mu , \zeta _{1}, \eta _{1})- \tilde{\varrho }(\mu , \zeta _{2}, \eta _{2}) \Vert ^{2}\leq C_{2} \{ \Vert \zeta _{1}-\zeta _{2} \Vert ^{2}+ \Vert \eta _{1}-\eta _{2} \Vert ^{2} \}$.

• (A2) $\forall \zeta _{1}, \zeta _{2}\in \mathcal{A}$, $\eta _{1}, \eta _{2}\in \mathbb{S}$ and $\mu \in \mathfrak{J}$, the coefficients ϑ̃ and $\mathfrak{R}$ satisfy that

  • (i)
    $$\begin{aligned}& \mathbb{E} \bigl\Vert \tilde{\vartheta }(\mu , \zeta _{1}, \eta _{1}) \bigr\Vert ^{2}\vee \mathbb{E} \biggl( \int _{ \Vert \mathfrak{X} \Vert < s} \bigl\Vert \mathfrak{R}(\mu , \zeta _{1}, \eta _{1}, \mathfrak{X}) \bigr\Vert ^{2} \lambda (d\mathfrak{X}) \biggr) \\& \quad \leq G_{1}( \mu ) \bigl\{ 1+ \Vert \zeta _{1} \Vert ^{2}+ \Vert \eta _{1} \Vert ^{2} \bigr\} , \end{aligned}$$
  • (ii)
    $$\begin{aligned}& \mathbb{E} \bigl\Vert \tilde{\vartheta }(\mu , \zeta _{1}, \eta _{1})- \tilde{\vartheta }(\mu , \zeta _{2}, \eta _{2}) \bigr\Vert ^{2} \\& \qquad {}\vee \mathbb{E} \biggl( \int _{ \Vert \mathfrak{X} \Vert < s} \bigl\Vert \mathfrak{R}(\mu , \zeta _{1}, \eta _{1}, \mathfrak{X})- \mathfrak{R}(\mu , \zeta _{2}, \eta _{2}, \mathfrak{X}) \bigr\Vert ^{2} \lambda (d\mathfrak{X}) \biggr) \\& \quad \leq G_{2}(\mu ) \bigl\{ \Vert \zeta _{1}- \zeta _{2} \Vert ^{2}+ \Vert \eta _{1}-\eta _{2} \Vert ^{2} \bigr\} , \end{aligned}$$

    where the functions $G_{1}(\mu )\in \mathcal{L}^{\frac{1}{2\alpha -1}}(\mathfrak{J})$ and $G_{2}(\mu )\in \mathcal{L}^{\frac{1}{2\alpha -1}}(\mathfrak{J})$, $\alpha \in (1/2,\iota )$.

• (A3)

  • 1. The functions $\mathfrak{g}_{\kappa}:\mathcal{A}\rightarrow \mathcal{A}$ are completely continuous and there exist constants $d_{\kappa}$ s.t.

    $$ \mathbb{E} \bigl\Vert \mathfrak{g}_{\kappa }(\zeta ) \bigr\Vert ^{2} \leq d_{\kappa},\quad \kappa =1,2,\ldots ,\mathcal{T}, \forall \zeta \in \mathcal{A}. $$

  • 2. ∃ is a constant $\ell ^{*}>0$ s.t.

    $$ \mathbb{E} \bigl\Vert \mathfrak{g}_{\kappa }(\zeta _{1})- \mathfrak{g}_{ \kappa }(\zeta _{2}) \bigr\Vert ^{2}\leq \ell ^{*}\mathbb{E} \Vert \zeta _{1}-\zeta _{2} \Vert ^{2} $$

    for each $\zeta _{1}, \zeta _{2}\in \mathcal{A}$ and $\kappa =1,2, \ldots ,\mathcal{T}$.

• (A4) ℘ and ℜ are bounded linear operators, ∃θ and ψ s.t. $\Vert \Re \Vert \leq \theta$ and $\Vert \wp \Vert \leq \psi$.

• (A5) ∃ is a function $\mathfrak{G}(\mu , \zeta ): \mathbb{R}^{+}\times \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ that is locally integrable with respect to μ, and nondecreasing, continuous, and concave with respect to ζ $\forall \mu \in [0,\rho ]$, $\int _{0^{+}} \frac{1}{\mathfrak{G}(\mu , \zeta )}\,d\zeta =\infty$. For any $\zeta _{1},\zeta _{2}\in \mathcal{A}$, $\eta _{1},\eta _{2}\in \mathbb{S}$, and $\mu \in \mathfrak{J}$, this inequality holds:

  $$\begin{aligned}& \bigl\Vert \tilde{\varrho }(\mu , \zeta _{1}, \eta _{1})- \tilde{\varrho }(\mu , \zeta _{2}, \eta _{2}) \bigr\Vert ^{2} \vee \bigl\Vert \tilde{\vartheta }( \mu , \zeta _{1}, \eta _{1})- \tilde{\vartheta }(\mu , \zeta _{2}, \eta _{2}) \bigr\Vert ^{2} \\& \qquad {} \vee \int _{ \Vert \mathfrak{X} \Vert < s} \bigl\Vert \mathfrak{R}(\mu , \zeta _{1}, \eta _{1}, \mathfrak{X})- \mathfrak{R}(\mu , \zeta _{2}, \eta _{2}, \mathfrak{X}) \bigr\Vert ^{2} \lambda (d\mathfrak{X}) \\& \quad \leq \mathfrak{G} \bigl(\mu , \Vert \zeta _{1}-\zeta _{2} \Vert ^{2}+ \Vert \eta _{1}- \eta _{2} \Vert ^{2} \bigr). \end{aligned}$$

• (A6) ∃ the measurable coefficient functions $\mathfrak{f}: \mathcal{A}\times \mathbb{S}\rightarrow \mathcal{A}$, $\mathfrak{H}: \mathcal{A}\times \mathbb{S}\rightarrow \mathcal{L}( \mathcal{B},\mathcal{A})$, $\mathfrak{C}: \mathcal{A}\times \mathbb{S}\times \mathcal{B} \rightarrow \mathcal{A}$, and $\Im : \mathbb{S}\rightarrow \mathcal{A}$, satisfying (A5) and for any $\tilde{\rho}\in \mathfrak{J}$, $\zeta \in \mathcal{A}$, $\eta \in \mathbb{S}$, ∃ is a positive bounded function $\hbar _{i}(\tilde{\rho})$, $i=1,2,3,4$, s.t.

  $$\begin{aligned}& \frac{1}{\tilde{\rho}} \int _{0}^{\tilde{\rho}} \bigl\Vert \tilde{\varrho }( \mathfrak{S}, \zeta , \eta )- \mathfrak{f}(\zeta , \eta ) \bigr\Vert ^{2}\,d\mathfrak{S} \leq \hbar _{1}(\tilde{\rho}) \bigl(1+ \Vert \zeta \Vert ^{2}+ \Vert \eta \Vert ^{2} \bigr), \\& \frac{1}{\tilde{\rho}} \int _{0}^{\tilde{\rho}} \bigl\Vert \tilde{\vartheta }( \mathfrak{S}, \zeta , \eta )- \mathfrak{H}(\zeta , \eta ) \bigr\Vert ^{4}\,d\mathfrak{S} \leq \hbar _{2}(\tilde{\rho}) \bigl(1+ \Vert \zeta \Vert ^{4}+ \Vert \eta \Vert ^{4} \bigr), \\& \frac{1}{\tilde{\rho}} \int _{0}^{\tilde{\rho}} \biggl\{ \int _{ \Vert \mathfrak{X} \Vert < s} \bigl\Vert \mathfrak{R}( \mathfrak{S}, \zeta , \eta , \mathfrak{X})- \mathfrak{C}(\zeta , \eta , \mathfrak{X}) \bigr\Vert ^{2}\lambda \,d\mathfrak{X} \biggr\} ^{2} \leq \hbar _{3}(\tilde{\rho}) \bigl(1+ \Vert \zeta \Vert ^{4}+ \Vert \eta \Vert ^{4} \bigr), \\& \frac{1}{\tilde{\rho}} \biggl\Vert \sum_{0< \mu _{\kappa }< \tilde{\rho }} \mathfrak{g}_{\kappa }(\zeta )- \tilde{\rho }\Im ( \zeta ) \biggr\Vert ^{2} \leq \hbar _{4}(\tilde{\rho}) \bigl(1+ \Vert \zeta \Vert ^{2} \bigr). \end{aligned}$$

Define the operator Φ as follows:

$$\begin{aligned} (\Phi \zeta ) (\mu ) =& \Re \mathfrak{N}_{\iota}(\mu )\Psi _{0} + \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \tilde{\varrho}\bigl(\mathfrak{S}, \zeta (\mathfrak{S}),\zeta ( \mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{S} \\ &{}+\frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \tilde{\vartheta}\bigl( \mathfrak{S},\zeta (\mathfrak{S}),\zeta ( \mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}(\mathfrak{S}) \\ &{}+\frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \biggl\{ \int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{R}\bigl(\mathfrak{S}, \zeta ( \mathfrak{S}), \zeta (\mathfrak{S}+\gamma ), \mathfrak{X}\bigr) \tilde{ \aleph}(d\mathfrak{S},d\mathfrak{X}) \biggr\} \\ &{}+\frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu}\mathfrak{Q}_{ \iota}(\mu - \mathfrak{S}) \bigl\{ \tilde{\varrho}\bigl(\mathfrak{S},\zeta ( \mathfrak{S}),\zeta (\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{S} \\ &{}+\frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu}\mathfrak{Q}_{ \iota}(\mu - \mathfrak{S}) \bigl\{ \tilde{\vartheta}\bigl(\mathfrak{S},\zeta ( \mathfrak{S}), \zeta (\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}( \mathfrak{S}) \\ &{}+\frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu}\mathfrak{Q}_{ \iota}(\mu - \mathfrak{S}) \biggl( \int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{R}\bigl(\mathfrak{S}, \zeta (\mathfrak{S}), \zeta ( \mathfrak{S}+\gamma ), \mathfrak{X}\bigr)\tilde{\aleph}(d \mathfrak{S},d \mathfrak{X}) \biggr) \\ &{}+\sum_{0< \mu _{\kappa}< \mu} \Re \mathfrak{N}_{\iota}( \mu -\mu _{ \kappa})\mathfrak{g}_{\kappa}\bigl(\zeta \bigl(\mu _{\kappa}^{-}\bigr)\bigr), \mu \in \mathfrak{J}. \end{aligned}$$

Theorem 3.1

Suppose that the assumptions (A1)–(A4) are satisfied, then the system (5) possesses a unique mild solution on $Q_{\sigma}$ provided that $\mathfrak{U}<1$, where

$$\begin{aligned} \mathfrak{U} =&14C_{2} \biggl( \frac{\psi \theta (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \frac{\mu ^{2\iota}}{2\iota -1}+ 28 \biggl(2 \frac{\psi \theta (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \mu ^{2\iota -2\alpha} \biggl(\frac{1-\alpha}{\iota -\alpha} \biggr)^{2-2 \alpha} \Vert G_{2} \Vert _{\mathcal{L}^{ \frac{1}{2\alpha -1}}} \\ &{}+14C_{2} \biggl( \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \frac{\mu ^{2\iota}}{2\iota -1} \\ &{}+ 28 \biggl(2 \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \mu ^{2\iota -2\alpha} \biggl(\frac{1-\alpha}{\iota -\alpha} \biggr)^{2-2 \alpha} \Vert G_{2} \Vert _{\mathcal{L}^{ \frac{1}{2\alpha -1}}} +7\bigl(\theta \mathfrak{J}^{*} \bigr)^{2}\ell ^{*}. \end{aligned}$$

Proof

In the next three steps, we demonstrate the evidence.

• Step 1. Φ is continuous on $Q_{\sigma}$. We can demonstrate the validity of this step using straightforward arguments from (A1), (A2), and (A3).

• Step 2. The operator Φ maps $Q_{\sigma}$ into itself.

  $\forall \zeta \in Q_{\sigma}$, $\mu \in \mathfrak{J}$, by the Hölder inequality, and the Burkholder–Davis–Gundy inequality, we can obtain

  $$\begin{aligned}& \Vert \Phi \zeta \Vert _{\tilde{\mathbb{S}}}^{2} \\& \quad \leq 8 \sup _{\mu \in \mathfrak{J}}\mathbb{E} \bigl\Vert \Re \mathfrak{N}_{ \iota }( \mu )\Psi _{0} \bigr\Vert ^{2} \\& \qquad {}+8\sup_{\mu \in \mathfrak{J}} \mathbb{E} \biggl\Vert \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu }(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \tilde{\varrho }\bigl( \mathfrak{S},\zeta (\mathfrak{S}),\zeta ( \mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{S} \biggr\Vert ^{2} \\& \qquad {}+8\sup_{\mu \in \mathfrak{J}}\mathbb{E} \biggl\Vert \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu }(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \tilde{\vartheta }\bigl( \mathfrak{S},\zeta (\mathfrak{S}),\zeta ( \mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}(\mathfrak{S}) \biggr\Vert ^{2} \\& \qquad {}+8\sup_{\mu \in \mathfrak{J}}\mathbb{E} \biggl\Vert \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \\& \qquad {}\times \int _{0}^{ \mu }(\mu -\mathfrak{S})^{\iota -1} \biggl\{ \int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{R}\bigl(\mathfrak{S}, \zeta ( \mathfrak{S}), \zeta (\mathfrak{S}+\gamma ), \mathfrak{X}\bigr) \tilde{\aleph }(d\mathfrak{S},d\mathfrak{X}) \biggr\} \biggr\Vert ^{2} \\& \qquad {}+8\sup_{\mu \in \mathfrak{J}}\mathbb{E} \biggl\Vert \frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu }\mathfrak{Q}_{ \iota }(\mu - \mathfrak{S}) \bigl\{ \tilde{\varrho }\bigl(\mathfrak{S}, \zeta (\mathfrak{S}), \zeta (\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{S} \biggr\Vert ^{2} \\& \qquad {}+8\sup_{\mu \in \mathfrak{J}}\mathbb{E} \biggl\Vert \frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu }\mathfrak{Q}_{ \iota }(\mu - \mathfrak{S}) \bigl\{ \tilde{\vartheta }\bigl(\mathfrak{S}, \zeta (\mathfrak{S}), \zeta (\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}(\mathfrak{S}) \biggr\Vert ^{2} \\& \qquad {}+8\sup_{\mu \in \mathfrak{J}}\mathbb{E} \biggl\Vert \frac{\iota \Re ^{2}}{\varpi (\iota )} \\& \qquad {}\times \int _{0}^{\mu }\mathfrak{Q}_{ \iota }(\mu - \mathfrak{S}) \biggl(\int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{R}\bigl(\mathfrak{S}, \zeta (\mathfrak{S}), \zeta (\mathfrak{S}+\gamma ), \mathfrak{X}\bigr)\tilde{\aleph }(d \mathfrak{S},d\mathfrak{X}) \biggr) \biggr\Vert ^{2} \\& \qquad {}+8\sup_{\mu \in \mathfrak{J}}\mathbb{E} \Biggl\Vert \sum _{0< \mu _{ \kappa }< \mu } \Re \mathfrak{N}_{\iota }(\mu -\mu _{\kappa }) \mathfrak{g}_{\kappa }\bigl(\zeta \bigl(\mu _{\kappa }^{-}\bigr)\bigr) \Biggr\Vert ^{2} \\& \quad \leq 8\bigl(\theta \mathfrak{J}^{*} \Vert \Psi _{0} \Vert \bigr)^{2}+8 \biggl(\frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \frac{\mu ^{2\iota}}{2\iota -1}C_{1} \bigl\{ 1+2 \Vert \zeta \Vert ^{2} \bigr\} \\& \qquad {}+64 \biggl( \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \biggl( \int _{0}^{\mu}(\mu -\mathfrak{S})^{ \frac{2(\iota -1)}{2-2\alpha}} \biggr)^{2-2\alpha} \biggl( \int _{0}^{\mu}G_{1}^{ \frac{1}{2\alpha -1}} \biggr)^{2\alpha -1} \bigg) \\& \qquad {}\times \bigl\{ 1+2 \Vert \zeta \Vert ^{2} \bigr\} \\& \qquad {}+8 \biggl( \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \frac{\mu ^{2\iota}}{2\iota -1}C_{1} \bigl\{ 1+2 \Vert \zeta \Vert ^{2} \bigr\} \\& \qquad {}+64 \biggl( \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \biggl( \int _{0}^{\mu}(\mu -\mathfrak{S})^{ \frac{2(\iota -1)}{2-2\alpha}} \biggr)^{2-2\alpha} \biggl( \int _{0}^{\mu}G_{1}^{ \frac{1}{2\alpha -1}} \biggr)^{2\alpha -1} \bigg) \\& \qquad {}\times \bigl\{ 1+2 \Vert \zeta \Vert ^{2} \bigr\} +8\bigl(\theta \mathfrak{J}^{*}\bigr)^{2}\sum _{ \kappa =1}^{\mathcal{T}}d_{\kappa} \\& \quad \leq 8\bigl(\theta \mathfrak{J}^{*} \Vert \Psi _{0} \Vert \bigr)^{2}+8 \biggl(\frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \frac{\mu ^{2\iota}}{2\iota -1}C_{1} \{1+2\sigma \} \\& \qquad {}+64 \biggl( \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \mu ^{2(\iota -\alpha )} \biggl(\frac{1-\alpha}{\iota -\alpha} \biggr)^{2-2 \alpha} \Vert G_{1} \Vert _{\mathcal{L}^{ \frac{1}{2\alpha -1}}} \{1+2\sigma \} \\& \qquad {}+8 \biggl( \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \frac{\mu ^{2\iota}}{2\iota -1}C_{1} \{1+2\sigma \} \\& \qquad {}+64 \biggl( \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \mu ^{2(\iota -\alpha )} \biggl(\frac{1-\alpha}{\iota -\alpha} \biggr)^{2-2 \alpha} \\& \qquad {}\times \Vert G_{1} \Vert _{\mathcal{L}^{ \frac{1}{2\alpha -1}}} \{1+2\sigma \} +8\bigl(\theta \mathfrak{J}^{*} \bigr)^{2} \sum_{\kappa =1}^{\mathcal{T}}d_{\kappa} \\& \quad = h_{1}+h_{2}\sigma \leq \sigma , \end{aligned}$$

  where

  $$\begin{aligned}& h_{1} = 8\bigl(\theta \mathfrak{J}^{*} \Vert \Psi _{0} \Vert \bigr)^{2}+8\bigl( \theta \mathfrak{J}^{*} \bigr)^{2}\sum_{\kappa =1}^{\mathcal{T}}d_{\kappa}+8C_{1} \biggl(\frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}\frac{\mu ^{2\iota}}{2\iota -1} \\& \hphantom{h_{1} =} {}+64 \biggl( \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \mu ^{2(\iota -\alpha )} \biggl(\frac{1-\alpha}{\iota -\alpha} \biggr)^{2-2 \alpha} \Vert G_{1} \Vert _{\mathcal{L}^{ \frac{1}{2\alpha -1}}} \\& \hphantom{h_{1} =} {}+8C_{1} \biggl( \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \frac{\mu ^{2\iota}}{2\iota -1}+64 \biggl( \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \mu ^{2(\iota -\alpha )} \biggl(\frac{1-\alpha}{\iota -\alpha} \biggr)^{2-2 \alpha} \Vert G_{1} \Vert _{\mathcal{L}^{ \frac{1}{2\alpha -1}}} \\& h_{2} = 16 \biggl( \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \frac{\mu ^{2\iota}}{2\iota -1} \\& \hphantom{h_{2} =} {}+32 \biggl(2 \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \mu ^{2(\iota -\alpha )} \biggl(\frac{1-\alpha}{\iota -\alpha} \biggr)^{2-2 \alpha} \Vert G_{1} \Vert _{\mathcal{L}^{ \frac{1}{2\alpha -1}}} \\& \hphantom{h_{2} =} {}+16 \biggl( \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \frac{\mu ^{2\iota}}{2\iota -1}+32 \biggl(2 \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \mu ^{2(\iota -\alpha )} \biggl( \frac{1-\alpha}{\iota -\alpha} \biggr)^{2-2 \alpha} \Vert G_{1} \Vert _{\mathcal{L}^{ \frac{1}{2\alpha -1}}}. \end{aligned}$$

  Thus, the operator Φ maps $Q_{\sigma}$ into itself.

• Step 3. Φ is a contraction on $Q_{\sigma}$.

  $\forall \zeta , \eta \in Q_{\sigma}$ and $\mu \in \mathfrak{J}$, hence, we can obtain

  $$\begin{aligned}& \bigl\Vert (\Phi \zeta )-(\Phi \eta ) \bigr\Vert _{ \tilde{\mathbb{S}}}^{2} \\& \quad \leq 7 \biggl( \frac{\psi \theta (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \frac{\mu ^{2\iota -1}}{2\iota -1} \\& \qquad {}\times \int _{0}^{\mu}\mathbb{E} \bigl\Vert \tilde{\varrho }\bigl(\mathfrak{S},\zeta (\mathfrak{S}),\zeta ( \mathfrak{S}+\gamma )\bigr)- \tilde{\varrho }\bigl(\mathfrak{S}, \eta ( \mathfrak{S}),\eta (\mathfrak{S}+\gamma )\bigr) \bigr\Vert ^{2}\,d\mathfrak{S} \\& \qquad {}+28 \biggl( \frac{\psi \theta (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \int _{0}^{\mu}(\mu -\mathfrak{S})^{2\iota -2} \\& \qquad {}\times \mathbb{E} \bigl\Vert \tilde{\vartheta }\bigl(\mathfrak{S},\zeta (\mathfrak{S}), \zeta ( \mathfrak{S}+\gamma )\bigr)-\tilde{\vartheta }\bigl(\mathfrak{S},\eta ( \mathfrak{S}),\eta (\mathfrak{S}+\gamma )\bigr) \bigr\Vert ^{2}\,d\mathfrak{S} \\& \qquad {}+28 \biggl( \frac{\psi \theta (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \int _{0}^{\mu}(\mu -\mathfrak{S})^{2\iota -2} \\& \qquad {} \times\mathbb{E} \int _{ \Vert \mathfrak{X} \Vert < s} \bigl\Vert \mathfrak{R}\bigl( \mathfrak{S}, \zeta ( \mathfrak{S}), \zeta (\mathfrak{S}+\gamma ), \mathfrak{X} \bigr) \\& \qquad {} - \mathfrak{R}\bigl(\mathfrak{S}, \eta (\mathfrak{S}), \eta (\mathfrak{S}+ \gamma ), \mathfrak{X}\bigr) \bigr\Vert ^{2}\tilde{\aleph}(d \mathfrak{S},d \mathfrak{X}) \\& \qquad {}+7 \biggl( \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \frac{\mu ^{2\iota -1}}{2\iota -1} \\& \qquad {}\times \int _{0}^{\mu}\mathbb{E} \bigl\Vert \tilde{\varrho }\bigl(\mathfrak{S},\zeta (\mathfrak{S}),\zeta ( \mathfrak{S}+\gamma )\bigr)- \tilde{\varrho }\bigl(\mathfrak{S}, \eta ( \mathfrak{S}),\eta (\mathfrak{S}+\gamma )\bigr) \bigr\Vert ^{2}\,d\mathfrak{S} \\& \qquad {}+28 \biggl( \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \int _{0}^{\mu}(\mu -\mathfrak{S})^{2\iota -2} \\& \qquad {}\times \mathbb{E} \bigl\Vert \tilde{\vartheta }\bigl(\mathfrak{S},\zeta (\mathfrak{S}), \zeta ( \mathfrak{S}+\gamma )\bigr)-\tilde{\vartheta }\bigl(\mathfrak{S},\eta ( \mathfrak{S}),\eta (\mathfrak{S}+\gamma )\bigr) \bigr\Vert ^{2}\,d\mathfrak{S} \\& \qquad {}+28 \biggl( \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \int _{0}^{\mu}(\mu -\mathfrak{S})^{2\iota -2} \\& \qquad {} \times \mathbb{E} \int _{ \Vert \mathfrak{X} \Vert < s} \bigl\Vert \mathfrak{R}\bigl( \mathfrak{S}, \zeta ( \mathfrak{S}), \zeta (\mathfrak{S}+\gamma ), \mathfrak{X} \bigr) \\& \qquad {} - \mathfrak{R}\bigl(\mathfrak{S}, \eta (\mathfrak{S}), \eta (\mathfrak{S}+ \gamma ), \mathfrak{X}\bigr) \bigr\Vert ^{2}\tilde{\aleph}(d \mathfrak{S},d \mathfrak{X}) \\& \qquad {}+7\bigl(\theta \mathfrak{J}^{*}\bigr)^{2}\mathbb{E} \Biggl\Vert \sum_{0< \mu _{\kappa }< \mu } \mathfrak{g}_{\kappa }\bigl(\zeta \bigl(\mu _{ \kappa }^{-} \bigr)\bigr)-\mathfrak{g}_{\kappa }\bigl(\eta \bigl(\mu _{\kappa }^{-} \bigr)\bigr) \Biggr\Vert ^{2} \\& \quad \leq 14C_{2} \biggl( \frac{\psi \theta (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2} \frac{\mu ^{2\iota}}{2\iota -1} \Vert \zeta -\eta \Vert ^{2} \\& \qquad {} + 28 \biggl(2 \frac{\psi \theta (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}\mu ^{2\iota -2\alpha} \biggl( \frac{1-\alpha}{\iota -\alpha} \biggr)^{2-2\alpha} \Vert G_{2} \Vert _{\mathcal{L}^{ \frac{1}{2\alpha -1}}} \Vert \zeta -\eta \Vert ^{2} \\& \qquad {}+14C_{2} \biggl( \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \frac{\mu ^{2\iota}}{2\iota -1} \Vert \zeta -\eta \Vert _{ \tilde{\mathbb{S}}}^{2} \\& \qquad {} + 28 \biggl(2 \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \mu ^{2\iota -2\alpha} \biggl( \frac{1-\alpha}{\iota -\alpha} \biggr)^{2-2 \alpha} \Vert G_{2} \Vert _{\mathcal{L}^{ \frac{1}{2\alpha -1}}} \Vert \zeta -\eta \Vert ^{2} \\& \qquad {}+7\bigl(\theta \mathfrak{J}^{*}\bigr)^{2}\ell ^{*} \Vert \zeta - \eta \Vert ^{2} \\& \quad =\mathfrak{U} \Vert \zeta -\eta \Vert ^{2} \leq \Vert \zeta - \eta \Vert ^{2}, \end{aligned}$$

which demonstrates that the mapping Φ is a contraction mapping. In light of this, Φ has a unique fixed point on the set $Q_{\sigma}$, which is consistent with the mild solution of system (5) having a single mild solution on $Q_{\sigma}$. □

The averaging principle to system (5) will be investigated in the following. The standard form of system (5) is defined as

$$\begin{aligned} \textstyle\begin{cases} {}^{\mathfrak{ABC}}D_{0+}^{\iota} \zeta _{\delta}(\mu ) \\ \quad = \mathfrak{E} \zeta _{\delta}(\mu )+\delta \tilde{\varrho}(\mu ,\zeta _{ \delta}(\mu ),\zeta _{\delta}(\mu +\gamma ))+\sqrt{\delta} \tilde{\vartheta}(\mu ,\zeta _{\delta}(\mu ),\zeta _{\delta}(\mu + \gamma ))\frac{d\mathfrak{W}(\mu )}{d\mu} \\ \qquad {} +\frac{\sqrt{\delta}}{d\mu}\int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{R}(\mu , \zeta _{\delta}(\mu ), \zeta _{ \delta}(\mu +\gamma ), \mathfrak{X})\tilde{\aleph}(d\mu ,d \mathfrak{X}) , \quad \mu \in \mathfrak{J}:=(0,\rho ], \mu \neq \mu _{\kappa} \\ \Delta \zeta | _{\mu =\mu _{\kappa}}=\mathfrak{g}_{\kappa}(\zeta ( \mu _{\kappa}^{-})), \quad \mu =\mu _{\kappa}, \kappa =1,2, \ldots ,\mathcal{T} \\ \zeta (\mu )=\Psi (\mu ), \quad -\xi \leq \mu \leq 0, \\ \zeta (0)=\Psi _{0}, \end{cases}\displaystyle \end{aligned}$$

(6)

where ϱ̃, ϑ̃, and $\mathfrak{R}$ have the same prerequisites as the system (5). $\delta \in (0,\delta _{1}]$ is a positive small parameter with $0<\delta _{1}\ll 1$ a fixed number.

The mild solution $\zeta _{\delta}(\mu )$ of system (6) is able to be offered by

$$\begin{aligned} \zeta _{\delta}(\mu ) :=& \Re \mathfrak{N}_{\iota}(\mu )\Psi _{0} + \delta \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \tilde{\varrho}\bigl(\mathfrak{S}, \zeta _{\delta}(\mathfrak{S}), \zeta _{\delta}(\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{S} \\ &{}+\sqrt{\delta} \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \tilde{\vartheta}\bigl( \mathfrak{S},\zeta _{\delta}(\mathfrak{S}), \zeta _{\delta}( \mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}( \mathfrak{S}) \\ &{}+\sqrt{\delta} \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \biggl\{ \int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{R}\bigl(\mathfrak{S}, \zeta _{ \delta}(\mathfrak{S}), \zeta _{\delta}(\mathfrak{S}+\gamma ), \mathfrak{X}\bigr)\tilde{\aleph}(d\mathfrak{S},d\mathfrak{X}) \biggr\} \\ &{}+\delta \frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu} \mathfrak{Q}_{\iota}(\mu - \mathfrak{S}) \bigl\{ \tilde{\varrho}\bigl( \mathfrak{S},\zeta _{\delta}( \mathfrak{S}),\zeta _{\delta}( \mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{S} \\ &{}+\sqrt{\delta}\frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu} \mathfrak{Q}_{\iota}(\mu - \mathfrak{S}) \bigl\{ \tilde{\vartheta}\bigl( \mathfrak{S},\zeta _{\delta}( \mathfrak{S}),\zeta _{\delta}( \mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}(\mathfrak{S}) \\ &{}+\sqrt{\delta}\frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu} \mathfrak{Q}_{\iota}(\mu - \mathfrak{S}) \biggl\{ \int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{R}\bigl(\mathfrak{S},\zeta _{ \delta}(\mathfrak{S}), \zeta _{\delta}(\mathfrak{S}+\gamma ), \mathfrak{X}\bigr)\tilde{\aleph}(d\mathfrak{S},d\mathfrak{X}) \biggr\} \\ &{}+\delta \sum_{0< \mu _{\kappa}< \mu} \Re \mathfrak{N}_{\iota}(\mu - \mu _{\kappa})\mathfrak{g}_{\kappa}( \zeta _{\delta ,\mu _{\kappa}^{-}}),\quad \mu \in \mathfrak{J}. \end{aligned}$$

Next, we must infer that when $\delta \rightarrow 0$, the original system solution $\zeta _{\delta}(\mu )$ converges to the solution process $\nu _{\delta}(\mu )$ of the averaged system

$$\begin{aligned} \nu _{\delta}(\mu ) :=& \Re \mathfrak{N}_{\iota}(\mu )\Psi _{0} + \delta \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \mathfrak{f}\bigl(\nu _{\delta}( \mathfrak{S}),\nu _{\delta}( \mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{S} \\ &{}+\sqrt{\delta} \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \mathfrak{H}\bigl(\nu _{\delta}( \mathfrak{S}),\nu _{\delta}( \mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}(\mathfrak{S}) \\ &{}+\sqrt{\delta} \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \biggl\{ \int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{C}\bigl( \nu _{\delta}( \mathfrak{S}), \nu _{\delta}(\mathfrak{S}+\gamma ), \mathfrak{X}\bigr) \tilde{\aleph}(d\mathfrak{S},d\mathfrak{X}) \biggr\} \\ &{}+\delta \frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu} \mathfrak{Q}_{\iota}(\mu - \mathfrak{S}) \bigl\{ \mathfrak{f}\bigl(\nu _{ \delta}(\mathfrak{S}),\nu _{\delta}(\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{S} \\ &{}+\sqrt{\delta}\frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu} \mathfrak{Q}_{\iota}(\mu - \mathfrak{S}) \bigl\{ \mathfrak{H}\bigl(\nu _{ \delta}(\mathfrak{S}),\nu _{\delta}(\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}(\mathfrak{S}) \\ &{}+\sqrt{\delta}\frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu} \mathfrak{Q}_{\iota}(\mu - \mathfrak{S}) \biggl\{ \int _{ \Vert \mathfrak{X} \Vert < s} \mathfrak{C}\bigl(\nu _{\delta}( \mathfrak{S}), \nu _{\delta}(\mathfrak{S}+\gamma ), \mathfrak{X}\bigr) \tilde{\aleph}(d\mathfrak{S},d\mathfrak{X}) \biggr\} \\ &{}+\delta \int _{0}^{\mu}\Re \mathfrak{N}_{\iota}(\mu - \mathfrak{S}) \Im (\nu _{\delta ,\mathfrak{S}})\,d\mathfrak{S},\quad \mu \in \mathfrak{J}. \end{aligned}$$

(7)

Theorem 3.2

Let the assumptions (A1)–(A6) be fulfilled. For a given arbitrary small number $\tau >0$, ∃ constants $h_{0}>0$, $\delta _{2}\in (0,\delta _{1}]$ and $\varepsilon \in (0,1)$, s.t. $\forall \delta \in (0, \delta _{1}]$, $3/4<\iota <1$,

$$ \mathbb{E} \Bigl\{ \sup_{\mu \in [-\xi , h_{0}\delta ^{-\varepsilon}]} \bigl\Vert \zeta _{\delta }(\mu )-\nu _{\delta }(\mu ) \bigr\Vert ^{2} \Bigr\} \leq \tau . $$

Proof

For $\mu \in \mathfrak{J}$, we obtain

$$\begin{aligned}& \zeta _{\delta}(\mu )-\nu _{\delta}(\mu ) \\& \quad = \delta \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \tilde{\varrho}\bigl(\mathfrak{S}, \zeta _{\delta}(\mathfrak{S}), \zeta _{\delta}(\mathfrak{S}+\gamma )\bigr)- \mathfrak{f}\bigl(\nu _{\delta}(\mathfrak{S}),\nu _{\delta}(\mathfrak{S}+ \gamma )\bigr) \bigr\} \,d\mathfrak{S} \\& \qquad {}+\sqrt{\delta} \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \\& \qquad {}\times \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \tilde{\vartheta}\bigl( \mathfrak{S},\zeta _{\delta}(\mathfrak{S}), \zeta _{\delta}( \mathfrak{S}+\gamma )\bigr)-\mathfrak{H}\bigl(\nu _{\delta}(\mathfrak{S}),\nu _{ \delta}(\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}(\mathfrak{S}) \\& \qquad {}+\sqrt{\delta} \frac{\wp \Re (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \int _{0}^{ \mu}(\mu -\mathfrak{S})^{\iota -1} \int _{ \Vert \mathfrak{X} \Vert < s} \\& \qquad {} \times \bigl\{ \mathfrak{R}\bigl(\mathfrak{S}, \zeta _{\delta}( \mathfrak{S}), \zeta _{\delta}(\mathfrak{S}+\gamma ), \mathfrak{X}\bigr)- \mathfrak{C}\bigl(\nu _{\delta}(\mathfrak{S}), \nu _{\delta}( \mathfrak{S}+ \gamma ), \mathfrak{X}\bigr) \bigr\} \tilde{\aleph}(d\mathfrak{S},d \mathfrak{X}) \\& \qquad {}+\delta \frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu} \mathfrak{Q}_{\iota}(\mu - \mathfrak{S}) \bigl\{ \tilde{\varrho}\bigl( \mathfrak{S},\zeta _{\delta}( \mathfrak{S}),\zeta _{\delta}( \mathfrak{S}+\gamma )\bigr)-\mathfrak{f} \bigl(\nu _{\delta}(\mathfrak{S}),\nu _{ \delta}(\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{S} \\& \qquad {}+\sqrt{\delta}\frac{\iota \Re ^{2}}{\varpi (\iota )} \\& \qquad {}\times \int _{0}^{\mu} \mathfrak{Q}_{\iota}(\mu - \mathfrak{S}) \bigl\{ \tilde{\vartheta}\bigl( \mathfrak{S},\zeta _{\delta}( \mathfrak{S}),\zeta _{\delta}( \mathfrak{S}+\gamma )\bigr)-\mathfrak{H} \bigl(\nu _{\delta}(\mathfrak{S}),\nu _{ \delta}(\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}(\mathfrak{S}) \\& \qquad {}+\sqrt{\delta}\frac{\iota \Re ^{2}}{\varpi (\iota )} \int _{0}^{\mu} \mathfrak{Q}_{\iota}(\mu - \mathfrak{S}) \int _{ \Vert \mathfrak{X} \Vert < s} \\& \qquad {} \times \bigl\{ \mathfrak{R}\bigl(\mathfrak{S}, \zeta _{\delta}( \mathfrak{S}), \zeta _{\delta}(\mathfrak{S}+\gamma ), \mathfrak{X}\bigr)- \mathfrak{C}\bigl( \nu _{\delta}(\mathfrak{S}), \nu _{\delta}( \mathfrak{S}+ \gamma ), \mathfrak{X}\bigr) \bigr\} \tilde{\aleph}(d\mathfrak{S},d \mathfrak{X}) \\& \qquad {}+\delta \Biggl(\sum_{0< \mu _{\kappa}< \mu} \Re \mathfrak{N}_{\iota}( \mu -\mu _{\kappa})\mathfrak{g}_{\kappa}( \zeta _{\delta ,\mu _{\kappa}^{-}})- \int _{0}^{\mu}\Re \mathfrak{N}_{\iota}(\mu - \mathfrak{S})\Im (\nu _{ \delta ,\mathfrak{S}})\,d\mathfrak{S} \Biggr). \end{aligned}$$

For any $\mu \in (0,\epsilon ]\subset \mathfrak{J}$, we obtain

$$\begin{aligned}& \mathbb{E}\sup_{0< \mu \leq \epsilon} \bigl\Vert \zeta _{\delta }( \mu )-\nu _{\delta }(\mu ) \bigr\Vert ^{2} \\& \quad =7 \biggl\{ \biggl( \delta \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}+ \biggl(\delta \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \biggr\} \\& \qquad {}\times \mathbb{E}\sup_{0< \mu \leq \epsilon} \biggl\Vert \int _{0}^{\mu }(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \tilde{\varrho }\bigl(\mathfrak{S},\zeta _{\delta }(\mathfrak{S}), \zeta _{ \delta }(\mathfrak{S}+\gamma )\bigr)-\mathfrak{f}\bigl(\nu _{\delta }( \mathfrak{S}),\nu _{\delta }(\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{S} \biggr\Vert ^{2} \\& \qquad {}+7 \biggl\{ \biggl(\sqrt{\delta} \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}+ \biggl(\sqrt{\delta} \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \biggr\} \\& \qquad {}\times \mathbb{E}\sup_{0< \mu \leq \epsilon} \biggl\Vert \int _{0}^{\mu }(\mu -\mathfrak{S})^{\iota -1} \bigl\{ \tilde{\vartheta }\bigl(\mathfrak{S},\zeta _{\delta }(\mathfrak{S}), \zeta _{ \delta }(\mathfrak{S}+\gamma )\bigr) \\& \qquad {}-\mathfrak{H}\bigl(\nu _{\delta }( \mathfrak{S}),\nu _{\delta }(\mathfrak{S}+\gamma )\bigr) \bigr\} \,d\mathfrak{W}(\mathfrak{S}) \biggr\Vert ^{2} \\& \qquad {}+7 \biggl\{ \biggl(\sqrt{\delta} \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}+ \biggl(\sqrt{\delta} \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \biggr\} \\& \qquad {}\times \mathbb{E}\sup_{0< \mu \leq \epsilon} \biggl\Vert \int _{0}^{\mu }(\mu -\mathfrak{S})^{\iota -1} \int _{ \Vert \mathfrak{X} \Vert < s} \bigl\{ \mathfrak{R}\bigl( \mathfrak{S}, \zeta _{\delta }(\mathfrak{S}), \zeta _{\delta }( \mathfrak{S}+\gamma ), \mathfrak{X}\bigr) \\& \qquad {}-\mathfrak{C}\bigl( \nu _{\delta }( \mathfrak{S}), \nu _{\delta }(\mathfrak{S}+\gamma ), \mathfrak{X}\bigr) \bigr\} \tilde{\aleph }(d\mathfrak{S},d\mathfrak{X}) \biggr\Vert ^{2} \\& \qquad {}+7 \bigl(\delta \mathfrak{J}^{*}\theta \bigr)^{2} \mathbb{E}\sup_{0< \mu \leq \epsilon} \Biggl\Vert \sum _{0< \mu _{ \kappa }< \mu } \mathfrak{g}_{\kappa }(\zeta _{\delta ,\mu _{\kappa }^{-}})- \int _{0}^{\mu }\Im (\nu _{\delta ,\mathfrak{S}})\,d\mathfrak{S} \Biggr\Vert ^{2} \\& \quad :=T_{1}+T_{2}+T_{3}+T_{4}. \end{aligned}$$

For $T_{1}$, we can obtain

$$\begin{aligned} T_{1} \leq & \Upsilon _{1}\epsilon \int _{0}^{\epsilon}(\epsilon - \mathfrak{S})^{2\iota -2} \mathfrak{G} \Bigl(\mathfrak{S},\mathbb{E} \sup_{0< \mathfrak{S}_{1}\leq \mathfrak{S}} \bigl\Vert \zeta _{ \delta }(\mathfrak{S}_{1})-\nu _{\delta }( \mathfrak{S}_{1}) \bigr\Vert ^{2} \\ &{}+\mathbb{E}\sup_{0< \mathfrak{S}_{1}\leq \mathfrak{S}} \bigl\Vert \zeta _{\delta }( \mathfrak{S}_{1}+\gamma )- \nu _{\delta }(\mathfrak{S}_{1}+ \gamma ) \bigr\Vert ^{2} \Bigr)\,d\mathfrak{S}+\Upsilon _{2} \epsilon ^{2\iota}, \end{aligned}$$

where

$$\begin{aligned}& \Upsilon _{1} = 14 \biggl\{ \biggl(\delta \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}+ \biggl(\delta \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \biggr\} \\& \Upsilon _{2} = \frac{14}{2\iota -1} \biggl\{ \biggl(\delta \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}+ \biggl(\delta \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \biggr\} \\& \hphantom{\Upsilon _{2} =}{}\times \sup_{0< \mu \leq \epsilon}\hbar _{1}( \mu ) \Bigl(1+\mathbb{E}\sup_{0< \mu \leq \epsilon} \bigl\Vert \zeta _{\delta }(\mu ) \bigr\Vert ^{2}+\mathbb{E}\sup _{0< \mu \leq \epsilon} \bigl\Vert \zeta _{ \delta }(\mu +\gamma ) \bigr\Vert ^{2} \Bigr). \end{aligned}$$

For the term $T_{2}$, we can obtain

$$\begin{aligned} T_{2} \leq & \Upsilon _{21}\epsilon \int _{0}^{\epsilon}(\epsilon - \mathfrak{S})^{2\iota -2} \mathfrak{G} \Bigl(\mathfrak{S},\mathbb{E} \sup_{0< \mathfrak{S}_{1}\leq \mathfrak{S}} \bigl\Vert \zeta _{ \delta }(\mathfrak{S}_{1})-\nu _{\delta }( \mathfrak{S}_{1}) \bigr\Vert ^{2} \\ &{}+\mathbb{E}\sup_{0< \mathfrak{S}_{1}\leq \mathfrak{S}} \bigl\Vert \zeta _{\delta }( \mathfrak{S}_{1}+\gamma )- \nu _{\delta }(\mathfrak{S}_{1}+ \gamma ) \bigr\Vert ^{2} \Bigr)\,d\mathfrak{S}+\Upsilon _{22}\epsilon ^{2\iota -1}, \end{aligned}$$

where

$$\begin{aligned}& \Upsilon _{21} = 64 \biggl\{ \biggl(\sqrt{\delta} \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}+ \biggl(\sqrt{\delta} \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \biggr\} \\& \Upsilon _{22} = \frac{64\sqrt{4\iota -3}}{4\iota -3} \biggl\{ \biggl( \delta \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}+ \biggl(\delta \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \biggr\} \\& \hphantom{\Upsilon _{22} =}{} \times \Bigl\{ \sup_{0< \mu \leq \epsilon}\hbar _{2}( \mu ) \Bigl(1+\mathbb{E}\sup_{0< \mu \leq \epsilon} \bigl\Vert \zeta _{ \delta }(\mu ) \bigr\Vert ^{4}+\mathbb{E}\sup _{0< \mu \leq \epsilon} \bigl\Vert \zeta _{\delta }(\mu +\gamma ) \bigr\Vert ^{4} \Bigr) \Bigr\} ^{1/2}. \end{aligned}$$

For $T_{3}$, we have

$$\begin{aligned} T_{3} \leq & \Upsilon _{31}\epsilon \int _{0}^{\epsilon}(\epsilon - \mathfrak{S})^{2\iota -2} \mathfrak{G} \Bigl(\mathfrak{S},\mathbb{E} \sup_{0< \mathfrak{S}_{1}\leq \mathfrak{S}} \bigl\Vert \zeta _{ \delta }(\mathfrak{S}_{1})-\nu _{\delta }( \mathfrak{S}_{1}) \bigr\Vert ^{2} \\ &{}+\mathbb{E}\sup_{0< \mathfrak{S}_{1}\leq \mathfrak{S}} \bigl\Vert \zeta _{\delta }( \mathfrak{S}_{1}+\gamma )- \nu _{\delta }(\mathfrak{S}_{1}+ \gamma ) \bigr\Vert ^{2} \Bigr)\,d\mathfrak{S}+\Upsilon _{32}\epsilon ^{2\iota -1}, \end{aligned}$$

where

$$\begin{aligned}& \Upsilon _{31} = 64 \biggl\{ \biggl(\sqrt{\delta} \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}+ \biggl(\sqrt{\delta} \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \biggr\} \\& \Upsilon _{32} = \frac{64\sqrt{4\iota -3}}{4\iota -3} \biggl\{ \biggl( \delta \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}+ \biggl(\delta \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \biggr\} \\& \hphantom{\Upsilon _{32} =}{} \times \Bigl\{ \sup_{0< \mu \leq \epsilon}\hbar _{2}( \mu ) \Bigl(1+\mathbb{E}\sup_{0< \mu \leq \epsilon} \bigl\Vert \zeta _{ \delta }(\mu ) \bigr\Vert ^{4}+\mathbb{E}\sup _{0< \mu \leq \epsilon} \bigl\Vert \zeta _{\delta }(\mu +\gamma ) \bigr\Vert ^{4} \Bigr) \Bigr\} ^{1/2}. \end{aligned}$$

Remark 3.1

• $$ \sum_{0< \mu _{\kappa}< \mu} \mathfrak{g}_{\kappa}( \zeta _{\delta , \mu _{\kappa}^{-}})=\sum_{i=1}^{\beta} \Biggl(\sum_{0< \mu _{\kappa}< \mu _{i+1}} \mathfrak{g}_{\kappa}( \zeta _{\delta ,\mu _{\kappa}^{-}})- \sum_{0< \mu _{\kappa}< \mu _{i}} \mathfrak{g}_{\kappa}(\zeta _{ \delta ,\mu _{\kappa}^{-}}) \Biggr); $$
• $$ \int _{0}^{\mu}\Im (\zeta _{\delta ,\mu _{\kappa}^{-}})\,d\mathfrak{S}= \sum_{i=1}^{\beta}(\mu _{i+1}-\mu _{i}) \int _{0}^{\mu}\Im (\zeta _{ \delta ,\mu _{\kappa}^{-}}), $$

  where β is the specified number of impulses on $[0,\rho ]$;

• ℑ satisfies the Lipschitz condition, ∃ a constant $M>0$ involving $\ell ^{*}$ s.t.

  $$ \bigl\Vert \Im (\eta _{1})-\Im (\eta _{2}) \bigr\Vert ^{2}\leq M \Vert \eta _{1}-\eta _{2} \Vert ^{2}, \quad \forall \eta _{1}, \eta _{2}\in \mathbb{S}. $$

For the last term, we obtain the following estimation:

$$\begin{aligned} T_{4} \leq &28 \bigl(\delta \mathfrak{J}^{*}\theta \bigr)^{2}\mathbb{E} \sup_{0< \mu \leq \epsilon} \Biggl\Vert \sum _{0< \mu _{\kappa }< \mu } \mathfrak{g}_{\kappa }( \zeta _{\delta ,\mu _{\kappa }^{-}})- \int _{0}^{ \mu }\Im (\zeta _{\delta ,\mu _{\kappa }^{-}})\,d\mathfrak{S} \Biggr\Vert ^{2} \\ &{}+28 \bigl(\delta \mathfrak{J}^{*}\theta \bigr)^{2} \mathbb{E}\sup_{0< \mu \leq \epsilon} \biggl\Vert \int _{0}^{\mu }\Im ( \zeta _{\delta ,\mu _{\kappa }^{-}})- \int _{0}^{\mu }\Im (\nu _{ \delta ,\mu _{\kappa }^{-}})\,d\mathfrak{S} \biggr\Vert ^{2} \\ &{}+28 \bigl(\delta \mathfrak{J}^{*}\theta \bigr)^{2} \mathbb{E}\sup_{0< \mu \leq \epsilon} \biggl\Vert \int _{0}^{\mu }\Im ( \nu _{\delta ,\mu _{\kappa }^{-}})- \int _{0}^{\mu }\Im (\nu _{\delta , \mathfrak{S}})\,d\mathfrak{S} \biggr\Vert ^{2} \\ \leq &28 \bigl(2\delta \mathfrak{J}^{*}\theta \epsilon \bigr)^{2}( \beta +1)\sup_{0< \mu \leq \epsilon}\hbar _{4} \bigl(1+\mathbb{E} \Vert \zeta _{\delta ,\mu _{\kappa }^{-}} \Vert ^{2} \bigr) \\ &{}+28 \bigl(\delta \mathfrak{J}^{*}\theta \bigr)^{2}\epsilon M \int _{0}^{ \epsilon}\mathbb{E} \Vert \zeta _{\delta ,\mathfrak{S}}-\nu _{ \delta ,\mathfrak{S}} \Vert ^{2}\,d\mathfrak{S} \\ &{}+28 \bigl(\delta \mathfrak{J}^{*}\theta \epsilon ^{ \iota +1} \bigr)^{2} M P_{1}+56 \bigl(\delta \mathfrak{J}^{*} \theta \bigr)^{2} \epsilon ^{2\iota +1}M P_{2}, \end{aligned}$$

where

$$\begin{aligned}& P_{1} = \biggl\{ \biggl(\delta \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}+ \biggl(\delta \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \biggr\} F_{1} \\& P_{2} = \biggl\{ \biggl(\sqrt{\delta} \frac{\theta \psi (1-\iota )}{\varpi (\iota )\Gamma (\iota )} \biggr)^{2}+ \biggl(\sqrt{\delta} \frac{\iota \theta ^{2}\mathfrak{J}_{1}^{*}}{\varpi (\iota )} \biggr)^{2} \biggr\} F_{2} \end{aligned}$$

and $F_{1}$, $F_{2}$ are two positive constants. Since $\mathfrak{G}(\mu , \zeta )$ is a concave function, ∃ two functions: $\mathcal{X}(\mu )>0$ and $\mathcal{Z}(\mu )>0$, s.t.

$$ \mathfrak{G}(\mu , \zeta )\leq \mathcal{X}(\mu )+\mathcal{Z}(\mu ) \zeta ,\qquad \int _{0}^{\epsilon}\mathcal{X}(\mu )\,d\mu < \infty ,\qquad \int _{0}^{\epsilon}\mathcal{Z}(\mu )\,d\mu < \infty . $$

Hence,

$$\begin{aligned}& \mathbb{E}\sup_{0< \mu \leq \epsilon} \bigl\Vert \zeta _{\delta }( \mu )-\nu _{\delta }(\mu ) \bigr\Vert ^{2} \\& \quad = \biggl( \frac{\Upsilon _{1}\epsilon ^{2\iota}+(\Upsilon _{21}+\Upsilon _{31})\epsilon ^{2\iota -1}}{2\iota -1} \biggr)\mathcal{X}^{*}+ (\Upsilon _{1} \epsilon +\Upsilon _{21}+ \Upsilon _{31} ) \mathcal{Z}^{*} \int _{0}^{\epsilon}(\epsilon - \mathfrak{S})^{2\iota -2} \\& \qquad {}\times \Bigl(\mathbb{E}\sup_{0< \mathfrak{S}_{1} \leq \mathfrak{S}} \bigl\Vert \zeta _{\delta }(\mathfrak{S}_{1})-\nu _{ \delta }( \mathfrak{S}_{1}) \bigr\Vert ^{2}+ \mathbb{E}\sup _{0< \mathfrak{S}_{1}\leq \mathfrak{S}} \bigl\Vert \zeta _{\delta }( \mathfrak{S}_{1}+\gamma )-\nu _{\delta }(\mathfrak{S}_{1}+ \gamma ) \bigr\Vert ^{2} \Bigr)\,d\mathfrak{S} \\& \qquad {}+\Upsilon _{2}\epsilon ^{2\iota}+(\Upsilon _{22}+ \Upsilon _{32}) \epsilon ^{2\iota -1}+28 \bigl(2\delta \mathfrak{J}^{*}\theta \epsilon \bigr)^{2}(\beta +1)\tilde{ \Upsilon} \\& \qquad {}+28 \bigl(\delta \mathfrak{J}^{*} \theta \bigr)^{2} \epsilon M \int _{0}^{\epsilon}\mathbb{E} \Vert \zeta _{\delta ,\mathfrak{S}}-\nu _{\delta ,\mathfrak{S}} \Vert ^{2}\,d\mathfrak{S} \\& \qquad {}+28 \bigl(\delta \mathfrak{J}^{*}\theta \epsilon ^{ \iota +1} \bigr)^{2} M P_{1}+56 \bigl(\delta \mathfrak{J}^{*} \theta \bigr)^{2} \epsilon ^{2\iota +1}M P_{2}, \end{aligned}$$

where $\mathcal{X}^{*}=\sup_{0<\mu \leq \epsilon}\mathcal{X}(\mu )$, $\mathcal{Z}^{*}=\sup_{0<\mu \leq \epsilon}\mathcal{Z}(\mu )$, and $\tilde{\Upsilon}=\sup_{0<\mu \leq \epsilon}\hbar _{4} (1+ \mathbb{E} \Vert \zeta _{\delta ,\mu _{\kappa }^{-}} \Vert ^{2} )$. Using the fact that $\mathbb{E} (\sup_{-\xi \leq \mu <0} \Vert \zeta _{\delta }( \mu )-\nu _{\delta }(\mu ) \Vert ^{2} )=0$, and letting $\Lambda (\epsilon )=\mathbb{E} (\sup_{0<\mu \leq \epsilon} \Vert \zeta _{\delta }(\mu )-\nu _{\delta }(\mu ) \Vert ^{2} )$, we obtain

$$ \mathbb{E} \Bigl(\sup_{0< \mathfrak{S}_{1}\leq \mathfrak{S}} \bigl\Vert \zeta _{\delta }(\mathfrak{S}_{1})-\nu _{\delta }( \mathfrak{S}_{1}) \bigr\Vert ^{2} \Bigr)=\Lambda ( \mathfrak{S}-\psi ),\quad 0< \psi \leq \xi . $$

Then,

$$\begin{aligned} \Lambda (\epsilon ) \leq & \biggl( \frac{\Upsilon _{1}\epsilon ^{2\iota}+(\Upsilon _{21}+\Upsilon _{31})\epsilon ^{2\iota -1}}{2\iota -1} \biggr) \mathcal{X}^{*} \\ &{}+ (\Upsilon _{1}\epsilon +\Upsilon _{21}+ \Upsilon _{31} )\mathcal{Z}^{*} \int _{0}^{\epsilon}(\epsilon - \mathfrak{S})^{2\iota -2} \bigl(\Lambda (\mathfrak{S})-\Lambda ( \mathfrak{S}-\psi ) \bigr)\,d\mathfrak{S} \\ &{}+\Upsilon _{2}\epsilon ^{2\iota}+(\Upsilon _{22}+ \Upsilon _{32}) \epsilon ^{2\iota -1}+28 \bigl(2\delta \mathfrak{J}^{*}\theta \epsilon \bigr)^{2}(\beta +1)\tilde{ \Upsilon} \\ &{}+28 \bigl(\delta \mathfrak{J}^{*} \theta \bigr)^{2} \epsilon M \int _{0}^{\epsilon}\mathbb{E} \Vert \zeta _{\delta ,\mathfrak{S}}-\nu _{\delta ,\mathfrak{S}} \Vert ^{2}\,d\mathfrak{S} \\ &{}+28 \bigl(\delta \mathfrak{J}^{*}\theta \epsilon ^{ \iota +1} \bigr)^{2} M P_{1}+56 \bigl(\delta \mathfrak{J}^{*} \theta \bigr)^{2} \epsilon ^{2\iota +1}M P_{2}. \end{aligned}$$

By setting $\Delta _{\epsilon}=\sup_{-\xi \leq \mu \leq \epsilon}\Lambda (\mu )$, $\forall \mu \in \mathfrak{J}$, then $\Lambda (\mathfrak{S})\leq \Delta _{\mathfrak{S}}$, and $\Lambda (\mathfrak{S}-\psi )\leq \Delta _{\mathfrak{S}}$, $0<\psi \leq \xi$. Hence, we obtain

$$\begin{aligned} \Lambda (\epsilon ) \leq & \biggl( \frac{\Upsilon _{1}\epsilon ^{2\iota}+(\Upsilon _{21}+\Upsilon _{31})\epsilon ^{2\iota -1}}{2\iota -1} \biggr) \mathcal{X}^{*}+ (\Upsilon _{1}\epsilon +\Upsilon _{21}+ \Upsilon _{31} )\mathcal{Z}^{*} \int _{0}^{\epsilon}(\epsilon - \mathfrak{S})^{2\iota -2}2 \Delta _{\mathfrak{S}}\,d\mathfrak{S} \\ &{}+\Upsilon _{2}\epsilon ^{2\iota}+(\Upsilon _{22}+ \Upsilon _{32}) \epsilon ^{2\iota -1}+28 \bigl(2\delta \mathfrak{J}^{*}\theta \epsilon \bigr)^{2}(\beta +1)\tilde{ \Upsilon} \\ &{}+28 \bigl(\delta \mathfrak{J}^{*} \theta \bigr)^{2} \epsilon M \int _{0}^{\epsilon}\mathbb{E} \Vert \zeta _{\delta ,\mathfrak{S}}-\nu _{\delta ,\mathfrak{S}} \Vert ^{2}\,d\mathfrak{S} \\ &{}+28 \bigl(\delta \mathfrak{J}^{*}\theta \epsilon ^{ \iota +1} \bigr)^{2} M P_{1}+56 \bigl(\delta \mathfrak{J}^{*} \theta \bigr)^{2} \epsilon ^{2\iota +1}M P_{2}. \end{aligned}$$

Due to the generalized Gronwall inequality, $\exists h_{0}>0$ and $\varepsilon \in (0,1)$, s.t.

$$ \mathbb{E}\sup_{-\xi \leq \mu < h_{0}\delta ^{-\varepsilon}} \bigl\Vert \zeta _{\delta }(\mu )-\nu _{\delta }(\mu ) \bigr\Vert ^{2} \leq M_{0} \delta ^{1-\varepsilon} $$

holds $\forall \mu \in (0, h_{0}\delta ^{-\varepsilon}]\subset \mathfrak{J}$, where

$$\begin{aligned} M_{0} =& \biggl\{ \biggl[\frac{\Upsilon _{1}\mathcal{X}^{*}}{2\iota -1}+ \Upsilon _{2} \biggr]h_{0}^{2\iota}\delta ^{\varepsilon -2\varepsilon \iota -1}+ \biggl[ \frac{(\Upsilon _{21}+\Upsilon _{31})}{2\iota -1}+ \Upsilon _{22}+\Upsilon _{32} \biggr]h_{0}^{2\iota -1}\delta ^{2 \varepsilon -2\varepsilon \iota -1} \\ &{}+56 \bigl(\mathfrak{J}^{*}\theta h_{0} \bigr)^{2}(\beta +1) \tilde{\Upsilon}\delta ^{1-\varepsilon}+28\bigl( \mathfrak{J}^{*}\theta \bigr)^{2}MP_{1}h_{0}^{2 \iota +2} \delta ^{1-\varepsilon -2\varepsilon \iota} \\ &{}+58\bigl(\mathfrak{J}^{*} \theta \bigr)^{2}MP_{2}h_{0}^{2\iota +1}\delta ^{1-2\varepsilon \iota} \biggr\} \\ &{} \times \mathbb{W}_{2\iota -1} \bigl( \bigl[28\bigl(\delta \mathfrak{J}^{*}\theta \bigr)^{2}h_{0}^{2} \delta ^{-2\varepsilon}M+2\bigl( \Upsilon _{1}\mathcal{Z}^{*}h_{0}^{2\iota} \delta ^{-2\iota \varepsilon}\bigr)+(\Upsilon _{21}+\Upsilon _{31}) \mathcal{Z}^{*} \bigr] \Gamma (2\iota -1) \bigr) \end{aligned}$$

is a constant. According to the aforementioned analysis, for an arbitrary small number $\tau >0$, $\exists \delta _{2}\in (0,\delta _{1}]$ s.t. for $\iota \in (3/4,1)$, for every $\delta \in (0,\delta _{2}]$, and $\forall \mu \in [-\xi , h_{0}\delta ^{-\varepsilon}]$,

$$ \mathbb{E} \Bigl(\sup_{\mu \in [-\xi , h_{0}\delta ^{-\varepsilon}]} \bigl\Vert \zeta _{\delta }(\mu )-\nu _{\delta }(\mu ) \bigr\Vert ^{2} \Bigr)\leq \tau . $$

 □

4 Applications

Consider the impulsive ABC-fractional stochastic partial differential as follows:

$$\begin{aligned} \textstyle\begin{cases} {}^{\mathfrak{ABC}}D_{0+}^{7/8} \zeta _{\delta}(\mu , \imath ) \\ \quad = \mathfrak{E} \zeta _{\delta}(\mu , \imath )+2\delta \hat{\varrho}_{1} \zeta _{\delta}(\mu , \imath )\sin ^{2}(\mu )+\sqrt{\delta} \hat{\varrho}_{2}\cos ^{2}(\mu )\zeta _{\delta}(\mu , \imath ) \frac{d\mathfrak{W}(\mu )}{d\mu} \\ \qquad {} +\sqrt{\delta}\frac{1}{d\mu}\int _{ \Vert \mathfrak{X} \Vert < s} 2\hat{\varrho}_{3}^{4}\cos ^{2}(\mu )\zeta _{ \delta}(\mu , \imath )\tilde{\aleph}(d\mu ,d\mathfrak{X}), \\ \quad \mu \in \mathfrak{J}:=(0,\pi ], \imath \in [0, \pi ], \mu \neq \mu _{\kappa} \\ \zeta _{\delta}(\mu , 0)=\zeta _{\delta}(\mu , \pi )=0, \\ \Delta \zeta _{\delta}| _{\mu =\mu _{\kappa}}=\mathfrak{g}_{\kappa}( \zeta _{\delta}(\mu _{\kappa}^{-}))=\delta \tan ^{-1}(\zeta _{\delta}( \mu _{\kappa}^{-})), \quad \mu =\mu _{\kappa}, \kappa =1,2, \ldots ,\rho \\ \zeta _{\delta}(x,\imath )=\Psi (x,\imath ), \quad -\xi \leq x\leq 0, \\ \zeta _{\delta}(0)=\Psi _{0}, \end{cases}\displaystyle \end{aligned}$$

(8)

where ${}^{\mathfrak{ABC}}D_{0+}^{7/8}$ is an ABC-fractional derivative of order $\xi =7/8$. $\mathcal{A}=\mathcal{L}^{2}([0,\pi ])$ is the separable Hilbert space and $\mathfrak{E} : \mathbf{D}(\mathfrak{E})\rightarrow \mathcal{A}$ is defined by $\mathfrak{E}\zeta =\zeta ^{\prime \prime }$, with $\mathbf{D}(\mathfrak{E})= \{\zeta \in \mathcal{A}: \zeta , \zeta ^{\prime }, are absolutely continuous , \zeta ^{\prime \prime }\in \mathcal{A}, \zeta (0)=\zeta (\pi )=0 \}$. Then, $\mathfrak{E}$ generates the strongly continuous semigroup $\{\mathbb{A}(\mu )\}_{\mu \geq 0}$ on a separable Hilbert space $\mathcal{A}$ that is given by $\mathbb{A}(\mu )\zeta :=\sum_{m=1}^{\infty}e^{ \frac{-m^{2}}{1+m^{2}}\mu}\langle \zeta ,\zeta _{m}\rangle \zeta _{m}$, and $\mathfrak{Q}_{\frac{7}{8}}(\mu ):=\mu ^{\frac{-1}{8}}\int _{0}^{ \infty}\frac{7}{8}\mathfrak{S}\mathbf{M}_{\frac{7}{8}}(\mathfrak{S}) \mathbb{A}(\mathfrak{S}\mu ^{\frac{7}{8}})\,d\mathfrak{S}$. Thus,

$$ \mathfrak{E}\zeta =\sum_{m=1}^{\infty}m^{2} \langle \zeta ,\zeta _{m} \rangle \zeta _{m},\quad \zeta \in \mathbf{D}(\mathfrak{E}), $$

where $\zeta _{m}(\mathfrak{S})=\sqrt{2/\pi}\sin m \mathfrak{S}$, $m=1,2,3, \ldots$ is an orthogonal set of eigenvectors of $\mathfrak{E}$. Let $\tilde{\rho}=\pi$ and let

$$\begin{aligned}& \frac{1}{\pi} \int _{0}^{\pi}\tilde{\varrho}\bigl(\mathfrak{S}, \zeta _{ \delta}(\mathfrak{S}), \zeta _{\delta ,\mathfrak{S}}\bigr)\,d\mathfrak{S} = \mathfrak{f}(\zeta _{\delta}, \zeta _{\delta ,\mu})=\hat{ \varrho}_{1} \zeta _{\delta} , \\& \frac{1}{\pi} \int _{0}^{\pi}\tilde{\vartheta}\bigl(\mathfrak{S}, \zeta _{ \delta}(\mathfrak{S}), \zeta _{\delta ,\mathfrak{S}}\bigr)\,d\mathfrak{S} = \mathfrak{H}(\zeta _{\delta}, \zeta _{\delta ,\mu})= \frac{\hat{\varrho}_{2}\zeta _{\delta}}{2} \\& \frac{1}{\pi} \int _{0}^{\pi}\mathfrak{R}\bigl(\mathfrak{S}, \zeta _{\delta}( \mathfrak{S}), \zeta _{\delta ,\mathfrak{S}}, \mathfrak{X}\bigr)\,d\mathfrak{S} = \mathfrak{C}(\zeta _{\delta}, \zeta _{\delta ,\mu}, \mathfrak{X})=\hat{\varrho}_{3}^{4}\zeta _{\delta}. \end{aligned}$$

Therefore, with the aforementioned choices, the fulfillment of all the requirements stated in Theorem 3.2 may be simply verified. Accordingly, system (7) may be used to define the averaged equation for (8). It is obvious that the time-averaged system is far simpler than the original system (8). Furthermore, their solutions are guaranteed to have a very small error by Theorem 3.2.

5 Conclusions

An averaging principle for a family of impulsive Atangana–Baleanu fractional stochastic delay differential equations driven by the Lévy process was derived in this study. In contrast to earlier research, we jointly consider the ABC-fractional derivative, evolution equations, and Lévy process. As a consequence, ABC-fractional differential equations may be solved using the stochastic averaging approach thanks to our suggested results. Furthermore, it is crucial that future studies look at the averaging principle for noninstantaneous impulsive Atangana–Baleanu fractional SDDE driven by both the Brownian motion process and fractional Brownian motion as these processes are highly prevalent in real life.