Fractional stochastic delay differential systems (FSDDs) are mathematical models that involve fractional derivatives, stochastic noise, and time delays. The fractional derivatives represent the memory effects and long-range dependence in the system, while the stochastic noise and delays account for the random fluctuations and time delays, respectively. FSDDs find applications in many fields, including physics, biology, finance, and engineering. They can be used to model systems with memory and randomness, such as anomalous diffusion processes, fractional-order control systems with stochastic disturbances, and biological systems with fractional-order kinetics and stochastic effects. They provide a powerful framework for understanding and predicting the behavior of complex systems with memory, randomness, and time delays. See for examples [1-6], and the references cited therein.

The averaging principle is a mathematical tool used to simplify the analysis of dynamical systems with fast and slow time scales. It provides an approximate description of the system’s behavior. In 1968, Khasminskii [7] first used the average principle to prove that the solution of the average equation can converge to the solution of the complex system. In [8], the authors presented an averaging method for stochastic differential equations with non-Gaussian L$\acute{\mathrm{e}}$vy noise. With the development of fractional calculus, many works have emerged that apply the averaging principle to fractional stochastic differential equations (FSDEs). In [9], Xu, et.al. presents an averaging principle for Caputo FSDEs driven by Brown motion. In [10], Luo, et.al. established an averaging principle for the solution of the a class of FSDEs with time-delays. In [11], Ahmed and Zhu investigated the averaging principle for the Hilfer fractional stochastic delay differential equation with Poisson jumps in the sense of mean square. The periodic averaging method for impulsive conformable fractional stochastic differential equations with Poisson jumps are discussed in [12] by Ahmed. In [13], Wang and Lin extended the averaging principle of the following FSDEs in the sense of mean square ($L^2$ convergence) to $L^p$ convergence ($p\ge 2$), which generated some works on the averaging principle for FSDES [9,10,14]. In [15], Yang, et.al. studied the averaging principle for a class of $\psi$-Capuo fractional stochastic delay differential equations with Poisson jumps.

Recently, Li and Wang in [16] studied the following Caputo type FSDDEs: the existence, uniqueness and the averaging principle for (1.2) are established.

In the present paper, motivated by [11,13,16], we study the following Caputo FSDDEs with Poisson jumps where ${ }^C{D}_0^{\alpha }$ is the left Caputo fractional derivative with $\frac{1}{2}<\alpha <1$, $J=[0,T]$, $A,B\in {\mathbb{R}}^{n\times n}$ are two constant matrices, the state vector $x\in {\mathbb{R}}^n$ is a stochastic process, $f:J\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, $\sigma :J\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^{n\times m}$ and $g:J\times {\mathbb{R}}^n\times {\mathbb{R}}^n\times V\to {\mathbb{R}}^n$ are measurable continuous functions. Let $(\Omega ,\mathcal{F},P)$ be a complete probability space equipped with some filtration ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$ satisfying the usual condition, $W\left(t\right)$ is an m-dimensional Brownian motion on the probability space $(\Omega ,\mathcal{F},P)$ adapted to the filtration ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$. Let $(V,\Phi ,\lambda (dv\left)\right)$ be a $\sigma$-finite measurable space. Given stationary Poisson point process ${\left(p_t\right)}_{t\ge 0}$, which is defined on $(\Omega ,\mathcal{F},P)$ with values in V and with characteristic measure $\lambda$. We denote by $N(t,dv)$ the counting measure of $p_t$ such that $\overline{N}(t,\Theta ):=\mathbb{E}\left(N(t,\Theta )\right)=t\lambda (\Theta )$ for $\Theta \in \Phi$. Define $\overline{N}(t,dv):=N(t,dv)-t\lambda \left(dv\right)$, and the Poisson martingale measure generated by $p_t$.

In this paper, we first prove the existence and uniqueness of solutions of Caputo type FSDDEs (1.3) by using delayed perturbation of Mittag-Leffler function and Banach fixed point theorem; Secondly, we prove the averaging principle for Caputo FSDDEs (1.3) in the sense of $L_p$ (pth moment) with inequality techniques. The main contributions and advantages of this paper are as follows:

(1) The solution of the averaged FSDDEs converges to that of the standard FSDDEs in the sense of $L_p$, which is a generalization of the existing result ($p=2$) of the averaging principle for FSDDEs,

(2) The fractional calculus, stochastic inequality and H$\ddot{\mathrm{o}}$lder inequality are effectively used to establish our result.

(3) our work in this paper is novel and more technical. Our result extends the main results of [17].

This paper will be organized as follows. In Section 2, we will briefly recall some definitions and preliminaries. In Section 3, we prove the existence and uniqueness of solutions for Caputo FSDDEs (1.3) with Poisson jumps. In Section 4, we prove that the solution of the FSDDEs (1.3) converges to that of the standard one in $L_p$ sense. In Section 5, an example is presented to illustrate our theoretical results. Finally, the paper is concluded in Section 6.

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

Let ${\rm Y}={\mathbb{L}}^p(\Omega ,\mathcal{F},\mathbb{P})$ denote the space of all $\mathcal{F}\left(t\right)$ measurable, p squqre integrable functions $x:\Omega \to {\mathbb{R}}^n$ with ${\parallel x\left(t\right)\parallel }_{ps}:={\left({\displaystyle \sum _{i=1}^n}\mathbb{E}\left(\right|x_i\left(t\right){|}^p)\right)}^{1/p}$, and $\parallel x\parallel ={\displaystyle \sum _{i=1}^n}\left|x_i\right|$ and $\parallel A\parallel =\underset{1\le i\le n}{max}{\displaystyle \sum _{i=1}^n}\left|a_{ij}\right|$ be the vector norm and matrix norm, respectively. A process $x:[-\tau ,T]\to {\mathbb{L}}^p(\Omega ,\mathcal{F},\mathbb{P})$ is said to be $\mathcal{F}\left(t\right)$-adapted if $x\left(t\right)\in {\rm Y}$.

Definition 2.1 [17]. Let $\alpha >0$, and f be an integrable function defined on $[a,b]$. The left Riemann-Liouville fractional integral operator of order $\alpha$ of a function f is defined by

Definition 2.2 [17]. Let $n-1<\alpha <n$, and $f\in C^n\left([a,b]\right)$. The left Caputo fractional derivative of order $\alpha$ of a function f is defined by where $n=\left[\alpha \right]+1$.

Definition 2.3 [18]. The coefficient matrices $Q_k\left(s\right)$, $k=0,1,2,...$, satisfy the following multivariate determining matrix equation where I is an identity matrix and $\Theta$ is a zero matrix.

Definition 2.4 [18]. Delayed perturbation of two parameter Mittag-Leffler type matrix function $X_{\tau ,\alpha ,\beta }^{A,B}$ generated by $A,B$ is defined by

From [17], we can easily obtain the following definition.

Definition 2.5. A ${\mathbb{R}}^n$-value stochastic process $\left\{x\right(t):-\tau \le t\le T\}$ is called a solution of (1.3) if $x\left(t\right)$ satisfies the integral equation of the following form: where $x\left(t\right)$ is $\mathcal{F}\left(t\right)$-adapted and $\mathbb{E}({\int }_{-\tau }^T\parallel x\left(t\right){\parallel }^{p}dt)<\infty$.

Lemma 2.1 ([19]). For any $t\ge 0$, $0<\alpha <1$, $0<\beta \le 1$ and $\alpha +\beta \ge 1$, we have

Lemma 2.2For any $p\ge 2$, $\alpha \in \left(1-\frac{1}{p},1\right)$ and $\gamma >0$, one has

Proof. Let $\gamma >0$ be arbitrary. Consifer the corresponding linear Caputo fractional differential equation of the following form

From [20], it is easy to know that the Mittag-Leffler function $E_{p\alpha -p+1,1}\left(\gamma t^{p\alpha -p+1}\right)$ is a solution of (2.7). So, the following equality holds: which completes the proof.

Lemma 2.3 ([21, 22]). Let $\varphi :R_{+}\times V\to R^n$ and assume that

Then there exists $D_p>0$ such that

Lemma 2.4 ([23]). Let $u,v$ be two integrable functions and g be continuous defined on domain $[a,b]$. Moreover, assume that

If

then

To study the qualitative properties of solution for (1.3), we impose the following conditions on data of the problem.

(H1) For any $x_1,x_2,y_1,y_2\in R^n$ and $t\in J$, there exist two constants $C_1,C_2>0$ such that

where $\parallel ·\parallel$ is the norm of ${\mathbb{R}}^n$, $x\vee y=max\{x,y\}$.

(H2) Let $\sigma (·,0,0)$ and $g(·,0,0)$ are essentially bounded, i.e. and $f(·,0,0)$ is ${\mathbb{L}}^p$ integrable, i.e.

Let ${\mathbb{H}}^p\left([0,T]\right)$ be the space of all the processes x which are measurable, $\mathcal{F}\left(t\right)$-adapted, and satisfied that ${\parallel x\parallel }_{{\mathbb{H}}^p}:=\underset{0\le t\le T}{sup}{\parallel x\left(t\right)\parallel }_{ps}<\infty$. Obviously, $({\mathbb{H}}^p\left([0,T]\right),\parallel ·{\parallel }_{{\mathbb{H}}^p})$ is a Banach space. Set $\mu =\parallel A\parallel +\parallel B\parallel$. For any $t\in [-\tau ,T]$ and $\varphi \in C([-\tau ,0],{\mathbb{R}}^n)$, we define an operator $\mathcal{T}:{\mathbb{H}}^p\left([0,T]\right)\to {\mathbb{H}}^p\left([0,T]\right)$ as follows :

Lemma 3.1Let $1-\frac{1}{p}<\alpha <1$. Assume that (H1) and (H2) hold. Then the operator $\mathcal{T}$ is well-defined.

Proof. For any $x\in {\mathbb{H}}^p\left([0,T]\right)$, by (3.1) and the following elementary inequality we have

For $I_1$, from Lemma 2.1, one has

For $I_2$, by Lemma 2.1, H$\ddot{\mathrm{o}}$lder inequality and $\alpha >1-\frac{1}{p}$, we obtain where $\frac{1}{p}+\frac{1}{q}=1$ and $\Xi ={\left({\int }_{-\tau }^0{\parallel {(}^C{D}_{-{\tau }^{+}}^{\alpha }\varphi )\left(s\right)-A\varphi \left(s\right)\parallel }^{q}ds\right)}^{p-1}<\infty$.

For $I_3$, applying (H1), (H2), H$\ddot{\mathrm{o}}$lder inequality, Lemma 2.1 and Jensen inequality, one has since

For $I_4$, by using (H1), (H2), Cauchy-Schwarz inequality, Ito’s isometry, Lemma 2.1 and Jensen inequality, we have

For $I_5$, by using (H1), (H2), Lemmas 2.1, 2.3 and Jensen inequality, we obtain

Submitting (3.4)-(3.8) into (3.3), which implies that ${\parallel \mathcal{T}x\parallel }_{{\mathbb{H}}^p}<\infty$. Thus, the operator $\mathcal{T}$ is well-defined.

Theorem 3.1Let $1-\frac{1}{p}<\alpha <1$. Assume that $\left(H1\right)$ and $\left(H2\right)$ hold, then (1.3) has a unique solution $x\in {\mathbb{H}}^p\left([0,T]\right)$.

Proof. For $T>0$, we choosing and fix a constant $\gamma >0$ such that

On the space ${\mathbb{H}}^p\left([0,T]\right)$, we define a weighted norm ${\parallel ·\parallel }_{\gamma }$ as below

Similar to the Theorem 1 in [18], It is easy to know that the norms ${\parallel ·\parallel }_{{\mathbb{H}}^p}$ and ${\parallel ·\parallel }_{\gamma }$ are equivalent. Hence, $({\mathbb{H}}^p\left([0,T]\right),\parallel ·{\parallel }_{\gamma })$ is a Banach space. We can easily prove that $\mathcal{T}:{\mathbb{H}}^p\left([0,T]\right)\to {\mathbb{H}}^p\left([0,T]\right)$ defined in (3.1) is uniformly bounded operator by Lemma 3.1. Next, we only check that $\mathcal{T}$ is a contraction operator.

Firstly, by using H$\ddot{\mathrm{o}}$lder inequality, (H1) and Lemma 2.1, we obtain

Secondly, similar to the proof of (3.7), one has

Thirdly, similar to the proof of (3.8), we obtain

(3.12)

For each $x,y\in {\mathbb{H}}^p\left([0,T]\right)$, from (3.1), (3.2), and (3.10)-(3.12), we have