On Averaging Principle for Caputo–Hadamard Fractional Stochastic Differential Pantograph Equation

Abstract

In this paper, we studied an averaging principle for Caputo–Hadamard fractional stochastic differential pantograph equation (FSDPEs) driven by Brownian motion. In light of some suggestions, the solutions to FSDPEs can be approximated by solutions to averaged stochastic systems in the sense of mean square. We expand the classical Khasminskii approach to Caputo–Hadamard fractional stochastic equations by analyzing systems solutions before and after applying averaging principle. We provided an applied example that explains the desired results to us.

1. Introduction

The nature of solutions for fractional stochastic differential pantograph equations (FSDPEs) in Euclidean space n-dimensional ${\mathbb{R}}^n$ [1,2], is particularly interesting in practical applications. In general, the systems take the form

$$\left\{\begin{array}{c}{\mathfrak{D}}_{\varsigma }^{\alpha }\mathcal{X}\left(\varsigma \right)=b(\varsigma ,\mathcal{X}\left(\varsigma \right),\mathcal{X}(1+\eta \varsigma ))+{\sigma }_1(\varsigma ,\mathcal{X}\left(\varsigma \right),\mathcal{X}(1+\eta \varsigma )){\displaystyle \frac{d\mathfrak{B}\left(\varsigma \right)}{d\varsigma }}\hfill \\ \mathcal{X}\left(1\right)={\mathcal{X}}_0,\hfill \end{array}\right.$$

(1)

where $\eta \in \left(0,\frac{T-1}{T}\right)$, ${\mathfrak{D}}_{\varsigma }^{\alpha }$ is the Caputo–Hadamard fractional derivative (CHFD), $\alpha \in (\frac{1}{2},1),$ for each $\varsigma \ge 1$, $b:\left[1,T\right]\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ and ${\sigma }_1:\left[1,T\right]\times {\mathbb{R}}^n\to {\mathbb{R}}^{n\times m}$ are measurable continuous functions (CF), $\mathfrak{B}\left(\varsigma \right)$ is a m-dimensional standard Brownian motion on $\left\{\Omega ,\mathfrak{F},P\right\}$ probability space. The initial value ${\mathcal{X}}_0$ is an ${\mathfrak{F}}_0$-measurable ${\mathbb{R}}^n$-value random variable, satisfying $E{\left|{\mathcal{X}}_0\right|}^2<\infty .$

Solutions of non-linear FSDPEs are almost impossible to solve and very difficult. For this reason we used symmetrical methods and techniques in the widest field. It plays very important in modernity of partial calculus [3,4].

In [5], Khasminiskii was interested in studying the convergence of idle systems on the drag time scale $\epsilon \to 0,$ in resolving intermediate arguments. He concluded that averaging principle lay in the study of equations lost in terms of the relevant average. So, we have an easy way to solve these equations, as it is known that such equations have been applied to many numerical algorithms to different models, including FSDEs see [6,7].

The generalized pantograph equation has a variety of applications. Only applications in number theory are mentioned [8], in electrodynamics [9] and in the absorption of energy by the pantograph of an electronic locomotive [10,11,12,13].

We rely on this article, which aims to expand Khasminskii’s classic argument into random fractional differential equations with CHFD. For our goal, with the help of rigorous mathematical deduction, which here accurately illustrates the fractional averaging principle mean square that has been reached. This means that an easy and effective way has been given to solve the FSDPEs (1) accurately. We have arranged the organization of this article as follows. We present in the Section 2 some basic ideas, definitions, lemmas and arguments. In Section 3, we explain an averaging principle obtained first, and complete with a main result. To explain this, we give a specific illustrative example.

2. Preliminaries

In this section, we introduce some basic techniques, definitions, lemmas and theorems (see [14,15,16,17,18,19]).

Definition 1

([2,19]). The Riemann–Liouville fractional integral (RLFI) of order $\alpha >0$ for a function $x:\left[0,+\infty \right)\to \mathbb{R}$ is defined as

$$I^{\alpha }x\left(\varsigma \right)=\frac{1}{{\Gamma }\left(\alpha \right)}{\int }_0^{\varsigma }{(\varsigma -s)}^{\alpha -1}x\left(s\right)ds,$$

where Γ is the Euler gamma function and it is defined by

$${\Gamma }\left(\alpha \right)={\int }_0^{\infty }{e}^{-\varsigma }{\varsigma }^{\alpha -1}d\varsigma .$$

Definition 2

([2,19]). The Hadamard fractional integral of order $\alpha >0$ for a CF $x:\left[1,+\infty \right)\to \mathbb{R}$ is defined as

$${\mathfrak{I}}_1^{\alpha }x\left(\varsigma \right)=\frac{1}{{\Gamma }\left(\alpha \right)}{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}x\left(s\right)\frac{ds}{s}.$$

Definition 3

([2,19]). The Riemann–Liouville fractional derivative (RLFD) of order $\alpha >0$ for a CF $x:\left[0,+\infty \right)\to \mathbb{R}$ is defined as

$$D^{\alpha }x\left(\varsigma \right)=\frac{1}{{\Gamma }(n-\alpha )}{\int }_0^{\varsigma }{\left(\varsigma -s\right)}^{n-\alpha -1}{x}^{\left(n\right)}\left(s\right)ds,n-1<\alpha <n,n\in \mathbb{N}.$$

Definition 4

([2,19]). The CHFD of order $\alpha >0$ for a CF $x:\left[1,+\infty \right)\to \mathbb{R}$ is defined as

$${\mathfrak{D}}_1^{\alpha }x\left(\varsigma \right)=\frac{1}{{\Gamma }\left(n-\alpha \right)}{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{n-\alpha -1}{\delta }^{n}x\left(s\right)\frac{ds}{s},n-1<\alpha <n,$$

where ${\delta }^n={\left(\varsigma \frac{d}{d\varsigma }\right)}^n, n\in \mathbb{N}$.

Lemma 1

([2,19]). Let $n-1<\alpha \le n$, $n\in \mathbb{N}$. The equality $\left({\mathfrak{I}}_1^{\alpha }{\mathfrak{D}}_1^{\alpha }x\right)\left(\varsigma \right)=0$ is true if and only if

$$x\left(\varsigma \right)=\stackrel{n}{\sum _{k=1}}{c}_k{\left(log\varsigma \right)}^{\alpha -k}foreach\varsigma \in \left[1,\infty \right),$$

where $c_k\in \mathbb{R}$, $k=1,\dots ,n$ are arbitrary constants.

Lemma 2

([2,19]). Let $m-1<\alpha \le m$, $m\in \mathbb{N}$ and $x\in C^{n-1}\left[1,\infty \right)$. Then

$${\mathfrak{I}}_1^{\alpha }\left[{\mathfrak{D}}_1^{\alpha }x\left(\varsigma \right)\right]=x\left(\varsigma \right)-\stackrel{m-1}{\sum _{k=0}}\frac{\left({\delta }^{k}x\right)\left(1\right)}{{\Gamma }\left(k+1\right)}{\left(log\varsigma \right)}^k.$$

Lemma 3

([2,19]). For all $\mu >0$ and $\nu >-1,$

$$\frac{1}{{\Gamma }\left(\mu \right)}{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\mu -1}{\left(logs\right)}^{\nu }\frac{ds}{s}=\frac{{\Gamma }\left(\nu +1\right)}{{\Gamma }\left(\mu +\nu +1\right)}{\left(log\varsigma \right)}^{\mu +\nu }.$$

Lemma 4

([2,19]). Let $x\left(\varsigma \right)={\left(log\varsigma \right)}^{\mu }$, where $\mu \ge 0$ and let $m-1<\alpha \le m$, $m\in \mathbb{N}$. Then

$${\mathfrak{D}}_1^{\alpha }x\left(\varsigma \right)=\left\{\begin{array}{c}0\mathrm{if}\mu \in \{0,1,\dots ,m-1\},\\ \frac{{\Gamma }\left(\nu +1\right)}{{\Gamma }\left(\mu +\nu +1\right)}{\left(log\varsigma \right)}^{\mu -\nu }\mathrm{if}\mu \in \mathbb{N},\mu \ge m\mathrm{or}\mu \notin \mathbb{N},\mu >m-1.\end{array}\right.$$

Here we put some conditions on coefficient functions, to study the qualitative properties of solving Equation (1), which will help us solve it.

$\left({\Lambda }1\right)$ For every $x,y,z,w\in {\mathbb{R}}^n$ and $\varsigma \in \left[1,T\right]$, there exist three constants $C_1,$$C_2$ and $C_3$ are positive, so that

$$\begin{array}{cc}\hfill {\left|b(\varsigma ,x,y)\right|}^2\vee {\left|{\sigma }_1(\varsigma ,x,y)\right|}^2& \le C_1^2\left(1+{\left|x\right|}^2+{\left|y\right|}^2\right)\hfill \\ \hfill \left|b\left(\varsigma ,x,y\right)-b\left(\varsigma ,w,z\right)\right|\vee \left|{\sigma }_1\left(\varsigma ,x,y\right)-{\sigma }_1\left(\varsigma ,w,z\right)\right|& \le C_2\left|x-w\right|+C_3\left|y-z\right|\hfill \end{array}$$

where $\left|.\right|$ is the norm of ${\mathbb{R}}^n$, $x_1\vee x_2=max\left\{x_{1,}{x}_2\right\}.$

In coordination with pivotal research of Zone [20], Zhang and Agarwal [21], as we recognize that by proposal $\left({\Lambda }1\right)$, FSDPEs (1) has a unique solution

$$\begin{array}{cc}\hfill \mathcal{X}\left(\varsigma \right)& ={\mathcal{X}}_0+\frac{1}{{\Gamma }\left(\alpha \right)}{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}b\left(s,\mathcal{X}\left(s\right),\mathcal{X}(1+\eta s)\right)\frac{ds}{s}\hfill \\ \hfill & +\frac{1}{{\Gamma }\left(\alpha \right)}{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}{\sigma }_1\left(s,\mathcal{X}\left(s\right),\mathcal{X}(1+\eta s)\right)\frac{d\mathfrak{B}\left(s\right)}{s},\hfill \end{array}$$

(2)

$\mathcal{X}\left(\varsigma \right)$ is $\mathfrak{F}\left(\varsigma \right)$-adapted and E$\left({\int }_1^T{\left|\mathcal{X}\left(\varsigma \right)\right|}^{2}d\varsigma \right)<\infty .$

3. An Averaging Principle

In this part we investigated the averaging principle for FSDPEs, combining the results of existence and uniqueness. Let us consider the standard form of Equation (1):

$$\begin{array}{cc}\hfill {\mathcal{X}}_{ϵ}\left(\varsigma \right)& ={\mathcal{X}}_0+\frac{ϵ}{{\Gamma }\left(\alpha \right)}{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}b\left(s,{\mathcal{X}}_{ϵ}\left(s\right),{\mathcal{X}}_{ϵ}(1+\eta s)\right)\frac{ds}{s}\hfill \\ \hfill & +\frac{\sqrt{ϵ}}{{\Gamma }\left(\alpha \right)}{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}{\sigma }_1\left(s,{\mathcal{X}}_{ϵ}\left(s\right),{\mathcal{X}}_{ϵ}(1+\eta s)\right)\frac{d\mathfrak{B}\left(s\right)}{s},\hfill \end{array}$$

(3)

where the initial value ${\mathcal{X}}_0$, coefficients b and ${\sigma }_1$ it has the same meaning as in Equation (1). We also denote by ${ϵ}_0$ a fixed number, and $ϵ\in \left[0,{ϵ}_0\right]$ is a positive small parameter.

Before we continue with the averaging principle, we impose some measurable coefficients, $\overline{b}:{\mathbb{R}}^n\to {\mathbb{R}}^n,\overline{\sigma }:{\mathbb{R}}^n\to {\mathbb{R}}^n,$ satisfying $\left({\Lambda }1\right)$ and the additional inequalities:

$\left({\Lambda }2\right)$ For any $T_1\in \left[1,T\right],$$x,y\in {\mathbb{R}}^n$, there exist two positive bounded functions ${{\Psi }}_i\left(T_1\right),i=1,2$ such that

$$\begin{array}{cc}\hfill \frac{1}{log{T}_1}{\int }_1^{T_1}\left|b(s,x,y)-\overline{b}(x,y)\right|\frac{ds}{s}& \le {{\Psi }}_1\left(T_1\right)(1+\left|x\right|+\left|y\right|),\hfill \\ \hfill \frac{1}{log{T}_1}{\int }_1^{T_1}{\left|{\sigma }_1(s,x,y)-{\overline{\sigma }}_1(x,y)\right|}^2\frac{ds}{s}& \le {{\Psi }}_2\left(T_1\right)(1+{\left|x\right|}^2+{\left|y\right|}^2),\hfill \end{array}$$

where $\underset{T_1\to \infty }{lim}{{\Psi }}_i\left(T_1\right)=0.$

With sufficient help above, we will explain that the exact solution ${\mathcal{X}}_{ϵ}\left(\varsigma \right)$ converges, as $ϵ$$\to 0$, tend to $Z_{ϵ}\left(\varsigma \right)$ of the averaged system

$$\begin{array}{cc}\hfill Z_{ϵ}\left(\varsigma \right)& ={\mathcal{X}}_0+\frac{ϵ}{{\Gamma }\left(\alpha \right)}{\int }_1^{\varsigma }{(log\frac{\varsigma }{s})}^{\alpha -1}\overline{b}(Z_{ϵ}\left(s\right),Z_{ϵ}(1+\eta s))\frac{ds}{s}\hfill \\ \hfill & +\frac{\sqrt{ϵ}}{{\Gamma }\left(\alpha \right)}{\int }_1^{\varsigma }{(log\frac{\varsigma }{s})}^{\alpha -1}{\overline{\sigma }}_1(Z_{ϵ}\left(s\right),Z_{ϵ}(1+\eta s))\frac{d\mathfrak{B}\left(s\right)}{s}.\hfill \end{array}$$

(4)

We come now and present the main result of this research.

Theorem 1.

Suggest that $\left({\Lambda }1\right)-\left({\Lambda }2\right)$ are satisfied. For ${\delta }_1>0$ there exists $L>1,{ϵ}_1\in \left(0,{ϵ}_0\right]$ and $\beta \in \left(0,1\right)$ us such for every $ϵ\in \left(0,{ϵ}_1\right],$

$$E\left(\underset{\varsigma ϵ\left[1,L^{ϵ{-}^{\beta }}\right]}{sup}{\left|{\mathcal{X}}_{ϵ}\left(\varsigma \right)-Z_{ϵ}\left(\varsigma \right)\right|}^2\right)\le {\delta }_1.$$

(5)

Proof. 

For any $\varsigma \in \left[1,u\right]\subset \left[1,T\right],$

$$\begin{array}{cc}\hfill & {\mathcal{X}}_{ϵ}\left(\varsigma \right)-Z_{ϵ}\left(\varsigma \right)\hfill \\ \hfill & =\frac{ϵ}{{\Gamma }\left(\alpha \right)}{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}\left[b\left(s,{\mathcal{X}}_{ϵ}\left(s\right),{\mathcal{X}}_{ϵ}\left(1+\eta s\right)\right)-\overline{b}\left(Z_{ϵ}\left(s\right),Z_{ϵ}(1+\eta s)\right)\right]\frac{ds}{s}\hfill \\ \hfill & +\frac{\sqrt{ϵ}}{{\Gamma }\left(\alpha \right)}{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}\left[{\sigma }_1\left(s,{\mathcal{X}}_{ϵ}\left(s\right),{\mathcal{X}}_{ϵ}\left(1+\eta s\right)\right)-{\overline{\sigma }}_1\left(Z_{ϵ}\left(s\right),Z_{ϵ}(1+\eta s)\right)\right]\frac{d\mathfrak{B}\left(s\right)}{s}.\hfill \end{array}$$

(6)

Using the elementary inequality

$${\left|x_1+x_2\right|}^2\le 2({\left|x_1\right|}^2+{\left|x_2\right|}^2),$$

(7)

we have

$$\begin{array}{cc}\hfill & E\left(\underset{1\le \varsigma \le u}{sup}{\left|{\mathcal{X}}_{ϵ}\left(\varsigma \right)-Z_{ϵ}\left(\varsigma \right)\right|}^2\right)\hfill \\ \hfill & \le \frac{2{ϵ}^2}{{\Gamma }{\left(\alpha \right)}^2}E\underset{1\le \varsigma \le u}{sup}\left|{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}\left[b\left(s,{\mathcal{X}}_{ϵ}\left(s\right),{\mathcal{X}}_{ϵ}\left(1+\eta s\right)\right)\right.\right.\hfill \\ \hfill & -{\left.\left.\overline{b}\left(Z_{ϵ}\left(s\right),Z_{ϵ}(1+\eta s)\right)\frac{ds}{s}\right]\right|}^2\hfill \\ \hfill & +\frac{2ϵ}{{\Gamma }{\left(\alpha \right)}^2}E\underset{1\le \varsigma \le u}{sup}\left|{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}\left[{\sigma }_1\left(s,{\mathcal{X}}_{ϵ}\left(s\right),{\mathcal{X}}_{ϵ}\left(1+\eta s\right)\right)\right.\right.\hfill \\ \hfill & -{\left.\left.{\overline{\sigma }}_1\left(Z_{ϵ}\left(s\right),Z_{ϵ}(1+\eta s)\right)\frac{d\mathfrak{B}\left(s\right)}{s}\right]\right|}^2\hfill \\ \hfill & =I_1+I_2.\hfill \end{array}$$

(8)

Recalling inequality (7), we obtain

$$\begin{array}{cc}\hfill I_1& \le \frac{4{ϵ}^2}{{\Gamma }{\left(\alpha \right)}^2}E\underset{1\le \varsigma \le u}{sup}\left|{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}\left[b\left(s,{\mathcal{X}}_{ϵ}\left(s\right),{\mathcal{X}}_{ϵ}\left(1+\eta s\right)\right)\right.\right.\hfill \\ \hfill & -{\left.\left.b\left(Z_{ϵ}\left(s\right),Z_{ϵ}(1+\eta s)\right)\right]\frac{ds}{s}\right|}^2\hfill \\ \hfill & +\frac{4ϵ}{{\Gamma }{\left(\alpha \right)}^2}E\underset{1\le \varsigma \le u}{sup}\left|{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}\left[b\left(s,Z_{ϵ}\left(s\right),Z_{ϵ}\left(1+\eta s\right)\right)\right.\right.\hfill \\ \hfill & -{\left.\left.\overline{b}\left(Z_{ϵ}\left(s\right),Z_{ϵ}(1+\eta s)\right)\right]\frac{ds}{s}\right|}^2\hfill \\ \hfill & =I_{11}+I_{12}.\hfill \end{array}$$

(9)

Using the Cauchy–Schwarz inequality and condition $\left({\Lambda }1\right),$ we obtain

$$I_{11}\le K_{11}{ϵ}^{2}logu{\int }_1^u{(log\frac{u}{s})}^{2\alpha -2}E\left(\underset{1\le s_1\le s}{sup}{\left|{\mathcal{X}}_{ϵ}\left(s_1\right)-Z_{ϵ}\left(s_1\right)\right|}^2\frac{ds}{s}\right),$$

(10)

where $K_{11}={\displaystyle \frac{8\left(C_2^2+C_3^2\right)}{{\Gamma }{\left(\alpha \right)}^2}}.$ By the definition of variable upper limit integration,

$$\begin{array}{cc}\hfill I_{12}& \le \frac{4{ϵ}^2}{{\Gamma }{\left(\alpha \right)}^2}E\underset{1\le \varsigma \le u}{sup}\left|{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}d\left[{\int }_1^{s}b\left(\tau ,Z_{ϵ}\left(\tau \right),Z_{ϵ}\left(1+\eta \tau \right)\right)\right.\right.\hfill \\ \hfill & -{\left.\left.\overline{b}\left(Z_{ϵ}\left(\tau \right),Z_{ϵ}\left(1+\eta \tau \right)\right)\frac{d\tau }{\tau }\right]\right|}^2,\hfill \end{array}$$

(11)

integration by parts is used,

$$\begin{array}{cc}\hfill I_{12}& \le \frac{4{ϵ}^2{(\alpha -1)}^2}{{\Gamma }{\left(\alpha \right)}^2}E\underset{1\le \varsigma \le u}{sup}\left|{\int }_1^{\varsigma }\left({\int }_1^{s}b\left(\tau ,Z_{ϵ}\left(\tau \right),Z_{ϵ}\left(1+\eta \tau \right)\right)\right.\right.\hfill \\ \hfill & -{\left.\left.\overline{b}\left(Z_{ϵ}\left(\tau \right),Z_{ϵ}\left(1+\eta \tau \right)\right)\frac{d\tau }{\tau }\right){(log\frac{\varsigma }{s})}^{\alpha -2}\frac{ds}{s}\right|}^2,\hfill \end{array}$$

(12)

then together with the hypothesis $\left({\Lambda }2\right)$ and the Cauchy–Schwarz inequality, we obtain

$$\begin{array}{cc}\hfill I_{12}& \le \frac{4{ϵ}^2{\left(\alpha -1\right)}^2{\left(logu\right)}^{2\alpha -3}}{\left(2\alpha -3\right){\Gamma }{\left(\alpha \right)}^2}\hfill \\ \hfill & \times E{\int }_1^u{\left|{\int }_1^{s}b\left(\tau ,Z_{ϵ}\left(\tau \right),Z_{ϵ}\left(1+\eta \tau \right)\right)-\overline{b}\left(Z_{ϵ}\left(\tau \right),Z_{ϵ}\left(1+\eta \tau \right)\right)\frac{d\tau }{\tau }\right|}^2\frac{ds}{s}\hfill \\ \hfill & \le K_{12}{ϵ}^2{\left(logu\right)}^{2\alpha },\hfill \end{array}$$

(13)

in which

$$\begin{array}{cc}\hfill K_{12}& ={\displaystyle \frac{4{(\alpha -1)}^2}{(2\alpha -3){\Gamma }{\left(\alpha \right)}^2}}\underset{1\le \varsigma \le u}{sup}{{\Psi }}_1{\left(\varsigma \right)}^2\left[1+E\left(\underset{1\le \tau \le u}{sup}{\left|Z_{ϵ}\left(\tau \right)\right|}^2\right)\right.\hfill \\ \hfill & +\left.E\left(\underset{1\le \tau \le u}{sup}{\left|Z_{ϵ}\left(1+\eta \tau \right)\right|}^2\right)\right].\hfill \end{array}$$

(14)

With the same technique we look forward to the second term,

$$\begin{array}{cc}\hfill I_2& \le \frac{4{ϵ}^2}{{\Gamma }{\left(\alpha \right)}^2}E\underset{1\le \varsigma \le u}{sup}\left|{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}\left[{\sigma }_1\left(s,{\mathcal{X}}_{ϵ}\left(s\right),{\mathcal{X}}_{ϵ}\left(1+\eta s\right)\right)\right.\right.\hfill \\ \hfill & -{\left.\left.{\sigma }_1\left(s,Z_{ϵ}\left(s\right),Z_{ϵ}\left(1+\eta s\right)\right)\right]\frac{d\mathfrak{B}\left(s\right)}{s}\right|}^2\hfill \\ \hfill & +\frac{4ϵ}{{\Gamma }{\left(\alpha \right)}^2}E\underset{1\le \varsigma \le u}{sup}\left|{\int }_1^{\varsigma }{\left(log\frac{\varsigma }{s}\right)}^{\alpha -1}\left[{\sigma }_1\left(s,Z_{ϵ}\left(s\right),Z_{ϵ}\left(1+\eta s\right)\right)\right.\right.\hfill \\ \hfill & -{\left.\left.{\overline{\sigma }}_1\left(Z_{ϵ}\left(s\right),Z_{ϵ}\left(1+\eta s\right)\right)\right]\frac{d\mathfrak{B}\left(s\right)}{s}\right|}^2\hfill \\ \hfill & =I_{21}+I_{22}.\hfill \end{array}$$

(15)

By applying Doob’s martingale inequality, Itô’s formula and condition $\left({\Lambda }1\right)$,

$$\begin{array}{cc}\hfill I_{21}& \le \frac{4ϵ}{{\Gamma }{\left(\alpha \right)}^2}E{\int }_1^u{\left(log\frac{u}{s}\right)}^{2\alpha -2}\left|{\sigma }_1\left(s,{\mathcal{X}}_{ϵ}\left(s\right),{\mathcal{X}}_{ϵ}\left(1+\eta s\right)\right)\right.\hfill \\ \hfill & -{\left.{\sigma }_1\left(s,Z_{ϵ}\left(s\right),Z_{ϵ}\left(1+\eta s\right)\right)\right|}^2\frac{ds}{s}\hfill \\ \hfill & \le K_{21}ϵ{\int }_1^u{\left(log\frac{u}{s}\right)}^{2a-2}E\left(\underset{1\le s_1\le s}{sup}{\left|{\mathcal{X}}_{ϵ}\left(s_1\right)-Z_{ϵ}\left(s_1\right)\right|}^2\right)\frac{ds}{s},\hfill \end{array}$$

(16)

where $K_{21}={\displaystyle \frac{8\left(C_2^2+C_3^2\right)}{{\Gamma }{\left(\alpha \right)}^2}}.$ Applying Doob’s martingale inequality and Itô’s formula again,

$$\begin{array}{cc}\hfill I_{22}& \le \frac{4ϵ}{{\Gamma }{\left(\alpha \right)}^2}E{\int }_1^u{\left(log\frac{u}{s}\right)}^{2\alpha -2}\left|{\sigma }_1\left(s,{\mathcal{X}}_{ϵ}\left(s\right),{\mathcal{X}}_{ϵ}\left(1+\eta s\right)\right)\right.\hfill \\ \hfill & -{\left.{\overline{\sigma }}_1\left(Z_{ϵ}\left(s\right),Z_{ϵ}\left(1+\eta s\right)\right)\right|}^2\frac{ds}{s}.\hfill \end{array}$$

(17)

Integrating by parts, produces

$$\begin{array}{cc}\hfill I_{22}& \le \frac{4ϵ}{{\Gamma }{\left(\alpha \right)}^2}E{\int }_1^u{\left(log\frac{u}{s}\right)}^{2\alpha -2}d\left[{\int }_1^s\left|{\sigma }_1\left(\tau ,Z_{ϵ}\left(\tau \right),Z_{ϵ}\left(1+\eta \tau \right)\right)\right.\right.\hfill \\ \hfill & -\left.{\left.{\overline{\sigma }}_1\left(Z_{ϵ}\left(\tau \right),Z_{ϵ}\left(1+\eta \tau \right)\right)\right|}^2\frac{d\tau }{\tau }\right]\hfill \\ \hfill & \le \frac{4ϵ(2\alpha -2)}{{\Gamma }{\left(\alpha \right)}^2}E{\int }_1^u\left({\int }_1^s\left|{\sigma }_1\left(\tau ,Z_{ϵ}\left(\tau \right),Z_{ϵ}\left(1+\eta \tau \right)\right)\right.\right.\hfill \\ \hfill & -\left.{\left.{\overline{\sigma }}_1\left(Z_{ϵ}\left(\tau \right),Z_{ϵ}\left(1+\eta \tau \right)\right)\right|}^2\frac{d\tau }{\tau }\right){\left(log\frac{u}{s}\right)}^{2\alpha -3}\frac{ds}{s},\hfill \end{array}$$

(18)

thanks to the hypothesis $\left({\Lambda }2\right),$ we can conclude

$$\begin{array}{cc}\hfill I_{22}& \le \frac{4ϵ(2\alpha -2)}{{\Gamma }{\left(\alpha \right)}^2}E{\int }_1^u\left(\underset{1\le s_1\le s}{sup}{{\Psi }}_2\left(s_1\right)\left[1+E\left(\underset{1\le \tau \le s}{sup}{\left|Z_{ϵ}\left(\tau \right)\right|}^2\right)\right.\right.\hfill \\ \hfill & +\left.\left.E\left(\underset{1\le \tau \le s}{sup}{\left|Z_{ϵ}\left(1+\eta \tau \right)\right|}^2\right)\right]\right)\left(logs\right){\left(log\frac{u}{s}\right)}^{2\alpha -3}\frac{ds}{s}\hfill \\ \hfill & \le K_{22}ϵ{\left(logu\right)}^{2\alpha -1},\hfill \end{array}$$

(19)

where

$$\begin{array}{cc}\hfill K_{22}& =\frac{3(2\alpha -2)}{\alpha (2\alpha -1){\Gamma }{\left(\alpha \right)}^2}\underset{1\le \varsigma \le u}{sup}{{\Psi }}_2\left(\varsigma \right)\left[1+E\left(\underset{1\le \varsigma \le u}{sup}{\left|Z_{ϵ}\left(\tau \right)\right|}^2\right)\right.\hfill \\ \hfill & +\left.E\left(\underset{1\le \varsigma \le u}{sup}{\left|Z_{ϵ}\left(1+\eta \tau \right)\right|}^2\right)\right].\hfill \end{array}$$

(20)

Now, substituting Equations (10)–(19) into (8), for any $u\in \left[1,T\right],$ we find

$$\begin{array}{cc}\hfill & E\left(\underset{1\le \varsigma \le u}{sup}{\left|{\mathcal{X}}_{ϵ}\left(\varsigma \right)\right|}^2\right)\hfill \\ \hfill & \le K_{12}{ϵ}^2{u}^{2\alpha }+K_{22}ϵu^{2\alpha -1}\hfill \\ \hfill & +\left(K_{11}{ϵ}^{2}u+K_{21}ϵ\right){\int }_1^u{\left(log\frac{u}{s}\right)}^{\left(2\alpha -1\right)-1}E\left(\underset{1\le s_1\le s}{sup}{\left|{\mathcal{X}}_{ϵ}\left(s_1\right)-Z_{ϵ}\left(s_1\right)\right|}^2\right)\frac{ds}{s},\hfill \end{array}$$

(21)

depending on the Gronwall–Bellman inequality [22], we find

$$\begin{array}{cc}\hfill & E\left(\underset{1\le \varsigma \le u}{sup}{\left|{\mathcal{X}}_{ϵ}\left(\varsigma \right)-Z_{ϵ}\left(\varsigma \right)\right|}^2\right)\hfill \\ \hfill & \le \left(K_{12}{ϵ}^2{\left(logu\right)}^{2\alpha }+K_{22}ϵ{\left(logu\right)}^{2\alpha -1}\right)\hfill \\ \hfill & \times \sum _{k=0}^{\infty }\frac{{\left(\left(K_{11}{ϵ}^2{\left(logu\right)}^{2\alpha }+K_{21}ϵ{\left(logu\right)}^{2\alpha -1}\right){\Gamma }\left(2\alpha -1\right)\right)}^k}{{\Gamma }\left(k\left(2\alpha -1\right)+1\right)}.\hfill \end{array}$$

(22)

This implies that we can select $\beta \in \left(0,1\right)$ and $L>1,$ such that for every $\varsigma \in \left[1,L^{{ϵ}^{-\beta }}\right]\subseteq \left[1,T\right]$ having

$$E\left(\underset{1\le \varsigma \le L^{{ϵ}^{-\beta }}}{sup}{\left|{\mathcal{X}}_{ϵ}\left(\varsigma \right)-Z_{ϵ}\left(\varsigma \right)\right|}^2\right)\le C{ϵ}^{1-\beta },$$

(23)

where

$$\begin{array}{cc}\hfill C& =\left(K_{12}{\left(logL\right)}^{2\alpha }{ϵ}^{1+\beta -2\alpha \beta }+K_{22}{\left(logL\right)}^{2\alpha -1}{ϵ}^{2\beta \left(1-\alpha \right)}\right)\hfill \\ \hfill & \times \sum _{k=0}^{\infty }\frac{{\left(\left(K_{11}{\left(logL\right)}^{2\alpha }{ϵ}^{2\left(1-\alpha \beta \right)}+K_{21}{\left(logL\right)}^{2\alpha -1}{ϵ}^{1+\beta \left(1-2\alpha \right)}\right){\Gamma }\left(2\alpha -1\right)\right)}^k}{{\Gamma }\left(k\left(2\alpha -1\right)+1\right)},\hfill \end{array}$$

(24)

is a constant. Hence, for any given number ${\delta }_1$, there exists ${ϵ}_1\in \left(0,{ϵ}_0\right]$ such that for each $ϵ\in \left(0,{ϵ}_1\right]$ and $\varsigma \in \left[1,L^{{ϵ}^{-\beta }}\right]$ having

$$E\left(\underset{1\le \varsigma \le L^{{ϵ}^{-\beta }}}{sup}{\left|{\mathcal{X}}_{ϵ}\left(\varsigma \right)-Z_{ϵ}\left(\varsigma \right)\right|}^2\right)\le {\delta }_1.$$

(25)

finished the proof. □

4. Example

We present the following equation FSDPEs

$$\left\{\begin{array}{c}{\mathfrak{D}}_1^{\alpha }{\mathcal{X}}_{ϵ}\left(\varsigma \right)=3ϵ\left({\mathcal{X}}_{ϵ}\left(\varsigma \right)+{\mathcal{X}}_{ϵ}\left(1+\eta \varsigma \right)\right){log}^2\left(\varsigma \right)+\sqrt{ϵ}{\displaystyle \frac{d\mathfrak{B}\left(\varsigma \right)}{d\varsigma }},\hfill \\ \mathcal{X}\left(1\right)=0,\hfill \end{array}\right.$$

(26)

where $\eta \in \left(0,\frac{\pi -1}{\pi }\right),\alpha \in \left(\frac{1}{2},1\right).$ The coefficients $b(\varsigma ,{\mathcal{X}}_{ϵ},Y_{ϵ})=3\left({\mathcal{X}}_{ϵ}+Y_{ϵ}\right){log}^2\left(\varsigma \right)$ and ${\sigma }_1(\varsigma ,{\mathcal{X}}_{ϵ},Y_{ϵ})=1$ verify the conditions $\left({\Lambda }1\right),$ so there has a unique solution to FSDPEs (26).

Define

$$\overline{b}({\mathcal{X}}_{ϵ},Y_{ϵ})=\frac{1}{log\pi }{\int }_1^{\pi }b(\varsigma ,{\mathcal{X}}_{ϵ},Y_{ϵ})\frac{d\varsigma }{\varsigma }=\left({\mathcal{X}}_{ϵ}+Y_{ϵ}\right){log}^2\left(\pi \right),{\overline{\sigma }}_1({\mathcal{X}}_{ϵ},Y_{ϵ})=1,$$

it is easily seen $\left({\Lambda }2\right)$ holds, so the averaging form of (26) is

$${\mathfrak{D}}_1^{\alpha }{Z}_{ϵ}\left(\varsigma \right)=ϵ\left(Z_{ϵ}\left(\varsigma \right)+Z_{ϵ}(1+\eta \varsigma )\right){log}^2\left(\pi \right)+\sqrt{ϵ}{\displaystyle \frac{d\mathfrak{B}\left(\varsigma \right)}{d\varsigma }},Z_{ϵ}\left(1\right)={\mathcal{X}}_0.$$

(27)

Depending to Theorem 1, as $ϵ\to 0$, the solution ${\mathcal{X}}_{ϵ}\left(\varsigma \right)$ and $Z_{ϵ}\left(\varsigma \right)$ to Equations (26) and (27) are equivalent in the sense of mean square.

5. Conclusions

Previously, many researchers studied the averaging principle for Caputo fractional stochastic differential equations approximated by solutions to averaged stochastic systems in the sense of mean square. The new idea in our research in (1) is a discussion of a special kind of Caputo–Hadamard fractional stochastic differential pantograph equations driven by Brownian motion. We have also made two commitments, the solutions to FSDPEs can be approximated by solutions to averaged stochastic systems in the sense of mean square. Moreover, we extend the classical Khasminskii approach to Caputo–Hadamard fractional stochastic differential pantograph equations.