Proportional Itô–Doob Stochastic Fractional Order Systems

Abstract

In this article, we discuss the existence and uniqueness of proportional Itô–Doob stochastic fractional order systems (PIDSFOS) by using the Picard iteration method. The paper provides new results using the proportional fractional integral and stochastic calculus techniques. We have shown the convergence of the solution of the averaged PIDSFOS to that of the standard PIDSFOS in the context of the mean square and also in probability. One example is given to illustrate our results.

1. Introduction

Fractional calculus has attracted much more attention over the past three decades, due to its applications in many fields of science and engineering. It provides several potentially useful tools for solving differential and integral equations.

The concept of fractional calculus occurred from a question posed in 1695 by L’Hôpital to Gottfried Wilhelm Leibniz, who wanted to understand the meaning of Leibniz’s notation $\frac{d^m}{d{\varsigma }^m}G\left(\varsigma \right)$ if $m=\frac{1}{2}$. In his answer, dated 30 September 1695, Leibniz wrote to L’Hôpital as follows: “It is an apparent paradox from which, one day, useful consequences will be drawn.” Later, many definitions of fractional derivatives were formulated by Euler, Lagrange, Laplace, Lacroix, Fourier, Liouville, and other research scientists. The theory of fractional analysis has many applications, such as the circulation of fluids, dynamic processes, telegraph equations, electrical networks, signal processing, etc. (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]). Furthermore, fractional calculus can be applied to solve some real-world problems in the engineering, biological, and physical disciplines (see [2,4,6]).

One of the most important classes of fractional derivative is the proportional and Caputo-proportional fractional derivatives. Many researchers have investigated the proportional and Caputo-proportional fractional differential equations; for more details, see [17,18,19,20,21,22]. Some inequalities were studied within the proportional fractional operators [19,21], and [23] discussed the proportional derivatives of a function with respect to another function. Itô–Doob stochastic fractional order systems (IDSFOS) are an important type of fractional integral equations; this class of equations receives much more attention (see [24,25,26,27]). In [25], Abouagwa et al. have studied the averaging principle of IDSFOS with non-Lipschitz coefficients. In [24], the authors discussed the criterion for well-posedness (existence and uniqueness) and mean square stability of solutions to non-Lipschitz IDSFOS. The averaging principle is an important tool to approximate the solutions of the systems driven by differential equations which result from mathematics, mechanics, control, and other fields of sciences. The general idea of the averaging principle theorem is as follows: the solutions of the original system can be approximated by the corresponding solutions to the standard system, in both the senses of the mean square and probability (see [25,28,29,30,31,32,33,34]).

Taking into account the discussion presented above, this article is devoted to the analysis and study of the averaging principle for PIDSFOS. In the literature, several works which are interested in the averaging principle focus on the field of stochastic differential equations. In the literature, the averaging principle of fractional stochastic equations is a new theory, and in particular, there is no existing paper which studies the averaging principle for PIDSFOS. Motivated by the previous work in [25], this paper investigated the problem of the averaging principle of the solution for the proportional fractional integral. The main contributions of our work are as follows:

⋄

Unlike other works in the literature, this paper presents a new theory by the proportional fractional integral.

⋄

We study the existence and uniqueness of the solutions of PIDSFOS using the Picard iteration technique.

⋄

We discuss the convergence of the solution of the averaged PIDSFOS towards that of the standard PIDSFOS in the sense of the mean square.

This paper is organized as follows. Some preliminary notions are presented in Section 2. Section 3 is devoted to the existence and uniqueness results using the Picard iteration method. Section 4 investigated the convergence of the solution of the averaged PIDSFOS to the solution of the standard PIDSFOS in the sense of the mean square and also in probability. Finally, in Section 5, we illustrate our results with a theoretical example.

2. Preliminaries and Definitions

Let $\mathcal{R}=\{\mathcal{X},\mathfrak{F},{\left({\mathfrak{F}}_{\varpi }\right)}_{\varpi \ge 1},\mathfrak{P}\}$ be a complete probability space.

Definition 1 

([22]). For $\nu \in \left(0,1\right],$ $\delta >0,$ the GPF integral of f of order δ is

$$\begin{array}{cc}\hfill {\mathcal{I}}_a^{\delta ,\nu }f\left(\mu \right)& =\frac{1}{{\nu }^{\delta }\Gamma \left(\delta \right)}{\int }_a^{x}exp\left(\frac{\nu -1}{\nu }\left(\mu -\kappa \right)\right){\left(\mu -\kappa \right)}^{\delta -1}f\left(\kappa \right)d\kappa .\hfill \end{array}$$

Consider the following PIDSFOS:

$$z\left(\varpi \right)=\theta +{\int }_0^{\varpi }\tau \left(q,z\left(q\right)\right)dq+{\int }_0^{\varpi }{\vartheta }_1\left(q,z\left(q\right)\right)d\mathcal{W}\left(q\right)+\frac{\varkappa }{a^{\varkappa }}{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -q\right)\right){\left(\varpi -q\right)}^{\varkappa -1}{\vartheta }_2\left(q,z\left(q\right)\right)dq.$$

(1)

with $\theta \in {\mathbb{R}}^n$, $\frac{1}{2}<\varkappa <1$, $\varpi \in [0,T]$.

Set the following assumptions:

• ${\mathcal{H}}_1$: There is $\overline{K}>0$ such that

$$|\tau (\eta ,{\theta }_1)-\tau (\eta ,{\theta }_2){|}^2\vee |{\vartheta }_1(\eta ,{\theta }_1)-{\vartheta }_1(\eta ,{\theta }_2){|}^2\vee |{\vartheta }_2(\eta ,{\theta }_1)-{\vartheta }_2(\eta ,{\theta }_2){|}^2\le \overline{K}{|{\theta }_1-{\theta }_2|}^2,$$

(2)

for all $(\eta ,{\theta }_1,{\theta }_2)\in [0,T]\times {\mathbb{R}}^n\times {\mathbb{R}}^n$.

• ${\mathcal{H}}_2$: There is $K>0$ such that

$${|\tau (\eta ,\theta )|}^2\vee |{\vartheta }_1{(\eta ,\theta )|}^2\vee |{\vartheta }_2{(\eta ,\theta )|}^2\le {K(1+|\theta |}^2),$$

(3)

for all $(\eta ,\theta )\in [0,T]\times {\mathbb{R}}^n$.

3. Existence and Uniqueness Results

Theorem 1. 

Suppose that ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ hold. Then, there is a unique solution $z\left(\varpi \right)$ of Equation (1) such that $z\left(\varpi \right)\in {\mathcal{M}}_2\left([0,T];{\mathbb{R}}^n\right)$.

Before showing Theorem 1, we present the following useful lemma.

Lemma 1. 

Suppose that ${\mathcal{H}}_2$ holds. If $z\left(\varpi \right)$ is a solution of Equation (1), then

$$\mathbb{E}(\underset{0\le \varpi \le T}{sup}{|z(\varpi )|}^2)\le (1+{3\mathbb{E}|\theta |}^2)e^{3KT(T+4+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)})}.$$

(4)

In particular, $z\left(\varpi \right)\in {\mathcal{M}}_2\left([0,T];{\mathbb{R}}^n\right)$.

Proof. 

For any integer $n\in \mathbb{N}$, let the stopping time ${\varsigma }_n=T\wedge inf\{t\in [0,T];|z\left(\varpi \right)|\ge n\}$. It is easy to see that ${\varsigma }_n\uparrow T$ a.s. Set $z_n\left(\varpi \right)=z(\varpi \wedge {\varsigma }_n)$, for $\varpi \in [0,T]$. Then, $z_n\left(\varpi \right)$ verifies

$$\begin{array}{ccc}\hfill z_n\left(\varpi \right)& =& \theta +{\int }_0^{\varpi }\tau \left(s,z_n\left(s\right)\right)1_{[0,{\varsigma }_n]}\left(s\right)ds+{\int }_0^{\varpi }{\vartheta }_1\left(s,z_n\left(s\right)\right)1_{[0,{\varsigma }_n]}\left(s\right)d\mathcal{W}\left(s\right)+\hfill \\ & & \frac{\varkappa }{a^{\varkappa }}{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi \wedge {\varsigma }_n-s\right)\right){\left(\varpi \wedge {\varsigma }_n-s\right)}^{\varkappa -1}{\vartheta }_2\left(s,z_n\left(s\right)\right)1_{[0,{\varsigma }_n]}\left(s\right)ds.\hfill \end{array}$$

(5)

By the Cauchy–Schwartz inequality and assumption ${\mathcal{H}}_2$, we derive

$$\begin{array}{ccc}\hfill |z_n{\left(\varpi \right)|}^2& \le & {3|\theta |}^2+3\varpi {\int }_0^{\varpi }{|\tau \left(s,z_n\left(s\right)\right)|}^{2}ds\hfill \\ & +& 3|{\int }_0^{\varpi }{\vartheta }_1\left(s,z_n\left(s\right)\right)1_{[0,{\varsigma }_n]}{\left(s\right)d\mathcal{W}\left(s\right)|}^2\hfill \\ & +& 3\frac{{\varkappa }^2}{a^{2\varkappa }}\left({\int }_0^{\varpi }exp\left(\frac{2\left(a-1\right)}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{2\varkappa -2}ds\right)\left({\int }_0^{\varpi }{|{\vartheta }_2\left(s,z_n\left(s\right)\right)|}^{2}ds\right)\hfill \\ & \le & {3|\theta |}^2+3\varpi {\int }_0^{\varpi }{|\tau \left(s,z_n\left(s\right)\right)|}^{2}ds\hfill \\ & +& 3|{\int }_0^{\varpi }{\vartheta }_1\left(s,z_n\left(s\right)\right)1_{[0,{\varsigma }_n]}{\left(s\right)d\mathcal{W}\left(s\right)|}^2\hfill \\ & +& 3\frac{{\varkappa }^2}{a^{2\varkappa }}\left({\int }_0^{\varpi }{\left(\varpi -s\right)}^{2\varkappa -2}ds\right)\left({\int }_0^{\varpi }{|{\vartheta }_2\left(s,z_n\left(s\right)\right)|}^{2}ds\right)\hfill \\ & \le & {3|\theta |}^2+3K\varpi {\int }_0^{\varpi }(1+|z_n\left(s\right){|)}^{2}ds\hfill \\ & +& 3|{\int }_0^{\varpi }{\vartheta }_1\left(s,z_n\left(s\right)\right)1_{[0,{\varsigma }_n]}{\left(s\right)d\mathcal{W}\left(s\right)|}^2\hfill \\ & +& \frac{3{\varkappa }^{2}K}{a^{2\varkappa }\left(2\varkappa -1\right)}{\varpi }^{2\varkappa -1}{\int }_0^{\varpi }(1+|z_n\left(s\right){|)}^{2}ds.\hfill \end{array}$$

(6)

Hence, using Theorem $7.2$ (p. 40 in [35]) and assumption ${\mathcal{H}}_2$, we have

$$\begin{array}{ccc}\hfill \mathbb{E}(\underset{0\le s\le \varpi }{sup}{|z_n\left(s\right)|}^2)& \le & {3\mathbb{E}|\theta |}^2+3KT{\int }_0^{\varpi }(1+\mathbb{E}|z_n\left(s\right){|}^2)ds\hfill \\ & +& 12\mathbb{E}{\int }_0^{\varpi }{|{\vartheta }_1\left(s,z_n\left(s\right)\right)|}^2{1}_{[0,{\varsigma }_n]}\left(s\right)ds\hfill \\ & +& \frac{3{\varkappa }^{2}K}{a^{2\varkappa }\left(2\varkappa -1\right)}{T}^{2\varkappa -1}{\int }_0^{\varpi }(1+\mathbb{E}|z_n\left(s\right){|}^2)ds\hfill \\ & \le & {3\mathbb{E}|\theta |}^2+3K(T+4+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}){\int }_0^{\varpi }(1+\mathbb{E}|z_n\left(s\right){|}^2)ds.\hfill \end{array}$$

(7)

Therefore,

$$1+\mathbb{E}\left(\underset{0\le s\le \varpi }{sup}{|z_n\left(s\right)|}^2\right)\le 1+3\mathbb{E}{|\theta |}^2+3K\left(T+4+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right){\int }_0^{\varpi }\left(1+\mathbb{E}(\underset{0\le l\le s}{sup}{|z_n\left(l\right)|}^2)\right)ds.$$

By the Gronwall inequality, we obtain

$$1+\mathbb{E}\left(\underset{0\le s\le \varpi }{sup}{|z_n\left(s\right)|}^2\right)\le \left(1+{3\mathbb{E}|\theta |}^2\right)e^{3KT(T+4+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)})}.$$

Finally, we obtain the required inequality by letting $n⟶\infty$. □

Now, we will show Theorem 1.

Proof of Theorem 1 

Uniqueness:– Consider $z\left(\varpi \right)$ and $\overline{z}\left(\varpi \right)$ to be two solutions of Equation (1).

• According to Lemma 1, $z\left(\varpi \right)$ and $\overline{z}\left(\varpi \right)$ belong to ${\mathcal{M}}_2\left([0,T];{\mathbb{R}}^n\right)$. Thus,

$$z\left(\varpi \right)-\overline{z}\left(\varpi \right)={\int }_0^{\varpi }\left(\tau \left(s,z\left(s\right)\right)-\tau \left(s,\overline{z}\left(s\right)\right)\right)ds+{\int }_0^{\varpi }\left({\vartheta }_1\left(s,z\left(s\right)\right)-{\vartheta }_1\left(s,\overline{z}\left(s\right)\right)\right)d\mathcal{W}\left(s\right)$$

(8)

$$+\frac{\varkappa }{a^{\varkappa }}{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}\left({\vartheta }_2\left(s,z\left(s\right)\right)-{\vartheta }_2\left(s,\overline{z}\left(s\right)\right)\right)ds.$$

Using the Hölder inequality, Theorem $7.2$ (p. 50 in [35]), and ${\mathcal{H}}_1$, proceeding as the proof of Lemma 1, we derive

$$\mathbb{E}\left(\underset{0\le s\le \varpi }{sup}{|z\left(s\right)-\overline{z}\left(s\right)|}^2\right)\le 3\overline{K}\left(T+4+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right){\int }_0^{\varpi }\mathbb{E}\left(\underset{0\le l\le s}{sup}{|z\left(l\right)-\overline{z}\left(l\right)|}^2\right)ds.$$

• By the Gronwall inequality, we can obtain $\mathbb{E}\left({sup}_{0\le s\le \varpi }{|z\left(s\right)-\overline{z}\left(s\right)|}^2\right)=0$. Hence, $z\left(\varpi \right)=\overline{z}\left(\varpi \right)$, a.s.
• Existence:– Set $z_0\left(\varpi \right)=\theta$. The iterative Picard method is defined by the sequence ${\left(z_n\right)}_{n\ge 1}$ as follows:

$$\begin{array}{ccc}\hfill z_n\left(\varpi \right)& =& \theta +{\int }_0^{\varpi }\tau \left(s,z_{n-1}\left(s\right)\right)ds+{\int }_0^{\varpi }{\vartheta }_1\left(s,z_{n-1}\left(s\right)\right)d\mathcal{W}\left(s\right)\hfill \\ & +& \frac{\varkappa }{a^{\varkappa }}{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}{\vartheta }_2\left(s,z_{n-1}\left(s\right)\right)ds.\hfill \end{array}$$

(9)

It is obvious to see that $\theta \in {\mathcal{M}}_2\left([0,T];{\mathbb{R}}^n\right)$. Moreover, it is not hard to see that $z_n\left(\varpi \right)\in {\mathcal{M}}_2\left([0,T];{\mathbb{R}}^n\right)$ by induction, because by Equation (9), we have

$$\mathbb{E}|z_n{\left(\varpi \right)|}^2\le C_1+4K\left(T+1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right){\int }_0^{\varpi }\mathbb{E}{|z_{n-1}\left(s\right)|}^{2}ds,$$

(10)

where $C_1=4\mathbb{E}{|\theta |}^2+4KT\left(T+1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right)$.

• By Equation (10), it yields that for any $k\ge 1$:

  $$\begin{array}{ccc}\hfill \underset{1\le n\le k}{max}\mathbb{E}{|z_n\left(\varpi \right)|}^2& \le & C_1+4K\left(T+1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right){\int }_0^{\varpi }\underset{1\le n\le k}{max}\mathbb{E}{|z_{n-1}\left(s\right)|}^{2}ds\hfill \\ & \le & C_1+4K\left(T+1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right){\int }_0^{\varpi }\left({\mathbb{E}|\theta |}^2+\underset{1\le n\le k}{max}\mathbb{E}{|z_n\left(s\right)|}^2\right)ds\hfill \\ & \le & C_2+4K\left(T+1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right){\int }_0^{\varpi }\underset{1\le n\le k}{max}\mathbb{E}{|z_n\left(s\right)|}^{2}ds,\hfill \end{array}$$

  (11)

  where $C_2=C_1+4KT\left(T+1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right)\mathbb{E}{|\theta |}^2$.
• Using the Gronwall inequality, we have

  $$\underset{1\le n\le k}{max}\mathbb{E}{|z_n\left(\varpi \right)|}^2\le C_2{e}^{4KT\left(T+1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right)}.$$

  Using the fact that k is arbitrary, this implies

  $$\mathbb{E}|z_n{\left(\varpi \right)|}^2\le C_2{e}^{4KT\left(T+1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right)},$$

  (12)

  for all $0\le \varpi \le T$, $n\ge 1$.
• Note that

  $$\begin{array}{ccc}\hfill |z_1{\left(\varpi \right)-\theta |}^2& \le & 3|{\int }_0^{\varpi }\tau \left(s,z_0\left(s\right)\right){ds|}^2+3{|{\int }_0^{\varpi }{\vartheta }_1\left(s,z_0\left(s\right)\right)d\mathcal{W}\left(s\right)|}^2\hfill \\ & +& 3\frac{{\varkappa }^2}{a^{2\varkappa }}{|{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}{\vartheta }_2\left(s,z_0\left(s\right)\right)ds|}^2.\hfill \end{array}$$

  Applying the expectation and using assumption ${\mathcal{H}}_1$, we derive

  $$\begin{array}{ccc}\hfill \mathbb{E}|z_1{\left(\varpi \right)-\theta |}^2& \le & 3K{\varpi }^2\left(1+\mathbb{E}|z_0{\left(\varpi \right)|}^2\right)+3K\varpi \left(1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right)\left(1+\mathbb{E}|z_0{\left(\varpi \right)|}^2\right)\hfill \\ & \le & C,\hfill \end{array}$$

  (13)

  where

  $$C=3KT\left(T+1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right)\left(1+\mathbb{E}|z_0{\left(\varpi \right)|}^2\right).$$

  Let us claim that for $n\ge 0$,

  $$\mathbb{E}|z_{n+1}\left(\varpi \right)-z_n{\left(\varpi \right)|}^2\le \frac{C{\left[M\varpi \right]}^n}{n!},$$

  (14)

  for $1\le \varpi \le T$, where $M=3\overline{K}\left(T+1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right)$.
• We will show that Equation (14) holds true by induction. Using Equation (13), we can see that Equation (14) holds when $n=0$.
• Suppose that Equation (14) holds for some $n\ge 0$. We will show that Equation (14) still holds for $n+1$.
• We can see that

  $$\begin{array}{ccc}\hfill |z_{n+2}\left(\varpi \right)-z_{n+1}{\left(\varpi \right)|}^2& \le & 3|{\int }_0^{\varpi }\left(\tau \left(s,z_{n+1}\left(s\right)\right)-\tau \left(s,z_n\left(s\right)\right)\right)ds{|}^2\hfill \\ & +& 3|{\int }_0^{\varpi }\left({\vartheta }_1\left(s,z_{n+1}\left(s\right)\right)-{\vartheta }_1\left(s,z_n\left(s\right)\right)\right)d\mathcal{W}\left(s\right){|}^2\hfill \\ & +& 3\frac{{\varkappa }^2}{a^{2\varkappa }}|{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -q\right)\right){\left(\varpi -q\right)}^{\varkappa -1}\left({\vartheta }_2\left(q,z_{n+1}\left(q\right)\right)-{\vartheta }_2\left(q,z_n\left(q\right)\right)\right)dq{|}^2.\hfill \end{array}$$

  (15)

Applying the expectation to Equation (15), by ${\mathcal{H}}_1$ and Equation (14), we have

$$\begin{array}{ccc}\hfill \mathbb{E}|z_{n+2}\left(\varpi \right)-z_{n+1}{\left(\varpi \right)|}^2& \le & 3\overline{K}\left(\varpi +1+\frac{{\varkappa }^2{\varpi }^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right)\mathbb{E}{\int }_0^{\varpi }{|z_{n+1}\left(s\right)-z_n\left(s\right)|}^{2}ds\hfill \\ & \le & M{\int }_0^{\varpi }C\frac{{\left[Ms\right]}^n}{n!}ds\hfill \\ & \le & C\frac{{\left[M\varpi \right]}^{n+1}}{(n+1)!}.\hfill \end{array}$$

(16)

Thus, Equation (14) holds for all $n\ge 0$. Furthermore, if we replace n in Equation (15) with $n-1$, it yields that

$$\begin{array}{ccc}\hfill \underset{0\le \varpi \le T}{sup}{|z_{n+1}\left(\varpi \right)-z_n\left(\varpi \right)|}^2& \le & 3\overline{K}\left(T+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right){\int }_0^T{|z_n\left(s\right)-z_{n-1}\left(s\right)|}^{2}ds\hfill \\ & +& 3\underset{0\le \varpi \le T}{sup}{|{\int }_0^{\varpi }\left({\vartheta }_1\left(s,z_n\left(s\right)\right)-{\vartheta }_1\left(s,z_{n-1}\left(s\right)\right)\right)d\mathcal{W}\left(s\right)|}^2.\hfill \end{array}$$

(17)

Applying the expectation to Equation (17), using Theorem $7.2$ (p. 40 in [35]) and Equation (14), we obtain

$$\begin{array}{ccc}\hfill \mathbb{E}\underset{0\le \varpi \le T}{sup}{|z_{n+1}\left(\varpi \right)-z_n\left(\varpi \right)|}^2& \le & 3\overline{K}\left(T+4+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right){\int }_0^T\mathbb{E}{|z_n\left(s\right)-z_{n-1}\left(s\right)|}^{2}ds\hfill \\ & \le & 4M{\int }_0^{T}C\frac{{\left[Ms\right]}^{n-1}}{(n-1)!}ds=\frac{4C{\left(MT\right)}^n}{n!}.\hfill \end{array}$$

(18)

Hence,

$$\mathcal{P}\{\underset{0\le \varpi \le T}{sup}|z_{n+1}\left(\varpi \right)-z_n\left(\varpi \right)|>\frac{1}{2}\}\le \frac{4C{\left(4MT\right)}^n}{n!}.$$

Since ${\displaystyle \sum _{n=0}^{\infty }\frac{4C{\left(4MT\right)}^n}{n!}<\infty }$, by the Borel Cantelli Lemma, for almost all $\omega \in \Omega$, there is a $n_0=n_0\left(\omega \right)$ satisfying

$$\underset{0\le \varpi \le T}{sup}|z_{n+1}\left(\varpi \right)-z_n\left(\varpi \right)|\le \frac{1}{2},$$

for $n\ge n_0$. Therefore, if the probability is equal to 1, we have

$${\displaystyle \theta +\sum _{i=0}^{n-1}\left(z_{i+1}\left(\varpi \right)-z_i\left(\varpi \right)\right)=z_n\left(\varpi \right)}$$

These are convergent uniformly in $\varpi \in [0,T]$.

• Let $z\left(\varpi \right)$ be the limit of $z_n\left(\varpi \right)$. It is clear that $z\left(\varpi \right)$ is continuous and ${\mathfrak{F}}_{\varpi }$ adapted. Moreover, using Equation (14), we can see that for every $\varpi$, ${\left\{z_n\left(\varpi \right)\right\}}_{n\ge 1}$ is a Cauchy sequence in $L^2$. Hence, $z_n\left(\varpi \right)⟶z\left(\varpi \right)$ in $L^2$. Letting $n⟶\infty$ in Equation (12), we have

  $${\mathbb{E}|z\left(\varpi \right)|}^2\le C_2{e}^{4KT\left(T+1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right)},$$

  (19)

  for all $0\le \varpi \le T$.
• Therefore, $z\left(\varpi \right)\in {\mathcal{M}}_2\left([0,T];{\mathbb{R}}^n\right)$.
• We can see that

  $$\begin{array}{ccc}\hfill & & \mathbb{E}|{\int }_0^{\varpi }\left(\tau \left(s,z_n\left(s\right)\right)-\tau \left(s,z\left(s\right)\right)\right){ds|}^2+\mathbb{E}{|{\int }_0^{\varpi }\left({\vartheta }_1\left(s,z_n\left(s\right)\right)-{\vartheta }_1\left(s,z\left(s\right)\right)\right)d\mathcal{W}\left(s\right)|}^2\hfill \\ & +& \frac{{\varkappa }^2}{a^{2\varkappa }}\mathbb{E}{|{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}\left({\vartheta }_2(s,z_n\left(s\right)-{\vartheta }_2\left(s,z\left(s\right)\right)\right))ds|}^2\hfill \\ & \le & \overline{K}\left(T+1+\frac{{\varkappa }^2{T}^{2\varkappa -1}}{a^{2\varkappa }\left(2\varkappa -1\right)}\right){\int }_0^T\mathbb{E}{|z_n\left(s\right)-z\left(s\right)|}^{2}ds⟶0,asn⟶\infty .\hfill \end{array}$$

Hence, letting $n⟶\infty$ in Equation (9), we obtain

$$\begin{array}{ccc}\hfill z\left(\varpi \right)& =& \theta +{\int }_0^{\varpi }\tau \left(q,z\left(q\right)\right)dq+{\int }_0^{\varpi }{\vartheta }_1\left(q,z\left(q\right)\right)d\mathcal{W}\left(q\right)\hfill \\ & +& \frac{\varkappa }{a^{\varkappa }}{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -q\right)\right){\left(\varpi -q\right)}^{\varkappa -1}{\vartheta }_2\left(q,z\left(q\right)\right)dq,\hfill \end{array}$$

for all $\varpi \in [0,T]$, as desired. □

4. Averaging Principle

The standard form of Equation (1) is

$$\begin{array}{ccc}\hfill z_{ϵ}\left(\varpi \right)& =& \theta +ϵ{\int }_0^{\varpi }\tau \left(s,z_{ϵ}\left(s\right)\right)ds+\sqrt{ϵ}{\int }_0^{\varpi }{\vartheta }_1\left(s,z_{ϵ}\left(s\right)\right)d\mathcal{W}\left(s\right)\hfill \\ & +& {ϵ}^{\varkappa }\frac{\varkappa }{a^{\varkappa }}{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}{\vartheta }_2\left(s,z_{ϵ}\left(s\right)\right)ds,\hfill \end{array}$$

(20)

with $b,$${\vartheta }_1$, and ${\vartheta }_2$ satisfying ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, and $ϵ\in (0,{ϵ}_0]$, with ${ϵ}_0\in (0,\frac{1}{2})$ being a fixed constant. By Theorem 1, Equation (20) has a unique global solution $z_{ϵ}\left(\varpi \right),$$\varpi \in [0,T]$, for every $ϵ\in (0,{ϵ}_0]$.

Assume that the following assumption holds true:

• ${\mathcal{H}}_3$: Suppose that the measurable functions

$$\overline{\tau }\left(z\right),\overline{{\vartheta }_1}\left(z\right),\overline{{\vartheta }_2}\left(z\right):{\mathbb{R}}^n\to {\mathbb{R}}^n$$

exist and verify the assumptions ${\mathcal{H}}_1,{\mathcal{H}}_2$. For any $(z,T_1)\in {\mathbb{R}}^n\times [0,T]$, we obtain

(1)

${\displaystyle \frac{1}{T_1}{\int }_0^{T_1}|\tau (s,z)-\overline{\tau }{\left(z\right)|}^{2}ds\le {\psi }_1\left(T_1\right){(1+|z|}^2);}$

(2)

${\displaystyle \frac{1}{T_1}{\int }_0^{T_1}|{\vartheta }_1(s,z)-\overline{{\vartheta }_1}{\left(z\right)|}^{2}ds\le {\psi }_2\left(T_1\right){(1+|z|}^2);}$

(3)

${\displaystyle \frac{1}{T_1}{\int }_0^{T_1}|{\vartheta }_2(s,z)-\overline{{\vartheta }_2}{\left(z\right)|}^{2}ds\le {\psi }_3\left(T_1\right){(1+|z|}^2);}$

• where ${\psi }_i\left(T_1\right)$ are bounded positive functions such that ${\displaystyle \underset{T_1⟶\infty }{lim}{\psi }_i\left(T_1\right)=0}$ for $i=1,2,3$.
• We will show that the solution $z_{ϵ}\left(\varpi \right)$ can be approximated by the solution $y_{ϵ}\left(\varpi \right)$ of the standard equation

  $$\begin{array}{ccc}\hfill y_{ϵ}\left(\varpi \right)& =& \theta +ϵ{\int }_0^{\varpi }\overline{\tau }\left(y_{ϵ}\left(s\right)\right)ds+\sqrt{ϵ}{\int }_0^{\varpi }{\overline{\vartheta }}_1\left(y_{ϵ}\left(s\right)\right)d\mathcal{W}\left(s\right)\hfill \\ & +& {ϵ}^{\varkappa }\frac{\varkappa }{a^{\varkappa }}{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}{\overline{\vartheta }}_2\left(y_{ϵ}\left(s\right)\right)ds,\hfill \end{array}$$

  (21)

  for $\varpi \in [0,T]$.

Let us now give our main results about the relationship between $z_{ϵ}\left(\varpi \right)$ and $y_{ϵ}\left(\varpi \right)$.

Theorem 2. 

Suppose that ${\mathcal{H}}_1$, ${\mathcal{H}}_2$, and ${\mathcal{H}}_3$ hold. Let ${\delta }_1>0$ be an arbitrary small constant; thus, there exist $L>0$ and ${ϵ}_1\in (0,{ϵ}_0]$ such that $\forall ϵ\in (0,{ϵ}_1]$,

$$\mathbb{E}(\underset{\varpi \in [0,\frac{L}{ϵ}]}{sup}{|z_{ϵ}\left(\varpi \right)-y_{ϵ}\left(\varpi \right)|}^2)\le {\delta }_1.$$

Proof. 

We obtain

$$\begin{array}{ccc}\hfill z_{ϵ}\left(\varpi \right)-y_{ϵ}\left(\varpi \right)& =& ϵ{\int }_0^{\varpi }\left[\tau \left(s,z_{ϵ}\left(s\right)\right)-\overline{\tau }\left(y_{ϵ}\left(s\right)\right)\right]ds\hfill \\ & +& \sqrt{ϵ}{\int }_0^{\varpi }\left[{\vartheta }_1\left(s,z_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_1\left(y_{ϵ}\left(s\right)\right)\right]d\mathcal{W}\left(s\right)\hfill \\ & +& {ϵ}^{\varkappa }\frac{\varkappa }{a^{\varkappa }}{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}\left[{\vartheta }_2\left(s,z_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_2\left(y_{ϵ}\left(s\right)\right)\right]ds.\hfill \end{array}$$

(22)

By the elementary inequality, we have

$$\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|z_{ϵ}\left(\varpi \right)-y_{ϵ}\left(\varpi \right)|}^2$$

$$\begin{array}{ccc}\hfill & \le & 3{ϵ}^2\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[\tau \left(s,z_{ϵ}\left(s\right)\right)-\overline{\tau }\left(y_{ϵ}\left(s\right)\right)\right]ds|}^2\hfill \\ & +& 3ϵ\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[{\vartheta }_1\left(s,z_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_1\left(y_{ϵ}\left(s\right)\right)\right]d\mathcal{W}\left(s\right)|}^2\hfill \\ & +& 3\frac{{\varkappa }^2}{a^{2\varkappa }}{ϵ}^{2\varkappa }\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}\left[{\vartheta }_2\left(s,z_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_2\left(y_{ϵ}\left(s\right)\right)\right]|}^2\hfill \\ & =& J_1+J_2+J_3.\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill J_1& =& 3{ϵ}^2\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[\tau \left(s,z_{ϵ}\left(s\right)\right)-\overline{\tau }\left(y_{ϵ}\left(s\right)\right)\right]ds|}^2\hfill \\ & \le & 6{ϵ}^2\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[\tau \left(s,z_{ϵ}\left(s\right)\right)-\tau \left(s,y_{ϵ}\left(s\right)\right)\right]ds|}^2\hfill \\ & +& 6{ϵ}^2\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[\tau \left(s,y_{ϵ}\left(s\right)\right)-\overline{\tau }\left(y_{ϵ}\left(s\right)\right)\right]ds|}^2\hfill \\ & \le & 6{ϵ}^2{J}_{11}+6{ϵ}^2{J}_{12}.\hfill \end{array}$$

(24)

By the Hölder inequality and ${\mathcal{H}}_1$, we derive

$$\begin{array}{ccc}\hfill J_{11}& =& \mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[\tau \left(s,z_{ϵ}\left(s\right)\right)-\tau \left(s,y_{ϵ}\left(s\right)\right)\right]ds|}^2\hfill \\ & \le & u\mathbb{E}{\int }_0^u{|\tau \left(s,z_{ϵ}\left(s\right)\right)-\tau \left(s,y_{ϵ}\left(s\right)\right)|}^{2}ds\hfill \\ & \le & \overline{K}u\mathbb{E}{\int }_0^u{|z_{ϵ}\left(s\right)-y_{ϵ}\left(s\right)|}^{2}ds\hfill \\ & \le & \overline{K}u{\int }_0^u\mathbb{E}\underset{v\in [0,s]}{sup}{|z_{ϵ}\left(v\right)-y_{ϵ}\left(v\right)|}^{2}ds.\hfill \end{array}$$

(25)

By ${\mathcal{H}}_3$, we have

$$\begin{array}{ccc}\hfill J_{12}& =& \mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[\tau \left(s,y_{ϵ}\left(s\right)\right)-\overline{\tau }\left(y_{ϵ}\left(s\right)\right)\right]ds|}^2\hfill \\ & \le & u\mathbb{E}\underset{\varpi \in [0,u]}{sup}{\int }_0^{\varpi }{|\tau \left(s,y_{ϵ}\left(s\right)\right)-\overline{\tau }\left(y_{ϵ}\left(s\right)\right)|}^{2}ds\hfill \\ & \le & u^2\mathbb{E}\underset{\varpi \in [0,u]}{sup}\frac{1}{\varpi }{\int }_0^{\varpi }{|\tau \left(s,y_{ϵ}\left(s\right)\right)-\overline{\tau }\left(y_{ϵ}\left(s\right)\right)|}^{2}ds.\hfill \end{array}$$

(26)

Proceeding as the proof of Theorem 1 in [34], we can obtain

$$J_{12}\le u^2\left(\underset{\varpi \in [0,u]}{sup}{\psi }_1\left(\varpi \right)\right)\left(1+\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|y_{ϵ}\left(\varpi \right)|}^2\right).$$

(27)

Therefore,

$$\begin{array}{ccc}\hfill J_1& \le & 6{ϵ}^2\overline{K}u{\int }_0^u\mathbb{E}\underset{v\in [0,s]}{sup}{|z_{ϵ}\left(v\right)-y_{ϵ}\left(v\right)|}^{2}ds\hfill \\ & +& 6{ϵ}^2{u}^2\left(\underset{\varpi \in [0,u]}{sup}{\psi }_1\left(\varpi \right)\right)\left(1+\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|y_{ϵ}\left(\varpi \right)|}^2\right).\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill J_2& =& 3ϵ\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[{\vartheta }_1\left(s,z_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_1\left(y_{ϵ}\left(s\right)\right)\right]d\mathcal{W}\left(s\right)|}^2\hfill \\ & =& 3ϵ\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[\left({\vartheta }_1\left(s,z_{ϵ}\left(s\right)\right)-{\vartheta }_1\left(s,y_{ϵ}\left(s\right)\right)\right)+\left({\vartheta }_1\left(s,y_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_1\left(y_{ϵ}\left(s\right)\right)\right)\right]d\mathcal{W}\left(s\right)|}^2\hfill \\ & \le & 6ϵ\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[{\vartheta }_1\left(s,z_{ϵ}\left(s\right)\right)-{\vartheta }_1\left(s,y_{ϵ}\left(s\right)\right)\right]d\mathcal{W}\left(s\right)|}^2\hfill \\ & +& 6ϵ\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[{\vartheta }_1\left(s,y_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_1\left(y_{ϵ}\left(s\right)\right)\right]d\mathcal{W}\left(s\right)|}^2\hfill \\ & \le & 6ϵJ_{21}+6ϵJ_{22}.\hfill \end{array}$$

(29)

According to Theorem $7.2$ (p. 40 in [35]) and ${\mathcal{H}}_1$, we have

$$\begin{array}{ccc}\hfill J_{21}& =& \mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[{\vartheta }_1\left(s,z_{ϵ}\left(s\right)\right)-{\vartheta }_1\left(s,y_{ϵ}\left(s\right)\right)\right]d\mathcal{W}\left(s\right)|}^2\hfill \\ & \le & 4\mathbb{E}{\int }_0^u{|{\vartheta }_1\left(s,z_{ϵ}\left(s\right)\right)-{\vartheta }_1\left(s,y_{ϵ}\left(s\right)\right)|}^{2}ds\hfill \\ & \le & 4\overline{K}\mathbb{E}{\int }_0^u{|z_{ϵ}\left(s\right)-y_{ϵ}\left(s\right)|}^{2}ds\hfill \\ & \le & 4\overline{K}{\int }_0^u\mathbb{E}\left(\underset{v\in [0,s]}{sup}{|z_{ϵ}\left(v\right)-y_{ϵ}\left(v\right)|}^2\right)ds.\hfill \end{array}$$

(30)

By ${\mathcal{H}}_3$, we derive

$$\begin{array}{ccc}\hfill J_{22}& =& \mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }\left[{\vartheta }_1\left(s,y_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_1\left(y_{ϵ}\left(s\right)\right)\right]d\mathcal{W}\left(s\right)|}^2\hfill \\ & \le & 4\mathbb{E}{\int }_0^u{|{\vartheta }_1\left(s,y_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_1\left(y_{ϵ}\left(s\right)\right)|}^{2}ds\hfill \\ & \le & 4u\mathbb{E}\frac{1}{u}{\int }_0^u{|{\vartheta }_1\left(s,y_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_1\left(y_{ϵ}\left(s\right)\right)|}^{2}ds\hfill \\ & \le & 4u\left(\underset{\varpi \in [0,u]}{sup}{\psi }_2\left(\varpi \right)\right)\left(1+\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|y_{ϵ}\left(\varpi \right)|}^2\right).\hfill \end{array}$$

(31)

Hence,

$$\begin{array}{ccc}\hfill J_2& \le & 24ϵ\overline{K}{\int }_0^u\mathbb{E}\left(\underset{v\in [0,s]}{sup}{|z_{ϵ}\left(v\right)-y_{ϵ}\left(v\right)|}^2\right)ds\hfill \\ & +& 24ϵu\left(\underset{\varpi \in [0,u]}{sup}{\psi }_2\left(\varpi \right)\right)\left(1+\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|y_{ϵ}\left(\varpi \right)|}^2\right).\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill J_3& =& 3\frac{{\varkappa }^2}{a^{2\varkappa }}{ϵ}^{2\varkappa }\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}\left[{\vartheta }_2\left(s,z_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_2\left(y_{ϵ}\left(s\right)\right)\right]|}^2\hfill \\ & \le & 6\frac{{\varkappa }^2}{a^{2\varkappa }}{ϵ}^{2\varkappa }\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}\left[{\vartheta }_2\left(s,z_{ϵ}\left(s\right)\right)-{\vartheta }_2\left(s,y_{ϵ}\left(s\right)\right)\right]ds|}^2\hfill \\ & +& 6\frac{{\varkappa }^2}{a^{2\varkappa }}{ϵ}^{2\varkappa }\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}\left[{\vartheta }_2\left(s,y_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_2\left(y_{ϵ}\left(s\right)\right)\right]ds|}^2\hfill \\ & \le & 6\frac{{\varkappa }^2}{a^{2\varkappa }}{ϵ}^{2\varkappa }{J}_{31}+6\frac{{\varkappa }^2}{a^{2\varkappa }}{ϵ}^{2\varkappa }{J}_{32}.\hfill \end{array}$$

(33)

By the Hölder inequality and ${\mathcal{H}}_1$, we have

$$\begin{array}{ccc}\hfill J_{31}& =& \mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}\left[{\vartheta }_2\left(s,z_{ϵ}\left(s\right)\right)-{\vartheta }_2\left(s,y_{ϵ}\left(s\right)\right)\right]ds|}^2\hfill \\ & \le & \underset{\varpi \in [0,u]}{sup}\left({\int }_0^{\varpi }exp\left(\frac{2\left(a-1\right)}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{2\varkappa -2}ds\right)\mathbb{E}{\int }_0^u{|{\vartheta }_2\left(s,z_{ϵ}\left(s\right)\right)-{\vartheta }_2\left(s,y_{ϵ}\left(s\right)\right)|}^{2}ds\hfill \\ & \le & \underset{\varpi \in [0,u]}{sup}\left({\int }_0^{\varpi }{\left(\varpi -s\right)}^{2\varkappa -2}ds\right)\mathbb{E}{\int }_0^u{|{\vartheta }_2\left(s,z_{ϵ}\left(s\right)\right)-{\vartheta }_2\left(s,y_{ϵ}\left(s\right)\right)|}^{2}ds\hfill \\ & \le & \frac{u^{2\varkappa -1}}{2\varkappa -1}\overline{K}\mathbb{E}{\int }_0^u{|z_{ϵ}\left(s\right)-y_{ϵ}\left(s\right)|}^{2}ds\hfill \\ & \le & \frac{u^{2\varkappa -1}}{2\varkappa -1}\overline{K}{\int }_0^u\mathbb{E}\left(\underset{\upsilon \in [0,s]}{sup}{|z_{ϵ}\left(v\right)-y_{ϵ}\left(v\right)|}^2\right)ds.\hfill \end{array}$$

(34)

By the Hölder inequality and ${\mathcal{H}}_3$, we obtain

$$\begin{array}{ccc}\hfill J_{32}& =& \mathbb{E}\underset{\varpi \in [0,u]}{sup}{|{\int }_0^{\varpi }exp\left(\frac{a-1}{a}\left(\varpi -s\right)\right){\left(\varpi -s\right)}^{\varkappa -1}\left[{\vartheta }_2\left(s,y_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_2\left(y_{ϵ}\left(s\right)\right)\right]ds|}^2\hfill \\ & \le & \frac{u^{2\varkappa }}{2\varkappa -1}\mathbb{E}\frac{1}{u}{\int }_0^u{|{\vartheta }_2\left(s,y_{ϵ}\left(s\right)\right)-{\overline{\vartheta }}_2\left(y_{ϵ}\left(s\right)\right)|}^{2}ds\hfill \\ & \le & \frac{u^{2\varkappa }}{2\varkappa -1}\left({\psi }_3\left(u\right)\right)(1+\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|y_{ϵ}\left(\varpi \right)|}^2).\hfill \end{array}$$

(35)

Consequently,

$$\begin{array}{ccc}\hfill J_3& \le & \frac{6{\varkappa }^2{ϵ}^{2\varkappa }}{a^{2\varkappa }(2\varkappa -1)}{u}^{2\varkappa -1}\overline{K}{\int }_0^u\mathbb{E}(\underset{v\in [0,s]}{sup}{|z_{ϵ}\left(v\right)-y_{ϵ}\left(v\right)|}^2)ds\hfill \\ & +& \frac{6{\varkappa }^2{ϵ}^{2\varkappa }}{a^{2\varkappa }(2\varkappa -1)}{u}^{2\varkappa }({\psi }_3\left(u\right))(1+\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|y_{ϵ}\left(\varpi \right)|}^2).\hfill \end{array}$$

(36)

Substituting Equations (28)–(36) into Equation (23) and using Lemma 1, we can derive that

$$\begin{array}{ccc}\hfill \mathbb{E}\underset{\varpi \in [0,u]}{sup}{|z_{ϵ}\left(\varpi \right)-y_{ϵ}\left(\varpi \right)|}^2& \le & 6ϵ\overline{K}(ϵu+4+\frac{{\varkappa }^2{ϵ}^{2\varkappa -1}}{a^{2\varkappa }(2\varkappa -1)}{u}^{2\varkappa -1}){\int }_0^u\mathbb{E}(\underset{\upsilon \in [0,s]}{sup}{|z_{ϵ}\left(\upsilon \right)-y_{ϵ}\left(\upsilon \right)|}^2)ds\hfill \\ & +& 6ϵu(ϵu(\underset{l\in [0,u]}{sup}{\psi }_1\left(l\right))+4(\underset{l\in [0,u]}{sup}{\psi }_2\left(l\right))+\frac{{\varkappa }^2{ϵ}^{2\varkappa -1}}{a^{2\varkappa }(2\varkappa -1)}{u}^{2\varkappa -1}{\psi }_3\left(u\right))\hfill \\ & \times & (1+\mathbb{E}\underset{\varpi \in [0,u]}{sup}{|y_{ϵ}\left(\varpi \right)|}^2)\hfill \\ & \le & ϵL_3+ϵL_2{\int }_0^u\mathbb{E}(\underset{\upsilon \in [0,s]}{sup}{|z_{ϵ}\left(\upsilon \right)-y_{ϵ}\left(\upsilon \right)|}^2)ds,\hfill \end{array}$$

(37)

where

$$L_2=6\overline{K}(ϵT+4+\frac{{\varkappa }^2{ϵ}^{2\varkappa -1}}{a^{2\varkappa }(2\varkappa -1)}{T}^{2\varkappa -1}),$$

and

$$\begin{array}{ccc}\hfill L_3& =& 6T(ϵT\underset{l\in [0,T]}{sup}{\psi }_1\left(l\right)+4\underset{l\in [0,T]}{sup}{\psi }_2\left(l\right)+\frac{{\varkappa }^2{ϵ}^{2\varkappa -1}}{a^{2\varkappa }(2\varkappa -1)}{T}^{2\varkappa -1}\underset{l\in [0,T]}{sup}{\psi }_3\left(l\right))(1+L_1).\hfill \end{array}$$

(38)

Using the Gronwall inequality, we derive

$$\mathbb{E}\left(\underset{\varpi \in [0,u]}{sup}{|z_{ϵ}\left(\varpi \right)-y_{ϵ}\left(\varpi \right)|}^2\right)\le ϵL_3{e}^{ϵL_{2}u}.$$

Therefore, given any number ${\delta }_1>0$, there exists $L>0$ and ${ϵ}_1\in (0,{ϵ}_0]$ satisfies for every $ϵ\in (0,{ϵ}_1]$,

$$\mathbb{E}\left(\underset{\varpi \in [0,\frac{L}{ϵ}]}{sup}{|z_{ϵ}\left(\varpi \right)-y_{ϵ}\left(\varpi \right)|}^2\right)\le {\delta }_1.$$

□

Next, Theorem 2 shows the convergence, in probability, between the states $z_{ϵ}\left(\varpi \right)$, $y_{ϵ}\left(\varpi \right)$.

Theorem 3. 

Suppose that the PIDSFOS Equations (20) and (21) both satisfy ${\mathcal{H}}_1$-${\mathcal{H}}_3$. Let ${\delta }_1>0$ be an arbitrary small constant; thus, there exist $L>0$ and ${ϵ}_1\in (0,{ϵ}_0]$ such that $\forall ϵ\in (0,{ϵ}_1]$,

$$\underset{ϵ\to 0}{lim}\mathbb{P}(\underset{\varpi \in [0,\frac{L}{ϵ}]}{sup}|z_{ϵ}\left(\varpi \right)-y_{ϵ}\left(\varpi \right)|>{\delta }_1)=0.$$

(39)

Proof. 

Given any constant ${\delta }_1>0$, using Theorem 2 and Chebyshev’s inequality, we can derive that

$$\begin{array}{ccc}\hfill \mathbb{P}(\underset{\varpi \in [0,\frac{L}{ϵ}]}{sup}|z_{ϵ}\left(\varpi \right)-y_{ϵ}\left(\varpi \right)|>{\delta }_1)& \le & \frac{1}{{\delta }_1^2}\mathbb{E}(\underset{\varpi \in [0,\frac{L}{ϵ}]}{sup}{|z_{ϵ}\left(\varpi \right)-y_{ϵ}\left(\varpi \right)|}^2)\hfill \\ & \le & \frac{1}{{\delta }_1^2}ϵL_3{e}^{ϵL_{2}T}.\hfill \end{array}$$

(40)

Setting $ϵ\to 0$, Equation (39) holds. □

Remark 1. 

For $a=1$, we obtain the results given in [25].

5. Illustrative Example

In the following, we will exhibit an example to prove the interest of our results.

Example 1. 

Consider Equation (20), for any $ϵ\in (0,\frac{1}{4}]$ and $\varpi \in [0,T]$, where

$$\begin{array}{ccc}\hfill z_{ϵ}\left(\varpi \right)& \in & {\mathcal{M}}_2\left([0,T];\mathbb{R}\right)\hfill \\ \hfill \tau (\varpi ,z_{ϵ}\left(\varpi \right))& =& \frac{\varpi +2}{\varpi +3}sin\left(z_{ϵ}\left(\varpi \right)\right).\hfill \\ \hfill {\vartheta }_1(\varpi ,z\left(\varpi \right))& =& \left(1+e^{-\varpi }\right)arctan\left(z_{ϵ}\left(\varpi \right)\right)\hfill \\ \hfill {\vartheta }_2(\varpi ,z\left(\varpi \right))& =& \left(1+\sqrt{\frac{e^{-\varpi }}{1+e^{-\varpi }}}\right)cos\left(z_{ϵ}\left(\varpi \right)\right)\hfill \end{array}$$

Set $(\eta ,{\theta }_1,{\theta }_2)\in [0,T]\times \mathbb{R}\times \mathbb{R}$; thus

$$|\tau (\eta ,{\theta }_1)-\tau (\eta ,{\theta }_2){|}^2\vee |{\vartheta }_1(\eta ,{\theta }_1)-{\vartheta }_1(\eta ,{\theta }_2){|}^2\vee |{\vartheta }_2(\eta ,{\theta }_1)-{\vartheta }_2(\eta ,{\theta }_2){|}^2\le \overline{K}{|{\theta }_1-{\theta }_2|}^2,$$

(41)

$${|\tau (\eta ,\theta )|}^2\vee |{\vartheta }_1{(\eta ,\theta )|}^2\vee |{\vartheta }_2{(\eta ,\theta )|}^2\le {K(1+|\theta |}^2).$$

(42)

Let ${\displaystyle \overline{\tau }\left(z\right)=sin\left(z\right)}$, ${\displaystyle \overline{{\vartheta }_1}\left(z\right)=arctan\left(z\right)}$, and ${\displaystyle \overline{{\vartheta }_2}\left(z\right)=cos\left(z\right)}$. Thus, for any $T_1\in [0,T]$, we obtain

$$\begin{array}{ccc}\hfill \frac{1}{T_1}{\int }_0^{T_1}{|\tau (s,z)-\overline{\tau }\left(z\right)|}^{2}ds& \le & {sin}^2\left(z\right)\frac{1}{T_1}{\int }_0^{T_1}\frac{1}{{(s+3)}^2}ds\hfill \\ & \le & \frac{1}{T_1+3}{z}^2\hfill \\ & \le & {\psi }_1\left(T_1\right)\left(1+{|z|}^2\right),\hfill \end{array}$$

(43)

$$\begin{array}{ccc}\hfill \frac{1}{T_1}{\int }_0^{T_1}{|{\vartheta }_1(s,z)-\overline{{\vartheta }_1}\left(z\right)|}^{2}ds& \le & \frac{1}{T_1}{arctan}^2\left(z\right){\int }_0^{T_1}{e}^{-2s}ds\hfill \\ & \le & \frac{e^{-2}-e^{-2T_1}}{2T_1}{z}^2\hfill \\ & \le & {\psi }_2\left(T_1\right)\left(1+{|z|}^2\right),\hfill \end{array}$$

(44)

and

$$\begin{array}{ccc}\hfill \frac{1}{T_1}{\int }_0^{T_1}{|{\vartheta }_2(s,z)-\overline{{\vartheta }_2}\left(z\right)|}^{2}ds& \le & \frac{1}{T_1}{cos}^2\left(z\right){\int }_0^{T_1}\frac{e^{-s}}{1+e^{-s}}ds\hfill \\ & \le & \left(ln\left(2\right)-ln(1+e^{-T_1})\right)\frac{{cos}^2\left(z\right)}{T_1}\hfill \\ & \le & {\psi }_3\left(T_1\right)\left(1+{|z|}^2\right),\hfill \end{array}$$

(45)

where ${\displaystyle {\psi }_1\left(T_1\right)=\frac{1}{T_1+3}}$, ${\displaystyle {\psi }_2\left(T_1\right)=\frac{e^{-2}-e^{-2T_1}}{2T_1}}$ and ${\displaystyle {\psi }_3\left(T_1\right)=\frac{1}{T_1}\left(ln\left(2\right)-ln(1+e^{-T_1})\right)}$.

• Consequently, ${\mathcal{H}}_1$-${\mathcal{H}}_3$ hold with $K=\overline{K}=4$. Finally, Theorem 2 is satisfied.

6. Conclusions

In this article, we have investigated the existence and uniqueness of PIDSFOS of order $\varpi \in (\frac{1}{2},1)$ using the Picard iteration technique and the fractional proportional integral properties. We have proved the convergence of the solution of the averaged PIDSFOS to the solution of the standard PIDSFOS in the sense of the mean square and also in probability by the moment and the Gronwall inequalities. Finally, we have presented an example to illustrate our results. In future work, we can extend this paper to a time-delay system with a non-Lipschitz condition.