Theoretical Results on the pth Moment of ϕ-Hilfer Stochastic Fractional Differential Equations with a Pantograph Term

Abstract

Here, we establish significant results on the well-posedness of solutions to stochastic pantograph fractional differential equations (SPFrDEs) with the $\varphi$-Hilfer fractional derivative. Additionally, we prove the smoothness theorem for the solution and present the averaging principle result. Firstly, the contraction mapping principle is applied to determine the existence and uniqueness of the solution. Secondly, continuous dependence findings are presented under the condition that the coefficients satisfy the global Lipschitz criteria, along with regularity results. Thirdly, we establish results for the averaging principle by applying inequalities and interval translation techniques. Finally, we provide numerical examples and graphical results to support our findings. We make two generalizations of these findings. First, in terms of the fractional derivative, our established theorems and lemmas are consistent with the Caputo operator for $\varphi \left(\mathfrak{t}\right)$ = $\mathfrak{t}$, $\mathfrak{a}=1$. Our findings match the Riemann–Liouville fractional operator for $\varphi \left(\mathfrak{t}\right)=\mathfrak{t}$, $\mathfrak{a}=0$. They agree with the Hadamard and Caputo–Hadamard fractional operators when $\varphi \left(\mathfrak{t}\right)=ln\left(\mathfrak{t}\right)$, $\mathfrak{a}=0$ and $\varphi \left(\mathfrak{t}\right)=ln\left(\mathfrak{t}\right)$, $\mathfrak{a}=1$, respectively. Second, regarding the space, we are make generalizations for the case $\mathfrak{p}=2$.

1. Introduction

The literature contains various types of fractional operators; the most important and commonly used are the Caputo and Riemann–Liouville derivatives. In 2000, Hilfer generalized the Riemann–Liouville derivative, which is known as the Hilfer fractional derivative (HFrD). In 2018, Sousa and Oliveira further generalized HFrD, defining it with respect to an increasing function $\varphi \left(\mathfrak{t}\right)$ to enhance the precision of objective modeling.

Various authors have utilized the $\varphi$-HFrD with fractional differential equations (FDEs) and fractional stochastic differential equations (FSDEs) to analyze multiple concepts. For example, Raheem et al. [1] presented controllability results for FSDEs and discussed the existence of a solution. Lavanya et al. [2] also explored controllability for FSDEs with the Rosenblatt process. Gokul and Udhayakumar [3] examined approximate controllability through sectorial operators. The authors [4] have presented stability results for FDEs with $\varphi$-HFrD and established various concepts related to the solutions of fractional integro-differential equations. Kucche and Mali [5] proved the existence of solutions for FDEs with $\varphi$-HFrD, while Bonilla et al. [6] investigated the solvability of such equations. Additionally, Lima et al. [7] provided results on Ulam–Hyers stability for FDEs with delay in the context of $\varphi$-HFrD. Abdo et al. [8] studied a fractional system in the sense of $\varphi$-HFrD and explored various concepts. Finally, the authors [9] established the approximate controllability of FDEs with $\varphi$-HFrD.

The Riemann–Liouville fractional integral of order $\mathfrak{q}$ with respect to an increasing, non-vanishing, and monotonic function $\varphi \left(\mathfrak{t}\right)$ for a continuous function $\mathrm{f}:[\mathrm{u},\mathfrak{g}]\to \mathfrak{R}$, as given below [10]:

$${\mathcal{I}}_{\mathrm{u}}^{\mathfrak{q},\varphi }\mathrm{f}\left(\mathfrak{t}\right)={\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_{\mathrm{u}}^{\mathfrak{t}}{\varphi }^{\prime }\left(\varpi \right){\left(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right)\right)}^{\mathfrak{q}-1}\mathrm{f}\left(\varpi \right)d\varpi .$$

(1)

The Riemann–Liouville fractional operator of order $\mathfrak{q}$ for $\varphi \left(\mathfrak{t}\right)$ is given as follows [10]:

$$\begin{array}{cc}\hfill {\mathcal{T}}_{{\mathrm{u}}^{+}}^{\mathfrak{q},\varphi }\mathrm{f}\left(\mathfrak{t}\right)& ={\left({\displaystyle \frac{1}{{\varphi }^{\prime }\left(\mathfrak{t}\right)}}{\displaystyle \frac{d}{d\mathfrak{t}}}\right)}^{\delta }{\mathcal{I}}_a^{\delta -\mathfrak{q},\varphi }\mathrm{f}\left(\mathfrak{t}\right)\hfill \\ & ={\displaystyle \frac{1}{\Gamma (\delta -\mathfrak{q})}}{\left({\displaystyle \frac{1}{{\varphi }^{\prime }\left(\mathfrak{t}\right)}}{\displaystyle \frac{d}{d\mathfrak{t}}}\right)}^{\delta }{\int }_{\mathrm{u}}^{\mathfrak{t}}{\varphi }^{\prime }\left(\varpi \right){\left(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right)\right)}^{\delta -\mathfrak{q}-1}\mathrm{f}\left(\varpi \right)d\varpi .\hfill \end{array}$$

(2)

where $\mathfrak{q}\in (\delta -1,\delta )$.

The $\varphi$-HFrD of order $\mathfrak{q}$ and type $\mathfrak{a}$ for the function $\mathrm{f}\left(\mathfrak{t}\right)$ is defined as [11]:

$${}^{\mathrm{H}}\mathcal{T}_{{\mathrm{u}}^{+}}^{\mathfrak{q},\mathfrak{a},\varphi }\mathrm{f}\left(\mathfrak{t}\right)={\mathcal{I}}_{{\mathrm{u}}^{+}}^{\mathfrak{a}(\delta -\mathfrak{q}),\varphi }{\left({\displaystyle \frac{1}{{\varphi }^{\prime }\left(\mathfrak{t}\right)}}{\displaystyle \frac{d}{d\mathfrak{t}}}\right)}^{\delta }{\mathcal{I}}_{{\mathrm{u}}^{+}}^{(1-\mathfrak{a})(\delta -\mathfrak{q}),\varphi }\mathrm{f}\left(\mathfrak{t}\right),$$

(3)

where $\delta =\left[\mathfrak{q}\right]+1$, $\delta \in \mathbf{N}$, and $\varphi$ is an increasing function.

Stochastic differential equations with fractional calculus have drawn a lot of interest and have been used in a number of study fields, such as quantitative finance, option pricing, and disease transmission. Numerous scholars have investigated various important results related to FSDEs, such as the existence and uniqueness (EU) of the solution, stability, regularity, controllability, and averaging principle (AP). Some important works include the use of the Picard approach by the authors in [12] to examine the stability and EU of solutions for FSDEs. Their results were developed in the ${\mathcal{L}}^2$ space, and they established the results with the Caputo fractional derivative (CFD). The Hadamard operator was used in [13] to establish results on a number of different features, including stability and EU, with findings expressed in the ${\mathcal{L}}^2$ space. The EU of FSDEs containing fractional derivatives of a variable order was the main emphasis of Moualkia and Xu [14]. Babaei et al. [15] used the collocation approach with CFD to find solutions for FSDEs. FSDEs were solved using the Taylor series method by Asai and Kloeden [16]. The Picard approach with CFD was used by the authors in [17] to establish the EU results. In their study, Zhang et al. [18] investigated EU for FSDEs with delay while taking the CFD into account. Using the Hadamard operator, Lavanya and Vadivoo [19] examined the controllability of FSDEs with fractional Brownian motion. The Euler–Maruyama method was used to develop solutions for FSDEs in [20], along with stability and EU results. The Euler–Maruyama method’s strong convergence was examined by Yang et al. [21]. In [22], the authors looked at EU and controllability for FSDEs with CFD. A large portion of the literature has been developed inside the framework of ${\mathcal{L}}^2$ space. The authors of [23] presented some important results for the stochastic Burgers system.

The concept of the AP was introduced by Khasminskii [24], who demonstrated that the average and original solutions of a system coincided under suitable restrictions. Using the AP tool, complex systems can be studied by approximating an average system, making it easier to analyze systems that are otherwise difficult to study directly. In the literature, AP results have been proven for various types of systems under appropriate conditions, with some of the most significant being: Mao et al. [25] proved a result for the AP using jumps for delay stochastic systems. Xu et al. [26] discussed AP for FSDEs with the Caputo operator with a mean square. Guo et al. [27] established a theorem for AP under weak conditions. Liu and Xu [28] studied AP for impulsive FSDEs with the Caputo operator. Liu et al. [29] presented a result on AP for partial FSDEs with the Caputo operator with a mean square. Ahmed and Zhu [30] also proved a theorem for FSDEs with Poisson jumps under the Hilfer operator. Xu et al. [31] further explored AP for FSDEs with a mean square. Additionally, the authors investigated AP for backward stochastic system in ${\mathcal{L}}^2$ space. Jing and Li [32] discussed the AP for the stochastic model in the mean square sense. Guo et al. [33] also presented a result for the AP in the mean square sense. In the sense of the Caputo fractional operator, Mouy et al. [34] also worked on AP with the Caputo–Hadamard fractional operator in the mean square sense. In [35], Shen et al. discussed the AP for the fractional heat system. The authors [36,37] also established results regarding AP in the ${\mathcal{L}}^2$ space. For further results on AP regarding fractional operators and space, see [38,39,40].

In this research, we present significant results concerning the EU, continuous dependence, regularity, and AP for SPFrDEs with $\varphi$-HFrD in the $\mathfrak{p}$th moment. Using the contraction mapping principle, we establish the EU result for the solutions of SPFrDEs, then prove the results for continuous dependence under the condition that the coefficients satisfy the global Lipschitz criteria. Additionally, we prove regularity results and, by applying various inequalities and the interval translation approach, derive results for the AP. Finally, we provide numerical examples and graphical representations to support our findings.

This study makes notable contributions in the following ways:

• To the best of our knowledge, this is the first research work to establish results regarding well-posedness, regularity, and AP for SPFrDEs concerning $\varphi$-HFrD.
• The majority of results in the literature concerning the EU and AP for FSDEs were developed in the mean square sense; however, we derived these results using the $\mathfrak{p}$th moment. Consequently, our research expanded the findings about regularity, well-posedness, and the AP for SPFrDEs to $\mathfrak{p}=2$.
• For $\varphi \left(\mathfrak{t}\right)=\mathfrak{t}$, $\mathfrak{a}=1$ our established results align with the FSDEs of CFD. For $\varphi \left(\mathfrak{t}\right)=\mathfrak{t}$, $\mathfrak{a}=0$ our results correspond to the FSDEs of the Riemann–Liouville fractional operator. When $\varphi \left(\mathfrak{t}\right)=ln\left(\mathfrak{t}\right)$$\mathfrak{a}=0$, and $\varphi \left(\mathfrak{t}\right)=ln\left(\mathfrak{t}\right)$$\mathfrak{a}=1$, they align with the Hadamard and Caputo–Hadamard, respectively.
• We present some numerical problems and their graphical results to prove the validity of our established theoretical results.

We examine the following SPFrDEs:

$$\{\begin{array}{cc}{}^{\mathrm{H}}\mathcal{T}_{\mathfrak{t}}^{\mathfrak{q},\mathfrak{a},\varphi }\mathrm{f}\left(\mathfrak{t}\right)=\mathcal{D}(\mathfrak{t},\mathrm{f}\left(\mathfrak{t}\right),\mathrm{f}\left(\mathrm{s}\mathfrak{t}\right))+\Xi (\mathfrak{t},\mathrm{f}\left(\mathfrak{t}\right),\mathrm{f}\left(\mathrm{s}\mathfrak{t}\right)){\displaystyle \frac{d\mathcal{W}\left(\mathfrak{t}\right)}{d\mathfrak{t}}},\hfill & \\ {\mathcal{I}}_{0^{+}}^{(1-\mathfrak{q})(1-\mathfrak{a}),\varphi }\mathrm{f}\left(0\right)=\mathfrak{c}.\hfill \end{array}$$

(4)

where $\mathrm{s}\in (0,1)$, $\varphi$ is an increasing function, and ${}^{\mathrm{H}}\mathcal{T}_{\mathfrak{t}}^{\mathfrak{q},\mathfrak{a},\varphi }$ represents the $\varphi$-HFrD with $\mathfrak{q}\in ({\displaystyle \frac{1}{2}},1)$ and $0\le \mathfrak{a}\le 1$, the functions $\mathcal{D}:[0,\mathfrak{g}]\times {\mathfrak{R}}^{\varkappa }\times {\mathfrak{R}}^{\varkappa }\to {\mathfrak{R}}^{\varkappa }$ and $\Xi :[0,\mathfrak{g}]\times {\mathfrak{R}}^{\varkappa }\times {\mathfrak{R}}^{\varkappa }\to {\mathfrak{R}}^{\varkappa \times \mathfrak{b}}$ are measurable continuous mappings.

This research includes several essential components: In the following section, we outline the fundamental concepts and assumptions. Subsequently, we discuss the well-posedness and regularity of solutions and the AP results for SPFrDEs concerning $\varphi$-HFrD in Section 3 and Section 4, respectively. The next four examples, along with the graphical results, are presented. Finally, our conclusions are summarized in Section 6.

2. Preliminaries

We now outline the definitions and assumptions that underpin the findings of this research study.

Lemma 1 

([41]). Let ${\mathrm{f}}_1\left(\mathfrak{t}\right)\in {\mathcal{L}}^{\mathfrak{p}}$ and ${\mathrm{f}}_2\left(\mathfrak{t}\right)\in {\mathcal{L}}^{\mathrm{q}}$. Then, Hölder’s inequality is defined as follows:

$$\parallel {\mathrm{f}}_1{\mathrm{f}}_2{\parallel }_1\le \parallel {\mathrm{f}}_1{\parallel }_{\mathfrak{p}}{\parallel {\mathrm{f}}_2\parallel }_{\mathrm{q}},$$

where ${\displaystyle \frac{1}{\mathfrak{p}}}+{\displaystyle \frac{1}{\mathrm{q}}}=1$.

Lemma 2 

([42]). Assume that there are real numbers ${\beta }_1,{\beta }_2,{\beta }_3,\cdots ,{\beta }_{\mathfrak{n}}(\mathfrak{n}\in \mathbf{N})$ with ${\beta }_{\iota }\ge 0$, $(\iota =1,2,3,\cdots \mathfrak{n})$. Then, Jensen’s inequality is as follows:

$${\left(\sum _{\iota =1}^{\mathfrak{n}}{\beta }_{\iota }\right)}^{\mathfrak{p}}\le {\mathfrak{n}}^{\mathfrak{p}-1}\sum _{\iota =1}^{\mathfrak{n}}{\beta }_{\iota }^{\mathfrak{p}},\forall \mathfrak{p}>1.$$

where ${\displaystyle \frac{1}{\mathfrak{p}}}+{\displaystyle \frac{1}{\mathrm{q}}}=1$.

Definition 1 

([43]). Let $\mathbb{G}:\mathbb{X}\to \mathbb{R}$ be a measurable function on a measure space $(\mathbb{X},\Sigma ,\mathbb{U})$. The essential supremum of $\mathbb{G}$, denoted as is the smallest real number $\mathbb{Q}$, such that $\mathbb{G}\left(\tau \right)\le \mathbb{Q}$ almost everywhere, meaning:

$$\mathbb{U}\left(\{\tau \in \mathbb{X}\setminus \mathbb{G}(\tau )>\mathbb{Q}\}\right)=0.$$

A measurable procedure $\mathrm{f}\left(\mathfrak{t}\right):[0,\mathfrak{g}]\to {\mathcal{L}}^{\mathfrak{p}}(\Omega ,\mathrm{F},\mathfrak{p})$ is an $\mathbb{F}-$adapted process when $\mathrm{f}\left(\mathfrak{t}\right)\in {\mathcal{Z}}_{\mathfrak{t}}^{\mathfrak{p}}$ with $\mathfrak{t}\ge 0$. The integral form of (4) is as follows:

$$\begin{array}{cc}\hfill \mathrm{f}\left(\mathfrak{t}\right)& =\mathfrak{c}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}+{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}{\varphi }^{\prime }\left(\varpi \right)\mathcal{D}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)d\varpi \hfill \\ & +{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}{\varphi }^{\prime }\left(\varpi \right)\Xi \left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)d\mathcal{W}\left(\varpi \right).\hfill \end{array}$$

(5)

For $\mathfrak{M}$ and $\mathfrak{B}$, we assume the following Lipschitz conditions, which are a fundamental criterion for proving the EU of solutions to differential equations.

• $\left({\xi }_1\right)$$\forall {\mathfrak{A}}_1,{\mathfrak{A}}_2,{\nu }_1,{\nu }_2\in {\mathfrak{R}}^{\varkappa }$ there are ${\mathfrak{T}}_1>0$ and ${\mathfrak{T}}_2>0$, such as

  $$\parallel \mathcal{D}(\mathfrak{t},{\mathfrak{A}}_1,{\mathfrak{A}}_2)-\mathcal{D}(\mathfrak{t},{\nu }_1,{\nu }_2){\parallel }_{\mathfrak{p}}\le {\mathfrak{T}}_1\left(\parallel {\mathfrak{A}}_1-{\nu }_1{\parallel }_{\mathfrak{p}}+{\parallel {\mathfrak{A}}_2-{\nu }_2\parallel }_{\mathfrak{p}}\right).$$

  $$\parallel \Xi (\mathfrak{t},{\mathfrak{A}}_1,{\mathfrak{A}}_2)-\Xi (\mathfrak{t},{\nu }_1,{\nu }_2){\parallel }_{\mathfrak{p}}\le {\mathfrak{T}}_2\left(\parallel {\mathfrak{A}}_1-{\nu }_1{\parallel }_{\mathfrak{p}}+{\parallel {\mathfrak{A}}_2-{\nu }_2\parallel }_{\mathfrak{p}}\right).$$
• $\left({\xi }_2\right)$ The $\mathcal{D}(\mathfrak{t},0,0)$ and $\Xi (\mathfrak{t},0,0)$ are essential bounded, so

  $$\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}{\parallel \mathcal{D}(\mathfrak{t},0,0)\parallel }_{\mathfrak{p}}<\gamma ,\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}{\parallel \Xi (\mathfrak{t},0,0)\parallel }_{\mathfrak{p}}<\gamma .$$

  (6)

Now, assume the following:

• $\left({\xi }_3\right):$$\forall {\mathfrak{A}}_1,{\mathfrak{A}}_2,{\nu }_1,{\nu }_2\in {\mathfrak{R}}^{\varkappa }$, $\mathfrak{t}\in [0,\mathfrak{g}]$ there is a ${\mathfrak{T}}_3>0$, such that:

  $$\begin{array}{cc}\hfill \parallel \mathcal{D}(\mathfrak{t},{\mathfrak{A}}_1,{\mathfrak{A}}_2)-& \mathcal{D}(\mathfrak{t},{\nu }_1,{\nu }_2)\parallel \vee \parallel \Xi (\mathfrak{t},{\mathfrak{A}}_1,{\mathfrak{A}}_2)-\Xi (\mathfrak{t},{\nu }_1,{\nu }_2)\parallel \hfill \\ & \le {\mathfrak{T}}_3\left(\parallel {\mathfrak{A}}_1-{\nu }_1\parallel +\parallel {\mathfrak{A}}_2-{\nu }_2\parallel \right).\hfill \end{array}$$
• $\left({\xi }_4\right):$ For $\forall \mathfrak{A}$, $\nu \in {\mathfrak{R}}^{\varkappa }$, $\mathfrak{t}\in [0,\mathfrak{g}]$, there is ${\mathfrak{T}}_4>0$, such as satisfy the following:

  $$\begin{array}{c}\hfill \parallel \mathcal{D}(\mathfrak{t},\mathfrak{A},\nu )\parallel \vee \parallel \Xi (\mathfrak{t},\mathfrak{A},\nu )\parallel \le {\mathfrak{T}}_4\left(1+\parallel \mathfrak{A}\parallel +\parallel \nu \parallel \right).\end{array}$$
• $\left({\xi }_5\right):$ For ${\mathfrak{g}}_1\in [0,\mathfrak{g}]$, $\mathfrak{t}\in [0,\mathfrak{g}]$, and $\mathfrak{p}\ge 2$, we have

  $$\begin{array}{cc}\hfill {\displaystyle \frac{1}{{\mathfrak{g}}_1}}{\int }_0^{{\mathfrak{g}}_1}\parallel \mathcal{D}(\mathfrak{t},\mathfrak{A},\nu )& -\tilde{\mathcal{D}}{(\mathfrak{t},\mathfrak{A},\nu )\parallel }^{\mathfrak{p}}d\mathfrak{t}\le {\mathbb{D}}_1\left({\mathfrak{g}}_1\right)\left(1+{\parallel \mathfrak{A}\parallel }^{\mathfrak{p}}+{\parallel \nu \parallel }^{\mathfrak{p}}\right),\hfill \end{array}$$

  $$\begin{array}{cc}\hfill {\displaystyle \frac{1}{{\mathfrak{g}}_1}}{\int }_0^{{\mathfrak{g}}_1}\parallel \Xi (\mathfrak{t},\mathfrak{A},\nu )& -\tilde{\Xi }{(\mathfrak{t},\mathfrak{A},\nu )\parallel }^{\mathfrak{p}}d\mathfrak{t}\le {\mathbb{D}}_2\left({\mathfrak{g}}_1\right)\left(1+{\parallel \mathfrak{A}\parallel }^{\mathfrak{p}}+{\parallel \nu \parallel }^{\mathfrak{p}}\right),\hfill \end{array}$$

where ${lim}_{{\mathfrak{g}}_1\to \infty }{\mathbb{D}}_1\left({\mathfrak{g}}_1\right)=0$, ${lim}_{{\mathfrak{g}}_1\to \infty }{\mathbb{D}}_2\left({\mathfrak{g}}_1\right)=0$ and ${\mathbb{D}}_1\left({\mathfrak{g}}_1\right)$, ${\mathbb{D}}_2\left({\mathfrak{g}}_1\right)$ are positively bounded functions.

3. Significant Outcomes

We prove the well-posedness and regularity for the solution of SPFrDEs in this section.

3.1. Well-Posedness

In this sub-section, we prove EU for the solution of SPFrDEs by using the contraction mapping principle, and then we prove that the solution depends continuously on the fractional and initial values. Assume that ${\mathcal{H}}^{\mathfrak{p}}(0,\mathfrak{g})$ is the space of the ${\mathbb{F}}_{\mathfrak{g}}-$adapted process, where ${\mathbb{F}}_{\mathfrak{g}}={\left({\mathrm{F}}_{\mathfrak{t}}\right)}_{\mathfrak{t}\in [0,\mathfrak{g}]}$, and we have

$${\parallel \mathrm{f}\parallel }_{{\mathcal{H}}^{\mathfrak{p}}}=\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}{\parallel \mathrm{f}\left(\mathfrak{t}\right)\parallel }_{\mathfrak{p}}<\infty .$$

Clearly, $\left({\mathcal{H}}^{\mathfrak{p}}(0,\mathfrak{g}),{\parallel ·\parallel }_{{\mathcal{H}}^{\mathfrak{p}}}\right)$ is a Banach space. Let ${\psi }_{\mathfrak{c}}:{\mathcal{H}}^{\mathfrak{p}}(0,\mathfrak{g})\to {\mathcal{H}}^{\mathfrak{p}}(0,\mathfrak{g})$ with ${\psi }_{\mathfrak{c}}\left(\mathrm{f}\left(0\right)\right)=\mathfrak{c}$ and

$$\begin{array}{cc}\hfill {\psi }_{\mathfrak{c}}\left(\mathrm{f}\left(\mathfrak{t}\right)\right)& =\mathfrak{c}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}{\varphi }^{\prime }\left(\varpi \right)\mathcal{D}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)d\varpi \hfill \\ & +{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}{\varphi }^{\prime }\left(\varpi \right)\Xi \left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)d\mathcal{W}\left(\varpi \right).\hfill \end{array}$$

(7)

The following lemma is very important in order to prove various results.

$$\parallel {\mathrm{f}}_1+{\mathrm{f}}_2{\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le 2^{\mathfrak{p}-1}({\parallel {\mathrm{f}}_1\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+\left(\parallel {\mathrm{f}}_2{\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right)),\forall {\mathrm{f}}_1,{\mathrm{f}}_2\in {\mathfrak{R}}^{\varkappa }.$$

(8)

Lemma 3. 

Let ${\xi }_1$ and ${\xi }_2$ hold. Then, ${\psi }_{\mathfrak{c}}$ is well-defined.

Proof. 

Suppose $\mathrm{f}\left(\mathfrak{t}\right)\in {\mathcal{H}}^{\mathfrak{p}}[0,\mathfrak{g}]$ and $\forall \mathfrak{t}\in [0,\mathfrak{g}]$. From (7) and (8), we have

$$\begin{array}{cc}\hfill \parallel {\psi }_{\mathfrak{c}}\left(\mathrm{f}\left(\mathfrak{t}\right)\right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le & 2^{\mathfrak{p}-1}\parallel \mathfrak{c}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}{\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill +& {\displaystyle \frac{2^{2\mathfrak{p}-2}}{{(\Gamma \left(\mathfrak{q}\right))}^{\mathfrak{p}}}}\parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\mathcal{D}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){\varphi }^{\prime }\left(\varpi \right)\mathrm{d}\varpi {\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill +& {\displaystyle \frac{2^{2\mathfrak{p}-2}}{{(\Gamma \left(\mathfrak{q}\right))}^{\mathfrak{p}}}}\parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\Xi \left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){\varphi }^{\prime }\left(\varpi \right)\mathrm{d}\mathcal{W}\left(\varpi \right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}.\hfill \end{array}$$

(9)

By Hölder’s inequality (HI) we have

$$\begin{array}{cc}\hfill & \parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\mathcal{D}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){\varphi }^{\prime }\left(\varpi \right)\mathrm{d}\varpi {\parallel }_{\mathfrak{p}}^{\mathfrak{p}}=\hfill \\ & {\sum }_{ı=1}^m\mathbf{E}{\left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left|{\mathcal{D}}_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)\right|{\varphi }^{\prime }\left(\varpi \right)\mathrm{d}\varpi \right)}^{\mathfrak{p}}\hfill \\ & \le {\sum }_{ı=1}^m\mathbf{E}\left({\left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{{\displaystyle \frac{(\mathfrak{q}-1)\mathfrak{p}}{(\mathfrak{p}-1)}}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{\mathfrak{p}-1}}}\mathrm{d}\varpi \right)}^{\mathfrak{p}-1}{\int }_0^{\mathfrak{t}}\right|{\mathcal{D}}_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){|}^{\mathfrak{p}}\mathrm{d}\varpi )\hfill \\ & \le {\sum }_{ı=1}^m\mathbf{E}({\left(\underset{0<\varpi \le \mathfrak{t}}{sup}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}-1}}}\right)}^{\mathfrak{p}-1}{\left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{{\displaystyle \frac{(\mathfrak{q}-1)\mathfrak{p}}{(\mathfrak{p}-1)}}}{\varphi }^{\prime }\left(\varpi \right)\mathrm{d}\varpi \right)}^{\mathfrak{p}-1}\hfill \\ & {\int }_0^{\mathfrak{t}}\left|{\mathcal{D}}_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){|}^{\mathfrak{p}}\mathrm{d}\varpi \right)\hfill \\ & \le {\Im }^{\mathfrak{p}-1}{\left({\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{q}\mathfrak{p}-1}{\mathfrak{p}-1}}}\right)}^{\mathfrak{p}-1}{\left({\displaystyle \frac{\mathfrak{p}-1}{\mathfrak{q}\mathfrak{p}-1}}\right)}^{\mathfrak{p}-1}{\int }_0^{\mathfrak{t}}\parallel \mathcal{D}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\mathrm{d}\varpi .\hfill \end{array}$$

(10)

where $\Im ={\displaystyle \underset{0<\varpi \le \mathfrak{t}}{sup}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}-1}}}$.

The detailed methodology used to simplify (10) is outlined in Appendix A.1. So, from (10), we have

$$\begin{array}{cc}\hfill & \parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\mathcal{D}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){\varphi }^{\prime }\left(\varpi \right)\mathrm{d}\varpi {\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le {\Im }^{\mathfrak{p}-1}\hfill \\ & {\left({\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{{\displaystyle \frac{(\mathfrak{q}\mathfrak{p}-1)}{\mathfrak{p}-1}}}\right)}^{\mathfrak{p}-1}{\left({\displaystyle \frac{\mathfrak{p}-1}{\mathfrak{q}\mathfrak{p}-1}}\right)}^{\mathfrak{p}-1}{2}^{\mathfrak{p}-1}\left({\mathfrak{T}}_1^{\mathfrak{p}}\mathfrak{g}\right(\parallel \mathrm{f}\left(\varpi \right){\parallel }_{{\mathcal{H}}^{\mathfrak{p}}}^{\mathfrak{p}}+\parallel \mathrm{f}\left(\mathrm{s}\varpi \right){\parallel }_{{\mathcal{H}}^{\mathfrak{p}}}^{\mathfrak{p}})+\mathfrak{g}{\gamma }^{\mathfrak{p}}).\hfill \end{array}$$

(11)

By the Burkholder–Davis–Gundy inequality (BDGI) and HI, we obtain

$$\begin{array}{cc}\hfill & \parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\Xi \left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){\varphi }^{\prime }\left(\varpi \right)\mathrm{d}\mathcal{W}\left(\varpi \right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ & ={\sum }_{ı=1}^m\mathbf{E}|{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left({\Xi }_{ı}(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){\varphi }^{\prime }\left(\varpi \right)\mathrm{d}\mathcal{W}\left(\varpi \right){|}^{\mathfrak{p}}\hfill \\ & \le {\sum }_{ı=1}^m{\mathcal{C}}_{\mathfrak{p}}\mathbf{E}\left|{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\right|{\Xi }_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){|}^2{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^2\mathrm{d}\varpi {|}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & \le {\sum }_{ı=1}^m{\mathcal{C}}_{\mathfrak{p}}\mathbf{E}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}|{\Xi }_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){|}^{\mathfrak{p}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^2\mathrm{d}\varpi \hfill \\ & {\left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^2\mathrm{d}\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}\hfill \\ & \le {\sum }_{ı=1}^m{\mathcal{C}}_{\mathfrak{p}}\mathbf{E}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}|{\Xi }_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){|}^{\mathfrak{p}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^2\mathrm{d}\varpi \hfill \\ & {\left(\underset{0<\varpi \le \mathfrak{t}}{sup}{\varphi }^{\prime }\left(\varpi \right){\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}{\varphi }^{\prime }\left(\varpi \right)\mathrm{d}\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}\hfill \\ & \le {\mathbf{G}}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\mathcal{C}}_{\mathfrak{p}}{\left({\displaystyle \frac{{\left(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right)\right)}^{2\mathfrak{q}-1}}{2\mathfrak{q}-1}}\right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}\hfill \\ & {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\parallel \Xi \left(\varpi ,\mathrm{f}(\varpi ,\mathrm{f}(\mathrm{s}\varpi )\right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}{\left({\varphi }^{\prime }\left(\mathfrak{t}\right)\right)}^2\mathrm{d}\varpi ,\hfill \end{array}$$

(12)

where $\mathbf{G}=\underset{0<\varpi \le \mathfrak{t}}{sup}{\varphi }^{\prime }\left(\varpi \right)$ and ${\mathcal{C}}_{\mathfrak{p}}={\left({\displaystyle \frac{{\mathfrak{p}}^{\mathfrak{p}+1}}{2{(\mathfrak{p}-1)}^{\mathfrak{p}-1}}}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}$.

The detailed methodology used to simplify (12) is outlined in Appendix A.2. Thus, from (12), we obtain

$$\begin{array}{cc}\hfill {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}& \parallel \Xi \left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^2\mathrm{d}\varpi \le {\displaystyle \frac{2^{\mathfrak{p}-1}{\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}\mathbf{G}\hfill \\ & ({\mathfrak{T}}_2^{\mathfrak{p}}\left({\parallel \mathrm{f}\left(\varpi \right)\parallel }_{{\mathcal{H}}_{\mathfrak{p}}}^{\mathfrak{p}}+{\parallel \mathrm{f}\left(\mathrm{s}\varpi \right)\parallel }_{{\mathcal{H}}_{\mathfrak{p}}}^{\mathfrak{p}}\right)+{\gamma }^{\mathfrak{p}}).\hfill \end{array}$$

(13)

Using (13) in (12), we obtain

$$\begin{array}{cc}\hfill & \parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\Xi \left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right){\varphi }^{\prime }\left(\varpi \right)\mathrm{d}\mathcal{W}\left(\varpi \right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le {\mathbf{G}}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}\hfill \\ & {\mathcal{C}}_{\mathfrak{p}}{\left({\displaystyle \frac{{\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}\right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\displaystyle \frac{2^{\mathfrak{p}-1}{\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}\hfill \\ & \mathbf{G}({\mathfrak{T}}_2^{\mathfrak{p}}\left({\parallel \mathrm{f}\left(\varpi \right)\parallel }_{{\mathcal{H}}_{\mathfrak{p}}}^{\mathfrak{p}}+{\parallel \mathrm{f}\left(\mathrm{s}\varpi \right)\parallel }_{{\mathcal{H}}_{\mathfrak{p}}}^{\mathfrak{p}}\right)+{\gamma }^{\mathfrak{p}}).\hfill \end{array}$$

(14)

Hence ${\parallel \psi \left(\mathrm{f}\left(\mathfrak{t}\right)\right)\parallel }_{\mathcal{H}\mathfrak{p}}<\infty$. So, ${\psi }_{\mathfrak{c}}$ is well-defined. □

The following lemma is important for EU.

Lemma 4. 

Assume $\mathfrak{q},{\rm Y}>0$, and $\forall \mathfrak{t}\in [0,\mathfrak{g}]$, then

$${\mathbb{I}}_{0^{+}}^{\mathfrak{q},\varphi }exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)\le {\displaystyle \frac{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}{{{\rm Y}}^{\mathfrak{q}}}}.$$

Proof. 

From (1), we obtain:

$${\mathbb{I}}_{0^{+}}^{\mathfrak{q},\varphi }exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)={\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right){\varphi }^{\prime }\left(\varpi \right)\mathrm{d}\varpi .$$

By $\mathbb{K}=\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right)$,

$${\mathbb{I}}_{0^{+}}^{\mathfrak{q},\varphi }exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)={\displaystyle \frac{\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right)}{\mathbb{K}}^{\mathfrak{q}-1}exp(-{\rm Y}\mathbb{K})\mathrm{d}\mathbb{K}.$$

(15)

Apply $\mathcal{V}={\rm Y}\mathbb{K}$ in (15),

$$\begin{array}{cc}\hfill {\mathbb{I}}_{0^{+}}^{\mathfrak{q},\varphi }exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)& ={\displaystyle \frac{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}{{{\rm Y}}^{\mathfrak{q}}\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{{\rm Y}\left(\varphi \right(\varpi )-\varphi (0\left)\right)}{\mathcal{V}}^{\mathfrak{q}-1}exp(-\mathcal{V})d\mathcal{V}\hfill \\ & \le {\displaystyle \frac{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}{{{\rm Y}}^{\mathfrak{q}}\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\infty }{\mathcal{V}}^{\mathfrak{q}-1}exp(-\mathcal{V})d\mathcal{V}\hfill \\ & ={\displaystyle \frac{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}{{{\rm Y}}^{\mathfrak{q}}}}.\hfill \end{array}$$

Thus, we have

$${\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)d\varpi \le {\displaystyle \frac{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}{{{\rm Y}}^{\mathfrak{q}}}}.$$

(16)

□

Now, the EU of solution is demonstrated by proving that the operator ${\psi }_{\mathfrak{c}}$ is contractive with respect to an appropriate weighted norm.

Theorem 1. 

Suppose $\left({\xi }_1\right)$ and $\left({\xi }_2\right)$ hold; then, system (4) has a unique solution.

Proof. 

Taking ${\rm Y}$ as

$${{\rm Y}}^{2\mathfrak{q}-1}>2\mathbb{Z}\Gamma (2\mathfrak{q}-1),$$

(17)

where

$$\begin{array}{cc}\hfill \mathbb{Z}& ={\displaystyle \frac{2^{\mathfrak{p}-1}}{{(\Gamma \left(\mathfrak{q}\right))}^{\mathfrak{p}}}}({\mathsf{ϝ}}^{\mathfrak{p}-1}{\displaystyle \frac{{\mathfrak{T}}_1^{\mathfrak{p}}{\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{(\mathfrak{p}\mathfrak{q}-2\mathfrak{q}+1)}{(\mathfrak{p}-1)}^{\mathfrak{p}-1}}{{\left(\mathfrak{p}\mathfrak{q}-2\mathfrak{q}+1\right)}^{\mathfrak{p}-1}}}\mathbf{G}+\hfill \\ & {\mathbf{G}}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\left({\displaystyle \frac{{\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}\right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\mathfrak{T}}_2^{\mathfrak{p}}{\mathcal{C}}_{\mathfrak{p}}\mathbf{G}).\hfill \end{array}$$

(18)

The ${\parallel ·\parallel }_{{\rm Y}}$ is

$${\parallel \mathrm{f}\left(\mathfrak{t}\right)\parallel }_{{\rm Y}}=\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}{\left({\displaystyle \frac{{\parallel \mathrm{f}\left(\mathfrak{t}\right)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}}{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}},\forall \mathrm{f}\left(\mathfrak{t}\right)\in {\mathcal{H}}^{\mathfrak{p}}\left([0,\mathfrak{g}]\right).$$

(19)

As the norms ${\parallel ·\parallel }_{{\mathcal{H}}^{\mathfrak{p}}}$ and ${\parallel ·\parallel }_{{\rm Y}}$ are equivalent. So, $\left({\mathcal{H}}^{\mathfrak{p}}(0,\mathfrak{g}),{\parallel ·\parallel }_{{\rm Y}}\right)$ is also a Banach space.

For $\mathrm{f}\left(\mathfrak{t}\right)$ and $\tilde{\mathrm{f}}\left(\mathfrak{t}\right)$, we obtain

$$\begin{array}{cc}\hfill \parallel {\psi }_{\mathfrak{c}}\left(\mathrm{f}\left(\mathfrak{t}\right)\right)& -{\psi }_{\mathfrak{c}}(\tilde{\mathrm{f}}\left(\mathfrak{t}\right){)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le \hfill \\ & {\displaystyle \frac{2^{\mathfrak{p}-1}}{{(\Gamma \left(\mathfrak{q}\right))}^{\mathfrak{p}}}}\parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-\mathcal{D}\left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\right)\right){\varphi }^{\prime }\left(\varpi \right)d\varpi {\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+\hfill \\ & {\displaystyle \frac{2^{\mathfrak{p}-1}}{{(\Gamma \left(\mathfrak{q}\right))}^{\mathfrak{p}}}}\parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left(\Xi \left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-\Xi \left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\right)\right){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}.\hfill \end{array}$$

(20)

Using the HI and $\left({\xi }_1\right)$, we obtain

$$\begin{array}{cc}\hfill & \parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-\mathcal{D}\left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\right)\right){\varphi }^{\prime }\left(\varpi \right)d\varpi {\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill =& {\sum }_{ı=1}^m\mathbf{E}{\left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left({\mathcal{D}}_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-{\mathcal{D}}_{ı}\left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\right)\right){\varphi }^{\prime }\left(\varpi \right)d\varpi \right)}^{\mathfrak{p}}\hfill \\ \hfill \le & {\sum }_{ı=1}^m\mathbf{E}({\left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{{\displaystyle \frac{(\mathfrak{q}-1)(\mathfrak{p}-2)}{\mathfrak{p}-1}}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{{\displaystyle \frac{\mathfrak{p}-2}{\mathfrak{p}-1}}}d\varpi \right)}^{\mathfrak{p}-1}\hfill \\ & \left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\right|{\mathcal{D}}_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-{\mathcal{D}}_{ı}\left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\left)\right|{\varphi }^{\prime }\left(\varpi \right)d\varpi \right))\hfill \\ \hfill \le & {\sum }_{ı=1}^m\mathbf{E}({\left(\underset{0<\varpi \le \mathfrak{t}}{sup}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{{\displaystyle \frac{1}{1-\mathfrak{p}}}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{{\displaystyle \frac{(\mathfrak{q}-1)(\mathfrak{p}-2)}{\mathfrak{p}-1}}}{\varphi }^{\prime }\left(\varpi \right)d\varpi \right)}^{\mathfrak{p}-1}\hfill \\ & \left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\right|{\mathcal{D}}_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-{\mathcal{D}}_{ı}\left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\left)\right|{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \right))\hfill \\ \hfill \le & {\mathsf{ϝ}}^{\mathfrak{p}-1}{\displaystyle \frac{{\mathfrak{T}}_1^{\mathfrak{p}}{\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{(\mathfrak{p}\mathfrak{q}-2\mathfrak{q}+1)}{(\mathfrak{p}-1)}^{\mathfrak{p}-1}}{{\left(\mathfrak{p}\mathfrak{q}-2\mathfrak{q}+1\right)}^{\mathfrak{p}-1}}}\hfill \\ & \underset{0<\varpi \le \mathfrak{t}}{sup}{\varphi }^{\prime }\left(\varpi \right){\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left(\parallel \mathrm{f}\left(\varpi \right)-\tilde{\mathrm{f}}\left(\varpi \right)){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+\parallel \mathrm{f}\left(\mathrm{s}\varpi \right)-\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right){\varphi }^{\prime }\left(\varpi \right)d\varpi .\hfill \end{array}$$

(21)

where $\mathsf{ϝ}=\underset{0<\varpi \le \mathfrak{t}}{sup}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{{\displaystyle \frac{1}{1-\mathfrak{p}}}}$.

Hence, we have

$$\begin{array}{cc}\hfill & \parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-\mathcal{D}\left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\right)\right){\varphi }^{\prime }\left(\varpi \right)d\varpi {\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill \le & \mathbf{G}{\mathsf{ϝ}}^{\mathfrak{p}-1}{\displaystyle \frac{{\mathfrak{T}}_1^{\mathfrak{p}}{\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{(\mathfrak{p}\mathfrak{q}-2\mathfrak{q}+1)}{(\mathfrak{p}-1)}^{\mathfrak{p}-1}}{{\left(\mathfrak{p}\mathfrak{q}-2\mathfrak{q}+1\right)}^{\mathfrak{p}-1}}}\hfill \\ & {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left(\parallel \mathrm{f}\left(\varpi \right)-\tilde{\mathrm{f}}\left(\varpi \right)){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+\parallel \mathrm{f}\left(\mathrm{s}\varpi \right)-\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right){\varphi }^{\prime }\left(\varpi \right)d\varpi .\hfill \end{array}$$

(22)

Now, using $\left({\xi }_1\right)$ and the BDGI, we have

$$\begin{array}{cc}\hfill & \parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left(\Xi \left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-\Xi \left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\right)\right){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill =& {\sum }_{ı=1}^m\mathbf{E}|{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left({\Xi }_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-{\Xi }_{ı}\left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\right)\right){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){|}^{\mathfrak{p}}\hfill \\ \hfill \le & {\sum }_{ı=1}^m{\mathcal{C}}_{\mathfrak{p}}\mathbf{E}\left|{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\right|{\Xi }_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-{\Xi }_{ı}\left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\right){|}^2{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi {|}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill \le & {\sum }_{ı=1}^m{\mathcal{C}}_{\mathfrak{p}}\mathbf{E}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}|{\Xi }_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-{\Xi }_{ı}\left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\right){|}^{\mathfrak{p}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \hfill \\ & {\left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}\hfill \\ \hfill \le & {\sum }_{ı=1}^m{\mathcal{C}}_{\mathfrak{p}}\mathbf{E}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}|{\Xi }_{ı}\left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-{\Xi }_{ı}\left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\right){|}^{\mathfrak{p}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \hfill \\ & {\left(\underset{0<\varpi \le \mathfrak{t}}{sup}\left({\varphi }^{\prime }\left(\varpi \right)\right){\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}{\varphi }^{\prime }\left(\varpi \right)d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}\hfill \\ \hfill \le & {\mathbf{G}}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\left({\displaystyle \frac{{\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}\right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\mathfrak{T}}_2^{\mathfrak{p}}{\mathcal{C}}_{\mathfrak{p}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left(\parallel \mathrm{f}\left(\varpi \right)-\tilde{\mathrm{f}}\left(\varpi \right){{\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+\parallel \mathrm{f}\left(\mathrm{s}\varpi \right)-\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right)\hfill \\ & {\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi .\hfill \\ \hfill \le & {\mathbf{G}}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\left({\displaystyle \frac{\left(\varphi \right(\varpi )-\varphi (0){)}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}\right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\mathfrak{T}}_2^{\mathfrak{p}}{\mathcal{C}}_{\mathfrak{p}}\underset{0<\varpi \le \mathfrak{t}}{sup}{\varphi }^{\prime }\left(\varpi \right){\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\hfill \\ \hfill & \left(\parallel \mathrm{f}\left(\varpi \right)-\tilde{\mathrm{f}}\left(\varpi \right){{\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+\parallel \mathrm{f}\left(\mathrm{s}\varpi \right)-\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right){\varphi }^{\prime }\left(\varpi \right)d\varpi .\hfill \end{array}$$

From the above, we have

$$\begin{array}{cc}\hfill & \parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left(\Xi \left(\varpi ,\mathrm{f}\left(\varpi \right),\mathrm{f}\left(\mathrm{s}\varpi \right)\right)-\Xi \left(\varpi ,\tilde{\mathrm{f}}\left(\varpi \right),\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\right)\right){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill \le & {\mathbf{G}}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\left({\displaystyle \frac{{\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}\right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\mathfrak{T}}_2^{\mathfrak{p}}{\mathcal{C}}_{\mathfrak{p}}\mathbf{G}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left(\parallel \mathrm{f}\left(\varpi \right)-\tilde{\mathrm{f}}\left(\varpi \right){{\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+\parallel \mathrm{f}\left(\mathrm{s}\varpi \right)-\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right)\hfill \\ \hfill & {\varphi }^{\prime }\left(\varpi \right)d\varpi .\hfill \end{array}$$

(23)

Thus, $\forall \mathfrak{t}\in [0,\mathfrak{g}]$, we have

$$\begin{array}{c}\hfill \parallel {\psi }_{\mathfrak{c}}\left(\mathrm{f}\left(\mathfrak{t}\right)\right)-{\psi }_{\mathfrak{c}}\left(\tilde{\mathrm{f}}\left(\mathfrak{t}\right)\right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le \mathbb{Z}{\int }_0^{\mathfrak{t}}\left(\parallel \mathrm{f}\left(\varpi \right)-\tilde{\mathrm{f}}\left(\varpi \right){{\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+\parallel \mathrm{f}\left(\mathrm{s}\varpi \right)-\tilde{\mathrm{f}}\left(\mathrm{s}\varpi \right)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right){(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}{\varphi }^{\prime }\left(\varpi \right)d\varpi ,\end{array}$$

(24)

So,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel {\psi }_{\mathfrak{c}}\mathrm{f}\left(\mathfrak{t}\right)-{\psi }_{\mathfrak{c}}\tilde{\mathrm{f}}{\left(\mathfrak{t}\right)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}}{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}}\le {\displaystyle \frac{1}{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}}\mathbb{Z}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\hfill \\ & \left(exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)){\displaystyle \frac{\parallel \mathrm{f}\left(\varpi \right)-\tilde{\mathrm{f}}{\left(\varpi \right)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}}{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}}+exp\left({\rm Y}\mathrm{s}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right){\displaystyle \frac{\parallel \mathrm{f}\left(\mathrm{s}\varpi \right)-\tilde{\mathrm{f}}{\left(\mathrm{s}\varpi \right)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}}{exp\left({\rm Y}\mathrm{s}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}}\right)\hfill \\ & {\varphi }^{\prime }\left(\varpi \right)d\varpi \hfill \\ \hfill \le & {\displaystyle \frac{1}{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}}\mathbb{Z}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}(exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)\underset{\varpi \in [0,\mathfrak{g}]}{esssup}\left({\displaystyle \frac{\parallel \mathrm{f}\left(\varpi \right)-\tilde{\mathrm{f}}{\left(\varpi \right)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}}{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}}\right)\hfill \\ \hfill +& exp\left({\rm Y}\mathrm{s}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)\underset{\varpi \in [0,\mathfrak{g}]}{esssup}\left({\displaystyle \frac{\parallel \mathrm{f}\left(\mathrm{s}\varpi \right)-\tilde{\mathrm{f}}{\left(\mathrm{s}\varpi \right)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}}{exp\left({\rm Y}\mathrm{s}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}}\right)){\varphi }^{\prime }\left(\varpi \right)d\varpi \hfill \\ \hfill \le & {\displaystyle \frac{\parallel \mathrm{f}\left(\varpi \right)-\tilde{\mathrm{f}}{\left(\varpi \right)\parallel }_{{\rm Y}}^{\mathfrak{p}}}{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}}\mathbb{Z}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left(exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)+exp\left({\rm Y}\mathrm{s}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)\right)\hfill \\ & {\varphi }^{\prime }\left(\varpi \right)d\varpi \hfill \\ \hfill \le & {\displaystyle \frac{2\parallel \mathrm{f}\left(\varpi \right)-\tilde{\mathrm{f}}{\left(\varpi \right)\parallel }_{{\rm Y}}^{\mathfrak{p}}}{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}}\mathbb{Z}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right){\varphi }^{\prime }\left(\varpi \right)d\varpi .\hfill \end{array}$$

(25)

Now, replace $\mathfrak{q}$ by $2\mathfrak{q}-1$ in (16).

$${\displaystyle \frac{1}{\Gamma (2\mathfrak{q}-1)}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right){\varphi }^{\prime }\left(\varpi \right)d\varpi \le {\displaystyle \frac{exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}{{{\rm Y}}^{2\mathfrak{q}-1}}}.$$

From above

$${\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right){\varphi }^{\prime }\left(\varpi \right)d\varpi \le {\displaystyle \frac{\Gamma (2\mathfrak{q}-1)exp\left({\rm Y}\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)\right)}{{{\rm Y}}^{2\mathfrak{q}-1}}}.$$

So,

$$\begin{array}{c}\hfill \parallel {\psi }_{\mathfrak{c}}\left(\mathrm{f}\left(\mathfrak{t}\right)\right)-{\psi }_{\mathfrak{c}}\left(\tilde{\mathrm{f}}\left(\mathfrak{t}\right)\right){\parallel }_{{\rm Y}}\le {\left({\displaystyle \frac{2\mathbb{Z}\Gamma (2\mathfrak{q}-1)}{{{\rm Y}}^{2\mathfrak{q}-1}}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}{\parallel \mathrm{f}\left(\varpi \right)-\tilde{\mathrm{f}}\left(\varpi \right)\parallel }_{{\rm Y}}.\end{array}$$

(26)

From (17), we have ${\displaystyle \frac{\mathbb{Z}\Gamma (2\mathfrak{q}-1)}{{{\rm Y}}^{2\mathfrak{q}-1}}}<1$. □

Theorem 2. 

Suppose ${\xi }_1$ and ${\xi }_2$ hold, then

$$\underset{\mathfrak{q}\to \tilde{\mathfrak{q}}}{lim}\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}{\parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\mathfrak{t},\mathfrak{c})\parallel }_{\mathfrak{p}}=0,$$

(27)

where ${\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})$ is the solution.

Proof. 

Suppose $\mathfrak{q}$, $\tilde{\mathfrak{q}}\in ({\displaystyle \frac{1}{2}},1)$. We obtain the following:

$$\begin{array}{cc}\hfill & {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\mathfrak{c},\mathfrak{t})={\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-\mathcal{D}(\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c}))\right){\varphi }^{\prime }\left(\varpi \right)d\varpi \hfill \\ \hfill +& {\int }_0^{\mathfrak{t}}\left({\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}}{\Gamma \left(\mathfrak{q}\right)}}-{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\tilde{\mathfrak{q}}-1}}{\Gamma \left(\tilde{\mathfrak{q}}\right)}}\right)\mathcal{D}(\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})){\varphi }^{\prime }\left(\varpi \right)d\varpi \hfill \\ \hfill +& {\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left(\Xi (\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-\Xi (\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c}))\right){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right)\hfill \\ \hfill +& {\int }_0^{\mathfrak{t}}\left({\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}}{\Gamma \left(\mathfrak{q}\right)}}-{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\tilde{\mathfrak{q}}-1}}{\Gamma \left(\tilde{\mathfrak{q}}\right)}}\right)\Xi (\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right).\hfill \end{array}$$

(28)

We extract the subsequent outcome from (28) employing (8).

$$\begin{array}{cc}\hfill & \parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\mathfrak{t},\mathfrak{c}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le 2^{\mathfrak{p}-1}\mathbb{Z}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\mathfrak{c},\mathfrak{t}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}d\varpi \hfill \\ \hfill +& 2^{2\mathfrak{p}-2}\parallel {\int }_0^{\mathfrak{t}}\left({\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}}{\Gamma \left(\mathfrak{q}\right)}}-{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\tilde{\mathfrak{q}}-1}}{\Gamma \left(\tilde{\mathfrak{q}}\right)}}\right)\mathcal{D}(\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})){\varphi }^{\prime }\left(\varpi \right)d\varpi {\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill +& 2^{2\mathfrak{p}-2}\parallel {\int }_0^{\mathfrak{t}}\left({\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}}{\Gamma \left(\mathfrak{q}\right)}}-{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\tilde{\mathfrak{q}}-1}}{\Gamma \left(\tilde{\mathfrak{q}}\right)}}\right)\Xi (\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}.\hfill \end{array}$$

(29)

Assume

$$\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )=|{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}}{\Gamma \left(\mathfrak{q}\right)}}-{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\tilde{\mathfrak{q}}-1}}{\Gamma \left(\tilde{\mathfrak{q}}\right)}}|{\varphi }^{\prime }\left(\varpi \right).$$

(30)

By HI, $\left({\xi }_1\right)$, $\left({\xi }_2\right)$, and (8), we have

$$\begin{array}{cc}\hfill & \parallel {\int }_0^{\mathfrak{t}}\left({\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}}{\Gamma \left(\mathfrak{q}\right)}}-{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\tilde{\mathfrak{q}}-1}}{\Gamma \left(\tilde{\mathfrak{q}}\right)}}\right)\mathcal{D}(\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c}){\varphi }^{\prime }\left(\varpi \right)d\varpi {\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill =& {\sum }_{\iota =1}^m\mathbf{E}{\left({\int }_0^{\mathfrak{t}}\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\left|{\mathcal{D}}_{ı}(\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c}))\right|d\varpi \right)}^{\mathfrak{p}}\hfill \\ \hfill \le & {\sum }_{\iota =1}^m\mathbf{E}({\left({\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^{{\displaystyle \frac{\mathfrak{p}}{\mathfrak{p}-1}}}\right)}^{\mathfrak{p}-1}{\int }_0^{\mathfrak{t}}\left|{\mathcal{D}}_{ı}(\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c}){|}^{\mathfrak{p}}d\varpi \right)\hfill \\ \hfill \le & {\left({\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^2\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{\left({\int }_0^{\mathfrak{t}}1d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\int }_0^{\mathfrak{t}}\parallel \mathcal{D}(\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}d\varpi \hfill \\ \hfill \le & {\left({\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^2\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{\mathfrak{g}}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\int }_0^{\mathfrak{t}}{2}^{\mathfrak{p}-1}\left({\mathfrak{T}}_1^{\mathfrak{p}}\parallel {\mathcal{B}}_{\tilde{\mathfrak{q}}}{(\varpi ,\mathfrak{c})\parallel }_{\mathfrak{p}}^{\mathfrak{p}}{)+\parallel \mathcal{D}(\varpi ,0)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right)\hfill \\ \hfill \le & {\left({\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^2\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{\mathfrak{g}}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{2}^{\mathfrak{p}-1}\left({\mathfrak{T}}_1^{\mathfrak{p}}\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}\parallel {\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}})+{\gamma }^{\mathfrak{p}}\right).\hfill \end{array}$$

(31)

Utilizing (30), $\left({\xi }_1\right)$, and $\left({\xi }_2\right)$ in (29).

$$\begin{array}{cc}\hfill & \parallel {\int }_0^{\mathfrak{t}}\left({\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}}{\Gamma \left(\mathfrak{q}\right)}}-{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\tilde{\mathfrak{q}}-1}}{\Gamma \left(\tilde{\mathfrak{q}}\right)}}\right)\Xi (\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill =& {\sum }_{\iota =1}^m\mathbf{E}|{\int }_0^{\mathfrak{t}}\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi ){\Xi }_{ı}(\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c}))d\mathcal{W}\left(\varpi \right){|}^{\mathfrak{p}}\hfill \\ \hfill \le & {\sum }_{\iota =1}^m{\mathcal{C}}_{\mathfrak{p}}\mathbf{E}\left|{\int }_0^{\mathfrak{t}}\mathcal{A}{(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )}^2\right|{\Xi }_{ı}(\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})){|}^{2}d\mathcal{W}\left(\varpi \right){|}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill \le & {\sum }_{\iota =1}^m{\mathcal{C}}_{\mathfrak{p}}\mathbf{E}{\left[{\left({\int }_0^{\mathfrak{t}}\mathcal{A}{(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )}^2|{\Xi }_{ı}(\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})){|}^{\mathfrak{p}}d\varpi \right)}^{{\displaystyle \frac{2}{\mathfrak{p}}}}{\left({\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}-2}{\mathfrak{p}}}}\right]}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill =& {\mathcal{C}}_{\mathfrak{p}}{\int }_0^{\mathfrak{t}}\mathcal{A}{(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )}^2\parallel \Xi (\varpi ,{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}d\varpi {\left({\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}-2}{\mathfrak{p}}}}\hfill \\ \hfill \le & {\mathcal{C}}_{\mathfrak{p}}{\left({\int }_0^{\mathfrak{t}}\mathcal{A}{(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{2}^{\mathfrak{p}-1}\left({\mathfrak{T}}_2^{\mathfrak{p}}\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}{\parallel {\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+{\gamma }^{\mathfrak{p}}\right).\hfill \end{array}$$

(32)

Thus, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\mathfrak{t},\mathfrak{c}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}}{exp\left({\rm Y}\left(\varphi \right(\varpi )-\varphi (0\left)\right)\right)}}\hfill \\ \hfill \le & {\displaystyle \frac{\mathbb{Z}{2}^{\mathfrak{p}-1}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\frac{\parallel {\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})-{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}}{exp\left({\rm Y}\left(\varphi \right(\varpi )-\varphi (0\left)\right)\right)}exp\left({\rm Y}\left(\varphi \right(\varpi )-\varphi (0\left)\right)\right)}{exp\left({\rm Y}\left(\varphi \right(\varpi )-\varphi (0\left)\right)\right)}}\hfill \\ \hfill +& 2^{3\mathfrak{p}-3}\left({\mathfrak{T}}_1^{\mathfrak{p}}\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}{\parallel {\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+{\gamma }^{\mathfrak{p}}\right){\left({\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{\mathfrak{g}}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill +& 2^{3\mathfrak{p}-3}\left({\mathfrak{T}}_2^{\mathfrak{p}}\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}{\parallel {\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+{\gamma }^{\mathfrak{p}}\right){\mathcal{C}}_{\mathfrak{p}}{\left({\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill \le & {\displaystyle \frac{\mathbb{Z}{2}^{\mathfrak{p}-1}\Gamma (2\mathfrak{q}-1)}{{\rm Y}}}\parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\mathfrak{t},\mathfrak{c}){\parallel }_{{\rm Y}}^{\mathfrak{p}}\hfill \\ \hfill +& 2^{3\mathfrak{p}-3}\left({\mathfrak{T}}_1^{\mathfrak{p}}\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}{\parallel {\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+{\gamma }^{\mathfrak{p}}\right){\left({\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{\mathfrak{g}}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill +& 2^{3\mathfrak{p}-3}\left({\mathfrak{T}}_2^{\mathfrak{p}}\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}{\parallel {\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+{\gamma }^{\mathfrak{p}}\right){\mathcal{C}}_{\mathfrak{p}}{\left({\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}.\hfill \end{array}$$

(34)

So

$$\begin{array}{cc}\hfill (1-& {\displaystyle \frac{\mathbb{Z}{2}^{\mathfrak{p}-1}\Gamma (2\mathfrak{q}-1)}{{\rm Y}}})\parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\tilde{\mathfrak{q}}}(\mathfrak{t},\mathfrak{c}){\parallel }_{{\rm Y}}^{\mathfrak{p}}\le 2^{3\mathfrak{p}-3}\left({\mathfrak{T}}_1^{\mathfrak{p}}\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}{\parallel {\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+{\gamma }^{\mathfrak{p}}\right)\hfill \\ & {\left({\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{\mathfrak{g}}^{{\displaystyle \frac{\mathfrak{p}}{2}}}+2^{3\mathfrak{p}-3}\left({\mathfrak{T}}_2^{\mathfrak{p}}\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{esssup}{\parallel {\mathcal{B}}_{\tilde{\mathfrak{q}}}(\varpi ,\mathfrak{c})\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+{\gamma }^{\mathfrak{p}}\right)\hfill \\ & {\mathcal{C}}_{\mathfrak{p}}{\left({\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}.\hfill \end{array}$$

(35)

Now we prove

$$\underset{\tilde{\mathfrak{q}}\to \mathfrak{q}}{lim}\underset{\mathfrak{t}\in [0,\mathfrak{g}]}{sup}{\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^{2}d\varpi =0.$$

To see this note

$$\begin{array}{cc}\hfill & {\int }_0^{\mathfrak{t}}{\left(\mathcal{A}(\mathfrak{t},\varpi ,\mathfrak{q},\tilde{\mathfrak{q}},\varphi )\right)}^{2}d\varpi ={\int }_0^{\mathfrak{t}}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}}{{\Gamma }^2\left(\mathfrak{q}\right)}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi +{\int }_0^{\mathfrak{t}}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\tilde{\mathfrak{q}}-2}}{{\Gamma }^2\left(\tilde{\mathfrak{q}}\right)}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \hfill \\ \hfill -& 2{\int }_0^{\mathfrak{t}}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}+\tilde{\mathfrak{q}}-2}}{\Gamma \left(\mathfrak{q}\right)\Gamma \left(\tilde{\mathfrak{q}}\right)}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \hfill \\ \hfill \le & \underset{0<\varpi \le \mathfrak{t}}{sup}{\varphi }^{\prime }\left(\varpi \right){\int }_0^{\mathfrak{t}}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}}{{\Gamma }^2\left(\mathfrak{q}\right)}}{\varphi }^{\prime }\left(\varpi \right)d\varpi +\underset{0<\varpi \le \mathfrak{t}}{sup}{\varphi }^{\prime }\left(\varpi \right){\int }_0^{\mathfrak{t}}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\tilde{\mathfrak{q}}-2}}{{\Gamma }^2\left(\tilde{\mathfrak{q}}\right)}}{\varphi }^{\prime }\left(\varpi \right)d\varpi \hfill \\ \hfill -& 2\underset{0<\varpi \le \mathfrak{t}}{sup}{\varphi }^{\prime }\left(\varpi \right){\int }_0^{\mathfrak{t}}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}+\tilde{\mathfrak{q}}-2}}{\Gamma \left(\mathfrak{q}\right)\Gamma \left(\tilde{\mathfrak{q}}\right)}}{\varphi }^{\prime }\left(\varpi \right)d\varpi \hfill \\ \hfill =& \mathbf{G}\left({\displaystyle \frac{{(\varphi \left(\varpi \right)-\varphi \left(0\right))}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}\right){\displaystyle \frac{1}{{\Gamma }^2\left(\mathfrak{q}\right)}}+\mathbf{G}\left({\displaystyle \frac{{(\varphi \left(\varpi \right)-\varphi \left(0\right))}^{(2\tilde{\mathfrak{q}}-1)}}{(2\tilde{\mathfrak{q}}-1)}}\right){\displaystyle \frac{1}{{\Gamma }^2\left(\tilde{\mathfrak{q}}\right)}}-\mathbf{G}{\displaystyle \frac{2{(\varphi \left(\varpi \right)-\varphi \left(0\right))}^{(\mathfrak{q}+\tilde{\mathfrak{q}}-1)}}{(\mathfrak{q}+\tilde{\mathfrak{q}}-1)\Gamma \left(\mathfrak{q}\right)\Gamma \left(\tilde{\mathfrak{q}}\right)}}.\hfill \end{array}$$

(36)

This therefore demonstrates the necessary outcome. □

Theorem 3. 

For any $\mathfrak{c},\mathfrak{u}$, we have

$$\parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}{(\mathfrak{t},\mathfrak{u})|}_{\mathfrak{p}}\le \mathfrak{T}{\parallel \mathfrak{c}-\mathfrak{u}\parallel }_{\mathfrak{p}},\forall \mathfrak{t}\in [0,\mathfrak{g}],$$

(37)

where ${\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})$ is the solution.

Proof. 

We have

$$\begin{array}{cc}\hfill {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-& {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{u})=\mathfrak{c}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}-\mathfrak{u}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-\mathcal{D}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}))\right){\varphi }^{\prime }\left(\varpi \right)d\varpi \hfill \\ & +{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left(\Xi (\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-\Xi (\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}))\right){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right).\hfill \end{array}$$

(38)

Apply (8) and we have

$$\begin{array}{cc}\hfill \parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},& \mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{u}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le 2^{\mathfrak{p}-1}\parallel \mathfrak{c}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}-\mathfrak{u}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}{\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill +& {\displaystyle \frac{2^{2\mathfrak{p}-2}}{{(\Gamma \left(\mathfrak{q}\right))}^{\mathfrak{p}}}}\parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}\right(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-\mathcal{D}\left(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}))\right){\varphi }^{\prime }\left(\varpi \right)d\varpi {\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill +& {\displaystyle \frac{2^{2\mathfrak{p}-2}}{{(\Gamma \left(\mathfrak{q}\right))}^{\mathfrak{p}}}}\parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}(\Xi \left(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))\right)-\Xi \left(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))\right){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}.\hfill \end{array}$$

(39)

By HI and $\left({\xi }_1\right)$.

$$\begin{array}{cc}\hfill & \parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}\right(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-\mathcal{D}\left(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}))\right){\varphi }^{\prime }\left(\varpi \right)d\varpi {\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill =& {\sum }_{ı=1}^m\mathbf{E}\left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\right({\mathcal{D}}_{ı}{\left(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-{\mathcal{D}}_{ı}\left(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}))\right){\varphi }^{\prime }\left(\varpi \right)d\varpi \right)}^{\mathfrak{p}}\hfill \\ \hfill \le & {\sum }_{ı=1}^m\mathbf{E}({\left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{{\displaystyle \frac{(\mathfrak{q}-1)(\mathfrak{p}-2)}{\mathfrak{p}-1}}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{{\displaystyle \frac{\mathfrak{p}-2}{\mathfrak{p}-1}}}d\varpi \right)}^{\mathfrak{p}-1}\hfill \\ & \left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\right|{\mathcal{D}}_{ı}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-{\mathcal{D}}_{ı}\left(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u})\left)\right|{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \right))\hfill \\ \hfill \le & {\sum }_{ı=1}^m\mathbf{E}({\left(\underset{0<\varpi \le \mathfrak{t}}{sup}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{{\displaystyle \frac{1}{1-\mathfrak{p}}}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{{\displaystyle \frac{(\mathfrak{q}-1)(\mathfrak{p}-2)}{\mathfrak{p}-1}}}{\varphi }^{\prime }\left(\varpi \right)d\varpi \right)}^{\mathfrak{p}-1}\hfill \\ \hfill & \left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\right|{\mathcal{D}}_{ı}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-{\mathcal{D}}_{ı}\left(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u})\left)\right|{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \right))\hfill \\ \hfill \le & {\mathsf{ϝ}}^{\mathfrak{p}-1}\left({\displaystyle \frac{{\mathfrak{T}}_1^{\mathfrak{p}}{\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{(\mathfrak{p}\mathfrak{q}-2\mathfrak{q}+1)}{(\mathfrak{p}-1)}^{\mathfrak{p}-1}}{{\left(\mathfrak{p}\mathfrak{q}-2\mathfrak{q}+1\right)}^{\mathfrak{p}-1}}}\right){\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left(\parallel {\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right){\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi .\hfill \\ \hfill \le & {\mathsf{ϝ}}^{\mathfrak{p}-1}\left({\displaystyle \frac{{\mathfrak{T}}_1^{\mathfrak{p}}{\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{(\mathfrak{p}\mathfrak{q}-2\mathfrak{q}+1)}{(\mathfrak{p}-1)}^{\mathfrak{p}-1}}{{\left(\mathfrak{p}\mathfrak{q}-2\mathfrak{q}+1\right)}^{\mathfrak{p}-1}}}\right)\underset{0<\varpi \le \mathfrak{t}}{sup}{\varphi }^{\prime }\left(\varpi \right){\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left(\parallel {\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right)\hfill \\ \hfill & {\varphi }^{\prime }\left(\varpi \right)d\varpi \hfill \\ \hfill =& {\mathsf{ϝ}}^{\mathfrak{p}-1}\left({\displaystyle \frac{{\mathfrak{T}}_1^{\mathfrak{p}}{\left(\varphi \left(\varpi \right)-\varphi \left(0\right)\right)}^{(\mathfrak{p}\mathfrak{q}-2\mathfrak{q}+1)}{(\mathfrak{p}-1)}^{\mathfrak{p}-1}}{{\left(\mathfrak{p}\mathfrak{q}-2\mathfrak{q}+1\right)}^{\mathfrak{p}-1}}}\right)\mathbf{G}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left(\parallel {\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right){\varphi }^{\prime }\left(\varpi \right)d\varpi .\hfill \end{array}$$

(40)

Through $\left({\xi }_1\right)$, HI, and BDGI, we have

$$\begin{array}{cc}\hfill & \parallel {\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}(\Xi (\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-\Xi \left(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}))\right){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ \hfill =& {\sum }_{ı=1}^m\mathbf{E}\left|{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\right({\Xi }_{ı}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-{\Xi }_{ı}\left(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}))\right){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){|}^{\mathfrak{p}}\hfill \\ \hfill \le & {\sum }_{ı=1}^m{\mathcal{C}}_{\mathfrak{p}}\mathbf{E}\left|{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\right|{\Xi }_{ı}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-{\Xi }_{ı}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u})){|}^2{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi {|}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill \le & {\sum }_{ı=1}^m{\mathcal{C}}_{\mathfrak{p}}\mathbf{E}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left|{\Xi }_{ı}\right(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-{\Xi }_{ı}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u})){|}^{\mathfrak{p}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \hfill \\ \hfill & {\left({\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}\hfill \\ \hfill \le & {\sum }_{ı=1}^m{\mathcal{C}}_{\mathfrak{p}}\mathbf{E}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left|{\Xi }_{ı}\right(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))-{\Xi }_{ı}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u})){|}^{\mathfrak{p}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \hfill \\ \hfill & {\left(\underset{0<\varpi \le \mathfrak{t}}{sup}{\varphi }^{\prime }\left(\varpi \right){\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}{\varphi }^{\prime }\left(\varpi \right)d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}\hfill \\ \hfill \le & {\mathbf{G}}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\mathfrak{T}}_2^{\mathfrak{p}}{\mathcal{C}}_{\mathfrak{p}}{\left({\displaystyle \frac{{\left(\varphi \right(\varpi )-\varphi (0\left)\right)}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}\right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left(\parallel {\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right){\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi .\hfill \\ \hfill \le & {\mathbf{G}}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\mathfrak{T}}_2^{\mathfrak{p}}{\mathcal{C}}_{\mathfrak{p}}{\left({\displaystyle \frac{{\left(\varphi \right(\varpi )-\varphi (0\left)\right)}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}\right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}\underset{0<\varpi \le \mathfrak{t}}{sup}{\varphi }^{\prime }\left(\varpi \right){\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left(\parallel {\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right){\varphi }^{\prime }\left(\varpi \right)d\varpi .\hfill \\ \hfill =& {\mathbf{G}}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}{\mathfrak{T}}_2^{\mathfrak{p}}{\mathcal{C}}_{\mathfrak{p}}{\left({\displaystyle \frac{{\left(\varphi \right(\varpi )-\varphi (0\left)\right)}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}\right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}\mathbf{G}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left(\parallel {\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right){\varphi }^{\prime }\left(\varpi \right)d\varpi .\hfill \end{array}$$

(41)

From (39) using (40) and (41), we obtain

$$\begin{array}{cc}\hfill \parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})& -{\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{u}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le 2^{\mathfrak{p}-1}\parallel \mathfrak{c}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}-\mathfrak{u}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}{\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ & +2^{\mathfrak{p}-1}\mathbb{Z}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\left(\parallel {\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{u}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right){\varphi }^{\prime }\left(\varpi \right)d\varpi .\hfill \end{array}$$

(42)

Now, the Grönwall inequality, gives

$$\begin{array}{cc}\hfill \parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})& -{\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{u}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le 2^{\mathfrak{p}-1}exp\left(2^{\mathfrak{p}-1}\mathbb{Z}{\int }_0^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}{\varphi }^{\prime }\left(\varpi \right)d\varpi \right)\hfill \\ & \parallel \mathfrak{c}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}-\mathfrak{u}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}{\parallel }_{\mathfrak{p}}^{\mathfrak{p}}.\hfill \end{array}$$

By ([44], [Lemma 7.1.1]), we have

$$\begin{array}{cc}\hfill \parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})& -{\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{u}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le 2^{\mathfrak{p}-1}{\mathbb{E}}_{2\mathfrak{q}-1}\left(2^{\mathfrak{p}-1}\mathbb{Z}\Gamma (2\mathfrak{q}-1){\varphi }^{(2\mathfrak{q}-1)}\right)\hfill \\ & \parallel \mathfrak{c}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}-\mathfrak{u}{\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}{\parallel }_{\mathfrak{p}}^{\mathfrak{p}}.\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill \underset{\mathfrak{c}\to \mathfrak{u}}{lim}\parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{u}){\parallel }_{\mathfrak{p}}=0.\end{array}$$

□

3.2. Regularity

Next, we prove the smoothness of the solution of SPFrDEs.

Theorem 4. 

Let $\left({\xi }_1\right)$ and $\left({\xi }_2\right)$ hold. Then, for $\mathcal{U}>0$ that depends on $\mathfrak{q},\mathfrak{T},\gamma ,\mathfrak{p},\mathfrak{g},\varphi$.

$$\parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{c},\mathfrak{t})-{\mathcal{B}}_{\mathfrak{q}}{(\mathfrak{c},\mathfrak{r})\parallel }_{\mathfrak{p}}\le \mathcal{U}{|\mathfrak{t}-\mathfrak{r}|}^{\mathfrak{q}-{\displaystyle \frac{1}{2}}},\forall \mathfrak{t},\mathfrak{r}\in [0,\mathfrak{g}].$$

(43)

Proof. 

For $\mathfrak{t}>\mathfrak{r}$, we achieve

$$\begin{array}{cc}\hfill & \parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\mathfrak{r},\mathfrak{c}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)2^{2-2\mathfrak{p}}}}\parallel {\int }_{\mathfrak{r}}^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\mathcal{D}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})){\varphi }^{\prime }\left(\varpi \right)d\varpi {\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ & +{\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)2^{2-2\mathfrak{p}}}}\parallel {\int }_{\mathfrak{r}}^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\Xi (\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ & +{\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)2^{2-2\mathfrak{p}}}}\parallel {\int }_0^{\mathfrak{r}}|{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}-{(\varphi \left(\mathfrak{r}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}|\mathcal{D}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})){\varphi }^{\prime }\left(\varpi \right)d\varpi {\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\hfill \\ & +{\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)2^{2-2\mathfrak{p}}}}\parallel {\int }_0^{\mathfrak{r}}|{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}-{(\varphi \left(\mathfrak{r}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}|\Xi (\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})){\varphi }^{\prime }\left(\varpi \right)d\mathcal{W}{\parallel }_{\mathfrak{p}}^{\mathfrak{p}}.\hfill \end{array}$$

(44)

Through HI and BDGI.

$$\begin{array}{cc}\hfill & {\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)2^{2-2\mathfrak{p}}\parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\mathfrak{r},\mathfrak{c}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le {\displaystyle \frac{{(\mathfrak{p}-1)}^{\mathfrak{p}-1}}{{(\mathfrak{p}\mathfrak{q}-1)}^{\mathfrak{p}-1}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\mathfrak{r}\right))}^{1-\mathfrak{q}\mathfrak{p}}}}{\int }_{\mathfrak{r}}^{\mathfrak{t}}\parallel \mathcal{D}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}d\varpi \hfill \\ \hfill +& {\mathcal{C}}_{\mathfrak{p}}\left({\int }_{\mathfrak{r}}^{\mathfrak{t}}\parallel \Xi (\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \right){\left({\int }_{\mathfrak{r}}^{\mathfrak{t}}{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}\hfill \\ \hfill +& {\displaystyle \frac{1}{{\left(\varphi \left(\mathfrak{g}\right)-\varphi \left(0\right)\right)}^{\frac{2-\mathfrak{p}}{2}}}}{\int }_0^{\mathfrak{r}}{\parallel \mathcal{D}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))\parallel }_{\mathfrak{p}}^{\mathfrak{p}}d\varpi {\left({\int }_0^{\mathfrak{r}}|{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}-{(\varphi \left(\mathfrak{r}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}{|}^2{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill +& {\mathcal{C}}_{\mathfrak{p}}{\int }_0^{\mathfrak{r}}{\left({(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}-{(\varphi \left(\mathfrak{r}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\right)}^2{\parallel \Xi (\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c}))\parallel }_{\mathfrak{p}}^{\mathfrak{p}}{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \hfill \\ \hfill \times & {\left({\int }_0^{\mathfrak{r}}{\left({(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}-{(\varphi \left(\mathfrak{r}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}\right)}^2{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \right)}^{{\displaystyle \frac{\mathfrak{p}-2}{2}}}.\hfill \end{array}$$

We obtain the following:

$$\parallel \mathcal{D}(\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le 2^{\mathfrak{p}-1}\left({\mathfrak{T}}_1^{\mathfrak{p}}\parallel {\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+{\parallel \mathcal{D}(\varpi ,0)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right)\le 2^{\mathfrak{p}-1}\left({\mathfrak{T}}_1^{\mathfrak{p}}{\gamma }_1+{\gamma }^{\mathfrak{p}}\right).$$

$$\parallel \Xi (\varpi ,{\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le 2^{\mathfrak{p}-1}\left({\mathfrak{T}}_2^{\mathfrak{p}}\parallel {\mathcal{B}}_{\mathfrak{q}}(\varpi ,\mathfrak{c})){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}+{\parallel \Xi (\varpi ,0)\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\right)\le 2^{\mathfrak{p}-1}\left({\mathfrak{T}}_2^{\mathfrak{p}}{\gamma }_1+{\gamma }^{\mathfrak{p}}\right).$$

Furthermore,

$$\begin{array}{cc}\hfill {\int }_0^{\mathfrak{r}}\left(\right(\varphi \left(\mathfrak{t}\right)-& {\varphi \left(\varpi \right))}^{\mathfrak{q}-1}-{(\varphi \left(\mathfrak{r}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}{)}^2{\left({\varphi }^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \hfill \\ \hfill & \le \underset{0<\varpi \le \mathfrak{t}}{sup}{\varphi }^{\prime }\left(\varpi \right){\int }_0^{\mathfrak{r}}{\left({(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{\mathfrak{q}-1}-{\left(\varphi \left(\mathfrak{r}\right)-\varphi \left(\varpi \right)\right)}^{\mathfrak{q}-1}\right)}^2{\varphi }^{\prime }\left(\varpi \right)d\varpi \hfill \\ \hfill & \le \mathbf{G}{\int }_0^{\mathfrak{r}}\left({(\varphi \left(\mathfrak{r}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}-{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\varpi \right))}^{2\mathfrak{q}-2}\right){\varphi }^{\prime }\left(\varpi \right)d\varpi \hfill \\ \hfill & ={\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\mathfrak{r}\right))}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}+{\displaystyle \frac{{\left(\varphi \left(\mathfrak{r}\right)\right)}^{(2\mathfrak{q}-1)}-{\left(\varphi \left(\mathfrak{t}\right)\right)}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}\hfill \\ \hfill & \le {\displaystyle \frac{{(\varphi \left(\mathfrak{t}\right)-\varphi \left(\mathfrak{r}\right))}^{(2\mathfrak{q}-1)}}{(2\mathfrak{q}-1)}}.\hfill \end{array}$$

(45)

So,

$$\begin{array}{cc}\hfill {\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)2^{2-2\mathfrak{p}}\parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\mathfrak{r},\mathfrak{c}){\parallel }_{\mathfrak{p}}^{\mathfrak{p}}\le & {\displaystyle \frac{{(2\mathfrak{p}-2)}^{\mathfrak{p}-1}}{{(\mathfrak{p}\mathfrak{q}-1)}^{\mathfrak{p}-1}}}{\left(\varphi \left(\mathfrak{t}\right)-\varphi \left(\mathfrak{r}\right)\right)}^{{\displaystyle \frac{(2\mathfrak{q}-1)\mathfrak{p}}{2}}}\left({\mathfrak{T}}_1^{\mathfrak{p}}{\gamma }_1+{\gamma }^{\mathfrak{p}}\right){\left(\varphi \left(\mathfrak{g}\right)-\varphi \left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill +& {\displaystyle \frac{1}{{(2\mathfrak{q}-1)}^{\frac{\mathfrak{p}}{2}}}}{\left(\varphi \left(\mathfrak{t}\right)-\varphi \left(\mathfrak{r}\right)\right)}^{{\displaystyle \frac{(2\mathfrak{q}-1)\mathfrak{p}}{2}}}\left({\mathfrak{T}}_2^{\mathfrak{p}}{\gamma }_1+{\gamma }^{\mathfrak{p}}\right)2^{\mathfrak{p}-1}{\mathcal{C}}_{\mathfrak{p}}\hfill \\ \hfill +& {\displaystyle \frac{2^{\mathfrak{p}-1}}{{(2\mathfrak{q}-1)}^{\mathfrak{p}-1}}}{\left(\varphi \left(\mathfrak{t}\right)-\varphi \left(\mathfrak{r}\right)\right)}^{{\displaystyle \frac{(2\mathfrak{q}-1)\mathfrak{p}}{2}}}\left({\mathfrak{T}}_1^{\mathfrak{p}}{\gamma }_1+{\gamma }^{\mathfrak{p}}\right){\left(\varphi \left(\mathfrak{g}\right)-\varphi \left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill +& {\displaystyle \frac{1}{{(2\mathfrak{q}-1)}^{\frac{\mathfrak{p}}{2}}}}{\left(\varphi \left(\mathfrak{t}\right)-\varphi \left(\mathfrak{r}\right)\right)}^{{\displaystyle \frac{(2\mathfrak{q}-1)\mathfrak{p}}{2}}}\left({\mathfrak{T}}_2^{\mathfrak{p}}{\gamma }_1+{\gamma }^{\mathfrak{p}}\right)2^{\mathfrak{p}-1}{\mathcal{C}}_{\mathfrak{p}}.\hfill \end{array}$$

Hence,

$$\parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\mathfrak{r},\mathfrak{c}){\parallel }_{\mathfrak{p}}\le \mathcal{U}{\left(\varphi \left(\mathfrak{t}\right)-\varphi \left(\mathfrak{r}\right)\right)}^{\mathfrak{p}\mathfrak{q}-{\displaystyle \frac{1}{2}}\mathfrak{p}},$$

where

$$\begin{array}{cc}\hfill {\mathcal{U}}^{\mathfrak{p}}=& 2^{2\mathfrak{p}-2}\left({\displaystyle \frac{{(2\mathfrak{p}-2)}^{\mathfrak{p}-1}}{{(\mathfrak{p}\mathfrak{q}-1)}^{\mathfrak{p}-1}}}\left({\mathfrak{T}}_1^{\mathfrak{p}}{\gamma }_1+{\gamma }^{\mathfrak{p}}\right){\left(\varphi \left(\mathfrak{g}\right)-\varphi \left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}+{\displaystyle \frac{1}{{(2\mathfrak{q}-1)}^{\frac{\mathfrak{p}}{2}}}}\left({\mathfrak{T}}_2^{\mathfrak{p}}{\gamma }_1+{\gamma }^{\mathfrak{p}}\right)2^{\mathfrak{p}-1}{\mathcal{C}}_{\mathfrak{p}}\right){\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}\hfill \\ \hfill +& 2^{2\mathfrak{p}-2}\left({\displaystyle \frac{2^{\mathfrak{p}-1}}{{(2\mathfrak{q}-1)}^{\mathfrak{p}-1}}}\left({\mathfrak{T}}_1^{\mathfrak{p}}{\gamma }_1+{\gamma }^{\mathfrak{p}}\right){\left(\varphi \left(\mathfrak{g}\right)-\varphi \left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}+{\displaystyle \frac{1}{{(2\mathfrak{q}-1)}^{\frac{\mathfrak{p}}{2}}}}\left({\mathfrak{T}}_2^{\mathfrak{p}}{\gamma }_1+{\gamma }^{\mathfrak{p}}\right)2^{\mathfrak{p}-1}{\mathcal{C}}_{\mathfrak{p}}\right){\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right).\hfill \end{array}$$

Thus, we obtain the following:

$$\underset{\mathfrak{r}\to \mathfrak{t}}{lim}\parallel {\mathcal{B}}_{\mathfrak{q}}(\mathfrak{t},\mathfrak{c})-{\mathcal{B}}_{\mathfrak{q}}(\mathfrak{r},\mathfrak{c}){\parallel }_{\mathfrak{p}}=0.$$

□

4. Averaging Principle

Now, we establish generalized results in the $\mathfrak{p}$th moment concerning the AP for SPFrDEs within the framework of the $\varphi$-HFrD.

Lemma 5. 

By utilizing $\left({\xi }_4\right)$ and $\left({\xi }_5\right)$, we have

$$\parallel \tilde{\Xi }{(\mathfrak{A},\nu )\parallel }^{\mathfrak{p}}\le {\mathfrak{T}}_6\left(1+{\parallel \mathfrak{A}\parallel }^{\mathfrak{p}}+{\parallel \nu \parallel }^{\mathfrak{p}}\right),$$

where ${\mathfrak{T}}_6=\left(2^{\mathfrak{p}-1}{\mathbb{D}}_2\left({\mathfrak{g}}_1\right)+6^{\mathfrak{p}-1}{\mathfrak{T}}_4^{\mathfrak{p}}\right)$.

Proof. 

By $\left({\xi }_4\right),\left({\xi }_5\right)$ and (8).

$$\begin{array}{cc}\hfill \parallel \tilde{\Xi }{(\mathfrak{A},\nu )\parallel }^{\mathfrak{p}}& \le \parallel \Xi (\mathfrak{t},\mathfrak{A},\nu )-\tilde{\Xi }{(\mathfrak{A},\nu )\parallel }^{\mathfrak{p}}+2^{\mathfrak{p}-1}{\parallel \Xi (\mathfrak{t},\mathfrak{A},\nu )\parallel }^{\mathfrak{p}}{2}^{\mathfrak{p}-1}\hfill \\ & \le 2^{\mathfrak{p}-1}{\mathbb{D}}_2\left({\mathfrak{g}}_1\right)\left(1+{\parallel \mathfrak{A}\parallel }^{\mathfrak{p}}+{\parallel \nu \parallel }^{\mathfrak{p}}\right)+2^{\mathfrak{p}-1}{\mathfrak{T}}_4^{\mathfrak{p}}{(1+\parallel \mathfrak{A}\parallel +\parallel \nu \parallel )}^{\mathfrak{p}}\hfill \\ & \le \left({\mathbb{D}}_2\left({\mathfrak{g}}_1{2}^{\mathfrak{p}-1}\right)+6^{\mathfrak{p}-1}{\mathfrak{T}}_4^{\mathfrak{p}}\right)\left(1+{\parallel \mathfrak{A}\parallel }^{\mathfrak{p}}+{\parallel \nu \parallel }^{\mathfrak{p}}\right).\hfill \end{array}$$

□

Now, we prove the time-scale change property to obtain the standard form for the AP of (4).

Lemma 6. 

Assume $\mathfrak{t}=\phi \hslash$, then

$${}^{\mathrm{H}}\mathcal{T}_{\hslash }^{\mathfrak{q},\mathfrak{a},\varphi }\mathrm{f}(\phi \hslash )={}^{\mathrm{H}}\mathcal{T}_{\mathfrak{t}}^{\mathfrak{q},\mathfrak{a},\tilde{\varphi }}\mathrm{f}\left(\mathfrak{t}\right).$$

Proof. 

From (2), we have

$$\begin{array}{cc}\hfill {}^{\mathrm{H}}\mathcal{T}_{\hslash }^{\mathfrak{q},\mathfrak{a},\varphi }\mathrm{f}(\phi \hslash )=& {\displaystyle \frac{1}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q}\left)\right)}}{\int }_0^{\hslash }{\displaystyle \frac{d}{d\varpi }}\varphi \left(\varpi \right){(\varphi (\hslash )-\varphi \left(\varpi \right))}^{\mathfrak{a}(1-\mathfrak{q})-1}{\displaystyle \frac{1}{\frac{d}{d\varpi }\varphi \left(\varpi \right)}}{\displaystyle \frac{d}{d\varpi }}d\varpi \hfill \\ & {\displaystyle \frac{1}{\Gamma \left(\right(1-\mathfrak{a}\left)\right(1-\mathfrak{q}\left)\right)}}{\int }_0^{\hslash }{\varphi }^{\prime }\left(\varpi \right){(\varphi (\hslash )-\varphi \left(\varpi \right))}^{(1-\mathfrak{a})(1-\mathfrak{q})-1}\hfill \\ & \mathrm{f}\left(\phi \varpi \right)d\varpi .\hfill \end{array}$$

Let $\phi \varpi =\theta$, $\varpi ={\displaystyle \frac{\theta }{\phi }}$, and by the chain rule ${\displaystyle \frac{d}{d\varpi }}={\displaystyle \frac{d}{d\theta }}.{\displaystyle \frac{d\theta }{d\varpi }}={\displaystyle \frac{d}{d\theta }}.{\displaystyle \frac{d}{d\varpi }}\left(\phi \varpi \right)=\phi {\displaystyle \frac{d}{\mathrm{d}\theta }}$.

So, we have

$$\begin{array}{cc}\hfill {}^{\mathrm{H}}\mathcal{T}_{\hslash }^{\mathfrak{q},\mathfrak{a},\varphi }\mathrm{f}(\phi \hslash )=& {\displaystyle \frac{1}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q}\left)\right)}}{\int }_0^{\phi \hslash }\phi {\displaystyle \frac{d}{d\theta }}\varphi \left({\displaystyle \frac{\theta }{\phi }}\right){(\varphi (\hslash )-\varphi \left({\displaystyle \frac{\theta }{\phi }}\right))}^{\mathfrak{a}(1-\mathfrak{q})-1}{\displaystyle \frac{1}{\phi \frac{d}{d\theta }\varphi \left(\frac{\theta }{\phi }\right)}}\phi {\displaystyle \frac{d}{d\theta }}\hfill \\ & {\displaystyle \frac{1}{\Gamma \left(\right(1-\mathfrak{a}\left)\right(1-\mathfrak{q}\left)\right)}}{\int }_0^{\phi \hslash }\phi {\displaystyle \frac{d}{d\theta }}\varphi \left({\displaystyle \frac{\theta }{\phi }}\right){(\varphi (\hslash )-\varphi \left({\displaystyle \frac{\theta }{\phi }}\right))}^{(1-\mathfrak{a})(1-\mathfrak{q})-1}\hfill \\ & \mathrm{f}\left(\theta \right){\displaystyle \frac{d\theta }{\phi }}{\displaystyle \frac{d\theta }{\phi }}.\hfill \end{array}$$

From above, we have $\phi \hslash =\mathfrak{t}$ and $\hslash ={\displaystyle \frac{\mathfrak{t}}{\phi }}$

$$\begin{array}{cc}\hfill {}^{\mathrm{H}}\mathcal{T}_{\hslash }^{\mathfrak{q},\mathfrak{a},\varphi }\mathrm{f}(\phi \hslash )=& {\displaystyle \frac{1}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q}\left)\right)}}{\int }_0^{\mathfrak{t}}{\displaystyle \frac{d}{d\theta }}\varphi \left({\displaystyle \frac{\theta }{\phi }}\right){(\varphi \left({\displaystyle \frac{\mathfrak{t}}{\phi }}\right)-\varphi \left({\displaystyle \frac{\theta }{\phi }}\right))}^{\mathfrak{a}(1-\mathfrak{q})-1}{\displaystyle \frac{1}{\frac{d}{d\theta }\varphi \left(\frac{\theta }{\phi }\right)}}{\displaystyle \frac{d}{d\theta }}\hfill \\ & {\displaystyle \frac{1}{\Gamma \left(\right(1-\mathfrak{a}\left)\right(1-\mathfrak{q}\left)\right)}}{\int }_0^{\mathfrak{t}}{\displaystyle \frac{d}{d\theta }}\varphi \left({\displaystyle \frac{\theta }{\phi }}\right){(\varphi \left({\displaystyle \frac{\mathfrak{t}}{\phi }}\right)-\varphi \left({\displaystyle \frac{\theta }{\phi }}\right))}^{(1-\mathfrak{a})(1-\mathfrak{q})-1}\hfill \\ & \mathrm{f}\left(\theta \right)d\theta d\theta .\hfill \end{array}$$

$\varphi \left({\displaystyle \frac{\theta }{\phi }}\right)=\tilde{\varphi }\left(\theta \right)$ and $\varphi \left({\displaystyle \frac{\mathfrak{t}}{\phi }}\right)=\tilde{\varphi }\left(\mathfrak{t}\right)$ or, equivalently,

$$\begin{array}{cc}\hfill {}^{\mathrm{H}}\mathcal{T}_{\hslash }^{\mathfrak{q},\mathfrak{a},\varphi }\mathrm{f}(\phi \hslash )=& {\displaystyle \frac{1}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q}\left)\right)}}{\int }_0^{\mathfrak{t}}{\displaystyle \frac{d}{d\theta }}\tilde{\varphi }\left(\theta \right){(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\theta \right))}^{\mathfrak{a}(1-\mathfrak{q})-1}{\displaystyle \frac{1}{\frac{d}{d\theta }\tilde{\varphi }\left(\theta \right)}}{\displaystyle \frac{d}{d\theta }}\hfill \\ & {\displaystyle \frac{1}{\Gamma \left(\right(1-\mathfrak{a}\left)\right(1-\mathfrak{q}\left)\right)}}{\int }_0^{\mathfrak{t}}{\displaystyle \frac{d}{d\theta }}\tilde{\varphi }\left(\theta \right){(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\theta \right))}^{(1-\mathfrak{a})(1-\mathfrak{q})-1}\hfill \\ & \mathrm{f}\left(\theta \right)d\theta d\theta .\hfill \end{array}$$

So, we have the following result:

$${}^{\mathrm{H}}\mathcal{T}_{\hslash }^{\mathfrak{q},\mathfrak{a},\varphi }\mathrm{f}(\phi \hslash )={}^{\mathrm{H}}\mathcal{T}_{\mathfrak{t}}^{\mathfrak{q},\mathfrak{a},\tilde{\varphi }}\mathrm{f}\left(\mathfrak{t}\right).$$

□

Remark 1. 

For $\varphi \left(\mathfrak{t}\right)=\mathfrak{t}$ and $\mathfrak{a}=1$, our established time-scale change property aligns with CFD. Similarly, for $\varphi \left(\mathfrak{t}\right)=\mathfrak{t}$ and $\mathfrak{a}=0$, it aligns with the Riemann–Liouville fractional operator. When $\varphi \left(\mathfrak{t}\right)=ln\left(\mathfrak{t}\right)$ with $\mathfrak{a}=0$, and $\varphi \left(\mathfrak{t}\right)=ln\left(\mathfrak{t}\right)$ with $\mathfrak{a}=1$, the formulation corresponds to the Hadamard and Caputo–Hadamard fractional operators, respectively.

Now, we establish the result regarding the AP of SPFrDEs for the generalized case. For this, first consider the following:

$$\{\begin{array}{cc}{}^{\mathrm{H}}\mathcal{T}_{\mathfrak{t}}^{\mathfrak{q},\mathfrak{a},\varphi }\mathrm{f}\left(\mathfrak{t}\right)=\mathcal{D}({\displaystyle \frac{\mathfrak{t}}{\epsilon }},\mathrm{f}\left(\mathfrak{t}\right),\mathrm{f}\left(\mathfrak{t}\mathrm{s}\right))+\Xi ({\displaystyle \frac{\mathfrak{t}}{\epsilon }},\mathrm{f}\left(\mathfrak{t}\right),\mathrm{f}\left(\mathrm{s}\mathfrak{t}\right)){\displaystyle \frac{d\mathcal{W}\left(\mathfrak{t}\right)}{d\mathfrak{t}}},\hfill & \\ {\mathcal{I}}_{0^{+}}^{(1-\mathfrak{q})(1-\mathfrak{a}),\varphi }\mathrm{f}\left(0\right)=\mathfrak{c}.\hfill \end{array}$$

(46)

Utilizing ${\displaystyle \frac{\mathfrak{t}}{\epsilon }}=\Theta$, Lemma 6 in (46):

$$\begin{array}{cc}\hfill & {}^{\mathrm{H}}\mathcal{T}_{\mathfrak{t}}^{\mathfrak{q},\mathfrak{a},\tilde{\varphi }}\mathrm{f}\left(\epsilon \Theta \right)=\mathcal{D}(\Theta ,\mathrm{f}(\epsilon \Theta ),\mathrm{f}(\mathrm{s}\epsilon \Theta ))+\Xi (\Theta ,\mathrm{f}\left(\epsilon \Theta \right),\mathrm{f}\left(\mathrm{s}\epsilon \Theta \right)){\displaystyle \frac{d\mathcal{W}(\epsilon \Theta )}{\epsilon d\Theta }}.\hfill \end{array}$$

Hence

$$\begin{array}{cc}\hfill & {}^{\mathrm{H}}\mathcal{T}_{\mathfrak{t}}^{\mathfrak{q},\mathfrak{a},\tilde{\varphi }}{\mathrm{f}}_{\epsilon }(\Theta )=\mathcal{D}(\Theta ,{\mathrm{f}}_{\epsilon }(\Theta ),{\mathrm{f}}_{\epsilon }(\mathrm{s}\Theta ))+{\epsilon }^{-{\displaystyle \frac{1}{2}}}\Xi (\Theta ,{\mathrm{f}}_{\epsilon }(\Theta ),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\Theta \right)){\displaystyle \frac{d\mathcal{W}(\Theta )}{d\Theta }}.\hfill \end{array}$$

So, we obtain

$$\left\{\begin{array}{cc}\hfill & {}^{\mathrm{H}}\mathcal{T}_{\mathfrak{t}}^{\mathfrak{q},\mathfrak{a},\tilde{\varphi }}{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)=\mathcal{D}\left(\mathfrak{t},{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\mathfrak{t}\right)\right)+{\epsilon }^{-{\displaystyle \frac{1}{2}}}\Xi \left(\mathfrak{t},{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\mathfrak{t}\right)\right){\displaystyle \frac{\mathrm{d}\mathcal{W}\left(\mathfrak{t}\right)}{\mathrm{d}\mathfrak{t}}},\hfill \\ & {\mathcal{I}}_{0^{+}}^{(1-\mathfrak{q})(1-\mathfrak{a}),\tilde{\varphi }}{\mathrm{f}}_{\epsilon }\left(0\right)=\mathfrak{c}.\hfill \end{array}\right.$$

(47)

Thus, (47) can be expressed integrally as

$$\begin{array}{cc}\hfill {\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)& =\mathfrak{c}{\displaystyle \frac{{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{\tilde{\varphi }}^{\prime }\left(\varpi \right){(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{1-\mathfrak{q}}\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)\mathrm{d}\varpi \hfill \\ & +{\epsilon }^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{\tilde{\varphi }}^{\prime }\left(\varpi \right){(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{1-\mathfrak{q}}\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)\mathrm{d}\mathcal{W}\left(\varpi \right),\hfill \end{array}$$

(48)

when $\epsilon \in (0,{\epsilon }_0]$ with fixed point ${\epsilon }_0$. From (48), we have

$$\begin{array}{cc}\hfill {\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right)& =\mathfrak{c}{\displaystyle \frac{{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{\tilde{\varphi }}^{\prime }\left(\varpi \right){(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{1-\mathfrak{q}}\tilde{\mathcal{D}}\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\mathrm{d}\varpi \hfill \\ & +{\epsilon }^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{\tilde{\varphi }}^{\prime }\left(\varpi \right){(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{1-\mathfrak{q}}\tilde{\Xi }\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\mathrm{d}\mathcal{W}\left(\varpi \right),\hfill \end{array}$$

(49)

where $\tilde{\mathcal{D}}:{\mathfrak{R}}^{\varkappa }\times {\mathfrak{R}}^{\varkappa }\to {\mathfrak{R}}^{\varkappa },\tilde{\Xi }:{\mathfrak{R}}^{\varkappa }\times {\mathfrak{R}}^{\varkappa }\to {\mathfrak{R}}^{\varkappa \times \mathfrak{b}}$.

Theorem 5. 

When $\left({\xi }_3\right)$ to $\left({\xi }_5\right)$ are valid. When $\mho >0$, $\varrho >0$, and ${\epsilon }_1\in \left(0,{\epsilon }_0\right]$ with $\eta \in (0,\mathfrak{q}\mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}})$, then

$$\mathbf{E}\left[\underset{\mathfrak{t}\in [0,\varrho {\epsilon }^{-\eta }]}{sup}\parallel {\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right){\parallel }^{\mathfrak{p}}\right]\le \mho ,\epsilon \in (0,{\epsilon }_1].$$

(50)

Proof. 

Using (48) and (49),

$$\begin{array}{cc}\hfill & {\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right)\hfill \\ & =\mathfrak{c}{\displaystyle \frac{{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}-\mathfrak{c}{\displaystyle \frac{{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(0\right))}^{(1-\mathfrak{q})(1-\mathfrak{a})}}{\Gamma \left(\mathfrak{a}\right(1-\mathfrak{q})+\mathfrak{a})}}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\tilde{\mathcal{D}}\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\right){\tilde{\varphi }}^{\prime }\left(\varpi \right)d\varpi \hfill \\ & +{\epsilon }^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{\mathfrak{q}-1}\left(\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\tilde{\Xi }\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\right)\hfill \\ & {\tilde{\varphi }}^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right).\hfill \end{array}$$

(51)

Via Jensen’s inequality (JI), we have

$$\begin{array}{cc}\hfill & \parallel {\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right){\parallel }^{\mathfrak{p}}\le 2^{\mathfrak{p}-1}\hfill \\ & \parallel {\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\tilde{\mathcal{D}}\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\right){\tilde{\varphi }}^{\prime }\left(\varpi \right)d\varpi {\parallel }^{\mathfrak{p}}\hfill \\ & +2^{\mathfrak{p}-1}\hfill \\ & \parallel {\epsilon }^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{1}{\Gamma \left(\mathfrak{q}\right)}}{\int }_0^{\mathfrak{t}}{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{\mathfrak{q}-1}\left(\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\tilde{\Xi }\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\right){\tilde{\varphi }}^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }^{\mathfrak{p}}\hfill \\ & \le {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{\mathfrak{p}-1}\hfill \\ & \parallel {\int }_0^{\mathfrak{t}}{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\tilde{\mathcal{D}}\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\right){\tilde{\varphi }}^{\prime }\left(\varpi \right)d\varpi {\parallel }^{\mathfrak{p}}\hfill \\ & +{\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{\mathfrak{p}-1}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}\hfill \\ & \parallel {\int }_0^{\mathfrak{t}}{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{\mathfrak{q}-1}\left(\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\tilde{\Xi }\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\right){\tilde{\varphi }}^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }^{\mathfrak{p}}.\hfill \end{array}$$

(52)

Utilizing (52) in (50).

$$\begin{array}{cc}\hfill \mathbf{E}[\underset{0\le \mathfrak{t}\le \vartheta }{sup}\parallel & {\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right){\parallel }^{\mathfrak{p}}]\hfill \\ & \le {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{\mathfrak{p}-1}\hfill \\ & \mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}\parallel {\int }_0^{\mathfrak{t}}{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\tilde{\mathcal{D}}\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\right){\tilde{\varphi }}^{\prime }\left(\varpi \right)d\varpi {\parallel }^{\mathfrak{p}}\right]\hfill \\ & +{\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{\mathfrak{p}-1}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}\hfill \\ & \mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}\parallel {\int }_0^{\mathfrak{t}}{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{\mathfrak{q}-1}\left(\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\tilde{\Xi }\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\right){\tilde{\varphi }}^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }^{\mathfrak{p}}\right]\hfill \\ & =j_1+j_2.\hfill \end{array}$$

(53)

From $j_1$

$$\begin{array}{cc}\hfill j_1\le & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}\hfill \\ & \mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}\parallel {\int }_0^{\mathfrak{t}}{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\right){\tilde{\varphi }}^{\prime }\left(\varpi \right)d\varpi {\parallel }^{\mathfrak{p}}\right]\hfill \\ & +{\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}\hfill \\ & \mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}\parallel {\int }_0^{\mathfrak{t}}{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{\mathfrak{q}-1}\left(\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)-\tilde{\mathcal{D}}\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\right){\tilde{\varphi }}^{\prime }\left(\varpi \right)d\varpi {\parallel }^{\mathfrak{p}}\right]\hfill \\ & =j_{11}+j_{12}.\hfill \end{array}$$

(54)

By utilizing HI, JI, and $\left({\xi }_3\right)$ on $j_{11}$:

$$\begin{array}{cc}\hfill j_{11}\le & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}{\left({\int }_0^{\vartheta }{(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(\varpi \right))}^{{\displaystyle \frac{(\mathfrak{q}-1)\mathfrak{p}}{\mathfrak{p}-1}}}{\left({\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{\mathfrak{p}-1}}}d\varpi \right)}^{\mathfrak{p}-1}\hfill \\ & \mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}{\int }_0^{\mathfrak{t}}{∥\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)∥}^{\mathfrak{p}}d\varpi \right]\hfill \\ \hfill \le & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}{\left(\underset{0<\varpi \le \vartheta }{sup}{\left({\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}-1}}}{\int }_0^{\vartheta }{(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(\varpi \right))}^{{\displaystyle \frac{(\mathfrak{q}-1)\mathfrak{p}}{\mathfrak{p}-1}}}{\tilde{\varphi }}^{\prime }\left(\varpi \right)d\varpi \right)}^{\mathfrak{p}-1}\hfill \\ & \mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}{\int }_0^{\mathfrak{t}}{∥\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)∥}^{\mathfrak{p}}d\varpi \right]\hfill \\ \hfill \le & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{3\mathfrak{p}-3}{\Im }^{\mathfrak{p}-1}{\mathfrak{T}}_3^{\mathfrak{p}}{\left({(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right))}^{{\displaystyle \frac{(\mathfrak{q}\mathfrak{p}-1)}{\mathfrak{p}-1}}}\right)}^{\mathfrak{p}-1}{\left({\displaystyle \frac{\mathfrak{p}-1}{\mathfrak{q}\mathfrak{p}-1}}\right)}^{\mathfrak{p}-1}\hfill \\ & \left(\mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}{\int }_0^{\mathfrak{t}}{∥{\mathrm{f}}_{\epsilon }\left(\varpi \right)-{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right)∥}^{\mathfrak{p}}d\varpi \right]+\mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}{\int }_0^{\mathfrak{t}}{∥{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)-{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)∥}^{\mathfrak{p}}d\varpi \right]\right)\hfill \\ \hfill =& {\mathbb{Y}}_{11}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{(\mathfrak{q}\mathfrak{p}-1)}({\int }_0^{\vartheta }\mathbf{E}\left[\underset{0\le \rho \le \varpi }{sup}{∥{\mathrm{f}}_{\epsilon }\left(\rho \right)-{\mathrm{f}}_{\epsilon }^{*}\left(\rho \right)∥}^{\mathfrak{p}}\right]d\varpi \hfill \\ & +{\int }_0^{\vartheta }\mathbf{E}\left[\underset{0\le \rho \le \varpi }{sup}{\parallel {\mathrm{f}}_{\epsilon }\left(\rho \mathrm{s}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\rho \mathrm{s}\right)\parallel }^{\mathfrak{p}}\right]d\varpi ),\hfill \end{array}$$

(55)

where ${\mathbb{Y}}_{11}={\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{3\mathfrak{p}-3}{\mathfrak{T}}_3^{\mathfrak{p}}{\left({\displaystyle \frac{\mathfrak{p}-1}{\mathfrak{q}\mathfrak{p}-1}}\right)}^{\mathfrak{p}-1}{\Im }^{\mathfrak{p}-1}$.

From $j_{12}$ by applying HI, JI, and $\left({\xi }_5\right)$:

$$\begin{array}{cc}\hfill j_{12}\le & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}{\left({\int }_0^{\vartheta }{(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(\varpi \right))}^{{\displaystyle \frac{(\mathfrak{q}-1)\mathfrak{p}}{\mathfrak{p}-1}}}{\left({\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{\mathfrak{p}-1}}}d\varpi \right)}^{\mathfrak{p}-1}\hfill \\ & \mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}{\int }_0^{\mathfrak{t}}\parallel \mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)-\tilde{\mathcal{D}}\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right){\parallel }^{\mathfrak{p}}d\varpi \right]\hfill \\ \hfill \le & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}{\left(\underset{0<\varpi \le \mathfrak{t}}{sup}{\left({\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}-1}}}{\int }_0^{\vartheta }{(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(\varpi \right))}^{{\displaystyle \frac{(\mathfrak{q}-1)\mathfrak{p}}{\mathfrak{p}-1}}}{\tilde{\varphi }}^{\prime }\left(\varpi \right)d\varpi \right)}^{\mathfrak{p}-1}\hfill \\ & \mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}{\int }_0^{\mathfrak{t}}\parallel \mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)-\tilde{\mathcal{D}}\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right){\parallel }^{\mathfrak{p}}d\varpi \right]\hfill \\ \hfill \le & {\Im }^{\mathfrak{p}-1}{\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}{\left({\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{{\displaystyle \frac{(\mathfrak{q}\mathfrak{p}-1)}{\mathfrak{p}-1}}}\right)}^{\mathfrak{p}-1}{\left({\displaystyle \frac{\mathfrak{p}-1}{\mathfrak{q}\mathfrak{p}-1}}\right)}^{\mathfrak{p}-1}{\mathbb{D}}_1\left(\vartheta \right)\hfill \\ & \vartheta \left(1+\mathbf{E}\parallel {\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right){\parallel }^{\mathfrak{p}}+\mathbf{E}\parallel {\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right){\parallel }^{\mathfrak{p}}\right)\hfill \\ \hfill =& {\mathbb{Y}}_{12}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{(\mathfrak{q}\mathfrak{p}-1)},\hfill \end{array}$$

(56)

where ${\mathbb{Y}}_{12}=\vartheta {\Im }^{\mathfrak{p}-1}{\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}{\left({\displaystyle \frac{\mathfrak{p}-1}{\mathfrak{q}\mathfrak{p}-1}}\right)}^{\mathfrak{p}-1}{\mathbb{D}}_1\left(\vartheta \right)\left(1+\mathbf{E}\parallel {\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right){\parallel }^{\mathfrak{p}}+\mathbf{E}\parallel {\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right){\parallel }^{\mathfrak{p}}\right)$.

The following is provided by $j_2$ via JI:

$$\begin{array}{cc}\hfill j_2\le & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}\hfill \\ & \left(\mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}\parallel {\int }_0^{\mathfrak{t}}{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{\mathfrak{q}-1}\left[\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\right]{\tilde{\varphi }}^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }^{\mathfrak{p}}\right]\right)\hfill \\ \hfill +& {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}{\epsilon }^{-{\displaystyle \frac{1}{2}})}\hfill \\ & \left(\mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}\parallel {\int }_0^{\mathfrak{t}}{(\tilde{\varphi }\left(\mathfrak{t}\right)-\tilde{\varphi }\left(\varpi \right))}^{\mathfrak{q}-1}\left[\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)-\tilde{\Xi }\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)\right]{\tilde{\varphi }}^{\prime }\left(\varpi \right)d\mathcal{W}\left(\varpi \right){\parallel }^{\mathfrak{p}}\right]\right)\hfill \\ \hfill =& j_{21}+j_{22}.\hfill \end{array}$$

(57)

By employing $\left({\xi }_3\right)$, HI, and BDGI on $j_{21}$:

$$\begin{array}{cc}\hfill j_{21}\le & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(2{(\mathfrak{p}-1)}^{1-\mathfrak{p}}{\mathfrak{p}}^{\mathfrak{p}+1}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & \mathbf{E}{\left[{\int }_0^{\vartheta }{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(\varpi \right)\right)}^{2\mathfrak{q}-2}{∥\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)∥}^2{\left({\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \right]}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & \le {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\left({\mathfrak{p}}^{\mathfrak{p}+1}2{(\mathfrak{p}-1)}^{1-\mathfrak{p}}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & \mathbf{E}\left[{\int }_0^{\vartheta }{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(\varpi \right)\right)}^{(\mathfrak{q}-1)\mathfrak{p}}{∥\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }\left(\varpi \right),{\mathrm{f}}_{\epsilon }\left(\mathrm{s}\varpi \right)\right)-\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)∥}^{\mathfrak{p}}{\left({\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)}^{\mathfrak{p}}d\varpi \right]\hfill \\ & \le {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{3\mathfrak{p}-3}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\mathfrak{T}}_3^{\mathfrak{p}}{\left({\mathfrak{p}}^{\mathfrak{p}+1}2{(1-\mathfrak{p})}^{\mathfrak{p}-1}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{\int }_0^{\vartheta }{(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(\varpi \right))}^{(\mathfrak{q}-1)\mathfrak{p}}\hfill \\ & \mathbf{E}\left[\underset{0\le \rho \le \varpi }{sup}\left[{∥{\mathrm{f}}_{\epsilon }\left(\rho \right)-{\mathrm{f}}_{\epsilon }^{*}\left(\rho \right)∥}^{\mathfrak{p}}+{∥{\mathrm{f}}_{\epsilon }\left(\rho \mathrm{s}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\rho \mathrm{s}\right)∥}^{\mathfrak{p}}\right]{\left({\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)}^{\mathfrak{p}}d\varpi \right]\hfill \\ & \le {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{3\mathfrak{p}-3}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\mathfrak{T}}_3^{\mathfrak{p}}{\left({\mathfrak{p}}^{\mathfrak{p}+1}2{(1-\mathfrak{p})}^{\mathfrak{p}-1}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\underset{0<\varpi \le \vartheta }{sup}{\left({\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)}^{\mathfrak{p}-1}{\int }_0^{\vartheta }{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(\varpi \right)\right)}^{(\mathfrak{q}-1)\mathfrak{p}}\hfill \\ & \mathbf{E}\left[\underset{0\le \rho \le \varpi }{sup}\left[{∥{\mathrm{f}}_{\epsilon }\left(\rho \right)-{\mathrm{f}}_{\epsilon }^{*}\left(\rho \right)∥}^{\mathfrak{p}}+{∥{\mathrm{f}}_{\epsilon }\left(\rho \mathrm{s}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\rho \mathrm{s}\right)∥}^{\mathfrak{p}}\right]{\tilde{\varphi }}^{\prime }\left(\varpi \right)d\varpi \right]\hfill \\ & ={\mathbb{Y}}_{21}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}\left({\int }_0^{\vartheta }\right(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }{\left(\varpi \right))}^{(\mathfrak{q}-1)\mathfrak{p}}\mathbf{E}\left[\underset{0\le \rho \le \varpi }{sup}{∥{\mathrm{f}}_{\epsilon }\left(\rho \right)-{\mathrm{f}}_{\epsilon }^{*}\left(\rho \right)∥}^{\mathfrak{p}}\right]{\tilde{\varphi }}^{\prime }\left(\varpi \right)d\varpi \hfill \\ & +{\int }_0^{\vartheta }\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }{\left(\varpi \right))}^{(\mathfrak{q}-1)\mathfrak{p}}\mathbf{E}\left[\underset{0\le \rho \le \varpi }{sup}{∥{\mathrm{f}}_{\epsilon }\left(\rho \mathrm{s}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\rho \mathrm{s}\right)∥}^{\mathfrak{p}}\right]{\tilde{\varphi }}^{\prime }\left(\varpi \right)d\varpi \right),\hfill \end{array}$$

(58)

where ${\mathbb{Y}}_{21}=2^{3\mathfrak{p}-3}{\mathfrak{T}}_3^{\mathfrak{p}}{\left({\displaystyle \frac{{\mathfrak{p}}^{\mathfrak{p}+1}}{2{(\mathfrak{p}-1)}^{\mathfrak{p}-1}}}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}\mathcal{Y}$ and $\mathcal{Y}=\underset{0<\varpi \le \vartheta }{sup}{\left({\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)}^{(\mathfrak{p}-1)}$.

Utilizing HI and BDGI on $j_{22}$:

$$\begin{array}{cc}\hfill j_{22}\le & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{2}^{2\mathfrak{p}-2}{\left(2{(\mathfrak{p}-1)}^{1-\mathfrak{p}}{\mathfrak{p}}^{\mathfrak{p}+1}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & \mathbf{E}{\left[{\int }_0^{\vartheta }{∥\mathcal{D}\left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)-\tilde{\Xi }\left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)∥}^2{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(\varpi \right)\right)}^{2\mathfrak{q}-2}{\left({\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)}^{2}d\varpi \right]}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill \le & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\left(2{(\mathfrak{p}-1)}^{\mathfrak{p}-1}{\mathfrak{p}}^{\mathfrak{p}+1}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\mathbf{E}\left[{\int }_0^{\vartheta }\right(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }{\left(\varpi \right))}^{(\mathfrak{q}-1)\mathfrak{p}}\hfill \\ & \left({∥\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)∥}^{\mathfrak{p}}+\parallel \tilde{\Xi }\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right){\parallel }^{\mathfrak{p}}\right){\left({\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)}^{\mathfrak{p}}d\varpi ]\hfill \\ \hfill \le & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{2}^{2\mathfrak{p}-2}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\left(2{(\mathfrak{p}-1)}^{\mathfrak{p}-1}{\mathfrak{p}}^{\mathfrak{p}+1}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\mathbf{E}[\underset{0<\varpi \le \vartheta }{sup}{\left({\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)}^{(\mathfrak{p}-1)}\hfill \\ & {\int }_0^{\vartheta }\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }{\left(\varpi \right))}^{(\mathfrak{q}-1)\mathfrak{p}}\left({∥\Xi \left(\varpi ,{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right)∥}^{\mathfrak{p}}+\parallel \tilde{\Xi }\left({\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right),{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)\right){\parallel }^{\mathfrak{p}}\right){\tilde{\varphi }}^{\prime }\left(\varpi \right)d\varpi \right]\hfill \\ \hfill \le & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}{\displaystyle \frac{2^{3\mathfrak{p}-3}{3}^{\mathfrak{p}-1}{\epsilon }^{\mathfrak{p}(-\frac{1}{2})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{((\mathfrak{q}-1)\mathfrak{p}+1)+\frac{\mathfrak{p}}{2}-1}{\mathfrak{T}}_4^{\mathfrak{p}}{\left({\mathfrak{T}}_4^{\mathfrak{p}}+{\mathfrak{T}}_6\right)}^{\mathfrak{p}}}{\left(\right(\mathfrak{q}-1)\mathfrak{p}+1)}}\hfill \\ & \mathcal{Y}{\left(2{(\mathfrak{p}-1)}^{1-\mathfrak{p}}{\mathfrak{p}}^{\mathfrak{p}+1}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\left(1+\mathbf{E}\left[{∥{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right)∥}^{\mathfrak{p}}\right]+\mathbf{E}\left[{∥{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)∥}^{\mathfrak{p}}\right]\right)\hfill \\ \hfill =& {\mathbb{Y}}_{22}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{((\mathfrak{q}-1)\mathfrak{p}+1)+{\displaystyle \frac{\mathfrak{p}}{2}}-1},\hfill \end{array}$$

(59)

here

$$\begin{array}{cc}\hfill & {\mathbb{Y}}_{22}=2^{3\mathfrak{p}-3}{3}^{\mathfrak{p}-1}{\mathfrak{T}}_4^{\mathfrak{p}}{\left({\mathfrak{T}}_4^{\mathfrak{p}}+{\mathfrak{T}}_6\right)}^{\mathfrak{p}}{\displaystyle \frac{1}{\left(\right(\mathfrak{q}-1)\mathfrak{p}+1)}}\mathcal{Y}{\left(2{(\mathfrak{p}-1)}^{1-\mathfrak{p}}{\mathfrak{p}}^{\mathfrak{p}+1}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\left(1+\mathbf{E}\left[{∥{\mathrm{f}}_{\epsilon }^{*}\left(\varpi \right)∥}^{\mathfrak{p}}\right]+\mathbf{E}\left[{∥{\mathrm{f}}_{\epsilon }^{*}\left(\mathrm{s}\varpi \right)∥}^{\mathfrak{p}}\right]\right)\hfill \\ & {\displaystyle \frac{1}{{\Gamma }^{\mathfrak{p}}\left(\mathfrak{q}\right)}}.\hfill \end{array}$$

From (54), (59), and (53).

$$\begin{array}{cc}\hfill & \mathbf{E}\left[\underset{0\le \mathfrak{t}\le \vartheta }{sup}{∥{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right)∥}^{\mathfrak{p}}\right]\le {\mathbb{Y}}_{12}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{(\mathfrak{q}\mathfrak{p}-1)}+{\mathbb{Y}}_{22}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{((\mathfrak{q}-1)\mathfrak{p}+1)+{\displaystyle \frac{\mathfrak{p}}{2}}-1}\hfill \\ \hfill & +{\int }_0^{\vartheta }({\mathbb{Y}}_{11}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{(\mathfrak{q}\mathfrak{p}-1)}+{\mathbb{Y}}_{21}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }{\left(\varpi \right))}^{(\mathfrak{q}-1)\mathfrak{p}}{\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)\hfill \\ \hfill & \mathbf{E}\left[\underset{0\le \rho \le \varpi }{sup}{∥{\mathrm{f}}_{\epsilon }\left(\rho \right)-{\mathrm{f}}_{\epsilon }^{*}\left(\rho \right)∥}^{\mathfrak{p}}\right]d\varpi +{\int }_0^{\vartheta }({\mathbb{Y}}_{11}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{(\mathfrak{q}\mathfrak{p}-1)}+{\mathbb{Y}}_{21}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}\hfill \\ & \left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }{\left(\varpi \right))}^{(\mathfrak{q}-1)\mathfrak{p}}{\tilde{\varphi }}^{\prime }\left(\varpi \right)\right)\mathbf{E}\left[\underset{0\le \rho \le \varpi }{sup}{∥{\mathrm{f}}_{\epsilon }\left(\rho \mathrm{s}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\rho \mathrm{s}\right)∥}^{\mathfrak{p}}\right]d\varpi .\hfill \end{array}$$

(60)

From (60), we have

$$\begin{array}{cc}\hfill \mathbf{E}[\underset{0\le \mathfrak{t}\le \vartheta }{sup}\parallel {\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)-& {\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right){\parallel }^{\mathfrak{p}}]\le \left({\mathbb{Y}}_{12}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{(\mathfrak{q}\mathfrak{p}-1)}+{\mathbb{Y}}_{22}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{((\mathfrak{q}-1)\mathfrak{p}+1)+{\displaystyle \frac{\mathfrak{p}}{2}}-1}\right)\hfill \\ & exp\left(2{\mathbb{Y}}_{11}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{(\mathfrak{q}\mathfrak{p}-1)}\vartheta +{\displaystyle \frac{2{\mathbb{Y}}_{21}}{\left(\right(\mathfrak{q}-1)\mathfrak{p}+1)}}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\vartheta \right)-\tilde{\varphi }\left(0\right)\right)}^{((\mathfrak{q}-1)\mathfrak{p}+1)+{\displaystyle \frac{\mathfrak{p}}{2}}-1}\right).\hfill \end{array}$$

So, for $\varrho >0$, $\eta \in (0,\mathfrak{q}\mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}})$, and $\mathfrak{t}\in \left[0,\varrho {\epsilon }^{-\eta }\right]\subseteq [0,\mathfrak{g}]$, we have

$$\mathbf{E}\left[\underset{0\le \mathfrak{t}\le \varrho {\epsilon }^{-\eta }}{sup}{∥{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right)∥}^{\mathfrak{p}}\right]\le \mathbb{A}{\epsilon }^{1-\eta },$$

(61)

where

$$\begin{array}{cc}\hfill \mathbb{A}=& {\epsilon }^{\eta -1}\left({\mathbb{Y}}_{12}{\left(\tilde{\varphi }\left(\varrho {\epsilon }^{-\eta }\right)-\tilde{\varphi }\left(0\right)\right)}^{(\mathfrak{q}\mathfrak{p}-1)}+{\mathbb{Y}}_{22}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\varrho {\epsilon }^{-\eta }\right)-\tilde{\varphi }\left(0\right)\right)}^{((\mathfrak{q}-1)\mathfrak{p}+1)+{\displaystyle \frac{\mathfrak{p}}{2}}-1}\right)\hfill \\ & exp\left(2{\mathbb{Y}}_{11}{\left(\tilde{\varphi }\left(\varrho {\epsilon }^{-\eta }\right)-\tilde{\varphi }\left(0\right)\right)}^{(\mathfrak{q}\mathfrak{p}-1)}\varrho {\epsilon }^{-\eta }+{\displaystyle \frac{2{\mathbb{Y}}_{21}}{\left(\right(\mathfrak{q}-1)\mathfrak{p}+1)}}{\epsilon }^{\mathfrak{p}(-{\displaystyle \frac{1}{2}})}{\left(\tilde{\varphi }\left(\varrho {\epsilon }^{-\eta }\right)-\tilde{\varphi }\left(0\right)\right)}^{((\mathfrak{q}-1)\mathfrak{p}+1)+{\displaystyle \frac{\mathfrak{p}}{2}}-1}\right).\hfill \end{array}$$

This concludes the proof. □

Corollary 1. 

Suppose that the assumptions $\left({\xi }_3\right)$ to $\left({\xi }_4\right)$ are satisfied. When ${\mho }_1>0$ and ${\epsilon }_1\in \left(0,{\epsilon }_0\right)$, $\eta \in (0,\mathfrak{q}\mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}})$, and $\varrho >0$, then $\forall \epsilon \in \left(0,{\epsilon }_1\right]$, we have:

$$\underset{\epsilon \to 0}{lim}\mathbb{P}\left[\underset{\mathfrak{t}\in [0,\varrho {\epsilon }^{-\eta }]}{sup}\parallel {\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right)\parallel >{\mho }_1\right]=0.$$

(62)

Proof. 

For ${\mho }_1>0$ by the Chebyshev–Markov inequality and Theorem 5,

$$\begin{array}{cc}\hfill \mathbb{P}\left[\underset{\mathfrak{t}\in [0,\varrho {\epsilon }^{-\eta }]}{sup}\parallel {\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right)\parallel >{\mho }_1\right]\le & {\displaystyle \frac{1}{{\mho }_1^2}}\mathbf{E}\left[\underset{\mathfrak{t}\in [0,\varrho {\epsilon }^{-\eta }]}{sup}\parallel {\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)-{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right){\parallel }^2\right]\hfill \\ & \le {\displaystyle \frac{\mathbb{A}{\epsilon }^{1-\eta }}{{\mho }_1^2}}\hfill \\ & \le 0\mathrm{as}\epsilon \to 0.\hfill \end{array}$$

□

5. Numerical Problems

To better understand the theoretical results established in this research, we present numerical problems along with graphical comparisons of the original and averaged solutions. The Figure 1, Figure 2, Figure 3 and Figure 4 show that the solutions of the original system and the average system overlap, demonstrating the effectiveness of our research study.

Problem 1. 

Take the system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{\mathrm{H}}\mathcal{T}_{\mathfrak{t}}^{0.8,\mathfrak{a},\tilde{\varphi }}{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)=6{sin}^2\left(\mathfrak{t}\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)+{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right){cos}^2\left({\displaystyle \frac{1}{2}}\mathfrak{t}\right)+\hfill \\ 3{\epsilon }^{-{\displaystyle \frac{1}{2}}}{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right){cos}^2\left(\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\displaystyle \frac{d\mathcal{W}\left(\mathfrak{t}\right)}{d\mathfrak{t}}},\mathfrak{t}\in [0,\pi ],\hfill \\ {\mathcal{I}}_{0^{+}}^{(1-0.8)(1-0.85),{\mathfrak{t}}^{{\displaystyle \frac{1}{2}}}}\mathrm{f}\left(0\right)=\mathfrak{c},\hfill \end{array}\right.\end{array}$$

(63)

where $\mathfrak{q}=0.8$, $\mathrm{s}={\displaystyle \frac{1}{2}}$, and

$$\begin{array}{cc}\hfill \mathcal{D}(\mathfrak{t},\mathrm{f}(\mathfrak{t}),\mathrm{f}(\mathfrak{t}\mathrm{s}\left)\right)& =6{sin}^2\left(\mathfrak{t}\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)+{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right){cos}^2\left({\displaystyle \frac{1}{2}}\mathfrak{t}\right),\hfill \\ \hfill \Xi (\mathfrak{t},\mathrm{f}(\mathfrak{t}),\mathrm{f}(\mathfrak{t}\mathrm{s}\left)\right)& =3{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right){cos}^2\left(\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right).\hfill \end{array}$$

The criteria of the EU are fulfilled by $6{sin}^2\left(\mathfrak{t}\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)+{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right){cos}^2\left({\displaystyle \frac{1}{2}}\mathfrak{t}\right)$ and $3{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right){cos}^2\left(\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)$.

The $\mathcal{D}$ and $\Xi$ are shown by the following expressions:

$$\begin{array}{cc}\hfill \tilde{\mathcal{D}}(\mathrm{f}\left(\mathfrak{t}\right),\mathrm{f}\left(\mathfrak{t}\mathrm{s}\right))& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }\left(6{sin}^2\left(\mathfrak{t}\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)+{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right){cos}^2\left({\displaystyle \frac{1}{2}}\mathfrak{t}\right)\right)d\mathfrak{t}={\displaystyle \frac{7}{2}}{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right),\hfill \\ \hfill \tilde{\Xi }(\mathrm{f}\left(\mathfrak{t}\right),\mathrm{f}\left(\mathfrak{t}\mathrm{s}\right))& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }3{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right){cos}^2\left(\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)d\mathfrak{t}={\displaystyle \frac{3}{2}}{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right).\hfill \end{array}$$

Hence

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{T}}_{\mathfrak{t}}^{0.80,\mathfrak{a},\tilde{\varphi }}{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right)={\displaystyle \frac{7}{2}}{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)+{\displaystyle \frac{3}{2}}{\epsilon }^{-{\displaystyle \frac{1}{2}}}{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\displaystyle \frac{d\mathcal{W}\left(\mathfrak{t}\right)}{d\mathfrak{t}}},\hfill \\ {\mathcal{I}}_{0^{+}}^{(1-0.80)(1-0.85),{\mathfrak{t}}^{{\displaystyle \frac{1}{2}}}}\mathrm{f}\left(0\right)=\mathfrak{c}.\hfill \end{array}\right.\end{array}$$

(64)

Problem 2. 

Consider SPFrDEs:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{\mathrm{H}}\mathcal{T}_{\mathfrak{t}}^{0.95,\mathfrak{a},\tilde{\varphi }}{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)=3sin\left({\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{1}{2}}\mathfrak{t}\right)\right){sin}^2\left(\mathfrak{t}\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)+{\epsilon }^{-{\displaystyle \frac{1}{2}}}sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)\hfill \\ {\displaystyle \frac{d\mathcal{W}\left(\mathfrak{t}\right)}{d\mathfrak{t}}},\mathfrak{t}\in [0,\pi ],\hfill \\ {\mathcal{I}}_{0^{+}}^{(1-0.95)(1-\mathfrak{a}),\tilde{\varphi }}\mathrm{f}\left(0\right)=\mathfrak{c},\hfill \end{array}\right.\end{array}$$

(65)

where $\mathfrak{q}=0.95$, $\mathrm{s}={\displaystyle \frac{1}{2}}$, and

$$\begin{array}{cc}\hfill \mathcal{D}(\mathfrak{t},\mathrm{f}(\mathfrak{t}),\mathrm{f}(\mathfrak{t}\mathrm{s}\left)\right)& =3sin\left({\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{1}{2}}\mathfrak{t}\right)\right){sin}^2\left(\mathfrak{t}\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right),\hfill \\ \hfill \Xi (\mathfrak{t},\mathrm{f}(\mathfrak{t}),\mathrm{f}(\mathfrak{t}\mathrm{s}\left)\right)& =sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right).\hfill \end{array}$$

The criteria of EU are fulfilled by $3sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){sin}^2\left(\mathfrak{t}\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)$ and $sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)$.

So,

$$\begin{array}{cc}\hfill \tilde{\mathcal{D}}(\mathrm{f}\left(\mathfrak{t}\right),\mathrm{f}\left(\mathfrak{t}\mathrm{s}\right))& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }3sin\left({\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{1}{2}}\mathfrak{t}\right)\right){sin}^2\left(\mathfrak{t}\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)d\mathfrak{t}={\displaystyle \frac{3}{2}}sin\left({\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{1}{2}}\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right),\hfill \\ \hfill \tilde{\Xi }(\mathrm{f}\left(\mathfrak{t}\right),\mathrm{f}\left(\mathfrak{t}\mathrm{s}\right))& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)d\mathfrak{t}=sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right).\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{T}}_{\mathfrak{t}}^{0.95,\mathfrak{a},\tilde{\varphi }}{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right)={\displaystyle \frac{3}{2}}sin\left({\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{1}{2}}\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)+{\epsilon }^{-{\displaystyle \frac{1}{2}}}sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)d\mathfrak{t},\hfill \\ {\mathcal{I}}_{0^{+}}^{(1-0.95)(1-\mathfrak{a}),\tilde{\varphi }}\mathrm{f}\left(0\right)=\mathfrak{c}.\hfill \end{array}\right.\end{array}$$

(66)

Problem 3. 

Consider SPFrDEs:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{\mathrm{H}}\mathcal{T}_{\mathfrak{t}}^{0.9,\mathfrak{a},\tilde{\varphi }}{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)={\displaystyle \frac{1}{3}}{\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{1}{4}}\mathfrak{t}\right)cos({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)+{\displaystyle \frac{3\pi }{4}}{\epsilon }^{-{\displaystyle \frac{1}{2}}}{sin}^3\mathfrak{t}cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\hfill \\ {\displaystyle \frac{d\mathcal{W}\left(\mathfrak{t}\right)}{d\mathfrak{t}}},\mathfrak{t}\in [0,\pi ],\hfill \\ {\mathcal{I}}_{0^{+}}^{(1-0.90)(1-\mathfrak{a}),\tilde{\varphi }}\mathrm{f}\left(0\right)=\mathfrak{c},\hfill \end{array}\right.\end{array}$$

(67)

where $\mathfrak{q}=0.9$, $\mathrm{s}={\displaystyle \frac{1}{4}}$, and

$$\begin{array}{cc}\hfill \mathcal{D}(\mathfrak{t},\mathrm{f}(\mathfrak{t}),\mathrm{f}(\mathfrak{t}\mathrm{s}\left)\right)& ={\displaystyle \frac{1}{3}}{\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{1}{4}}\mathfrak{t}\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right),\hfill \\ \hfill \Xi (\mathfrak{t},\mathrm{f}(\mathfrak{t}),\mathrm{f}(\mathfrak{t}\mathrm{s}\left)\right)& ={\displaystyle \frac{3\pi }{4}}{sin}^3\mathfrak{t}cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right).\hfill \end{array}$$

The criteria of EU are fulfilled by ${\displaystyle \frac{1}{3}}{\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{1}{4}}\mathfrak{t}\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)$ and ${\displaystyle \frac{3\pi }{4}}{sin}^3\mathfrak{t}cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)$.

The following equations display the $\mathcal{D}$ and $\Xi$ averages:

$$\begin{array}{cc}\hfill \tilde{\mathcal{D}}(\mathrm{f}\left(\mathfrak{t}\right),\mathrm{f}\left(\mathfrak{t}\mathrm{s}\right))& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }{\displaystyle \frac{1}{3}}{\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{1}{4}}\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)d\mathfrak{q}={\displaystyle \frac{1}{3}}{\mathrm{f}}_{\epsilon }^{*}\left({\displaystyle \frac{1}{4}}\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right),\hfill \\ \hfill \tilde{\Xi }(\mathrm{f}\left(\mathfrak{t}\right),\mathrm{f}\left(\mathfrak{t}\mathrm{s}\right))& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }{\displaystyle \frac{3\pi }{4}}{sin}^3\mathfrak{t}cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)d\mathfrak{q}=sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right).\hfill \end{array}$$

Hence

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{T}}_{\mathfrak{t}}^{0.9,\mathfrak{a},\tilde{\varphi }}{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right)={\displaystyle \frac{1}{3}}{\mathrm{f}}_{\epsilon }^{*}\left({\displaystyle \frac{1}{4}}\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)+{\epsilon }^{-{\displaystyle \frac{1}{2}}}cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right)sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\displaystyle \frac{d\mathcal{W}\left(\mathfrak{t}\right)}{d\mathfrak{t}}},\hfill \\ {\mathcal{I}}_{0^{+}}^{(1-0.90)(1-\mathfrak{a}),\tilde{\varphi }}\mathrm{f}\left(0\right)=\mathfrak{c}.\hfill \end{array}\right.\end{array}$$

(68)

Problem 4. 

Consider SPFrDEs:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{\mathrm{H}}\mathcal{T}_{\mathfrak{t}}^{0.95,\mathfrak{a},\tilde{\varphi }}{\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)={\displaystyle \frac{9}{2}}sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){exp}^{-\mathfrak{t}}+{\epsilon }^{-{\displaystyle \frac{1}{2}}}sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)\hfill \\ {\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{5}{3}}\mathfrak{t}\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\displaystyle \frac{d\mathcal{W}\left(\mathfrak{t}\right)}{d\mathfrak{t}}},\mathfrak{t}\in [0,\pi ],\hfill \\ {\mathcal{I}}_{0^{+}}^{(1-0.95)(1-\mathfrak{a}),\tilde{\varphi }}\mathrm{f}\left(0\right)=\mathfrak{c},\hfill \end{array}\right.\end{array}$$

(69)

where $\mathfrak{q}=0.95$, $\mathrm{s}={\displaystyle \frac{5}{3}}$, and

$$\begin{array}{cc}\hfill \mathcal{D}(\mathfrak{t},\mathrm{f}(\mathfrak{t}),\mathrm{f}(\mathfrak{t}\mathrm{s}\left)\right)& ={\displaystyle \frac{9}{2}}sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){exp}^{-\mathfrak{t}},\hfill \\ \hfill \Xi (\mathfrak{t},\mathrm{f}(\mathfrak{t}),\mathrm{f}(\mathfrak{t}\mathrm{s}\left)\right)& =sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{5}{3}}\mathfrak{t}\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right).\hfill \end{array}$$

The ${\displaystyle \frac{9}{2}}sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){exp}^{-\mathfrak{t}}$ and $sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{5}{3}}\mathfrak{t}\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)$ satisfies the needs of EU.

Hence,

$$\begin{array}{cc}\hfill \tilde{\mathcal{D}}(\mathrm{f}\left(\mathfrak{t}\right),\mathrm{f}\left(\mathfrak{t}\mathrm{s}\right))& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }\left({\displaystyle \frac{9}{2}}sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){exp}^{-\mathfrak{t}}\right)d\mathfrak{t}\hfill \\ & ={\displaystyle \frac{9}{2\pi }}sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)(1-{exp}^{-\pi }),\hfill \\ \hfill \tilde{\Xi }(\mathrm{f}\left(\mathfrak{t}\right),\mathrm{f}\left(\mathfrak{t}\mathrm{s}\right))& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{5}{3}}\mathfrak{t}\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)d\mathfrak{t}\hfill \\ & =sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{5}{3}}\mathfrak{t}\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right).\hfill \end{array}$$

Thus, we have the corresponding averaged SPFrDEs

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{T}}_{\mathfrak{t}}^{0.95,\mathfrak{a},\tilde{\varphi }}{\mathrm{f}}_{\epsilon }^{*}\left(\mathfrak{t}\right)={\displaystyle \frac{9}{2\pi }}sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right)(1-{exp}^{-\pi })+\hfill \\ {\epsilon }^{-{\displaystyle \frac{1}{2}}}sin\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\mathrm{f}}_{\epsilon }\left({\displaystyle \frac{5}{3}}\mathfrak{t}\right)cos\left({\mathrm{f}}_{\epsilon }\left(\mathfrak{t}\right)\right){\displaystyle \frac{d\mathcal{W}\left(\mathfrak{t}\right)}{d\mathfrak{t}}},\hfill \\ {\mathcal{I}}_{0^{+}}^{(1-0.95)(1-\mathfrak{a}),\tilde{\varphi }}\mathrm{f}\left(0\right)=\mathfrak{c}.\hfill \end{array}\right.\end{array}$$

(70)

6. Conclusions

We prove the EU theorem using the contraction mapping principle and demonstrate that the solutions of SPFrDEs have continuous dependence on the fractional and initial values. Additionally, we establish that the solution satisfies the smoothness property by proving the regularity theorem. Using various inequalities, we prove the AP theorem. The primary tools employed in our proofs include the BDGI, JI, and HI. We prove the theorems and lemmas for FFrPSDEs under the $\varphi$-HFrD.

Our research work is important, for the following reasons: First, by proving the results of the EU, continuous dependence, regularity, and AP in the $\mathfrak{p}$th moment, we extend the outcomes for $\mathfrak{p}=2$. Secondly, we construct theorems and lemmas in the context of $\varphi$-HFrD. In this way, we generalize the results of Caputo, Riemann–Liouville, and Hadamard and Caputo–Hadamard fractional operators. Third, we take into account SPFrDEs, which are a more widespread subclass of FSDEs.