Revised and Generalized Results of Averaging Principles for the Fractional Case

Abstract

The averaging principle involves approximating the original system with a simpler system whose behavior can be analyzed more easily. Recently, numerous scholars have begun exploring averaging principles for fractional stochastic differential equations. However, many previous studies incorrectly defined the standard form of these equations by placing $\epsilon$ in front of the drift term and $\sqrt{\epsilon }$ in front of the diffusion term. This mistake results in incorrect estimates of the convergence rate. In this research work, we explain the correct process for determining the standard form for the fractional case, and we also generalize the result of the averaging principle and the existence and uniqueness of solutions to fractional stochastic delay differential equations in two significant ways. First, we establish the result in ${\mathrm{L}}^{\mathrm{p}}$ space, generalizing the case of $\mathrm{p}=2$. Second, we establish the result using the Caputo–Katugampola operator, which generalizes the results of the Caputo and Caputo–Hadamard derivatives.

1. Introduction

Fractional order derivatives (FODs) extend traditional calculus by allowing differentiation and integration to non-integer orders. This flexibility provides several significant benefits across various scientific and engineering disciplines [1,2,3].

• Modeling Memory and Hereditary Properties:
  FODs inherently consider the entire history of a function, making them ideal for modeling systems with memory and hereditary effects. Traditional integer-order derivatives are local, depending only on the function’s current state. In contrast, FODs capture long-range temporal dependencies, providing more accurate and realistic models for systems where past states influence current behavior.
  • Viscoelastic materials: Fractional models describe how materials like polymers exhibit both elastic and viscous behaviors over time.
  • Biological systems: They capture the memory effects in biological processes, such as gene regulation and neural activity.
• Enhanced control systems: Fractional-order controllers, such as fractional proportional integral derivative controllers, provide additional tuning parameters beyond traditional integer-order controllers. The extra degree of freedom in FODs allows for finer control and better performance in dynamic systems, particularly those with uncertain or time-varying characteristics.
  • Robotics: Improve stability and response time in robotic systems.
  • Automotive systems: Enhance control in vehicle dynamics and stability systems.
• Superior signal processing: FODs improve techniques for analyzing and filtering signals, especially those exhibiting non-stationary behavior. They provide more robust algorithms for detecting features and trends in complex signals, improving the accuracy and reliability of signal-processing tasks.
  • Biomedical engineering: Enhance analysis of electrocardiogram signals for detecting cardiac abnormalities.
  • Seismology: Better detect and characterize seismic events.
• Accurate modeling of anomalous diffusion: FODs effectively describe anomalous diffusion processes, where particle movement deviates from classical Brownian motion. They capture the irregular and complex diffusion patterns observed in heterogeneous or disordered media, providing more accurate models for these processes.
  • Environmental engineering: Model pollutant transport in porous media.
  • Biophysics: Describe the diffusion of molecules within cellular environments.
• Improved financial models: FODs capture memory and path-dependent behaviors in financial markets, leading to more accurate models. They allow for better risk assessment and option pricing by incorporating long memory effects and non-local interactions into market data.
  • Option pricing: Fractional stochastic models provide more accurate pricing by considering historical volatility.
  • Risk management: Improve modeling of market risks and asset returns.
• Flexibility and generalization: FODs generalize classical calculus, providing a continuum of derivative orders between integers. This flexibility allows for more precise and adaptable modeling across a wide range of applications, accommodating the unique characteristics of various systems.
  • Physics: Involves modeling phenomena that cannot be adequately described by integer-order equations.
  • Engineering: Involves designing systems with specific dynamic responses tailored through fractional modeling.
• Better fit for experimental data: FODs often provide a better fit for experimental data compared to integer-order models. They reduce the discrepancy between theoretical models and observed data, enhancing the accuracy and predictive power of the models.
  • Material science: Accurate characterization of material properties under various stress and strain conditions.
  • Biological studies: Better representation of complex biological processes and their responses to stimuli.
• Versatility in numerical methods: Various numerical methods have been developed to solve fractional differential equations (FDEs), expanding the applicability of FODs. Numerical methods enable the practical application of FODs to complex real-world problems where analytical solutions are not feasible.
  • Finite difference methods: Approximating FODs for solving engineering problems.
  • Spectral methods: Using orthogonal polynomials to solve FDEs in physics.

Researchers have been actively engaged in studying various types of FODs. For example, the authors of [4,5] established significant results with Caputo-Fabrizio derivatives. References [6,7] introduced new approaches to solving FDEs using conformable fractional operators. Zhang et al. [8] found solutions to time-fractional partial differential equations within the framework of Caputo fractional derivatives (Cap-FD). Zhang and Xiong [9] demonstrated global exponential stability and the existence of unique solutions to the periodic solutions of FDEs with semilinear impulses using the Cap-FD. Syam and Al-Refai [10] established several useful results regarding solutions to FDEs using Atangana–Baleanu derivatives. Li and Wang [11] found solutions to systems of the fractional Rössler chaotic using Grünwald–Letnikov fractional derivatives. Reference [12] addressed various kinds of stability results for FDEs with Brownian motion concerning the Caputo–Hadamard fractional derivative (Cap-HFD). For further study, refer to [13,14,15,16,17].

Among various fractional operators, the most important is the Caputo–Katugampola fractional (Cap-KGF) derivative, defined as follows [18]:

$${\mathcal{D}}_{\mathfrak{b}}^{\phi ,\delta }\mathbb{Y}\left(\eta \right)={\displaystyle \frac{{\delta }^{\phi }}{\mathrm{\Gamma }(1-\phi )}}{\int }_{\mathfrak{b}}^{\eta }{\displaystyle \frac{{\mathbb{Y}}^{*}\left(\xi \right)}{{({\eta }^{\delta }-{\xi }^{\delta })}^{\phi }}}\mathrm{d}\xi .$$

Remark 1. 

With $\delta =1$, the Cap-KGF derivative simplifies to the Cap-FD, while with $\delta \to 0^{+}$, it results in the Cap-HFD [19,20].

The definition of the Cap-KGF integral is [21]:

$${\mathcal{I}}_{\mathfrak{b}}^{\phi ,\delta }\mathbb{Y}\left(\eta \right)={\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}{\int }_{\mathfrak{b}}^{\eta }{\displaystyle \frac{{\xi }^{\delta -1}\mathbb{Y}\left(\xi \right)}{{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }}}\mathrm{d}\xi .$$

Many researchers have recently been actively engaged in studying the Cap-KGF derivative. For example, Li et al. [22] developed a new approach using the Cap-KGF derivative to solve various fractional-order models. They also established well-posedness and obtained approximate solutions using the new approach. Kahouli et al. [23] developed a useful method involving the Cap-KGF derivative and solved some problems. The authors of [24] studied the existence and uniqueness (Ex-Un) of solutions to the fractional models and also established some stability results. Sweilam et al. [18] created a novel approach based on the spectral method using shifted Chebyshev polynomials to solve various problems. Nazeer et al. [25] developed various useful concepts related to the Cap-KGF derivative and established several inequalities. Hoa et al. [26] studied the Ex-Un of the initial value fractional system and successfully solved it. Zeng et al. [27] solved a system of fractional models using the Cap-KGF operator. The authors of [28,29] worked on the Ex-Un of the solutions to the fractional-order model using various fixed-point theories with the Cap-KGF operator. Omaba [30] worked on the Ex-Un of solutions to the stochastic model of the Cap-KGF derivative and also studied its asymptotic behavior.

Fractional calculus (Fr-Cal) is used in deterministic FDEs to model systems with memory effects and non-local interactions. The behaviors of these equations are entirely determined by the initial conditions and the deterministic nature of the equations themselves; they do not involve random processes or noise. On the other hand, fractional stochastic differential equations (FSDEs) combine stochastic processes and Fr-Cal to model systems that exhibit both random noise or fluctuations and fractional-order dynamics. Complex systems with memory effects and random disturbances, which are prevalent in many real-world applications, are particularly well-suited for the FSDE description.

Delay fractional stochastic differential equations (DFSDEs) are part of an advanced mathematical framework designed to model systems that exhibit memory effects, randomness, and delays. They combine elements of delay differential equations, Fr-Cal, and stochastic processes. The following is a detailed explanation of each component.

• Delay:
  The inclusion of delay means that the evolution of the system depends not only on its current state but also on its state at previous times. This is useful for modeling systems where past events influence future behavior. In population dynamics, the birth rate at a given time may depend on the population size at some earlier time due to gestation periods.
• Fractional derivatives:
  Fractional derivatives are generalizations of ordinary derivatives to non-integer orders. They are useful for describing processes with memory and hereditary properties. In viscoelastic materials, the stress–strain relationship can be more accurately described using fractional derivatives.
• Stochastic Processes:
  This component incorporates randomness into the equations, typically using terms that represent random noise, such as Brownian motion or Wiener processes. In finance, stock prices can be modeled as stochastic processes due to the randomness of market movements.

DFSDEs are used in various fields to model complex systems that exhibit these combined effects. Here are some examples [31,32]:

• Biology:
  DFSDEs are used to model population dynamics, where the delay represents the time lag in the response of the population to changes in the environment. This can include predator–prey interactions and the spread of diseases.
• Finance:
  In financial mathematics, DFSDEs help model stock prices and interest rates, incorporating memory effects and stochastic volatility. This is crucial for pricing derivatives and managing financial risks.
• Engineering:
  These equations are applied in control systems and signal processing, where delays and stochastic perturbations are common. They help in designing systems that can withstand random disturbances and delays.
• Physics:
  DFSDEs model various physical phenomena, such as viscoelastic materials and thermal processes, where the delay represents the time-dependent response of materials to external forces.
• Environmental Science:
  They are used to model climate systems and ecological processes, where delays can represent the time lag in the response of the environment to changes in external factors like pollution or climate change.

Researchers have actively been working on different aspects of FSDEs. Among these, Chen et al. [33] worked on the Ex-Un results and the stability of solutions to FSDEs. The authors of [34] investigated the Ex-Un of solutions to Riemann–Liouville FSDEs and examined Ulam–Hyers (UH) stability. Kahouli et al. [35] studied the Ex-Un of FSDEs using the Banach fixed-point theorem (BFPT) and demonstrated their UH stability based on generalized Gronwall inequalities. Rhaima [36] explored the Ex-Un and UH stability of Caputo–Hadamard-type FSDEs. The author established the Ex-Un of solutions via the BFPT. The authors of [37] further investigated the Ex-Un and UH stability of FSDEs by utilizing Hadamard and Riemann-type fractional operators. Their primary goal was to prove the Ex-Un results of solutions using the BFPT. Luo et al. [38] examined the Ex-Un and UH stability results of Caputo-type FSDEs with delays. Tian and Luo [39] discussed the Ex-Un and finite-time stability results of solutions to FSDEs. For more on this, see [40,41,42,43].

The study of Ex-Un for solutions of DFSDEs is fundamental for several reasons. Understanding these properties ensures that the mathematical models we use to describe complex systems are both valid and reliable. To ensure that the DFSDE has at least one solution, meaning the model we propose actually corresponds to a possible real-world behavior or phenomenon. Without a guarantee of existence, the equations we set up may not correspond to any real scenario, rendering the model useless. To ensure that the solution is unique, meaning that given the same initial conditions, the system will behave in a predictable and consistent manner. Without uniqueness, the same initial conditions could lead to multiple different outcomes, making the model unpredictable and unreliable.

A common approach to investigating systems of complex differential equations (DEs) is the averaging principle (Ave-P), which can be used to reduce the complexity or improve the computational efficiency of a problem by approximating it. It was first put forward by Krylov and Bogolyubov [44], then by Gikhman [45] and Volosov [46] for non-linear ordinary DEs. With the development of the theory of stochastic analysis, many authors began to study Ave-P for FSDEs. For instance, Ave-P for Caputo FSDEs in the ${\mathrm{L}}^2$ space was examined in reference [47]. Xu et al. [48] developed the Ave-P result for FSDEs using the Lévy procedure. They proved their findings both in the sense of Cap-FD and ${\mathrm{L}}^2$ space. Wang and Lin [49] researched Ave-P for Caputo FSDEs in the sense of ${\mathrm{L}}^{\mathfrak{p}}$. Yang et al. [50] investigated Ave-P for DFSDEs in ${\mathrm{L}}^{\mathfrak{p}}$ space. The authors of [51] worked on Ave-P for backward stochastic DE in ${\mathrm{L}}^2$ space. The authors of [52] studied Ave-P for Caputo FSDEs in ${\mathrm{L}}^2$ space. Mouy et al. [53] focused on Ave-P for Hadamard FSDEs in ${\mathrm{L}}^2$ space. Xu et al. [54] worked on Ave-P for FSDEs with Cap-FD in ${\mathrm{L}}^2$ space. Zou and Luo [55] studied Ave-P for DFSDEs with Cap-FD in ${\mathrm{L}}^2$ space. The authors of [56] studied Ave-P for DFSDEs using the Hilfer fractional operator in ${\mathrm{L}}^2$ space.

Although Ave-P is an essential concept, in most studies of FSDEs, the standard form of the original equations is defined by placing $\epsilon$ before the drift term and $\sqrt{\epsilon }$ before the diffusion term. For fractional-order cases, the standard form of the original equation must be established through a time scale change and cannot be described directly by preceding the drift term with $\epsilon$ and the diffusion term with $\sqrt{\epsilon }$ due to its physical relevance.

In this study, we establish Ex-Un and Ave-P for DFSDEs in the framework of the Cap-KGF derivative. In the first stage, we establish the Ex-Un results for the solutions of DFSDEs by using the BFPT. In the second stage, using inequality and interval translation techniques, we demonstrate the result of Ave-P. Lastly, we provide two examples to help illustrate the results we establish. The main part of the proof of our established results involves the use of the Burkholder–Davis–Gundy inequality (B-D-G-I), Jensen inequality (J-I), Hölder’s inequality (H-I), Grönwall–Bellman inequality (G-B-I), and Chebyshev–Markov inequality (C-M-I).

Our research work makes significant contributions in the following ways:

• We corrected the mistakes in the proof of Ave-P found in various publications. In most studies, the standard form of FSDEs is incorrectly established by adding $\epsilon$ in front of the drift term and $\sqrt{\epsilon }$ in front of the diffusion term, which is not correct for the fractional case. We derive the correct standard form for FSDEs using the time-scale change property of Cap-KGF derivatives.
• We generalize the Ex-Un results of the solution for DFSDEs concerning fractional derivatives by establishing the results in the sense of the Cap-KGF derivative. When using condition $\delta =1$ in our established results, we obtain the results in the sense of the Cap-FD, and under condition $\delta \to 0^{+}$, we obtain the results in the sense of the Cap-HFD.
• As most of the results regarding Ex-Un and Ave-P in the literature are established in ${\mathrm{L}}^2$ space, we also generalize these results by establishing them in ${\mathrm{L}}^{\mathfrak{p}}$ space. In this way, our research generalizes the results of Ex-Un and Ave-P for $\mathfrak{p}=2$.
  We examined the following DFSDEs:

  $$\left\{\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }^{\phi ,\delta }\varrho \left(\eta \right)={\mathcal{Z}}_1\left(\eta ,\varrho \left(\eta \right),\varrho (\eta -\mathrm{\Phi })\right)+{\mathcal{Z}}_2\left(\eta ,\varrho \left(\eta \right),\varrho (\eta -\mathrm{\Phi })\right){\displaystyle \frac{\mathrm{d}{\mathcal{W}}_{\eta }}{\mathrm{d}\eta }},\eta \in [0,\Im ],\hfill \\ & \varrho \left(\eta \right)=\gamma \left(\eta \right),\eta \in [-\mathrm{\Phi },0],\hfill \end{array}\right.$$

  (1)

  here, ${\mathcal{W}}_{\eta }$ is a Brownian motion (B-M), ${\mathcal{D}}_{\eta }^{\phi ,\delta }$ represents the Cap-KGF operator with $\phi \in ({\displaystyle \frac{1}{2}},1]$, $\delta >0$. The functions ${\mathcal{Z}}_1[0,\Im ]\times {\mathfrak{R}}^{\mathfrak{a}}\times {\mathfrak{R}}^{\mathfrak{a}}\to {\mathfrak{R}}^{\mathfrak{a}}$ and ${\mathcal{Z}}_2:[0,\Im ]\times {\mathfrak{R}}^{\mathfrak{a}}\times {\mathfrak{R}}^{\mathfrak{a}}\to {\mathfrak{R}}^{\mathfrak{a}\times \mathfrak{r}}$ are measurable continuous. For $\mathfrak{r}$-dimensional B-M, a probability space $\left(\mathrm{\Omega },\mathcal{F},\mathcal{P}\right)$ is constructed. The function $\gamma \left(\eta \right):[-\mathrm{\Phi },0]\to {\mathfrak{R}}^{\mathfrak{a}}$ is continuous.

The research work includes the following components: In the next section, we present a definition, prove an important lemma, and assume several assumptions that are the foundation for our findings about DFSDEs. We demonstrate the Ex-Un of solutions to DFSDEs and Ave-P in Section 3. We illustrate our findings with two examples in Section 4. Finally, in Section 5, we present our conclusions.

2. Preliminaries

In this section, we present a definition, prove an important lemma, and make some assumptions that will be used throughout the work to establish useful results regarding Ex-Un and Ave-P for the DFSDEs.

Definition 1. 

The considered problem has a unique solution as an ${\mathfrak{R}}^{\mathfrak{a}}$-valued stochastic process ${\left\{\varrho \left(\eta \right)\right\}}_{-\mathrm{\Phi }\le \eta \le \Im }$ if $\varrho \left(\eta \right)$ is $\mathcal{F}\left(\eta \right)$-adapted, with $\mathcal{E}\left[{\int }_{-\mathrm{\Phi }}^{\Im }\parallel \varrho \left(\eta \right)\parallel \mathrm{d}\eta \right]<\infty$, $\varrho \left(0\right)={\varrho }_0$, fulfilling the following:

$$\begin{array}{cc}\hfill \varrho \left(\eta \right)=& \gamma \left(0\right)+{\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}{\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }{\mathcal{Z}}_1(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))\mathrm{d}\xi \hfill \\ & +{\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}{\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }{\mathcal{Z}}_2(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))\mathrm{d}\mathcal{W}\left(\xi \right),\eta \in [0,\Im ],\hfill \end{array}$$

(2)

with $\varrho \left(\eta \right)=\gamma \left(\eta \right),\eta \in [-\mathrm{\Phi },0].$

The ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ in Equation (1) when $\forall {\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2,{\mathcal{B}}_1,{\mathcal{B}}_2\in {\mathfrak{R}}^{\mathfrak{a}}$, $\eta \in [0,\Im ]$ satisfy the following requirements for ${\zeta }_1>0$ and ${\zeta }_1>0$:

• $\left({\mathfrak{T}}_1\right)$:

$$\begin{array}{cc}\hfill \parallel {\mathcal{Z}}_1(\eta ,{\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)-& {\mathcal{Z}}_1(\eta ,{\mathcal{B}}_1,{\mathcal{B}}_2)\parallel \vee \parallel {\mathcal{Z}}_2(\eta ,{\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)-{\mathcal{Z}}_2(\eta ,{\mathcal{B}}_1,{\mathcal{B}}_2)\parallel \hfill \\ & \le {\zeta }_1(\parallel {\mathrm{\Xi }}_1-{\mathcal{B}}_1\parallel +\parallel {\mathrm{\Xi }}_2-{\mathcal{B}}_2\parallel ),\hfill \end{array}$$

where ${\mathcal{Z}}_1\vee {\mathcal{Z}}_2=max({\mathcal{Z}}_1,{\mathcal{Z}}_2)$.

• $\left({\mathfrak{T}}_2\right)$:

$$\begin{array}{c}\hfill \parallel {\mathcal{Z}}_1(\eta ,{\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)\parallel \vee \parallel {\mathcal{Z}}_2(\eta ,{\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)\parallel \le {\zeta }_2(1+\parallel {\mathrm{\Xi }}_1\parallel +\parallel {\mathrm{\Xi }}_2\parallel ).\end{array}$$

• $\left({\mathfrak{T}}_3\right)$: Functions ${\tilde{\mathcal{Z}}}_1$ and ${\tilde{\mathcal{Z}}}_2$ exist, and for ${\Im }_1\in [0,\Im ]$, $\eta \in [0,\Im ]$, and $\mathfrak{p}\ge 2$, the bounded functions ${\mathsf{F}}_1\left({\Im }_1\right)>0$ and ${\mathsf{F}}_2\left({\Im }_2\right)>0$ exist, such that we have the following:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{{\Im }_1}}{\int }_0^{{\Im }_1}{\parallel {\mathcal{Z}}_1(\eta ,{\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)-{\tilde{\mathcal{Z}}}_1({\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)\parallel }^{\mathfrak{p}}\mathrm{d}\eta \le {\mathsf{F}}_1\left({\Im }_1\right)\left(1+\parallel {\mathrm{\Xi }}_1{\parallel }^{\mathfrak{p}}+{\parallel {\mathrm{\Xi }}_2\parallel }^{\mathfrak{p}}\right),\hfill \\ \hfill & {\displaystyle \frac{1}{{\Im }_1}}{\int }_0^{{\Im }_1}{\parallel {\mathcal{Z}}_2(\eta ,{\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)-{\tilde{\mathcal{Z}}}_2({\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)\parallel }^{\mathfrak{p}}\mathrm{d}\eta \le {\mathsf{F}}_2\left({\Im }_1\right)\left(1+\parallel {\mathrm{\Xi }}_1{\parallel }^{\mathfrak{p}}+{\parallel {\mathrm{\Xi }}_2\parallel }^{\mathfrak{p}}\right),\hfill \\ \hfill & with\underset{{\Im }_1\to \infty }{lim}{\mathsf{F}}_1\left({\Im }_1\right)=0 and\underset{{\Im }_1\to \infty }{lim}{\mathsf{F}}_2\left({\Im }_1\right)=0.\hfill \end{array}$$

We now outline the conditions for the growth of ${\tilde{\mathcal{Z}}}_2({\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)$.

Lemma 1. 

For ${\Im }_1\in [0,\Im ]$, the growth conditions for ${\tilde{\mathcal{Z}}}_2$ with $\left({\mathfrak{T}}_2\right),\left({\mathfrak{T}}_3\right)$ are as follows:

$$\parallel {\tilde{\mathcal{Z}}}_2({\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2){\parallel }^{\mathfrak{p}}\le {\zeta }_3\left(1+\parallel {\mathrm{\Xi }}_1{\parallel }^{\mathfrak{p}}+{\parallel {\mathrm{\Xi }}_2\parallel }^{\mathfrak{p}}\right),$$

where ${\zeta }_3=\left(2^{\mathfrak{p}-1}{\mathsf{F}}_2\left({\Im }_1\right)+6^{\mathfrak{p}-1}{\zeta }_2^{\mathfrak{p}}\right)$.

Proof. 

Taking into account Je-In and assumptions $\left({\mathfrak{T}}_2\right),\left({\mathfrak{T}}_3\right)$, we obtain the following outcome:

$$\begin{array}{cc}\hfill \parallel {\tilde{\mathcal{Z}}}_2({\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2){\parallel }^{\mathfrak{p}}& \le 2^{\mathfrak{p}-1}{\displaystyle \frac{1}{{\Im }_1}}{\int }_0^{{\Im }_1}{\parallel {\mathcal{Z}}_2(\eta ,{\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)-{\tilde{\mathcal{Z}}}_2({\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)\parallel }^{\mathfrak{p}}\mathrm{d}\eta +\hfill \\ & 2^{\mathfrak{p}-1}{\displaystyle \frac{1}{{\Im }_1}}{\int }_0^{{\Im }_1}{\parallel {\mathcal{Z}}_2(\eta ,{\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)\parallel }^{\mathfrak{p}}\mathrm{d}\eta \hfill \\ & \le 2^{\mathfrak{p}-1}{\mathsf{F}}_2\left({\Im }_1\right)\left(1+\parallel {\mathrm{\Xi }}_1{\parallel }^{\mathfrak{p}}+{\parallel {\mathrm{\Xi }}_2\parallel }^{\mathfrak{p}}\right)+2^{\mathfrak{p}-1}{\zeta }_2^{\mathfrak{p}}(1+\parallel {\mathrm{\Xi }}_1\parallel +\parallel {\mathrm{\Xi }}_2{\parallel )}^{\mathfrak{p}}\hfill \\ & \le \left(1+\parallel {\mathrm{\Xi }}_1{\parallel }^{\mathfrak{p}}+{\parallel {\mathrm{\Xi }}_2\parallel }^{\mathfrak{p}}\right)2^{\mathfrak{p}-1}{\mathsf{F}}_2\left({\Im }_1\right)+(1+\parallel {\mathrm{\Xi }}_1{\parallel }^{\mathfrak{p}}+\parallel {\mathrm{\Xi }}_2{\parallel }^{\mathfrak{p}})6^{\mathfrak{p}-1}{\zeta }_2^{\mathfrak{p}}\hfill \\ & \le \left(2^{\mathfrak{p}-1}{\mathsf{F}}_2\left({\Im }_1\right)+6^{\mathfrak{p}-1}{\zeta }_2^{\mathfrak{p}}\right)\left(1+\parallel {\mathrm{\Xi }}_1{\parallel }^{\mathfrak{p}}+\parallel {\mathrm{\Xi }}_2{\parallel }^{\mathfrak{p}})\right..\hfill \end{array}$$

□

3. Main Findings

We develop generalized results for the Ex-Un and Ave-P concepts of DFSDEs in the ${\mathrm{L}}^{\mathfrak{p}}$ space using the Cap-KGF operator. First, we present the Ex-Un results using the BFPT. In the second stage, we employ inequality and interval translation techniques to demonstrate the Ave-P result.

3.1. Existence and Uniqueness of the Solutions to DFSDEs

Using the BFPT, we present the Ex-Un results for the DFSDEs in this subsection. The BFPT, also known as the contraction mapping theorem, offers several advantages over sequence approximation methods for finding fixed points.

• Guaranteed convergence: The BFPT guarantees the Ex-Un of a fixed point for a contraction mapping on a complete metric space. This provides a stronger assurance compared to sequence approximation methods, which may not always converge or may converge to different points depending on the initial guess.
• Rate of convergence: The theorem provides a clear rate of convergence. Specifically, it states that the sequence of iterates converges to the fixed point at a geometric rate, which is often faster and more predictable than the convergence rates of general sequence approximation methods.
• Clear and simple conditions: BFPT requires only that the mapping be a contraction and that the space is complete. These are relatively straightforward conditions and are often easier to verify. Sequence approximation methods, however, may involve more complicated conditions or assumptions about the nature of the sequence or the underlying space.
• Broad applicability: The theorem applies to any complete metric space, making it versatile for various types of problems. Sequence approximation methods may require specific conditions or modifications to work effectively in different contexts.

Theorem 1. 

If $\left(\mathfrak{T}1\right)$ and $\left(\mathfrak{T}2\right)$ hold true, then the considered problem has a unique solution provided that the following condition is satisfied:

$$\begin{array}{cc}\hfill \mathrm{\Lambda }=& 2\varsigma \eta {\zeta }_1^{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{4}^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\eta }^{\left(\frac{\delta \phi \mathfrak{p}-1}{\mathfrak{p}-1}\right)}}{\delta }}\right)}^{\mathfrak{p}-1}+{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\displaystyle \frac{2^{2\mathfrak{p}-1}{\zeta }_1^{\mathfrak{p}}{\eta }^{\frac{(2\phi \delta -1)\mathfrak{p}}{2}}}{{(2\phi \delta -\delta )}^{\frac{\mathfrak{p}}{2}}}}{\varphi }_{\mathfrak{p}},\hfill \end{array}$$

(3)

where $0<\mathrm{\Lambda }<1$, ${\varphi }_{\mathfrak{p}}={\left({\mathfrak{p}}^{\mathfrak{p}+1}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{2}^{{\displaystyle \frac{-\mathfrak{p}}{2}}}{\left({(\mathfrak{p}-1)}^{1-\mathfrak{p}}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}$, and $\varsigma ={\left({\displaystyle \frac{\mathfrak{p}-1}{\phi \mathfrak{p}-1}}\right)}^{\mathfrak{p}-1}$.

Proof. 

We have $\chi :\rho \to \rho$ with $\varrho \left(0\right)=\gamma \left(0\right)$. Then,

$$\begin{array}{cc}\hfill \chi \left(\varrho \left(\eta \right)\right)& =\gamma \left(0\right)+{\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}{\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }{\mathcal{Z}}_1\left(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi })\right)\mathrm{d}\xi \hfill \\ & +{\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}{\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }{\mathcal{Z}}_2\left(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi })\right)\mathrm{d}\mathcal{W}\left(\xi \right).\hfill \end{array}$$

(4)

• Step 1: First, we will demonstrate that $\chi$ maps $\rho$ into itself. Thus, by J-I,

$$\begin{array}{cc}\hfill & \mathcal{E}\left[{\parallel \chi \left(\varrho \left(\eta \right)\right)\parallel }^{\mathfrak{p}}\right]\le 3^{\mathfrak{p}-1}\mathcal{E}\left[{\parallel \gamma \left(0\right)\parallel }^{\mathfrak{p}}\right]\hfill \\ & +{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{3}^{\mathfrak{p}-1}\mathcal{E}\left[\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{\left({\eta }^{\delta }-{\xi }^{\delta }\right)}^{\phi -1}{\mathcal{Z}}_1(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))\mathrm{d}\xi {\parallel }^{\mathfrak{p}}\right]\hfill \\ & +3^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\mathcal{E}\left[\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{\left({\eta }^{\delta }-{\xi }^{\delta }\right)}^{\phi -1}{\mathcal{Z}}_2(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))\mathrm{d}\mathcal{W}\left(\xi \right){\parallel }^{\mathfrak{p}}\right]\hfill \\ & ={\mathfrak{C}}_1+{\mathfrak{C}}_2.\hfill \end{array}$$

(5)

By H-I, $\left({\mathfrak{T}}_2\right)$, and J-I, we have the following:

$$\begin{array}{cc}\hfill & {\mathfrak{C}}_1=3^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\mathcal{E}\left[\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{\left({\eta }^{\delta }-{\xi }^{\delta }\right)}^{\phi -1}{\mathcal{Z}}_1(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))\mathrm{d}\xi {\parallel }^{\mathfrak{p}}\right]\hfill \\ & \le 3^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\left({\int }_0^{\eta }{\xi }^{{\displaystyle \frac{\mathfrak{p}(\delta -1)}{\mathfrak{p}-1}}}{({\eta }^{\delta }-{\xi }^{\delta })}^{{\displaystyle \frac{\mathfrak{p}(\phi -1)}{\mathfrak{p}-1}}}\mathrm{d}\xi \right)}^{\mathfrak{p}-1}\mathcal{E}\left[{\int }_0^{\eta }{\parallel {\mathcal{Z}}_1(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))\parallel }^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ & \le 3^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\left(\underset{0<\xi \le \eta }{sup}{\xi }^{{\displaystyle \frac{\delta -1}{\mathfrak{p}-1}}}{\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{{\displaystyle \frac{\mathfrak{p}(\phi -1)}{\mathfrak{p}-1}}}\mathrm{d}\xi \right)}^{\mathfrak{p}-1}\mathcal{E}\left[{\int }_0^{\eta }{\parallel {\mathcal{Z}}_1(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))\parallel }^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ & \le 3^{\mathfrak{p}-1}\varsigma {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\left({\displaystyle \frac{{\eta }^{\delta \phi \mathfrak{p}-1}}{{\delta }^{\mathfrak{p}-1}}}\right)\mathcal{E}\left[{\int }_0^{\eta }{\zeta }_2^{\mathfrak{p}}{\left(1+\parallel \varrho \left(\xi \right)\parallel +\parallel \varrho (\xi -\mathrm{\Phi })\parallel \right)}^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ & \le 6^{\mathfrak{p}-1}\varsigma {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\left({\displaystyle \frac{{\eta }^{\delta \phi \mathfrak{p}-1}}{{\delta }^{\mathfrak{p}-1}}}\right)\mathcal{E}\left[{\int }_0^{\eta }{\zeta }_2^{\mathfrak{p}}\left(1+{\left(\parallel \varrho \left(\xi \right)\parallel +\parallel \varrho (\xi -\mathrm{\Phi })\parallel \right)}^{\mathfrak{p}}\right)\mathrm{d}\xi \right]\hfill \\ & \le 6^{\mathfrak{p}-1}\varsigma {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\left({\displaystyle \frac{{\eta }^{\delta \phi \mathfrak{p}-1}}{{\delta }^{\mathfrak{p}-1}}}\right)\hfill \\ & \mathcal{E}\left[{\int }_0^{\eta }{\zeta }_2^{\mathfrak{p}}\left(1+2^{\mathfrak{p}-1}\left({\parallel \varrho \left(\xi \right)\parallel }^{\mathfrak{p}}+{\parallel \varrho (\xi -\mathrm{\Phi })\parallel }^{\mathfrak{p}}\right)\right)\mathrm{d}\xi \right]\hfill \\ & \le 6^{\mathfrak{p}-1}\eta \varsigma {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\zeta }_2^{\mathfrak{p}}\left({\displaystyle \frac{{\eta }^{\delta \phi \mathfrak{p}-1}}{{\delta }^{\mathfrak{p}-1}}}\right)\left(1+2^{\mathfrak{p}}\mathcal{E}\left[{\parallel \varrho \parallel }^{\mathfrak{p}}\right]\right).\hfill \end{array}$$

(6)

Through B-D-G-I, $\left({\mathfrak{T}}_2\right)$, and J-I, we have the following:

$$\begin{array}{cc}\hfill & {\mathfrak{C}}_2\le 3^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\mathcal{E}\left[\underset{\eta \in [0,\Im ]}{sup}\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{\left({\eta }^{\delta }-{\xi }^{\delta }\right)}^{\phi -1}{\mathcal{Z}}_2(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))\mathrm{d}\mathcal{W}\left(\xi \right){\parallel }^{\mathfrak{p}}\right]\hfill \\ & \le {\phi }_{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\mathcal{E}{\left[{\int }_0^{\eta }{\xi }^{2\delta -2}{\left({\eta }^{\delta }-{\xi }^{\delta }\right)}^{2\phi -2}\parallel {\mathcal{Z}}_2(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi })){\parallel }^2\mathrm{d}\xi \right]}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{3}^{\mathfrak{p}-1}\hfill \\ & \le {\phi }_{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}{\left[\underset{0<\xi \le \eta }{sup}{\xi }^{\delta -1}{\int }_0^{\eta }{\xi }^{\delta -1}{\left({\eta }^{\delta }-{\xi }^{\delta }\right)}^{2\phi -2}\parallel {\mathcal{Z}}_2(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi })){\parallel }^2\mathrm{d}\xi \right]}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{3}^{\mathfrak{p}-1}\hfill \\ & \le 3^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\left({\displaystyle \frac{{\eta }^{(2\phi \delta -1)}}{\delta (2\phi -1)}}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{\zeta }_2^{\mathfrak{p}}{\phi }_{\mathfrak{p}}\mathcal{E}{\left(1+\left(\parallel \varrho \left(\xi \right)\parallel +\parallel \varrho (\xi -\mathrm{\Phi })\parallel \right)\right)}^{\mathfrak{p}}\hfill \\ & \le 6^{\mathfrak{p}-1}{\phi }_{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\left({\displaystyle \frac{{\eta }^{(2\phi \delta -1)}}{\delta (2\phi -1)}}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}{\zeta }_2^{\mathfrak{p}}\mathcal{E}\left(1+{\left(\parallel \varrho \left(\xi \right)\parallel +\parallel \varrho (\xi -\mathrm{\Phi })\parallel \right)}^{\mathfrak{p}}\right)\hfill \\ & \le 6^{\mathfrak{p}-1}{\zeta }_2^{\mathfrak{p}}{\phi }_{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\left({\displaystyle \frac{{\eta }^{(2\phi \delta -1)}}{\delta (2\phi -1)}}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\left(1+2^{\mathfrak{p}}\mathcal{E}\left[{\parallel \varrho \parallel }^{\mathfrak{p}}\right]\right).\hfill \end{array}$$

(7)

Using Equations (6) and (7) in Equation (5), we have the following:

$$\begin{array}{cc}\hfill \mathcal{E}\left[{\parallel \chi \left(\varrho \left(\eta \right)\right)\parallel }^{\mathfrak{p}}\right]& \le 3^{\mathfrak{p}-1}\mathcal{E}\left[\parallel \gamma \left(0\right){\parallel }^{\mathfrak{p}}\right]+6^{\mathfrak{p}-1}\varsigma \eta {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\zeta }_2^{\mathfrak{p}}\left({\displaystyle \frac{{\eta }^{\delta \phi \mathfrak{p}-1}}{{\delta }^{\mathfrak{p}-1}}}\right)\hfill \\ & \left(1+2^{\mathfrak{p}}\mathcal{E}\left[{\parallel \varrho \parallel }^{\mathfrak{p}}\right]\right)+6^{\mathfrak{p}-1}{\zeta }_2^{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\phi }_{\mathfrak{p}}{\left({\displaystyle \frac{{\eta }^{(2\phi \delta -1)}}{\delta (2\phi -1)}}\right)}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & \left(1+2^{\mathfrak{p}}\mathcal{E}\left[{\parallel \varrho \parallel }^{\mathfrak{p}}\right]\right).\hfill \end{array}$$

(8)

We have the required outcomes:

$$\mathcal{E}\left[{\parallel \chi \left(\varrho \left(\eta \right)\right)\parallel }^{\mathfrak{p}}\right]\le \mathbb{V}\left(1+\mathcal{E}\left[{\parallel \varrho \parallel }^{\mathfrak{p}}\right]\right).$$

• Step 2: We now demonstrate the contractivity of $\chi$. Thus, by using J-I, $\varrho \left(\eta \right)$, and $\sigma \left(\eta \right)$, we have the following:

$$\begin{array}{cc}\hfill \mathcal{E}& \left[{\parallel \chi \left(\varrho \left(\eta \right)\right)-\chi \left(\sigma \left(\eta \right)\right)\parallel }^{\mathfrak{p}}\right]\hfill \\ \hfill \le & 2^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_1(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))-{\mathcal{Z}}_1(\xi ,\sigma \left(\xi \right),\sigma (\xi -\mathrm{\Phi }))\right)\mathrm{d}\xi {\parallel }^{\mathfrak{p}}\right]\hfill \\ \hfill +& 2^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_2(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))-{\mathcal{Z}}_2(\xi ,\sigma \left(\xi \right),\sigma (\xi -\mathrm{\Phi }))\right)\mathrm{d}\mathcal{W}\left(\xi \right){\parallel }^{\mathfrak{p}}\right]\hfill \\ \hfill =& {\mathfrak{C}}_3+{\mathfrak{C}}_4.\hfill \end{array}$$

(9)

By H-I and $\left({\mathfrak{T}}_1\right)$, we have the following:

$$\begin{array}{cc}\hfill {\mathfrak{C}}_3\le & 2^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\left({\int }_0^{\eta }{\xi }^{{\displaystyle \frac{\mathfrak{p}(\delta -1)}{\mathfrak{p}-1}}}{({\eta }^{\delta }-{\xi }^{\delta })}^{{\displaystyle \frac{\mathfrak{p}(\phi -1)}{\mathfrak{p}-1}}}d\xi \right)}^{\mathfrak{p}-1}\hfill \\ & \mathcal{E}\left[{\int }_0^{\eta }{\parallel {\mathcal{Z}}_1(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))-{\mathcal{Z}}_1(\xi ,\sigma \left(\xi \right),\sigma (\xi -\mathrm{\Phi }))\parallel }^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ \hfill \le & 2^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\left(\underset{0<\xi \le \eta }{sup}{\xi }^{{\displaystyle \frac{\delta -1}{\mathfrak{p}-1}}}{\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{{\displaystyle \frac{\mathfrak{p}(\phi -1)}{\mathfrak{p}-1}}}d\xi \right)}^{\mathfrak{p}-1}\hfill \\ & \mathcal{E}\left[{\int }_0^{\eta }{\parallel {\mathcal{Z}}_1(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))-{\mathcal{Z}}_1(\xi ,\sigma \left(\xi \right),\sigma (\xi -\mathrm{\Phi }))\parallel }^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ \hfill \le & 2^{\mathfrak{p}-1}\varsigma {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\left({\displaystyle \frac{{\eta }^{\delta \phi \mathfrak{p}-1}}{{\delta }^{\mathfrak{p}-1}}}\right)\hfill \\ & {\int }_0^{\eta }{\zeta }_1^{\mathfrak{p}}\mathcal{E}{\left[\parallel \varrho \left(\xi \right)-\sigma \left(\xi \right)\parallel +\parallel \varrho (\xi -\mathrm{\Phi })-\sigma (\xi -\mathrm{\Phi })\parallel \right]}^{\mathfrak{p}}\mathrm{d}\xi \hfill \\ \hfill \le & 4^{\mathfrak{p}-1}\varsigma {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\left({\displaystyle \frac{{\eta }^{\delta \phi \mathfrak{p}-1}}{{\delta }^{\mathfrak{p}-1}}}\right)\hfill \\ & {\int }_0^{\eta }{\zeta }_1^{\mathfrak{p}}\mathcal{E}\left[{\parallel \varrho \left(\xi \right)-\sigma \left(\xi \right)\parallel }^{\mathfrak{p}}+{\parallel \varrho (\xi -\mathrm{\Phi })-\sigma (\xi -\mathrm{\Phi })\parallel }^{\mathfrak{p}}\right]\mathrm{d}\xi \hfill \\ \hfill \le & 2\varsigma \eta {\zeta }_1^{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{4}^{\mathfrak{p}-1}\left({\displaystyle \frac{{\eta }^{\delta \phi \mathfrak{p}-1}}{{\delta }^{\mathfrak{p}-1}}}\right)\underset{\eta \in [0,\Im ]}{sup}\mathcal{E}\left[{\parallel \varrho \left(\eta \right)-\sigma \left(\eta \right)\parallel }^{\mathfrak{p}}\right].\hfill \end{array}$$

(10)

By B-D-G-I and $\left({\mathfrak{T}}_1\right)$, we have the following:

$$\begin{array}{cc}\hfill & {\mathfrak{C}}_4=2^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\underset{\eta \in [0,\Im ]}{sup}\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_2(\xi ,\varrho \left(\xi \right),\varrho (\xi -\mathrm{\Phi }))-{\mathcal{Z}}_2(\xi ,\sigma \left(\xi \right),\sigma (\xi -\mathrm{\Phi }))\right)\mathrm{d}\mathcal{W}\left(\xi \right){\parallel }^{\mathfrak{p}}\right]\hfill \\ \hfill \le & 2^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\phi }_{\mathfrak{p}}\mathcal{E}{\left[{\int }_0^{\eta }{\xi }^{2\delta -2}{({\eta }^{\delta }-{\xi }^{\delta })}^{2\phi -2}\parallel {\mathcal{Z}}_2(\xi ,\varrho \left(\xi \right))-{\mathcal{Z}}_2(\xi ,\sigma \left(\xi \right)){\parallel }^2\mathrm{d}\xi \right]}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill \le & {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\displaystyle \frac{2^{\mathfrak{p}-1}{\zeta }_1^{\mathfrak{p}}{\eta }^{\frac{(2\phi \delta -1)\mathfrak{p}}{2}}}{{(2\phi \delta -\delta )}^{\frac{\mathfrak{p}}{2}}}}{\phi }_{\mathfrak{p}}\mathcal{E}{\left[\parallel \varrho \left(\eta \right)-\sigma \left(\eta \right)\parallel +\parallel \varrho \left(\mathrm{\Phi }\eta \right)-\sigma \left(\mathrm{\Phi }\eta \right)\parallel \right]}^{\mathfrak{p}}.\hfill \\ \hfill \le & {\phi }_{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\displaystyle \frac{2^{2\mathfrak{p}-1}{\zeta }_1^{\mathfrak{p}}{\eta }^{\frac{(2\phi \delta -1)\mathfrak{p}}{2}}}{{(2\phi \delta -\delta )}^{\frac{\mathfrak{p}}{2}}}}\underset{\eta \in [0,\Im ]}{sup}\mathcal{E}\left[{\parallel \varrho \left(\eta \right)-\sigma \left(\eta \right)\parallel }^{\mathfrak{p}}\right].\hfill \end{array}$$

(11)

Employing Equations (10) and (11) in Equation (9), we have the following:

$$\begin{array}{cc}\hfill \mathcal{E}& \left[{\parallel \chi \left(\varrho \left(\eta \right)\right)-\chi \left(\sigma \left(\eta \right)\right)\parallel }^{\mathfrak{p}}\right]\hfill \\ & \le 2\eta {\zeta }_1^{\mathfrak{p}}\varsigma {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{4}^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\eta }^{\left(\frac{\delta \phi \mathfrak{p}-1}{\mathfrak{p}-1}\right)}}{\delta }}\right)}^{\mathfrak{p}-1}\underset{\eta \in [0,\Im ]}{sup}\mathcal{E}\left[{\parallel \varrho \left(\eta \right)-\sigma \left(\eta \right)\parallel }^{\mathfrak{p}}\right]\hfill \\ \hfill +& {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\displaystyle \frac{2^{2\mathfrak{p}-1}{\zeta }_1^{\mathfrak{p}}{\eta }^{\frac{(2\phi \delta -1)\mathfrak{p}}{2}}}{{(2\phi \delta -\delta )}^{\frac{\mathfrak{p}}{2}}}}{\phi }_{\mathfrak{p}}\underset{\eta \in [0,\Im ]}{sup}\mathcal{E}\left[{\parallel \varrho \left(\eta \right)-\sigma \left(\eta \right)\parallel }^{\mathfrak{p}}\right].\hfill \\ & =(2\varsigma \eta {\zeta }_1^{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{4}^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\eta }^{\left(\frac{\delta \phi \mathfrak{p}-1}{\mathfrak{p}-1}\right)}}{\delta }}\right)}^{\mathfrak{p}-1}\hfill \\ \hfill +& {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\displaystyle \frac{2^{2\mathfrak{p}-1}{\zeta }_1^{\mathfrak{p}}{\eta }^{\frac{(2\phi \delta -1)\mathfrak{p}}{2}}}{{(2\phi \delta -\delta )}^{\frac{\mathfrak{p}}{2}}}}{\phi }_{\mathfrak{p}})\underset{\eta \in [0,\Im ]}{sup}\mathcal{E}\left[{\parallel \varrho \left(\eta \right)-\sigma \left(\eta \right)\parallel }^{\mathfrak{p}}\right].\hfill \end{array}$$

(12)

Therefore, we have the following:

$$\left[{\parallel \chi \left(\varrho \left(\eta \right)\right)-\chi \left(\sigma \left(\eta \right)\right)\parallel }^{\mathfrak{p}}\right]\le \mathrm{\Lambda }{\parallel \varrho \left(\eta \right)-\sigma \left(\eta \right)\parallel }^{\mathfrak{p}}.$$

We thereby demonstrate the necessary outcome. □

3.2. Averaging Principle Result

In this section, we construct generalized results in the ${\mathrm{L}}^{\mathfrak{p}}$ space related to Ave-P for DFSDEs within the Cap-KGF derivative. Our work presents the standard form of DFSDEs within the framework of Cap-KGF derivatives, and its derivation relies on the techniques outlined in the mentioned research and the Cap-KGF integral. By establishing the results of Ave-P in the context of the Cap-KGF operator, we generalized the results regarding Cap-FD and Cap-HFD. We attained the results of Ave-P under Cap-FD [57] when we substituted $\delta =1$ in our result; we can obtain the result regarding Cap-HFD [58] when we put $\delta \to 0^{+}$ in our established outcomes. As we established our result in ${\mathrm{L}}^{\mathfrak{p}}$ space, we generalized the result of $\mathfrak{p}=2$. The results established in [59] have aided us in extending the work on Ave-P in the context of the Cap-KGF derivative.

Unfortunately, in most FSDE studies, the standard form of the original equations is defined by adding $\epsilon$ in front of the drift term and $\sqrt{\epsilon }$ in front of the diffusion term. For the fractional order case, the standard form of the original equation must still be given through a time scale change and cannot be directly defined by adding $\epsilon$ in front of the drift term and $\sqrt{\epsilon }$ in front of the diffusion term. Developing the correct standard form is necessary due to its physical significance. So, in this study, we correct this mistake by establishing the result of Ave-P by utilizing inequality and interval translation techniques.

To clarify our point, we first explain the procedure for obtaining the standard form for integer-order stochastic differential equations (SDEs). So, for the integer-order case, when we study the Ave-P of the following:

$$\left\{\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }\varrho \left(\eta \right)={\mathcal{Z}}_1\left(\eta ,\varrho \left(\eta \right)\right)+{\mathcal{Z}}_2(\eta ,\varrho \left(\eta \right)){\displaystyle \frac{\mathrm{d}\mathcal{W}\left(\eta \right)}{\mathrm{d}\eta }},\eta \in [0,\Im ],\hfill \\ & \varrho \left(0\right)={\varrho }_0,\hfill \end{array}\right.$$

(13)

it means that when $\epsilon \to 0^{+}$, the solution to the following:

$$\left\{\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }\varrho \left(\eta \right)={\mathcal{Z}}_1\left({\displaystyle \frac{\eta }{\epsilon }},\varrho \left(\eta \right)\right)+{\mathcal{Z}}_2({\displaystyle \frac{\eta }{\epsilon }},\varrho \left(\eta \right)){\displaystyle \frac{\mathrm{d}\mathcal{W}\left(\eta \right)}{\mathrm{d}\eta }},\hfill \\ & \varrho \left(0\right)={\varrho }_0,\hfill \end{array}\right.$$

(14)

converges to that of the averaged equation. But the standard form of the Equation (13) be defined as follows:

$$\left\{\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }{\varrho }_{\epsilon }\left(\eta \right)={\mathcal{Z}}_1\left(\eta ,{\varrho }_{\epsilon }\left(\eta \right)\right)+{\mathcal{Z}}_2(\eta ,{\varrho }_{\epsilon }\left(\eta \right)){\displaystyle \frac{\mathrm{d}\mathcal{W}\left(\eta \right)}{\mathrm{d}\eta }},\hfill \\ & {\varrho }_{\epsilon }\left(0\right)={\varrho }_0.\hfill \end{array}\right.$$

(15)

This is because when ${\displaystyle \frac{\eta }{\epsilon }}=\theta$, Equation (14) can be rewritten as follows:

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }\varrho \left(\epsilon \theta \right)={\mathcal{Z}}_1\left(\theta ,\varrho \left(\epsilon \theta \right)\right)+{\mathcal{Z}}_2(\theta ,\varrho \left(\epsilon \theta \right)){\displaystyle \frac{\mathrm{d}\mathcal{W}\left(\epsilon \theta \right)}{\epsilon \mathrm{d}\theta }}.\hfill \end{array}$$

We have $d\mathcal{W}\left(\epsilon \theta \right)=\sqrt{\epsilon }d\mathcal{W}\left(\theta \right)$ and denote $\varrho \left(\epsilon \theta \right)={\varrho }_{\epsilon }\left(\theta \right)$, and we obtain the following:

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }{\varrho }_{\epsilon }\left(\theta \right)=\epsilon {\mathcal{Z}}_1\left(\theta ,{\varrho }_{\epsilon }\left(\theta \right)\right)+\sqrt{\epsilon }{\mathcal{Z}}_2(\theta ,{\varrho }_{\epsilon }\left(\theta \right)){\displaystyle \frac{\mathrm{d}\mathcal{W}\left(\theta \right)}{\mathrm{d}\theta }}.\hfill \end{array}$$

When $\theta =\eta$, the standard form of the Equation (13) can be defined as follows:

$$\left\{\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }{\varrho }_{\epsilon }\left(\eta \right)=\epsilon {\mathcal{Z}}_1\left(\eta ,{\varrho }_{\epsilon }\left(\eta \right)\right)+\sqrt{\epsilon }{\mathcal{Z}}_2(\eta ,{\varrho }_{\epsilon }\left(\eta \right)){\displaystyle \frac{\mathrm{d}\mathcal{W}\left(\eta \right)}{\mathrm{d}\eta }},\eta \in [0,\Im ],\hfill \\ & {\varrho }_{\epsilon }\left(0\right)={\varrho }_0.\hfill \end{array}\right.$$

From the above discussion, we have observed that—in most publications on the fractional case—authors used the same approach for constructing the standard form that is correct for integer-order SDEs rather than for the fractional case, placing $\epsilon$ with the deterministic part and $\sqrt{\epsilon }$ with the diffusion part. This approach results in an incorrect standard form. In the following, we develop the correct standard form for the fractional case. To do this, we first prove a lemma that presents the time scale change property for the Cap-KGF operator, an essential tool for obtaining the standard form for the fractional case.

Lemma 2. 

(Time scale change property). Suppose the time scale is $\varpi \varkappa$, then we have the following:

$${\mathcal{D}}_{\varkappa }^{\phi ,\delta }\mathbb{Y}\left(\varpi \varkappa \right)={\varpi }^{\delta \phi }{\mathcal{D}}_{\eta }^{\phi ,\delta }\mathbb{Y}\left(\eta \right)$$

Proof. 

From the Cap-KGF integral, we have the following:

$${\mathcal{D}}_{\varkappa }^{\phi ,\delta }\mathbb{Y}\left(\varpi \varkappa \right)={\displaystyle \frac{{\delta }^{\phi }}{\mathrm{\Gamma }(1-\phi )}}{\int }_0^{\varkappa }{\displaystyle \frac{{\mathbb{Y}}^{*}\left(\varpi \xi \right)}{{({\varkappa }^{\delta }-{\xi }^{\delta })}^{\phi }}}\mathrm{d}\xi .$$

Let $\varpi \xi =\vartheta$ and by chain rule ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\xi }}={\displaystyle \frac{\mathrm{d}}{\mathrm{d}\vartheta }}.{\displaystyle \frac{\mathrm{d}\vartheta }{\mathrm{d}\xi }}={\displaystyle \frac{\mathrm{d}}{\mathrm{d}\vartheta }}.{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\xi }}\left(\varpi \xi \right)=\varpi {\displaystyle \frac{\mathrm{d}}{\mathrm{d}\vartheta }}$.

So, we have the following:

$${\mathcal{D}}_{\varkappa }^{\phi ,\delta }\mathbb{Y}\left(\varpi \varkappa \right)={\displaystyle \frac{{\delta }^{\phi }}{\mathrm{\Gamma }(1-\phi )}}{\int }_0^{\varpi \varkappa }{\displaystyle \frac{\varpi {\mathbb{Y}}^{*}\left(\vartheta \right)}{{({\varkappa }^{\delta }-{\left(\frac{\vartheta }{\varpi }\right)}^{\delta })}^{\phi }}}{\displaystyle \frac{1}{\varpi }}\mathrm{d}\vartheta .$$

From the above, we have the following:

$${\mathcal{D}}_{\varkappa }^{\phi ,\delta }\mathbb{Y}\left(\varpi \varkappa \right)={\varpi }^{\delta \phi }{\displaystyle \frac{{\delta }^{\phi }}{\mathrm{\Gamma }(1-\phi )}}{\int }_0^{\varpi \varkappa }{\displaystyle \frac{{\mathbb{Y}}^{*}\left(\vartheta \right)}{{\left({\left(\varpi \varkappa \right)}^{\delta }-{\vartheta }^{\delta }\right)}^{\phi }}}\mathrm{d}\vartheta .$$

or

$${\mathcal{D}}_{\varkappa }^{\phi ,\delta }\mathbb{Y}\left(\varpi \varkappa \right)={\varpi }^{\delta \phi }{\displaystyle \frac{{\delta }^{\phi }}{\mathrm{\Gamma }(1-\phi )}}{\int }_0^{\eta }{\displaystyle \frac{{\mathbb{Y}}^{*}\left(\vartheta \right)}{{\left({\eta }^{\delta }-{\vartheta }^{\delta }\right)}^{\phi }}}\mathrm{d}\vartheta .$$

So, we have the following results:

$${\mathcal{D}}_{\varkappa }^{\phi ,\delta }\mathbb{Y}\left(\varpi \varkappa \right)={\varpi }^{\delta \phi }{\mathcal{D}}_b^{\phi ,\delta }\mathbb{Y}\left(\eta \right).$$

or

$${\mathcal{D}}_{\eta }^{\phi ,\delta }\mathbb{Y}\left(\eta \right)={\varpi }^{-\delta \phi }{\mathcal{D}}_{\varkappa }^{\phi ,\delta }\mathbb{Y}\left(\varpi \varkappa \right).$$

□

Now, we examine the Ave-P of DFSDEs. For this, we first consider the following:

$$\left\{\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }^{\phi ,\delta }\varrho \left(\eta \right)={\mathcal{Z}}_1\left({\displaystyle \frac{\eta }{\epsilon }},\varrho \left(\eta \right),\varrho (\eta -\mathrm{\Phi })\right)+{\mathcal{Z}}_2\left({\displaystyle \frac{\eta }{\epsilon }},\varrho \left(\eta \right),\varrho (\eta -\mathrm{\Phi })\right){\displaystyle \frac{\mathrm{d}{\mathcal{W}}_{\eta }}{\mathrm{d}\eta }},\hfill \\ & \varrho \left(\eta \right)=\gamma \left(\eta \right),\eta \in [-\mathrm{\Phi },0].\hfill \end{array}\right.$$

(16)

When we suppose ${\displaystyle \frac{\eta }{\epsilon }}=\theta$ and employ Lemma 2, we have the following from Equation (16):

$$\begin{array}{cc}\hfill & {\epsilon }^{-\phi \delta }{\mathcal{D}}_{\theta }^{\phi ,\delta }\varrho \left(\epsilon \theta \right)={\mathcal{Z}}_1\left(\theta ,\varrho \left(\epsilon \theta \right),\varrho (\epsilon \theta -\mathrm{\Phi })\right)+{\mathcal{Z}}_2\left(\theta ,\varrho \left(\epsilon \theta \right),\varrho (\epsilon \theta -\mathrm{\Phi })\right){\displaystyle \frac{\mathrm{d}\mathcal{W}\left(\epsilon \theta \right)}{\epsilon \mathrm{d}\theta }}.\hfill \end{array}$$

By considering $d\mathcal{W}\left(\epsilon \theta \right)=\sqrt{\epsilon }d\mathcal{W}\left(\theta \right)$ and representing $\varrho \left(\epsilon \theta \right)={\varrho }_{\epsilon }\left(\theta \right)$ and $\varrho (\epsilon \theta -\mathrm{\Phi })={\varrho }_{\epsilon }(\theta -\mathrm{\Phi })$, we obtain the following:

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{\theta }^{\phi ,\delta }{\varrho }_{\epsilon }\left(\theta \right)={\epsilon }^{\phi \delta }{\mathcal{Z}}_1\left(\theta ,{\varrho }_{\epsilon }\left(\theta \right),{\varrho }_{\epsilon }(\theta -\mathrm{\Phi })\right)+{\epsilon }^{\phi \delta -{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_2\left(\theta ,{\varrho }_{\epsilon }\left(\theta \right),{\varrho }_{\epsilon }(\theta -\mathrm{\Phi })\right){\displaystyle \frac{\mathrm{d}\mathcal{W}\left(\theta \right)}{\mathrm{d}\theta }}.\hfill \end{array}$$

Despite losing generality, $\theta =\eta$ can be expressed. Equation (1) can be obtained in standard form by applying the natural time scaling ${\displaystyle \frac{\eta }{\epsilon }}\mapsto \eta$:

$$\left\{\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }^{\phi ,\delta }{\varrho }_{\epsilon }\left(\eta \right)={\epsilon }^{\phi \delta }{\mathcal{Z}}_1\left(\eta ,{\varrho }_{\epsilon }\left(\eta \right),{\varrho }_{\epsilon }(\eta -\mathrm{\Phi })\right)+{\epsilon }^{\phi \delta -{\displaystyle \frac{1}{2}}}{\mathcal{Z}}_2\left(\eta ,{\varrho }_{\epsilon }\left(\eta \right),{\varrho }_{\epsilon }(\eta -\mathrm{\Phi })\right){\displaystyle \frac{\mathrm{d}\mathcal{W}\left(\eta \right)}{\mathrm{d}\eta }},\hfill \\ & {\varrho }_{\epsilon }\left(\eta \right)=\gamma \left(\eta \right),\eta \in [-\mathrm{\Phi },0],\hfill \end{array}\right.$$

(17)

Thus, Equation (17) can be expressed integrally as follows:

$$\begin{array}{cc}\hfill {\varrho }_{\epsilon }\left(\eta \right)& =\gamma \left(0\right)+{\epsilon }^{\delta \phi }{\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}{\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }{\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)\mathrm{d}\xi \hfill \\ & +{\epsilon }^{\delta \phi -{\displaystyle \frac{1}{2}}}{\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}{\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }{\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)\mathrm{d}\mathcal{W}\left(\xi \right),\hfill \end{array}$$

(18)

for $\epsilon$ belonging to $(0,{\epsilon }_0]$ with a fixed point at ${\epsilon }_0$. Furthermore, $\left({\mathfrak{T}}_1\right)$ and $\left({\mathfrak{T}}_2\right)$ specify conditions that are met by ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$. This leads to the following averaged representation of Equation (18):

$$\begin{array}{cc}\hfill {\varrho }_{\epsilon }^{*}\left(\eta \right)& =\gamma \left(0\right)+{\epsilon }^{\delta \phi }{\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}{\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }{\tilde{\mathcal{Z}}}_1\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\mathrm{d}\xi \hfill \\ & +{\epsilon }^{\delta \phi -{\displaystyle \frac{1}{2}}}{\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}{\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }{\tilde{\mathcal{Z}}}_2\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\mathrm{d}\mathcal{W}\left(\xi \right),\hfill \end{array}$$

(19)

where ${\tilde{\mathcal{Z}}}_1:{\mathfrak{R}}^{\mathfrak{a}}\times {\mathfrak{R}}^{\mathfrak{a}}\to {\mathfrak{R}}^{\mathfrak{a}},{\tilde{\mathcal{Z}}}_2:{\mathfrak{R}}^{\mathfrak{a}}\times {\mathfrak{R}}^{\mathfrak{a}}\to {\mathfrak{R}}^{\mathfrak{a}\times r}$.

Theorem 2. 

When $\left({\mathfrak{T}}_1\right)$ to $\left({\mathfrak{T}}_3\right)$ are true, and given $\mu >0$ and $\mathcal{P}>0$, ${\epsilon }_1$ belongs to $\left(0,{\epsilon }_0\right]$ with $\lambda \in (0,\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}})$, then we have the following:

$$\mathcal{E}\left[\underset{\eta \in [-\mathrm{\Phi },\mathcal{P}{\epsilon }^{-\lambda }]}{sup}\parallel {\varrho }_{\epsilon }\left(\eta \right)-{\varrho }_{\epsilon }^{*}\left(\eta \right){\parallel }^{\mathfrak{p}}\right]\le \mu ,\epsilon \in \left(0,{\epsilon }_1\right].$$

(20)

Proof. 

When $\eta \in [0,\Im ]$, we have the following:

$$\begin{array}{cc}\hfill {\varrho }_{\epsilon }\left(\eta \right)-{\varrho }_{\epsilon }^{*}\left(\eta \right)& ={\epsilon }^{\delta \phi }{\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\hfill \\ & {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_1\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right)\mathrm{d}\xi \hfill \\ & +{\epsilon }^{\delta \phi -{\displaystyle \frac{1}{2}}}{\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\hfill \\ & {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_2\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right)\mathrm{d}\mathcal{W}\left(\xi \right).\hfill \end{array}$$

(21)

We obtain the following via J-I:

$$\begin{array}{cc}\hfill & \parallel {\varrho }_{\epsilon }\left(\eta \right)-{\varrho }_{\epsilon }^{*}\left(\eta \right){\parallel }^{\mathfrak{p}}\le 2^{\mathfrak{p}-1}\hfill \\ & \parallel {\epsilon }^{\delta \phi }{\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}{\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_1\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right)\mathrm{d}\xi {\parallel }^{\mathfrak{p}}\hfill \\ & +2^{\mathfrak{p}-1}\hfill \\ & \parallel {\epsilon }^{\delta \phi -{\displaystyle \frac{1}{2}}}{\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}{\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_2\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right)\mathrm{d}\mathcal{W}\left(\xi \right){\parallel }^{\mathfrak{p}}\hfill \\ & =2^{\mathfrak{p}-1}{\epsilon }^{\delta \phi \mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_1\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right)\mathrm{d}\xi {\parallel }^{\mathfrak{p}}\hfill \\ & +2^{\mathfrak{p}-1}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_2\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right)\mathrm{d}\mathcal{W}\left(\xi \right){\parallel }^{\mathfrak{p}}.\hfill \end{array}$$

(22)

From the above, we obtain the following:

$$\begin{array}{cc}\hfill \parallel {\varrho }_{\epsilon }\left(\eta \right)-& {\varrho }_{\epsilon }^{*}\left(\eta \right){\parallel }^{\mathfrak{p}}\le 2^{\mathfrak{p}-1}{\epsilon }^{\delta \phi \mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_1\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right)\mathrm{d}\xi {\parallel }^{\mathfrak{p}}+\hfill \\ & 2^{\mathfrak{p}-1}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_2\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right)\mathrm{d}\mathcal{W}\left(\xi \right){\parallel }^{\mathfrak{p}}.\hfill \end{array}$$

(23)

For all $\mathfrak{u}\in [0,\Im ]$, we have $\mathcal{E}$.

$$\begin{array}{cc}\hfill & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}\parallel {\varrho }_{\epsilon }\left(\eta \right)-{\varrho }_{\epsilon }^{*}\left(\eta \right){\parallel }^{\mathfrak{p}}\right]\le 2^{\mathfrak{p}-1}{\epsilon }^{\delta \phi \mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-\widetilde{{\mathcal{Z}}_1}\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right)\mathrm{d}\xi {\parallel }^{\mathfrak{p}}\right]\hfill \\ & +2^{\mathfrak{p}-1}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-\widetilde{{\mathcal{Z}}_2}\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right)\mathrm{d}\mathcal{W}\left(\xi \right){\parallel }^{\mathfrak{p}}\right]\hfill \\ & ={\mathfrak{V}}_1+{\mathfrak{V}}_2.\hfill \end{array}$$

(24)

From ${\mathfrak{V}}_1$, we have the following:

$$\begin{array}{cc}\hfill {\mathfrak{V}}_1\le & 2^{2\mathfrak{p}-2}{\epsilon }^{\delta \phi \mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right)\mathrm{d}\xi {\parallel }^{\mathfrak{p}}\right]\hfill \\ \hfill +& 2^{2\mathfrak{p}-2}{\epsilon }^{\delta \phi \mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left({\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_1\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right)\mathrm{d}\xi {\parallel }^{\mathfrak{p}}\right]\hfill \\ \hfill =& {\mathfrak{V}}_{11}+{\mathfrak{V}}_{12}.\hfill \end{array}$$

(25)

The following outcome is obtained by applying H-I and $\left({\mathfrak{T}}_1\right)$ to ${\mathfrak{V}}_{11}$:

$$\begin{array}{cc}\hfill {\mathfrak{V}}_{11}\le & 2^{2\mathfrak{p}-2}{\epsilon }^{\delta \phi \mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\left({\int }_0^{\mathfrak{u}}{\xi }^{{(1-\mathfrak{p})}^{-1}\mathfrak{p}(\delta -1)}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{{\mathfrak{p}(\phi -1)(\mathfrak{p}-1)}^{-1}}\mathrm{d}\xi \right)}^{\mathfrak{p}-1}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}{\int }_0^{\eta }{∥{\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)∥}^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ \hfill \le & 2^{2\mathfrak{p}-2}{\epsilon }^{\delta \phi \mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\left({\int }_0^{\mathfrak{u}}{\xi }^{{(1-\mathfrak{p})}^{-1}\mathfrak{p}(\delta -1)}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{{\mathfrak{p}(\phi -1)(\mathfrak{p}-1)}^{-1}}\mathrm{d}\xi \right)}^{\mathfrak{p}-1}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}{\int }_0^{\eta }{\left(∥{\varrho }_{\epsilon }\left(\xi \right)-{\varrho }_{\epsilon }^{*}\left(\xi \right)∥+∥{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })-{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })∥\right)}^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ \hfill \le & 2^{2\mathfrak{p}-2}{\epsilon }^{\delta \phi \mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\left(\underset{0\le \xi \le \mathfrak{u}}{sup}{\xi }^{{\displaystyle \frac{\delta -1}{\mathfrak{p}-1}}}{\int }_0^{\mathfrak{u}}{\xi }^{\delta -1}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{{(1-\mathfrak{p})}^{-1}\mathfrak{p}(\phi -1)}\mathrm{d}\xi \right)}^{\mathfrak{p}-1}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}{\int }_0^{\eta }{\left(∥{\varrho }_{\epsilon }\left(\xi \right)-{\varrho }_{\epsilon }^{*}\left(\xi \right)∥+∥{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })-{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })∥\right)}^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ \hfill =& 2^{2\mathfrak{p}-2}{\epsilon }^{\delta \phi \mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\left({\mathfrak{u}}^{{\displaystyle \frac{\delta -1}{\mathfrak{p}-1}}}{\int }_0^{\mathfrak{u}}{\xi }^{\delta -1}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{{(1-\mathfrak{p})}^{-1}\mathfrak{p}(\phi -1)}\mathrm{d}\xi \right)}^{\mathfrak{p}-1}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}{\int }_0^{\eta }{\left(∥{\varrho }_{\epsilon }\left(\xi \right)-{\varrho }_{\epsilon }^{*}\left(\xi \right)∥+∥{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })-{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })∥\right)}^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ \hfill =& 2^{3\mathfrak{p}-3}\varsigma {\epsilon }^{\delta \phi \mathfrak{p}}{\zeta }_1^{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\left({\displaystyle \frac{{\mathfrak{u}}^{\delta \phi \mathfrak{p}-1}}{{\delta }^{\mathfrak{p}-1}}}\right)\hfill \\ & \left(\mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}{\int }_0^{\eta }{∥{\varrho }_{\epsilon }\left(\xi \right)-{\varrho }_{\epsilon }^{*}\left(\xi \right)∥}^{\mathfrak{p}}\mathrm{d}\xi \right]+\mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}{\int }_0^{\eta }{∥{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })-{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })∥}^{\mathfrak{p}}\mathrm{d}\xi \right]\right)\hfill \\ \hfill =& {\beth }_{11}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-1}\hfill \\ & \left({\int }_0^{\mathfrak{u}}\mathcal{E}\left[\underset{0\le \mathfrak{m}\le \xi }{sup}{∥{\varrho }_{\epsilon }\left(\mathfrak{m}\right)-{\varrho }_{\epsilon }^{*}\left(\mathfrak{m}\right)∥}^{\mathfrak{p}}\right]\mathrm{d}\xi +{\int }_0^{\mathfrak{u}}\mathcal{E}\left[\underset{0\le \mathfrak{m}\le \xi }{sup}{∥{\varrho }_{\epsilon }(m-\mathrm{\Phi })-{\varrho }_{\epsilon }^{*}(\mathfrak{m}-\mathrm{\Phi })∥}^{\mathfrak{p}}\right]\mathrm{d}\xi \right),\hfill \end{array}$$

(26)

where ${\beth }_{11}=2^{3\mathfrak{p}-3}\varsigma {\zeta }_1^{\mathfrak{p}}{\left({\displaystyle \frac{1}{\delta }}\right)}^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}.$

The subsequent consequence can be achieved by applying H-I and $\left({\mathfrak{T}}_3\right)$ to ${\mathfrak{V}}_{12}$:

$$\begin{array}{cc}\hfill {\mathfrak{V}}_{12}\le & 2^{2\mathfrak{p}-2}{\epsilon }^{\delta \phi \mathfrak{p}}{\left({\int }_0^{\mathfrak{u}}{\xi }^{{\displaystyle \frac{(\delta -1)\mathfrak{p}}{\mathfrak{p}-1}}}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{{\displaystyle \frac{(\phi -1)\mathfrak{p}}{\mathfrak{p}-1}}}\mathrm{d}\xi \right)}^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}{\int }_0^{\eta }\parallel {\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_1\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right){\parallel }^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ \hfill \le & 2^{2\mathfrak{p}-2}{\epsilon }^{\delta \phi \mathfrak{p}}{\left(\underset{0\le \xi \le \mathfrak{u}}{sup}{\xi }^{{\displaystyle \frac{\delta -1}{\mathfrak{p}-1}}}{\int }_0^{\mathfrak{u}}{\xi }^{\delta -1}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{{\displaystyle \frac{(\phi -1)\mathfrak{p}}{\mathfrak{p}-1}}}\mathrm{d}\xi \right)}^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}{\int }_0^{\eta }\parallel {\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_1\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right){\parallel }^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ \hfill =& 2^{2\mathfrak{p}-2}{\epsilon }^{\delta \phi \mathfrak{p}}{\left({\mathfrak{u}}^{{\displaystyle \frac{\delta -1}{\mathfrak{p}-1}}}{\int }_0^{\mathfrak{u}}{\xi }^{\delta -1}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{{\displaystyle \frac{(\phi -1)\mathfrak{p}}{\mathfrak{p}-1}}}\mathrm{d}\xi \right)}^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}{\int }_0^{\eta }\parallel {\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_1\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right){\parallel }^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ \hfill \le & 2^{2\mathfrak{p}-2}\mathfrak{u}\varsigma {\epsilon }^{\delta \phi \mathfrak{p}}\left({\displaystyle \frac{{\mathfrak{u}}^{\delta \phi \mathfrak{p}-1}}{{\delta }^{\mathfrak{p}-1}}}\right){\mathsf{F}}_1\left(\mathfrak{u}\right)\left(1+\mathcal{E}\parallel {\varrho }_{\epsilon }^{*}\left(\xi \right){\parallel }^{\mathfrak{p}}+\mathcal{E}\parallel {\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi }){\parallel }^{\mathfrak{p}}\right){\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ \hfill =& {\beth }_{12}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}},\hfill \end{array}$$

(27)

where ${\beth }_{12}=2^{2\mathfrak{p}-2}{\mathsf{F}}_1\left(\mathfrak{u}\right)\varsigma \left(1+\mathcal{E}{∥{\varrho }_{\epsilon }^{*}\left(\xi \right)∥}^{\mathfrak{p}}+\mathcal{E}{∥{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })∥}^{\mathfrak{p}}\right){\left({\displaystyle \frac{1}{\delta }}\right)}^{\mathfrak{p}-1}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}.$

By J-I, from ${\mathfrak{V}}_2$, we have the following:

$$\begin{array}{cc}\hfill & {\mathfrak{V}}_2\le {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{2}^{2\mathfrak{p}-2}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left[{\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right]\mathrm{d}\mathcal{W}\left(\xi \right){\parallel }^{\mathfrak{p}}\right]\hfill \\ \hfill +& {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{2}^{2\mathfrak{p}-2}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}\parallel {\int }_0^{\eta }{\xi }^{\delta -1}{({\eta }^{\delta }-{\xi }^{\delta })}^{1-\phi }\left[{\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_2\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)\right]\mathrm{d}\mathcal{W}\left(\xi \right){\parallel }^{\mathfrak{p}}\right]\hfill \\ \hfill =& {\mathfrak{V}}_{21}+{\mathfrak{V}}_{22}.\hfill \end{array}$$

(28)

We obtain the following results by using B-D-G-I, H-I, and $\left({\mathfrak{T}}_1\right)$ on ${\mathfrak{V}}_{21}$:

$$\begin{array}{cc}\hfill & {\mathfrak{V}}_{21}\le 2^{2\mathfrak{p}-2}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\phi }_{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}{\left[{\int }_0^{\mathfrak{u}}{\xi }^{2\delta -2}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{2\phi -2}{∥{\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)∥}^2\mathrm{d}\xi \right]}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill \le & 2^{2\mathfrak{p}-2}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\phi }_{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}\hfill \\ & \mathcal{E}\left[{\int }_0^{\mathfrak{u}}{\xi }^{(\delta -1)\mathfrak{p}}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{(\phi -1)\mathfrak{p}}{∥{\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }\left(\xi \right),{\varrho }_{\epsilon }(\xi -\mathrm{\Phi })\right)-{\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)∥}^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ \hfill \le & {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{2}^{2\mathfrak{p}-2}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\zeta }_1^{\mathfrak{p}}{\phi }_{\mathfrak{p}}{\int }_0^{\mathfrak{u}}{\xi }^{(\delta -1)\mathfrak{p}}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{(\phi -1)\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\underset{0\le \mathfrak{m}\le \xi }{sup}{\left[∥{\varrho }_{\epsilon }\left(\mathfrak{m}\right)-{\varrho }_{\epsilon }^{*}\left(\mathfrak{m}\right)∥+∥{\varrho }_{\epsilon }(\mathfrak{m}-\mathrm{\Phi })-{\varrho }_{\epsilon }^{*}(\mathfrak{m}-\mathrm{\Phi })∥\right]}^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ \hfill \le & {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{2}^{3\mathfrak{p}-3}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\zeta }_1^{\mathfrak{p}}{\phi }_{\mathfrak{p}}{\int }_0^{\mathfrak{u}}{\xi }^{(\delta -1)\mathfrak{p}}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{(\phi -1)\mathfrak{p}}\hfill \\ & \mathcal{E}\left[\underset{0\le \mathfrak{m}\le \xi }{sup}\left[{∥{\varrho }_{\epsilon }\left(\mathfrak{m}\right)-{\varrho }_{\epsilon }^{*}\left(\mathfrak{m}\right)∥}^{\mathfrak{p}}+{∥{\varrho }_{\epsilon }(\mathfrak{m}-\mathrm{\Phi })-{\varrho }_{\epsilon }^{*}(\mathfrak{m}-\mathrm{\Phi })∥}^{\mathfrak{p}}\right]\mathrm{d}\xi \right]\hfill \\ & ={\beth }_{21}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}({\int }_0^{\mathfrak{u}}{\xi }^{(\delta -1)\mathfrak{p}}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{(\phi -1)\mathfrak{p}}\mathcal{E}\left[\underset{0\le \mathfrak{m}\le \xi }{sup}{∥{\varrho }_{\epsilon }\left(\mathfrak{m}\right)-{\varrho }_{\epsilon }^{*}\left(\mathfrak{m}\right)∥}^{\mathfrak{p}}\mathrm{d}\xi \right]\hfill \\ \hfill +& {\int }_0^{\mathfrak{u}}{\xi }^{(\delta -1)\mathfrak{p}}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{(\phi -1)\mathfrak{p}}\mathcal{E}\left[\underset{0\le \mathfrak{m}\le \xi }{sup}{∥{\varrho }_{\epsilon }(\mathfrak{m}-\mathrm{\Phi })-{\varrho }_{\epsilon }^{*}(\mathfrak{m}-\mathrm{\Phi })∥}^{\mathfrak{p}}\mathrm{d}\xi \right]),\hfill \end{array}$$

(29)

where ${\beth }_{21}=2^{3\mathfrak{p}-3}{\zeta }_1^{\mathfrak{p}}{\phi }_{\mathfrak{p}}{\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}.$

Again, using B-D-G-I, H-I, and $\left({\mathfrak{T}}_1\right)$ on ${\mathfrak{V}}_{22}$, we have the following:

$$\begin{array}{cc}\hfill & {\mathfrak{V}}_{22}\le {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\displaystyle \frac{1}{2^{-2\mathfrak{p}}}}{\displaystyle \frac{1}{4}}{\phi }_{\mathfrak{p}}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}\hfill \\ & \mathcal{E}{\left[{\int }_0^{\mathfrak{u}}{∥{\mathcal{Z}}_1\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)-{\tilde{\mathcal{Z}}}_2\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)∥}^2{\xi }^{2\delta -2}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{2\phi -2}\mathrm{d}\xi \right]}^{{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ \hfill \le & {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{2}^{2\mathfrak{p}-2}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\phi }_{\mathfrak{p}}\mathcal{E}[{\int }_0^{\mathfrak{u}}{\xi }^{(\delta -1)\mathfrak{p}}{({\mathfrak{u}}^{\delta }-{\xi }^{\delta })}^{(\phi -1)\mathfrak{p}}\hfill \\ & \left({∥{\mathcal{Z}}_2\left(\xi ,{\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right)∥}^{\mathfrak{p}}+\parallel {\tilde{\mathcal{Z}}}_2\left({\varrho }_{\epsilon }^{*}\left(\xi \right),{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })\right){\parallel }^{\mathfrak{p}}\right)\mathrm{d}\xi ]\hfill \\ \hfill \le & {\left({\displaystyle \frac{{\delta }^{1-\phi }}{\mathrm{\Gamma }\left(\phi \right)}}\right)}^{\mathfrak{p}}{\displaystyle \frac{2^{3\mathfrak{p}-3}{\epsilon }^{(\delta \phi -\frac{1}{2})\mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-\frac{\mathfrak{p}}{2}}\left({\zeta }_2^{\mathfrak{p}}+{\zeta }_3^{\mathfrak{p}}\right)}{\delta \left((\phi -1)\mathfrak{p}+1\right)}}{\phi }_{\mathfrak{p}}\hfill \\ & \left(1+\mathcal{E}\left[{∥{\varrho }_{\epsilon }^{*}\left(\xi \right)∥}^{\mathfrak{p}}\right]+\mathcal{E}\left[{∥{\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi })∥}^{\mathfrak{p}}\right]\right)\hfill \\ \hfill =& {\beth }_{22}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}},\hfill \end{array}$$

(30)

where ${\beth }_{22}={\left({\delta }^{1-\phi }\right)}^{\mathfrak{p}}{\left(\mathrm{\Gamma }\left(\phi \right)\right)}^{-\mathfrak{p}}{2}^{3\mathfrak{p}-3}\left({\zeta }_2^{\mathfrak{p}}{\displaystyle \frac{1}{\delta \left((\phi -1)\mathfrak{p}+1\right)}}{\phi }_{\mathfrak{p}}+{\zeta }_4^{\mathfrak{p}}{\displaystyle \frac{1}{\delta \left((\phi -1)\mathfrak{p}+1\right)}}{\phi }_{\mathfrak{p}}\right)$$\left(1+\mathcal{E}[\parallel {\varrho }_{\epsilon }^{*}{\left(\xi \right)\parallel }^{\mathfrak{p}}]+\mathcal{E}[\parallel {\varrho }_{\epsilon }^{*}(\xi -\mathrm{\Phi }){\parallel }^{\mathfrak{p}}]\right).$

So, we have the following:

$$\begin{array}{cc}\hfill & \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}{∥{\varrho }_{\epsilon }\left(\eta \right)-{\varrho }_{\epsilon }^{*}\left(\eta \right)∥}^{\mathfrak{p}}\right]\le {\beth }_{12}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}}+{\beth }_{22}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & +{\int }_0^{\mathfrak{u}}\left[{\beth }_{11}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-1}+{\beth }_{21}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\left({\mathfrak{u}}^{\delta }-{\xi }^{\delta }\right)}^{(\phi -1)\mathfrak{p}}{\xi }^{(\delta -1)\mathfrak{p}}\right]\hfill \\ & \left(\mathcal{E}\left[\underset{0\le \mathfrak{m}\le \xi }{sup}{∥{\varrho }_{\epsilon }\left(\mathfrak{m}\right)-{\varrho }_{\epsilon }^{*}\left(\mathfrak{m}\right)∥}^{\mathfrak{p}}\mathrm{d}\xi \right]+\mathcal{E}\left[\underset{0\le \mathfrak{m}\le \xi }{sup}{∥{\varrho }_{\epsilon }(\mathfrak{m}-\mathrm{\Phi })-{\varrho }_{\epsilon }^{*}(\mathfrak{m}-\mathrm{\Phi })∥}^{\mathfrak{p}}\mathrm{d}\xi \right]\right).\hfill \end{array}$$

(31)

Taking $\mathcal{M}\left(\mathfrak{u}\right)=\mathcal{E}\left[{sup}_{0\le \eta \le \mathfrak{u}}{∥{\varrho }_{\epsilon }\left(\eta \right)-{\varrho }_{\epsilon }^{*}\left(\eta \right)∥}^{\mathfrak{p}}\right]$ and $\mathcal{E}[{sup}_{-\aleph \le \eta \le 0}\parallel {\varrho }_{\epsilon }\left(\eta \right)-$${\varrho }_{\epsilon }^{*}{\left(\eta \right)\parallel }^{\mathfrak{p}}]=0$.

So,

$$\mathcal{E}\left[\underset{0\le \mathfrak{m}\le \xi }{sup}\parallel {\varrho }_{\epsilon }(\mathfrak{m}-\aleph )-{\varrho }_{\epsilon }^{*}(\mathfrak{m}-\aleph ){\parallel }^{\mathfrak{p}}\right]=\mathcal{M}(\xi -\aleph ).$$

Consequently, we have the following:

$$\begin{array}{cc}\hfill \mathcal{M}\left(\mathfrak{u}\right)\le & {\beth }_{12}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}}+{\beth }_{22}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & +{\int }_0^{\mathfrak{u}}\left[{\beth }_{11}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-1}+{\beth }_{21}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\left({\mathfrak{u}}^{\delta }-{\xi }^{\delta }\right)}^{(\phi -1)\mathfrak{p}}{\xi }^{(\delta -1)\mathfrak{p}}\right]\hfill \\ & \left(\mathcal{M}\right(\xi )+\mathcal{M}(\xi -\aleph \left)\right)\mathrm{d}\xi .\hfill \end{array}$$

(32)

Let $\mho \left(\mathfrak{u}\right)={sup}_{\mathcal{T}\in [-\aleph ,\mathfrak{u}]}\mathcal{M}\left(\mathcal{T}\right)$. So, $\forall \mathfrak{u}\in [0,{\rm Y}]$; we have $\mathcal{M}\left(\xi \right)\le \mho \left(\xi \right)$ and $\mathcal{M}(\xi -\aleph )\le \mho \left(\xi \right)$.

So, we have the following:

$$\begin{array}{cc}\hfill \mathcal{M}\left(\mathfrak{u}\right)& \le {\beth }_{12}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}}+{\beth }_{22}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & +2{\int }_0^{\mathfrak{u}}\left[{\beth }_{11}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-1}+{\beth }_{21}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\left({\mathfrak{u}}^{\delta }-{\xi }^{\delta }\right)}^{(\phi -1)\mathfrak{p}}{\xi }^{(\delta -1)\mathfrak{p}}\right]\mho \left(\xi \right)\mathrm{d}\xi .\hfill \end{array}$$

For all $\mathcal{T}\in [0,\mathfrak{u}]$, we have the following:

$$\begin{array}{cc}\hfill \mathcal{M}\left(\mathcal{T}\right)\le & {\beth }_{12}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathcal{T}}^{\delta \phi \mathfrak{p}}+{\beth }_{22}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathcal{T}}^{\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & +{\int }_0^{\mathcal{T}}\left[{\beth }_{11}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathcal{T}}^{\delta \phi \mathfrak{p}-1}+{\beth }_{21}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathcal{T}}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\left({\mathcal{T}}^{\delta }-{\xi }^{\delta }\right)}^{(\phi -1)\mathfrak{p}}{\xi }^{(\delta -1)\mathfrak{p}}\right]\mho \left(\xi \right)\mathrm{d}\xi \hfill \\ \hfill \le & {\beth }_{12}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}}+{\beth }_{22}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & +2{\int }_0^{\mathfrak{u}}\left[{\beth }_{11}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-1}+{\beth }_{21}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\left({\mathfrak{u}}^{\delta }-{\xi }^{\delta }\right)}^{(\phi -1)\mathfrak{p}}{\xi }^{(\delta -1)\mathfrak{p}}\right]\mho \left(\xi \right)\mathrm{d}\xi .\hfill \end{array}$$

As a result, we have the following:

$$\begin{array}{cc}\hfill \mho \left(\mathfrak{u}\right)=& \underset{\mathcal{T}\in [-\aleph ,\mathfrak{u}]}{sup}\mathcal{M}\left(\mathcal{T}\right)\hfill \\ \hfill \le & max\left\{\underset{\mathcal{T}\in [-\aleph ,0]}{sup}\mathcal{M}\left(\mathcal{T}\right),\underset{\mathcal{T}\in [0,\mathfrak{u}]}{sup}\mathcal{M}\left(\mathcal{T}\right)\right\}\hfill \\ \hfill \le & {\beth }_{12}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}}+{\beth }_{22}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & +2{\int }_0^{\mathfrak{u}}\left[{\beth }_{11}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-1}+{\beth }_{21}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{{\displaystyle \frac{\mathfrak{p}}{2}}-1}{\left({\mathfrak{u}}^{\delta }-{\xi }^{\delta }\right)}^{(\phi -1)\mathfrak{p}}{\xi }^{(\delta -1)\mathfrak{p}}\right]\mho \left(\xi \right)\mathrm{d}\xi .\hfill \end{array}$$

By G-B-I, we have the following:

$$\begin{array}{cc}\hfill \mho \left(\mathfrak{u}\right)\le & {\beth }_{12}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}}+{\beth }_{22}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & exp\left(2{\beth }_{11}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}}+{\displaystyle \frac{2{\beth }_{21}}{\delta \left((\phi -1)\mathfrak{p}+1\right)}}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}+1-\mathfrak{p}}\right).\hfill \end{array}$$

From the above, we have the following:

$$\begin{array}{cc}\hfill \mathcal{E}\left[\underset{0\le \eta \le \mathfrak{u}}{sup}{∥{\varrho }_{\epsilon }\left(\eta \right)-{\varrho }_{\epsilon }^{*}\left(\eta \right)∥}^{\mathfrak{p}}\right]\le & {\beth }_{12}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}}+{\beth }_{22}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}}\hfill \\ & exp\left(2{\beth }_{11}{\epsilon }^{\delta \phi \mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}}+{\displaystyle \frac{2{\beth }_{21}}{\delta \left((\phi -1)\mathfrak{p}+1\right)}}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}}{\mathfrak{u}}^{\delta \phi \mathfrak{p}+1-\mathfrak{p}}\right).\hfill \end{array}$$

This implies that $\forall \eta \in \left[0,\mathcal{P}{\epsilon }^{-\lambda }\right]\subseteq [0,\Im ]$, there are $\mathcal{P}>0$ and $\lambda \in (0,\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}})$ as well.

$$\mathcal{E}\left[\underset{0\le \eta \le \mathcal{P}{\epsilon }^{-\lambda }}{sup}{∥{\varrho }_{\epsilon }\left(\eta \right)-{\varrho }_{\epsilon }^{*}\left(\eta \right)∥}^{\mathfrak{p}}\right]\le \mathcal{Z}{\epsilon }^{1-\lambda },$$

where $\mathcal{Z}={\beth }_{12}{\epsilon }^{-\delta \phi \mathfrak{p}\lambda +\delta \phi \mathfrak{p}-\left(\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}\right)+\lambda }{\mathcal{P}}^{\delta \phi \mathfrak{p}}+{\beth }_{22}{\epsilon }^{-\lambda (\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}})+(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}+\lambda -\left(\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}\right)}$${\mathcal{P}}^{\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}}exp\left(2{\beth }_{11}{\epsilon }^{-\delta \phi \mathfrak{p}\lambda +\delta \phi \mathfrak{p}}{\mathcal{P}}^{\delta \phi \mathfrak{p}}+2{\beth }_{21}{\left(\delta \left((\phi -1)\mathfrak{p}+1\right)\right)}^{-1}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}-\lambda (\delta \phi \mathfrak{p}+1-\mathfrak{p})}{\mathcal{P}}^{\delta \phi \mathfrak{p}+1-\mathfrak{p}}\right).$ So, we have the following:

$$\mathcal{E}\left[\underset{-\mathrm{\Phi }\le \eta \le \mathcal{P}{\epsilon }^{-\lambda }}{sup}{∥{\varrho }_{\epsilon }\left(\eta \right)-{\varrho }_{\epsilon }^{*}\left(\eta \right)∥}^{\mathfrak{p}}\right]\le \mu .$$

□

Corollary 1. 

Assuming that the $\left({\Im }_1\right)$ to $\left({\Im }_3\right)$ are met. When $\lambda \in (0,\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}})$, $\mathcal{P}>0$, and ${\epsilon }_1\in \left(0,{\epsilon }_0\right)$, taking into account ${\mu }_1>0$ when ε belong to $\left(0,{\epsilon }_1\right)$, we possess

$$\underset{\epsilon \to 0}{lim}\mathcal{P}\left(\underset{\eta \in [-\mathrm{\Phi },\mathcal{P}{\epsilon }^{-\lambda }]}{sup}\parallel {\varrho }_{\epsilon }\left(\eta \right)-{\varrho }_{\epsilon }^{*}\left(\eta \right)\parallel >{\mu }_1\right)=0.$$

Proof. 

Using C-M-I together with Theorem 2, the following is proven for ${\mu }_1$.

$$\begin{array}{cc}\hfill \mathcal{P}\left[\underset{\eta \in [-\mathrm{\Phi },\mathcal{P}{\epsilon }^{-\lambda }]}{sup}\parallel {\varrho }_{\epsilon }\left(\eta \right)-{\varrho }_{\epsilon }^{*}\left(\eta \right)\parallel >{\mu }_1\right]\le & {\displaystyle \frac{1}{{\mu }_1^2}}\mathcal{E}\left[\underset{\xi \in [-\mathrm{\Phi },\mathcal{P}{\epsilon }^{-\lambda }]}{sup}\parallel {\varrho }_{\epsilon }\left(\xi \right)-{\varrho }_{\epsilon }^{*}\left(\xi \right){\parallel }^2\right]\hfill \\ & \le {\displaystyle \frac{\mathcal{H}{\epsilon }^{1-\lambda }}{{\mu }_1^2}}\hfill \\ & \le 0\mathrm{as}\epsilon \to 0,\hfill \end{array}$$

where $\mathcal{H}={\beth }_{12}{\epsilon }^{-\delta \phi \mathfrak{p}\lambda +\delta \phi \mathfrak{p}-\left(\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}\right)+\lambda }{\mathcal{P}}^{\delta \phi \mathfrak{p}}+{\beth }_{22}{\epsilon }^{-\lambda (\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}})+(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}+\lambda -\left(\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}\right)}{\mathcal{P}}^{\delta \phi \mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{2}}}exp\left(2{\beth }_{11}{\epsilon }^{-\delta \phi \mathfrak{p}\lambda +\delta \phi \mathfrak{p}}{\mathcal{P}}^{\delta \phi \mathfrak{p}}+{\displaystyle \frac{2{\beth }_{21}}{\delta \left((\phi -1)\mathfrak{p}+1\right)}}{\epsilon }^{(\delta \phi -{\displaystyle \frac{1}{2}})\mathfrak{p}-\lambda (\delta \phi \mathfrak{p}+1-\mathfrak{p})}{\mathcal{P}}^{\delta \phi \mathfrak{p}+1-\mathfrak{p}}\right).$

The proof is now complete. □

4. Examples

Two examples are given in this section to illustrate our findings.

Example 1. 

We take into consideration the subsequent DFSDEs:

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }^{0.8,0.9}={\epsilon }^{\delta \phi }\left(3sin\left(\eta \right){cos}^2\left({\varrho }_{\epsilon }(\eta -\mathrm{\Phi })\right)sin\left({\varrho }_{\epsilon }\left(\eta \right)\right)\right)+{\epsilon }^{\delta \phi -{\displaystyle \frac{1}{2}}}sin\left(\eta \right){\varrho }_{\epsilon }(\eta -\mathrm{\Phi }){\displaystyle \frac{\mathrm{d}{\mathcal{W}}_{\eta }}{\mathrm{d}\eta }},\eta \in [0,\Im ],\hfill \end{array}$$

(33)

with $\varrho \left(\eta \right)=\gamma \left(\eta \right),\eta \in [-\mathrm{\Phi },0]$.

Equation (33) fulfills the criteria for the Ex-Un of the solution to the above system. Consequently, only one solution exists for the above problem. Considering the previously mentioned system, we obtain $\phi =0.8$, $\delta =0.9$, and the following:

$$\begin{array}{cc}{\mathcal{Z}}_1(\eta ,\varrho \left(\eta \right))\hfill & =3sin\left(\eta \right){cos}^2\left({\varrho }_{\epsilon }(\eta -\mathrm{\Phi })\right)sin\left({\varrho }_{\epsilon }\left(\eta \right)\right),\hfill \\ {\mathcal{Z}}_2(\eta ,\varrho \left(\eta \right))\hfill & =sin\left(\eta \right){\varrho }_{\epsilon }(\eta -\mathrm{\Phi }).\hfill \end{array}$$

The averages of ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ are shown by the following expressions:

$$\begin{array}{cc}\hfill {\tilde{\mathcal{Z}}}_1\left({\varrho }_{\epsilon }\left(\eta \right)\right)& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }\left(3sin\left(\eta \right){cos}^2\left({\varrho }_{\epsilon }(\eta -\mathrm{\Phi })\right)sin\left({\varrho }_{\epsilon }\left(\eta \right)\right)\right)\mathrm{d}\eta \hfill \\ \hfill & ={\displaystyle \frac{6}{\pi }}{cos}^2\left({\varrho }_{\epsilon }^{*}(\eta -\mathrm{\Phi })\right)sin\left({\varrho }_{\epsilon }^{*}\left(\eta \right)\right),\hfill \\ \hfill {\tilde{\mathcal{Z}}}_2\left({\varrho }_{\epsilon }\left(\eta \right)\right)& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }sin\left(\eta \right){\varrho }_{\epsilon }(\eta -\mathrm{\Phi })\mathrm{d}\eta \hfill \\ & ={\varrho }_{\epsilon }^{*}(\eta -\mathrm{\Phi }){\displaystyle \frac{2}{\pi }}.\hfill \end{array}$$

Replace ${\varrho }_{\epsilon }\left(\eta \right)$ with ${\varrho }_{\epsilon }^{*}\left(\eta \right)$ and ${\varrho }_{\epsilon }(\eta -\mathrm{\Phi })$ with ${\varrho }_{\epsilon }^{*}(\eta -\mathrm{\Phi })$ to obtain the average formulation associated with Equation (33). As a result, the simplified averaged equation can be written as follows:

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }^{0.8,0.9}{\varrho }_{\epsilon }^{*}\left(\eta \right)={\epsilon }^{\delta \phi }\left({\displaystyle \frac{6}{\pi }}{cos}^2\left({\varrho }_{\epsilon }^{*}(\eta -\mathrm{\Phi })\right)sin\left({\varrho }_{\epsilon }^{*}\left(\eta \right)\right)\right)\hfill \\ & +{\epsilon }^{\delta \phi -{\displaystyle \frac{1}{2}}}{\varrho }_{\epsilon }^{*}(\eta -\mathrm{\Phi }){\displaystyle \frac{2}{\pi }}\mathrm{d}\left(\eta \right),\eta \in [0,\Im ],\hfill \end{array}$$

(34)

when $\varrho \left(\eta \right)=\gamma \left(\eta \right),\eta \in [-\mathrm{\Phi },0]$.

Therefore, every need mentioned in Theorem 2 is met. Thus, ${\varrho }_{\epsilon }\left(\eta \right)$ and ${\varrho }_{\epsilon }^{*}\left(\eta \right)$ are identical in the pth moment in the limit as $\epsilon \to 0$. Figure 1 provides a numerical comparison between solution ${\varrho }_{\epsilon }^{*}\left(\eta \right)$ of the averaged Equation (34) and solution ${\varrho }_{\epsilon }\left(\eta \right)$ of the original Equation (33). It highlights a notable agreement between ${\varrho }_{\epsilon }^{*}\left(\eta \right)$ and ${\varrho }_{\epsilon }\left(\eta \right)$, validating the accuracy of our theoretical findings.

Example 2. 

Consider the subsequent DFSDEs:

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }^{0.9,0.7}{\varrho }_{\epsilon }\left(\eta \right)={\epsilon }^{\delta \phi }\left({\displaystyle \frac{1}{9}}{\varrho }_{\epsilon }(\eta -\mathrm{\Phi })sin\left({\varrho }_{\epsilon }\left(\eta \right)\right)cos\left({\varrho }_{\epsilon }\left(\eta \right)\right)sin\left({\varrho }_{\epsilon }(\eta -\mathrm{\Phi })\right)\right)\hfill \\ & +{\epsilon }^{\delta \phi -{\displaystyle \frac{1}{2}}}{\displaystyle \frac{1}{3}}cos\left({\varrho }_{\epsilon }(\eta -\mathrm{\Phi })\right)sin\left({\varrho }_{\epsilon }\left(\eta \right)\right){cos}^2\left(\eta \right){\varrho }_{\epsilon }\left(\eta \right){\displaystyle \frac{\mathrm{d}{\mathcal{W}}_{\eta }}{\mathrm{d}\eta }},\eta \in [0,\Im ],\hfill \end{array}$$

(35)

with $\varrho \left(\eta \right)=\gamma \left(\eta \right),\eta \in [-\mathrm{\Phi },0]$.

The criteria of the Ex-Un for the solution to Equation (35) as stated in Theorem 1 are clearly fulfilled. As a result, there is a unique solution to Equation (35). Furthermore, from Equation (35), we obtain the following: $\phi =0.9$, $\delta =0.7$, and

$$\begin{array}{cc}{\mathcal{Z}}_1(\eta ,\varrho \left(\eta \right))\hfill & ={\displaystyle \frac{1}{9}}{\varrho }_{\epsilon }(\eta -\mathrm{\Phi })sin\left({\varrho }_{\epsilon }\left(\eta \right)\right)cos\left({\varrho }_{\epsilon }\left(\eta \right)\right)sin\left({\varrho }_{\epsilon }(\eta -\mathrm{\Phi })\right),\hfill \\ {\mathcal{Z}}_2(\eta ,\varrho \left(\eta \right))\hfill & ={\displaystyle \frac{1}{3}}sin\left({\varrho }_{\epsilon }\left(\eta \right)\right){cos}^2\left(\eta \right){\varrho }_{\epsilon }\left(\eta \right)cos\left({\varrho }_{\epsilon }(\eta -\mathrm{\Phi })\right).\hfill \end{array}$$

The averages of ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ are presented by the subsequent expressions:

$$\begin{array}{cc}\hfill {\tilde{\mathcal{Z}}}_1\left({\varrho }_{\epsilon }\left(\eta \right)\right)& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }\left({\displaystyle \frac{1}{9}}{\varrho }_{\epsilon }(\eta -\mathrm{\Phi })sin\left({\varrho }_{\epsilon }\left(\eta \right)\right)cos\left({\varrho }_{\epsilon }\left(\eta \right)\right)sin\left({\varrho }_{\epsilon }(\eta -\mathrm{\Phi })\right)\right)\mathrm{d}\eta \hfill \\ \hfill & ={\displaystyle \frac{1}{9}}{\varrho }_{\epsilon }^{*}(\eta -\mathrm{\Phi })sin\left({\varrho }_{\epsilon }^{*}\left(\eta \right)\right)cos\left({\varrho }_{\epsilon }^{*}\left(\eta \right)\right)sin\left({\varrho }_{\epsilon }^{*}(\eta -\mathrm{\Phi })\right),\hfill \\ \hfill {\tilde{\mathcal{Z}}}_2\left({\varrho }_{\epsilon }\left(\eta \right)\right)& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }{\displaystyle \frac{1}{3}}sin\left({\varrho }_{\epsilon }\left(\eta \right)\right){cos}^2\left(\eta \right){\varrho }_{\epsilon }\left(\eta \right)cos\left({\varrho }_{\epsilon }(\eta -\mathrm{\Phi })\right)\mathrm{d}\eta \hfill \\ & ={\displaystyle \frac{1}{6}}{\varrho }_{\epsilon }^{*}\left(\eta \right)sin\left({\varrho }_{\epsilon }^{*}\left(\eta \right)\right)cos\left({\varrho }_{\epsilon }^{*}(\eta -\mathrm{\Phi })\right).\hfill \end{array}$$

Replace ${\varrho }_{\epsilon }\left(\eta \right)$ with ${\varrho }_{\epsilon }^{*}\left(\eta \right)$ and ${\varrho }_{\epsilon }(\eta -\mathrm{\Phi })$ with ${\varrho }_{\epsilon }^{*}(\eta -\mathrm{\Phi })$ to obtain the average formulation associated with Equation (35). As a result, the simplified averaged equation can be written as follows:

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{\eta }^{0.9,0.7}{\varrho }_{\epsilon }^{*}\left(\eta \right)={\epsilon }^{\delta \phi }\left({\displaystyle \frac{1}{9}}{\varrho }_{\epsilon }^{*}(\eta -\mathrm{\Phi })sin\left({\varrho }_{\epsilon }^{*}\left(\eta \right)\right)cos\left({\varrho }_{\epsilon }^{*}\left(\eta \right)\right)sin\left({\varrho }_{\epsilon }^{*}(\eta -\mathrm{\Phi })\right)\right)\hfill \\ & +{\epsilon }^{\delta \phi -{\displaystyle \frac{1}{2}}}{\displaystyle \frac{1}{6}}{\varrho }_{\epsilon }^{*}\left(\eta \right)sin\left({\varrho }_{\epsilon }^{*}\left(\eta \right)\right)cos\left({\varrho }_{\epsilon }^{*}(\eta -\mathrm{\Phi })\right)\mathrm{d}\eta ,\eta \in [0,\Im ],\hfill \end{array}$$

(36)

for $\varrho \left(\eta \right)=\gamma \left(\eta \right),\eta \in [-\mathrm{\Phi },0]$.

Therefore, every need mentioned in Theorem 2 is met. Thus, ${\varrho }_{\epsilon }\left(\eta \right)$ and ${\varrho }_{\epsilon }^{*}\left(\eta \right)$ are identical in the pth moment in the limit as $\epsilon \to 0$. Figure 2 offers a numerical comparison between solution ${\varrho }_{\epsilon }^{*}\left(\eta \right)$ of the averaged Equation (36) and solution ${\varrho }_{\epsilon }\left(\eta \right)$ of the original Equation (35). It illustrates a strong alignment between ${\varrho }_{\epsilon }^{*}\left(\eta \right)$ and ${\varrho }_{\epsilon }\left(\eta \right)$, supporting the accuracy of our theoretical findings.

5. Conclusions

In this research, we demonstrate the results of the Ex-Un using BFPT and Ave-P using inequality and interval translation techniques. We also provide examples that clarify our theoretical findings.

We establish the results of the Ex-Un and Ave-P in four significant ways: First, we develop the correct standard form for DFSDEs to build results for Ave-P. Second, we generalize the results for $\mathfrak{p}=2$ by establishing results in ${\mathrm{L}}^{\mathfrak{p}}$ space. Third, we establish results within the framework of the Cap-KGF derivative, which generalizes the results for Cap-FD and Had-FD. Fourth, we consider DFSDEs, which represent a more generalized class of FSDEs.

In the future, we will focus on developing a finance model in the form of DFSDEs.