Some Results for a Class of Pantograph Integro-Fractional Stochastic Differential Equations

Abstract

Symmetrical fractional differential equations have been explored through a variety of methods in recent years. In this paper, we analyze the existence and uniqueness of a class of pantograph integro-fractional stochastic differential equations (PIFSDEs) using the Banach fixed-point theorem (BFPT). Also, Gronwall inequality is used to demonstrate the Ulam–Hyers stability (UHS) of PIFSDEs. The results are illustrated by two examples.

1. Introduction

Fractional derivatives were first introduced by Leibnitz and L’Hôpital in 1695. Numerous scientists have investigated this theory; see [1]. This idea will be demonstrated by an overview of the work carried out in this field. Abel’s work in fractional calculation fields, Euler’s studies from 1730, Lagrange’s applications from 1772, and Laplace’s idea of the fractal derivative from 1812 are all important. For additional information about the extensions made to fractional calculus by Atangana, Baleanu, and other scientists, see [2,3,4].

In recent decades, fractional calculus has grown in popularity. Its significance is shown in numerous domains, including biology, physics, economics, and automatic control. Fictitious calculus can provide more flexibility and precision when studying a variety of phenomena because it can differentiate and integrate non-integer orders; see [5,6,7,8]. For instance, Laskin in [5] introduced a new fractional Langevin-type stochastic differential equation. Koeller in [6] presented the applications of fractional calculus to the theory of viscoelasticity. Petras et al. in [7] studied the numerical solution of a two-compartmental pharmacokinetic model for oral drug administration. Yang et al. in [8] presented a fractional damped vibration problem. The methods and ideas for using fractional derivatives to solve symmetrical differential equations are covered in this area.

Fractional stochastic differential equations (FSDEs) have been the subject of an increasing amount of research in recent years [9,10,11]. These equations combine fractional calculus with stochastic processes to yield an impotent framework for modeling and evaluating complicated systems determined by memory and randomness. The study of FSDEs is very beneficial to many domains, including engineering and physics [6,12], providing valuable insight into the dynamics of complex systems.

The existence, uniqueness, and stability of pantograph stochastic differential equations are important for a variety of reasons. First, they provide a robust theoretical framework, assuring that solutions are well defined and predictable under specified initial conditions. This is critical for accurately simulating real-world processes in domains such as biology, economics, and engineering, where these equations are frequently used. Furthermore, stability analysis allows us to understand how solutions react to disturbances, which is critical for applications in control systems and other fields. Finally, these studies provide useful insights that might improve both theoretical knowledge and practical applications across a variety of disciplines.

UHS, or Ulam stability, is vital for various equations [13,14,15] as it offers analytical approximate solutions for many problems where exact solutions are inaccessible. It is important to note that stability is crucial; if a system is stable in the UHS sense, essential properties will hold near the exact solution. This is evident in fields such as optimization, economics, and biology. UHS emerged after Ulam’s notable talk at a conference in 1940 [16]. The UHS idea has been extensively studied [17,18,19]. For instance, Ben Makhlouf et al. in [17] proved the UHS of pantograph FSDEs by using generalized Gronwall inequality. Rhaima et al. in [18] investigated UHS for a class of integro-FSDEs. The authors in [19] proved UHS for a class of neutral FSDEs.

Motivated by the works [17,18], this paper presents some results for a class of PIFSDEs. The main findings of the article are as follows: (1) it proves the existence and uniqueness of the solution of PIFSDEs; (2) it studies the Ulam stability of PIFSDEs.

2. Notational Preliminaries

Let $\{\mathcal{F},\mathcal{E},{\mathfrak{F}}_T,\mathbb{P}\}$ be a complete probability space, with ${\mathfrak{F}}_T={\left\{{\mathfrak{F}}_s\right\}}_{s\in [1,T]}$, $T>1$ and $W\left(s\right)$ represents a standard Brownian motion. Let ${\mathcal{X}}_s=L^2(\mathsf{\Omega },{\mathfrak{F}}_s,\mathbb{P})$ be the space of all mean square integrable and ${\mathfrak{F}}_s$-measurable functions $h:\mathcal{F}\to \mathbb{R}$. Let

$${∥h∥}_{ms}=\sqrt{\mathbb{E}{\left|h\right|}^2}.$$

Definition 1

([1]). The Riemann–Liouville fractional integral of order $a>0$ for $X\in L^1([1,b],\mathbb{R})$ is given by

$${}^{RL}I^{\upsilon }X\left(u\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(a\right)}}{\int }_1^u{\left(u-r\right)}^{a-1}X\left(r\right)dr.$$

Definition 2

([1]). The Hadamard fractional integral of order $a>0$ for $X\in L^1([1,b],\mathbb{R})$ is given by

$${}^{H}I^{a}X\left(u\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(a\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{r}}\right)}^{a-1}{\displaystyle \frac{X\left(r\right)}{r}}dr.$$

Definition 3

([1]). Let $X:[1,b]\to \mathbb{R}$. The Caputo–Hadamard fractional derivative of order $a\in (0,1)$ of X is given by

$${}^{H_C}D_1^{a}G\left(u\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-a)}}\left(u{\displaystyle \frac{d}{du}}\right){\int }_1^u{\left(log{\displaystyle \frac{u}{r}}\right)}^{-a}{\displaystyle \frac{\left(G\left(r\right)-G\left(1\right)\right)}{r}}dr.$$

Definition 4

([1]). The Mittag-Leffler function with one-parameter is given by

$${\displaystyle E_a\left(x\right)=\sum _{m=0}^{\infty }\frac{x^m}{\mathsf{\Gamma }(ma+1)},}$$

where $a>0$ and $x\in \mathbb{C}$.

Consider the following PIFSDE:

$${}^{H_C}D_1^{\mathfrak{a}}\left(\beta \left(s\right)-{\sum _{i=1}^m}^{RL}{I}^{{\mathfrak{b}}_i}{G}_i(s,\beta \left(s\right),\beta \left(\theta s\right))\right)={\zeta }_1(s,\beta \left(s\right),\beta \left(\theta s\right))+{\zeta }_2(s,\beta \left(s\right),\beta \left(\theta s\right)){\displaystyle \frac{dW\left(s\right)}{ds}},$$

(1)

$s\in [1,T],$ where $\theta \in (0,1)$, ${\displaystyle \frac{1}{2}}<\mathfrak{a}<1$, ${\displaystyle \frac{1}{2}}<{\mathfrak{b}}_{\kappa }<1$, $\kappa =1,\cdots ,m$, $\beta \left(s\right)=\chi \left(s\right)$ for $s\in [\theta ,1]$ and $G_{\kappa },{\zeta }_j:[1,T]\times \mathbb{R}\to \mathbb{R}$, $\kappa =1,\cdots ,m$, $j=1,2$ are measurable.

Consider the hypothesis:

$\left({\mathcal{H}}_1\right)$

There is $L>0$ such that

$$\sum _{\kappa =1}^m|G_{\kappa }(s,{\beta }_1,{\tilde{\beta }}_1)-G_{\kappa }(s,{\beta }_2,{\tilde{\beta }}_2)|+\sum _{\kappa =1}^2|{\zeta }_{\kappa }(s,{\beta }_1,{\tilde{\beta }}_1)-{\zeta }_{\kappa }(s,{\beta }_2,{\tilde{\beta }}_2)|\le L\left(|{\beta }_1-{\beta }_2|+|{\tilde{\beta }}_1-{\tilde{\beta }}_2|\right),$$

for every $(s,{\beta }_1,{\beta }_2)\in [1,T]\times \mathbb{R}\times \mathbb{R}$.

$\left({\mathcal{H}}_2\right)$

The functions ${\zeta }_2(·,0,0)$, ${\zeta }_1(·,0,0)$ and $G_i$, $i=1,\cdots ,m$ satisfy the following conditions:

$$||{\zeta }_2{(·,0,0)||}_{\infty }=ess\underset{s\in [1,T]}{sup}|{\zeta }_2(s,0,0)|<\infty ,{\int }_1^T|G_i{(s,0,0)|}^{2}ds<\infty ,{\int }_1^T{|{\zeta }_1(s,0,0)|}^{2}ds<\infty .$$

3. Existence and Uniqueness Results

Consider the Banach space ${\mathbb{D}}^2\left([\theta ,T]\right)$ of all processes $\beta$ that are ${\mathfrak{F}}_T$-adapted and measurable with

$${∥\beta ∥}_{{\mathbb{D}}^2\left([\theta ,T]\right)}=\underset{\theta \le l\le T}{sup}{∥\beta \left(l\right)∥}_{ms}<\infty ,$$

Consider the operator ${\mathcal{P}}_{\chi }:{\mathbb{D}}^2\left([\theta ,T]\right)\to {\mathbb{D}}^2\left([\theta ,T]\right)$ defined as follows:

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{\chi }\beta \left(s\right)& =& \chi \left(1\right)+\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{b}}_j\right)}}{\int }_1^s{(s-l)}^{{\mathfrak{b}}_j-1}{G}_j(l,\beta \left(l\right),\beta \left(\theta l\right))dl\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathfrak{a}\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{\mathfrak{a}-1}}{l}}{\zeta }_1(l,\beta \left(l\right),\beta \left(\theta l\right))dl\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathfrak{a}\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{\mathfrak{a}-1}}{l}}{\zeta }_2(l,\beta \left(l\right),\beta \left(\theta l\right))dW\left(l\right),s\in [1,T]\hfill \end{array}$$

and

$${\mathcal{P}}_{\chi }\beta \left(s\right)=\chi \left(s\right),s\in [\theta ,1].$$

Lemma 1.

∀$\chi \in {\mathbb{D}}^2\left([\theta ,1]\right)$, the operator ${\mathcal{P}}_{\chi }$ is well defined.

Proof. 

Given $\beta \in {\mathbb{D}}^2\left([\theta ,T]\right)$, then

$$\begin{array}{ccc}\hfill {∥{\mathcal{P}}_{\chi }\beta \left(s\right)∥}_{ms}^2& =& \mathbb{E}{\left|{\mathcal{P}}_{\chi }\beta \left(s\right)\right|}^2\hfill \\ & \le & (m+3)\left({∥\chi \left(1\right)∥}_{ms}^2+\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }{\left({\mathfrak{b}}_j\right)}^2}}\mathbb{E}\left({\left|{\int }_1^s{(s-l)}^{{\mathfrak{b}}_j-1}{G}_i(l,\beta \left(l\right),\beta \left(\theta l\right))dl\right|}^2\right)\right.\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}\mathbb{E}\left({\left|{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{\mathfrak{a}-1}}{l}}{\zeta }_1(l,\beta \left(l\right),\beta \left(\theta l\right))dl\right|}^2\right)\hfill \\ & & \left.+{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}\mathbb{E}\left({\left|{\int }_1^s{\left(log{\displaystyle \frac{s}{l}}\right)}^{\mathfrak{a}-1}{\displaystyle \frac{{\zeta }_2(l,\beta \left(l\right),\beta \left(\theta l\right))}{l}}dW\left(l\right)\right|}^2\right)\right)\hfill \end{array}$$

It follows from the Cauchy–Schwarz inequality that

$$\begin{array}{ccc}\hfill \mathbb{E}\left({\left|{\int }_1^s{(s-l)}^{{\mathfrak{b}}_i-1}{G}_j(l,\beta \left(l\right),\beta \left(\theta l\right))dl\right|}^2\right)& \le & \left({\int }_1^s{(s-l)}^{2{\mathfrak{b}}_j-2}dl\right)\mathbb{E}\left({\int }_1^s{\left|G_j(l,\beta \left(l\right),\beta \left(\theta l\right))\right|}^{2}dl\right)\hfill \\ & \le & {\displaystyle \frac{1}{2{\mathfrak{b}}_j-1}}{(s-1)}^{2{\mathfrak{b}}_j-1}\mathbb{E}\left({\int }_1^s{\left|G_j(l,\beta \left(l\right),\beta \left(\theta l\right))\right|}^{2}dl\right).\hfill \end{array}$$

It follows from $\left({\mathcal{H}}_1\right)$ that

$${\left|G_j(l,\beta \left(l\right),\beta \left(\theta l\right))\right|}^2\le 2L^2{\left(\left|\beta \left(l\right)\right|+\left|\beta \left(\theta l\right)\right|\right)}^2+2{\left|G_j(l,0,0)\right|}^2.$$

Therefore,

$$\mathbb{E}\left({\int }_1^s{\left|G_j(l,\beta \left(l\right),\beta \left(\theta l\right))\right|}^{2}dl\right)\le 8(T-1)L^2\underset{s\in [\theta ,T]}{sup}\mathbb{E}{\left|\beta \left(s\right)\right|}^2+2{\int }_1^s{\left|G_j(l,0,0)\right|}^{2}dl.$$

Thus,

$$\mathbb{E}\left({\left|{\int }_1^s{(s-l)}^{{\mathfrak{b}}_j-1}{G}_j(l,\beta \left(l\right),\beta \left(\theta l\right))dl\right|}^2\right)$$

$$\le {\displaystyle \frac{{(T-1)}^{2{\mathfrak{b}}_j-1}}{2{\mathfrak{b}}_j-1}}\left(8(T-1)L^2\underset{s\in [\theta ,T]}{sup}\mathbb{E}{\left|\beta \left(s\right)\right|}^2+2{\int }_1^s{\left|G_j(l,0,0)\right|}^{2}dl\right).$$

It follows from the Cauchy–Schwarz inequality that

$$\begin{array}{ccc}\hfill \mathbb{E}\left({\left|{\int }_1^s{\displaystyle \frac{{(log\frac{s}{l})}^{\mathfrak{a}-1}}{l}}{\zeta }_1(l,\beta \left(l\right),\beta \left(\theta l\right))dl\right|}^2\right)& \le & \left({\int }_1^s{\displaystyle \frac{{(log\frac{s}{l})}^{2\mathfrak{a}-2}}{l^2}}dl\right)\mathbb{E}\left({\int }_1^s{\left|{\zeta }_1(l,\beta \left(l\right),\beta \left(\theta l\right))\right|}^{2}dl\right)\hfill \\ & \le & {\displaystyle \frac{1}{2\mathfrak{a}-1}}{(logT)}^{2\mathfrak{a}-1}\mathbb{E}\left({\int }_1^s{\left|{\zeta }_1(l,\beta \left(l\right),\beta \left(\theta l\right))\right|}^{2}dl\right).\hfill \end{array}$$

Using $\left({\mathcal{H}}_1\right)$, we obtain:

$${\left|{\zeta }_1(l,\beta \left(l\right),\beta \left(\theta l\right))\right|}^2\le 2L^2{\left(\left|\beta \left(l\right)\right|+\left|\beta \left(\theta l\right)\right|\right)}^2+2{\left|{\zeta }_1(l,0,0)\right|}^2.$$

Therefore,

$$\mathbb{E}\left({\left|{\int }_1^s{\displaystyle \frac{{(log\frac{s}{l})}^{\mathfrak{a}-1}}{l}}{\zeta }_1(l,\beta \left(l\right),\beta \left(\theta l\right))dl\right|}^2\right)$$

$$\le {\displaystyle \frac{{(logT)}^{2\mathfrak{a}-1}}{2\mathfrak{a}-1}}\left(8L^2(T-1)\underset{u\in [\theta ,T]}{sup}\mathbb{E}{\left|\beta \left(u\right)\right|}^2+2{\int }_1^s{\left|{\zeta }_1(l,0,0)\right|}^{2}dl\right).$$

By applying the Itô isometry, we obtain

$$\mathbb{E}\left({\left|{\int }_1^s{\left(log{\displaystyle \frac{s}{l}}\right)}^{\mathfrak{a}-1}{\displaystyle \frac{{\zeta }_2(l,\beta \left(l\right),\beta \left(\theta l\right))}{l}}dW\left(l\right)\right|}^2\right)=\mathbb{E}\left({\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{2\mathfrak{a}-2}}{l^2}}{\left|{\zeta }_2(l,\beta \left(l\right),\beta \left(\theta l\right))\right|}^{2}dl\right).$$

It follows from Assumption $\left({\mathcal{H}}_1\right)$ that

$${\left|{\zeta }_2(l,\beta \left(l\right),\beta \left(\theta l\right))\right|}^2\le 2L^2{\left(\left|\beta \left(l\right)\right|+\left|\beta \left(\theta l\right)\right|\right)}^2+2{∥{\zeta }_2(l,0,0)∥}_{\infty }^2.$$

Then,

$$\begin{array}{ccc}& & \mathbb{E}\left({\left|{\int }_1^s{\left(log{\displaystyle \frac{s}{l}}\right)}^{\mathfrak{a}-1}{\displaystyle \frac{{\zeta }_2(l,\beta \left(l\right),\beta \left(\theta l\right))}{l}}dW\left(l\right)\right|}^2\right)\hfill \\ & \le & 2L^2\mathbb{E}\left({\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{2\mathfrak{a}-2}}{l^2}}{\left(\left|\beta \left(l\right)\right|+\left|\beta \left(\theta l\right)\right|\right)}^{2}dl\right)+2{∥{\zeta }_2(l,0,0)∥}_{\infty }^2{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{2\mathfrak{a}-2}}{l^2}}dl\hfill \\ & \le & {\displaystyle \frac{8L^2}{2\mathfrak{a}-1}}{\left(logT\right)}^{2\mathfrak{a}-1}{∥\beta ∥}_{{\mathbb{D}}^2\left([\theta ,T]\right)}+{\displaystyle \frac{2}{2\mathfrak{a}-1}}{\left(logT\right)}^{2\mathfrak{a}-1}{∥{\zeta }_2(l,0,0)∥}_{\infty }^2.\hfill \end{array}$$

Therefore ${\mathcal{P}}_{\chi }$ is well defined. □

Theorem 1.

If $\left({\mathcal{H}}_1\right)$ and $\left({\mathcal{H}}_2\right)$ are satisfied, then Equation (1) possesses a unique solution.

Proof. 

Let us consider the norm $∥·∥$ on ${\mathbb{D}}^2\left([\theta ,T]\right)$ by:

$${∥\beta ∥}_{\{\lambda ,{\lambda }_1,\cdots ,{\lambda }_m\}}=\underset{t\in [\theta ,T]}{sup}\sqrt{{\displaystyle \frac{\mathbb{E}\left({\left|\beta \left(t\right)\right|}^2\right)}{h\left(t\right)}}},\forall \beta \in {\mathbb{D}}^2\left([\theta ,T]\right),$$

where

$$h\left(t\right)=t^{\lambda }\prod _{i=1}^m{E}_{2{\mathfrak{b}}_i-1}\left({\lambda }_i{(t-1)}^{2{\mathfrak{b}}_i-1}\right)\mathrm{for}t\in [1,T],$$

with $\lambda ,{\lambda }_1,\cdots ,{\lambda }_m>0$ and

$$(4m+8)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\mathfrak{b}}_i\right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2{\mathfrak{b}}_i-1)}{{\lambda }_i}}+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2\mathfrak{a}-1)}{{\lambda }^{2\mathfrak{a}-1}}}\right)<1$$

and

$$h\left(t\right)=1\mathrm{for}t\in [\theta ,1].$$

We know that $\left({\mathbb{D}}^2\left([\theta ,T]\right),{∥·∥}_{\{\lambda ,{\lambda }_1,\cdots ,{\lambda }_m\}}\right)$ is a Banach space because ${∥·∥}_{{\mathbb{D}}^2\left([\theta ,T]\right)}$ and ${∥·∥}_{\{\lambda ,{\lambda }_1,\cdots ,{\lambda }_m\}}$ are equivalent.

Let ${\beta }_1,{\beta }_2\in {\mathbb{D}}^2\left([\theta ,T]\right)$.

For $t\in [\theta ,1]$, we have ${\mathcal{P}}_{\chi }{\beta }_1\left(t\right)-{\mathcal{P}}_{\chi }{\beta }_2\left(t\right)=0$.

For $t\in [1,T]$, we obtain

$$\begin{array}{ccc}& & \mathbb{E}\left({\left|{\mathcal{P}}_{\chi }{\beta }_1\left(t\right)-{\mathcal{P}}_{\chi }{\beta }_2\left(t\right)\right|}^2\right)\hfill \\ & & \le (m+2)\left(\sum _{i=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }{\left({\mathfrak{b}}_i\right)}^2}}\mathbb{E}\left({\left|{\int }_1^t{(t-s)}^{{\mathfrak{b}}_i-1}[G_i(s,{\beta }_1\left(s\right),{\beta }_1\left(\theta s\right))-G_i(s,{\beta }_2\left(s\right),{\beta }_2\left(\theta s\right))]ds\right|}^2\right)\right.\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}\mathbb{E}\left({\left|{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{s}\right)}^{\mathfrak{a}-1}}{s}}\left({\zeta }_1(s,{\beta }_1\left(s\right),{\beta }_1\left(\theta s\right))-{\zeta }_1(s,{\beta }_2\left(s\right),{\beta }_2\left(\theta s\right))\right)ds\right|}^2\right)\hfill \\ & & +\left.{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}\mathbb{E}\left({\left|{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{s}\right)}^{\mathfrak{a}-1}}{s}}\left({\zeta }_2(s,{\beta }_1\left(s\right),{\beta }_1\left(\theta s\right))-{\zeta }_2(s,{\beta }_2\left(s\right),{\beta }_2\left(\theta s\right))\right)dW\left(s\right)\right|}^2\right)\right).\hfill \end{array}$$

Using the Cauchy–Schwartz inequality, we obtain

$$\begin{array}{ccc}& & \mathbb{E}\left(|{\int }_1^t{(t-s)}^{{\mathfrak{b}}_i-1}[G_i(s,{\beta }_1\left(s\right),{\beta }_1\left(\theta s\right))-G_i(s,{\beta }_2\left(s\right),{\beta }_2\left(\theta s\right))]ds{|}^2\right)\hfill \\ & & \le 2L^2(T-1){\int }_1^t{(t-s)}^{2{\mathfrak{b}}_i-2}\mathbb{E}\left(\right|{\beta }_1\left(s\right)-{\beta }_2{\left(s\right)|}^2+|{\beta }_1\left(\theta s\right)-{\beta }_2\left(\theta s\right){|}^2)ds\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \mathbb{E}\left({\left|{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{s}\right)}^{\mathfrak{a}-1}}{s}}\left({\zeta }_1(s,{\beta }_1\left(s\right),{\beta }_1\left(\theta s\right))-{\zeta }_1(s,{\beta }_2\left(s\right),{\beta }_2\left(\theta s\right))\right)ds\right|}^2\right)\hfill \\ & & \le 2L^2(T-1){\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{s}\right)}^{2\mathfrak{a}-2}}{s^2}}\mathbb{E}(|{\beta }_1\left(s\right)-{\beta }_2{\left(s\right)|}^2+|{\beta }_1\left(\theta s\right)-{\beta }_2\left(\theta s\right){|}^2)ds.\hfill \end{array}$$

It follows from the isometry of Itô that

$$\begin{array}{ccc}& & \mathbb{E}\left({\left|{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{s}\right)}^{\mathfrak{a}-1}}{s}}\left({\zeta }_2(s,{\beta }_1\left(s\right),{\beta }_1\left(\theta s\right))-{\zeta }_2(s,{\beta }_2\left(s\right),{\beta }_2\left(\theta s\right))\right)dW\left(s\right)\right|}^2\right)\hfill \\ & & =\mathbb{E}\left({\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{s}\right)}^{2\mathfrak{a}-2}}{s^2}}{\left|{\zeta }_2(s,{\beta }_1\left(s\right),{\beta }_1\left(\theta s\right))-{\zeta }_2(s,{\beta }_2\left(s\right),{\beta }_2\left(\theta s\right))\right|}^{2}ds\right)\hfill \\ & & \le 2L^2{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{s}\right)}^{2\mathfrak{a}-2}}{s^2}}\mathbb{E}(|{\beta }_1\left(s\right)-{\beta }_2{\left(s\right)|}^2+|{\beta }_1\left(\theta s\right)-{\beta }_2\left(\theta s\right){|}^2)ds.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}& & \mathbb{E}\left({\left|{\mathcal{P}}_{\chi }{\beta }_1\left(t\right)-{\mathcal{P}}_{\chi }{\beta }_2\left(t\right)\right|}^2\right)\hfill \\ & & \le (m+2)\left(\sum _{i=1}^m{\displaystyle \frac{2L^2(T-1)}{\mathsf{\Gamma }{\left({\mathfrak{b}}_i\right)}^2}}{\int }_1^t{(t-s)}^{2{\mathfrak{b}}_i-2}\mathbb{E}(|{\beta }_1\left(s\right)-{\beta }_2{\left(s\right)|}^2+|{\beta }_1\left(\theta s\right)-{\beta }_2\left(\theta s\right){|}^2)ds\right.\hfill \\ & & +{\displaystyle \frac{2L^2(T-1)}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{s}\right)}^{2\mathfrak{a}-2}}{s^2}}\mathbb{E}(|{\beta }_1\left(s\right)-{\beta }_2{\left(s\right)|}^2+|{\beta }_1\left(\theta s\right)-{\beta }_2\left(\theta s\right){|}^2)ds\hfill \\ & & \left.+{\displaystyle \frac{2L^2}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{s}\right)}^{2\mathfrak{a}-2}}{s^2}}\mathbb{E}(|{\beta }_1\left(s\right)-{\beta }_2{\left(s\right)|}^2+|{\beta }_1\left(\theta s\right)-{\beta }_2\left(\theta s\right){|}^2)ds\right)\hfill \\ & & \le (2m+4)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\mathfrak{b}}_i\right)}^2}}{\int }_1^t{(t-s)}^{2{\mathfrak{b}}_i-2}\mathbb{E}(|{\beta }_1\left(s\right)-{\beta }_2{\left(s\right)|}^2+|{\beta }_1\left(\theta s\right)-{\beta }_2\left(\theta s\right){|}^2)ds\right.\hfill \\ & & \left.+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{s}\right)}^{2\mathfrak{a}-2}}{s^2}}\mathbb{E}(|{\beta }_1\left(s\right)-{\beta }_2{\left(s\right)|}^2+|{\beta }_1\left(\theta s\right)-{\beta }_2\left(\theta s\right){|}^2)ds\right)\hfill \\ & & \le (2m+4)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\mathfrak{b}}_i\right)}^2}}{\int }_1^t{(t-s)}^{2{\mathfrak{b}}_i-2}[{\displaystyle \frac{h\left(s\right)\mathbb{E}(|{\beta }_1\left(s\right)-{\beta }_2\left(s\right){|}^2)}{h\left(s\right)}}\right.\hfill \\ & & +{\displaystyle \frac{h\left(\theta s\right)\mathbb{E}(|{\beta }_1\left(\theta s\right)-{\beta }_2\left(\theta s\right){|}^2)}{h\left(\theta s\right)}}]ds\hfill \\ & & \left.+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{s}\right)}^{2\mathfrak{a}-2}}{s^2}}\left[{\displaystyle \frac{h\left(s\right)\mathbb{E}(|{\beta }_1\left(s\right)-{\beta }_2\left(s\right){|}^2)}{h\left(s\right)}}+{\displaystyle \frac{h\left(\theta s\right)\mathbb{E}(|{\beta }_1\left(\theta s\right)-{\beta }_2\left(\theta s\right){|}^2)}{h\left(\theta s\right)}}\right]ds\right)\hfill \\ & & \le (4m+8)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\mathfrak{b}}_i\right)}^2}}{∥{\beta }_1-{\beta }_2∥}_{\lambda ,{\lambda }_1,\cdots ,{\lambda }_m}^2{\int }_1^t{(t-s)}^{2{\mathfrak{b}}_i-2}h\left(s\right)ds\right.\hfill \\ & & \left.+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}{∥{\beta }_1-{\beta }_2∥}_{\lambda ,{\lambda }_1,\cdots ,{\lambda }_m}^2{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{s}\right)}^{2\mathfrak{a}-2}}{s^2}}h\left(s\right)ds\right).\hfill \end{array}$$

Similar to the proof of Theorem 3.2 in [18], we obtain

$$\begin{array}{ccc}& & \mathbb{E}\left({\left|{\mathcal{P}}_{\chi }{\beta }_1\left(t\right)-{\mathcal{P}}_{\chi }{\beta }_2\left(t\right)\right|}^2\right)\hfill \\ & & \le (4m+8)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\mathfrak{b}}_i\right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2{\mathfrak{b}}_i-1)}{{\lambda }_i}}+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2\mathfrak{a}-1)}{{\lambda }^{2\mathfrak{a}-1}}}\right)h\left(t\right){∥{\beta }_1-{\beta }_2∥}_{\lambda ,{\lambda }_1,\cdots ,{\lambda }_m}^2.\hfill \end{array}$$

Consequently, we have

$${∥{\mathcal{P}}_{\chi }{\beta }_1-{\mathcal{P}}_{\chi }{\beta }_2∥}_{\lambda ,{\lambda }_1,\cdots ,{\lambda }_m}\le K{∥{\beta }_1-{\beta }_2∥}_{\lambda ,{\lambda }_1,\cdots ,{\lambda }_m}$$

where

$$K^2=(4m+8)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\mathfrak{b}}_i\right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2{\mathfrak{b}}_i-1)}{{\lambda }_i}}+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2\mathfrak{a}-1)}{{\lambda }^{2\mathfrak{a}-1}}}\right).$$

Consequently, (1) has a unique solution $\beta$ with $\beta \left(t\right)=\chi \left(t\right)$ for $t\in [\theta ,1]$. □

4. Stability Results

Firstly, we present a relevant definition before discussing our results.

Definition 5

([18]). Equation (1) is UHS with respect to ϖ if $\tilde{K}>0$ exists so that ∀$\varpi >0$ and $y\in {\mathbb{D}}^2\left([\theta ,T]\right)$, with $y\left(s\right)=\xi \left(s\right)$ for $s\in [\theta ,1]$ which satisfies

$$\mathbb{E}{\left|y\left(s\right)-{\mathcal{P}}_{\xi }y\left(s\right)\right|}^2\le \varpi ;\forall s\in [1,T],$$

(2)

there exists a solution $\beta \in {\mathbb{D}}^2\left([\theta ,T]\right)$ of (1), with $\beta \left(s\right)=\xi \left(s\right)$ for $s\in [\theta ,1]$ which satisfies

$$\mathbb{E}{\left|y\left(s\right)-\beta \left(s\right)\right|}^2\le \tilde{K}\varpi ,\forall s\in [1,T].$$

Theorem 2.

If Assumptions $\left({\mathcal{H}}_1\right)$ and $\left({\mathcal{H}}_2\right)$ are satisfied, then Equation (1) is UHS with respect to ϖ.

Proof. 

Consider $\varpi >0$. Suppose $y\left(s\right)$ is a solution of (2). Let $\beta \left(s\right)$ be the solution of (1) so that $\beta \left(s\right)=\xi \left(s\right)$ when $s\in [\theta ,1]$. Therefore,

$$\begin{array}{ccc}\hfill \beta \left(s\right)& =& \xi \left(1\right)+\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{b}}_j\right)}}{\int }_1^s{(s-l)}^{{\mathfrak{b}}_j-1}{G}_j(l,\beta \left(l\right),\beta \left(\theta l\right))dl\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathfrak{a}\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{\mathfrak{a}-1}}{l}}{\zeta }_1(l,\beta \left(l\right),\beta \left(\theta l\right))dl\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathfrak{a}\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{\mathfrak{a}-1}}{l}}{\zeta }_2(l,\beta \left(l\right),\beta \left(\theta l\right))dW\left(l\right).\hfill \end{array}$$

Hence, we have

$$\begin{array}{ccc}\hfill y\left(s\right)-\beta \left(s\right)& =& (y\left(s\right)-\xi \left(1\right)-\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{b}}_j\right)}}{\int }_1^s{(s-l)}^{{\mathfrak{b}}_j-1}{G}_j(l,y\left(l\right),y\left(\theta l\right))dl\hfill \\ & & -{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathfrak{a}\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{\mathfrak{a}-1}}{l}}{\zeta }_1(l,y\left(l\right),y\left(\theta l\right))dl\hfill \\ & & -{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathfrak{a}\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{\mathfrak{a}-1}}{l}}{\zeta }_2(l,y\left(l\right),y\left(\theta l\right))dW\left(l\right))\hfill \\ & & +\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{b}}_j\right)}}{\int }_1^s{(s-l)}^{{\mathfrak{b}}_j-1}\left(G_j(l,y\left(l\right),y\left(\theta l\right))-G_j(l,\beta \left(l\right),\beta \left(\theta l\right))\right)dl\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathfrak{a}\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{\mathfrak{a}-1}}{l}}\left({\zeta }_1(l,y\left(l\right),y\left(\theta l\right))-{\zeta }_1(l,\beta \left(l\right),\beta \left(\theta l\right))\right)dl\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathfrak{a}\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{\mathfrak{a}-1}}{l}}\left({\zeta }_2(l,y\left(l\right),y\left(\theta l\right))-{\zeta }_2(l,\beta \left(l\right),\beta \left(\theta l\right))\right)dW\left(l\right).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}& & \mathbb{E}{\left|y\left(s\right)-\beta \left(s\right)\right|}^2\hfill \\ & & \le (m+3)\left(\varpi +\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }{\left({\mathfrak{b}}_j\right)}^2}}\mathbb{E}{\left|{\int }_1^s{(s-l)}^{{\mathfrak{b}}_j-1}[G_j(l,y\left(l\right),y\left(\theta l\right))-G_j(l,\beta \left(l\right),\beta \left(\theta l\right))]dl\right|}^2\right.\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}\mathbb{E}{\left|{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{\mathfrak{a}-1}}{l}}[{\zeta }_1(l,y\left(l\right),y\left(\theta l\right))-{\zeta }_1(l,\beta \left(l\right),\beta \left(\theta l\right))]dl\right|}^2\hfill \\ & & \left.+{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}\mathbb{E}{\left|{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{\mathfrak{a}-1}}{l}}[{\zeta }_2(l,y\left(l\right),y\left(\theta l\right))-{\zeta }_2(l,\beta \left(l\right),\beta \left(\theta l\right))]dW\left(l\right)\right|}^2\right).\hfill \end{array}$$

It follows from the Cauchy–Schwarz inequality and the isometry of Itô that

$$\begin{array}{ccc}& & \mathbb{E}{\left|y\left(s\right)-\beta \left(s\right)\right|}^2\hfill \\ & & \le (m+3)\left(\varpi +\sum _{j=1}^{m}2{\displaystyle \frac{{(T-1)}^{2{\mathfrak{b}}_j-1}{L}^2}{(2{\mathfrak{b}}_j-1)\mathsf{\Gamma }{\left({\mathfrak{b}}_j\right)}^2}}{\int }_1^s\left(\mathbb{E}{\left|y\left(l\right)-\beta \left(l\right)\right|}^2+\mathbb{E}{\left|y\left(\theta l\right)-\beta \left(\theta l\right)\right|}^2\right)dl\right.\hfill \\ & & +2{\displaystyle \frac{{\left(logT\right)}^{2\mathfrak{a}-1}{L}^2}{(2\mathfrak{a}-1)\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}{\int }_1^s\left(\mathbb{E}{\left|y\left(l\right)-\beta \left(l\right)\right|}^2+\mathbb{E}{\left|y\left(\theta l\right)-\beta \left(\theta l\right)\right|}^2\right)dl\hfill \\ & & \left.+2{\displaystyle \frac{L^2}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{2\mathfrak{a}-2}}{l^2}}\left(\mathbb{E}{\left|y\left(l\right)-\beta \left(l\right)\right|}^2+\mathbb{E}{\left|y\left(\theta l\right)-\beta \left(\theta l\right)\right|}^2\right)dl\right).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill \mathbb{E}{\left|y\left(s\right)-\beta \left(s\right)\right|}^2& \le & q_1\varpi +q_2{\int }_1^s\left(\mathbb{E}{\left|y\left(l\right)-\beta \left(l\right)\right|}^2+\mathbb{E}{\left|y\left(\theta l\right)-\beta \left(\theta l\right)\right|}^2\right)dl\hfill \\ & & +q_3{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{2\mathfrak{a}-2}}{l}}\left(\mathbb{E}{\left|y\left(l\right)-\beta \left(l\right)\right|}^2+\mathbb{E}{\left|y\left(\theta l\right)-\beta \left(\theta l\right)\right|}^2\right)dl,\hfill \end{array}$$

where

$$q_1=m+3,q_2=2L^2(m+3)\left(\sum _{j=1}^m{\displaystyle \frac{{(T-1)}^{2{\mathfrak{b}}_j-1}}{(2{\mathfrak{b}}_j-1)\mathsf{\Gamma }{\left({\mathfrak{b}}_j\right)}^2}}+{\displaystyle \frac{{\left(logT\right)}^{2\mathfrak{a}-1}}{(2\mathfrak{a}-1)\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}\right)\mathrm{and}{q}_3=2{\displaystyle \frac{(m+3)L^2}{\mathsf{\Gamma }{\left(\mathfrak{a}\right)}^2}}.$$

Let ${\displaystyle f\left(s\right)=\underset{l\in [\theta ,s]}{sup}\mathbb{E}{|y\left(l\right)-\beta \left(l\right)|}^2}$ for $s\in [1,T]$.

We obtain ${\mathbb{E}|y\left(s\right)-\beta \left(s\right)|}^2\le f\left(s\right)$ and ${\mathbb{E}|y\left(\theta s\right)-\beta \left(\theta s\right)|}^2\le f\left(s\right)$, for all $s\in [1,T].$

Therefore, for $s\in [1,T]$, we obtain

$$\begin{array}{ccc}\hfill \mathbb{E}{\left|y\left(s\right)-\beta \left(s\right)\right|}^2& \le & q_1\varpi +2q_2{\int }_1^{s}f\left(l\right)dl\hfill \\ & & +2q_3{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{2\mathfrak{a}-2}}{l}}f\left(l\right)dl.\hfill \end{array}$$

Then, for every $l\in [1,s]$,

$$\begin{array}{ccc}\hfill \mathbb{E}{\left|y\left(l\right)-\beta \left(l\right)\right|}^2& \le & q_1\varpi +2q_2{\int }_1^{s}f\left(u\right)du\hfill \\ & & +2q_3{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{u}\right)}^{2\mathfrak{a}-2}}{u}}f\left(u\right)du.\hfill \end{array}$$

Therefore, for each $s\in [1,T]$,

$$\begin{array}{ccc}\hfill f\left(s\right)& \le & q_1\varpi +2q_2{\int }_1^{s}f\left(l\right)dl\hfill \\ & & +2q_3{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{2\mathfrak{a}-2}}{l}}f\left(l\right)dl.\hfill \end{array}$$

We obtain from Corollary 13.2 in [20]

$$\begin{array}{ccc}\hfill f\left(s\right)& \le & \left(q_1\varpi +2q_2{\int }_1^{s}f\left(l\right)dl\right)E_{2\mathfrak{a}-1}\left(2q_3\mathsf{\Gamma }(2\mathfrak{a}-1){\left(logs\right)}^{2\mathfrak{a}-1}\right)\hfill \\ & \le & \varpi q_4+q_5{\int }_1^{s}f\left(l\right)dl,\hfill \end{array}$$

with

$$q_4=q_1{E}_{2\mathfrak{a}-1}\left(2q_3\mathsf{\Gamma }(2\mathfrak{a}-1){\left(logT\right)}^{2\mathfrak{a}-1}\right)$$

and

$$q_5=2q_2{E}_{2\mathfrak{a}-1}\left(2q_3\mathsf{\Gamma }(2\mathfrak{a}-1){\left(logT\right)}^{2\mathfrak{a}-1}\right).$$

It follows from the Gronwall inequality that

$$f\left(s\right)\le \varpi q_4{e}^{q_5(T-1)}.$$

Thus,

$$\mathbb{E}{\left|y\left(s\right)-\beta \left(s\right)\right|}^2\le \varpi M,\forall s\in [1,T],$$

with $M=q_4{e}^{q_5(T-1)}$. Thus, Equation (1) is UHS with respect to $\varpi$. □

5. Examples

Example 1.

Consider the PIFSDE:

$${}^{H_C}D_1^{\mathfrak{a}}\left(\beta \left(s\right)-{\sum _{i=1}^2}^{RL}{I}^{{\displaystyle \frac{4+i}{9}}}{G}_i(s,\beta \left(s\right),\beta \left(\theta s\right))\right)={\zeta }_1(s,\beta \left(s\right),\beta \left(\theta s\right))+{\zeta }_2(s,\beta \left(s\right),\beta \left(\theta s\right)){\displaystyle \frac{dW\left(s\right)}{ds}},$$

(3)

$1<s\le 5,$ where

$$\begin{array}{ccc}\hfill G_i(s,\beta \left(s\right),\beta \left(\theta s\right))& =& {\displaystyle \frac{1}{18}}sin\left(\beta \left(\theta s\right)\right),i=1,2\hfill \\ \hfill {\zeta }_1(s,\beta \left(s\right),\beta \left(\theta s\right))& =& {\displaystyle \frac{1}{15}}{\displaystyle \frac{sin\left(\beta \right(s\left)\right)}{\left(s^2+1\right)}},\hfill \\ \hfill {\zeta }_2(s,\beta \left(s\right),\beta \left(\theta s\right))& =& {\displaystyle \frac{cos\left(\beta \right(\theta s\left)\right)}{s^2+17}}.\hfill \end{array}$$

The assumptions $\left({\mathcal{H}}_1\right)$ and $\left({\mathcal{H}}_2\right)$ are satisfied with $L={\displaystyle \frac{1}{2}}$. Using Theorem 1, we obtain the existence and uniqueness of the solutions of Equation (4). In addition, it follows from Theorem 2 that (4) is UHS with respect to ϖ.

Example 2.

Consider the PIFSDE:

$${}^{H_C}D_1^{\mathfrak{a}}\left(\beta \left(s\right)-{\sum _{i=1}^2}^{RL}{I}^{{\displaystyle \frac{3+i}{7}}}{G}_i(s,\beta \left(s\right),\beta \left(\theta s\right))\right)={\zeta }_1(s,\beta \left(s\right),\beta \left(\theta s\right))+{\zeta }_2(s,\beta \left(s\right),\beta \left(\theta s\right)){\displaystyle \frac{dW\left(s\right)}{ds}},$$

(4)

$1<s\le 3,$ where

$$\begin{array}{ccc}\hfill G_i(s,\beta \left(s\right),\beta \left(\theta s\right))& =& {\displaystyle \frac{1}{28}}cos\left(\beta \left(\theta s\right)\right),i=1,2\hfill \\ \hfill {\zeta }_1(s,\beta \left(s\right),\beta \left(\theta s\right))& =& {\displaystyle \frac{1}{25}}{\displaystyle \frac{cos\left(\beta \right(s\left)\right)}{\left(s^6+7\right)}},\hfill \\ \hfill {\zeta }_2(s,\beta \left(s\right),\beta \left(\theta s\right))& =& {\displaystyle \frac{sin\left(\beta \right(\theta s\left)\right)}{s^2+27}}.\hfill \end{array}$$

The assumptions $\left({\mathcal{H}}_1\right)$ and $\left({\mathcal{H}}_2\right)$ are satisfied with $L={\displaystyle \frac{1}{5}}$. Using Theorem 1, we obtain the existence and uniqueness of the solutions of Equation (4). In addition, it follows from Theorem 2 that (4) is UHS with respect to ϖ.

6. Conclusions

In this work, we have explored the existence, uniqueness and UHS of PIFSDEs using a combination of valuable mathematical techniques, stochastic analysis and Banach fixed-point methods. We have demonstrated how our results are applicable with two examples. In a future paper, it would be fascinating to expand this research to include the effects of infinite time delays.