Existence and Uniqueness Results for a Class of Fractional Integro-Stochastic Differential Equations

Abstract

The objective of this paper is to demonstrate the existence and uniqueness (EU) of solutions to a class of Fractional Integro-Stochastic Differential Equations (FISDEs) by utilizing the fixed-point technique (FPT) and stochastic techniques. Additionally, the paper proves the continuous dependence (CD) of solutions on the initial data. We examine the Hyers–Ulam stability (HUS) of FISDEs by applying Gronwall inequalities. Two theoretical examples are presented to demonstrate our findings.

1. Introduction

It is feasible to explain the behavior of a broad category of infinite-dimensional dynamical systems using fractional-order differentiation and integration [1]. Their applications extend across various fields such as mechanics, control theory, electricity, signal processing, chemistry, and biological systems [2,3,4]. Extensions to fractional calculus have been developed by Atangana, Baleanu, and other researchers. For instance, B. Ahmad et al. in [5] focus on the recent advancements in Fractional Differential Equations (FDEs) as well as inclusions and inequalities that involve the Hadamard derivative and integral. R. Almeida in [6] presented Caputo–Hadamard derivatives of variable fractional order. D. Baleanu et al. in [7] discussed the potential of fractional-order modeling in addressing complex systems within engineering and science. The authors in [8] presented a study on the Caputo modification of Hadamard fractional derivatives. Hadamard, in [9], provided an essay on the study of functions described by their Taylor series expansion. A. A. Kilbas et al. in [10] presented various results on ordinary and partial FDEs and their applications. C. Li et al. in [11] investigated various numerical methods for fractional calculus. I. Podlubny in [12], provides readers with all the essentials for a foundational study and the practical application of fractional derivatives and FDEs.

The use of non-integer-order derivatives has made FDEs a commonly used tool for studying systems with hereditary and memory characteristics. Usually using fixed-point theorems, the study of FDEs encompasses crucial aspects such as the EU of solutions, where specific conditions related to continuity and Lipschitz continuity are established. For instance, the authors in [13] investigated some stability results for a class of neutral integro-differential equations with delay of fractional order. Jia et al. in [14] presented the EU of the solutions of uncertain FDEs with Jump. The authors in [15] investigated the EU for uncertain Differential Equations (DEs) with time delay and V − jump. A. Zada et al. in [16] studied the EU and stability results for a class of higher-order DEs with fractional integrable impulses. S. Abbas in [17] presented the stability results for partial FDEs with multiple delays and impulses. The Ulam stability also plays an important role in ensuring the model’s reliability in practical applications by assessing how small changes in initial conditions or parameters affect the solutions [18,19].

In recent years, an increasing amount of research has been dedicated to Fractional Stochastic Differential Equations (FSDEs). For instance, M. Ahmad et al. in [20] studied the EU for a class of weighted impulsive FSDEs. S. Shahid et al. in [21] presented the EU and stability results for implicit random FDEs. S. Saifullah et al. in [22] investigated the EU for a class of FSDEs with Retarded and Advanced Arguments. S. Begum et al. in [23] presented some results about the EU and Ulam stability for a class of random FDEs via Hilfer derivative. A. Ben Makhlouf et al. in [24] investigated the EU and stability results for FSDEs via Caputo-Hadamard derivative. By integrating fractional calculus with stochastic processes, these equations offer a vital framework for analyzing and modeling complex systems affected by memory and randomness. The exploration of FSDEs offers significant advantages to fields like physics and engineering, as they provide valuable insight into complex systems’ dynamics; see ([25,26,27]).

A wide range of applications are covered by numerous research publications on the HUS concept, making finding precise solutions difficult. For example, with the help of a generalized Gronwall inequality, authors in [18] demonstrated the HUS of pantograph equations. An investigation of HUS for mixed-FSDEs was conducted by Rhaima et al. in [19]. M. Ahmad et al. in [28] investigated the HUS for neutral FSDEs. Authors in [29] studied the HUS for proportional Caputo neutral FSDEs. Y. Guo et al. in [30] proved the HUS for a class of impulsive Riemann—Liouville neutral FSDEs. Authors in [31] explored the HUS for FSDEs with fBm.

This paper builds on these foundational studies by presenting novel results tailored to a specific class of FISDEs. The analysis focuses on systems that incorporate the Hadamard fractional integral and the Caputo fractional derivative, two essential tools in fractional calculus that offer greater flexibility in modeling memory and hereditary properties in dynamic systems. The innovative contributions of this work include:

(1)

Establishing the EU of solutions for FISDEs.

(2)

Investigating the CD of solutions on initial data for the FISDEs.

(3)

Proving the HUS of the FISDEs.

2. Notational Preliminaries

Set $\{\mathsf{\Omega },\mathbb{F},\mathcal{F}:={\left({\mathbb{F}}_{\mu }\right)}_{1\le \mu \le T},\mathbb{P}\}$ with $T>1$, a complete probability space, and $W\left(\mu \right)$ is a d-dimensional Brownian motion.

Let ${\mathcal{X}}_{\mu }=L^2(\mathsf{\Omega },{\mathbb{F}}_{\mu },\mathbb{P})$ (for each $\mu \in [1,T]$) denote the collection of all ${\mathbb{F}}_{\mu }$-measurable ${\displaystyle g:\mathsf{\Omega }\to \mathbb{R}}$ that are mean square integrable with

$${||g||}_{ms}=\sqrt{{\mathbb{E}|g|}^2}.$$

Definition 1 

([10]). Let $r>0$. The Fractional Integral (FI) in the sense of Riemann–Liouville for $H\in L^1[1,a]$ is defined by

$$I^{r}H\left(y\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(r\right)}}{\int }_1^y{\left(y-s\right)}^{r-1}H\left(s\right)ds.$$

Definition 2 

([10]). The Caputo fractional derivative of order $0<r<1$ of H is given by

$${}^C\mathit{D}_1^{r}H\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-r)}}{\displaystyle \frac{d}{dt}}{\int }_1^t{\left(t-s\right)}^{-r}\left(H\left(s\right)-H\left(1\right)\right)ds.$$

Definition 3 

([10]). Let $r>0$. The FI in the sense of Hadamard for $H\in L^1[1,a]$ is defined by

$${}^H\mathit{I}^{r}H\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(r\right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{s}}\right)}^{r-1}{\displaystyle \frac{H\left(s\right)}{s}}ds.$$

Definition 4 

([10]). The Mittag–Leffler function is expressed as

$${\displaystyle E_r\left(z\right)=\sum _{\iota =0}^{+\infty }\frac{z^{\iota }}{\mathsf{\Gamma }(\iota r+1)},}$$

where $r>0$ and $z\in \mathbb{C}$.

Let us consider the following class of FISDEs:

$${}^C\mathit{D}_1^{\vartheta }\left(x\left(s\right)-\sum _{i=1}^m{}^H\mathit{I}^{{\xi }_i}{\mathsf{\Phi }}_i(s,x\left(s\right))\right)=F_1(s,x\left(s\right))+F_2(s,x\left(s\right)){\displaystyle \frac{dW\left(s\right)}{ds}},s\in [1,T],$$

(1)

where ${\displaystyle \frac{1}{2}}<\vartheta <1$, ${\displaystyle \frac{1}{2}}<{\xi }_{\iota }<1$, $1\le \iota \le m$, $x\left(1\right)=a\in \mathbb{R}$ and ${\mathsf{\Phi }}_{\iota },F_j:[1,T]\times \mathbb{R}\to \mathbb{R}$ are measurable such that

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

There is $L>0$ such that

$$\sum _{\iota =1}^m|{\mathsf{\Phi }}_{\iota }(s,{\varpi }_1)-{\mathsf{\Phi }}_{\iota }(s,{\varpi }_2)|+\sum _{\iota =1}^2|F_{\iota }(s,{\varpi }_1)-F_{\iota }(s,{\varpi }_2)|\le L|{\varpi }_1-{\varpi }_2|,$$

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

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

$F_j(·,0)$ and ${\mathsf{\Phi }}_k(·,0)$, $j=1,2$, $k=1,\dots ,m$ fulfills the following conditions:

$$||F_2{(·,0)||}_{\infty }=ess\underset{s\in [1,T]}{sup}|F_2(s,0)|<\infty ,{\int }_1^T|{\mathsf{\Phi }}_i{(s,0)|}^{2}ds<\infty ,{\int }_1^T{|F_1(s,0)|}^{2}ds<\infty .$$

3. Results on Existence and Uniqueness

Let ${\mathbb{D}}^2$ be the Banach space of all processes $\varpi$ that are ${\mathbb{F}}_T$-adapted, measurable and satisfy:

$${∥\varpi ∥}_{{\mathbb{D}}^2}=\underset{1\le y\le T}{sup}{∥\varpi \left(y\right)∥}_{ms}<\infty .$$

Consider the operator $R_a:{\mathbb{D}}^2\to {\mathbb{D}}^2$ defined for $a\in \mathbb{R}$ as follows:

$$\begin{array}{ccc}\hfill R_a\varpi \left(s\right)& =& a+\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\xi }_j\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{{\xi }_j-1}}{l}}{\mathsf{\Phi }}_j(l,\varpi \left(l\right))dl+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\vartheta \right)}}{\int }_1^s{(s-l)}^{\vartheta -1}{F}_1(l,\varpi \left(l\right))dl\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\vartheta \right)}}{\int }_1^s{(s-l)}^{\vartheta -1}{F}_2(l,\varpi \left(l\right))dW\left(l\right).\hfill \end{array}$$

Lemma 1. 

For $a\in \mathbb{R}$, $R_a$ is well-defined.

Proof. 

Let $\varpi \in {\mathbb{D}}^2$; therefore

$$\begin{array}{ccc}\hfill {∥R_a\varpi \left(s\right)∥}_{ms}^2& =& \mathbb{E}{\left|R_a\varpi \left(s\right)\right|}^2\hfill \\ & \le & (m+3)[{∥a∥}_{ms}^2+\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }{\left({\xi }_j\right)}^2}}\mathbb{E}\left({\left|{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{y}\right)}^{{\xi }_j-1}}{y}}{\mathsf{\Phi }}_j(y,\varpi \left(y\right))dy\right|}^2\right)\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}\mathbb{E}\left({\left|{\int }_1^s{(s-y)}^{\vartheta -1}{F}_1(y,\varpi \left(y\right))dy\right|}^2\right)\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}\mathbb{E}\left({\left|{\int }_1^s{(s-y)}^{\vartheta -1}{F}_2(y,\varpi \left(y\right))dW\left(y\right)\right|}^2\right)].\hfill \end{array}$$

Using Cauchy–Schwarz inequality, we obtain

$$\begin{array}{ccc}\hfill \mathbb{E}\left({\left|{\int }_1^s{\displaystyle \frac{{(log\frac{s}{y})}^{{\xi }_j-1}}{y}}{\mathsf{\Phi }}_j(y,\varpi \left(y\right))dy\right|}^2\right)& \le & \left({\int }_1^s{\displaystyle \frac{{(log\frac{s}{y})}^{2{\xi }_j-2}}{y^2}}dy\right)\mathbb{E}\left({\int }_1^s{\left|{\mathsf{\Phi }}_j(y,\varpi \left(y\right))\right|}^{2}dy\right)\hfill \\ & \le & {\displaystyle \frac{1}{2{\xi }_j-1}}{(logT)}^{2{\xi }_j-1}\mathbb{E}\left({\int }_1^s{\left|{\mathsf{\Phi }}_j(y,\varpi \left(y\right))\right|}^{2}dy\right).\hfill \end{array}$$

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

$${\left|{\mathsf{\Phi }}_j(y,\varpi \left(y\right))\right|}^2\le 2L^2{\left|\varpi \left(y\right)\right|}^2+2{\left|{\mathsf{\Phi }}_j(y,0)\right|}^2.$$

Therefore,

$$\mathbb{E}\left({\int }_1^s{\left|{\mathsf{\Phi }}_j(y,\varpi \left(y\right))\right|}^{2}dy\right)\le 2(T-1)L^2\underset{y\in [1,T]}{sup}\mathbb{E}{\left|\varpi \left(y\right)\right|}^2+2{\int }_1^T{\left|{\mathsf{\Phi }}_j(y,0)\right|}^{2}dy.$$

Thus,

$$\mathbb{E}\left({\left|{\int }_1^s{\displaystyle \frac{{(log\frac{s}{y})}^{{\xi }_j-1}}{y}}{\mathsf{\Phi }}_j(y,\varpi \left(y\right))dy\right|}^2\right)\le$$

$${\displaystyle \frac{{(logT)}^{2{\xi }_j-1}}{2{\xi }_j-1}}\left(2L^2(T-1)\underset{y\in [1,T]}{sup}\mathbb{E}{\left|\varpi \left(y\right)\right|}^2+2{\int }_1^T{\left|{\mathsf{\Phi }}_j(y,0)\right|}^{2}dy\right).$$

By applying the Cauchy–Schwarz inequality, it can be deduced that

$$\begin{array}{ccc}\hfill \mathbb{E}\left({\left|{\int }_1^s{(s-y)}^{\vartheta -1}{F}_1(y,\varpi \left(y\right))dy\right|}^2\right)& \le & \left({\int }_1^s{(s-y)}^{2\vartheta -2}dy\right)\mathbb{E}\left({\int }_1^s{\left|F_1(y,\varpi \left(y\right))\right|}^{2}dy\right)\hfill \\ & \le & {\displaystyle \frac{1}{2\vartheta -1}}{(s-1)}^{2\vartheta -1}\mathbb{E}\left({\int }_1^s{\left|F_1(y,\varpi \left(y\right))\right|}^{2}dy\right).\hfill \end{array}$$

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

$${\left|F_1(y,\varpi \left(y\right))\right|}^2\le 2L^2{\left|\varpi \left(y\right)\right|}^2+2{\left|F_1(y,0)\right|}^2.$$

Therefore,

$$\mathbb{E}\left({\left|{\int }_1^s{(s-y)}^{\vartheta -1}{F}_1(y,\varpi \left(y\right))dy\right|}^2\right)\le$$

$${\displaystyle \frac{{(T-1)}^{2\vartheta -1}}{2\vartheta -1}}\left(2L^2(T-1)\underset{y\in [1,T]}{sup}\mathbb{E}{\left|\varpi \left(y\right)\right|}^2+2{\int }_1^T{\left|F_1(y,0)\right|}^{2}dy\right).$$

Using the isometry of Itô, we obtain

$$\mathbb{E}\left({\left|{\int }_1^s{(s-y)}^{\vartheta -1}{F}_2(y,\varpi \left(y\right))dW\left(y\right)\right|}^2\right)=\mathbb{E}\left({\int }_1^s{(s-y)}^{2\vartheta -2}{\left|F_2(y,\varpi \left(y\right))\right|}^{2}dy\right).$$

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

$${\left|F_2(y,\varpi \left(y\right))\right|}^2\le 2L^2{\left|\varpi \left(y\right)\right|}^2+2{∥F_2(.,0)∥}_{\infty }^2.$$

Then,

$$\begin{array}{ccc}& & \mathbb{E}\left({\left|{\int }_1^s{(s-y)}^{\vartheta -1}{F}_2(y,\varpi \left(y\right))dW\left(y\right)\right|}^2\right)\hfill \\ & \le & 2L^2\mathbb{E}\left({\int }_1^s{(s-y)}^{2\vartheta -2}{\left|\varpi \left(y\right)\right|}^{2}dy\right)+2{∥F_2(y,0)∥}_{\infty }^2{\int }_1^s{(s-y)}^{2\vartheta -2}dy\hfill \\ & \le & {\displaystyle \frac{2L^2}{2\vartheta -1}}{\left(T-1\right)}^{2\vartheta -1}{∥\varpi ∥}_H^2+{\displaystyle \frac{2}{2\vartheta -1}}{\left(T-1\right)}^{2\vartheta -1}{∥F_2(.,0)∥}_{\infty }^2.\hfill \end{array}$$

Thus, $R_a$ is well-defined. □

Theorem 1. 

Suppose that $\left({\mathcal{H}}_1\right)$ and $\left({\mathcal{H}}_2\right)$ hold, therefore Equation (1) admits a unique solution $x\in {\mathbb{D}}^2$ satisfying $x\left(1\right)=a$.

Proof. 

Let us consider the norm ${∥·∥}_{{\mu }_1,{\mu }_2}$ on ${\mathbb{D}}^2$ given by:

$${∥\varpi ∥}_{{\mu }_1,{\mu }_2}=\underset{y\in [1,T]}{sup}\mathbb{E}\left({\left|\varpi \left(y\right)\right|}^2\right)h\left(y\right),\forall \varpi \in {\mathbb{D}}^2,$$

where

$$h\left(y\right)=y^{{\mu }_1}{E}_{2\vartheta -1}\left({\mu }_2{(y-1)}^{2\vartheta -1}\right)$$

with ${\mu }_1,{\mu }_2>0$ and

$$(m+2)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2{\xi }_i-1)}{{\mu }_1^{2{\xi }_i-1}}}+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2\vartheta -1)}{{\mu }_2}}\right)<1.$$

The space $\left({\mathbb{D}}^2,{∥·∥}_{{\mu }_1,{\mu }_2}\right)$ is a Banach space because the norms ${∥·∥}_{{\mathbb{D}}^2}$ and ${∥·∥}_{{\mu }_1,{\mu }_2}$ are equivalent.

We will prove that $R_a$ is a contraction by using the weighted norm ${∥.∥}_{{\mu }_1,{\mu }_2}.$

Let $c\in \mathbb{R}$ and ${\varpi }_1,{\varpi }_2\in {\mathbb{D}}^2$. Taking the expectation, we get:

$$\begin{array}{ccc}& & \mathbb{E}\left({\left|R_a{\varpi }_1\left(t\right)-R_a{\varpi }_2\left(t\right)\right|}^2\right)\hfill \\ & & \le (m+2)\left(\sum _{i=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}}\mathbb{E}\left({\left|{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{y}\right)}^{{\xi }_i-1}}{y}}[{\mathsf{\Phi }}_i(y,{\varpi }_1\left(y\right))-{\mathsf{\Phi }}_i(y,{\varpi }_2\left(y\right))]dy\right|}^2\right)\right.\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}\mathbb{E}\left({\left|{\int }_1^t{\left(t-y\right)}^{\vartheta -1}\left(F_1(y,{\varpi }_1\left(y\right))-F_1(y,{\varpi }_2\left(y\right))\right)dy\right|}^2\right)\hfill \\ & & +\left.{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}\mathbb{E}\left({\left|{\int }_1^t{\left(t-y\right)}^{\vartheta -1}\left(F_2(y,{\varpi }_1\left(y\right))-F_2(y,{\varpi }_1\left(y\right))\right)dW\left(y\right)\right|}^2\right)\right).\hfill \end{array}$$

Using Cauchy–Schwartz inequality, we obtain

$$\mathbb{E}\left(|{\int }_1^t{(t-y)}^{\vartheta -1}[F_1(y,{\varpi }_1\left(y\right))-F_1(y,{\varpi }_2\left(y\right))]dy{|}^2\right)\le$$

$$L^2(T-1){\int }_1^t{(t-y)}^{2\vartheta -2}\mathbb{E}(|{\varpi }_1\left(y\right)-{\varpi }_2\left(y\right){|}^2)dy$$

and

$$\mathbb{E}\left({\left|{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{y}\right)}^{{\xi }_i-1}}{y}}\left({\mathsf{\Phi }}_i(y,{\varpi }_1\left(y\right))-{\mathsf{\Phi }}_i(y,{\varpi }_2\left(y\right))\right)dy\right|}^2\right)\le$$

$$L^2(T-1){\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{y}\right)}^{2{\xi }_i-2}}{y^2}}\mathbb{E}\left({\left|{\varpi }_1\left(y\right)-{\varpi }_2\left(y\right)\right|}^2\right)dy.$$

Using the isometry of Itô, we obtain

$$\begin{array}{ccc}& & \mathbb{E}\left({\left|{\int }_1^t{(t-y)}^{\vartheta -1}\left(F_2(y,{\varpi }_1\left(y\right))-F_2(y,{\varpi }_1\left(y\right))\right)dW\left(y\right)\right|}^2\right)\hfill \\ & & =\mathbb{E}\left({\int }_1^t{(t-y)}^{2\vartheta -2}{\left|F_2(y,{\varpi }_1\left(y\right))-F_2(y,{\varpi }_1\left(y\right))\right|}^{2}dy\right)\hfill \\ & & \le L^2{\int }_1^t{(t-y)}^{2\vartheta -2}\mathbb{E}\left({\left|{\varpi }_1\left(y\right)-{\varpi }_2\left(y\right)\right|}^2\right)dy.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}& & \mathbb{E}\left({\left|R_a{\varpi }_1\left(t\right)-R_a{\varpi }_2\left(t\right)\right|}^2\right)\hfill \\ & & \le (m+2)\left(\sum _{i=1}^m{\displaystyle \frac{L^2(T-1)}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}}{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{y}\right)}^{2{\xi }_i-2}}{y^2}}\mathbb{E}\left({\left|{\varpi }_1\left(y\right)-{\varpi }_2\left(y\right)\right|}^2\right)dy\right.\hfill \\ & & +{\displaystyle \frac{L^2(T-1)}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\int }_1^t{\left(t-y\right)}^{2\vartheta -2}\mathbb{E}\left({\left|{\varpi }_1\left(y\right)-{\varpi }_2\left(y\right)\right|}^2\right)dy\hfill \\ & & \left.+{\displaystyle \frac{L^2}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\int }_1^t{\left(t-y\right)}^{2\vartheta -2}\mathbb{E}\left({\left|{\varpi }_1\left(y\right)-{\varpi }_2\left(y\right)\right|}^2\right)dy\right)\hfill \\ & & \le (m+2)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}}{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{y}\right)}^{2{\xi }_i-2}}{y}}\mathbb{E}\left({\left|{\varpi }_1\left(y\right)-{\varpi }_2\left(y\right)\right|}^2\right)dy\right.\hfill \\ & & \left.+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\int }_1^t{\left(t-y\right)}^{2\vartheta -2}\mathbb{E}\left({\left|{\varpi }_1\left(y\right)-{\varpi }_2\left(y\right)\right|}^2\right)dy\right)\hfill \\ & & \le (m+2)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}}{∥{\varpi }_1-{\varpi }_2∥}_{{\mu }_1,{\mu }_2}^2{\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{y}\right)}^{2{\xi }_i-2}}{y}}h\left(y\right)dy\right.\hfill \\ & & \left.+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{∥{\varpi }_1-{\varpi }_2∥}_{{\mu }_1,{\mu }_2}^2{\int }_1^t{\left(t-y\right)}^{2\vartheta -2}h\left(y\right)dy\right).\hfill \end{array}$$

From Remark 1 in [13], we obtain

$$\begin{array}{ccc}\hfill {\int }_1^t{(t-y)}^{2\vartheta -2}h\left(y\right)dy& =& {\int }_1^t{(t-y)}^{2\vartheta -2}{y}^{{\mu }_1}{E}_{2\vartheta -1}\left({\mu }_2{(y-1)}^{2\vartheta -1}\right)dy\hfill \\ & \le & t^{{\mu }_1}{\int }_1^t{(t-y)}^{2\vartheta -2}{E}_{2\vartheta -1}\left({\mu }_2{(y-1)}^{2\vartheta -1}\right)dy\hfill \\ & \le & {\displaystyle \frac{\mathsf{\Gamma }(2\vartheta -1)}{{\mu }_2}}{t}^{{\mu }_1}{E}_{2\vartheta -1}\left({\mu }_2{(t-1)}^{2\vartheta -1}\right).\hfill \end{array}$$

Using the following inequality (see [13]),

$${\int }_1^t{\left(log{\displaystyle \frac{t}{y}}\right)}^{2{\xi }_i-2}{\displaystyle \frac{y^{{\mu }_1}}{y}}dy\le {\displaystyle \frac{t^{{\mu }_1}}{{\mu }_1^{2{\xi }_i-1}}}\mathsf{\Gamma }(2{\xi }_i-1),$$

we get

$$\begin{array}{ccc}\hfill {\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{y}\right)}^{2{\xi }_i-2}}{y}}h\left(y\right)dy& \le & E_{2\vartheta -1}\left({\mu }_2{(t-1)}^{2\vartheta -1}\right){\int }_1^t{\displaystyle \frac{{\left(log\frac{t}{y}\right)}^{2{\xi }_i-2}}{y}}{y}^{{\mu }_1}dy\hfill \\ & \le & h\left(t\right){\displaystyle \frac{\mathsf{\Gamma }(2{\xi }_i-1)}{{\mu }_1^{2{\xi }_i-1}}}.\hfill \end{array}$$

Consequently, based on the previously obtained inequalities, it follows that

$$\begin{array}{ccc}& & \mathbb{E}\left({\left|R_a{\varpi }_1\left(t\right)-R_a{\varpi }_2\left(t\right)\right|}^2\right)\hfill \\ & & \le (m+2)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2{\xi }_i-1)}{{\mu }_1^{2{\xi }_i-1}}}+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2\vartheta -1)}{{\mu }_2}}\right)h\left(t\right){∥{\varpi }_1-{\varpi }_2∥}_{{\mu }_1,{\mu }_2}^2.\hfill \end{array}$$

Consequently, we have

$${∥R_a{\varpi }_1-R_a{\varpi }_2∥}_{{\mu }_1,{\mu }_2}\le K{∥{\varpi }_1-{\varpi }_2∥}_{{\mu }_2,{\mu }_2}$$

where

$$K^2=(m+2)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2{\xi }_i-1)}{{\mu }_1^{2{\xi }_i-1}}}+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2\vartheta -1)}{{\mu }_2}}\right).$$

Hence, $R_a$ is a contraction. Thus, the desired result is established. □

Remark 1. 

In the previous theorem, we have used the weighted norm ${∥.∥}_{{\mu }_1,{\mu }_2}$ to obtain that $R_a$ is a contraction without any condition on the Lipschitz constant L.

Theorem 2. 

Let $a_1,a_2\in \mathbb{R}$ such that $a_1\ne a_2$. Therefore,

$$\underset{a_1\to a_2}{lim}\underset{s\in [1,T]}{sup}{\parallel x_1\left(s\right)-x_2\left(s\right)\parallel }_{ms}=0,$$

where $x_1\left(s\right)$ and $x_2\left(s\right)$ represent the solutions of (1) corresponding to the initial data $a_1$ and $a_2$, respectively.

Proof. 

We have

$$\begin{array}{cc}\hfill x_1\left(s\right)-x_2\left(s\right)& =(a_1-a_2)+\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\xi }_j\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{l}\right)}^{{\xi }_j-1}}{l}}[{\mathsf{\Phi }}_j(l,x_1\left(l\right))-{\mathsf{\Phi }}_j(l,x_2\left(l\right))]dl\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\vartheta \right)}}{\int }_1^s{(s-l)}^{\vartheta -1}[F_1(l,x_1\left(l\right))-F_1(l,x_2\left(l\right))]dl\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\vartheta \right)}}{\int }_1^s{(s-l)}^{\vartheta -1}[F_2(l,x_1\left(l\right))-F_2(l,x_2\left(l\right))]dW\left(l\right).\hfill \end{array}$$

Taking the expectation, we obtain:

$$\begin{array}{cc}\hfill & \parallel x_1\left(s\right)-x_2{\left(s\right)\parallel }_{ms}^2⩽(m+3)[{\parallel a_1-a_2\parallel }_{ms}^2\hfill \\ \hfill & +\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }{\left({\xi }_j\right)}^2}}\mathbb{E}\left({\left|{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{y}\right)}^{{\xi }_j-1}}{y}}[{\mathsf{\Phi }}_j(y,x_1\left(y\right))-{\mathsf{\Phi }}_j(y,x_2\left(y\right))]dy\right|}^2\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}\mathbb{E}\left({\left|{\int }_1^s{(s-y)}^{\vartheta -1}[F_1(y,x_1\left(y\right))-F_1(y,x_2\left(y\right))]dy\right|}^2\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}\mathbb{E}\left({\left|{\int }_1^s{(s-y)}^{\vartheta -1}[F_2(y,x_1\left(y\right))-F_2(y,x_2\left(y\right))]dW\left(y\right)\right|}^2\right)].\hfill \end{array}$$

By applying the Cauchy–Schwarz inequality, we obtain

$$\mathbb{E}\left(|{\int }_1^s{(s-y)}^{\vartheta -1}[F_1(y,x_1\left(y\right))-F_1(y,x_2\left(y\right))]dy{|}^2\right)\le$$

$$L^2(T-1){\int }_1^s{(s-y)}^{2\vartheta -2}\mathbb{E}(|x_1\left(y\right)-x_2\left(y\right){|}^2)dy$$

and

$$\mathbb{E}\left({\left|{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{y}\right)}^{{\xi }_i-1}}{y}}\left({\mathsf{\Phi }}_j(y,x_1\left(y\right))-{\mathsf{\Phi }}_j(y,x_2\left(y\right))\right)dy\right|}^2\right)\le$$

$$L^2(T-1){\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{y}\right)}^{2{\xi }_i-2}}{y^2}}\mathbb{E}\left({\left|x_1\left(y\right)-x_2\left(y\right)\right|}^2\right)dy.$$

Using the isometry of Itô, we obtain

$$\begin{array}{ccc}& & \mathbb{E}\left({\left|{\int }_1^s{(s-y)}^{\vartheta -1}\left(F_2(y,x_1\left(y\right))-F_2(y,x_1\left(y\right))\right)dW\left(y\right)\right|}^2\right)\hfill \\ & & \le L^2{\int }_1^s{(s-y)}^{2\vartheta -2}\mathbb{E}\left({\left|x_1\left(y\right)-x_2\left(y\right)\right|}^2\right)dy.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill \parallel x_1\left(s\right)-x_2{\left(s\right)\parallel }_{ms}^2& & \le (m+3)[\parallel a_1-a_2{\parallel }_{ms}^2\hfill \\ & & +\sum _{i=1}^m{\displaystyle \frac{L^2(T-1)}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{y}\right)}^{2{\xi }_i-2}}{y^2}}\mathbb{E}\left({\left|x_1\left(y\right)-x_2\left(y\right)\right|}^2\right)dy\hfill \\ & & +{\displaystyle \frac{L^2(T-1)}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\int }_1^s{\left(s-y\right)}^{2\vartheta -2}\mathbb{E}\left({\left|x_1\left(y\right)-x_2\left(y\right)\right|}^2\right)dy\hfill \\ & & +{\displaystyle \frac{L^2}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\int }_1^s{\left(s-y\right)}^{2\vartheta -2}\mathbb{E}\left({\left|x_1\left(y\right)-x_2\left(y\right)\right|}^2\right)dy]\hfill \\ & & \le (m+3)[\parallel a_1-a_2{\parallel }_{ms}^2\hfill \\ & & +L^2\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{y}\right)}^{2{\xi }_i-2}}{y}}\mathbb{E}\left({\left|x_1\left(y\right)-x_2\left(y\right)\right|}^2\right)dy\hfill \\ & & +L^2{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\int }_1^s{\left(s-y\right)}^{2\vartheta -2}\mathbb{E}\left({\left|x_1\left(y\right)-x_2\left(y\right)\right|}^2\right)dy]\hfill \\ & & \le (m+3)[\parallel a_1-a_2{\parallel }_{ms}^2\hfill \\ & & +L^2\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}}{∥x_1-x_2∥}_{{\mu }_1,{\mu }_2}^2{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{y}\right)}^{2{\xi }_i-2}}{y}}h\left(y\right)dy\hfill \\ & & +L^2{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{∥x_1-x_2∥}_{{\mu }_1,{\mu }_2}^2{\int }_1^s{\left(s-y\right)}^{2\vartheta -2}h\left(y\right)dy].\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill & \parallel x_1\left(s\right)-x_2{\left(s\right)\parallel }_{ms}^2⩽(m+3){\parallel a_1-a_2\parallel }_{ms}^2\hfill \\ \hfill & +(m+3)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2{\xi }_i-1)}{{\mu }_1^{2{\xi }_i-1}}}+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2\vartheta -1)}{{\mu }_2}}\right)h\left(s\right){∥x_1-x_2∥}_{{\mu }_1,{\mu }_2}^2.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill & {∥x_1-x_2∥}_{{\mu }_1,{\mu }_2}^2⩽(m+3){\parallel a_1-a_2\parallel }_{ms}^2\hfill \\ \hfill & +(m+3)L^2\left(\sum _{i=1}^m{\displaystyle \frac{(T-1)}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2{\xi }_i-1)}{{\mu }_1^{2{\xi }_i-1}}}+{\displaystyle \frac{T}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\displaystyle \frac{\mathsf{\Gamma }(2\vartheta -1)}{{\mu }_2}}\right){∥x_1-x_2∥}_{{\mu }_1,{\mu }_2}^2.\hfill \end{array}$$

Let ${\mu }_1,{\mu }_2>0$ such that

$$C({\mu }_1,{\mu }_2)=(m+3)L^2\left({\displaystyle \sum _{i=1}^m\frac{(T-1)}{\mathsf{\Gamma }{\left({\xi }_i\right)}^2}\frac{\mathsf{\Gamma }(2{\xi }_i-1)}{{\mu }_1^{2{\xi }_i-1}}+\frac{T}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}\frac{\mathsf{\Gamma }(2\vartheta -1)}{{\mu }_2}}\right)<1.$$

Then,

$$\left(1-C({\mu }_1,{\mu }_2)\right){∥x_1-x_2∥}_{{\mu }_1,{\mu }_2}^2⩽(m+3){\parallel a_1-a_2\parallel }_{ms}^2.$$

Therefore,

$$\underset{a_1\to a_2}{lim}\underset{s\in [1,T]}{sup}{\parallel x_1\left(s\right)-x_2\left(s\right)\parallel }_{ms}=0.$$

□

4. Stability Results

Definition 5. 

Equation (1) is UHS with regard to ε if there exists $\tilde{M}>0$ so that $\forall \epsilon >0$ and each solution $z\in {\mathbb{D}}^2$, with $z\left(1\right)=a_0$, of

$$\mathbb{E}{\left|z\left(t\right)-R_{a_0}z\left(t\right)\right|}^2\le \epsilon ;\forall t\in [1,T],$$

(2)

there exists a solution $\varpi \in {\mathbb{D}}^2$ of (1), with $\varpi \left(1\right)=a_0$ that satisfies

$$\mathbb{E}{\left|z\left(t\right)-\varpi \left(t\right)\right|}^2\le \tilde{M}\epsilon ,\forall t\in [1,T].$$

(3)

For the UHS of (1), we present the following result.

Theorem 3. 

Assume that $\left({\mathcal{H}}_1\right)$ and $\left({\mathcal{H}}_2\right)$ hold; then, (1) is UHS with regard to ε.

Proof. 

Let $\epsilon >0$ and $z\left(s\right)$ be a solution of (2). Let $\varpi \left(s\right)$ be the solution of (1) with $\varpi \left(1\right)=a_0$, so

$$\begin{array}{ccc}\hfill \varpi \left(s\right)& =& a_0+\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\xi }_j\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{y}\right)}^{{\xi }_j-1}}{y}}{\mathsf{\Phi }}_j(y,\varpi \left(y\right))dy+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\vartheta \right)}}{\int }_1^s{\left(s-y\right)}^{\vartheta -1}{F}_1(y,\varpi \left(y\right))dy\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\vartheta \right)}}{\int }_1^s{\left(s-y\right)}^{\vartheta -1}{F}_2(y,\varpi \left(y\right))dW\left(y\right).\hfill \end{array}$$

Hence, we have

$$\begin{array}{ccc}\hfill z\left(s\right)-\varpi \left(s\right)& =& \left(z\left(s\right)-a_0-\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\xi }_j\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{y}\right)}^{{\xi }_j-1}}{y}}{\mathsf{\Phi }}_j(y,z\left(y\right))dy\right.\hfill \\ & & -{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\vartheta \right)}}{\int }_1^s{\left(s-y\right)}^{\vartheta -1}{F}_1(y,z\left(y\right))dy\hfill \\ & & \left.-{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\vartheta \right)}}{\int }_1^s{\left(s-y\right)}^{\vartheta -1}{F}_2(y,z\left(y\right))dW\left(y\right)\right)\hfill \\ & & +\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\xi }_j\right)}}{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{y}\right)}^{{\xi }_j-1}}{y}}\left({\mathsf{\Phi }}_j(y,z\left(y\right))-{\mathsf{\Phi }}_j(y,\varpi \left(y\right))\right)dy\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\vartheta \right)}}{\int }_1^s{\left(s-y\right)}^{\vartheta -1}\left(F_1(y,z\left(y\right))-F_1(y,\varpi \left(y\right))\right)dy\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\vartheta \right)}}{\int }_1^s{\left(s-y\right)}^{\vartheta -1}\left(F_2(y,z\left(y\right))-F_2(y,\varpi \left(y\right))\right)dW\left(y\right).\hfill \end{array}$$

The inequality (2) leads to the conclusion that

$$\begin{array}{ccc}\hfill \mathbb{E}{\left|z\left(s\right)-\varpi \left(s\right)\right|}^2& \le & (m+3)\left(\epsilon +\sum _{j=1}^m{\displaystyle \frac{1}{\mathsf{\Gamma }{\left({\xi }_j\right)}^2}}\mathbb{E}{\left|{\int }_1^s{\displaystyle \frac{{\left(log\frac{s}{y}\right)}^{{\xi }_j-1}}{y}}[{\mathsf{\Phi }}_j(y,z\left(y\right))-{\mathsf{\Phi }}_j(y,\varpi \left(y\right))]dy\right|}^2\right.\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}\mathbb{E}{\left|{\int }_1^s{\left(s-y\right)}^{\vartheta -1}[F_1(y,z\left(y\right))-F_1(y,\varpi \left(y\right))]dy\right|}^2\hfill \\ & & \left.+{\displaystyle \frac{1}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}\mathbb{E}{\left|{\int }_1^s{\left(s-y\right)}^{\vartheta -1}[F_2(y,z\left(y\right))-F_2(y,\varpi \left(y\right))]dW\left(y\right)\right|}^2\right).\hfill \end{array}$$

We obtain

$$\begin{array}{ccc}\hfill \mathbb{E}{\left|z\left(s\right)-\varpi \left(s\right)\right|}^2& \le & (m+3)\left(\epsilon +\sum _{j=1}^m{\displaystyle \frac{{\left(logT\right)}^{2{\xi }_j-1}{L}^2}{(2{\xi }_j-1)\mathsf{\Gamma }{\left({\xi }_j\right)}^2}}{\int }_1^s\mathbb{E}{\left|z\left(y\right)-\varpi \left(y\right)\right|}^{2}dy\right.\hfill \\ & & +{\displaystyle \frac{{\left(T-1\right)}^{2\vartheta -1}{L}^2}{(2\vartheta -1)\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\int }_1^s\mathbb{E}{\left|z\left(y\right)-\varpi \left(y\right)\right|}^{2}dy\hfill \\ & & \left.+{\displaystyle \frac{L^2}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}{\int }_1^s{\left(s-y\right)}^{2\vartheta -2}\mathbb{E}{\left|z\left(y\right)-\varpi \left(y\right)\right|}^{2}dy\right)\hfill \\ & \le & d_1\epsilon +d_2{\int }_1^s\mathbb{E}{\left|z\left(y\right)-\varpi \left(y\right)\right|}^{2}dy+d_3{\int }_1^s{\left(s-y\right)}^{2\vartheta -2}\mathbb{E}{\left|z\left(y\right)-\varpi \left(y\right)\right|}^{2}dy,\hfill \end{array}$$

where

$$d_1=m+3,d_2=L^2(m+3)\left(\sum _{j=1}^m{\displaystyle \frac{{\left(logT\right)}^{2{\xi }_j-1}}{(2{\xi }_j-1)\mathsf{\Gamma }{\left({\xi }_j\right)}^2}}+{\displaystyle \frac{{\left(T-1\right)}^{2\vartheta -1}}{(2\vartheta -1)\mathsf{\Gamma }{\left(\vartheta \right)}^2}}\right)\mathrm{and}{d}_3={\displaystyle \frac{(m+3)L^2}{\mathsf{\Gamma }{\left(\vartheta \right)}^2}}.$$

Using Generalized Gronwall inequality (see [32]), we obtain

$$\begin{array}{ccc}\hfill \mathbb{E}{\left|z\left(s\right)-\varpi \left(s\right)\right|}^2& \le & \left(d_1\epsilon +d_2{\int }_1^s\mathbb{E}{\left|z\left(y\right)-\varpi \left(y\right)\right|}^{2}dy\right)E_{2\vartheta -1}\left(d_3\mathsf{\Gamma }(2\vartheta -1){\left(s-1\right)}^{2\vartheta -1}\right)\hfill \\ & \le & \epsilon d_4+d_5{\int }_1^s\mathbb{E}{\left|z\left(y\right)-\varpi \left(y\right)\right|}^{2}dy,\hfill \end{array}$$

where

$$d_4=d_1{E}_{2\vartheta -1}\left(d_3\mathsf{\Gamma }(2\vartheta -1){\left(T-1\right)}^{2\vartheta -1}\right)\mathrm{and}{d}_5=d_2{E}_{2\vartheta -1}\left(d_3\mathsf{\Gamma }(2\vartheta -1){\left(T-1\right)}^{2\vartheta -1}\right).$$

It follows from inequality of Gronwall that

$$\mathbb{E}{\left|z\left(s\right)-\varpi \left(s\right)\right|}^2\le \epsilon d_4{e}^{d_5(T-1)}.$$

Thus,

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

where $\tilde{M}=d_4{e}^{d_5(T-1)}$.

Thus, we obtain the UHS with regard to $\epsilon$ of Equation (1). □

5. Examples

Example 1. 

Consider the FISDEs:

$${}^C\mathit{D}_1^{\vartheta }\left(\varpi \left(s\right)-\sum _{i=1}^2{}^H\mathit{I}^{{\displaystyle \frac{3+i}{8}}}{\mathsf{\Phi }}_i(s,\varpi \left(s\right))\right)=F_1(s,\varpi \left(s\right))+F_2(s,\varpi \left(s\right)){\displaystyle \frac{dW\left(s\right)}{ds}},1<s\le 10,$$

(4)

where,

$$\begin{array}{ccc}\hfill {\mathsf{\Phi }}_j(s,\varpi \left(s\right))& =& {\displaystyle \frac{1}{7}}sin\left(2\varpi \left(s\right)\right),j=1,2\hfill \\ \hfill F_1(s,\varpi \left(s\right))& =& {\displaystyle \frac{cos\left(3\varpi \right(s\left)\right)}{\left(s^4+5\right)}},\hfill \\ \hfill F_2(s,\varpi \left(s\right))& =& {\displaystyle \frac{\varpi \left(s\right)}{\left(s^6+7\right)}}.\hfill \end{array}$$

The assumptions $\left({\mathcal{H}}_1\right)$ and $\left({\mathcal{H}}_2\right)$ are satisfied for $L=3$. According to Theorem 1, Equation (4) admits a unique solution ϖ. In addition, we obtain the UHS with regard to ε of the equation from Theorem 3.

Example 2. 

Consider the FISDEs:

$${}^C\mathit{D}_1^{\vartheta }\left(\varpi \left(s\right)-\sum _{i=1}^2{}^H\mathit{I}^{{\displaystyle \frac{2+i}{7}}}{\mathsf{\Phi }}_i(s,\varpi \left(s\right))\right)=F_1(s,\varpi \left(s\right))+F_2(s,\varpi \left(s\right)){\displaystyle \frac{dW\left(s\right)}{ds}},1<s\le 8,$$

(5)

where

$$\begin{array}{ccc}\hfill {\mathsf{\Phi }}_j(s,\varpi \left(s\right))& =& {\displaystyle \frac{1}{5}}cos\left(j\varpi \left(s\right)\right),j=1,2\hfill \\ \hfill F_1(s,\varpi \left(s\right))& =& {\displaystyle \frac{\varpi \left(s\right)}{\left(s^2+1\right)}},\hfill \\ \hfill F_2(s,\varpi \left(s\right))& =& {\displaystyle \frac{sin\left(\varpi \right(s\left)\right)}{\left(s^6+4\right)}}.\hfill \end{array}$$

The assumptions $\left({\mathcal{H}}_1\right)$ and $\left({\mathcal{H}}_2\right)$ are satisfied for $L=1$. According to Theorem 1, Equation (5) admits a unique solution ϖ. Furthermore, we obtain the UHS with regard to ε of the equation from Theorem 3.

6. Conclusions

This work examines the EU of FISDEs using the FPT method and a new norm on ${\mathbb{D}}^2$. The CD of solutions on initial data is also established, demonstrating that small changes in the initial conditions result in correspondingly small variations in the solution. To achieve this, a generalized Gronwall inequality was utilized, providing a powerful tool for handling integral inequalities that arise in the analysis. Additionally, stochastic analysis techniques were employed to thoroughly investigate the HUS of FISDEs. We conclude the paper with two theoretical examples that illustrate our main findings. In future work, we intend to extend our results to the case of pantograph FISDEs.