Ulam–Hyers Stability of Pantograph Hadamard Fractional Stochastic Differential Equations

Abstract

In this article, we investigate the existence and uniqueness Theorem of Pantograph Hadamard fractional stochastic differential equations (PHFSDE) using the fixed-point Theorem of Banach (BFPT). According to the generalized Gronwall inequalities, we prove the stability in the sense of Ulam–Hyers (UHS) of PHFSDE. We give some examples to show the effectiveness of our results.

1. Introduction

The origin of fractional derivation theory dates back to the end of the 17th century; the century when Newton and Leibniz developed the foundations of differential and integral calculus. The first question that led to the fractional calculus was: can the derivative with an integer order $\frac{d^{n}f}{d{x}^n}$ be extended to a fractional order derivative? The answer to the question was yes.

The first conference on the subject of the fractional derivative was organized in June 1974 by B.Ross, entitled “First conference on fractional calculus And its application” at the University of New Haven. The first book published by K.B.Oldham and J. Spanier (see [1]) was devoted to the fractional calculus in 1974 after a collaborative work begun in 1968. The fractional differential equation has been the subject of significant investigation with growing interest. Indeed, many studies can be found in various fields of science and engineering (see [2,3,4,5,6,7,8,9,10,11]). The Hadamard fractional derivatives, whose kernels are defined in terms of logarithmic functions, are an important class of fractional derivative and serve as a natural choice for modeling ultraslow diffusion processes and, thus, attract wide attention (see [2,4,5,11,12,13,14,15,16]).

Dynamic systems may not only depend on present states but also the past states. Stochastic differential equations with delay (SDED) and stochastic pantograph differential equations (SPDEs) are often used to model these phenomena, whose systems depend on the past state $x(t-\theta )$ and $x\left(pt\right)$, where $\theta >0$ and $0<p<1$, respectively. The first paper in this axis appears in [17]; in particular, the SPDEs have much more real-world applications in biology, economy, the sciences, engineering, control and electrodynamics (see [18,19,20,21,22]).

The stability theory of the solution is the most popular topic in this field of stochastic systems and control. Many researchers have investigated a special theory, the Ulam–Hyers stability concept and its applications. We refer the reader to [12,23,24,25,26,27,28,29] and references therein.

To our knowledge, there are no existing works on the stability of PHFSDE; our work is an extension of [12] to the PHFSDE. They are different from the previous results in the literature, and the main highlights of this work are:

(1)

Study the existence and uniqueness Theorem of PHFSDE by using the BFPT.

(2)

Prove the UHS of PHFSDE by employing the stochastic calculus techniques and the generalized Gronwall inequalities.

(3)

Different from the results in [12], due to the influence of the pantograph term in the system, which makes our results more interesting.

The style of this article is organized as follows: Section 2 is devoted to exhibit some fundamental results and definitions of the fractional analysis. In Section 3, we present the existence and uniqueness Theorem EUT and the UHS of PHFSDE. In Section 4, we illustrate our results by examples.

2. Preliminaries

Let $\Phi >1$. Denote by $\{\Omega ,\mathbf{F},{\left({\mathbf{F}}_{\zeta }\right)}_{1\le \zeta \le \Phi },\mathbb{P}\}$ the complete probability space and $W\left(\zeta \right)$ is the standard Brownian motion.

Let ${\mathbb{X}}_{\zeta }=L^2(\Omega ,{\mathbf{F}}_{\zeta },\mathbb{P})$ (for each $1\le \zeta \le \Phi$) be the space of all ${\mathbf{F}}_{\zeta }$-measurable, integrable and mean square functions ${\displaystyle o={(o_1,...,o_k)}^T:\Omega \to {\mathbb{R}}^k}$ with

$${\left|\right|o\left|\right|}_{ms}=\sqrt{\sum _{l=1}^k\mathbb{E}\left(|o_l{|}^2\right)}=\sqrt{{\mathbb{E}\left|\right|o\left|\right|}^2}.$$

Definition 1

([7]). The Hadamard fractional integral of order $\vartheta >0$ for a locally integrable function $g:[1,\infty )\to \mathbb{R}$ is given by

$$I^{\vartheta }g\left(\xi \right)=\frac{1}{\Gamma \left(\vartheta \right)}{\int }_1^{\xi }{\left(log\frac{\xi }{\mu }\right)}^{\vartheta -1}\frac{g\left(\mu \right)}{\mu }d\mu .$$

Definition 2

([7]). The Caputo–Hadamard derivative of fractional order $\vartheta \in (0,1)$ for a locally integrable function $g:[1,\infty )\to \mathbb{R}$ is given by

$${}^{C_H}{D}_1^{\vartheta }g\left(\xi \right)=\frac{1}{\Gamma (1-\vartheta )}\left(\xi \frac{d}{d\xi }\right){\int }_1^{\xi }{\left(log\frac{\xi }{\mu }\right)}^{-\vartheta }\frac{\left(g\left(\mu \right)-g\left(1\right)\right)}{\mu }d\mu .$$

Definition 3

([7]). The Mittag-Leffler function $E_{\xi ,\zeta }:\mathbb{C}\to \mathbb{C}$ is given by

$${\displaystyle E_{\xi ,\zeta }\left(y\right)=\sum _{m=0}^{+\infty }\frac{y^m}{\Gamma (m\xi +\zeta )},}$$

where $\xi >0$, $\zeta >0$, $y\in \mathbb{C}$.

Consider the following CPHFSDE:

$${}^{C_H}{D}_1^{\pi }\delta \left(\zeta \right)=h_1\left(\zeta ,\delta \left(\zeta \right),\delta \left(\rho \zeta \right)\right)+h_2\left(\zeta ,\delta \left(\zeta \right),\delta \left(\rho \zeta \right)\right)\frac{dW\left(\zeta \right)}{d\zeta },$$

(1)

with, $\delta \left(\zeta \right)=\phi \left(\zeta \right)$, for $\zeta \in [\rho ,1]$, is the initial condition, $\rho \in (0,1),$$\pi \in (\frac{1}{2},1)$, $h_1:[1,\Phi ]\times {\mathbb{R}}^k\times {\mathbb{R}}^k⟶{\mathbb{R}}^k$ and $h_2:[1,\Phi ]\times {\mathbb{R}}^k\times {\mathbb{R}}^k⟶{\mathbb{R}}^k$ are measurable functions. Set the following hypothesis:

${\mathcal{H}}_1$: There is $L>0$ such that

$$\left|\right|h_1(\zeta ,{\delta }_1,{\tilde{\delta }}_1)-h_2(\zeta ,{\delta }_2,{\tilde{\delta }}_2)\left|\right|+\left|\right|h_1(\zeta ,{\delta }_1,{\tilde{\delta }}_1)-h_2(\zeta ,{\delta }_2,{\tilde{\delta }}_2)\left|\right|\le L\left(\left|\right|{\delta }_1-{\delta }_2\left|\right|+\left|\right|{\tilde{\delta }}_1-{\tilde{\delta }}_2\left|\right|\right),$$

(2)

for all $(\zeta ,{\delta }_1,{\delta }_2,{\tilde{\delta }}_1,{\tilde{\delta }}_2)\in [1,\Phi ]\times {\mathbb{R}}^k\times {\mathbb{R}}^k\times {\mathbb{R}}^k\times {\mathbb{R}}^k$.

${\mathcal{H}}_2$: $h_1(·,0,0)$ and $h_2(·,0,0)$ verifies

$$\left|\right|h_2{(·,0,0)\left|\right|}_{\infty }=ess\underset{l\in [1,\Phi ]}{sup}\left|\right|h_2(l,0,0)\left|\right|<\infty ,$$

(3)

$${\int }_1^{\Phi }\left|\right|h_1(l,0,0){\left|\right|}^{2}dl<\infty .$$

3. Main Results

Denote by ${\mathbb{H}}^2\left([\rho ,\Phi ]\right)$ the set of all processes $\delta$ which are measurable and ${\mathcal{F}}_{\Phi }$-adapted, where ${\mathcal{F}}_{\Phi }={\left\{{\mathbf{F}}_l\right\}}_{l\in [\rho ,\Phi ]}$, such that

$${\left|\right|\delta \left|\right|}_{{\mathbb{H}}^2}=\underset{\rho \le l\le \Phi }{sup}{\left|\right|\delta \left(l\right)\left|\right|}_{ms}<\infty .$$

It is not hard to prove that ${\displaystyle ({\mathbb{H}}^2\left([\rho ,\Phi ]\right),||·|{|}_{{\mathbb{H}}^2})}$ is a Banach space. For $\phi \in {\mathbb{H}}^2\left([\rho ,1]\right)$, let $R_{\phi }:{\mathbb{H}}^2\left([\rho ,\Phi ]\right)\to {\mathbb{H}}^2\left([\rho ,\Phi ]\right)$ be the operator defined by:

$$R_{\phi }B\left(\zeta \right)=\phi \left(1\right)+\frac{1}{\Gamma \left(\pi \right)}\left[{\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\frac{h_1(s,B\left(s\right),B\left(\rho s\right))}{s}ds+{\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\frac{h_2(s,B\left(s\right),B\left(\rho s\right))}{s}dW\left(s\right)\right]$$

(4)

for $\zeta \in [1,\Phi ]$ and $R_{\phi }B\left(\zeta \right)=\phi \left(\zeta \right)$ for $\zeta \in [\rho ,1]$.

Proposition 1.

For each $\phi \in {\mathbb{H}}^2\left([\rho ,1]\right)$, $R_{\phi }$ is well-defined.

Proof. 

Let $B\in {\mathbb{H}}^2\left([\rho ,\Phi ]\right)$. We have

$$\begin{array}{ccc}\hfill \left|\right|R_{\phi }B\left(\zeta \right){\left|\right|}_{ms}^2& \le & {3\left|\right|\phi \left(1\right)\left|\right|}_{ms}^2+\frac{3}{\Gamma {\left(\pi \right)}^2}\mathbb{E}\left({\left|\left|{\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\frac{h_1(s,B\left(s\right),B\left(\rho s\right))}{s}ds\right|\right|}^2\right)\hfill \\ & +& \frac{3}{\Gamma {\left(\pi \right)}^2}\mathbb{E}\left({\left|\left|{\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\frac{h_2(s,B\left(s\right),B\left(\rho s\right))}{s}dW\left(s\right)\right|\right|}^2\right).\hfill \end{array}$$

(5)

Using ${\mathcal{H}}_1$, we get

$${\left|\left|h_1(l,B\left(l\right),B\left(\rho l\right))\right|\right|}^2\le 2L^2{\left(\left|\left|B\left(l\right)\right|\right|+\left|\left|B\left(\rho l\right)\right|\right|\right)}^2+2{\left|\left|h_1(l,0,0)\right|\right|}^2.$$

(6)

Then,

$$\begin{array}{ccc}\hfill E\left({\int }_1^{\zeta }{\left|\left|h_1(l,B\left(l\right),B\left(\rho l\right))\right|\right|}^{2}dl\right)& \le & 4L^2\left(\Phi -1\right)\underset{l\in [1,\Phi ]}{sup}E\left({\left|\left|B\left(l\right)\right|\right|}^2+{\left|\left|B\left(\rho l\right)\right|\right|}^2\right)+2{\int }_1^{\Phi }{\left|\left|h_1(l,0,0)\right|\right|}^{2}dl\hfill \\ & \le & 8L^2\left(\Phi -1\right)\underset{l\in [\rho ,\Phi ]}{sup}E\left({\left|\left|B\left(l\right)\right|\right|}^2\right)+2{\int }_1^{\Phi }{\left|\left|h_1(l,0,0)\right|\right|}^{2}dl.\hfill \end{array}$$

(7)

By employing the Cauchy–Schwartz inequality (see [30]), we have

$$\mathbb{E}\left({\left|\left|{\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\frac{h_1(s,B\left(s\right),B\left(\rho s\right))}{s}ds\right|\right|}^2\right)$$

$$\begin{array}{ccc}& \le & \left({\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}ds\right)\mathbb{E}\left({\int }_1^{\zeta }{\left|\left|h_1(s,B\left(s\right),B\left(\rho s\right))\right|\right|}^{2}ds\right)\hfill \\ & \le & \frac{1}{2\pi -1}{\left(ln\Phi \right)}^{2\pi -1}[8L^2\left(\Phi -1\right)\underset{l\in [\rho ,\Phi ]}{sup}E\left({\left|\left|B\left(l\right)\right|\right|}^2\right)\hfill \\ & +& 2{\int }_1^{\Phi }{\left|\left|h_1(l,0,0)\right|\right|}^{2}dl].\hfill \end{array}$$

(8)

By Itô’s isometry formula (see [30]), we get

$$\mathbb{E}\left({\left|\left|{\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\frac{h_2(s,B\left(s\right),B\left(\rho s\right))}{s}dW\left(s\right)\right|\right|}^2\right)=\mathbb{E}\left({\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}{\left|\left|\frac{h_2(s,B\left(s\right),B\left(\rho s\right))}{s}\right|\right|}^{2}ds\right).$$

(9)

Using ${\mathcal{H}}_1$, we get

$${\left|\left|h_2(s,B\left(s\right),B\left(\rho s\right))\right|\right|}^2\le 4L^2\left({\left|\left|B\left(s\right)\right|\right|}^2+{\left|\left|B\left(\rho s\right)\right|\right|}^2\right)+2{\left|\left|h_2(·,0,0)\right|\right|}_{\infty }^2.$$

(10)

Therefore,

$$\mathbb{E}\left({\left|\left|{\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\frac{h_2(s,B\left(s\right),B\left(\rho s\right))}{s}dW\left(s\right)\right|\right|}^2\right)$$

$$\begin{array}{ccc}& \le & \mathbb{E}\left({\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}\left[4L^2\left({\left|\left|B\left(s\right)\right|\right|}^2+{\left|\left|B\left(\rho s\right)\right|\right|}^2\right)+2{\left|\left|h_2(·,0,0)\right|\right|}_{\infty }^2\right]ds\right)\hfill \\ & \le & \frac{8L^2}{2\pi -1}{\left(ln\Phi \right)}^{2\pi -1}{\left|\left|T\right|\right|}_{{\mathbb{H}}_2}^2+\frac{2}{2\pi -1}{\left(ln\Phi \right)}^{2\pi -1}{\left|\left|h_2(·,0)\right|\right|}_{\infty }^2,\hfill \end{array}$$

(11)

and the proof is completed. □

Theorem 1.

Assume that ${\mathcal{H}}_1$–${\mathcal{H}}_2$ hold. Then, Equation (1) has a unique global solution.

Proof. 

Let $\nu$ with ${\displaystyle {\nu }^{2\pi -1}>\frac{8L^2\Phi \Gamma (2\pi -1)}{\Gamma {\left(\pi \right)}^2}}$. Let $\left|\right|·{\left|\right|}_{\nu }$ be the norm on ${\mathbb{H}}^2\left([\rho ,\Phi ]\right)$ defined by

$${\left|\right|\delta \left|\right|}_{\nu }=\underset{\zeta \in [\rho ,\Phi ]}{sup}\sqrt{\frac{\mathbb{E}\left({\left|\left|\delta \left(\zeta \right)\right|\right|}^2\right)}{k\left(\zeta \right)}},\forall \delta \in {\mathbb{H}}^2\left([\rho ,\Phi ]\right),$$

(12)

with ${\displaystyle k\left(\zeta \right)={\zeta }^{\nu }}$ for $\zeta \in [\rho ,\Phi ]$. It is not hard to show that $\left|\right|·{\left|\right|}_{{\mathbb{H}}^2}$ and $\left|\right|·{\left|\right|}_{\nu }$ are equivalent. Hence, $({\mathbb{H}}^2\left([\rho ,\Phi ]\right),||·|{|}_{\nu })$ is a Banach space.

Let ${\delta }_1$, ${\delta }_2\in {\mathbb{H}}^2\left([\rho ,\Phi ]\right)$, we have $\forall \zeta \in [\rho ,1]$, $R_{\phi }{\delta }_1\left(\zeta \right)-R_{\phi }{\delta }_2\left(\zeta \right)=0$.

For $\zeta \in [1,\Phi ]$, we get

$$\mathbb{E}\left({\left|\left|R_{\phi }{\delta }_1\left(\zeta \right)-R_{\phi }{\delta }_2\left(\zeta \right)\right|\right|}^2\right)$$

$$\begin{array}{ccc}& \le & \frac{2}{\Gamma {\left(\pi \right)}^2}\mathbb{E}\left({\left|\left|{\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\frac{\left(h_1(s,{\delta }_1\left(s\right),{\delta }_1\left(\rho s\right))-h_1(s,{\delta }_2\left(s\right),{\delta }_2\left(\rho s\right))\right)}{s}ds\right|\right|}^2\right)\hfill \\ & +& \frac{2}{\Gamma {\left(\pi \right)}^2}\mathbb{E}\left({\left|\left|{\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\frac{\left(h_2(s,{\delta }_1\left(s\right),{\delta }_1\left(\rho s\right))-h_2(s,{\delta }_2\left(s\right),{\delta }_2\left(\rho s\right))\right)}{s}dW\left(s\right)\right|\right|}^2\right).\hfill \end{array}$$

By the Cauchy–Schwartz inequality (see [30]), we get

$$\mathbb{E}\left({\left|\left|{\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\frac{\left(h_1(l,{\delta }_1\left(l\right),{\delta }_1\left(\rho l\right))-h_1(l,{\delta }_2\left(l\right),{\delta }_2\left(\rho l\right))\right)}{s}dl\right|\right|}^2\right)$$

$$\le 2L^2(\Phi -1){\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}\mathbb{E}\left({\left|\left|{\delta }_1\left(s\right)-{\delta }_2\left(s\right)\right|\right|}^2+{\left|\left|{\delta }_1\left(\rho s\right)-{\delta }_2\left(\rho s\right)\right|\right|}^2\right)ds.$$

Using the Itô isometry formula (see [30]), we get

$$\mathbb{E}\left({\left|\left|{\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\frac{\left(h_2(s,{\delta }_1\left(s\right),{\delta }_1\left(\rho s\right))-h_2(s,{\delta }_2\left(s\right),{\delta }_2\left(\rho s\right))\right)}{s}dW\left(s\right)\right|\right|}^2\right)$$

$$\begin{array}{ccc}& =& \mathbb{E}\left({\int }_1^{\zeta }\frac{1}{s^2}{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}{\left|\left|h_2(s,{\delta }_1\left(s\right),{\delta }_1\left(\rho s\right))-h_2(s,{\delta }_2\left(s\right),{\delta }_2\left(\rho s\right))\right|\right|}^{2}ds\right)\hfill \\ & \le & 2L^2{\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}\left(\mathbb{E}\left({\left|\left|{\delta }_1\left(s\right)-{\delta }_2\left(s\right)\right|\right|}^2\right)+\mathbb{E}\left({\left|\left|{\delta }_1\left(\rho s\right)-{\delta }_2\left(\rho s\right)\right|\right|}^2\right)\right)ds.\hfill \end{array}$$

(13)

Then,

$$\mathbb{E}\left({\left|\left|R_{\phi }{\delta }_1\left(\zeta \right)-R_{\phi }{\delta }_2\left(\zeta \right)\right|\right|}^2\right)\le c{\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}\left[\mathbb{E}\left({\left|\left|{\delta }_1\left(s\right)-{\delta }_2\left(s\right)\right|\right|}^2\right)+\mathbb{E}\left({\left|\left|{\delta }_1\left(\rho s\right)-{\delta }_2\left(\rho s\right)\right|\right|}^2\right)\right]ds,$$

where $c=\frac{4L^2\Phi }{\Gamma {\left(\pi \right)}^2}$.

Then,

$$\begin{array}{ccc}\hfill \frac{\mathbb{E}\left({\left|\left|R_{\phi }{\delta }_1\left(\zeta \right)-R_{\phi }{\delta }_2\left(\zeta \right)\right|\right|}^2\right)}{{\zeta }^{\nu }}& \le & \frac{c}{{\zeta }^{\nu }}{\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}[\frac{k\left(s\right)\mathbb{E}\left({\left|\left|{\delta }_1\left(s\right)-{\delta }_2\left(s\right)\right|\right|}^2\right)}{k\left(s\right)}\hfill \\ & +& \frac{k\left(\rho s\right)\mathbb{E}\left({\left|\left|{\delta }_1\left(\rho s\right)-{\delta }_2\left(\rho s\right)\right|\right|}^2\right)}{k\left(\rho s\right)}]ds\hfill \\ & \le & \frac{c}{{\zeta }^{\nu }}{\left|\left|{\delta }_1-{\delta }_2\right|\right|}_{\nu }^2{\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}\left(k\left(s\right)+k\left(\rho s\right)\right)ds\hfill \\ & \le & \frac{2c}{{\zeta }^{\nu }}{\left|\left|{\delta }_1-{\delta }_2\right|\right|}_{\nu }^2{\int }_0^{\nu ln\zeta }{\left(\frac{t}{\nu }\right)}^{\pi -1}{\zeta }^{\nu }\frac{e^{-t}}{\nu }dt\hfill \\ & \le & \frac{2c\Gamma (2\pi -1)}{{\nu }^{2\pi -1}}{\left|\left|{\delta }_1-{\delta }_2\right|\right|}_{\nu }^2.\hfill \end{array}$$

(14)

Therefore,

$${\left|\left|R_{\phi }{\delta }_1-R_{\phi }{\delta }_2\right|\right|}_{\nu }\le \sqrt{\frac{2c\Gamma (2\pi -1)}{{\nu }^{2\pi -1}}}{\left|\left|{\delta }_1-{\delta }_2\right|\right|}_{\nu }.$$

(15)

Then, there is a unique solution of (1). □

Definition 4.

Equation (1) is stable with respect to ε in the sense of Ulam–Hyers (UHSwrε) if there exists $N>0$ such that for every $\epsilon >0$ and $z\in {\mathbb{H}}^2\left([\rho ,\Phi ]\right)$ of the following inequality: $\forall \zeta \in [1,\Phi ]$

$$\mathbb{E}{\left|\left|z\left(\zeta \right)-z\left(1\right)-\frac{1}{\Gamma \left(\pi \right)}\left({\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\left[h_1(s,z\left(s\right),z\left(\rho s\right))ds+h_2(s,z\left(s\right),z\left(\rho s\right))dW\left(s\right)\right]\right)\right|\right|}^2\le \epsilon ,$$

(16)

there exists a solution $\delta \in {\mathbb{H}}^2\left([\rho ,\Phi ]\right)$ of (1), with $\delta \left(t\right)=z\left(t\right)$ for $t\in [\rho ,1]$, satisfies ${\displaystyle \mathbb{E}\left|\right|z\left(\zeta \right)-{\delta \left(\zeta \right)\left|\right|}^2\le N\epsilon }$, $\forall \zeta \in [1,\Phi ]$.

Theorem 2.

Under ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, the PHFSDE (1) is UHSwrε on $[1,\Phi ]$.

Proof. 

Set $\epsilon >0$ and $z\in {\mathbb{H}}^2\left([\rho ,\Phi ]\right)$ satisfies (16). Denote by $\delta \in {\mathbb{H}}^2\left([\rho ,\Phi ]\right)$ be the solution of (1) with $\delta \left(t\right)=z\left(t\right)$ for $t\in [\rho ,1]$, then

$$\delta \left(\zeta \right)=z\left(1\right)+\frac{1}{\Gamma \left(\pi \right)}\left({\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}{h}_1(s,\delta \left(s\right),\delta \left(\rho s\right))ds+{\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}{h}_2(s,\delta \left(s\right),\delta \left(\rho s\right))dW\left(s\right)\right).$$

(17)

Thus,

$$\begin{array}{ccc}& & \mathbb{E}\left|\right|z\left(\zeta \right)-{\delta \left(\zeta \right)\left|\right|}^2\hfill \\ & \le & 2\mathbb{E}{\left|\left|z\left(\zeta \right)-z\left(1\right)-\frac{1}{\Gamma \left(\pi \right)}\left({\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}{h}_1(s,z\left(s\right),z\left(\rho s\right))ds+{\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}{h}_2(s,z\left(s\right),z\left(\rho s\right))dW\left(s\right)\right)\right|\right|}^2\hfill \\ & +& 2\mathbb{E}{\left|\left|\frac{1}{\Gamma \left(\pi \right)}\left({\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\left(h_1(s,z\left(s\right),z\left(\rho s\right))-h_1(s,\delta \left(s\right),\delta \left(\rho s\right))\right)ds+{\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\left(h_2(s,z\left(s\right),z\left(\rho s\right))-h_2(s,\delta \left(s\right),\delta \left(\rho s\right))\right)dW\left(s\right)\right)\right|\right|}^2.\hfill \end{array}$$

(18)

Then, according to the Cauchy–Schwartz inequality (see [30]) and ${\mathcal{H}}_1$–${\mathcal{H}}_2$, we obtain

$$\mathbb{E}\left|\right|z\left(\zeta \right)-{\delta \left(\zeta \right)\left|\right|}^2$$

$$\begin{array}{ccc}& \le & 2\epsilon +4\mathbb{E}{\left|\left|\frac{1}{\Gamma \left(\pi \right)}{\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\left(h_1(s,z\left(s\right),z\left(\rho s\right))-h_1(s,\delta \left(s\right),\delta \left(\rho s\right))\right)ds\right|\right|}^2\hfill \\ & +& 4\mathbb{E}{\left|\left|\frac{1}{\Gamma \left(\pi \right)}{\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\left(h_2(s,z\left(s\right),z\left(\rho s\right))-h_2(s,\delta \left(s\right),\delta \left(\rho s\right))\right)dW\left(s\right)\right|\right|}^2\hfill \\ & \le & 2\epsilon +\frac{8L^2{\left(ln\Phi \right)}^{2\pi -1}}{(2\pi -1)\Gamma {\left(\pi \right)}^2}\mathbb{E}\left({\int }_1^{\zeta }\left(\left|\right|z\left(s\right)-{\delta \left(s\right)\left|\right|}^2+\left|\right|z\left(\rho s\right)-{\delta \left(\rho s\right)\left|\right|}^2\right)ds\right)\hfill \\ & +& \frac{8L^2}{\Gamma {\left(\pi \right)}^2}\mathbb{E}\left({\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}\left(\left|\right|z\left(s\right)-{\delta \left(s\right)\left|\right|}^2+\left|\right|z\left(\rho s\right)-{\delta \left(\rho s\right)\left|\right|}^2\right)ds\right).\hfill \end{array}$$

Let ${\displaystyle \varphi \left(\zeta \right)=\underset{s\in [\rho ,\zeta ]}{sup}\mathbb{E}\left|\right|z\left(s\right)-{\delta \left(s\right)\left|\right|}^2}$ for $\zeta \in [1,\Phi ]$.

We derive that $\mathbb{E}\left|\right|z\left(s\right)-{\delta \left(s\right)\left|\right|}^2\le \varphi \left(s\right)$ and $\mathbb{E}\left|\right|z\left(\rho s\right)-{\delta \left(\rho s\right)\left|\right|}^2\le \varphi \left(s\right)$, for all $s\in [1,\Phi ].$

Then, by Fubini’s Theorem, for $\zeta \in [1,\Phi ]$, we obtain

$$\mathbb{E}\left|\right|z\left(\zeta \right)-{\delta \left(\zeta \right)\left|\right|}^2\le 2\epsilon +\frac{16L^2{\left(ln\Phi \right)}^{2\pi -1}}{(2\pi -1)\Gamma {\left(\pi \right)}^2}{\int }_1^{\zeta }\varphi \left(s\right)ds+\frac{16L^2}{\Gamma {\left(\pi \right)}^2}{\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}\varphi \left(s\right)ds.$$

Hence, for all $l\in [1,\zeta ]$,

$$\mathbb{E}\left|\right|z\left(l\right)-{\delta \left(l\right)\left|\right|}^2\le 2\epsilon +\frac{16L^2{\left(ln\Phi \right)}^{2\pi -1}}{(2\pi -1)\Gamma {\left(\pi \right)}^2}{\int }_1^{\zeta }\varphi \left(s\right)ds+\frac{16L^2}{\Gamma {\left(\pi \right)}^2}{\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}\varphi \left(s\right)ds.$$

Then,

$$\varphi \left(\zeta \right)\le 2\epsilon +c_1{\int }_1^{\zeta }\varphi \left(s\right)ds+c_2{\int }_1^{\zeta }\frac{1}{s}{\left(ln\frac{\zeta }{s}\right)}^{2\pi -2}\varphi \left(s\right)ds,$$

(19)

for all $\zeta \in [1,\Phi ]$, where ${\displaystyle c_1=\frac{16L^2{\left(ln\Phi \right)}^{2\pi -1}}{(2\pi -1)\Gamma {\left(\pi \right)}^2}}$ and ${\displaystyle c_2=\frac{16L^2}{\Gamma {\left(\pi \right)}^2}}$.

Using generalized Gronwall inequality, we get

$$\begin{array}{ccc}\hfill \varphi \left(\zeta \right)& \le & \left[2\epsilon +c_1{\int }_1^{\zeta }\varphi \left(s\right)ds\right]E_{2\pi -1}\left(c_2\Gamma (2\pi -1){\left(ln\zeta \right)}^{2\pi -1}\right)\hfill \\ & \le & c_3\epsilon +c_4{\int }_1^{\zeta }\varphi \left(s\right)ds,\hfill \end{array}$$

(20)

where ${\displaystyle c_3=2E_{2\pi -1}\left(c_2\Gamma (2\pi -1){\left(ln\Phi \right)}^{2\pi -1}\right)}$ and ${\displaystyle c_4=c_1{E}_{2\pi -1}\left(c_2\Gamma (2\pi -1){\left(ln\Phi \right)}^{2\pi -1}\right)}$.

Using Gronwall inequality, we get

$$\varphi \left(\zeta \right)\le c_3\epsilon e^{c_4(\zeta -1)}.$$

(21)

Hence,

$$\mathbb{E}\left|\right|z\left(\zeta \right)-{\delta \left(\zeta \right)\left|\right|}^2\le N\epsilon ,\forall \zeta \in [1,\Phi ],$$

(22)

where ${\displaystyle N=c_3{e}^{c_4(\Phi -1)}}$.

Thus, Equation (1) is UHSwr$\epsilon$. □

Remark 1.

The authors in [31] studied the existence, uniqueness and the Hyers–Ulam stability of stochastic differential equations with the Caputo fractional derivative.

4. Examples

This section is devoted to illustrate our results with two examples.

Example 1.

Let the PHFSDE for each $\epsilon >0$ and for $\zeta \in [1,3]$ given by

$$\left\{\begin{array}{c}{}^C{D}_1^{\pi }\delta \left(\zeta \right)=h_1\left(\zeta ,\delta \left(\zeta \right),\delta \left(0.3\zeta \right)\right)+h_2\left(\zeta ,\delta \left(\zeta \right),\delta \left(0.3\zeta \right)\right)\frac{dW\left(\zeta \right)}{d\zeta },\hfill \\ \\ \mathbb{E}{\left|\delta \left(\zeta \right)-\delta \left(1\right)-\frac{1}{\Gamma \left(\pi \right)}\left({\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\left[h_1(s,\delta \left(s\right),\delta \left(0.3s\right))ds+h_2(s,\delta \left(s\right),\delta \left(0.3s\right))dW\left(s\right)\right]\right)\right|}^2\le \epsilon ,\hfill \\ \\ \delta \left(\zeta \right)=\phi \left(\zeta \right),\zeta \in [0.3,1],\hfill \end{array}\right.$$

(23)

where

$$\begin{array}{ccc}\hfill \delta \left(\zeta \right)& \in & {\mathbb{H}}^2([0.3,3],\mathbb{R})\hfill \\ \hfill h_1(\zeta ,\delta \left(\zeta \right),\delta \left(0.3\zeta \right))& =& e^{0.3\zeta }arctan\left(\delta \left(\zeta \right)\right)\hfill \\ \hfill h_2(\zeta ,\delta \left(\zeta \right),\delta \left(0.3\zeta \right))& =& \frac{{sin}^2\left(\delta \left(0.3\zeta \right)\right)}{\sqrt{1+0.3{\zeta }^2}}.\hfill \end{array}$$

We will show that Equation (23) is UHSwrε.

Let $(\zeta ,{\delta }_1,{\delta }_2)\in [1,3]\times \mathbb{R}\times \mathbb{R}$, then

$$|h_1(\zeta ,{\delta }_1,{\tilde{\delta }}_1)-h_1(\zeta ,{\delta }_2,{\tilde{\delta }}_2)|+|h_2(\zeta ,{\delta }_1,{\tilde{\delta }}_1)-h_2(\zeta ,{\delta }_2,{\tilde{\delta }}_2)|\le e^1\left(|{\delta }_1-{\delta }_2|+|{\tilde{\delta }}_1-{\tilde{\delta }}_2|\right),$$

(24)

Consequently, hypothesis ${\mathcal{H}}_1$ is satisfied. Moreover,

$$|h_2{(·,0,0)|}_{\infty }\le \frac{1}{2},$$

(25)

and

$${\int }_1^3{\left|h_1(\zeta ,0,0)\right|}^{2}d\zeta \le 1.$$

(26)

Therefore, assumptions ${\mathcal{H}}_1$-${\mathcal{H}}_2$ are fulfilled. Hence, using Theorem 2, Equation (23) is UHSwrε on $[1,3]$.

Example 2.

Let the PHFSDE for each $\epsilon >0$ and for $\zeta \in [1,5]$ given by

$$\left\{\begin{array}{c}{}^C{D}_1^{\pi }\delta \left(\zeta \right)=h_1\left(\zeta ,\delta \left(\zeta \right),\delta \left(0.75\zeta \right)\right)+h_2\left(\zeta ,\delta \left(\zeta \right),\delta \left(0.75\zeta \right)\right)\frac{dW\left(\zeta \right)}{d\zeta },\hfill \\ \\ \mathbb{E}{\left|\delta \left(\zeta \right)-\delta \left(1\right)-\frac{1}{\Gamma \left(\pi \right)}\left({\int }_1^{\zeta }{\left(ln\frac{\zeta }{s}\right)}^{\pi -1}\left[h_1(s,\delta \left(s\right),\delta \left(0.75s\right))ds+h_2(s,\delta \left(s\right),\delta \left(0.75s\right))dW\left(s\right)\right]\right)\right|}^2\le \epsilon ,\hfill \\ \\ \delta \left(\zeta \right)=\phi \left(\zeta \right),\zeta \in [0.75,1]\hfill \end{array}\right.$$

(27)

where

$$\begin{array}{ccc}\hfill \delta \left(\zeta \right)& \in & {\mathbb{H}}^2([0.75,5],\mathbb{R})\hfill \\ \hfill h_1(\zeta ,\delta \left(\zeta \right),\delta \left(0.75\zeta \right))& =& \frac{cos\delta \left(\zeta \right)+sin\delta \left(0.75\zeta \right)}{1+{\zeta }^2}\hfill \\ \hfill h_2(\zeta ,\delta \left(\zeta \right),\delta \left(0.75\zeta \right))& =& \frac{cos\delta \left(\zeta \right)+\delta \left(0.75\zeta \right)}{1+e^{\zeta }}.\hfill \end{array}$$

We will show that Equation (27) is UHSwrε.

Let $(\zeta ,{\delta }_1,{\delta }_2)\in [1,5]\times \mathbb{R}\times \mathbb{R}$, then

$$|h_1(\zeta ,{\delta }_1,{\tilde{\delta }}_1)-h_1(\zeta ,{\delta }_2,{\tilde{\delta }}_2)|+|h_2(\zeta ,{\delta }_1,{\tilde{\delta }}_1)-h_2(\zeta ,{\delta }_2,{\tilde{\delta }}_2)|\le 2\left(|{\delta }_1-{\delta }_2|+|{\tilde{\delta }}_1-{\tilde{\delta }}_2|\right),$$

(28)

Consequently, hypothesis ${\mathcal{H}}_1$ is satisfied. Moreover,

$$|h_2{(·,0,0)|}_{\infty }\le 1,$$

(29)

and

$${\int }_1^5{\left|h_1(\zeta ,0,0)\right|}^{2}d\zeta \le arctan5.$$

(30)

Therefore, assumptions ${\mathcal{H}}_1$–${\mathcal{H}}_2$ are fulfilled. Hence, using Theorem 2, Equation (27) is UHSwrε in $[1,5]$.

5. Conclusions

In this article, we study the existence and uniqueness Theorem of PHFSDE using the BFPT. We prove the stability in the sense of Ulam–Hyers of PHFSDE by using Gronwall inequalities and stochastic analysis methods. Finally, we exhibit two examples to make our results applicable.