The Cădariu-Radu Method for Existence, Uniqueness and Gauss Hypergeometric Stability of Ω-Hilfer Fractional Differential Equations

Abstract

Using the Cădariu–Radu method derived from the Diaz–Margolis theorem, we study the existence, uniqueness and Gauss hypergeometric stability of Ω-Hilfer fractional differential equations defined on compact domains. Next, we show the main results for unbounded domains. To illustrate the main result for a fractional system, we present an example.

1. Introduction

In 1941, Hyers proved that for each $\vartheta >0,$ there exists a $\sigma >0$ such that if $\parallel h(z+w)-h\left(z\right)-h\left(w\right)\parallel <\sigma$, then there exists an additive mapping $h^{\prime }\left(z\right)$ with $\parallel h\left(z\right)-h^{\prime }\left(z\right)\parallel \le \vartheta$. Next, the Hyers’ results has been developed by Th. M. Rassias. In fact, he attempted to weaken the condition as follows:

$$\parallel h(z+w)-h\left(z\right)-h\left(w\right)\parallel <\sigma (\parallel z\parallel +\parallel w\parallel ),$$

this led to the generalization of what is known as Hyers–Ulam–Rassias stability of functional equations [1,2].

On the other hand, in 1695, the question of the semi-derivative was raised. The first known references can be found in inventing of the concept of the derivative of nth order, belonging to Marquis de l’Hospital and Gottfried Leibniz. This question attracted the interest of many mathematicians like Liouville, Riemann, Euler, Laplace and many others. The theory of fractional calculus (FC), developed rapidly and its application is done very widely nowadays. For more details, see [3,4,5,6,7,8].

Motivated by [9], in this paper we replace Mittag–Leffler control functions by Hypergeometric control functions to investigate Hypergeometric stability of the following Ω-Hilfer fractional differential equation, through the Cădariu–Radu method derived from the Diaz–Margolis theorem,

$$\left\{\begin{array}{cc}{}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}g\left(\tau \right)=F(\tau ,g\left(\tau \right),g\left(\mu \left(\tau \right)\right)),& \tau \in (0,p],\\ {\mathsf{I}}_{0^{+}}^{1-{\mathfrak{S}}_3;\mathsf{\Omega }}g\left(0^{+}\right)={\mathcal{E}}_{\circ },& {\mathcal{E}}_{\circ }\in \mathbb{R},\end{array}\right.$$

(1)

in which ${}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}(.)$ is the $\mathsf{\Omega }$-Hilfer fractional derivative of order $0<{\mathfrak{S}}_1\le 1$ and type $0\le {\mathfrak{S}}_2\le 1$, ${\mathsf{I}}_{0^{+}}^{1-{\mathfrak{S}}_3;\mathsf{\Omega }}(.)$ is the Riemann–Liouville fractional integral of order $1-{\mathfrak{S}}_3$, ${\mathfrak{S}}_3={\mathfrak{S}}_1+{\mathfrak{S}}_2(1-{\mathfrak{S}}_1)$ w.r.t the mapping $\mathsf{\Omega }$ ([10]), and $F:(0,p]\times {\mathbb{R}}^2⟶\mathbb{R}$ is a given mapping.

2. Preliminaries

Let $[{\mathfrak{M}}_1,{\mathfrak{M}}_2](0\le {\mathfrak{M}}_1<{\mathfrak{M}}_2<\infty )$ be a finite interval on ${\mathbb{R}}^{+}$, and let $C[{\mathfrak{M}}_1,{\mathfrak{M}}_2]$ be the space of continuous functions $\mu :[{\mathfrak{M}}_1,{\mathfrak{M}}_2]⟶\mathbb{R}$ with norm

$${\parallel \mu \parallel }_{C[{\mathfrak{M}}_1,{\mathfrak{M}}_2]}=\underset{{\mathfrak{M}}_1\le {\mathfrak{B}}_{\circ }\le {\mathfrak{M}}_2}{sup}\left|\mu \left({\mathfrak{B}}_{\circ }\right)\right|.$$

The weighted space $C_{1-{\mathfrak{S}}_3;\mathsf{\Omega }}[{\mathfrak{M}}_1,{\mathfrak{M}}_2]$ of continuous $\mu$ on $({\mathfrak{M}}_1,{\mathfrak{M}}_2]$ is defined by (see [5]).

$C_{1-{\mathfrak{S}}_3;\mathsf{\Omega }}[{\mathfrak{M}}_1,{\mathfrak{M}}_2]=\left\{\mu :({\mathfrak{M}}_1,{\mathfrak{M}}_2]⟶\mathbb{R};{(\mathsf{\Omega }\left({\mathfrak{B}}_{\circ }\right)-\mathsf{\Omega }\left({\mathfrak{M}}_1\right))}^{1-{\mathfrak{S}}_3}\mu \left({\mathfrak{B}}_{\circ }\right)\in C[{\mathfrak{M}}_1,{\mathfrak{M}}_2]\right\},$ in which $0\le {\mathfrak{S}}_3<1$, with norm

$${\parallel \mu \parallel }_{C_{1-{\mathfrak{S}}_3;\mathsf{\Omega }}[{\mathfrak{M}}_1,{\mathfrak{M}}_2]}=\underset{{\mathfrak{B}}_{\circ }\in [{\mathfrak{M}}_1,{\mathfrak{M}}_2]}{sup}\left|{(\mathsf{\Omega }\left({\mathfrak{B}}_{\circ }\right)-\mathsf{\Omega }\left({\mathfrak{M}}_1\right))}^{1-{\mathfrak{S}}_3}\mu \left({\mathfrak{B}}_{\circ }\right)\right|.$$

Definition 1

([11]). Consider the Gamma function Γ. Let $({\mathfrak{M}}_1,{\mathfrak{M}}_2)(-\infty \le {\mathfrak{M}}_1<{\mathfrak{M}}_2\le \infty )$ be an interval on $\mathbb{R}$, and ${\mathfrak{S}}_1>0$. Furthermore, consider the increasing map $\mathsf{\Omega }\left({\mathfrak{B}}_{\circ }\right)>0$ on $({\mathfrak{M}}_1,{\mathfrak{M}}_2]$, which has a continuous derivative ${\mathsf{\Omega }}^{\prime }\left({\mathfrak{B}}_{\circ }\right)$ on $({\mathfrak{M}}_1,{\mathfrak{M}}_2)$. The fractional integrals of a function μ, w.r.t Ω, on $[{\mathfrak{M}}_1,{\mathfrak{M}}_2]$ are defined as

$${\mathsf{I}}_{{\mathfrak{M}}_1^{+}}^{{\mathfrak{S}}_1;\mathsf{\Omega }}\mu \left({\mathfrak{B}}_{\circ }\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_{{\mathfrak{M}}_1}^{{\mathfrak{B}}_{\circ }}{\mathsf{\Omega }}^{\prime }\left(\tau \right){(\mathsf{\Omega }\left({\mathfrak{B}}_{\circ }\right)-\mathsf{\Omega }\left(\tau \right))}^{{\mathfrak{S}}_1-1}\mu \left(\tau \right)d\tau .$$

Definition 2

([11]). Let ${\mathfrak{S}}_1\in (ı-1,ı)$ with $ı\in \mathbb{N}$, and suppose $F,\mathsf{\Omega }\in C^{ı}[{\mathfrak{M}}_1,{\mathfrak{M}}_2]$ are two mappings such that Ω is increasing and ${\mathsf{\Omega }}^{\prime }\left({\mathfrak{B}}_{\circ }\right)\ne 0$ for any ${\mathfrak{B}}_{\circ }\in [{\mathfrak{M}}_1,{\mathfrak{M}}_2]$. Then the Ω-Hilfer fractional derivative ${}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}(.)$ of a mapping μ of order ${\mathfrak{S}}_1$ and type $0\le {\mathfrak{S}}_2\le 1$ is defined as

$${}^{\mathcal{H}}{\mathsf{D}}_{{\mathfrak{M}}_1^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}\mu \left({\mathfrak{B}}_{\circ }\right)={\mathsf{I}}_{{\mathfrak{M}}_1^{+}}^{{\mathfrak{S}}_2(ı-{\mathfrak{S}}_1);\mathsf{\Omega }}{\left({\displaystyle \frac{1}{{\mathsf{\Omega }}^{\prime }\left({\mathfrak{B}}_{\circ }\right)}}{\displaystyle \frac{d}{d{\mathfrak{B}}_{\circ }}}\right)}^{ı}{\mathsf{I}}_{{\mathfrak{M}}_1^{+}}^{(1-{\mathfrak{S}}_2)(ı-{\mathfrak{S}}_1);\mathsf{\Omega }}\mu \left({\mathfrak{B}}_{\circ }\right)$$

.

Theorem 1

([11]). Let $\mu \in C^1[0,{\mathfrak{M}}_2]$, and ${\mathfrak{S}}_2\in [0,1]$. Then

$${}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}{\mathsf{I}}_{0^{+}}^{{\mathfrak{S}}_1;\mathsf{\Omega }}\mu \left({\mathfrak{B}}_{\circ }\right)=\mu \left({\mathfrak{B}}_{\circ }\right)$$

.

Theorem 2

([11]). Let $\mu \in C^1[0,{\mathfrak{M}}_2]$, and ${\mathfrak{S}}_2\in [0,1]$. Then

$${\mathsf{I}}_{0^{+}}^{{\mathfrak{S}}_1;\mathsf{\Omega }}{}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}\mu \left({\mathfrak{B}}_{\circ }\right)=\mu \left({\mathfrak{B}}_{\circ }\right)-{\displaystyle \frac{{(\mathsf{\Omega }\left({\mathfrak{B}}_{\circ }\right)-{\mathsf{\Omega }}^{\prime }\left(0\right))}^{{\mathfrak{S}}_3-1}}{\mathsf{\Gamma }\left({\mathfrak{S}}_3\right)}}{\mathsf{I}}_{0^{+}}^{(1-{\mathfrak{S}}_2)(1-{\mathfrak{S}}_1);\mathsf{\Omega }}\mu \left(0\right)$$

.

Lemma 1

([12]). Let ${\hslash }_1,{\hslash }_2>0$. If $\mathsf{\Psi }\left(\tau \right)={(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left({\mathfrak{M}}_1\right))}^{{\hslash }_2-1},$ then

$${\mathsf{I}}_{{\mathfrak{M}}_1^{+}}^{{\hslash }_1;\mathsf{\Omega }}\mathsf{\Psi }\left(\tau \right)=\frac{\mathsf{\Gamma }\left({\hslash }_2\right)}{\mathsf{\Gamma }({\hslash }_2+{\hslash }_1)}{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left({\mathfrak{M}}_1\right))}^{{\hslash }_1+{\hslash }_2-1}.$$

For $F\in C((0,p]\times {\mathbb{R}}^2,\mathbb{R})$ and $\theta >0$, consider the following equations

$${}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}g\left(\tau \right)=F(\tau ,g\left(\tau \right),g\left(\mu \left(\tau \right)\right)),\tau \in (0,p],$$

(2)

$${\mathsf{I}}_{0^{+}}^{1-{\mathfrak{S}}_3;\mathsf{\Omega }}g\left(0^{+}\right)={\mathcal{E}}_{\circ },{\mathcal{E}}_{\circ }\in \mathbb{R},$$

(3)

and consider the inequality

$$|{}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}f\left(\tau \right)-F(\tau ,f\left(\tau \right),f\left(\mu \left(\tau \right)\right))|\le \theta \mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1}),\tau \in (0,p],$$

(4)

where $\mathbf{B}(\alpha ,\beta ,\gamma ;\xi )$ is the Gauss Hypergeometric series (see [10]) defined by

$$\mathbf{B}(\alpha ,\beta ,\gamma ;\xi )=\sum _{s=0}^{\infty }\frac{{\left(\alpha \right)}_s{\left(\beta \right)}_s}{{\left(\gamma \right)}_s}\frac{{\xi }^s}{s!}=\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(b\right)}\sum _{s=0}^{\infty }\frac{\mathsf{\Gamma }(\alpha +s)\mathsf{\Gamma }(\beta +s)}{\mathsf{\Gamma }(\gamma +s)}\frac{{\xi }^s}{s!},$$

in which $\xi ,\alpha ,\beta ,\gamma \in \mathbb{C}$, and $\Re \left(\alpha \right),\Re \left(\beta \right),\Re \left(\gamma \right)>0$.

Motivated by [9], we define the following concept.

Definition 3.

Equations (2) and (3) have Gauss Hypergeometric stability w.r.t $\mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1})$ if we can find $c>0$ such that, for each $\theta >0$ and each solution $f\in C_{1-{\mathfrak{S}}_3;\mathsf{\Omega }}(0,p]$ to (4) and ${\mathsf{I}}_{0^{+}}^{1-{\mathfrak{S}}_3;\mathsf{\Omega }}g\left(0^{+}\right)={\mathcal{E}}_{\circ }$, there exists a solution $g\in C_{1-{\mathfrak{S}}_3;\mathsf{\Omega }}(0,p]$ to (2) and (3) with

$$|f\left(\tau \right)-g\left(\tau \right)|\le c\theta \mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1}),\tau \in (0,p].$$

Lemma 2

([5]). Consider the continuous mapping $F:(0,p]\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$. Then (2) and (3) are equivalent to

$$g\left(\tau \right)={\displaystyle \frac{{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_3-1}}{\mathsf{\Gamma }\left({\mathfrak{S}}_3\right)}}{\mathcal{E}}_{\circ }+{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left({\tau }^{\prime }\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left({\tau }^{\prime }\right))}^{{\mathfrak{S}}_1-1}F({\tau }^{\prime },g\left({\tau }^{\prime }\right),g\left(\mu \left({\tau }^{\prime }\right)\right))d{\tau }^{\prime }.$$

Remark 1.

Suppose $f\in C_{1-{\mathfrak{S}}_3;\mathsf{\Omega }}(0,p]$ is a solution of inequality (4) and ${\mathsf{I}}_{0^{+}}^{1-{\mathfrak{S}}_3;\mathsf{\Omega }}g\left(0^{+}\right)={\mathcal{E}}_{\circ }$. Then, f is a solution of the following integral inequality:

$$\begin{array}{cc}\hfill & |f\left(\tau \right)-{\displaystyle \frac{{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_3-1}}{\mathsf{\Gamma }\left({\mathfrak{S}}_3\right)}}{\mathcal{E}}_{\circ }-\frac{1}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left(s\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(s\right))}^{{\mathfrak{S}}_1-1}F(s,f\left(s\right),f\left(\mu \left(s\right)\right))ds|\hfill \\ \hfill & \le \frac{\theta }{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left(s\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(s\right))}^{{\mathfrak{S}}_1-1}\mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(s\right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1})ds\hfill \\ \hfill & \le \frac{\theta }{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left(s\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(s\right))}^{{\mathfrak{S}}_1-1}\frac{\mathsf{\Gamma }\left(\gamma \right)}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }(\alpha +k)\mathsf{\Gamma }(\beta +k)}{\mathsf{\Gamma }(\gamma +k)}\frac{{(\mathsf{\Omega }\left(s\right)-\mathsf{\Omega }\left(0\right))}^{k{\mathfrak{S}}_1}}{k!}ds\hfill \\ \hfill & =\frac{\theta }{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}\frac{\mathsf{\Gamma }\left(\gamma \right)}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }(\alpha +k)\mathsf{\Gamma }(\beta +k)}{\mathsf{\Gamma }(\gamma +k)}\frac{1}{k!}{\int }_0^{\tau }{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(s\right))}^{{\mathfrak{S}}_1-1}{(\mathsf{\Omega }\left(s\right)-\mathsf{\Omega }\left(0\right))}^{k{\mathfrak{S}}_1}d\mathsf{\Omega }\left(s\right)\hfill \\ \hfill & =\frac{\theta }{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}\frac{\mathsf{\Gamma }\left(\gamma \right)}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }(\alpha +k)\mathsf{\Gamma }(\beta +k)}{\mathsf{\Gamma }(\gamma +k)}\frac{1}{k!}{\int }_0^{\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right)}{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right)-u)}^{{\mathfrak{S}}_1-1}{\left(u\right)}^{k{\mathfrak{S}}_1}du\hfill \\ \hfill & (u=\mathsf{\Omega }(s)-\mathsf{\Omega }(0\left)\right)\hfill \\ \hfill & \le \frac{\theta }{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}\frac{\mathsf{\Gamma }\left(\gamma \right)}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }(\alpha +k)\mathsf{\Gamma }(\beta +k)}{\mathsf{\Gamma }(\gamma +k)}\frac{1}{k!}{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1-1}{\int }_0^{\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right)}{(1-{\displaystyle \frac{u}{\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right)}})}^{{\mathfrak{S}}_1-1}{u}^{k{\mathfrak{S}}_1}du\hfill \\ \hfill & =\frac{\theta }{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}\frac{\mathsf{\Gamma }\left(\gamma \right)}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }(\alpha +k)\mathsf{\Gamma }(\beta +k)}{\mathsf{\Gamma }(\gamma +k)}\frac{1}{k!}{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{(k+1){\mathfrak{S}}_1}{\int }_0^1{(1-v)}^{{\mathfrak{S}}_1-1}{v}^{k{\mathfrak{S}}_1}dv\hfill \\ \hfill & \left(v={\displaystyle \frac{u}{\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right)}}\right)\hfill \\ \hfill & =\frac{\theta }{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}\frac{\mathsf{\Gamma }\left(\gamma \right)}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }(\alpha +k)\mathsf{\Gamma }(\beta +k)}{\mathsf{\Gamma }(\gamma +k)}\frac{1}{k!}{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{(k+1){\mathfrak{S}}_1}{\displaystyle \frac{\mathsf{\Gamma }(k{\mathfrak{S}}_1+1)\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}{\mathsf{\Gamma }((k+1){\mathfrak{S}}_1+1)}}\hfill \\ \hfill & \le \theta \frac{\mathsf{\Gamma }\left(\gamma \right)}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\sum _{k=0}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }(\alpha +k)\mathsf{\Gamma }(\beta +k)}{\mathsf{\Gamma }(\gamma +k)}}{\displaystyle \frac{{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{k{\mathfrak{S}}_1}}{k!}}\hfill \\ \hfill & =\theta \mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1}).\hfill \end{array}$$

Now, we present an alternative fixed point theorem from the literature:

Theorem 3

([13]). Let $\mathcal{M}$ be a set with a complete $[0,\infty ]$-valued metric φ, and also let the self map δ on $\mathcal{M}$ satisfy

$$\phi (\delta y,\delta x)\le \kappa \phi (y,x),\kappa <1isaLipschitzconstant.$$

Let $y\in \mathcal{M}$. Then, we have two options:

$$\left(I\right)\phi ({\delta }^{m}y,{\delta }^{m+1}y)=\infty ,\forall m\in \mathbb{N},$$

or

(II) we can find $m_0\in \mathbb{N}$ such that:

(1) 

$\phi ({\delta }^{m}y,{\delta }^{m+1}y)<\infty ,\forall m\ge m_0$;

(2) 

the fixed point $x^{*}$ of δ is the convergence point of the sequence $\left\{{\delta }^{m}y\right\}$;

(3) 

in the set $V=\{x\in \mathcal{M}\mid \phi ({\delta }^{m_0}y,x)<\infty \}$, $x^{*}$ is the unique fixed point of δ;

(4) 

$(1-\kappa )\phi (x,x^{*})\le \phi (x,\delta x)$ for every $x\in \mathcal{M}$.

3. Hypergeometric Stability

Using the Diaz–Margolis theorem (i.e., Theorem 3), we obtain a new stability approximation result for (2), for more details, we refer to [13,14,15,16,17,18,19,20,21,22,23,24]. In these sources, one can find new problems.

We assume the following conditions hold:

$\left({\mathfrak{J}}_1\right)$$F\in C((0,p]\times {\mathbb{R}}^2,\mathbb{R}),\mu \in C([0,p],[0,p]),\mu \left(\tau \right)\le \tau ,\tau \ge 0$.

$\left({\mathfrak{J}}_2\right)$ There exists $0<\mathsf{\Theta }<\frac{1}{2}$ such that

$$|F(\tau ,S_1,S_2)-F(\tau ,R_1,R_2)|\le \mathsf{\Theta }\sum _{j=1}^2|S_j-R_j|\mathrm{for}\mathrm{all}\tau \in (0,p],S_j,R_j\in \mathbb{R},j=1,2.$$

Theorem 4.

Assume that $\left({\mathfrak{J}}_1\right)$ and $\left({\mathfrak{J}}_2\right)$ are satisfied. If f in $C_{1-{\mathfrak{S}}_3;\mathsf{\Omega }}(0,p]$ satisfies (4) and ${\mathsf{I}}_{0^{+}}^{1-{\mathfrak{S}}_3;\mathsf{\Omega }}g\left(0^{+}\right)={\mathcal{E}}_{\circ }$, thus, we can find a unique function g satisfying in (2) and (3) such that

$$|g\left(\tau \right)-f\left(\tau \right)|\le {\displaystyle \frac{\theta }{1-2\mathsf{\Theta }}}\mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1}),$$

(5)

for every $\tau \in (0,p]$.

Proof. 

Set $\mathsf{\Xi }=C_{1-{\mathfrak{S}}_3;\mathsf{\Omega }}(0,p]$ and define a mapping $\phi :\mathsf{\Xi }\times \mathsf{\Xi }\to [0,\infty ]$ by

$$\phi (\rho ,\varrho )=inf\left\{\complement \ge 0:|\rho \left(\tau \right)-\varrho \left(\tau \right)|\le \complement \theta \mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1})\right\}.$$

(6)

We prove that $(\mathsf{\Xi },\phi )$ is a complete $[0,\infty ]$-valued metric space. For the metric part, it is enough to assume that $\phi (\rho ,\varrho )>\phi (\rho ,\nu )+\phi (\nu ,\varrho )$, for some $\rho ,\varrho$ and $\nu \in \mathsf{\Xi }$. Thus, we can find ${\tau }_{\circ }\in (0,p]$ with

$$|\rho \left({\tau }_{\circ }\right)-\varrho \left({\tau }_{\circ }\right)|>(\phi (\rho ,\nu )+\phi (\nu ,\varrho ))\theta \mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left({\tau }_{\circ }\right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1}).$$

Hence, from the definition of $\phi$, we get

$$\begin{array}{ccc}& & |\rho \left({\tau }_{\circ }\right)-\varrho \left({\tau }_{\circ }\right)|\hfill \\ & >& |\rho \left({\tau }_{\circ }\right)-\nu \left({\tau }_{\circ }\right)|+|(\nu \left({\tau }_{\circ }\right)-\varrho \left({\tau }_{\circ }\right)|,\hfill \end{array}$$

which is a contradiction. Next, we prove that $(\mathsf{\Xi },\phi )$ is complete. Assume ${\omega }_n$ is a Cauchy sequence in $(\mathsf{\Xi },\phi )$. Thus, for each $ϵ>0$ we can find ${\aleph }_{ϵ}\in \mathbb{N}$ such that $\phi ({\omega }_m,{\omega }_n)\le ϵ$ for each $m,n\ge {\aleph }_{ϵ}$. According to (6), we have

$$\begin{array}{c}\hfill |{\omega }_m\left(\tau \right)-{\omega }_n\left(\tau \right)|\le ϵ\theta \mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1}),\end{array}$$

(7)

for any $\tau \in (0,p]$. If $\tau$ is fixed, (7) implies that $\left\{{\omega }_n\left(\tau \right)\right\}$ is a Cauchy sequence in $\mathbb{R}$. Since $\mathbb{R}$ is complete, $\left\{{\omega }_n\left(\tau \right)\right\}$ converges for any $\tau \in (0,p]$. Thus, we can define a mapping $\omega$ by

$$\omega \left(\tau \right):=\underset{n\to \infty }{lim}{\omega }_n\left(\tau \right),(\tau \in (0,p]).$$

It is easy to check $\omega \in \mathsf{\Xi }$. If we let $m\to \infty$, it infers from (7) that

$$\begin{array}{c}\hfill |\omega \left(\tau \right)-{\omega }_n\left(\tau \right)|\le ϵ\theta \mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1}).\end{array}$$

Considering (6), we receive

$$\phi (\omega ,{\omega }_n)\le ϵ.$$

This implies that the Cauchy sequence $\left\{{\omega }_n\right\}$ converges to $\omega$ in $(\mathsf{\Xi },\phi )$. Therefore, $(\mathsf{\Xi },\phi )$ is complete. According to Lemma 2, we have that (2) is equivalent to the following system:

$$g\left(\tau \right)={\displaystyle \frac{{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_3-1}}{\mathsf{\Gamma }\left({\mathfrak{S}}_3\right)}}{\mathcal{E}}_{\circ }+{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left({\tau }^{\prime }\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left({\tau }^{\prime }\right))}^{{\mathfrak{S}}_1-1}F({\tau }^{\prime },g\left({\tau }^{\prime }\right),g\left(\mu \left({\tau }^{\prime }\right)\right))d{\tau }^{\prime },$$

(8)

for every $\tau \in (0,p]$. To see this note applying the fractional integral operator ${\mathsf{I}}_{0^{+}}^{{\mathfrak{S}}_1;\mathsf{\Omega }}(.)$ on both sides of the fractional Equation (1) and using Theorem 2, we get (8).

On the other hand, if g satisfies (8), then g satisfies (1). However, applying the fractional derivative ${}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}(.)$ on both sides of (8), we have

$${}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}f\left(\tau \right)={}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}\left[{\displaystyle \frac{{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_3-1}}{\mathsf{\Gamma }\left({\mathfrak{S}}_3\right)}}{\mathcal{E}}_{\circ }\right]+{}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}{\mathsf{I}}_{0^{+}}^{{\mathfrak{S}}_1;\mathsf{\Omega }}F({\tau }^{\prime },f\left({\tau }^{\prime }\right),f\left(\mu \left({\tau }^{\prime }\right)\right)).$$

Applying Theorem 1 and

$${}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_3-1}=0,0<{\mathfrak{S}}_3<1,$$

we conclude that $g\left(\tau \right)$ satisfies the initial value problem Equation (1) if and only if $g\left(\tau \right)$ satisfies the integral Equation (8).

Consider $\delta :\mathsf{\Xi }⟶\mathsf{\Xi }$ such that for $\rho \in \mathsf{\Xi }$,

$$\begin{array}{ccc}\hfill & & \delta \left(\rho \right(\tau \left)\right)\hfill \\ & =& {\displaystyle \frac{{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_3-1}}{\mathsf{\Gamma }\left({\mathfrak{S}}_3\right)}}{\mathcal{E}}_{\circ }+{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left({\tau }^{\prime }\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left({\tau }^{\prime }\right))}^{{\mathfrak{S}}_1-1}F({\tau }^{\prime },\rho \left({\tau }^{\prime }\right),\rho \left(\mu \left({\tau }^{\prime }\right)\right))d{\tau }^{\prime },\hfill \end{array}$$

(9)

for every $\tau \in (0,p]$.

For $\rho \in \mathsf{\Xi }$, we have

$$\begin{array}{ccc}& & |\delta \left(\rho \left(\tau \right)\right)-\delta \left(\rho \left({\tau }_0\right)\right)|\hfill \\ & =& |{\displaystyle \frac{{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_3-1}}{\mathsf{\Gamma }\left({\mathfrak{S}}_3\right)}}{\mathcal{E}}_{\circ }+{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left({\tau }^{\prime }\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left({\tau }^{\prime }\right))}^{{\mathfrak{S}}_1-1}F({\tau }^{\prime },\rho \left({\tau }^{\prime }\right),\rho \left(\mu \left({\tau }^{\prime }\right)\right))d{\tau }^{\prime }\hfill \\ & -& {\displaystyle \frac{{(\mathsf{\Omega }\left({\tau }_0\right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_3-1}}{\mathsf{\Gamma }\left({\mathfrak{S}}_3\right)}}{\mathcal{E}}_{\circ }-{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_0^{{\tau }_0}{\mathsf{\Omega }}^{\prime }\left({\tau }^{\prime }\right){(\mathsf{\Omega }\left({\tau }_0\right)-\mathsf{\Omega }\left({\tau }^{\prime }\right))}^{{\mathfrak{S}}_1-1}F({\tau }^{\prime },\rho \left({\tau }^{\prime }\right),\rho \left(\mu \left({\tau }^{\prime }\right)\right))d{\tau }^{\prime }|\hfill \\ & ⟶& 0\hfill \end{array}$$

as $\tau ⟶{\tau }_0$, which implies the the continuity property of $\delta \rho$ and so $\delta \rho \in \mathsf{\Xi }$.

Next, we show that the self-mapping $\delta$ is contractive on $\mathsf{\Xi }$. Consider $\delta :\mathsf{\Xi }⟶\mathsf{\Xi }$ defined in (9).

Suppose $\rho ,\varrho \in C_{1-{\mathfrak{S}}_3;\mathsf{\Omega }}(0,p],\mathbb{k}\in [0,\infty ]$, and $\phi \left(\rho \right(\tau ),\varrho (\tau \left)\right)\le \mathbb{k}.$ Then for every $\tau \in (0,p]$,

$$|\rho \left(\tau \right)-\varrho \left(\tau \right)|\le \mathbb{k}\theta \mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1}).$$

Using Remark 1, for all $\tau \in (0,p]$, we have

$$\begin{array}{ccc}& & \left|\delta \right(\rho \left(\tau \right))-\delta (\varrho \left(\tau \right)\left)\right|\hfill \\ & \le & {\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left({\tau }^{\prime }\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left({\tau }^{\prime }\right))}^{{\mathfrak{S}}_1-1}|F({\tau }^{\prime },\rho \left({\tau }^{\prime }\right),\rho \left(\mu \left({\tau }^{\prime }\right)\right))-F({\tau }^{\prime },\varrho \left({\tau }^{\prime }\right),\varrho \left(\mu \left({\tau }^{\prime }\right)\right))|d{\tau }^{\prime }\hfill \\ & \le & {\displaystyle \frac{\mathsf{\Theta }}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left({\tau }^{\prime }\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left({\tau }^{\prime }\right))}^{{\mathfrak{S}}_1-1}\left[|\rho \left({\tau }^{\prime }\right)-\varrho \left({\tau }^{\prime }\right)|+|\rho \left(\mu \left({\tau }^{\prime }\right)\right)-\varrho \left(\mu \left({\tau }^{\prime }\right)\right)|\right]d{\tau }^{\prime }\hfill \\ & \le & {\displaystyle \frac{2\mathsf{\Theta }}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left({\tau }^{\prime }\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left({\tau }^{\prime }\right))}^{{\mathfrak{S}}_1-1}\mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left({\tau }^{\prime }\right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1})d{\tau }^{\prime }\hfill \\ & \le & 2\mathsf{\Theta }\mathbb{k}\theta \mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1}).\hfill \end{array}$$

Then we have

$$\phi (\delta \rho ,\delta \varrho )\le 2\mathsf{\Theta }\phi (\rho ,\varrho ).$$

Now, since $0<\mathsf{\Theta }<\frac{1}{2}$, we can conclude the contractively property of $\delta$.

Let $f\in \mathsf{\Xi }$. Since $\delta f\in \mathsf{\Xi }$, we have that

$$\begin{array}{ccc}& & \left|\delta \right(f\left(\tau \right))-f(\tau \left)\right|\hfill \\ & \le & |f\left(\tau \right)-{\displaystyle \frac{{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_3-1}}{\mathsf{\Gamma }\left({\mathfrak{S}}_3\right)}}{\mathcal{E}}_{\circ }-{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left({\tau }^{\prime }\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1-1}F({\tau }^{\prime },f\left({\tau }^{\prime }\right),f\left(\mu \left({\tau }^{\prime }\right)\right))d{\tau }^{\prime }|\hfill \\ & \le & \theta \mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1})\hfill \end{array}$$

for $\tau \in (0,p]$, which implies that

$$\phi (\delta f,f)\le 1.$$

(10)

and hence for every $n\in \mathbb{N}$, we have $\phi ({\delta }^{n}f,{\delta }^{n+1}f)\le +\infty$. Now, we can apply the second part of Theorem 3 and so we can find a unique map $g\in \{\tilde{\sigma }\in \mathsf{\Xi }:\phi (\delta f,\tilde{\sigma })<\infty \}$ such that $\delta g=g$ and so

$$g\left(\tau \right)={\displaystyle \frac{{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_3-1}}{\mathsf{\Gamma }\left({\mathfrak{S}}_3\right)}}{\mathcal{E}}_{\circ }+{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left({\tau }^{\prime }\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left({\tau }^{\prime }\right))}^{{\mathfrak{S}}_1-1}F({\tau }^{\prime },g\left({\tau }^{\prime }\right),g\left(\mu \left({\tau }^{\prime }\right)\right))d{\tau }^{\prime },$$

for every $\tau \in (0,p]$, where ${\mathsf{I}}_{0^{+}}^{1-{\mathfrak{S}}_3;\mathsf{\Omega }}g\left(0^{+}\right)={\mathcal{E}}_{\circ }\in \mathbb{R}.$

Using Theorem 3 and (10), we have

$$\begin{array}{ccc}\hfill \phi (g,f)& \le & {\displaystyle \frac{1}{1-2\mathsf{\Theta }}}\phi (\delta f,f)\hfill \\ & \le & {\displaystyle \frac{1}{1-2\mathsf{\Theta }}},\hfill \end{array}$$

which implies (5). □

In Theorem 4, we proved Gauss Hypergeometric stability of the $\mathsf{\Omega }$-Hilfer fractional differential Equation (1) on bounded domains. Now, we extend our result to unbounded domains. Let $\mathsf{S}=C_{1-{\mathfrak{S}}_3;\mathsf{\Omega }}\left(\mathbb{R}\right)$.

Theorem 5.

Let $\left({\mathfrak{J}}_1\right)$ and $\left({\mathfrak{J}}_2\right)$ be hold. If f in $\mathsf{S}$ satisfies (4) and ${\mathsf{I}}_{0^{+}}^{1-{\mathfrak{S}}_3;\mathsf{\Omega }}g\left(0^{+}\right)={\mathcal{E}}_{\circ },$ thus, we can find a unique function g satisfying in (2) and (3) such that satisfies (5) for every $\tau \in \mathbb{R}$.

Proof. 

We are going to show the result for $\mathbb{R}$. By the same way, we can prove the theorem for $(-\infty ,0]$ or $[0,+\infty )$.

For each $n\in \mathbb{N},$ we consider ${\mathsf{P}}_n=[\mathfrak{P}-n,\mathfrak{P}+n].$ As stated by Theorem 4, we can find a unique mapping $g_n\in C_{1-{\mathfrak{S}}_3;\mathsf{\Omega }}({\mathsf{P}}_n,\mathbb{R}),$ such that

$${}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{{\mathfrak{S}}_1,{\mathfrak{S}}_2;\mathsf{\Omega }}{g}_n\left(\tau \right)=F(\tau ,g_n\left(\tau \right),g_n\left(\mu \left(\tau \right)\right)),$$

(11)

$${\mathsf{I}}_{0^{+}}^{1-{\mathfrak{S}}_3;\mathsf{\Omega }}{g}_n\left(0^{+}\right)={\mathcal{E}}_{\circ },{\mathcal{E}}_{\circ }\in \mathbb{R},$$

(12)

and

$$|g_n\left(\tau \right)-f\left(\tau \right)|\le {\displaystyle \frac{\theta }{1-2\mathsf{\Theta }}}\mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1}),$$

(13)

for every $\tau \in {\mathsf{P}}_n$. The uniqueness of $g_n$ implies that if $\tau \in {\mathsf{P}}_n,$ then

$$g_n\left(\tau \right)=g_{n+1}\left(\tau \right)=g_{n+2}\left(\tau \right)=\cdots .$$

(14)

For all $\tau \in \mathbb{R},$ we define $n\left(\tau \right)\in \mathbb{N}$ as

$$n\left(\tau \right)=min\{n\in \mathbb{N}|\tau \in {\mathsf{P}}_n\}.$$

In addition, we consider a mapping g, as

$$g\left(\tau \right)=g_{n\left(\tau \right)}\left(\tau \right),\tau \in \mathbb{R},$$

and we claim that $g\in \mathsf{S}$. For a real number ${\tau }_1\in \mathbb{R},$ we consider $n_1=n\left({\tau }_1\right)\in \mathbb{N}$. Thus, ${\tau }_1\in {\left({\mathsf{P}}_{n_1+1}\right)}^{\circ }$ (the interior) and we can find an $ϵ>0$ such that $g\left(\tau \right)=g_{n+1}\left(\tau \right)$ for all $\tau$ with ${\tau }_1-ϵ<\tau <{\tau }_1+ϵ$.

Now, we want to show that g satisfies (2), (3) and (5) for any $\tau \in \mathbb{R}.$ For any $\tau \in \mathbb{R},$ we consider $n\left(\tau \right)\in \mathbb{N}$. Therefore, $\tau \in {\mathsf{P}}_{n\left(\tau \right)}$ and it concludes from (11) and (12) that

$$\begin{array}{ccc}& g\left(\tau \right)& =g_{n\left(\tau \right)}\left(\tau \right)\hfill \\ & & ={\displaystyle \frac{{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_3-1}}{\mathsf{\Gamma }\left({\mathfrak{S}}_3\right)}}{\mathcal{E}}_{\circ }\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left({\tau }^{\prime }\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left({\tau }^{\prime }\right))}^{{\mathfrak{S}}_1-1}F({\tau }^{\prime },g_{n\left({\tau }^{\prime }\right)}\left({\tau }^{\prime }\right),g_{n\left({\tau }^{\prime }\right)}\left(\mu \left({\tau }^{\prime }\right)\right))d{\tau }^{\prime },\hfill \\ & & ={\displaystyle \frac{{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_3-1}}{\mathsf{\Gamma }\left({\mathfrak{S}}_3\right)}}{\mathcal{E}}_{\circ }\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left({\mathfrak{S}}_1\right)}}{\int }_0^{\tau }{\mathsf{\Omega }}^{\prime }\left({\tau }^{\prime }\right){(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left({\tau }^{\prime }\right))}^{{\mathfrak{S}}_1-1}F({\tau }^{\prime },g\left({\tau }^{\prime }\right),g\left(\mu \left({\tau }^{\prime }\right)\right))d{\tau }^{\prime },\hfill \end{array}$$

where the above equality holds true because $n\left({\tau }^{\prime }\right)\le n\left(\tau \right)$ for each ${\tau }^{\prime }\in {\mathsf{P}}_{n\left(\tau \right)}$ and it concludes from (14) that

$$g\left({\tau }^{\prime }\right)=g_{n\left({\tau }^{\prime }\right)}\left({\tau }^{\prime }\right)=g_{n\left(\tau \right)}\left({\tau }^{\prime }\right).$$

Since $g\left(\tau \right)=g_{n\left(\tau \right)}\left(\tau \right)$ and $\tau \in {\mathsf{P}}_{n\left(\tau \right)}$ for all $\tau \in \mathbb{R},$ (13) implies that

$$\begin{array}{c}\hfill |g\left(\tau \right)-f\left(\tau \right)|=|g_{n\left(\tau \right)}\left(\tau \right)-f\left(\tau \right)|\le {\displaystyle \frac{\theta }{1-2\mathsf{\Theta }}}\mathbf{B}(\alpha ,\beta ,\gamma ;{(\mathsf{\Omega }\left(\tau \right)-\mathsf{\Omega }\left(0\right))}^{{\mathfrak{S}}_1}).\end{array}$$

In the end, we claim that g is unique. Let $g^{\prime }\in \mathsf{S}$ be another function, which satisfies (2), (3) and (5), for all $\tau \in \mathbb{R}.$ Since ${g|}_{{\mathsf{P}}_{n\left(\tau \right)}}(=g_{n\left(\tau \right)})$ and $g^{\prime }{|}_{{\mathsf{P}}_{n\left(\tau \right)}}$ both satisfy (2), (3) and (5), for all $\tau \in {\mathsf{P}}_{n\left(\tau \right)},$ the uniqueness of $g_{n\left(\tau \right)}{=g|}_{{\mathsf{P}}_{n\left(\tau \right)}}$ implies that

$${g\left(\tau \right)=g|}_{{\mathsf{P}}_{n\left(\tau \right)}}\left(\tau \right)=g^{\prime }{|}_{{\mathsf{P}}_{n\left(\tau \right)}}\left(\tau \right)=g^{\prime }\left(\tau \right)$$

as required. □

4. Application

In this section, as an application, we apply our main result to prove an example. Our main result can be applied for recent results presented in [25,26,27,28,29,30,31,32,33,34,35].

Example 1.

Consider the following fractional system

$$\left\{\begin{array}{cc}{}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{\frac{1}{4},\frac{3}{4};e^{\tau }}g\left(\tau \right)={\displaystyle \frac{1}{4}}sin\left(g\left({\displaystyle \frac{3}{4}}\tau \right)\right)+{\displaystyle \frac{1}{4}}arccot\left(g\left(\tau \right)\right),& \tau \in (0,1],\\ {\mathsf{I}}_{0^{+}}^{\frac{3}{16};e^{\zeta }}g\left(0^{+}\right)=g_0,& \end{array}\right.$$

(15)

and the inequality

$$|{}^{\mathcal{H}}{\mathsf{D}}_{0^{+}}^{\frac{1}{4},\frac{3}{4};e^{\tau }}g\left(\tau \right)-F(\tau ,g\left(\tau \right),g\left({\displaystyle \frac{3}{4}}\tau \right))|\le \theta \mathbf{B}(\alpha ,\beta ,\gamma ;{(e^{\tau }-1)}^{\frac{1}{4}}).$$

Let ${\mathfrak{S}}_1={\displaystyle \frac{1}{4}},{\mathfrak{S}}_2={\displaystyle \frac{3}{4}}$, then ${\mathfrak{S}}_3={\mathfrak{S}}_1+{\mathfrak{S}}_2(1-{\mathfrak{S}}_1)={\displaystyle \frac{13}{16}},p=1,\mathsf{\Omega }(.)=e^{.},\mu (.)={\displaystyle \frac{3}{4}}(.)$, $F(.,g(.),g\left(\mu (.)\right))={\displaystyle \frac{1}{4}}sin\left(g\left({\displaystyle \frac{3}{4}}(.)\right)\right)+{\displaystyle \frac{1}{4}}arccot\left(g(.)\right)$, and $\mathsf{\Theta }={\displaystyle \frac{1}{4}}$. Thus, $2\mathsf{\Theta }={\displaystyle \frac{1}{2}}$.

We want to show that $F(.,g(.),g(\mu (.)\left)\right)$ satisfies in $\left({\mathfrak{J}}_2\right)$:

$$\begin{array}{ccc}& & |{\displaystyle \frac{1}{4}}sin\left(g_1\right)+{\displaystyle \frac{1}{4}}arccot\left(g_1\right)-{\displaystyle \frac{1}{4}}sin\left(g_2\right)-{\displaystyle \frac{1}{4}}arccot\left(g_2\right)|\hfill \\ \hfill & \le & |{\displaystyle \frac{1}{4}}sin\left(g_1\right)-{\displaystyle \frac{1}{4}}sin\left(g_2\right)|+|{\displaystyle \frac{1}{4}}arccot\left(g_1\right)-{\displaystyle \frac{1}{4}}arccot\left(g_2\right)|\hfill \\ \hfill & \le & \left|{\displaystyle \frac{1}{4}}(g_1-g_2)\right|+\left|{\displaystyle \frac{1}{4}}{\left(arccot\left(\mathfrak{C}\right)\right)}^{\prime }(g_1-g_2)\right|\hfill \\ \hfill & \le & \left|{\displaystyle \frac{1}{4}}(g_1-g_2)\right|+\left|{\displaystyle \frac{1}{4}}{\displaystyle \frac{-1}{1+{\mathfrak{C}}^2}}(g_1-g_2)\right|\hfill \\ \hfill & \le & \left|{\displaystyle \frac{1}{4}}(g_1-g_2)\right|+\left|{\displaystyle \frac{1}{4}}(g_1-g_2)\right|\hfill \\ \hfill & \le & {\displaystyle \frac{1}{2}}|g_1-g_2|,\hfill \end{array}$$

in which$g_1\le \mathfrak{C}\le g_2$, thus F is a contraction.

Now, Theorem 4 implies that problem (15) has a unique solution and also is Hypergeometric stable with

$$|g\left(\tau \right)-f\left(\tau \right)|\le 2\theta \mathbf{B}(\alpha ,\beta ,\gamma ;{(e^{\tau }-1)}^{\frac{1}{4}}),\tau \in (0,1].$$

5. Conclusions

In this paper, we introduced a class of $\mathsf{\Omega }$-Hilfer fractional-order delay differential equations and through the Cădariu–Radu method derived from the Diaz–Margolis theorem, we studied the Hypergeometric stability for both bounded and unbounded domains. Finally, as an application, we investigated the Hypergeometric stability of a fractional system.