Analysis of Delay-Type Integro-Differential Systems Described by the Φ-Hilfer Fractional Derivative

Abstract

In this article, a novel type of equation, namely the $\mathsf{\Phi }$-Hilfer fractional-order integro-differential delay system ($\mathsf{\Phi }$-HFOIDDS), is proposed. Here, we study the existence and Hyers–Ulam–Mittag–Leffler (H-U-M-L) stability of the aforementioned equation which are obtained by using the multivariate Mittag–Leffler function, Banach contraction principle, and Picard operator method as well as generalized Gronwall inequality. Finally, we conclude this paper by constructing a suitable example to illustrate the applicability of the principal outcomes.

1. Introduction

Like regular calculus, fractional calculus (FC) has received significant recent scientific attention. This is because FC can provide better descriptions of phenomena that occur in the real world. Additionally, these derivatives have a wide range of uses in the description of systems with memory effects. The aforesaid calculus has been applied in a variety of frameworks to study several infectious disease models, for example, those in [1,2,3]. Additionally, FC has been widely used in the fields of physics, engineering, and cosmology. For illustration, we quote some citations [4,5,6]. Here we note that FC has important applications in physics which have recently been identified by numerous scientists (see some relevant results in [7,8]. Moreover, the said area has been used increasingly to investigate various viral infectious diseases, including Hepatitis B and others in [9,10,11,12]. Also, some interesting applications in fluid mechanics, vegetable oil processing, electromagnetic theory, and pine wilt disease can be read in the cited articles as [13,14,15,16,17].

In the literature, there are several definitions of fractional integrals and derivatives, the most well-known of which are the Riemann–Liouville (R-L) and Caputo-type fractional derivatives. Hilfer [5] introduced a generalization that encompasses both the R-L and Caputo derivatives, known as the Hilfer fractional derivative (HFD). Theoretical simulations of thermal reversibility in crystalline compounds, engineering, constitutive modeling of physics, chemical processing, and other fields have established the usefulness and applicability of the HFD. Building on previous work, Sousa and Oliveira [18] proposed the $\psi$-Hilfer fractional derivative ($\psi$-HFD), offering a more flexible framework for FC. The advantage of the fractional operator $\psi$-Hilfer proposed here is the freedom of choice of the classical differentiation operator and the choice of the function $\psi$, i.e., from the choice of the function $\psi$, the operator of classical differentiation, can act on the fractional integration operator or else the fractional integration operator can act on the classical differentiation operator. Some properties of this operator can be found in [18]. Several scientists [19,20,21,22,23,24,25] have used various fractional derivatives to investigate the existence and Ulam-type stability of fractional-order integro-differential equations (FOIDEs). In [26,27,28,29,30], the Ulam-type stability of FOIDEs with delay has been studied.

The authors in [31] considered the following initial value problem for the $\psi$-Hilfer fractional-order differential equation in delay form

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {}^H\mathbb{D}_{0^{+}}^{\alpha ,\beta ,\psi }x\left(\tau \right)=f(\tau ,x\left(\tau \right),x\left(g\left(\tau \right)\right)),\tau \in I:=(0,d],\hfill \\ \hfill & I_{0^{+}}^{1-\gamma ,\psi }x\left(0^{+}\right)=x_0,\hfill \\ \hfill & x\left(t\right)=\phi \left(t\right),t\in [-h,0],\hfill \end{array}\right.\end{array}$$

where ${}^H\mathbb{D}_{0^{+}}^{\alpha ,\beta ,\psi }(·)$ is the $\psi$-HFD of order $0<\alpha <1$, and type $0\le \beta \le 1$, $I_{0^{+}}^{1-\gamma ,\psi }(·)$ is the $\psi$-type R-L fractional integral of order $1-\gamma ,\gamma =\alpha +\beta -\alpha \beta$, by some properties of $\psi$-Hilfer fractional calculus, the standard Picard operator, and an abstract Gronwall lemma.

Li in [22], studied the existence and uniqueness of solutions to a new nonlinear Hilfer integro-differential equation with an initial condition

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^H\mathit{D}_{a^{+}}^{\alpha ,\beta }u\left(x\right)+\sum _{i=1}^m{\lambda }_i{I}_{a^{+}}^{{\beta }_i}u\left(x\right)=I_{a^{+}}^{\beta }g(x,u\left(x\right)),{\beta }_i\ge \beta ,x\in (a,b]}\hfill \\ I_{a^{+}}^{1-\gamma }u\left(a\right)=u_a,\gamma =\alpha +\beta -\alpha \beta ,\hfill \end{array}\right.\end{array}$$

where ${\lambda }_i\in \mathfrak{R}$ for $i=1,2,\cdots ,m.$ and ${}^H\mathit{D}_{a^{+}}^{\alpha ,\beta }(·)$ is the HFD of order $0<\alpha <1$ and type $0\le \beta \le 1$, $I_{a^{+}}^{{\beta }_i}(·)$ is the R-L fractional integral of the order ${\beta }_i$, and the nonlinear term $g:(a,b]\times \mathfrak{R}\to \mathfrak{R}$ is a function satisfying certain conditions.

In precise terms, here we study the following $\mathsf{\Phi }$-HFOIDDS which are motivated by the preceding studies [22,31]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^H\mathfrak{D}_{0^{+}}^{\sigma ,\xi ,\mathsf{\Phi }}v\left(t\right)+\sum _{i=1}^m{\mu }_i{\mathfrak{I}}_{0^{+}}^{{\xi }_i,\mathsf{\Phi }}v\left(t\right)={\mathfrak{I}}_{0^{+}}^{\xi ,\mathsf{\Phi }}f(t,v\left(t\right),v\left(g\left(t\right)\right)),t\in \theta :=(0,b],}\hfill \\ {\mathfrak{I}}_{0^{+}}^{1-\vartheta ,\mathsf{\Phi }}v\left(0^{+}\right)=v_0\in \mathfrak{R},\hfill \\ v\left(t\right)=\eta \left(t\right),t\in [-\rho ,0],\hfill \end{array}\right.\end{array}$$

(1)

where ${\mu }_i\in \mathfrak{R}$ for $i=1,2,\cdots ,m$, ${}^H\mathfrak{D}_{0^{+}}^{\sigma ,\xi ,\mathsf{\Phi }}(·)$ indicates the $\mathsf{\Phi }$-Hilfer fractional derivative (for short $\mathsf{\Phi }$-HFD) of order $\sigma \in (0,1)$ and type $\xi \in [0,1]$, ${\xi }_i\ge \xi$, ${\mathfrak{I}}_{0^{+}}^{1-\vartheta ,\mathsf{\Phi }}(·)$, ${\mathfrak{I}}_{0^{+}}^{{\xi }_i,\mathsf{\Phi }}(·)$ are $\mathsf{\Phi }$-R-L fractional integrals of orders $1-\vartheta (\vartheta =\sigma +\xi -\sigma \xi )$, and ${\xi }_i$, $f:\theta \times \mathfrak{R}\times \mathfrak{R}\to \mathfrak{R}$ is a given function.

The main contributions and advantages of this manuscript are as follows:

• For the first time in the literature, the existence and H-U-M-L stability for $\mathsf{\Phi }$-HFOIDDS is investigated.
• A new set of sufficient conditions is established for the existence and H-U-M-L stability for $\mathsf{\Phi }$-HFOIDDS 1.
• Our main technique relies on the Picard operator method, and the generalized Gronwall’s inequality for the $\mathsf{\Phi }$-Hilfer derivative is efficiently used to establish the novel results.
• An example is presented to verify the considered theoretical results.

2. Prerequisites

Let ${\theta }^{\ast }=[a,b](0<a<b<\infty )$ be a finite interval on ${\mathfrak{R}}^{+}$, and let $\mathcal{C}\left({\theta }^{\ast }\right)$ be the space of continuous function $f:{\theta }^{\ast }\to \mathfrak{R}$ with the norm

$$\begin{array}{c}\hfill {\parallel f\parallel }_{\mathcal{C}\left({\theta }^{\ast }\right)}=\underset{t\in {\theta }^{\ast }}{sup}\left|f\left(t\right)\right|.\end{array}$$

The weighted space ${\mathcal{C}}_{1-\vartheta ,\mathsf{\Phi }}\left({\theta }^{\ast }\right)$ of continuous f on $\theta$ is defined by (see [19])

$$\begin{array}{c}\hfill {\mathcal{C}}_{1-\vartheta ,\mathsf{\Phi }}\left({\theta }^{\ast }\right)=\{f:\theta \to \mathfrak{R};{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(a\right))}^{1-\vartheta }f\left(t\right)\in \mathcal{C}\left({\theta }^{\ast }\right)\},0\le \vartheta <1,\end{array}$$

with the norm

$$\begin{array}{c}\hfill {\parallel f\parallel }_{{\mathcal{C}}_{1-\vartheta ,\mathsf{\Phi }}\left({\theta }^{\ast }\right)}=\underset{t\in {\theta }^{\ast }}{sup}\left|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(a\right))}^{1-\vartheta }f\left(t\right)\right|.\end{array}$$

Definition 1 

([18]). Let $\sigma >0$, f be an integrable function defined on ${\theta }^{\ast }$ and $\mathsf{\Phi }\in {\mathcal{C}}^1\left({\theta }^{\ast }\right)$ be an increasing function with ${\mathsf{\Phi }}^{\prime }\left(t\right)\ne 0$, for all $t\in {\theta }^{\ast }$. The left-sided Φ-type R-L fractional integral operator of order σ of a function f is described by

$$\begin{array}{c}\hfill {\mathfrak{I}}_{a^{+}}^{\sigma ,\mathsf{\Phi }}f\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\sigma \right)}}{\int }_a^t{\mathsf{\Phi }}^{\prime }\left(s\right){(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(s\right))}^{\sigma -1}f\left(s\right)ds.\end{array}$$

Definition 2 

([18]). Let $n-1<\sigma <n$ with $n\in \mathbb{N}$, $f,\mathsf{\Phi }\in {\mathcal{C}}^n({\theta }^{\ast },\mathfrak{R})$ two functions such that Φ is increasing and ${\mathsf{\Phi }}^{\prime }\left(t\right)\ne 0$, for all $t\in {\theta }^{\ast }$. The left-sided Φ-HFD ${}^H\mathfrak{D}_{a^{+}}^{\sigma ,\xi ,\mathsf{\Phi }}$ of function f of order σ and type $\xi \in [0,1]$ is defined by

$$\begin{array}{c}\hfill {}^H\mathfrak{D}_{a^{+}}^{\sigma ,\xi ,\mathsf{\Phi }}f\left(t\right)={\mathfrak{I}}_{a^{+}}^{\xi (n-\sigma ),\mathsf{\Phi }}{\left({\displaystyle \frac{1}{{\mathsf{\Phi }}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{\mathfrak{I}}_{a^{+}}^{(1-\xi )(n-\sigma ),\mathsf{\Phi }}f\left(t\right).\end{array}$$

Definition 3 

([22]). The multivariate Mittag–Leffler function is described by

$$\begin{array}{c}\hfill {\displaystyle {\mathbb{E}}_{({\sigma }_1,\cdots ,{\sigma }_m),\xi }(x_1,\cdots ,x_m)=\sum _{l=0}^{\infty }\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,\cdots ,l_m}\right){\displaystyle \frac{x_1^{l_1}\cdots x_m^{l_m}}{\mathsf{\Gamma }({\sigma }_1{l}_1+\cdots +{\sigma }_m{l}_m+\xi )}},}\end{array}$$

where ${\sigma }_i,\xi >0$ and $x_i\in \mathbb{C}$ for $i=1,2,\cdots ,m$ and

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{l}{l_1,\cdots ,l_m}}\right)={\displaystyle \frac{l!}{l_1!\cdots l_m!}}.\end{array}$$

Definition 4 

([31]). Let $(X,\tau )$ be a metric space. Now $\mathcal{S}:X\to X$ is called a Picard operator if there exists $v^{\ast }\in X$ such that

$\left(i\right)$

${\mathbb{F}}_{\mathcal{S}}=\left\{v^{\ast }\right\}$, where ${\mathbb{F}}_{\mathcal{S}}=\{v\in X:\mathcal{S}\left(v\right)=v\}$ is the fixed point set of $\mathcal{S}$;

$\left(ii\right)$

the sequence ${\left({\mathcal{S}}^n\left(v_0\right)\right)}_{n\in \mathbb{N}}$ converges to $v^{\ast }$ for every $v_0\in X$.

Lemma 1 

([31]). Let $(X,\tau ,\le )$ be an ordered metric space, and let $\mathcal{S}:X\to X$ be an increasing Picard operator with ${\mathbb{F}}_{\mathcal{S}}=\left\{v_{\mathcal{S}}^{\ast }\right\}$. Then for $v\in X$, $v\le \mathcal{S}\left(v\right)$ implies $v\le v_{\mathcal{S}}^{\ast }$.

3. Existence and Uniqueness of Solutions

Using Lemma 2.9 in [31] and Theorem 3 in [22], we can rewrite the equation in its equivalent integral form as

$$\begin{array}{c}\hfill v\left(t\right)=\left\{\begin{array}{cc}\hfill & \eta \left(t\right),t\in [-\rho ,0],\hfill \\ \hfill & {\displaystyle {\displaystyle \frac{v_0}{\mathsf{\Gamma }\left(\vartheta \right)}}\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\vartheta -1}\hfill \\ & {\displaystyle +\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}f(t,v\left(t\right),v\left(g\left(t\right)\right)),t\in \theta .\hfill \end{array}\right.\end{array}$$

(2)

The following hypotheses are necessary in obtaining the main results.

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

The functions $f\in \mathcal{C}(\theta \times {\mathfrak{R}}^2,\mathfrak{R})$, $g\in \mathcal{C}(\theta ,[-\rho ,b\left]\right)$, with $g\left(t\right)\le t$ and $\rho >0$.

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

There is a constant ${\mathcal{L}}_f>0$ such that

$$\begin{array}{c}\hfill |f(t,v_1,v_2)-f(t,w_1,w_2)|\le {\mathcal{L}}_f\sum _{i=1}^2|v_i-w_i|,\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}t\in \theta ,v_i,w_i\in \mathfrak{R},i=1,2.\end{array}$$

$\left({\mathcal{H}}_3\right)$

We have the inequality

$$\begin{array}{cc}\hfill \mathrm{\Omega }:=& 2{\mathcal{L}}_f{(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +\xi }\mathsf{\Gamma }\left(\vartheta \right){\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m),2(\sigma +\xi )-\sigma \xi }\hfill \\ & \times \left(\right|{\mu }_1|{(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)<1.\hfill \end{array}$$

Theorem 1.

If $\left({\mathcal{H}}_i\right)(i=1,2,3)$ hold, then Equation (1) has a unique solution in $\mathcal{C}[-\rho ,b]\cap {\mathcal{C}}_{1-\vartheta ,\mathsf{\Phi }}\left({\theta }^{\ast }\right)$.

Proof.

In view of Equation (2), we have

$$\begin{array}{c}\hfill v\left(t\right)=\left\{\begin{array}{cc}\hfill & \eta \left(t\right),t\in [-\rho ,0],\hfill \\ \hfill & {\displaystyle {\displaystyle \frac{v_0}{\mathsf{\Gamma }\left(\vartheta \right)}}\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\vartheta -1}\hfill \\ & {\displaystyle +\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}f(t,v\left(t\right),v\left(g\left(t\right)\right)),t\in \theta .\hfill \end{array}\right.\end{array}$$

(3)

Let us define an operator ${\mathcal{S}}_f:\mathcal{C}[-\rho ,b]\to \mathcal{C}[-\rho ,b]$ as follows:

$$\begin{array}{c}\hfill {\mathcal{S}}_f\left(v\left(t\right)\right)=\left\{\begin{array}{cc}\hfill & \eta \left(t\right),t\in [-\rho ,0],\hfill \\ \hfill & {\displaystyle {\displaystyle \frac{v_0}{\mathsf{\Gamma }\left(\vartheta \right)}}\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\vartheta -1}\hfill \\ & {\displaystyle +\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}f(t,v\left(t\right),v\left(g\left(t\right)\right)),t\in \theta .\hfill \end{array}\right.\end{array}$$

(4)

We transform the existence of a solution for Equation (3) into a fixed point, $w={\mathcal{S}}_{f}v$, where ${\mathcal{S}}_f$ is mentioned by Equation (4). Clearly, the operator ${\mathcal{S}}_f$ is continuous:

$$\begin{array}{cc}\hfill |{\mathcal{S}}_f& \left(v\right)\left(t\right)-{\mathcal{S}}_f\left(v\right)\left(t_0\right)|\hfill \\ & {\displaystyle =\left|\right({\displaystyle \frac{v_0}{\mathsf{\Gamma }\left(\vartheta \right)}}\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\vartheta -1}\hfill \\ & {\displaystyle +\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}f(t,v\left(t\right),v\left(g\left(t\right)\right)))\hfill \\ & {\displaystyle -({\displaystyle \frac{v_0}{\mathsf{\Gamma }\left(\vartheta \right)}}\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}{(\mathsf{\Phi }\left(t_0\right)-\mathsf{\Phi }\left(0\right))}^{\vartheta -1}\hfill \\ & {\displaystyle +\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}f(t_0,v\left(t_0\right),v\left(g\left(t_0\right)\right))\left)\right|\hfill \\ & \to 0ast\to t_0.\hfill \end{array}$$

Next, we show that the operator ${\mathcal{S}}_f$ is a contraction mapping. For $t\in [-\rho ,0]$, we have

$$\begin{array}{c}\hfill |{\mathcal{S}}_f\left(v\right)\left(t\right)-{\mathcal{S}}_f\left(w\right)\left(t\right)|=0,v,w\in \mathcal{C}([-\rho ,0],\mathfrak{R}).\end{array}$$

For each $t\in \theta$, we have

$$\begin{array}{cc}\hfill |{\mathcal{S}}_f& \left(v\right)\left(t\right)-{\mathcal{S}}_f\left(w\right)\left(t\right)|\hfill \\ & {\displaystyle \le \sum _{l=0}^{\infty }\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}{\mathfrak{I}}_{0^{+}}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}}\hfill \\ & \times \left|f\right(t,v\left(t\right),v\left(g\right(t\left)\right))-f(t,w\left(t\right),w\left(g\right(t\left)\right)\left)\right|\hfill \\ & {\displaystyle \le \sum _{l=0}^{\infty }\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}(\frac{{\mathcal{L}}_f}{\mathsf{\Gamma }(\sigma +\xi +(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}\hfill \\ & \times {\int }_0^t{\mathsf{\Phi }}^{\prime }\left(s\right){(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(s\right))}^{\sigma +\xi +(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}{(\mathsf{\Phi }\left(s\right)-\mathsf{\Phi }\left(0\right))}^{\vartheta -1}\hfill \\ & \times \left\{{(\mathsf{\Phi }\left(s\right)-\mathsf{\Phi }\left(0\right))}^{1-\vartheta }\left[\left|v\right(s)-w(s\left)\right|+\left|v\right(g\left(s\right))-w(g\left(s\right)\left)\right|\right]\right\}ds)\hfill \\ & {\displaystyle \le \sum _{l=0}^{\infty }\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}(\frac{{\mathcal{L}}_f}{\mathsf{\Gamma }(\sigma +\xi +(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}\hfill \\ & \times {\int }_0^t{\mathsf{\Phi }}^{\prime }\left(s\right){(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(s\right))}^{\sigma +\xi +(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}{(\mathsf{\Phi }\left(s\right)-\mathsf{\Phi }\left(0\right))}^{\vartheta -1}\hfill \\ & \times \left\{\right[\underset{s\in [0,b]}{max}\left|{(\mathsf{\Phi }\left(s\right)-\mathsf{\Phi }\left(0\right))}^{1-\vartheta }(v\left(s\right)-w\left(s\right))\right|\hfill \\ & +\underset{s\in [0,b]}{max}\left|{(\mathsf{\Phi }\left(s\right)-\mathsf{\Phi }\left(0\right))}^{1-\vartheta }(v\left(g\left(s\right)\right)-w\left(g\left(s\right)\right))\right|\left]\right\}ds)\hfill \\ & {\displaystyle \le \sum _{l=0}^{\infty }\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}(\frac{2{\mathcal{L}}_f}{\mathsf{\Gamma }(\sigma +\xi +(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}\hfill \\ & {\times \parallel v-w\parallel }_{{\mathcal{C}}_{1-\vartheta ,\mathsf{\Phi }}[0,b]}{\int }_0^t{\mathsf{\Phi }}^{\prime }\left(s\right){(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(s\right))}^{\sigma +\xi +(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}{(\mathsf{\Phi }\left(s\right)-\mathsf{\Phi }\left(0\right))}^{\vartheta -1}ds\hfill \\ & {\displaystyle =2{\mathcal{L}}_f{(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +\xi }\mathsf{\Gamma }\left(\vartheta \right)\sum _{l=0}^{\infty }\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right)|{\mu }_1{|}^{l_1}|{\mu }_2{|}^{l_2}\cdots {\left|{\mu }_m\right|}^{l_m}}\hfill \\ & \times {\displaystyle \frac{{(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m}}{\mathsf{\Gamma }(2(\sigma +\xi )-\sigma \xi +(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}{\parallel v-w\parallel }_{{\mathcal{C}}_{1-\vartheta ,\mathsf{\Phi }}[0,b]}.\hfill \\ & ={\mathsf{\Omega }\parallel v-w\parallel }_{{\mathcal{C}}_{1-\vartheta ,\mathsf{\Phi }}[0,b]}.\hfill \end{array}$$

It follows that $\left({\mathcal{H}}_3\right)$, ${\mathcal{S}}_f$ is a contraction on $\mathcal{C}[-\rho ,b]$. Hence, thanks to the Banach contraction principle, ${\mathcal{S}}_f$ has a unique fixed point v which is a solution to Equation (1). □

4. Stability Theory

In this section, we will discuss the H-U-M-L stability results for $\mathsf{\Phi }$-HFOIDDSs (1).

Definition 5.

Equation (1) is H-U-M-L stable with respect to ${\mathbb{E}}_{(\sigma +{\xi }_1,\dots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1{|(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\dots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)$, if there exists ${\mathcal{C}}_{{\mathbb{E}}_{(\sigma +{\xi }_1,\dots ,\sigma +{\xi }_m)}}>0$ where for each $ϵ>0$ and for each solution $w\in \mathcal{C}\left(\right[-\rho ,b],\mathfrak{R})$ of

$$\begin{array}{cc}|{}^H\mathfrak{D}_{0^{+}}^{\sigma ,\xi ,\mathsf{\Phi }}\hfill & w\left(t\right)+\sum _{i=1}^m{\mu }_i{\mathfrak{I}}_{0^{+}}^{{\xi }_i,\mathsf{\Phi }}w\left(t\right)-{\mathfrak{I}}_{0^{+}}^{\xi ,\mathsf{\Phi }}f(t,w\left(t\right),w\left(g\left(t\right)\right))|\hfill \\ & \hfill \le ϵ{\mathbb{E}}_{(\sigma +{\xi }_1,\dots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1{|(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right),\end{array}$$

(5)

there exists a solution $v\in \mathcal{C}\left(\right[-\rho ,b],\mathfrak{R})$ of Equation (1) with

$$\begin{array}{cc}\hfill \left|w\right(t)-v(t\left)\right|& \le {\mathcal{C}}_{{\mathbb{E}}_{(\sigma +{\xi }_1,\dots ,\sigma +{\xi }_m)}}ϵ{\mathbb{E}}_{(\sigma +{\xi }_1,\dots ,\sigma +{\xi }_m)}\hfill \\ & \times \left(\right|{\mu }_1|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\dots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right),t\in [-\rho ,b].\hfill \end{array}$$

Remark 1.

A function $v\in \mathcal{C}\left(\right[-\rho ,b],\mathfrak{R})$ is a solution of inequality (5) iff there exists $f_1\in \mathcal{C}([-\rho ,b],\mathfrak{R})$ such that

$\left(i\right)$

$|f_1\left(t\right)|\le ϵ{\mathbb{E}}_{(\sigma +{\xi }_1,\dots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1{|(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\dots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)$, $t\in [-\rho ,b]$,

$\left(ii\right)$

${\displaystyle {}^H\mathfrak{D}_{0^{+}}^{\sigma ,\xi ,\mathsf{\Phi }}v\left(t\right)+\sum _{i=1}^m{\mu }_i{\mathfrak{I}}_{0^{+}}^{{\xi }_i,\mathsf{\Phi }}v\left(t\right)={\mathfrak{I}}_{0^{+}}^{\xi ,\mathsf{\Phi }}f(t,v\left(t\right),v\left(g\left(t\right)\right))+f_1\left(t\right)}$, $t\in \theta$.

Theorem 2.

Assume that $\left({\mathcal{H}}_i\right)(i=1,2,3)$ are fulfilled, then Equation (1) is H-U-M-L stable.

Proof. 

Let $v\in \mathcal{C}[-\rho ,0]\cap {\mathcal{C}}_{1-\vartheta ,\mathsf{\Phi }}[0,b]$ satisfy (5). It follows that

$$\begin{array}{cc}\hfill |& {\displaystyle v\left(t\right)-{\displaystyle \frac{v_0}{\mathsf{\Gamma }\left(\vartheta \right)}}\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\vartheta -1}\hfill \\ & {\displaystyle -\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}f(t,v\left(t\right),v\left(g\left(t\right)\right))|\hfill \\ & \hfill \le ϵ{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1{|(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right),\in t\in \theta .\end{array}$$

(6)

Observe that $\left|w\right(s)-v(s\left)\right|=0$, for $t\in [-\rho ,0]$.

For every $t\in \theta$, using $\left({\mathcal{H}}_2\right)$ and inequality (6), we have

$$\begin{array}{cc}\hfill \left|w\right(t)& -v\left(t\right)|\hfill \\ & {\displaystyle \le |w\left(t\right)-{\displaystyle \frac{w_0}{\mathsf{\Gamma }\left(\vartheta \right)}}\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\vartheta -1}\hfill \\ & {\displaystyle -\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}f(t,w\left(t\right),w\left(g\left(t\right)\right))|\hfill \\ & {\displaystyle +|\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}f(t,w\left(t\right),w\left(g\left(t\right)\right))\hfill \\ & {\displaystyle -\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\mathfrak{I}}_{0^{+}}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m,\mathsf{\Phi }}f(t,v\left(t\right),v\left(g\left(t\right)\right))|\hfill \\ & \le ϵ{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1{|(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)\hfill \\ & {\displaystyle +\sum _{l=0}^{\infty }\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\displaystyle \frac{{\mathcal{L}}_f}{\mathsf{\Gamma }((\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}\hfill \\ & \times {\int }_0^t{\mathsf{\Phi }}^{\prime }\left(s\right){(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(s\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}[|w\left(s\right)-v\left(s\right)|\hfill \\ & +\left|w\right(g\left(s\right))-v(g\left(s\right)\left)\right|]ds.\hfill \end{array}$$

(7)

For all $u\in \mathcal{C}([-\rho ,b],{\mathfrak{R}}_{+})$, define an operator ${\mathcal{S}}_1:\mathcal{C}([-\rho ,b],{\mathfrak{R}}_{+})\to \mathcal{C}([-\rho ,b],{\mathfrak{R}}_{+})$ by

$$\begin{array}{c}\hfill {\mathcal{S}}_1\left(u\right)\left(t\right)=\left\{\begin{array}{cc}\hfill & 0,t\in [-\rho ,0],\hfill \\ \hfill & ϵ{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1{|(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)\hfill \\ & {\displaystyle +\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\displaystyle \frac{{\mathcal{L}}_f}{\mathsf{\Gamma }((\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}\hfill \\ & {\displaystyle \times ({\int }_0^t{\mathsf{\Phi }}^{\prime }\left(s\right){(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(s\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}u\left(s\right)ds}\hfill \\ & {\displaystyle +{\int }_0^t{\mathsf{\Phi }}^{\prime }\left(s\right){(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(s\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}u\left(g\left(s\right)\right)ds),t\in \theta .}\hfill \end{array}\right.\end{array}$$

We want to prove that ${\mathcal{S}}_1$ is a Picard operator. Let $u,u_1\in \mathcal{C}([-\rho ,b],\mathfrak{R})$, and by using $\left({\mathcal{H}}_2\right)$, we have

$$\begin{array}{cc}\hfill \parallel {\mathcal{S}}_1& \left(u\right)\left(t\right)-{\mathcal{S}}_1\left(u_1\right)\left(t\right){\parallel }_{{\mathcal{C}}_{1-\vartheta ,\mathsf{\Phi }}}\hfill \\ & {\displaystyle \le 2{\mathcal{L}}_f{(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +\xi }\mathsf{\Gamma }\left(\vartheta \right)\sum _{l=0}^{\infty }\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right)|{\mu }_1{|}^{l_1}|{\mu }_2{|}^{l_2}\cdots {\left|{\mu }_m\right|}^{l_m}}\hfill \\ & \times {\displaystyle \frac{{(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m}}{\mathsf{\Gamma }(2(\sigma +\xi )-\sigma \xi +(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}{\parallel u-u_1\parallel }_{{\mathcal{C}}_{1-\vartheta ,\mathsf{\Phi }}[0,b]}.\hfill \end{array}$$

Hence ${\mathcal{S}}_1$ is a contraction on $\mathcal{C}\left(\right[-\rho ,b],\mathfrak{R})$.

Taking the Banach contraction principle to ${\mathcal{S}}_1$, we see that ${\mathcal{S}}_1$ is a Picard operator and ${\mathbb{F}}_{{\mathcal{S}}_1}=u^{\ast }$. Then, for any $t\in {\theta }^{\ast }$, we have

$$\begin{array}{cc}\hfill u^{\ast }\left(t\right)(=& {\mathcal{S}}_1{u}^{\ast }\left(t\right))\hfill \\ & =ϵ{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1{|(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)\hfill \\ & {\displaystyle +\sum _{l=0}^{\infty }\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\displaystyle \frac{{\mathcal{L}}_f}{\mathsf{\Gamma }((\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}\hfill \\ & {\displaystyle \times ({\int }_0^t{\mathsf{\Phi }}^{\prime }\left(s\right){(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(s\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}{u}^{\ast }\left(s\right)ds}\hfill \\ & {\displaystyle +{\int }_0^t{\mathsf{\Phi }}^{\prime }\left(s\right){(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(s\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}{u}^{\ast }\left(g\left(s\right)\right)ds).}\hfill \end{array}$$

Next, it will be shown that the solution $u^{\ast }$ is increasing. For every $0\le t_1<t_2\le b$ (letting ${\displaystyle p:=\underset{s\in [0,b]}{min}[u^{\ast }\left(s\right)+u^{\ast }\left(g\left(s\right)\right)]\in {\mathfrak{R}}_{+}}$), we have

$$\begin{array}{cc}\hfill u^{\ast }& \left(t_2\right)-u^{\ast }\left(t_1\right)\hfill \\ & =ϵ[{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1|{(\mathsf{\Phi }\left(t_2\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t_2\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)\hfill \\ & -{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1|{(\mathsf{\Phi }\left(t_1\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t_1\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)]\hfill \\ & +{\displaystyle \frac{{\mathcal{L}}_f}{\mathsf{\Gamma }((\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}\hfill \\ & \times {\int }_0^{t_1}{\mathsf{\Phi }}^{\prime }\left(s\right)[{(\mathsf{\Phi }\left(t_2\right)-\mathsf{\Phi }\left(s\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}\hfill \\ & -{(\mathsf{\Phi }\left(t_1\right)-\mathsf{\Phi }\left(s\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}](u^{\ast }\left(s\right)+u^{\ast }\left(g\left(s\right)\right))ds\hfill \\ & +{\displaystyle \frac{{\mathcal{L}}_f}{\mathsf{\Gamma }((\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}\hfill \\ & +{\int }_{t_1}^{t_2}{\mathsf{\Phi }}^{\prime }\left(s\right){(\mathsf{\Phi }\left(t_2\right)-\mathsf{\Phi }\left(s\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}(u^{\ast }\left(s\right)+u^{\ast }\left(g\left(s\right)\right))ds\hfill \\ & \ge ϵ[{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1|{(\mathsf{\Phi }\left(t_2\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t_2\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)\hfill \\ & -{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1|{(\mathsf{\Phi }\left(t_1\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t_1\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)]\hfill \\ & {\displaystyle +\sum _{l=0}^{\infty }\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\displaystyle \frac{p{\mathcal{L}}_f}{\mathsf{\Gamma }((\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}\hfill \\ & \times {\int }_0^{t_1}{\mathsf{\Phi }}^{\prime }\left(s\right)[{(\mathsf{\Phi }\left(t_2\right)-\mathsf{\Phi }\left(s\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}\hfill \\ & -{(\mathsf{\Phi }\left(t_1\right)-\mathsf{\Phi }\left(s\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}]ds\hfill \\ & {\displaystyle +\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\displaystyle \frac{p{\mathcal{L}}_f}{\mathsf{\Gamma }((\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}\hfill \\ & \times {\int }_{t_1}^{t_2}{\mathsf{\Phi }}^{\prime }\left(s\right){(\mathsf{\Phi }\left(t_2\right)-\mathsf{\Phi }\left(s\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}ds\hfill \\ & =ϵ[{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1|{(\mathsf{\Phi }\left(t_2\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t_2\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)\hfill \\ & -{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1|{(\mathsf{\Phi }\left(t_1\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t_1\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)]\hfill \\ & {\displaystyle +\sum _{l=0}^{\infty }\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\displaystyle \frac{p{\mathcal{L}}_f}{\mathsf{\Gamma }((\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m+1)}}\hfill \\ & \times \left[{(\mathsf{\Phi }\left(t_2\right)-\mathsf{\Phi }\left(0\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m}-{(\mathsf{\Phi }\left(t_1\right)-\mathsf{\Phi }\left(0\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m}\right]\hfill \\ & >0.\hfill \end{array}$$

Thus $u^{\ast }$ is increasing, so $u^{\ast }\left(g\left(t\right)\right)\le u^{\ast }\left(t\right)$ since $g\left(t\right)\le t$ and

$$\begin{array}{cc}\hfill u^{\ast }\left(t\right)& \le ϵ{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1{|(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)\hfill \\ & {\displaystyle +\sum _{l=0}^{\infty }{(-1)}^l\sum _{l1+l2+\cdots +lm=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2,\cdots ,l_m}\right){\mu }_1^{l_1}{\mu }_2^{l_2}\cdots {\mu }_m^{l_m}}\hfill \\ & \times {\displaystyle \frac{2{\mathcal{L}}_f}{\mathsf{\Gamma }((\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m)}}\hfill \\ & \times {\int }_0^t{\mathsf{\Phi }}^{\prime }\left(s\right){(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(s\right))}^{(\sigma +\xi )+(\sigma +{\xi }_1)l_1+\cdots +(\sigma +{\xi }_m)l_m-1}{u}^{\ast }\left(s\right)ds.\hfill \end{array}$$

By Lemma 2.11 in [31], we get

$$\begin{array}{cc}\hfill u^{\ast }\left(t\right)& \le ϵ{\mathbb{E}}_{(\sigma +{\xi }_1,\dots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1{|(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\dots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)\hfill \\ & \le {\mathcal{C}}_{{\mathbb{E}}_{(\sigma +{\xi }_1,\dots ,\sigma +{\xi }_m)}}ϵ{\mathbb{E}}_{(\sigma +{\xi }_1,\dots ,\sigma +{\xi }_m)}\left(\right|{\mu }_1{|(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\dots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\mathcal{C}}_{{\mathbb{E}}_{(\sigma +{\xi }_1,\dots ,\sigma +{\xi }_m)}}:={\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\left(2{\mathcal{L}}_f\left(\right|{\mu }_1{|(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(b\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right)\right).\end{array}$$

Specifically, if $u=|w-v|$, from Equation (7), $u\le {\mathcal{S}}_{1}u$ by Lemma 1, we deduce that $u\le u^{\ast }$, where ${\mathcal{S}}_1$ is increasing Picard operator. Then

$$\begin{array}{cc}\hfill \left|v\right(t)-w(t\left)\right|& \le {\mathcal{C}}_{{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}}ϵ{\mathbb{E}}_{(\sigma +{\xi }_1,\cdots ,\sigma +{\xi }_m)}\hfill \\ & \times \left(\right|{\mu }_1|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_1},\cdots ,|{\mu }_m\left|{(\mathsf{\Phi }\left(t\right)-\mathsf{\Phi }\left(0\right))}^{\sigma +{\xi }_m}\right),t\in [-\rho ,b].\hfill \end{array}$$

Consequently, Equation 1 is H-U-M-L stable. □

5. An Example

Consider the following $\mathsf{\Phi }$-HFOIDDS

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^H\mathfrak{D}_{0^{+}}^{0.4,0.7,t^2}v\left(t\right)+3{\mathfrak{I}}_{0^{+}}^{0.7,t^2}v\left(t\right)-3{\mathfrak{I}}_{0^{+}}^{0.8,t^2}v\left(t\right)={\displaystyle \frac{1}{25}}{\mathfrak{I}}_{0^{+}}^{0.7,t^2}\left[{\displaystyle \frac{v^2(t-1)}{1+v^2(t-1)}}+arctan\left(v\left(t\right)\right)\right]\hfill \\ {\mathfrak{I}}_{0^{+}}^{1-0.82,t^2}v\left(0\right)=v_0,\hfill \\ \hfill & v\left(t\right)=0,t\in [-1,0],\hfill \end{array}\right.\end{array}$$

(8)

and the inequality

$$\begin{array}{cc}\hfill & \left|{}^H\mathfrak{D}_{0^{+}}^{0.4,0.7,t^2}v\left(t\right)+3{\mathfrak{I}}_{0^{+}}^{0.7,t^2}v\left(t\right)-3{\mathfrak{I}}_{0^{+}}^{0.8,t^2}v\left(t\right)-{\displaystyle \frac{1}{25}}{\mathfrak{I}}_{0^{+}}^{0.7,t^2}f(t,v\left(t\right),v(t-1))\right|\hfill \\ & \le ϵ{\mathbb{E}}_{(1.1,1.2)}(3,3).\hfill \end{array}$$

Let $\sigma =0.4$, $\xi =0.7$. Then $\vartheta =\sigma +\xi (1-\sigma )=0.82$, $\mathsf{\Phi }\left(t\right)=t^2$, ${\xi }_1=0.7$, ${\xi }_2=0.8$, ${\mu }_1={\mu }_2=3$, $g(·)=-1$, $f(·,v(·),v\left(g(·)\right))={\displaystyle \frac{1}{25}}\left[{\displaystyle \frac{v^2(·-1)}{1+v^2(·-1)}}+arctan\left(v(·)\right)\right]$, and ${\mathcal{L}}_f={\displaystyle \frac{1}{25}}$. Thus $2{\mathcal{L}}_f={\displaystyle \frac{2}{25}}$.

We want to verify that $f(·,v(·),v(g(·)\left)\right)$ satisfies $\left({\mathcal{H}}_2\right)$:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{25}}{\displaystyle \frac{v_1^2(·-1)}{1+v_1^2(·-1)}}+{\displaystyle \frac{1}{25}}arctan\left(v_1\right)-{\displaystyle \frac{1}{25}}{\displaystyle \frac{v_2^2(·-1)}{1+v_2^2(·-1)}}-{\displaystyle \frac{1}{25}}arctan\left(v_2\right)\right|\hfill \\ & \le \left|{\displaystyle \frac{1}{25}}{\displaystyle \frac{v_1^2(·-1)}{1+v_1^2(·-1)}}-{\displaystyle \frac{1}{25}}{\displaystyle \frac{v_2^2(·-1)}{1+v_2^2(·-1)}}\right|+\left|{\displaystyle \frac{1}{25}}arctan\left(v_1\right)-{\displaystyle \frac{1}{25}}arctan\left(v_2\right)\right|\hfill \\ & \le \left|{\displaystyle \frac{1}{25}}(v_1-v_2)\right|+\left|{\displaystyle \frac{1}{25}}{(arctan\left(z\right))}^{\prime }(v_1-v_2)\right|\hfill \\ & \le \left|{\displaystyle \frac{1}{25}}(v_1-v_2)\right|+\left|{\displaystyle \frac{1}{25}}{\displaystyle \frac{1}{1+z^2}}(v_1-v_2)\right|\hfill \\ & \le \left|{\displaystyle \frac{1}{25}}(v_1-v_2)\right|+\left|{\displaystyle \frac{1}{25}}(v_1-v_2)\right|\hfill \\ & \le {\displaystyle \frac{2}{25}}|v_1-v_2|,\hfill \end{array}$$

in which $v_1\le z\le v_2$, and thus f is a contraction. In view of hypothesis $\left({\mathcal{H}}_3\right)$, we find that

$$\begin{array}{cc}\hfill \mathrm{\Omega }=& {\displaystyle \frac{2}{25}}\mathsf{\Gamma }\left(0.82\right){\mathbb{E}}_{(1.1,1.2),1.82}(3,3)\hfill \\ \hfill =& {\displaystyle \frac{2}{25}}\mathsf{\Gamma }\left(0.82\right)\sum _{l=0}^{\infty }\sum _{l1+l2=l}\left({\displaystyle \genfrac{}{}{0pt}{}{l}{l_1,l_2}}\right){\displaystyle \frac{3^{l_1}{3}^{l_2}}{\mathsf{\Gamma }(1.1l_1+1.2l_2+1.82)}}.\hfill \end{array}$$

Applying ${\displaystyle \sum _{l_1+l_2=l}\left(\genfrac{}{}{0pt}{}{l}{l_1,l_2}\right)=3^l}$ and ${\displaystyle \frac{3^{l_1}{3}^{l_2}}{\mathsf{\Gamma }(1.1l_1+1.2l_2+1.82)}}\le {\displaystyle \frac{3^l}{\mathsf{\Gamma }(l+1.82)}}$, we find that

$$\begin{array}{c}\hfill \mathsf{\Omega }\le {\displaystyle \frac{2}{25}}\mathsf{\Gamma }\left(0.82\right)\sum _{l=0}^{\infty }{\displaystyle \frac{3^l}{\mathsf{\Gamma }(l+1.82)}}<1.\end{array}$$

Thus, by Theorem 1, Equation (8) has a unique solution and also is H-U-M-L stable with

$$\begin{array}{c}\hfill |v\left(t\right)-w\left(t\right)|\le {\mathcal{C}}_{{\mathbb{E}}_{(1.1,1.2})}ϵ{\mathbb{E}}_{(1.1,1.2)}(3,3),t\in [-1,1],\end{array}$$

where ${\mathcal{C}}_{{\mathbb{E}}_{(1.1,1.2})}={\mathbb{E}}_{(1.1,1.2}\left({\displaystyle \frac{6}{25}},{\displaystyle \frac{6}{25}}\right)$.

6. Concluding Remarks

In this paper, we discuss the existence and uniqueness of the $\mathsf{\Phi }$-HFOIDDS. Furthermore, we study the H-U-M-L stability of $\mathsf{\Phi }$-HFOIDDS. The main findings are obtained through the multivariate Mittag–Leffler function, the Banach contraction principle, and the Picard operator method, as well as generalized Gronwall’s inequality. Moreover, an illustrative example is provided to demonstrate the effectiveness of the theoretical results. Our outcomes obtained are new and extend the existing literature on this topic. In the future, this result could be extended to investigate the H-U-M-L stability for a class of $\mathsf{\Phi }$-HFOIDDS with impulsive conditions.