On Ulam–Hyers–Mittag-Leffler Stability of Fractional Integral Equations Containing Multiple Variable Delays

Abstract

In recent decades, many researchers have pointed out that derivatives and integrals of the non-integer order are well suited for describing various real-world materials, for example, polymers. It has also been shown that fractional-order mathematical models are more effective than integer-order mathematical models. Thereby, given these considerations, the investigation of qualitative properties, in particular, Ulam-type stabilities of fractional differential equations, fractional integral equations, etc., has now become a highly attractive subject for mathematicians, as this represents an important field of study due to their extensive applications in various branches of aerodynamics, biology, chemistry, the electrodynamics of complex media, polymer science, physics, rheology, and so on. Meanwhile, the qualitative concepts called Ulam–Hyers–Mittag-Leffler (U-H-M-L) stability and Ulam–Hyers–Mittag-Leffler–Rassias (U-H-M-L-R) stability are well-suited for describing the characteristics of fractional Ulam-type stabilities. The Banach contraction principle is a fundamental tool in nonlinear analysis, with numerous applications in operational equations, fractal theory, optimization theory, and various other fields. In this study, we consider a nonlinear fractional Volterra integral equation (FrVIE). The nonlinear terms in the FrVIE contain multiple variable delays. We prove the U-H-M-L stability and U-H-M-L-R stability of the FrVIE on a finite interval. Throughout this article, new sufficient conditions are obtained via six new results with regard to the U-H-M-L stability or the U-H-M-L-R stability of the FrVIE. The proofs depend on Banach’s fixed-point theorem, as well as the Chebyshev and Bielecki norms. In the particular case of the FrVIE, an example is delivered to illustrate U-H-M-L stability.

1. Introduction

Fractional-order ordinary differential equations (FrODEs), fractional-order integro-differential equations (FrIDEs), fractional integral equations (FrIEs), etc., are mathematical models that can be utilized to more precisely describe certain events in diverse and widespread fields of science and engineering (Balachandran [1], Benchohra et al. [2], Diethelm [3], Kilbas et al. [4], Miller and Ross [5], Podlubny [6], Ray [7], Ray and Sahoo [8], Zhou [9]). Meanwhile, numerous processes with regard to these equations, both natural and human-made, in sciences, economics, medicine, engineering, etc., involve time delays. Thereby, in the relevant literature, a large number of very interesting and novel applications with regard to these kinds of equations are available. In fact, new and improved models of these kinds of equations are always being studied for their qualitative properties, especially for stability, in the relevant literature. In particular, FrVIEs are one such model that have lately received academic attention in qualitative studies. It should be noted that in the present literature, three types of stability are well known: stability in the sense of Lyapunov (Lyapunov [10]), stability in the sense of Ulam (Ulam [11]) and stability in the sense of U-H-M-L (Li et al. [12,13]). Thereby, the stabilities of ODEs, IEs, IDEs, FrODEs and FrIDEs in the sense of Lyapunov and Ulam have been investigated extensively and are still being studied in the related literature. However, according to the best available information from the pertinent literature, we found only one paper, the work of Eghbali et al. [14], regarding the U-H-M-L and the U-H-M-L-R stabilities of FrVIEs. Thereby, qualitative concepts such as the U-H-M-L stability and the U-H-M-L-R stability for FrVIEs containing multiple variable delays deserve discussion. As is well known, finding exact solutions to such equations is highly challenging, and even if an exact solution exists, its complexity often makes it impractical for numerical computation. Given this challenge, it is essential to investigate approximate solutions and determine whether they closely approximate the exact solution. A differential equation is generally said to be stable in the Ulam sense if, for every solution of the equation, there exists a nearby approximate solution to the perturbed equation. In particular, we would now like to outline some works with regard to U-H-M-L stability, U-H-M-L-R stability, M-L-U stability, Ulam type stabilities, etc., of some mathematical models: In 2009 and 2012, Li et al. [12,13] and Wang and Zhou [15] studied M-L stability and four types of M-L-U stabilities of some FrODEs using the second method of Lyapunov and a Gronwall-type inequality, respectively. Butt and Rehman [16], Houas and Samei [17] and Wang and Zhou [15] also introduced the concepts of M-L stability, G-M-L stability and four types M-L-U stabilities, respectively.

Wang and Zhang [18] studied the existence, uniqueness of solutions and U-H-M-L stability of the following nonlinear Caputo fractional-order differential equation (CFrDE) containing the delay function $g\left(t\right):$

$${}^{C}D_t^{\alpha }x\left(t\right)=f(t,x\left(t\right),x\left(g\left(t\right)\right)),\alpha \in (0,1).$$

Wang and Zhou [18] employed distinct norms and the Banach contraction principle to discuss the existence, uniqueness of solutions and U-H-M-L stability of the above CFrDE containing a variable delay.

Later on, in 2016, Eghbali et al. [14] discussed both U-H-M-L stability and U-H-M-L-R stability for the following FrVIE containing a variable delay:

$$y\left(x\right)=I_{c^{+}}^{q}f(x,x,y\left(x\right),y\left(\alpha \left(x\right)\right))={\displaystyle \frac{1}{\Gamma \left(q\right)}}\underset{c}{\overset{x}{\int }}{(x-\tau )}^{q-1}f(x,\tau ,y\left(\tau \right),y\left(\alpha \left(\tau \right)\right))d\tau ,$$

(1)

where $I_{c^{+}}^q$ is the fractional integral of the order $q,q\in (0,1),\Gamma \left(q\right)$ is the gamma function and $\alpha \left(x\right)$ is the variable delay function. In [14], Eghbali et al. obtained six qualitative results with respect to the U-H-M-L stability and U-H-M-L-R stability of FrVIE (1) via different norms and the Banach contraction principle.

In 2018, Niazi et al. [19] considered the nonlinear neutral-type CFrDE:

$${}^{C}D_0^{\alpha }x\left(t\right)=f(t,x_t,{}^{C}D_0^{\beta }{x}_t),t\in I,$$

where $I=[0,1],\alpha \in (1,2),\beta \in (0,1),{}^{C}D_0^{\alpha }x$ and ${}^{C}D_0^{\beta }$ are Caputo derivatives. Niazi et al. [19] established two results regarding U-H-M-L stability via Chebyshev and Bielecki norms and the Picard operator.

In 2018, Eghbali and Kalvandi [20] proved U-H-M-L stability for the ODE

$$y^{\prime }\left(x\right)=F(x,y\left(x\right))$$

by means of a fixed-point method.

In 2020, Ahmad et al. [21] were concerned with the existence of solutions for an implicit system of neutral CFrDEs according to the Chebyshev norm. In addition, Ahmad et al. [21] also proved the U-H-M-L stability of neutral CFrDEs along with the Picard operator.

Some other works have dealt with the qualitative properties of FrDEs, FrIDEs, FrIEs, etc., and they are described briefly as follows: For the stability of implicit FrDEs, see the book of Abbas et al. [22]; for the U-H-M-L stability of delay fractional difference equations, see the paper of Butt and Rehman [16]; for the existence of the solutions and M-L-U stability of FrDEs, see the paper of Houas and Samei [17]; for the qualitative properties and applications of FrDEs via various methods, look at the books of Balachandran [1], Benchohra et al. [2], Diethelm [3], Kilbas et al. [4], Miller and Ross [5], Podlubny [6], Ray [7], Ray and Sahoo [8], and Zhou [9]; for the U-H and U-H-R stability of various FrDEs, IEs, IDEs, etc., see the papers of Boucenna et al. [23], Brzdek et al. [24], Castro and Simões [25], Ciplea et al. [26], Graef et al. [27], Hyers [28], Jung [29,30], Liu et al. [31], Salim et al. [32], Tunç and Tunç [33,34,35], Tunç et al. [36,37,38] and Wang et al. [39]; for the M-L stability of impulsive FrDEs, see the paper of Mathiyalagan and Ma [40]; and for the M-L stability and synchronization of neutral FrDEs, see the paper of Popa [41]. Avazzadeh [42] presented an efficient stabilized meshless technique with a hybrid kernel to simulate the fractional Rayleigh–Stokes problem for an edge in a viscoelastic fluid. The semi-discretized approach in terms of the convergence and stability properties was also discussed theoretically. El-Sayed and El-Gendy [43] investigated the nonlocal stochastic-integral problem of the fractional order using a stochastic differential equation. The U-H stability of the problem was studied, and the existence of a solution and its continuous dependence on the Brownian motion were also proven.

Inspired by the above papers and books, especially by the paper of Eghbali et al. [14], we will take into account the following FrVIE containing multiple variable delays:

$$\vartheta \left(x\right)={\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{c}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\vartheta \left(s\right),\vartheta \left({\alpha }_k\left(s\right)\right))ds,$$

(2)

where the right hand side of (2) is the fractional integral of the order $q,q\in (0,1)$, and $\Gamma \left(q\right)$ is the gamma function. We assume that $a,b,c\in \mathbb{R}$ and they are fixed numbers such that $-\infty <a\le x\le b<\infty$, $c\in (a,b)$, $f_k\in C([a,b]\times [a,b]\times {\mathbb{R}}^2,\mathbb{R})$, ${\alpha }_k\in C([a,b],[a,b])$ and $0\le {\alpha }_k\left(x\right)\le x$ for each $x\in [a,b],k=1,2,\dots ,N$.

As has been outlined above, problems regarding the qualitative behaviors of fractional mathematical models in the sense of Ulam, U-H-M-L and U-H-M-L-R have gained considerable popularity and importance during the past four decades, mainly because they have demonstrated numerous applications in science, engineering, medicine and more. Therefore, it is necessary to provide several potentially essential tools, such as the second method of Lyapunov, Banach’s fixed-point theorem, Chebyshev norm, Bielecki norm, special functions of mathematical physics, etc., to determine the qualitative properties of these kinds of equations and various real-world problems related to them without solving the equation(s) under study. In light of the given information, we would like to outline the main contributions of this work as follows: To the best of our knowledge there has been no work regarding to the U-H-M-L stability and the U-H-M-L-R stability of FrVIEs containing multiple variable delays. This is the first work with regard to these qualitative properties of FrVIEs containing multiple variable delays. Our aim here is to generalize and enhance the results of Eghbali et al. [14] and Theorems 3.2–3.5, 4.2, 5.2 and 5.4 from FrVIE (1) to FrVIE (2). Hence, the results of Eghbali et al. [14] are extended and improved from one variable delay to N-multiple variable delays depending upon Banach’s fixed-point theorem and the Chebyshev and Bielecki norms. Additionally, this study also has the potential to allow a significant difference across various applications. Specifically, this study also has the potential to enhance the theory of artificial neural networks, particularly bidirectional associative memory networks with leakage delays, as well as numerous models in population dynamics, ecological problems, and more. These applications can be effectively represented by fractional differential equations and integral equations consisting of multiple variable delays. Considering the provided information, our FrVIE (2), being nonlinear with multiple variable delays, presents an attractive mathematical model for researchers in the mentioned fields. Moreover, in a certain case, we also give an example to illustrate applications of the results. Thereby, the new results of this study represent new and considerable contributions to the qualitative theory of the U-H-M-L stability and U-H-M-L-R stability of delay FrVIEs.

The rest of this work is structured as follows: Section 2 presents a few basic definitions, theorems and two remarks. In Section 3, the definition of U-H-M-L stability of the first type, and three results regarding U-H-M-L stability of the first type and a particular example, are presented. Section 4 includes the definition of U-H-M-L-R stability of the first type and a theorem with regard to U-H-M-L-R stability of the first type. In Section 5, the definitions of U-H-M-L stability and U-H-M-L-R stability of the second type are given, and two new results regarding U-H-M-L stability and the U-H-M-L-R stability of the second type are obtained. Lastly, at the end of the study, Section 6 presents the conclusion of the paper.

2. Basic Information

In Section 2, we offer some definitions, two theorems and two remarks, which will be utilized in this paper throughout.

Definition 1

(Kilbas et al. [4]). Let $[a,b]\subset \mathbb{R}.$ The fractional integral of order γ for a function $f\in L^1([a,b],\mathbb{R})$ is described by

$$I_t^{\gamma }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}\underset{a}{\overset{t}{\int }}{\displaystyle \frac{f\left(s\right)}{{(t-s)}^{1-\gamma }}}ds,t>0,\gamma >0,$$

provided that the right side is point-wise defined on ${\mathbb{R}}^{+},{\mathbb{R}}^{+}=[0,\infty )$, where $\Gamma \left(\gamma \right)$ is the gamma function.

Definition 2

(Kilbas et al. [4]). The Riemann–Liouville derivative of order γ for a function $f:{\mathbb{R}}^{+}\to \mathbb{R}$ is described by

$${}^{L}D_t^{\gamma }f\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\gamma )}}{\displaystyle \frac{d^n}{d{t}^n}}\underset{a}{\overset{t}{\int }}{\displaystyle \frac{f\left(s\right)}{{(t-s)}^{\gamma +1-n}}}ds,t>0,n-1<\gamma <n.$$

Definition 3

(Kilbas et al. [4]). The Caputo fractional derivative of order γ of the function $f:{\mathbb{R}}^{+}\to \mathbb{R}$ is described by

$${}_0{}^{C}D_t^{\gamma }f\left(t\right)={\displaystyle \frac{1}{\Gamma (\gamma -n)}}\underset{0}{\overset{t}{\int }}{\displaystyle \frac{f^{\left(n\right)}\left(s\right)ds}{{(t-s)}^{\gamma +1-n}}},t>0,n-1<\gamma <n,n\in N.$$

We will now give a basic result in Theorem 1.

Theorem 1

(Eghbali et al. [14]). Let $(X,d)$ be a generalized complete metric space and $T:X\to X$ be a strictly contractive operator with a Lipschitz constant $L<1$. If there exists a non-negative integer k such that $d(T^{k+1}x,T^{k}x)<\infty$ for some $x\in X$, then the following three outcomes hold true:

(a) 

The sequence ${\left(T^{n}x\right)}_{n\in N}$ converges to a fixed point $x^{*}$ of T;

(b) 

$x^{*}$ is the unique fixed point of T in the set $X^{*}=\left\{y\in X:d(T^{k}x,y)<\infty \right\};$

(c) 

If $y\in X^{*}$, then

$$d\left(y,x^{*}\right)\le {(1-L)}^{-1}d(Ty,y).$$

Theorem 2

(Ye et al. [44]). Let $\tilde{a}\ge 0$ be a locally integrable function on ${\mathbb{R}}^{+},$${\mathbb{R}}^{+}=[0,\infty ),$ and $\tilde{g}\left(t\right)\ge 0$ be a nondecreasing continuous function defined on ${\mathbb{R}}^{+}$ such that $\tilde{g}\left(t\right)\le M,\forall t\in {\mathbb{R}}^{+},$ where $M>0,M\in \mathbb{R},$ and also let $u\left(t\right)$ be a non-negative and locally integrable function on ${\mathbb{R}}^{+}$ with

$$u\left(t\right)\le \tilde{a}\left(t\right)+\tilde{g}\left(t\right)\underset{0}{\overset{t}{\int }}{(t-s)}^{q-1}u\left(s\right)ds,t\in {\Re }^{+}.$$

Then,

$$u\left(t\right)\le \tilde{a}\left(t\right)+\underset{0}{\overset{t}{\int }}\left[\sum _{n=1}^{\infty }{\displaystyle \frac{{\left(\tilde{g}\left(t\right)\gamma \left(q\right)\right)}^n}{\Gamma \left(nq\right)}}{(t-s)}^{nq-1}\tilde{a}\left(s\right)\right]ds,t\in {\mathbb{R}}^{+}.$$

Remark 1

(Ye et al. [44]). Depending upon the conditions of Theorem 2, let the function $\tilde{a}\left(t\right)$ be nondecreasing on ${\mathbb{R}}^{+}.$ Then, $u\left(t\right)\le \tilde{a}\left(t\right)E_q\left[\tilde{g}\left(t\right)\Gamma \left(q\right)t^q\right],$ where $E_q$ is the Mittag-Leffler function described by $E_q\left[z\right]={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{z^k}{\Gamma (kq+1)}},$ where $z\in \mathbb{C}.$

Remark 2.

Assume that $\vartheta \in C(I,\mathbb{R})$ and ϑ is a solution of the inequality

$$\left|{}^{C}D_t^{\alpha }\vartheta \left(t\right)-\sum _{k=1}^N{f}_k(x,t,\vartheta \left(t\right),\vartheta ({\alpha }_k\left(t\right)\left)\right)\right|\le \epsilon E_q\left(t^q\right).$$

Then, ϑ is a solution of the integral inequality

$$\left|\vartheta \left(x\right)-\vartheta \left(0\right)-{\displaystyle \frac{1}{\Gamma \left(q\right)}}\underset{0}{\overset{x}{\int }}\left[{(x-s)}^{q-1}\sum _{k=1}^N{f}_k(x,s,\vartheta \left(s\right),\vartheta \left({\alpha }_k\left(s\right)\right))\right]ds\right|\le \epsilon E_q\left(t^q\right).$$

Due to the first inequality, we deduce that

$$\begin{array}{c}\begin{array}{c} |\vartheta (x)-\vartheta \left(0\right)-{\displaystyle \frac{1}{\Gamma \left(q\right)}}\underset{0}{\overset{x}{\int }}\left[{(x-s)}^{q-1}\sum _{k=1}^N{f}_k(x,s,\vartheta \left(s\right),\vartheta \left({\alpha }_k\left(s\right)\right))\right]ds|\hfill \end{array}\\ \begin{array}{cc}\hfill \le & {\displaystyle \frac{\epsilon }{\Gamma \left(q\right)}}\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{E}_q\left(s^q\right)ds\hfill \\ \hfill =& {\displaystyle \frac{\epsilon }{\Gamma \left(q\right)}}\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}\left(\sum _{k=0}^{\infty }{\displaystyle \frac{s^{kq}}{\Gamma (kq+1)}}\right)ds\hfill \\ \hfill =& {\displaystyle \frac{\epsilon }{\Gamma \left(q\right)}}\sum _{k=0}^{\infty }{\displaystyle \frac{1}{\Gamma (kq+1)}}\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{s}^{kq}ds\hfill \\ \hfill =& {\displaystyle \frac{\epsilon }{\Gamma \left(q\right)}}\sum _{k=0}^{\infty }{\displaystyle \frac{t^{(k+1)q}}{\Gamma (kq+1)}}{\displaystyle \frac{\Gamma \left(q\right)\Gamma (kq+1)}{\Gamma \left((k+1)q+1)\right)}}\hfill \\ \hfill =& \epsilon \sum _{k=0}^{\infty }{\displaystyle \frac{t^{(k+1)q}}{\Gamma \left((k+1)q+1)\right)}}\hfill \\ \hfill \le & \epsilon \sum _{n=0}^{\infty }{\displaystyle \frac{t^{nq}}{\Gamma \left(nq+1)\right)}}\hfill \\ \hfill \le & \epsilon E_q\left(t^q\right).\hfill \end{array}\end{array}$$

Definition 4

(Berinde et al. [45]). Let $\left(X,d\right)$ be a complete metric space, $T:X\to X$ be a mapping and $F_T$ be the fixed-point set of $T,$ i.e., $F_T:=\left\{x\in X:x=T\left(x\right)\right\}$. If there exists $0<\ell <1$ such that

$$d\left(T\left(x\right),T\left(y\right)\right)\le \ell d\left(x,y\right)$$

for all $x,y\in X$, then the mapping T is called a strict contraction.

Theorem 3

(Burton [46], Banach contraction mapping principle). Let $\left(S,\rho \right)$ be a complete metric space and let $P:S\to S$. If there is a positive constant $\alpha <1$ such that for each pair ${\varphi }_1,{\varphi }_2\in S$, we have

$$\rho \left(P{\varphi }_1,P{\varphi }_2\right)\le \alpha \rho \left({\varphi }_1,{\varphi }_2\right).$$

Then, there is one and only one point $\varphi \in S$ with $P\varphi =\varphi$.

3. The First Type of U-H-M-L Stability

Definition 5.

FrVIE (2) is U-H-M-L-stable of the first type regarding the function $E_q$ if there exists a constant $c\in \mathbb{R}$, $c>0$, such that for each $\epsilon >0$ and solution ϑ of the inequality

$$\left|\vartheta \left(x\right)-{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{c}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\vartheta \left(s\right),\vartheta \left({\alpha }_k\left(s\right)\right))ds\right|\le \epsilon E_q\left(x^q\right),$$

(3)

there is a unique solution ${\vartheta }_0$ of FrVIE (2) satisfying the inequality

$$\left|\vartheta \left(x\right)-{\vartheta }_0\left(x\right)\right|\le c\epsilon E_q\left(x^q\right).$$

We will give the first new outcome of this study with regard to the first type of U-H-M-L stability in Theorem 4.

Theorem 4.

We suppose that

(As1) 

$f_k\in C([a,b]\times [a,b]\times {\mathbb{R}}^2,\mathbb{R}),{\alpha }_k\in C([a,b],[a,b])$, and $0\le {\alpha }_k\left(x\right)\le x$ for each $x\in [a,b]$ and $\vartheta \in \mathbb{R}$, $k=1,2,\dots ,N;$

(As2) 

There exists $L_k>0$, $L_k\in \mathbb{R}$, $k=1,2,\dots ,N$ such that

$$\left|f_k\left(x,s,{\vartheta }_1\left(s\right),{\vartheta }_1\left({\alpha }_k\left(s\right)\right)\right)-f_k\left(x,s,{\vartheta }_2\left(s\right),{\vartheta }_2\left({\alpha }_k\left(s\right)\right)\right)\right|\le \left(L_k\right)\left|{\vartheta }_1-{\vartheta }_2\right|,$$

for all $x,s\in [a,b]$, $a,b\in \Re$, ${\vartheta }_1,{\vartheta }_2\in \Re$, and the functions $f_k$ satisfy inequality (3), where

$$0<\sum _{k=1}^N\left(L_k\right)<1.$$

Then, FrVIE (2) admits the first type of U-H-M-L stability.

Proof. 

To prove this theorem, we consider the space

$$X=\{\nu :[a,b]\to \mathbb{R}|\nu \mathrm{is}\mathrm{continuous}\}$$

and the generalized metric described by

$$d(\nu ,\omega )=inf\left\{M\in [0,\infty ]\left|\left|\nu \left(x\right)-\omega \left(x\right)\right|\le M\epsilon E_q\left(x^q\right),\forall x\in [a,b]\right.\right\}.$$

We know that $(X,d)$ is a complete metric space. We define an operator T, $T:X\to X$, as

$$\left(T\nu \right)\left(x\right)={\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\nu \left(s\right),\nu \left({\alpha }_k\left(s\right)\right))ds,$$

(4)

for $\nu \in X$ and $x\in [a,b]$. Since $\nu$ is continuous, $T\nu$ is also continuous. Hence, T is a well-defined operator. Next, for any $\nu ,\omega \in X$, let $M_{\nu \omega }\in [0,\infty ]$ such that

$$\left|\nu \left(x\right)-\omega \left(x\right)\right|\le M_{\nu \omega }\epsilon E_q\left(x^q\right),$$

for each $x\in [a,b]$. From the definition of the operator T, (4) and (As2), we determine that

$$\begin{array}{cc}\hfill \left|\left(T\nu \right)\left(x\right)-\left(T\omega \right)\left(x\right)\right|=& {\displaystyle \frac{1}{\Gamma \left(q\right)}}\left|\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\nu \left(s\right),\nu \left({\alpha }_k\left(s\right)\right))ds\right.\hfill \\ \hfill & -\left.\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\omega \left(s\right),\omega \left({\alpha }_k\left(s\right)\right))ds\right|\hfill \\ \hfill \le & {\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\left(L_k\right)\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}\left|\nu \left(s\right)-\omega \left(s\right)\right|ds\hfill \\ \hfill \le & {\displaystyle \frac{M_{\nu \omega }\epsilon }{\Gamma \left(q\right)}}\sum _{k=1}^N\left(L_k\right)\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{E}_q\left(s^q\right)ds\hfill \\ \hfill =& {\displaystyle \frac{M_{\nu \omega }\epsilon }{\Gamma \left(q\right)}}\sum _{k=1}^N\left(L_k\right)\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}\sum _{k=0}^{\infty }{\displaystyle \frac{s^{kq}}{\Gamma (kq+1)}}ds.\hfill \end{array}$$

Let

$$\mathrm{S}=\sum _{k=1}^N\left(L_k\right).$$

Consequently, we deduce that

$$\begin{array}{cc}\hfill \left|\left(T\nu \right)\left(x\right)-\left(T\omega \right)\left(x\right)\right|\le & {\displaystyle \frac{M_{\nu \omega }\mathrm{S}\epsilon }{\Gamma \left(q\right)}}\sum _{k=0}^{\infty }{\displaystyle \frac{1}{\Gamma (kq+1)}}\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{s}^{kq}ds\hfill \\ \hfill =& {\displaystyle \frac{M_{\nu \omega }\mathrm{S}\epsilon }{\Gamma \left(q\right)}}\sum _{k=0}^{\infty }{\displaystyle \frac{x^{(k+1)q}}{\Gamma (kq+1)}}\underset{0}{\overset{x}{\int }}{(1-t)}^{q-1}{t}^{kq}dt\hfill \\ \hfill =& {\displaystyle \frac{M_{\nu \omega }\mathrm{S}\epsilon }{\Gamma \left(q\right)}}\sum _{k=0}^{\infty }{\displaystyle \frac{x^{(k+1)q}}{\Gamma (kq+1)}}{\displaystyle \frac{\Gamma \left(q\right)\Gamma (kq+1)}{\Gamma (q+(kq+1\left)\right)}}\hfill \\ \hfill =& \left(M_{\nu \omega }\mathrm{S}\epsilon \right)\sum _{k=0}^{\infty }{\displaystyle \frac{x^{(k+1)q}}{\Gamma \left(\right(k+1)q+1)}}\hfill \\ \hfill =& \left(M_{\nu \omega }\mathrm{S}\epsilon \right)\sum _{n=0}^{\infty }{\displaystyle \frac{x^{nq}}{\Gamma (nq+1)}}\le \left(M_{\nu \omega }\mathrm{S}\epsilon \right)E\left(x^q\right),\hfill \end{array}$$

(5)

for each $x\in [a,b]$. According to Equation (5), we obtain

$$d(T\nu ,T\omega )\le \left(M_{\nu \omega }\mathrm{S}\epsilon \right)E\left(x^q\right).$$

Thereby, we can draw the conclusion

$$d(T\nu ,T\omega )\le \sum _{k=1}^N\left(L_k\right)d(\nu ,\omega ),$$

for each $\nu ,\omega \in X$. Since $0<{\displaystyle \sum _{k=1}^N}\left(L_k\right)<1,$ T is a strict contraction.

Next, let us consider ${\nu }_0\in X$. Subsequently, due to the continuity of ${\nu }_0$ and $T{\nu }_0$, there exists a constant ${\mathrm{S}}_1\in \mathbb{R}$ with $0<{\mathrm{S}}_1<\infty$ such that

$$\begin{array}{cc}\hfill |\left(T{\nu }_0\right)\left(x\right)& - {\nu }_0\left(x\right)|\hfill \\ \hfill =& \left|{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,{\nu }_0\left(s\right),{\nu }_0\left({\alpha }_k\left(s\right)\right))ds-{\nu }_0\left(x\right)\right|\le {\mathrm{S}}_{1}E\left(x^q\right),\hfill \end{array}$$

(6)

for each $x\in [a,b]$. Since $f_k,k=1,2,\dots ,N$ and ${\nu }_0$ are bounded on the interval $[a,b]$ and $\underset{x\in [a,b]}{\min }E\left(x^q\right)>0$, from (6), we obtain $d(T{\nu }_0,{\nu }_0)<\infty$. Thereby, by virtue of Theorem 1 (a), we have a function ${\vartheta }_0\in C([a,b],\mathbb{R})$ such that $T^n{\vartheta }_0\to {\vartheta }_0$ in $(X,d)$ as $n\to \infty$ and $T{\vartheta }_0={\vartheta }_0$. Accordingly, ${\vartheta }_0$ satisfies FrVIE (2) for each $x\in [a,b]$.

Next, we will show that

$$\left\{\nu \in X\left|d({\nu }_0,\nu )<\infty \right.\right\}=X.$$

For any $\nu \in X$, since $\nu$ and ${\nu }_0$ are bounded on $[a,b]$ and $\underset{x\in [a,b]}{\min }E\left(x^q\right)>0$, it follows that there exists a constant $M_v,0<M_v<\infty$ such that

$$\left|{\nu }_0\left(x\right)-\nu \left(x\right)\right|\le {\mathrm{M}}_{\nu }{E}_q\left(x^q\right),$$

for each $x\in [a,b]$. Thereby, we obtain $d({\nu }_0,\nu )<\infty$ for each $\nu \in X$, i.e., $\left\{\nu \in X\left|d({\nu }_0,\nu )<\infty \right.\right\}=X$.

In turn, according to Theorem 1 (b), we determine that ${\vartheta }_0$ is continuous and the unique function which satisfies FrVIE (2). We now have

$$d(\vartheta ,T\vartheta )\le \epsilon E_q\left(x^q\right).$$

Finally, according to Theorem 1 (c) and the last inequality, we can determine that

$$d(\vartheta ,{\vartheta }_0)\le {\displaystyle \frac{1}{1-{\displaystyle \sum _{k=1}^N}\left(L_k\right)}}d(T\vartheta ,\vartheta )\le {\displaystyle \frac{1}{1-{\displaystyle \sum _{k=1}^N}\left(L_k\right)}}\epsilon E_q\left(x^q\right).$$

Thereby, this is the last step of the proof of Theorem 4. □

We give an example to illustrate the application of Theorem 3, which includes FrVIE (2).

Example 1.

Take into consideration the following FrVIE containing a variable delay:

$$\left\{\begin{array}{cc}\hfill & \vartheta \left(x\right)=\underset{0}{\overset{x}{\int }}{(x-s)}^{-{\displaystyle \frac{1}{2}}}\left[{\displaystyle \frac{1}{200}}{\displaystyle \frac{\vartheta \left(s\right)}{1+{\vartheta }^2\left(s\right)}}+{\displaystyle \frac{sin\left(\vartheta \left(s\right)\right)}{1000+x^2+{\vartheta }^2\left(2^{-1}s\right)}}\right]ds,x\in [0,1],\hfill \\ \hfill & \vartheta \left(0\right)=0.\hfill \end{array}\right.$$

(7)

and the inequality

$$\left|\vartheta \left(x\right)-{\displaystyle \frac{1}{\Gamma \left(q\right)}}\underset{0}{\overset{x}{\int }}{(x-s)}^{-2^{-1}}\left({\displaystyle \frac{1}{200}}{\displaystyle \frac{\vartheta \left(s\right)}{1+{\vartheta }^2\left(s\right)}}+{\displaystyle \frac{sin\left(\vartheta \left(s\right)\right)}{1000+x^2+{\vartheta }^2\left(2^{-1}s\right)}}\right)ds\right|\le \epsilon E_q\left(x^{2^{-1}}\right).$$

Clearly, FrVIE (7) has the same form as FrVIE (2) with the following outcomes:

$$q={\displaystyle \frac{1}{2}}\in (0,1),[a,b]=[0,1],0<{\alpha }_1\left(x\right)=2^{-1}x,$$

$$f_1(x,s,\vartheta \left(s\right),\vartheta \left({\alpha }_1\left(s\right)\right))={\displaystyle \frac{1}{200}}{\displaystyle \frac{\vartheta \left(s\right)}{1+{\vartheta }^2\left(s\right)}}+{\displaystyle \frac{sin\left(\vartheta \left(s\right)\right)}{1000+x^2+{\vartheta }^2\left(2^{-1}s\right)}}.$$

We will now demonstrate that (As1) and (As2) of Theorem 4 hold.

Evidently, $f_1\in C([0,1]\times [0,1]\times {\mathbb{R}}^2,\mathbb{R})$, $0<{\alpha }_1\left(x\right)=2^{-1}x\le x$ for each $x\in [0,1]$ and ${\alpha }_1\in C([0,1],[0,1])$. Thus, condition (As1) of Theorem 4 holds. We now let $L_{f_1}={\displaystyle \frac{11}{1000}},[0,1]=[a,b]$ and calculate

$$\begin{array}{cc}\hfill \left|f_1(x,s,{\ell }_1,{\ell }_1^{*})-f_1(x,s,{\rho }_1,{\rho }_1^{*})\right|\le & {\displaystyle \frac{1}{200}}\left|{\displaystyle \frac{{\ell }_1}{1+{\ell }_1^2}}-{\displaystyle \frac{{\rho }_1}{1+{\rho }_1^2}}\right|+{\displaystyle \frac{1}{1000}}\left|sin\left({\ell }_1\right)-sin\left({\rho }_1\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{1}{200}}\left|{\displaystyle \frac{{\ell }_1\left(1+{\rho }_1^2\right)-{\rho }_1\left(1+{\ell }_1^2\right)}{\left(1+{\ell }_1^2\right)\left(1+{\rho }_1^2\right)}}\right|+{\displaystyle \frac{1}{1000}}\left|{\ell }_1-{\rho }_1\right|\hfill \\ \hfill \le & {\displaystyle \frac{1}{200}}\left|{\ell }_1-{\rho }_1\right|+{\displaystyle \frac{1}{200}}\left|{\ell }_1-{\rho }_1\right|+{\displaystyle \frac{1}{1000}}\left|{\ell }_1-{\rho }_1\right|\hfill \\ \hfill \le & {\displaystyle \frac{11}{1000}}\left|{\ell }_1-{\rho }_1\right|,\hfill \end{array}$$

and

$$0<\sum _{k=1}^N\left(L_k\right)=L_1={\displaystyle \frac{11}{1000}}<1.$$

Thereby, condition (As2) of Theorem 4 holds. Thus, FrVIE (7) has the first type U-H-M-L stability and

$$\left|\vartheta \left(x\right)-{\vartheta }_0\left(x\right)\right|\le c\epsilon E_{2^{-1}}\left(x^{2^{-1}}\right),x\in [0,1].$$

Consequently, Example 1 satisfies (As1) and (As2) of Theorem 4.

We will now give the result of the first type of U-H-M-L stability by means of the Chebyshev norm in Theorem 5.

Remark 3.

Throughout Theorem 5, we will use the Chebyshev norm described by

$${∥\nu ∥}_C:=\underset{t\in J}{\max }\left|\nu \left(t\right)\right|,\nu \in J,J\subset {\mathbb{R}}^{+}.$$

Theorem 5.

Suppose that (As1) of Theorem 4 and the following condition hold:

(As3) 

There exist $L_k>0$, $L_k\in \mathbb{R}$, $k=1,2,\dots ,N$ such that

$$|f_k(x,s,{\vartheta }_1\left(s\right),{\vartheta }_1\left({\alpha }_k\left(s\right)\right))-f_k\left(x,s,{\vartheta }_2\left(s\right),{\vartheta }_2\left({\alpha }_k\left(s\right)\right)\right)|$$

$$\le \left(L_k\right)\left[\left|{\vartheta }_1-{\vartheta }_2\right|+\left|{\vartheta }_1\left({\alpha }_k\left(s\right)\right)-{\vartheta }_2\left({\alpha }_k\left(s\right)\right)\right|\right],$$

for each $x,s\in [a,b]$, $a,b\in \Re$, ${\vartheta }_1,{\vartheta }_2\in \Re$ and the functions $f_k$ satisfy inequality (3), where

$$0<2E_q\left(b\right)\sum _{k=1}^N\left(L_k\right)<1.$$

Then, FrVIE (2) admits the first type of U-H-M-L stability according to the Chebyshev norm.

Proof. 

Just as in Theorem 4, here, we only show that the operator T defined in (4) is a contraction mapping on the set X pertaining to the Chebyshev norm. Hence, according to the definition of the operator T, the conditions of Theorem 4 and the Chebyshev norm, we determine that

$$\begin{array}{cc}\hfill \left|T\left(\nu \right)\left(x\right)-T\left(\omega \right)\left(x\right)\right|=& {\displaystyle \frac{1}{\Gamma \left(q\right)}}\left|\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\nu \left(s\right),\nu \left({\alpha }_k\left(s\right)\right))ds\right.\hfill \\ \hfill & -\left.\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\omega \left(s\right),\omega \left({\alpha }_k\left(s\right)\right))ds\right|\hfill \\ \hfill \le & {\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\left(L_k\right)\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}\underset{s\in [a,b]}{\max }\left|\nu \left(s\right)-\omega \left(s\right)\right|ds\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\left(L_k\right)\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}\underset{s\in [a,b]}{\max }\left|\nu \left({\alpha }_k\left(s\right)\right)-\omega \left({\alpha }_k\left(s\right)\right)\right|ds\hfill \\ \hfill \le & {\displaystyle \frac{2{\displaystyle \sum _{k=1}^N}\left(L_k\right)}{\Gamma \left(q\right)}}{∥v-\omega ∥}_C\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}ds\hfill \\ \hfill \le & {\displaystyle \frac{2b^q{\displaystyle \sum _{k=1}^N}\left(L_k\right)}{\Gamma (q+1)}}{∥v-\omega ∥}_C\hfill \\ \hfill =& 2E_q\left(b\right)\sum _{k=1}^N\left(L_k\right){∥v-\omega ∥}_C,\hfill \end{array}$$

for each $x\in [a,b]$. Thereby, we obtain

$$d(T{\nu }_0,T\nu )\le 2E_q\left(b\right)\sum _{k=1}^N\left(L_k\right){∥v-\omega ∥}_C.$$

Consequently, we deduce that

$$d(T{\nu }_0,T\nu )\le 2E_q\left(b\right)\sum _{k=1}^N\left(L_k\right)d({\nu }_0,\nu ),$$

for ${\nu }_0,\nu \in X$. Let $0<2E_q\left(b\right){\displaystyle \sum _{k=1}^N}\left(L_k\right)<1$. The final inequality confirms that T is a strictly continuous operator. Just as in the proof of Theorem 4, we can readily obtain

$$d(\vartheta ,{\vartheta }_0)\le {\displaystyle \frac{1}{1-2E_q\left(b\right){\displaystyle \sum _{k=1}^N}\left(L_k\right)}}d(T\vartheta ,\vartheta )\le {\displaystyle \frac{1}{1-2E_q\left(b\right){\displaystyle \sum _{k=1}^N}\left(L_k\right)}}\epsilon C{E}_q\left(x^q\right).$$

Thus, FrVIE (2) admits the first type U-H-M-L stability according to the Chebyshev norm.

Subsequently, we use the Bielecki’s norm ${∥.∥}_B$, i.e., ${∥x∥}_B={max}_{t\in J}\left|x\left(t\right)\right|exp(-\theta t)$, $\theta >0$, $J\subset {\mathbb{R}}^{+}$, to derive similar results to the above for FrVIE (2). □

Remark 4.

Throughout Theorem 6, we will use the Bielecki norm described by

$${∥\nu ∥}_B:=\underset{t\in J}{\max }\left|\nu \left(t\right)\right|exp(-\theta t),\theta >0,J\subset {\mathbb{R}}^{+}.$$

Theorem 6.

Suppose that (As1) of Theorem 4 and the below condition hold:

(As4) 

there exists $L_k>0$, $L_k\in \mathbb{R}$, $k=1,2,\dots ,N$ such that

$$\left|f_k\left(x,s,{\vartheta }_1\left(s\right),{\vartheta }_1\left({\alpha }_k\left(s\right)\right)\right)-f_k\left(x,s,{\vartheta }_2\left(s\right),{\vartheta }_2\left({\alpha }_k\left(s\right)\right)\right)\right|$$

$$\le \left(L_k\right)\left[\left|{\vartheta }_1-{\vartheta }_2\right|+\left|{\vartheta }_1\left({\alpha }_k\left(s\right)\right)-{\vartheta }_2\left({\alpha }_k\left(s\right)\right)\right|\right],$$

for all $x,s\in [a,b]$, $a,b\in \Re$, ${\vartheta }_1,{\vartheta }_2\in \Re$, and the functions $f_k$ satisfy inequality (3), where

$$0<{\displaystyle \frac{2b^{q}exp\left(\theta b\right)}{\Gamma \left(q\right)\sqrt{2\theta (2q-1)}}}\sum _{k=1}^N\left(L_k\right)<1.$$

Then, FrVIE (2) admits the first type of U-H-M-L stability according to the Bielecki norm.

Proof. 

As was shown in the previous theorem, we will only show that the operator T described by (4) is a contraction mapping on X according to the Bielecki norm. Hence, by virtue of the definition of the operator T, (As1) and (As4) of Theorem 6, we determine that

$$\begin{array}{cc}\hfill |& T\left(\nu \right)\left(x\right)-T\left(\omega \right)\left(x\right)|\hfill \\ \hfill =& {\displaystyle \frac{1}{\Gamma \left(q\right)}}|\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\nu \left(s\right),\nu \left({\alpha }_k\left(s\right)\right))ds\hfill \\ \hfill & -\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\omega \left(s\right),\omega \left({\alpha }_k\left(s\right)\right))ds|\hfill \\ \hfill \le & {\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\left(L_k\right)\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}exp\left(\theta s\right)\left[\underset{t\in [a,b]}{\max }\left|\nu \left(s\right)-\omega \left(s\right)\right|exp(-\theta s)\right]ds\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\left(L_k\right)\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}\hfill \\ \hfill & \times exp\left(\theta s\right)\left[\underset{t\in [a,b]}{\max }\left|\nu \left({\alpha }_k\left(s\right)\right)-\omega \left({\alpha }_k\left(s\right)\right)\right|exp(-\theta s)\right]ds\hfill \\ \hfill \le & {\displaystyle \frac{2}{\Gamma \left(q\right)}}{∥\nu -\omega ∥}_B\sum _{k=1}^N\left(L_k\right)\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}exp\left(\theta s\right)ds\hfill \\ \hfill \le & {\displaystyle \frac{2}{\Gamma \left(q\right)}}{∥\nu -\omega ∥}_B\sum _{k=1}^N\left(L_k\right)\left[{\left(\underset{0}{\overset{x}{\int }}{(x-s)}^{2q-2}ds\right)}^{{\displaystyle \frac{1}{2}}}\times {\left(\underset{0}{\overset{x}{\int }}exp\left(2\theta s\right)ds\right)}^{{\displaystyle \frac{1}{2}}}\right]\hfill \\ \hfill \le & {\displaystyle \frac{2b^{q}exp\left(\theta b\right)}{\Gamma \left(q\right)\sqrt{2\theta (2q-1)}}}\sum _{k=1}^N\left(L_k\right){∥\nu -\omega ∥}_B,\hfill \end{array}$$

for each $x\in [a,b]$. Thereby, we have

$$d(T\nu ,T\omega )\le {\displaystyle \frac{2b^{q}exp\left(\theta b\right)}{\Gamma \left(q\right)\sqrt{2\theta (2q-1)}}}\sum _{k=1}^N\left(L_k\right){∥\nu -\omega ∥}_B.$$

According to the data above, we can determine that

$$d(T\nu ,T\omega )\le {\displaystyle \frac{2b^{q}exp\left(\theta b\right)}{\Gamma \left(q\right)\sqrt{2\theta (2q-1)}}}\sum _{k=1}^N\left(L_k\right)d(\nu ,\omega ),$$

for any $\nu ,\omega \in X$. Since

$$0<{\displaystyle \frac{2b^{q}exp\left(\theta b\right)}{\Gamma \left(q\right)\sqrt{2\theta (2q-1)}}}\sum _{k=1}^N\left(L_k\right)<1,$$

T is a strictly continuous contraction.

Finally, similarly to the procedure in the proof of Theorem 4 and the last inequality, we can conclude that

$$d(\vartheta ,{\vartheta }_0)\le {\displaystyle \frac{1}{1-\frac{2b^{q}exp\left(\theta b\right)}{\Gamma \left(q\right)\sqrt{2\theta (2q-1)}}{\displaystyle \sum _{k=1}^N}\left(L_k\right)}}d(T\vartheta ,\vartheta )$$

$$\le {\displaystyle \frac{1}{1-\frac{2b^{q}exp\left(\theta b\right)}{\Gamma \left(q\right)\sqrt{2\theta (2q-1)}}{\displaystyle \sum _{k=1}^N}\left(L_k\right)}}\le \epsilon C{E}_q\left(x^q\right).$$

Then, FrVIE (2) admits the first type of U-H-M-L stability according to the Bielecki norm. Thereby, we complete the proof. □

4. The First Type of U-H-M-L-R Stability

Definition 6.

FrVIE (2) admits the first type of U-H-M-L-R stability regarding the function $E_q$ if there exists a $\mathrm{C}\in \mathbb{R},$$C>0,$ such that for each $\epsilon >0$ and solution ϑ of the inequality

$$\left|\vartheta \left(x\right)-{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{c}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\vartheta \left(s\right),\vartheta \left({\alpha }_k\left(s\right)\right))ds\right|\le \varphi \left(x\right)\epsilon E_q\left(x^q\right),$$

(8)

there exists a unique solution ${\vartheta }_0$ of FrVIE (2) satisfying

$$\left|\vartheta \left(x\right)-{\vartheta }_0\left(x\right)\right|\le \mathrm{C}\epsilon \varphi \left(x\right)E_q\left(x^q\right),$$

where $\varphi \in C(X,{\mathbb{R}}^{+})$.

We will give the result of the new first type of U-H-M-L-R stability in Theorem 7.

Theorem 7.

Suppose that (As1), (As2) of Theorem 4 and the following condition hold:

(As5) 

There exist constants $M>0$ and $M_0>0$ such that

$$0<M{M}_0\sum _{k=1}^N\left(L_k\right)<1,$$

where $M_0={\left(b-a\right)}^{q-p}{\displaystyle \frac{1}{\Gamma \left(q\right)}}{\left({\displaystyle \frac{1-p}{q-p}}\right)}^{1-p}$ with $0<p<q$.

If a function $\vartheta \in C\left(\right[a,b],\mathbb{R})$ satisfies the inequality (8) for each $x\in [a,b]$ such that $\varphi :[a,b]\to [a,b]$ is an $L^{{\displaystyle \frac{1}{p}}}$ integrable function satisfying

$${\left(\underset{0}{\overset{x}{\int }}{\left(\varphi \left(s\right)\right)}^{{\displaystyle \frac{1}{p}}}ds\right)}^p\le M\varphi \left(x\right),$$

then there is a unique function ${\vartheta }_0\in C([a,b],\mathbb{R})$ such that ${\vartheta }_0$ satisfies FrVIE (2) and the inequality

$$\left|\vartheta \left(x\right)-{\vartheta }_0\left(x\right)\right|\le \mathrm{C}\epsilon \varphi \left(x\right)E_q\left(x^q\right),\mathit{for}\mathit{each}x\in [a,b].$$

This result means that FrVIE (2) admits the first type of U-H-M-R stability with respect to the function $E_q$.

Proof. 

We will take into account the space

$$X=\{\nu :[a,b]\to \mathbb{R}|\nu \mathrm{is}\mathrm{continuous}\}$$

(9)

and the generalized metric described by

$$d(\nu ,\omega )=inf\left\{M\in [0,\infty ]\left|\left|\nu \left(x\right)-\omega \left(x\right)\right|\le M\varphi \left(x\right),\forall x\in [a,b]\right.\right\}.$$

We note that $(X,d)$ is a complete metric space. We describe an operator T, $T:X\to X$, as

$$\left(T\nu \right)\left(x\right)={\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\nu \left(s\right),\nu \left({\alpha }_k\left(s\right)\right))ds,$$

(10)

for each $\nu \in X$ and $x\in [a,b]$. Since $\nu$ is a continuous function, $T\nu$ is also continuous. Hence, T is a well-defined operator. Then, for any $\nu ,\omega \in T$, let $M_{\nu \omega }\in [0,\infty ]$ such that the inequality

$$\left|\nu \left(x\right)-\omega \left(x\right)\right|\le M_{\nu \omega }\varphi \left(x\right)$$

(11)

holds, for each $x\in [a,b]$. From the definition of the operator T in (10), (As2), (As5) and (11), we determine that

$$\begin{array}{cc}\hfill \left|T\right(\nu \left)\right(x)-T(\omega \left)\right(x\left)\right|=& {\displaystyle \frac{1}{\Gamma \left(q\right)}}|\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\nu \left(s\right),\nu \left({\alpha }_k\left(s\right)\right))ds\hfill \\ \hfill & -\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\omega \left(s\right),\omega \left({\alpha }_k\left(s\right)\right))ds|\hfill \\ \hfill \le & {\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\left(L_k\right)\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}\left|\nu \left(s\right)-\omega \left(s\right)\right|ds\hfill \\ \hfill \le & {\displaystyle \frac{M_{\nu \omega }}{\Gamma \left(q\right)}}\sum _{k=1}^N\left(L_k\right)\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}\varphi \left(s\right)ds\hfill \\ \hfill \le & {\displaystyle \frac{M_{\nu \omega }}{\Gamma \left(q\right)}}{\sum _{k=1}^N\left(L_k\right)\left(\underset{0}{\overset{x}{\int }}{(x-s)}^{{\displaystyle \frac{q-1}{1-p}}}ds\right)}^{1-p}\times \underset{0}{\overset{x}{\int }}{\left({\left(\varphi \left(s\right)\right)}^{{\displaystyle \frac{1}{p}}}ds\right)}^p\hfill \\ \hfill \le & \left(M_{\nu \omega }\mathrm{M}{\mathrm{M}}_0\right)\sum _{k=1}^N\left(L_k\right)\varphi \left(x\right),\hfill \end{array}$$

(12)

for each $x\in [a,b]$. According to the inequality (12), we obtain

$$d(T\nu ,T\omega )\le \left(M_{\nu \omega }\mathrm{M}{\mathrm{M}}_0\sum _{k=1}^N\left(L_k\right)\right)\varphi \left(x\right).$$

Thereby, we can see that

$$d(T\nu ,T\omega )\le \left(\mathrm{M}{\mathrm{M}}_0\sum _{k=1}^N\left(L_k\right)\right)d(\nu ,\omega ),$$

for each $\nu ,\omega \in X$. From (As5), since $0<\mathrm{M}{\mathrm{M}}_0{\displaystyle \sum _{k=1}^N}\left(L_k\right)<1$, the operator T is a strict contraction.

Next, let us take ${\nu }_0\in X$. Thereby, from the continuity properties of ${\nu }_0$ and $T{\nu }_0$, we deduce that there exists a constant ${\mathrm{S}}_1$, $0<{\mathrm{S}}_1<\infty$, such that

$$\left|\left(T{\nu }_0\right)\left(x\right)-{\nu }_0\left(x\right)\right|$$

$$=\left|{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,{\nu }_0\left(s\right),{\nu }_0\left({\alpha }_k\left(s\right)\right))ds-{\nu }_0\left(x\right)\right|\le {\mathrm{S}}_1\varphi \left(x\right),$$

for each $x\in [a,b]$. Since $f_k$, $k=1,2,\dots ,N$ and ${\nu }_0$ are bounded on the interval $[a,b]$ and $\underset{x\in [a,b]}{\min }E\left(x^q\right)>0$, (9) implies that $d(T{\nu }_0,{\nu }_0)<\infty$. Thereby, in light of Theorem 1 (a), there exists a function ${\vartheta }_0\in C([a,b],\mathbb{R})$ such that $T^n{\vartheta }_0\to {\vartheta }_0$ in $(X,d)$ as $n\to \infty$ and $T{\vartheta }_0={\vartheta }_0$. Hence, ${\vartheta }_0$ satisfies FrVIE (2) for each $x\in [a,b]$.

Next, we will demonstrate that

$$\left\{\nu \in X\left|d({\nu }_0,\nu )<\infty \right.\right\}=X,$$

for any $\nu \in X$, since $\nu$ and ${\nu }_0$ are bounded on the interval $[a,b]$ and $\underset{x\in [a,b]}{\min }E\left(x^q\right)>0$, subsequently, we have a constant $M_v$, $0<M_v<\infty$, such that

$$\left|{\nu }_0\left(x\right)-\nu \left(x\right)\right|\le {\mathrm{M}}_{\nu }\varphi \left(x\right),$$

for any $x\in [a,b]$. Thereby, we obtain $d({\nu }_0,\nu )<\infty$ for each $\nu \in X$, i.e.,

$$\left\{\nu \in X\left|d({\nu }_0,\nu )<\infty \right.\right\}=X.$$

Hence, according to Theorem 1 (b), we can deduce that ${\vartheta }_0$ is continuous and the unique function satisfying FrVIE (2).

From (8), we now know that

$$d({\vartheta }_0,T{\vartheta }_0)\le \epsilon E_q\left(x^q\right)\varphi \left(x\right).$$

Finally, according to Theorem 1 (c) and the last inequality, we conclude that

$$d(\vartheta ,{\vartheta }_0)\le {\displaystyle \frac{1}{1-\mathrm{M}{\mathrm{M}}_0{\displaystyle \sum _{k=1}^N}\left(L_k\right)}}d(T\vartheta ,\vartheta )\le {\displaystyle \frac{1}{1-\mathrm{M}{\mathrm{M}}_0{\displaystyle \sum _{k=1}^N}\left(L_k\right)}}\varphi \left(x\right)\epsilon E_q\left(x^q\right).$$

Thus, the inequality

$$\left|\vartheta \left(x\right)-{\vartheta }_0\left(x\right)\right|\le \mathrm{C}\epsilon \varphi \left(x\right)E_q\left(x^q\right)$$

holds, for each $x\in [a,b]$. Hence, FrVIE (2) admits the the first type of U-H-M-L-R stability. We thus reach the conclusion of the proof. □

5. The Second Type of U-H-M-L and U-H-M-L-R Stability

We now consider FrVIE (2) for the interval $[0,b]$.

Definition 7.

FrVIE (2) admits the second type of U-H-M-L stability regarding the function $E_q$ if for every $\epsilon >0$ and solution $\vartheta \in C\left([0,b],\mathbb{R}\right)$ of the inequality

$$\left|\vartheta \left(x\right)-{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{c}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\vartheta \left(s\right),\vartheta \left({\alpha }_k\left(s\right)\right))ds\right|\le \epsilon ,$$

(13)

there exists a solution ${\vartheta }_0\in C([0,b],\mathbb{R})$ of FrVIE (2) satisfying

$$\left|\vartheta \left(x\right)-{\vartheta }_0\left(x\right)\right|\le \epsilon E_q\left(c{x}^q\right),$$

for each $x\in [0,b]$ and $c\in \mathbb{R}$.

We will give results of the new and second types of U-H-M-L stability in Theorem 8.

Theorem 8.

Let B be a Banach algebra and the following conditions hold:

(C1) 

$$f_k\in C([0,b]\times [0,b]\times B\times B,B),{\alpha }_k\in C([0,b],[0,b]),$$

$0\le {\alpha }_k\left(x\right)\le x$, for each $x\in [0,b]$, $b\in \mathbb{R}$, $b>0$, and $\vartheta \in B$, $k=1,2,\dots ,N;$

(C2) 

There exists $L_k>0$, $L_k\in \mathbb{R}$ such that the functions $f_k$ and $k=1,2,\dots ,N$ satisfy the Lipschitz condition

$$\left|f_k(x,s,\rho \left(s\right),\rho \left({\alpha }_k\left(s\right)\right))-f_k(x,s,{\rho }_0\left(s\right),{\rho }_0\left({\alpha }_k\left(s\right)\right))\right|\le \left(L_k\right)\left|\rho \left(s\right)-{\rho }_0\left(s\right)\right|,$$

for each $x,s\in [0,b]$, $b\in \mathbb{R}$, $b>0$, $\rho ,{\rho }_0\in B$ and the inequality (13). Then, FrVIE (2) admits the second type of U-H-M-L stability.

Proof. 

Let $\vartheta \in \mathrm{C}\in \left(\right[0,b],B)$ satisfy inequality (13) and ${\vartheta }_0\in \mathrm{C}\in ([0,b],B)$ denote the unique solution of FrVIE (2). Then, we have

$$\begin{array}{cc}\hfill |& \vartheta \left(x\right)-{\vartheta }_0\left(x\right)|\hfill \\ \hfill \le & \left|\vartheta \left(x\right)-{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\vartheta \left(s\right),\vartheta \left({\alpha }_k\left(s\right)\right))ds\right|\hfill \\ \hfill & +|{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\vartheta \left(s\right),\vartheta \left({\alpha }_k\left(s\right)\right))ds\hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,{\vartheta }_0\left(s\right),{\vartheta }_0\left({\alpha }_k\left(s\right)\right))ds|\hfill \\ \hfill \le & \epsilon +\left|{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}\left[f_k(x,s,\vartheta \left(s\right),\vartheta \left({\alpha }_k\left(s\right)\right))-f_k(x,s,{\vartheta }_0\left(s\right),{\vartheta }_0\left({\alpha }_k\left(s\right)\right))\right]ds\right|\hfill \\ \hfill \le & \epsilon +{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}\left|f_k(x,s,\vartheta \left(s\right),\vartheta \left({\alpha }_k\left(s\right)\right))-f_k(x,s,{\vartheta }_0\left(s\right),{\vartheta }_0\left({\alpha }_k\left(s\right)\right))\right|ds\hfill \\ \hfill \le & \epsilon +{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\left(L_k\right)\underset{0}{\overset{x}{\int }}{(x-s)}^{q-1}\left|\vartheta \left(s\right)-{\vartheta }_0\left(s\right)\right|ds.\hfill \end{array}$$

Hence, in light of Remark 1 and Theorem 2, we can conclude that

$$\left|\vartheta \left(x\right)-{\vartheta }_0\left(x\right)\right|\le \epsilon E_q\left(c{x}^q\right).$$

Thereby, the proof of Theorem 8 has been completed. □

Definition 8.

FrVIE (2) admits the second type of U-H-M-L-R stability pertaining to the function $E_q$ if for every $\epsilon >0$ and for each solution $\vartheta \in C\left([a,b],\mathbb{R}\right)$ of the inequality

$$\left|\vartheta \left(x\right)-{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{k=1}^N\underset{c}{\overset{x}{\int }}{(x-s)}^{q-1}{f}_k(x,s,\vartheta \left(s\right),\vartheta \left({\alpha }_k\left(s\right)\right))ds\right|\le \epsilon \varphi \left(x\right),$$

(14)

there exists a solution ${\vartheta }_0\in \left(C[a,b],\mathbb{R}\right)$ of FrVIE (2) satisfying

$$\left|\vartheta \left(x\right)-{\vartheta }_0\left(x\right)\right|\le \lambda E_q\left(c{x}^q\right),$$

for each $x\in [a,b]$, when $\varphi :X\to {\mathbb{R}}^{+}$ is a non-decreasing locally integrable function on ${\mathbb{R}}^{+}$ and $\lambda ,c\in \mathbb{R},\lambda >0$.

Theorem 9.

Suppose that B is a Banach algebra, and that (C1) of Theorem 8 and the following condition also hold:

(C3) 

There exists $L_k>0$, $L_k\in \mathbb{R}$ such that the function $f_k$, $k=1,2,\dots ,N$ satisfies the Lipschitz condition

$$\left|f_k(x,s,{\vartheta }_1\left(s\right),{\vartheta }_1\left({\alpha }_k\left(s\right)\right))-f_k(x,s,{\vartheta }_2\left(s\right),{\vartheta }_2\left({\alpha }_k\left(s\right)\right))\right|\le \left(L_k\right)\left|{\vartheta }_1\left(s\right)-{\vartheta }_2\left(s\right)\right|,$$

for each $x,s\in [0,b]$, $b\in \mathbb{R}$, $b>0$, ${\vartheta }_1,{\vartheta }_2\in B$ and the inequality (14). Then, FrVIE (2) admits the second type of U-H-M-L-R stability

Proof. 

In the proof of Theorem 8, taking $\epsilon \varphi \left(x\right)$ instead of $\epsilon$, we can complete the proof of Theorem 9. Thereby, we will not give the details of the calculations. □

Remark 5.

The new results of this study, i.e., Theorems 4–9, have been proven in light of the Banach contraction principle, the generalized metric, the Banach contraction principle, the Chebyshev norm, the Banach contraction principle and the Bielecki norm, respectively. Thereby, the conditions of Theorems 4–9 are partially different from the each other. Consequently, in the relevant literature, these tools are very interesting and common basic techniques to prove qualitative results, for example the first and second types of U-H-M-L-R and U-H-M-L-R stability, etc., on the same and different mathematical models (see, in particular, the papers of Eghbali et al. [14], Kh.Niazi et al. [19] and Wang and Zhang [18]).

6. Conclusions

This paper took into consideration a nonlinear FrVIE containing multiple variable delays. Throughout the paper, the U-H-M-L stability and the U-H-M-L-R stability of a nonlinear FrVIE were discussed via six new theorems containing sufficient conditions. The approaches used in the proofs of the theorems were based on Banach’s fixed-point theorem and the Chebyshev and the Bielecki norms. To the best of the authors’ knowledge, no result was found regarding the U-H-M-L stability and U-H-M-L-R stability of FrVIEs containing multiple retardations in the present literature. The outcomes of this paper provide new contributions to the qualitative theory of FrVIEs in the sense of U-H-M-L stability and U-H-M-L-R stability. Based on the new results of the present work, there are several potential directions for further research pertaining to the first and second types of U-H-M-L and U-H-M-L-R stability as open problems:

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U-H-M-L stability and U-H-M-L-R stability of partial differential equations with and without delay.

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U-H-M-L stability and U-H-M-L-R stability of Caputo fractional delay IDEs, Riemann–Liouville fractional delay IDEs and neutral-type FrODEs and neutral type FrIDEs.

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U-H-M-L stability and U-H-M-L-R stability of more general FrVIEs.