q-Fractional Langevin Differential Equation with q-Fractional Integral Conditions

Abstract

The major goal of this manuscript is to investigate the existence, uniqueness, and stability of a q-fractional Langevin differential equation with q-fractional integral conditions. We demonstrate the existence and uniqueness of the solution to the proposed q-fractional Langevin differential equation using the Banach contraction principle and Schaefer’s fixed-point theorem. We also elaborate on different kinds of Ulam stability. The theoretical outcomes are verified by examples.

1. Introduction

The most recently developed field of study is fractional calculus ($\mathbb{FC}$). Many scientists from different areas of research have become interested in the topic of $\mathbb{FC}$, which is based on integrals and derivatives of fractional order. It has numerous uses in a broad range of scientific and engineering disciplines, including optics, mathematical biology, plasma physics, electromagnetic theory, and many more. Currently, there are many significant results in the literature regarding the existence and uniqueness ($\mathbb{EU}$) of solutions of fractional boundary value problems ($\mathbb{BVP}\mathrm{s}$) [1,2,3]. Fractional differential equations ($\mathbb{FDE}\mathrm{s}$) have been successfully used in recent years to model issues that arise in science and industry. The research on $\mathbb{FDE}\mathrm{s}$ is interdisciplinary and occurs in many areas, including quantum mechanics, bio-engineering, control theory, and complex systems. As global operators, $\mathbb{FDE}\mathrm{s}$ are also used to describe a variety of systems that have arisen in fields such as fluid mechanics, control theory, and probability theory [4,5,6,7,8,9,10,11]. This field has drawn the interest of many researchers because of its significance and broad variety of applications. Many scholars have developed numerous methods to demonstrate the presence of solutions to challenging fractional $\mathbb{BVP}\mathrm{s}$ and benefited from various fixed-point theorems. For more on this topic, see [12,13,14,15] in detail.

Fractional q-$\mathbb{DE}\mathrm{s}$ and q-calculus have recently received a lot of interest from mathematical engineers. q-calculus, also known as quantum calculus, dates back to Jackson’s study in 1908 [16]. The q-$\mathbb{DE}\mathrm{s}$ are based on q-calculus and can describe some unique physical processes, e.g., in quantum mechanics, dynamical systems, stochastic processes, etc. For a basic introduction to the basic idea of q-calculus, the interested reader may consult [17,18,19]. The stability analysis of $\mathbb{DE}\mathrm{s}$ in the Ulam sense and their various varieties is another field of research that has drawn significant attention from scholars. In fact, in 1940, Ulam [20] challenged the problem of functional equations, which Hyers [21] enhanced in 1941 using Banach spaces. Because of this reason, this problem led to the notion of stability and is popular as Ulam–Hyers stability ($\mathbb{UHS}$). Rassias [22] then extended $\mathbb{UHS}$ to a generalized form and called it $\mathbb{UH}$–Rassias stability. Additionally, many scientists have developed a widespread understanding of $\mathbb{UHS}$ for various classes of functional equations [23]. Ulam’s type of stability is important for a variety of applications, including optimization, numerical analysis, economics, biology, etc. Numerous mathematicians and technologists have considered Ulam-type stability. For more information on recent investigations of the Ulam-type stability of $\mathbb{DE}\mathrm{s}$, see, for the ordinary form, $\mathbb{DE}\mathrm{s}$ [24], for the delayed form, $\mathbb{DE}\mathrm{s}$ [25], and for the Caputo form, $\mathbb{DE}\mathrm{s}$ [26].

Paul Langevin, a French physicist, introduced the Langevin equations ($\mathbb{LE}\mathrm{s}$) in 1908. The equation has the following form:

$$\mathrm{m}\frac{{\mathrm{d}}^2\mathrm{z}}{\mathrm{d}{\mathsf{\xi }}^2}=\mathsf{\lambda }\frac{\mathrm{d}\mathrm{z}}{\mathrm{d}\mathsf{\xi }}+\mathsf{\Phi }\left(\mathsf{\xi }\right).$$

The fractional Langevin equations ($\mathbb{FLE}\mathrm{s}$) have been a crucial topic in the field of physics, chemistry, and electrical engineering. In fact, the evolution of physical events in a changing environment is well described by $\mathbb{LE}$. It is important to note that random variations are regarded as Gaussian noise in the $\mathbb{LE}\mathrm{s}$ description of Brownian motion. In order to eliminate noise and staircase effects, many researchers used $\mathbb{FDE}$ to replace the ordinary $\mathbb{DE}\mathrm{s}$. The literature widely explores $\mathbb{FLE}\mathrm{s}$ from both theoretical and numerical perspectives. Readers may consult [27,28] for more information. In generalizing the conventional one, $\mathbb{FLE}\mathrm{s}$ provides a fractional Gaussian process parameterized by two indices that is more flexible for modeling fractal processes [29,30].

Etemad et al. [31] have studied the fractional q-integro-differential equation:

$$(\mathsf{\lambda }{\mathcal{D}}_{\mathit{q}}^{\alpha }+(1-\mathsf{\lambda }){\mathcal{D}}_{\mathit{q}}^{\mathsf{\beta }})\mathrm{z}\left(\mathsf{\xi }\right)=\mathrm{a}\mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right))+\mathrm{b}{\mathrm{I}}_q^{\mathsf{\delta }}\mathrm{g}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right)),\mathsf{\xi }\in [0,1],\mathrm{a},\mathrm{b}\in {\mathfrak{R}}^{+}$$

with $\mathbb{BC}\mathrm{s}$

$$\mathrm{z}\left(0\right)=0,\mathsf{\mu }{\int }_0^1\frac{{(1-q\mathrm{s})}^{({\gamma }_1-1)}}{{\mathsf{\Gamma }}_q\left({\gamma }_1\right)}\mathrm{z}\left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s},(1-\mathsf{\mu }){\int }_0^1\frac{{(1-q\mathrm{s})}^{({\gamma }_2-1)}}{{\mathsf{\Gamma }}_q\left({\gamma }_2\right)}\mathrm{z}\left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}=0,{\gamma }_1,{\gamma }_2>0,$$

where $q,\mathsf{\delta }\in (0,1)$, $\mathsf{\lambda }\in (0,1]$, $\mathsf{\mu }\in (0,1]$, $1<\alpha ,\mathsf{\beta }<2$, $\alpha -\mathsf{\beta }>1$, ${\mathcal{D}}_{\mathit{q}}^{\alpha }$ denotes the Reimann–Liouville fractional q-derivative of order $\alpha$, and f and g are appropriate functions.

Rizwan and Zada [32] studied the existence and stability analysis similar to the $\mathbb{UH}$ and $\mathbb{UHRS}$ of the $\mathbb{FLE}\mathrm{s}$ by utilizing the generalized Diaz–Margoli’s fixed-point ($\mathfrak{F}\mathfrak{P}$) theorem:

$${}^{\mathrm{C}}{\mathcal{D}}_{\mathsf{\xi }}^{\alpha }({}^{\mathrm{C}}{\mathcal{D}}_{\mathsf{\xi }}^{\mathsf{\beta }}+\mathsf{\lambda })\mathrm{z}\left(\mathsf{\xi }\right)=\mathrm{F}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right)),0<\alpha ,\mathsf{\beta }<1,$$

where ${}^{\mathrm{C}}{\mathcal{D}}_{\mathsf{\xi }}^{\alpha }$ and ${}^{\mathrm{C}}{\mathcal{D}}_{\mathsf{\xi }}^{\mathsf{\beta }}$ represents the classical Caputo derivative ($\mathbb{CD}$) [2] of order $\alpha$ and $\mathsf{\beta }$, respectively, and $\mathsf{\lambda }>0$.

Zhou and Liu [33] applied Monch’s $\mathfrak{FP}$ theorem and a measure of weak non-compactness to analysis the fractional q-difference equation:

$$\left\{\begin{array}{cc}\hfill & {}^{\mathrm{C}}{\mathcal{D}}_q^{\alpha }\mathrm{z}\left(\mathsf{\xi }\right)=\mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right)),\mathsf{\xi }\in [0,1],0<q<1,\hfill \\ \hfill & \mathrm{z}\left(0\right)={\mathcal{D}}_q^2\mathrm{z}\left(0\right)=0,\hfill \\ \hfill & \gamma \left({\mathcal{D}}_q\mathrm{z}\right)\left(0\right)+\mathsf{\beta }\left({\mathcal{D}}_q^2\mathrm{z}\right)\left(1\right)=0,\hfill \end{array}\right.$$

where $\alpha \in (2,3)$, $\gamma ,\mathsf{\beta }\ge 0$, and f is an appropriate function.

Alam et al. [34] demonstrated the implicit $\mathbb{LE}\mathrm{s}$ involving mixed derivatives and Stieltjes $\mathbb{IBC}\mathrm{s}$:

$$\left\{\begin{array}{cc}\hfill & {}^{\mathrm{C}}{\mathcal{D}}_{0,\mathsf{\xi }}^{\alpha }(\mathcal{D}+\mathsf{\lambda })\mathrm{z}\left(\mathsf{\xi }\right)=\mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right)),\mathsf{\xi }\in (0,1],\hfill \\ \hfill & \mathrm{z}\left(0\right)=0,\hfill \\ \hfill & {}^{\mathrm{C}}{\mathcal{D}}_{0,\mathsf{\xi }}^{{\mathsf{\beta }}_0}\mathrm{z}\left(1\right)=\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^1{}^{\mathrm{C}}{\mathcal{D}}_{0,\mathsf{\xi }}^{{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{z}\left(\mathsf{\xi }\right){\mathrm{d}}_{{\mathsf{\mu }}_{\mathrm{i}}}\mathsf{\xi },\hfill \end{array}\right.$$

where ${}^{\mathrm{C}}{\mathcal{D}}_{0,\mathsf{\xi }}^{\alpha }$ represents the $\mathbb{CD}$ of order $\alpha$ with zero as a lower bound, $\alpha \in (0,1)$, $\mathsf{\lambda }\in \mathfrak{R}\backslash \left\{0\right\}$, $\mathrm{p}\in \mathrm{N}$, ${\mathsf{\beta }}_{\mathrm{i}}\in \mathfrak{R}$ (∀$\mathrm{i}=0,\dots ,\mathrm{p}$), $0<{\mathsf{\beta }}_1,{\mathsf{\beta }}_2,\dots ,<{\mathsf{\beta }}_{\mathrm{p}}<\alpha$, ${\mathsf{\beta }}_0\in [0,\alpha )$, f is an appropriate function, and the Riemann–Stieltjes integrals are taken in the $\mathbb{BC}\mathrm{s}$ with ${\mathsf{\mu }}_{\mathrm{i}}(\mathrm{i}=1,\dots ,\mathrm{p})$ being a function of the bounded variation.

Zada and Waheed [35] studied the class of implicit $\mathbb{FDE}$ with implicit $\mathbb{IBC}\mathrm{s}$:

$$\left\{\begin{array}{cc}& {}^{\mathrm{C}}{\mathcal{D}}^{\alpha }\mathrm{z}\left(\mathsf{\xi }\right)=\mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}^{\alpha }\mathrm{z}\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-\mathrm{s})}^{\mathsf{\sigma }-1}}{\mathsf{\Gamma }\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}^{\alpha }\mathrm{z}\left(\mathrm{s}\right))\mathrm{d}\mathrm{s},\mathsf{\xi }\in \mathrm{J}=(0,\mathrm{T}),\alpha \in (0,1],\hfill \\ \hfill & \mathrm{z}\left(0\right)=-{\int }_0^{\mathsf{\xi }}\frac{{(\mathrm{T}-\mathsf{\eta })}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}\mathrm{K}(\mathsf{\eta },\mathrm{z}\left(\mathsf{\eta }\right),{}^{\mathrm{C}}{\mathcal{D}}^{\alpha }\mathrm{z}\left(\mathsf{\eta }\right))\mathrm{d}\mathsf{\eta },\hfill \end{array}\right.$$

where ${}^{\mathrm{C}}{\mathcal{D}}^{\alpha }$ is the Caputo fractional derivative of order $\alpha$ on J, $\mathrm{f},\mathrm{g},\mathrm{K}$ are appropriate functions, $\mathsf{\delta }>0$, and $\mathsf{\sigma }>0$.

From the above discussion and models, we are motivated to study the following implicit q-$\mathbb{FDE}\mathrm{s}$ with $\mathbb{FIBC}\mathrm{s}$:

$$\left\{\begin{array}{cc}\hfill & {}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }(\mathcal{D}+\mathsf{\lambda })\mathrm{z}\left(\mathsf{\xi }\right)=\mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s},\mathsf{\xi }\in \mathrm{J}=(0,\mathrm{T}),\alpha \in (1,2],\hfill \\ \hfill & \mathrm{z}\left(0\right)=-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathsf{\eta })}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\mathrm{K}(\mathsf{\eta },\mathrm{z}\left(\mathsf{\eta }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathsf{\eta }\right)){\mathrm{d}}_q\mathsf{\eta },\hfill \\ \hfill & {}^{\mathrm{C}}{\mathcal{D}}_q\mathrm{z}\left(0\right)=0,{}^{\mathrm{C}}{\mathcal{D}}_q^{{\mathsf{\beta }}_0}\mathrm{z}\left(\mathrm{T}\right)=\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{z}\left(\mathsf{\xi }\right){\mathrm{d}}_q\mathsf{\xi },\hfill \end{array}\right.$$

(1)

where ${}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }$ is the Caputo fractional q-derivative of order $\alpha$ on J, D denotes the ordinary differential operator, $\mathrm{f},\mathrm{g},\mathrm{K}$ are appropriate functions, $\mathsf{\lambda }\in \mathfrak{R}\backslash \left\{0\right\}$, $\mathrm{p}\in \mathrm{N}$, ${\mathsf{\beta }}_{\mathrm{i}}\in \mathfrak{R}$ (∀$\mathrm{i}=0,\dots ,\mathrm{p})$, $0<{\mathsf{\beta }}_1,{\mathsf{\beta }}_2,\dots ,<{\mathsf{\beta }}_{\mathrm{p}}<\alpha$, ${\mathsf{\beta }}_0\in [0,\alpha )$, and $\mathsf{\delta }>0$ and $\mathsf{\sigma }>0$ are constants. In comparison to the other studies on q-fractional Langevin $\mathbb{DE}\mathrm{s}$ that hav ebeen published in the literature, we deal with the extended version of a new Caputo q–Langevin $\mathbb{FDE}$ with q-fractional integral conditions in which the uniqueness property of the solution is derived in terms of the Banach and Schefer $\mathfrak{FP}$ theorems. These procedures on the proposed Langevin q-$\mathbb{FDE}$ $\left(1\right)$ have been implemented in research studies on q-fractional modelings. Furthermore, the Ulam stability of $\mathbb{FDE}$ is quite significant in the sense that it guarantees a bound between the exact and approximate solutions. Therefore, it can be required in a number of applications, such as optimization, approximation, and numerical analysis. Because of this importance of Ulam stability, we also investigate it for the proposed Langevin q-$\mathbb{FDE}$ $\left(1\right)$.

The manuscript is organized as follows: In Section 2, we give some basic definitions and theorems associated with both q-fractional derivatives and q-fractional integrals. The existence and uniqueness of a solution for the proposed q-fractional Langevin differential equation is obtained in Section 3 utilizing the Banach contraction principle and Schaefer’s fixed-point theorem. Different kinds of Ulam stability for the considered system are investigated in Section 4. In Section 5, the results are verified with the help of examples.

2. Preliminary

Let $\mathrm{J}=[0,\mathrm{T}]$, $\mathbb{X}$ be the space of all continuous functions, i.e., $\mathbb{X}=\{\mathrm{z}:\mathrm{J}\to \mathfrak{R};\mathrm{z}\in \mathfrak{C}(\mathrm{J},\mathfrak{R}\left)\right\}$. Introducing the norm

$${\parallel \mathrm{z}\parallel }_{\mathbb{X}}=sup\left\{\right|\mathrm{z}\left(\mathsf{\xi }\right)|:\mathsf{\xi }\in \mathrm{J}\},$$

it is obvious that $\mathbb{X}$ is a Banach space.

Definition 1

([2]). The $q$-$\mathbb{FI}$ of order α from 0 to ξ for $z$ is given by

$${\mathrm{I}}_q^{\alpha }z\left(\mathsf{\xi }\right)=\frac{1}{{\mathsf{\Gamma }}_q\left(\alpha \right)}{\int }_0^{\mathsf{\xi }}{(\mathsf{\xi }-q\mathrm{s})}^{\alpha -1}z\left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s},\mathsf{\xi }>0,\alpha >0,$$

where $\mathsf{\Gamma }(.)$ is the gamma function.

Definition 2

([36]). The Caputo–q-$\mathbb{FD}$ of order $\alpha \ge 0$ for $z$ on J is represented as

$${}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }z\left(\mathsf{\xi }\right)={\mathfrak{I}}_{\mathit{q}}^{\lceil \alpha \rceil -\alpha }{\mathcal{D}}_{\mathit{q}}^{\lceil \alpha \rceil }z\left(\mathsf{\xi }\right),$$

where $\lceil \alpha \rceil$ is the smallest integer greater than α. Equivalently,

$${}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }z\left(\mathsf{\xi }\right)=\frac{1}{{\mathsf{\Gamma }}_{\mathit{q}}(-\alpha )}{\int }_0^{\mathsf{\xi }}\frac{z\left(\mathrm{s}\right)}{{(\mathsf{\xi }-\mathit{q}\mathrm{s})}^{(1+\alpha )}}{\mathrm{d}}_{\mathit{q}}\mathrm{s}.$$

Lemma 1

([37]). Let $\alpha >0$ and $\mathrm{p}\in \mathrm{N}$. Then, the following holds:

$${{\mathrm{I}}_{\mathit{q}}^{\alpha }}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\mathrm{p}}z\left(\mathsf{\xi }\right)={}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\mathrm{p}}{\mathrm{I}}_{\mathit{q}}^{\alpha }z\left(\mathsf{\xi }\right)-\sum _{\mathrm{k}=0}^{\mathrm{p}-1}\frac{{\mathsf{\xi }}^{\alpha -\mathrm{p}-\mathrm{k}}}{{\mathsf{\Gamma }}_{\mathit{q}}(\alpha -\mathrm{p}-\mathrm{k}+1)}{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\mathrm{k}}z\left(0\right).$$

(2)

We recall the following from [19]. Let $q\in (0,1)$ and define

$${\left[\mathrm{a}\right]}_q=\frac{q^{\mathrm{a}}-1}{q-1}={\mathrm{a}}^{\mathrm{a}-1}+\dots +1,\mathrm{a}\in \mathbb{R}.$$

The q-analogue of the power ${(\mathrm{a}-\mathrm{b})}^{\mathrm{n}}$ is

$${(\mathrm{a}-\mathrm{b})}^{\left(0\right)}=1,{(\mathrm{a}-\mathrm{b})}^{\left(\mathrm{n}\right)}=\prod _{\mathrm{k}=0}^{\mathrm{n}-1}(\mathrm{a}-\mathrm{b}{q}^{\mathrm{k}}),\mathrm{a},\mathrm{b}\in \mathbb{R},\mathrm{n}\in \mathrm{N}.$$

If $\alpha$ is not a positive integer, then

$${(\mathrm{a}-\mathrm{b})}^{\left(\alpha \right)}={\mathrm{a}}^{\alpha }\prod _{\mathrm{i}=\mathrm{o}}^{\infty }\frac{(1-\left(\frac{\mathrm{b}}{\mathrm{a}}\right)q^{\mathrm{i}})}{(1-\left(\frac{\mathrm{b}}{\mathrm{a}}\right)q^{\alpha +\mathrm{i}})}.$$

Note that if $\mathrm{b}=0$, then ${\mathrm{a}}^{\left(\alpha \right)}={\mathrm{a}}^{\alpha }$. Using the above notations, we state the following definition.

Definition 3.

The $q$-Gamma function is defined as

$${\mathsf{\Gamma }}_q\left(\alpha \right)=\frac{{(1-q)}^{(\alpha -1)}}{{(1-q)}^{\alpha -1}},\alpha \in \mathfrak{R}\backslash \{0,-1,-2,\dots \},0<q<1$$

and satisfies

$${\mathsf{\Gamma }}_q(\alpha +1)={\lceil \alpha \rceil }_q{\mathsf{\Gamma }}_q\left(\alpha \right).$$

We also point out the formulas, taken from [18], that will be used in our results.

$$\begin{array}{cc}\hfill & {\left[\mathrm{a}(\mathsf{\xi }-\mathrm{s})\right]}^{\left(\alpha \right)}={\mathrm{a}}^{\alpha }{(\mathsf{\xi }-\mathrm{s})}^{\alpha },\hfill \\ \hfill & {}_{\mathsf{\xi }}{\mathcal{D}}_q{(\mathsf{\xi }-\mathrm{s})}^{\alpha }={\lceil \alpha \rceil }_q{(\mathsf{\xi }-\mathrm{s})}^{(\alpha -1)}.\hfill \end{array}$$

Theorem 1

([38]). [Theorem of Banach about $\mathfrak{FP}$] Consider a Banach space $\mathbb{X}$ and let a map $\mathsf{\rho }:\mathbb{X}\to \mathbb{X}$ be a contraction on $\mathbb{X}$. Then, ρ has precisely one $\mathfrak{FP}$.

Theorem 2

([37]). [Theorem of Arzela–Ascoli] Let $\mathbb{M}\subset \mathfrak{C}(\mathrm{J},\mathfrak{R})$; then, it is relatively compact in $\mathfrak{C}(\mathrm{J},\mathfrak{R})$ $\mathrm{i}\mathrm{f}$ the functions in $\mathbb{M}$ are uniformly bounded and equi-continuous on J.

Theorem 3

([37]). [Theorem of Schaefer about $\mathfrak{FP}$] Let $\mathbb{X}$ be a Banach space and let the operator $\mathsf{\rho }:\mathbb{X}\to \mathbb{X}$ be a continuous and compact mapping. Moreover, we assume that

$$\begin{array}{c}\hfill \mathfrak{S}=\{z\in \mathbb{X}|z=\mathsf{\varphi }\mathsf{\rho }z,0<\mathsf{\varphi }<1\}\end{array}$$

is bounded. Then, $\mathfrak{S}$ in $\mathbb{X}$ has at least one $\mathfrak{FP}$.

From [24], we state the following definition.

Definition 4.

(1) is $\mathbb{UHS}$ if there ∃ a real number ${\mathrm{C}}_{\mathrm{f},\mathrm{g}}>0$ in such a way that for each $\mathsf{ϵ}>0$ and $x\in \mathbb{X}$ satisfying

$$|{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathsf{\xi }\right)-\mathrm{f}(\mathsf{\xi },x\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathsf{\xi }\right))-{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},x\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}|⩽\mathsf{ϵ},\mathsf{\xi }\in \mathrm{J},$$

(3)

there ∃ a solution $z\in \mathbb{X}$ of (1) with

$$|x\left(\mathsf{\xi }\right)-z\left(\mathsf{\xi }\right)|⩽{\mathrm{C}}_{\mathrm{f},\mathrm{g}}\mathsf{ϵ},\mathsf{\xi }\in \mathrm{J}.$$

Definition 5.

(1) is generalized $\mathbb{UHS}$ $\left(\mathrm{G}\mathrm{U}\mathrm{H}\mathrm{S}\right)$ if there ∃ a function ${\mathrm{F}}_{\mathrm{f},\mathrm{g}}\in \mathfrak{C}({\mathfrak{R}}_{+},{\mathfrak{R}}_{+})$, ${\mathrm{F}}_{\mathrm{f},\mathrm{g}}\left(0\right)=0$ in such a way that for each solution $x\in \mathbb{X}$ of (3), there ∃ a solution $z\in \mathbb{X}$ of (1) with

$$|x\left(\mathsf{\xi }\right)-z\left(\mathsf{\xi }\right)|⩽{\mathrm{F}}_{\mathrm{f},\mathrm{g}}\mathsf{ϵ},\mathsf{\xi }\in \mathrm{J}.$$

Definition 6.

(1) is $\mathbb{UHRS}$ corresponding to $\mathsf{\psi }\in \mathfrak{C}(\mathbb{X},{\mathfrak{R}}_{+})$ if ∃${\mathrm{C}}_{\mathrm{f},\mathrm{g}},\mathsf{\psi }>0$ in such a way that for each $x\in \mathbb{X}$, satisfying

$$|{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathsf{\xi }\right)-\mathrm{f}(\mathsf{\xi },x\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathsf{\xi }\right))-{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},x\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}|⩽\mathsf{ϵ}\mathsf{\psi }\left(\mathsf{\xi }\right),\mathsf{\xi }\in \mathrm{J},$$

(4)

there ∃ a solution $z\in \mathbb{X}$ of (1) with

$$|x\left(\mathsf{\xi }\right)-z\left(\mathsf{\xi }\right)|⩽{\mathrm{C}}_{\mathrm{f},\mathrm{g}}\mathsf{ϵ}\mathsf{\psi }\left(\mathsf{\xi }\right),\mathsf{\xi }\in \mathrm{J}.$$

Definition 7.

(1) is $\mathbb{GUHRS}$ with respect to a function $\mathsf{\psi }\in \mathfrak{C}(\mathbb{X},{\mathfrak{R}}_{+})$ if there exists ${\mathrm{C}}_{\mathrm{f},\mathrm{g}},\mathsf{\psi }>0$ such that for each solution $x\in \mathbb{X}$ of

$$|{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathsf{\xi }\right)-\mathrm{f}(\mathsf{\xi },x\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathsf{\xi }\right))-{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},x\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}|⩽\mathsf{\psi }\left(\mathsf{\xi }\right),\mathsf{\xi }\in \mathrm{J},$$

(5)

there ∃ a solution $z\in \mathbb{X}$ of (1) with

$$|x\left(\mathsf{\xi }\right)-z\left(\mathsf{\xi }\right)|⩽{\mathrm{C}}_{\mathrm{f},\mathrm{g}}\mathsf{\psi }\left(\mathsf{\xi }\right),\mathsf{\xi }\in \mathrm{J}.$$

Remark 1.

It is obvious that

1. 

Definition 4 ⇒ Definition 5;

2. 

Definition 6 ⇒ Definition 7.

Remark 2.

$x\in \mathbb{X}$ is a solution of (3) if and only if there ∃ a function $\mathsf{\psi }\in \mathbb{X}$ depending on $x$ such that

1. 

$\left|\mathsf{\psi }\right(\mathsf{\xi }\left)\right|⩽\mathsf{ϵ},$$\forall \mathsf{\xi }\in \mathrm{J};$

2. 

${}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }(\mathcal{D}+\mathsf{\lambda })x\left(\mathsf{\xi }\right)=\mathrm{f}(\mathsf{\xi },x\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},x\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}+\mathsf{\psi }\left(\mathsf{\xi }\right),\mathsf{\xi }\in \mathrm{J}.$

Remark 3.

$x\in \mathbb{X}$ is a solution of (5) if and only if there ∃ a functions $\mathsf{\psi }\in \mathbb{X}$ depending on $x$ such that

1. 

$\left|\mathsf{\psi }\right(\mathsf{\xi }\left)\right|⩽\mathsf{ϵ}\mathsf{\psi }\left(\mathsf{\xi }\right),$$\forall \mathsf{\xi }\in \mathrm{J};$

2. 

${}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }(\mathcal{D}+\mathsf{\lambda })x\left(\mathsf{\xi }\right)=\mathrm{f}(\mathsf{\xi },x\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},x\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }x\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}+\mathsf{\psi }\left(\mathsf{\xi }\right),\mathsf{\xi }\in \mathrm{J}.$

Lemma 2.

Assume $\alpha \in (1,2]$ and $\mathsf{\omega }\in \mathfrak{C}[0,\mathrm{T}]$; then, the q-$\mathbb{FDE}$

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& {}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }(\mathcal{D}+\mathsf{\lambda })z\left(\mathsf{\xi }\right)=\mathsf{\omega }\left(\mathsf{\xi }\right),\mathsf{\xi }\in \mathrm{J}=(0,\mathrm{T}),\alpha \in (1,2],\hfill \\ & z\left(0\right)=-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathsf{\eta })}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\mathrm{K}(\mathsf{\eta },z\left(\mathsf{\eta }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }z\left(\mathsf{\eta }\right)){\mathrm{d}}_q\mathsf{\eta },\hfill \\ & {}^{\mathrm{C}}{\mathcal{D}}_{q}z\left(0\right)=0,{}^{\mathrm{C}}{\mathcal{D}}_q^{{\mathsf{\beta }}_0}z\left(\mathrm{T}\right)=\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{{\mathsf{\beta }}_{\mathrm{i}}}z\left(\mathsf{\xi }\right){\mathrm{d}}_q\mathsf{\xi },\hfill \end{array}\right.\end{array}$$

(6)

has a solution given by

$$z\left(\mathsf{\xi }\right)={\int }_0^{\mathrm{T}}{\mathbb{G}}_{\mathsf{\omega }}(\mathsf{\xi },\mathit{q}\mathrm{s})\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\int }_0^{\mathrm{T}}{\mathbb{G}}_{\chi }(\mathsf{\xi },\mathit{q}\mathrm{s})\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s},$$

where $\mathsf{\omega }\in \mathbb{X}$ is given by

$$\mathsf{\omega }\left(\mathsf{\xi }\right)=\mathrm{f}(\mathsf{\xi },z\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }z\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},z\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }z\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}$$

and

$$\chi \left(\mathsf{\xi }\right)=\mathrm{K}(\mathsf{\xi },z\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }z\left(\mathsf{\xi }\right)),$$

$$\begin{array}{cc}\hfill \mathbb{G}(\mathsf{\xi },\mathit{q}\mathrm{s})& ={\mathbb{G}}_{\mathsf{\omega }}(\mathsf{\xi },\mathit{q}\mathrm{s})+{\mathbb{G}}_{\chi }(\mathsf{\xi },\mathit{q}\mathrm{s}),\hfill \\ \hfill {\mathbb{G}}_{\mathsf{\omega }}(\mathsf{\xi },\mathit{q}\mathrm{s})=& {\mathbb{G}}_1(\mathsf{\xi },\mathit{q}\mathrm{s})+{\mathbb{G}}_2(\mathsf{\xi },\mathit{q}\mathrm{s}),\hfill \\ \hfill {\mathbb{G}}_{\chi }(\mathsf{\xi },\mathit{q}\mathrm{s})=& {\mathbb{G}}_3(\mathsf{\xi },\mathit{q}\mathrm{s}),\hfill \\ \hfill {\mathbb{G}}_1(\mathsf{\xi },\mathit{q}\mathrm{s})& =\left\{\begin{array}{cc}{\displaystyle \frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\left(\mathrm{T}\right)}{\Delta }+{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\left(\mathrm{T}\right),}\hfill & 0⩽\mathit{q}\mathrm{s}⩽\mathsf{\xi }⩽\mathrm{T},\hfill \\ {\displaystyle \frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\left(\mathrm{T}\right)}{\Delta },}\hfill & 0⩽\mathsf{\xi }⩽\mathit{q}\mathrm{s}⩽\mathrm{T},\hfill \end{array}\right.\hfill \\ \hfill {\mathbb{G}}_2(\mathsf{\xi },\mathit{q}\mathrm{s})& =\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\left(\mathrm{T}\right){\mathrm{d}}_q\mathsf{\xi },\hfill \\ \hfill {\mathbb{G}}_3(\mathsf{\xi },\mathit{q}\mathrm{s})& =\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{\Delta {\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right),\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \Delta =\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{(-\mathsf{\lambda })}^{{\mathsf{\beta }}_{\mathrm{i}}}{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}{\mathrm{d}}_q\mathsf{\xi }-{(-\mathsf{\lambda })}^{{\mathsf{\beta }}_0}{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}}\ne 0.\end{array}$$

Proof. 

Consider

$$\begin{array}{c}\hfill {}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }(\mathcal{D}+\mathsf{\lambda })\mathrm{z}\left(\mathsf{\xi }\right)=\mathsf{\omega }\left(\mathsf{\xi }\right),\mathsf{\xi }\in \mathrm{J}=(0,\mathrm{T}),\alpha \in (1,2].\end{array}$$

(7)

Applying the operator ${\mathrm{I}}_{\mathit{q}}^{\alpha }$ and using (2) with $\mathrm{p}=2$, we obtain

$$(\mathcal{D}+\mathsf{\lambda })\mathrm{z}\left(\mathsf{\xi }\right)={\mathrm{I}}_q^{\alpha }\mathsf{\omega }\left(\mathsf{\xi }\right)+{\mathrm{c}}_0+{\mathrm{c}}_1\mathsf{\xi },$$

(8)

where $\mathcal{D}$ denotes ordinary differential operator. This equation can also be rewritten as

$$\mathcal{D}\left({\mathrm{e}}^{\mathsf{\lambda }\mathsf{\xi }}\mathrm{z}\left(\mathsf{\xi }\right)\right)={\mathrm{e}}^{\mathsf{\lambda }\mathsf{\xi }}[{\mathrm{I}}_q^{\alpha }\mathsf{\omega }\left(\mathsf{\xi }\right)+{\mathrm{c}}_0+{\mathrm{c}}_1\mathsf{\xi }].$$

Integrating from 0 to $\mathsf{\xi }$, we obtain

$$\begin{array}{c}\hfill \mathrm{z}\left(\mathsf{\xi }\right)={\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\frac{{\mathrm{c}}_0}{\mathsf{\lambda }}(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})+\frac{{\mathrm{c}}_1}{{\mathsf{\lambda }}^2}(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})+{\mathrm{c}}_2{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}.\end{array}$$

(9)

Using the boundary condition

$$\mathrm{z}\left(0\right)=-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\mathrm{K}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s},$$

we obtain

$${\mathrm{c}}_2=-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s},$$

where $\chi \left(\mathrm{s}\right)=\mathrm{K}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathrm{s}\right))$.

By applying ${}^{\mathrm{C}}{\mathcal{D}}_q\mathrm{z}\left(0\right)=0$, we have

$$\mathrm{z}\left(\mathsf{\xi }\right)={\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\frac{{\mathrm{c}}_0}{\mathsf{\lambda }}(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})+\frac{{\mathrm{c}}_1}{{\mathsf{\lambda }}^2}(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}\left({\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}\right).$$

Therefore,

$${}^{\mathrm{C}}{\mathcal{D}}_q\mathrm{z}\left(\mathsf{\xi }\right)={\mathrm{c}}_0{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}+\frac{{\mathrm{c}}_1}{\mathsf{\lambda }}(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})+{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -1}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\mathsf{\lambda }{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}.$$

At $\mathsf{\xi }=0$, we obtain

$$\begin{array}{cc}\hfill {}^{\mathrm{C}}{\mathcal{D}}_q\mathrm{z}\left(0\right)=& {\mathrm{c}}_0+\mathsf{\lambda }{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}\hfill \\ \hfill \end{array}$$

and ${}^{\mathrm{C}}{\mathcal{D}}_q\mathrm{z}\left(0\right)=0$, we obtain

$$\begin{array}{cc}\hfill & {\mathrm{c}}_0=-\mathsf{\lambda }{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}.\hfill \end{array}$$

Using another boundary condition, we obtain

$${}^{\mathrm{C}}{\mathcal{D}}_q^{{\mathsf{\beta }}_0}\mathrm{z}\left(\mathrm{T}\right)=\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{z}\left(\mathsf{\xi }\right){\mathrm{d}}_q\mathsf{\xi }.$$

(10)

Fitting (9) in (10), we obtain

$$\begin{array}{cc}\hfill \mathrm{z}\left(\mathrm{T}\right)=& {\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\left({\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}\right)(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})+\frac{{\mathrm{c}}_1}{{\mathsf{\lambda }}^2}(\mathsf{\lambda }\mathrm{T}-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})\hfill \\ \hfill & -\left({\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}\right){\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}},\hfill \end{array}$$

i.e.,

$$\begin{array}{cc}\hfill \mathrm{z}\left(\mathrm{T}\right)=& {\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}+\frac{{\mathrm{c}}_1}{{\mathsf{\lambda }}^2}(\mathsf{\lambda }\mathrm{T}-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}}).\hfill \\ \hfill \end{array}$$

This implies

$$\begin{array}{cc}\hfill {}^{\mathrm{C}}{\mathcal{D}}_q^{{\mathsf{\beta }}_0}\mathrm{z}\left(\mathrm{T}\right)=& {\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}+\frac{{\mathrm{c}}_1}{{\mathsf{\lambda }}^2}\left((-{\mathsf{\lambda }}^{{\mathsf{\beta }}_0}){\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}}\right)\hfill \\ \hfill \end{array}$$

(11)

and

$$\begin{array}{cc}\hfill \mathrm{z}\left(\mathsf{\xi }\right)=& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\left({\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}\right)(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})+\frac{{\mathrm{c}}_1}{{\mathsf{\lambda }}^2}(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\hfill \\ \hfill & -\left({\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}\right){\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{z}\left(\mathsf{\xi }\right)=& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}+\frac{{\mathrm{c}}_1}{{\mathsf{\lambda }}^2}(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\hfill \\ \hfill \end{array}$$

Therefore, (10) becomes

$$\begin{array}{cc}\hfill \sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{z}\left(\mathsf{\xi }\right){\mathrm{d}}_q\mathsf{\xi }=& \sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }+\frac{{\mathrm{c}}_1}{{\mathsf{\lambda }}^2}\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{(-\mathsf{\lambda })}^{{\mathsf{\beta }}_{\mathrm{i}}}{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}{\mathrm{d}}_q\mathsf{\xi }.\hfill \end{array}$$

(12)

From (11) and (12), we have

$$\begin{array}{cc}\hfill {\mathrm{c}}_1=& \frac{{\mathsf{\lambda }}^2}{\Delta }\left[{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}\right].\hfill \end{array}$$

Putting all values of ${\mathrm{c}}_0$, ${\mathrm{c}}_1$, and ${\mathrm{c}}_2$ in (9), we obtain

$$\begin{array}{cc}\hfill \mathrm{z}\left(\mathsf{\xi }\right)=& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }({\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & -\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s})\hfill \\ \hfill =& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }{\int }_{\mathsf{\xi }}^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }\left(-\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\right)\hfill \\ \hfill & -{\int }_0^{\mathrm{T}}\left(\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{\Delta {\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}\right)\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}\hfill \\ \hfill =& {\int }_0^{\mathsf{\xi }}\left({\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}}{\Delta }\right)\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }{\int }_{\mathsf{\xi }}^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{\Delta {\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q)}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}\hfill \\ \hfill =& {\int }_0^{\mathrm{T}}{\mathbb{G}}_1(\mathsf{\xi },q\mathrm{s})\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\int }_0^{\mathrm{T}}{\mathbb{G}}_2(\mathsf{\xi },q\mathrm{s})\mathsf{\omega }\left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}+{\int }_0^{\mathrm{T}}{\mathbb{G}}_3(\mathsf{\xi },q\mathrm{s})\chi \left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill =& {\int }_0^{\mathrm{T}}({\mathbb{G}}_1(\mathsf{\xi },q\mathrm{s})+{\mathbb{G}}_2(\mathsf{\xi },q\mathrm{s}))\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\int }_0^{\mathrm{T}}{\mathbb{G}}_3(\mathsf{\xi },q\mathrm{s})\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}.\hfill \end{array}$$

Thus

$$\mathrm{z}\left(\mathsf{\xi }\right)={\int }_0^{\mathrm{T}}{\mathbb{G}}_{\mathsf{\omega }}(\mathsf{\xi },q\mathrm{s})\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+{\int }_0^{\mathrm{T}}{\mathbb{G}}_{\chi }(\mathsf{\xi },q\mathrm{s})\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}.$$

   ☐

3. Existence of Solution

Here, we build up some sufficient assumptions for the $\mathbb{EU}$ of the solution to Problem (1).

In view of Lemma 2.15, Problem (6) has the following solution:

$$\begin{array}{cc}\hfill \mathrm{z}\left(\mathsf{\xi }\right)=& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathsf{\omega }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{\Delta {\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)\chi \left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s},\hfill \end{array}$$

where

$$\mathsf{\omega }\left(\mathsf{\xi }\right)=\mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s},$$

and

$$\chi \left(\mathsf{\xi }\right)=\mathrm{K}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathsf{\xi }\right)).$$

Assume

$$\begin{array}{cc}\hfill \mathrm{p}\left(\mathsf{\xi }\right)=& \mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}\hfill \\ \hfill =& \mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),\mathrm{p}\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),\mathrm{p}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathrm{v}\left(\mathsf{\xi }\right)=& \mathrm{K}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathsf{\xi }\right))\hfill \\ \hfill =& \mathrm{K}(\mathsf{\xi },\mathrm{z}(\mathsf{\xi }),\mathrm{v}(\mathsf{\xi }\left)\right).\hfill \end{array}$$

Define an operator $\mathsf{\rho }:\mathbb{X}\to \mathbb{X}$ as

$$\begin{array}{cc}\hfill \mathsf{\rho }\mathrm{z}\left(\mathsf{\xi }\right)=& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{\Delta {\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)\mathrm{v}\left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s},\hfill \end{array}$$

(13)

where p and $\mathrm{v}\in \mathbb{X}$.

The following hypothesis will be used in further results.

$\left({\mathbf{A}}_{\mathbf{1}}\right)$

${\displaystyle \Delta =\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{(-\mathsf{\lambda })}^{{\mathsf{\beta }}_{\mathrm{i}}}{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}{\mathrm{d}}_q\mathsf{\xi }-{(-\mathsf{\lambda })}^{{\mathsf{\beta }}_0}{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}}\ne 0}$.

$\left({\mathbf{A}}_{\mathbf{2}}\right)$

$\mathrm{f},\mathrm{g},\mathrm{K}:\mathrm{J}\times \mathfrak{R}\times \mathfrak{R}\to \mathfrak{R}$ are continuous.

$\left({\mathbf{A}}_{\mathbf{3}}\right)$

There ∃${\mathbb{N}}_1>0$ and $0<{\mathbb{N}}_2<1$ in such way that $\mathrm{t}\in \mathrm{J}$, and ∀$\mathsf{\zeta },\widehat{\mathsf{\zeta }},\mathsf{\theta },\widehat{\mathsf{\theta }}\in \mathfrak{R}$; thus, we have

$$|\mathrm{f}(\mathsf{\xi },\mathsf{\zeta },\mathsf{\theta })-\mathrm{f}(\mathsf{\xi },\widehat{\mathsf{\zeta }},\widehat{\mathsf{\theta }})|⩽{\mathbb{N}}_1|\mathsf{\zeta }-\widehat{\mathsf{\zeta }}|+{\mathbb{N}}_2|\mathsf{\theta }-\widehat{\mathsf{\theta }}|.$$

$\left({\mathbf{A}}_{\mathbf{4}}\right)$

There ∃${\mathbb{N}}_3>0$ and $0<{\mathbb{N}}_4<1$ in such way that $\mathrm{t}\in \mathrm{J}$ and ∀$\mathsf{\zeta },\widehat{\mathsf{\zeta }},\mathsf{\theta },\widehat{\mathsf{\theta }}\in \mathfrak{R}$; thus, we have

$$|\mathrm{K}(\mathsf{\xi },\mathsf{\zeta },\mathsf{\theta })-\mathrm{K}(\mathsf{\xi },\widehat{\mathsf{\zeta }},\widehat{\mathsf{\theta }})|⩽{\mathbb{N}}_3|\mathsf{\zeta }-\widehat{\mathsf{\zeta }}|+{\mathbb{N}}_4|\mathsf{\theta }-\widehat{\mathsf{\theta }}|.$$

$\left({\mathbf{A}}_{\mathbf{5}}\right)$

There ∃${\mathbb{N}}_5>0$ and $0<{\mathbb{N}}_6<1$ in such a way that $\mathsf{\xi }\in \mathrm{J}$ and ∀$\mathsf{\zeta },\widehat{\mathsf{\zeta }},\mathsf{\theta },\widehat{\mathsf{\theta }}\in \mathfrak{R}$; we thus have

$$|\mathrm{g}(\mathsf{\xi },\mathsf{\zeta },\mathsf{\theta })-\mathrm{g}(\mathsf{\xi },\widehat{\mathsf{\zeta }},\widehat{\mathsf{\theta }})|⩽{\mathbb{N}}_5|\mathsf{\zeta }-\widehat{\mathsf{\zeta }}|+{\mathbb{N}}_6|\mathsf{\theta }-\widehat{\mathsf{\theta }}|.$$

$\left({\mathbf{A}}_{\mathbf{6}}\right)$

There ∃$\mathrm{l},\mathrm{m},\mathrm{n}\in \mathfrak{C}(\mathrm{J},{\mathfrak{R}}^{+})$, which is bounded and satisfies

$$\left|\mathrm{f}\right(\mathsf{\xi },\mathsf{\zeta }\left(\mathsf{\xi }\right),\mathsf{\theta }\left(\mathsf{\xi }\right)\left)\right|⩽\mathrm{l}\left(\mathsf{\xi }\right)+\mathrm{m}\left(\mathsf{\xi }\right)\left|\mathsf{\zeta }\right(\mathsf{\xi }\left)\right|+\mathrm{n}\left(\mathsf{\xi }\right)\left|\mathsf{\theta }\right(\mathsf{\xi }\left)\right|$$

with ${\mathrm{n}}^{*}={sup}_{\mathsf{\xi }\in \mathrm{J}}\mathrm{n}\left(\mathsf{\xi }\right)<1$.

$\left({\mathbf{A}}_{\mathbf{7}}\right)$

There ∃$\mathrm{b},\mathrm{c},\mathrm{e}\in \mathfrak{C}(\mathrm{J},{\mathfrak{R}}^{+})$, which is bounded and satisfies

$$\left|\mathrm{K}\right(\mathsf{\xi },\mathsf{\zeta }\left(\mathsf{\xi }\right),\mathsf{\theta }\left(\mathsf{\xi }\right)\left)\right|⩽\mathrm{b}\left(\mathsf{\xi }\right)+\mathrm{c}\left(\mathsf{\xi }\right)\left|\mathsf{\zeta }\right(\mathsf{\xi }\left)\right|+\mathrm{e}\left(\mathsf{\xi }\right)\left|\mathsf{\theta }\right(\mathsf{\xi }\left)\right|$$

with ${\mathrm{e}}^{*}={sup}_{\mathsf{\xi }\in \mathrm{J}}\mathrm{e}\left(\mathsf{\xi }\right)<1$.

$\left({\mathbf{A}}_{\mathbf{8}}\right)$

There ∃$\mathrm{i},\mathrm{j},\mathrm{k}\in \mathfrak{C}(\mathrm{J},{\mathfrak{R}}^{+})$, which is bounded and satisfies

$$\left|\mathrm{g}\right(\mathsf{\xi },\mathsf{\zeta }\left(\mathsf{\xi }\right),\mathsf{\theta }\left(\mathsf{\xi }\right)\left)\right|⩽\mathrm{i}\left(\mathsf{\xi }\right)+\mathrm{j}\left(\mathsf{\xi }\right)\left|\mathsf{\zeta }\right(\mathsf{\xi }\left)\right|+\mathrm{k}\left(\mathsf{\xi }\right)\left|\mathsf{\theta }\right(\mathsf{\xi }\left)\right|$$

with ${\mathrm{k}}^{*}={sup}_{\mathsf{\xi }\in \mathrm{J}}\mathrm{k}\left(\mathsf{\xi }\right)<1$.

$\left({\mathbf{A}}_{\mathbf{9}}\right)$

There ∃ a non-decreasing $\mathsf{\psi }\in \mathfrak{C}(\mathrm{J},{\mathfrak{R}}^{+})$ and ${\mathfrak{L}}_{\mathsf{\psi }}>0$ fulfilling

$${\mathrm{I}}_q^{\alpha }\mathsf{\psi }\left(\mathsf{\xi }\right)⩽{\mathfrak{L}}_{\mathsf{\psi }}\mathsf{\psi }\left(\mathsf{\xi }\right),\mathsf{\xi }\in \mathrm{J}.$$

Lemma 3.

The Green’s functions ${\mathbb{G}}_{\mathsf{\omega }}(\mathsf{\xi },\mathit{q}\mathrm{s})$ and ${\mathbb{G}}_{\chi }(\mathsf{\xi },\mathit{q}\mathrm{s})$, which are obtained in Lemma 2, fulfill the following conditions

(1) 

${\mathbb{G}}_{\mathsf{\omega }}(\mathsf{\xi },\mathit{q}\mathrm{s})$ and ${\mathbb{G}}_{\chi }(\mathsf{\xi },\mathit{q}\mathrm{s})$ are continuous on J×J,

(2) 

${\displaystyle \underset{\mathsf{\xi }\in \mathrm{J}}{max}\left|{\int }_0^{\mathrm{T}}{\mathbb{G}}_{\mathsf{\omega }}(\mathsf{\xi },\mathit{q}\mathrm{s})\mathrm{d}\mathrm{s}\right|⩽{\mathfrak{Y}}_{\mathsf{\omega }}}$   and    ${\displaystyle \underset{\mathsf{\xi }\in \mathrm{J}}{max}\left|{\int }_0^{\mathrm{T}}{\mathbb{G}}_{\chi }(\mathsf{\xi },\mathit{q}\mathrm{s}){\mathrm{d}}_{\mathit{q}}\mathrm{s}\right|⩽{\mathfrak{Y}}_{\chi }}$,

where

$$\begin{array}{cc}\hfill {\mathfrak{Y}}_{\mathsf{\omega }}=& \frac{{\mathrm{T}}^{\alpha }(1-{(1-q)}^{\alpha })}{\mathsf{\lambda }q{\mathsf{\Gamma }}_q(\alpha +1)}+\frac{|(\mathsf{\lambda }\mathrm{T}-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{\mathsf{\lambda }\nabla q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}\hfill \\ \hfill & +\sum _{\mathrm{i}=1}^{\mathrm{p}}\frac{|(1-\mathsf{\lambda }\mathrm{T}-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1})}{\mathsf{\lambda }\nabla q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_{\mathrm{i}}+2)},\hfill \\ \hfill {\mathfrak{Y}}_{\chi }=& \frac{|1-\mathsf{\lambda }\mathrm{T}-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}}|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{\nabla q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}+\frac{{\mathrm{T}}^{\alpha }(1-{(1-q)}^{\alpha })}{q{\mathsf{\Gamma }}_q\left(\alpha \right)}\hfill \\ \hfill \mathrm{a}\mathrm{n}\mathrm{d}\\ \hfill \nabla =& \sum _{\mathrm{i}=1}^{\mathrm{p}}{(-\mathsf{\lambda })}^{{\mathsf{\beta }}_{\mathrm{i}}}\frac{{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}}}{\mathsf{\lambda }}-\left({(-\mathsf{\lambda })}^{{\mathsf{\beta }}_0}{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}}\right).\hfill \end{array}$$

Proof. (1) The continuities of ${\mathbb{G}}_{\mathsf{\omega }}(\mathsf{\xi },\mathit{q}\mathrm{s})$ and ${\mathbb{G}}_{\chi }(\mathsf{\xi },\mathit{q}\mathrm{s})$ are obvious.

(2)

${\mathbb{G}}_{\mathsf{\omega }}(\mathsf{\xi },\mathit{q}\mathrm{s})$ and ${\mathbb{G}}_{\chi }(\mathsf{\xi },\mathit{q}\mathrm{s})$ have the following expression:

$$\begin{array}{cc}\hfill \left|{\int }_0^{\mathrm{T}}{\mathbb{G}}_{\mathsf{\omega }}(\mathsf{\xi },\mathit{q}\mathrm{s})\mathrm{d}\mathrm{s}\right|=& |{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}\left({\int }_0^{\mathsf{\xi }}\frac{1}{{\mathsf{\Gamma }}_q\left(\alpha \right)}{(\mathsf{\xi }-q\mathrm{s})}^{\alpha -1}{\mathrm{d}}_q\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}\left({\int }_0^{\mathsf{\xi }}\frac{1}{{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}{(\mathsf{\xi }-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}{\mathrm{d}}_q\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}\left({\int }_0^{\mathsf{\xi }}\frac{1}{{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_{\mathrm{i}})}{(\mathsf{\xi }-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}-1}{\mathrm{d}}_q\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }|\hfill \\ \hfill ⩽& \frac{|1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}|{\mathsf{\xi }}^{\alpha }(1-{(1-q)}^{\alpha })}{\mathsf{\lambda }q{\mathsf{\Gamma }}_q(\alpha +1)}+\frac{|(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathsf{\xi }}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}\hfill \\ \hfill & +\sum _{\mathrm{i}=1}^{\mathrm{p}}\frac{|(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_{\mathrm{i}}+2)}.\hfill \end{array}$$

Now taking ${max}_{\mathsf{\xi }\in \mathrm{J}}$, we obtain

$$\begin{array}{cc}\hfill \underset{\mathsf{\xi }\in \mathrm{J}}{max}\left|{\int }_0^{\mathrm{T}}{\mathbb{G}}_{\mathsf{\omega }}(\mathsf{\xi },\mathit{q}\mathrm{s})\mathrm{d}\mathrm{s}\right|⩽& \frac{|1-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}}|{\mathrm{T}}^{\alpha }(1-{(1-q)}^{\alpha })}{\mathsf{\lambda }q{\mathsf{\Gamma }}_q(\alpha +1)}\hfill \\ \hfill & +\frac{|(\mathsf{\lambda }\mathrm{T}-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{\mathsf{\lambda }\nabla q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}\hfill \\ \hfill & +\sum _{\mathrm{i}=1}^{\mathrm{p}}\frac{|(1-\mathsf{\lambda }\mathrm{T}-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1})}{\mathsf{\lambda }\nabla q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_{\mathrm{i}}+2)}\hfill \\ \hfill ⩽& \frac{{\mathrm{T}}^{\alpha }(1-{(1-q)}^{\alpha })}{\mathsf{\lambda }q{\mathsf{\Gamma }}_q(\alpha +1)}+\frac{|(\mathsf{\lambda }\mathrm{T}-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{\mathsf{\lambda }\nabla q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}\hfill \\ \hfill & +\sum _{\mathrm{i}=1}^{\mathrm{p}}\frac{|(1-\mathsf{\lambda }\mathrm{T}-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1})}{\mathsf{\lambda }\nabla q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_{\mathrm{i}}+2)}\hfill \\ \hfill =& {\mathfrak{Y}}_{\mathsf{\omega }}.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill \left|{\int }_0^{\mathrm{T}}{\mathbb{G}}_{\chi }(\mathsf{\xi },\mathit{q}\mathrm{s}){\mathrm{d}}_q\mathrm{s}\right|⩽& |\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}{\mathrm{d}}_q\mathrm{s}-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}{\mathrm{d}}_q\mathrm{s}|\hfill \\ \hfill ⩽& \frac{|1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}+\frac{{\mathrm{T}}^{\alpha }(1-{(1-q)}^{\alpha })}{q{\mathsf{\Gamma }}_q\left(\alpha \right)}.\hfill \end{array}$$

Now taking ${max}_{\mathsf{\xi }\in \mathrm{J}}$, we obtain

$$\begin{array}{cc}\hfill \underset{\mathsf{\xi }\in \mathrm{J}}{max}& \left|{\int }_0^{\mathrm{T}}{\mathbb{G}}_{\chi }(\mathsf{\xi },\mathit{q}\mathrm{s}){\mathrm{d}}_q\mathrm{s}\right|⩽\frac{|1-\mathsf{\lambda }\mathrm{T}-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}}|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{\nabla q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}+\frac{{\mathrm{T}}^{\alpha }(1-{(1-q)}^{\alpha })}{q{\mathsf{\Gamma }}_q\left(\alpha \right)}={\mathfrak{Y}}_{\chi }.\hfill \end{array}$$

   ☐

Theorem 4.

If the hypothesis $\left({\mathbf{A}}_{\mathbf{2}}\right)$–$\left({\mathbf{A}}_{\mathbf{5}}\right)$ and the inequality

$$\left[{\mathfrak{Y}}_{\mathsf{\omega }}\left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}+\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}\right)+{\mathfrak{Y}}_{\chi }\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)\right]<1$$

(14)

are satisfied, then (6) has a unique solution.

Proof. 

For $\mathrm{z},\mathrm{x}\in \mathbb{X}$, using (13), we have

$$\begin{array}{cc}\hfill \mathsf{\rho }\mathrm{z}\left(\mathsf{\xi }\right)=& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }|\mathrm{p}\left(\mathrm{s}\right)-\mathrm{p}\widehat{\left(\mathrm{s}\right)}|\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}|\mathrm{p}\left(\mathrm{s}\right)-\mathrm{p}\widehat{\left(\mathrm{s}\right)}|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}|\mathrm{p}\left(\mathrm{s}\right)-\mathrm{p}\widehat{\left(\mathrm{s}\right)}|\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{|\Delta |{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)|\mathrm{v}\left(\mathrm{s}\right)-\mathrm{v}\widehat{\left(\mathrm{s}\right)}|{\mathrm{d}}_q\mathrm{s},\hfill \end{array}$$

(15)

where $\widehat{\mathrm{p}}$ and $\widehat{\mathrm{v}}\in \mathbb{X}$ are given by

$$\begin{array}{cc}\hfill \mathrm{p}\widehat{\left(\mathsf{\xi }\right)}=& \mathrm{f}(\mathsf{\xi },\mathrm{x}\left(\mathsf{\xi }\right),\mathrm{p}\widehat{\left(\mathsf{\xi }\right)})+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{x}\left(\mathrm{s}\right),\mathrm{p}\widehat{\left(\mathrm{s}\right)}){\mathrm{d}}_q\mathrm{s}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathrm{v}\widehat{\left(\mathsf{\xi }\right)}=& \mathrm{K}(\mathsf{\xi },\mathrm{x}\left(\mathsf{\xi }\right),\mathrm{v}\widehat{\left(\mathsf{\xi }\right)}).\hfill \end{array}$$

Using $\left({\mathbf{A}}_{\mathbf{2}}\right),\left({\mathbf{A}}_{\mathbf{3}}\right)$ and $\left({\mathbf{A}}_{\mathbf{5}}\right)$, we have

$$\begin{array}{cc}\hfill |\mathrm{p}\left(\mathsf{\xi }\right)-\mathrm{p}\widehat{\left(\mathsf{\xi }\right)}|=& |\mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),\mathrm{p}\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),\mathrm{p}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}-\mathrm{f}(\mathsf{\xi },\mathrm{x}\left(\mathsf{\xi }\right),\mathrm{p}\widehat{\left(\mathsf{\xi }\right)})\hfill \\ \hfill & +{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{x}\left(\mathrm{s}\right),\mathrm{p}\widehat{\left(\mathrm{s}\right)}){\mathrm{d}}_q\mathrm{s}|\hfill \\ \hfill ⩽& |\mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),\mathrm{p}\left(\mathsf{\xi }\right))-\mathrm{f}(\mathsf{\xi },\mathrm{x}\left(\mathsf{\xi }\right),\mathrm{p}\widehat{\left(\mathsf{\xi }\right)})|+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\left|\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),\mathrm{p}\left(\mathrm{s}\right))\mathrm{g}(\mathrm{s},\mathrm{x}\left(\mathrm{s}\right),\mathrm{p}\widehat{\left(\mathrm{s}\right)})\right|{\mathrm{d}}_q\mathrm{s}\hfill \\ \hfill ⩽& {\mathrm{N}}_1|\mathrm{z}\left(\mathsf{\xi }\right)-\mathrm{x}\left(\mathsf{\xi }\right)|+{\mathrm{N}}_2|\mathrm{p}\left(\mathsf{\xi }\right)-\mathrm{p}\widehat{\left(\mathsf{\xi }\right)}|+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\left({\mathrm{N}}_5|\mathrm{z}\left(\mathrm{s}\right)-\mathrm{x}\left(\mathrm{s}\right)|+{\mathrm{N}}_6|\mathrm{p}\left(\mathrm{s}\right)-\mathrm{p}\widehat{\left(\mathrm{s}\right)}|\right){\mathrm{d}}_q\mathrm{s}\hfill \\ \hfill =& {\mathrm{N}}_1|\mathrm{z}\left(\mathsf{\xi }\right)-\mathrm{x}\left(\mathsf{\xi }\right)|+{\mathrm{N}}_2|\mathrm{p}\left(\mathsf{\xi }\right)-\mathrm{p}\widehat{\left(\mathsf{\xi }\right)}|+\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}{\mathrm{N}}_5|\mathrm{z}\left(\mathsf{\xi }\right)-\mathrm{x}\left(\mathsf{\xi }\right)|\hfill \\ \hfill & +\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}{\mathrm{N}}_6|\mathrm{p}\left(\mathsf{\xi }\right)-\mathrm{p}\widehat{\left(\mathsf{\xi }\right)}|.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill |\mathrm{p}\left(\mathsf{\xi }\right)-\mathrm{p}\widehat{\left(\mathsf{\xi }\right)}|⩽& \left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}\right)|\mathrm{z}\left(\mathsf{\xi }\right)-\mathrm{x}\left(\mathsf{\xi }\right)|\hfill \\ \hfill & +\frac{{\mathrm{N}}_5{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}|\mathrm{z}\left(\mathsf{\xi }\right)-\mathrm{x}\left(\mathsf{\xi }\right)|.\hfill \end{array}$$

(16)

Similarly,

$$\begin{array}{cc}\hfill |\mathrm{v}\left(\mathsf{\xi }\right)-\mathrm{v}\widehat{\left(\mathsf{\xi }\right)}|& =|\mathrm{K}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),\mathrm{v}\left(\mathsf{\xi }\right))-\mathrm{K}(\mathsf{\xi },\mathrm{x}\left(\mathsf{\xi }\right),\mathrm{v}\widehat{\left(\mathsf{\xi }\right)})|\hfill \\ \hfill & ⩽\left|\mathrm{K}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),\mathrm{v}\left(\mathsf{\xi }\right))\right|+|\mathrm{K}(\mathsf{\xi },\mathrm{x}\left(\mathsf{\xi }\right),\mathrm{v}\widehat{\left(\mathsf{\xi }\right)})|.\hfill \end{array}$$

Using $\left({\mathbf{A}}_{\mathbf{4}}\right)$, we obtain

$$\begin{array}{cc}\hfill |\mathrm{v}\left(\mathsf{\xi }\right)-\mathrm{v}\widehat{\left(\mathsf{\xi }\right)}|& ⩽{\mathrm{N}}_3|\mathrm{z}\left(\mathsf{\xi }\right)-\mathrm{x}\left(\mathsf{\xi }\right)|+{\mathrm{N}}_4|\mathrm{v}\left(\mathsf{\xi }\right)-\mathrm{v}\widehat{\left(\mathsf{\xi }\right)}|.\hfill \end{array}$$

Equivalently,

$$\begin{array}{cc}\hfill |\mathrm{v}\left(\mathsf{\xi }\right)-\mathrm{v}\widehat{\left(\mathsf{\xi }\right)}|& ⩽\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)|\mathrm{z}\left(\mathsf{\xi }\right)-\mathrm{x}\left(\mathsf{\xi }\right)|.\hfill \end{array}$$

(17)

Using (16) and (17) in (15), we have

$$\begin{array}{cc}\hfill \left|\mathsf{\rho }\mathrm{z}\right(\mathsf{\xi })-\mathsf{\rho }\mathrm{x}(\mathsf{\xi }\left)\right|⩽& [\frac{|1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}|{\mathsf{\xi }}^{\alpha }(1-{(1-q)}^{\alpha })}{\mathsf{\lambda }q{\mathsf{\Gamma }}_q(\alpha +1)}+\frac{|(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathsf{\xi }}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}\hfill \\ \hfill & +\sum _{\mathrm{i}=1}^{\mathrm{p}}\frac{|(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_{\mathrm{i}}+2)}[\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}\hfill \\ \hfill & +\frac{{\mathrm{N}}_5{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}]+\frac{|1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{|\Delta |q{\mathsf{\Gamma }}_q(\alpha +1)}\hfill \\ \hfill & +\frac{{\mathrm{T}}^{\alpha }(1-{(1-q)}^{\alpha })}{q{\mathsf{\Gamma }}_q\left(\alpha \right)}\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)]|\mathrm{z}\left(\mathsf{\xi }\right)-\mathrm{x}\left(\mathsf{\xi }\right)|.\hfill \end{array}$$

Since $\mathsf{\xi }\in [0,\mathrm{T}]$ and using Lemma 3, we obtain

$$\begin{array}{cc}\hfill \left|\mathsf{\rho }\mathrm{z}\right(\mathsf{\xi })-\mathsf{\rho }\mathrm{x}(\mathsf{\xi }\left)\right|⩽& [{\mathfrak{Y}}_{\mathsf{\omega }}\left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}+\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}\right)\hfill \\ \hfill & +{\mathfrak{Y}}_{\chi }\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)]|\mathrm{z}\left(\mathsf{\xi }\right)-\mathrm{x}\left(\mathsf{\xi }\right)|.\hfill \end{array}$$

Thus

$$\begin{array}{cc}\hfill \left|\right|\mathsf{\rho }\mathrm{z}-{\mathsf{\rho }\mathrm{x}\left|\right|}_{\mathbb{X}}⩽& [{\mathfrak{Y}}_{\mathsf{\omega }}\left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}+\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}\right)\hfill \\ \hfill & +{\mathfrak{Y}}_{\chi }\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)\left]\right||\mathrm{z}\left(\mathsf{\xi }\right)-{\mathrm{x}\left(\mathsf{\xi }\right)\left|\right|}_{\mathbb{X}}.\hfill \end{array}$$

Moreover,

$$\begin{array}{c}\hfill \left[{\mathfrak{Y}}_{\mathsf{\omega }}\left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}+\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}\right)+{\mathfrak{Y}}_{\chi }\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)\right]<1.\end{array}$$

Thus by the Banach theorem, $\mathsf{\rho }$ has a $\mathfrak{FP}$ (unique), i.e., (6) has a unique, solution.    ☐

Theorem 5.

Using the hypothesis $\left({\mathbf{A}}_{\mathbf{2}}\right)$–$\left({\mathbf{A}}_{\mathbf{8}}\right)$, (6) has at least one solution.

Proof. 

We use the predefined operator $\mathsf{\rho }$ and follow the steps of Schafer $\mathfrak{FP}$, as follows.

Step 1:

First, we prove that $\mathsf{\rho }$ is continuous. Consider a sequence ${\mathrm{z}}_{\mathrm{n}}\in \mathbb{X}$ such that ${\mathrm{z}}_{\mathrm{n}}\to \mathrm{z}\in \mathbb{X}$. For $\mathsf{\xi }\in \mathrm{J}$, we have

$$\begin{array}{cc}\hfill |\mathsf{\rho }{\mathrm{z}}_{\mathrm{n}}\left(\mathsf{\xi }\right)-\mathsf{\rho }\mathrm{z}\left(\mathsf{\xi }\right)|⩽& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }|{\mathrm{p}}_{\mathrm{n}}\left(\mathrm{s}\right)-\mathrm{p}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}|{\mathrm{p}}_{\mathrm{n}}\left(\mathrm{s}\right)-\mathrm{p}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}|{\mathrm{p}}_{\mathrm{n}}\left(\mathrm{s}\right)-\mathrm{p}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{|\Delta |{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)|{\mathrm{v}}_{\mathrm{n}}\left(\mathrm{s}\right)-\mathrm{v}\left(\mathrm{s}\right)|{\mathrm{d}}_q\mathrm{s},\hfill \end{array}$$

where ${\mathrm{p}}_{\mathrm{n}},{\mathrm{v}}_{\mathrm{n}}\in \mathbb{X}$

$$\begin{array}{cc}\hfill {\mathrm{p}}_{\mathrm{n}}\left(\mathsf{\xi }\right)=& \mathrm{f}(\mathsf{\xi },{\mathrm{z}}_{\mathrm{n}}\left(\mathsf{\xi }\right),{\mathrm{p}}_{\mathrm{n}}\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},{\mathrm{z}}_{\mathrm{n}}\left(\mathrm{s}\right),{\mathrm{p}}_{\mathrm{n}}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\mathrm{v}}_{\mathrm{n}}\left(\mathsf{\xi }\right)=& \mathrm{K}(\mathsf{\xi },{\mathrm{z}}_{\mathrm{n}}\left(\mathsf{\xi }\right),{\mathrm{v}}_{\mathrm{n}}\left(\mathsf{\xi }\right)).\hfill \end{array}$$

Hence, by $\left({\mathbf{A}}_{\mathbf{3}}\right)$–$\left({\mathbf{A}}_{\mathbf{5}}\right)$, we obtain

$$\begin{array}{cc}\hfill |{\mathrm{p}}_{\mathrm{n}}\left(\mathsf{\xi }\right)-\mathrm{p}\left(\mathsf{\xi }\right)|⩽& \left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}\right)|{\mathrm{z}}_{\mathrm{n}}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|\hfill \\ \hfill & +\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}|{\mathrm{z}}_{\mathrm{n}}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|.\hfill \end{array}$$

$$\begin{array}{c}\hfill |{\mathrm{v}}_{\mathrm{n}}\left(\mathsf{\xi }\right)-\mathrm{v}\left(\mathsf{\xi }\right)|⩽\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)|{\mathrm{z}}_{\mathrm{n}}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|\end{array}$$

$$\begin{array}{cc}\hfill & |\mathsf{\rho }{\mathrm{z}}_{\mathrm{n}}\left(\mathsf{\xi }\right)-\mathsf{\rho }\mathrm{z}\left(\mathsf{\xi }\right)|\hfill \\ \hfill ⩽& [\frac{|1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}|{\mathsf{\xi }}^{\alpha }(1-{(1-q)}^{\alpha })}{\mathsf{\lambda }q{\mathsf{\Gamma }}_q(\alpha +1)}+\frac{|(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathsf{\xi }}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}\hfill \\ \hfill & +\sum _{\mathrm{i}=1}^{\mathrm{p}}\frac{|(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_{\mathrm{i}}+2)}[\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}\hfill \\ \hfill & +\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}]+\frac{|1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{|\Delta |q{\mathsf{\Gamma }}_q(\alpha +1)}\hfill \\ \hfill & +\frac{{\mathrm{T}}^{\alpha }(1-{(1-q)}^{\alpha })}{q{\mathsf{\Gamma }}_q\left(\alpha \right)}\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)]|{\mathrm{z}}_{\mathrm{n}}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|.\hfill \end{array}$$

For each $\mathsf{\xi }\in \mathrm{J}$, the sequence ${\mathrm{z}}_{\mathrm{n}}\to \mathrm{z}$ as $\mathrm{n}\to \infty$; thus, by the Lebesgue dominated convergence theorem,

$$|\mathsf{\rho }{\mathrm{z}}_{\mathrm{n}}\left(\mathsf{\xi }\right)-\mathsf{\rho }\mathrm{z}\left(\mathsf{\xi }\right)|\to 0\mathrm{n}\to \infty .$$

This implies that

$$\left|\right|\mathsf{\rho }{\mathrm{z}}_{\mathrm{n}}-\mathsf{\rho }\mathrm{z}\left|\right|\to 0\mathrm{n}\to \infty ;$$

i.e., $\mathsf{\rho }$ is continuous on J.

Step 2:

Next, for each $\mathrm{z}\in {\pi }_{\mathrm{k}}=\{\mathrm{z}\in \mathbb{X}:|\left|\mathrm{z}\right||⩽\mathrm{k}\}$. We need to verify that $\left|\right|\mathsf{\rho }\mathrm{z}\left|\right|⩽\mathfrak{L}$ with some $\mathfrak{L}>0$. For $\mathsf{\xi }\in \mathrm{J}$, we obtain

$$\begin{array}{cc}\hfill \left|\mathsf{\rho }\mathrm{z}\right(\mathsf{\xi }\left)\right|⩽& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\left|\mathrm{p}\left(\mathrm{s}\right)\right|\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\left|\mathrm{p}\left(\mathrm{s}\right)\right|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\left|\mathrm{p}\left(\mathrm{s}\right)\right|\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{|\Delta |{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)\left|\mathrm{v}\left(\mathrm{s}\right)\right|{\mathrm{d}}_q\mathrm{s},\hfill \end{array}$$

(18)

where $\mathrm{p},\mathrm{v}\in \mathbb{X}$ are given by

$$\begin{array}{cc}\hfill \mathrm{p}\left(\mathsf{\xi }\right)=& \mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),\mathrm{p}\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),\mathrm{p}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \mathrm{v}\left(\mathsf{\xi }\right)=\mathrm{K}(\mathsf{\xi },\mathrm{z}(\mathsf{\xi }),\mathrm{v}(\mathsf{\xi }\left)\right).\end{array}$$

By $\left({\mathbf{A}}_{\mathbf{5}}\right)$ and $\left({\mathbf{A}}_{\mathbf{6}}\right)$, we have

$$\begin{array}{cc}\hfill \mathrm{p}\left(\mathsf{\xi }\right)=& \mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),\mathrm{p}\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),\mathrm{p}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \left|\mathrm{p}\right(\mathsf{\xi }\left)\right|& =|\mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),\mathrm{p}\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),\mathrm{p}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}|\hfill \\ \hfill & ⩽\left|\mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),\mathrm{p}\left(\mathsf{\xi }\right))\right|+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\left|\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),\mathrm{p}\left(\mathrm{s}\right))\right|{\mathrm{d}}_q\mathrm{s}\hfill \\ \hfill & ⩽\mathrm{l}\left(\mathsf{\xi }\right)+\mathrm{m}\left(\mathsf{\xi }\right)\left|\mathrm{z}\left(\mathsf{\xi }\right)\right|+\mathrm{n}\left(\mathsf{\xi }\right)\left|\mathrm{p}\right(\mathsf{\xi }\left)\right|+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}(\mathrm{i}\left(\mathrm{s}\right)+\mathrm{j}\left(\mathrm{s}\right)|\mathrm{z}\left(\mathrm{s}\right)|+\mathrm{k}\left(\mathrm{s}\right)|\mathrm{p}\left(\mathrm{s}\right)\left|\right){\mathrm{d}}_q\mathrm{s}\hfill \\ \hfill & ⩽{\mathrm{l}}^{*}+{\mathrm{m}}^{*}{\left|\right|\mathrm{z}\left|\right|}_{\mathbb{X}}+{\mathrm{n}}^{*}{\left|\right|\mathrm{p}\left|\right|}_{\mathbb{X}}+({\mathrm{i}}^{*}+{\mathrm{j}}^{*}{\left|\right|\mathrm{z}\left|\right|}_{\mathbb{X}}+{\mathrm{k}}^{*}{\left|\right|\mathrm{p}\left|\right|}_{\mathbb{X}})\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)},\hfill \end{array}$$

where ${\mathrm{l}}^{*}=\mathrm{s}\mathrm{u}{\mathrm{p}}_{\mathsf{\xi }\in \mathrm{J}}\mathrm{l}\left(\mathsf{\xi }\right)$, ${\mathrm{m}}^{*}=\mathrm{s}\mathrm{u}{\mathrm{p}}_{\mathsf{\xi }\in \mathrm{J}}\mathrm{m}\left(\mathsf{\xi }\right)$, ${\mathrm{n}}^{*}=\mathrm{s}\mathrm{u}{\mathrm{p}}_{\mathsf{\xi }\in \mathrm{J}}\mathrm{n}\left(\mathsf{\xi }\right)<1$, ${\mathrm{i}}^{*}=\mathrm{s}\mathrm{u}{\mathrm{p}}_{\mathsf{\xi }\in \mathrm{J}}\mathrm{i}\left(\mathsf{\xi }\right)$, ${\mathrm{j}}^{*}=\mathrm{s}\mathrm{u}{\mathrm{p}}_{\mathsf{\xi }\in \mathrm{J}}\mathrm{j}\left(\mathsf{\xi }\right)$, ${\mathrm{k}}^{*}=\mathrm{s}\mathrm{u}{\mathrm{p}}_{\mathsf{\xi }\in \mathrm{J}}\mathrm{k}\left(\mathsf{\xi }\right)<1$

$$\begin{array}{cc}\hfill \left|\mathrm{p}\right(\mathsf{\xi }\left)\right|& ⩽{\left|\right|\mathrm{p}\left|\right|}_{\mathbb{X}}⩽\left(\frac{{\mathrm{l}}^{*}+{\mathrm{m}}^{*}{\left|\right|\mathrm{z}\left|\right|}_{\mathbb{X}}}{1-{\mathrm{n}}^{*}-{\mathrm{k}}_{*}\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}\right)+\frac{\left({\mathrm{i}}^{*}+{\mathrm{j}}^{*}{\left|\right|\mathrm{z}\left|\right|}_{\mathbb{X}}\right){\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{n}}^{*}-{\mathrm{k}}^{*}\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}=\mathrm{h}.\hfill \end{array}$$

Similarly,

$$\begin{array}{c}\hfill \left|\mathrm{v}\right(\mathsf{\xi }\left)\right|=\left|\mathrm{K}\right(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),\mathrm{v}\left(\mathsf{\xi }\right)\left)\right|.\end{array}$$

By $\left({\mathbf{A}}_{\mathbf{7}}\right)$, we obtain

$$\begin{array}{cc}\hfill \left|\mathrm{v}\right(\mathsf{\xi }\left)\right|& ⩽\mathrm{b}\left(\mathsf{\xi }\right)+\mathrm{c}\left(\mathsf{\xi }\right)\left|\mathrm{z}\right(\mathsf{\xi }\left)\right|+\mathrm{e}\left(\mathsf{\xi }\right)\left|v\right(\mathsf{\xi }\left)\right|\hfill \\ \hfill & ⩽{\mathrm{b}}^{*}+{\mathrm{c}}^{*}{\left|\right|\mathrm{z}\left|\right|}_{\mathbb{X}}+{\mathrm{e}}^{*}{\left|\right|\mathrm{v}\left|\right|}_{\mathbb{X}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left|\mathrm{v}\right(\mathsf{\xi }\left)\right|& ⩽\frac{{\mathrm{b}}^{*}+{\mathrm{c}}^{*}\mathrm{k}}{1-{\mathrm{e}}^{*}}={\mathrm{h}}^{*}.\hfill \end{array}$$

Clearly, h and ${\mathrm{h}}^{*}$ are constants. Using Lemma 3 and (18), we obtain

$$\begin{array}{cc}\hfill {\left|\right|\mathsf{\rho }\left(\mathrm{z}\right)\left|\right|}_{\mathbb{X}}& ⩽{\mathfrak{Y}}_{\mathsf{\omega }}\left(\mathrm{h}\right)+{\mathfrak{Y}}_{\chi }\left({\mathrm{h}}^{*}\right)=\mathrm{L}.\hfill \end{array}$$

Step 3:

Take ${\mathsf{\xi }}_1,{\mathsf{\xi }}_2\in \mathbb{X}$ in such way that ${\mathsf{\xi }}_1<{\mathsf{\xi }}_2$ and assume that ${\pi }_{\mathrm{k}}$ is bounded. Then, for $\mathrm{z}\in {\mathfrak{o}}_{\mathrm{k}}$,

$$\begin{array}{cc}\hfill |\mathsf{\rho }\mathrm{z}\left({\mathsf{\xi }}_2\right)-\mathsf{\rho }\mathrm{z}\left({\mathsf{\xi }}_1\right)|=& |{\int }_0^{{\mathsf{\xi }}_2}{\mathrm{e}}^{-\mathsf{\lambda }({\mathsf{\xi }}_2-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-{\int }_0^{{\mathsf{\xi }}_1}{\mathrm{e}}^{-\mathsf{\lambda }({\mathsf{\xi }}_1-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(\mathsf{\lambda }{\mathsf{\xi }}_2-1+{\mathrm{e}}^{-\mathsf{\lambda }{\mathsf{\xi }}_2})}{\Delta }{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & -\frac{(\mathsf{\lambda }{\mathsf{\xi }}_1-1+{\mathrm{e}}^{-\mathsf{\lambda }{\mathsf{\xi }}_1})}{\Delta }{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }{\mathsf{\xi }}_2-{\mathrm{e}}^{-\mathsf{\lambda }{\mathsf{\xi }}_2})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{{\mathsf{\xi }}_2}{\mathrm{e}}^{-\mathsf{\lambda }({\mathsf{\xi }}_2-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q{\mathsf{\xi }}_2\hfill \\ \hfill & -\frac{(1-\mathsf{\lambda }{\mathsf{\xi }}_1-{\mathrm{e}}^{-\mathsf{\lambda }{\mathsf{\xi }}_1})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{{\mathsf{\xi }}_1}{\mathrm{e}}^{-\mathsf{\lambda }({\mathsf{\xi }}_1-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q{\mathsf{\xi }}_1|\hfill \\ \hfill \end{array}$$

In Step 2, we obtained that

$$\begin{array}{cc}\hfill \left|\mathrm{p}\right(\mathsf{\xi }\left)\right|& ⩽{\left|\right|\mathrm{p}\left|\right|}_{\mathbb{X}}⩽\left(\frac{{\mathrm{l}}^{*}+{\mathrm{m}}^{*}{\left|\right|\mathrm{z}\left|\right|}_{\mathbb{X}}}{1-{\mathrm{n}}^{*}-{\mathrm{k}}_{*}\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}\right)+\frac{\left({\mathrm{i}}^{*}+{\mathrm{j}}^{*}{\left|\right|\mathrm{z}\left|\right|}_{\mathbb{X}}\right){\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{n}}^{*}-{\mathrm{k}}^{*}\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}=\mathrm{h},\hfill \end{array}$$

$$\begin{array}{cc}\hfill |\mathsf{\rho }\mathrm{z}\left({\mathsf{\xi }}_2\right)-\mathsf{\rho }\mathrm{z}\left({\mathsf{\xi }}_1\right)|⩽& \mathrm{h}|{\int }_0^{{\mathsf{\xi }}_2}{\mathrm{e}}^{-\mathsf{\lambda }({\mathsf{\xi }}_2-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}-{\int }_0^{{\mathsf{\xi }}_1}{\mathrm{e}}^{-\mathsf{\lambda }({\mathsf{\xi }}_1-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(\mathsf{\lambda }{\mathsf{\xi }}_2-1+{\mathrm{e}}^{-\mathsf{\lambda }{\mathsf{\xi }}_2})}{\Delta }{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & -\frac{(\mathsf{\lambda }{\mathsf{\xi }}_1-1+{\mathrm{e}}^{-\mathsf{\lambda }{\mathsf{\xi }}_1})}{\Delta }{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }{\mathsf{\xi }}_2-{\mathrm{e}}^{-\mathsf{\lambda }{\mathsf{\xi }}_2})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{{\mathsf{\xi }}_2}{\mathrm{e}}^{-\mathsf{\lambda }({\mathsf{\xi }}_2-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q{\mathsf{\xi }}_2\hfill \\ \hfill & -\frac{(1-\mathsf{\lambda }{\mathsf{\xi }}_1-{\mathrm{e}}^{-\mathsf{\lambda }{\mathsf{\xi }}_1})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{{\mathsf{\xi }}_1}{\mathrm{e}}^{-\mathsf{\lambda }({\mathsf{\xi }}_1-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q{\mathsf{\xi }}_1|.\hfill \\ \hfill \end{array}$$

(19)

It is obvious that the right-hand side of (19) tends to zero as ${\mathsf{\xi }}_1\to {\mathsf{\xi }}_2$. Therefore, from Step 1 to Step 3, the Arzela–Ascoli theorem $\mathsf{\rho }:\mathbb{X}\to \mathbb{X}$ is a completely continuous mapping.

Step 4:

Consider the set

$$\mathfrak{S}=\{\mathrm{z}\in \mathbb{X}|\mathrm{z}=\mathsf{\varphi }\mathsf{\rho }\left(\mathrm{z}\right),0<\mathsf{\varphi }<1\}.$$

We need to show that $\mathfrak{S}$ is bounded. Let $\mathrm{z}\in \mathfrak{S}$. Then, for some $0<\mathfrak{S}<1$ with $\mathrm{z}=\mathsf{\varphi }\mathsf{\rho }\left(\mathrm{z}\right)$, we have

$$\begin{array}{cc}\hfill \left|\mathrm{z}\right(\mathsf{\xi }\left)\right|=& |\mathsf{\varphi }{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\frac{\mathsf{\varphi }(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{\mathsf{\varphi }(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +\mathsf{\varphi }{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{\Delta {\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)\mathrm{v}\left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s}|.\hfill \end{array}$$

This implies

$$\begin{array}{cc}\hfill \left|\mathrm{z}\right(\mathsf{\xi }\left)\right|⩽& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\left|\mathrm{p}\left(\mathrm{s}\right)\right|\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\left|\mathrm{p}\left(\mathrm{s}\right)\right|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\left|\mathrm{p}\left(\mathrm{s}\right)\right|\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{|\Delta |{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)\left|\mathrm{v}\left(\mathrm{s}\right)\right|{\mathrm{d}}_q\mathrm{s}.\hfill \end{array}$$

(20)

By $\left({\mathbf{A}}_{\mathbf{6}}\right)$–$\left({\mathbf{A}}_{\mathbf{8}}\right)$, we have

$$\begin{array}{cc}\hfill \left|\mathrm{p}\right(\mathsf{\xi }\left)\right|& ⩽{\left|\right|\mathrm{p}\left|\right|}_{\mathbb{X}}⩽\left(\frac{{\mathrm{l}}^{*}+{\mathrm{m}}^{*}{\left|\right|\mathrm{z}\left|\right|}_{\mathbb{X}}}{1-{\mathrm{n}}^{*}-{\mathrm{k}}_{*}\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}\right)+\frac{\left({\mathrm{i}}^{*}+{\mathrm{j}}^{*}{\left|\right|\mathrm{z}\left|\right|}_{\mathbb{X}}\right){\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{n}}^{*}-{\mathrm{k}}^{*}\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}=\mathrm{h}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left|\mathrm{v}\right(\mathsf{\xi }\left)\right|& ⩽\frac{{\mathrm{b}}^{*}+{\mathrm{c}}^{*}\mathrm{k}}{1-{\mathrm{e}}^{*}}={\mathrm{h}}^{*}.\hfill \end{array}$$

Now, from Lemma 3 and (20),

$$\begin{array}{cc}\hfill \left|\mathrm{z}\right(\mathsf{\xi }\left)\right|⩽& \mathrm{h}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathrm{d}\mathrm{s}+\frac{\mathrm{h}(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{\mathrm{h}(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\mathrm{h}}^{*}{\int }_0^{\mathsf{\xi }}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{|\Delta |{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right){\mathrm{d}}_q\mathrm{s}\hfill \\ \hfill ⩽& {\mathfrak{Y}}_{\mathsf{\omega }}\left(\mathrm{h}\right)+{\mathfrak{Y}}_{\chi }\left({\mathrm{h}}^{*}\right)=\mathrm{L}.\hfill \end{array}$$

i.e., $\left|\mathrm{z}\right(\mathsf{\xi }\left)\right|⩽\mathrm{L}$. This shows that $\mathfrak{S}$ is bounded. Therefore, by Theorem 3, $\mathsf{\rho }$ has at least one $\mathfrak{FP}$, i.e., (6) has at least one solution.

   ☐

4. Stability

Here, we discuss different types of Ulam stability for (6).

Theorem 6.

If the hypothesis $\left({\mathbf{A}}_{\mathbf{2}}\right)$–$\left({\mathbf{A}}_{\mathbf{5}}\right)$ with (6) holds then (6) is $\mathbb{UHS}$ and $\mathbb{GUHS}$.

Proof. 

Let $\mathrm{x}$ be an approximate solution of (3) and let $\mathrm{z}$ satisfy

$$\left\{\begin{array}{cc}& {}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }(\mathcal{D}+\mathsf{\lambda })\mathrm{z}\left(\mathsf{\xi }\right)=\mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathsf{\xi }\right))\hfill \\ \hfill & +{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s},\mathsf{\xi }\in \mathrm{J}=(0,\mathrm{T}),\alpha \in (1,2],\hfill \\ \hfill & \mathrm{z}\left(0\right)=-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathsf{\eta })}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\mathrm{K}(\mathsf{\eta },\mathrm{z}\left(\mathsf{\eta }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathsf{\eta }\right)){\mathrm{d}}_q\mathsf{\eta },\hfill \\ \hfill & {}^{\mathrm{C}}{\mathcal{D}}_q\mathrm{z}\left(0\right)=0,{}^{\mathrm{C}}{\mathcal{D}}_q^{{\mathsf{\beta }}_0}\mathrm{z}\left(\mathrm{T}\right)=\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{z}\left(\mathsf{\xi }\right){\mathrm{d}}_q\mathsf{\xi }.\hfill \end{array}\right.$$

By Lemma 2, we have

$$\begin{array}{cc}\hfill \mathrm{z}\left(\mathsf{\xi }\right)=& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{\Delta {\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)\mathrm{v}\left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s},\hfill \end{array}$$

where $\mathrm{p},\mathrm{v}\in \mathbb{X}$ are given by

$$\begin{array}{cc}\hfill \mathrm{p}\left(\mathsf{\xi }\right)=& \mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),\mathrm{p}\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),\mathrm{p}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathrm{v}\left(\mathsf{\xi }\right)=& \mathrm{K}(\mathsf{\xi },\mathrm{z}(\mathsf{\xi }),\mathrm{v}(\mathsf{\xi }\left)\right).\hfill \end{array}$$

As x is a solution of (3) and using Remark 2, we obtain

$$\left\{\begin{array}{cc}& {}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }(\mathcal{D}+\mathsf{\lambda })\mathrm{x}\left(\mathsf{\xi }\right)=\mathrm{f}(\mathsf{\xi },\mathrm{x}\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{x}\left(\mathsf{\xi }\right))\hfill \\ \hfill & +{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{x}\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{x}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}+\mathsf{\psi }\left(\mathsf{\xi }\right),\mathsf{\xi }\in \mathrm{J}=(0,\mathrm{T}),\alpha \in (1,2],\hfill \\ \hfill & \mathrm{x}\left(0\right)=-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathsf{\eta })}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\mathrm{K}(\mathsf{\eta },\mathrm{x}\left(\mathsf{\eta }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{x}\left(\mathsf{\eta }\right)){\mathrm{d}}_q\mathsf{\eta },\hfill \\ \hfill & {}^{\mathrm{C}}{\mathcal{D}}_q\mathrm{x}\left(0\right)=0,{}^{\mathrm{C}}{\mathcal{D}}_q^{{\mathsf{\beta }}_0}\mathrm{x}\left(\mathrm{T}\right)=\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{x}\left(\mathsf{\xi }\right){\mathrm{d}}_q\mathsf{\xi }.\hfill \end{array}\right.$$

(21)

Clearly the solution of (21) will be

$$\begin{array}{cc}\hfill \mathrm{x}\left(\mathsf{\xi }\right)=& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathrm{p}\widehat{\left(\mathrm{s}\right)}\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathrm{p}\widehat{\left(\mathrm{s}\right)}\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{p}\widehat{\left(\mathrm{s}\right)}\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathsf{\psi }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathsf{\psi }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathsf{\psi }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{\Delta {\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)\mathrm{v}\widehat{\left(\mathrm{s}\right)}{\mathrm{d}}_q\mathrm{s},\hfill \end{array}$$

where $\widehat{\mathrm{p}},\widehat{\mathrm{v}}\in \mathbb{X}$ are given by

$$\begin{array}{cc}\hfill \mathrm{p}\widehat{\left(\mathsf{\xi }\right)}=& \mathrm{f}(\mathsf{\xi },\mathrm{x}\left(\mathsf{\xi }\right),\mathrm{p}\widehat{\left(\mathsf{\xi }\right)})+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{x}\left(\mathrm{s}\right),\mathrm{p}\widehat{\left(\mathrm{s}\right)}){\mathrm{d}}_q\mathrm{s}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathrm{v}\widehat{\left(\mathsf{\xi }\right)}=& \mathrm{K}(\mathsf{\xi },\mathrm{x}\left(\mathsf{\xi }\right),\mathrm{v}\widehat{\left(\mathsf{\xi }\right)}).\hfill \end{array}$$

For each $\mathsf{\xi }\in \mathrm{J}$, we have

$$\begin{array}{cc}\hfill \left|\mathrm{x}\right(\mathsf{\xi })-\mathrm{z}(\mathsf{\xi }\left)\right|⩽& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }|\mathrm{p}\widehat{\left(\mathrm{s}\right)}-\mathrm{p}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}|\mathrm{p}\widehat{\left(\mathrm{s}\right)}-\mathrm{p}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}|\mathrm{p}\widehat{\left(\mathrm{s}\right)}-\mathrm{p}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\left|\mathsf{\psi }\left(\mathrm{s}\right)\right|\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\left|\mathsf{\psi }\left(\mathrm{s}\right)\right|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\left|\mathsf{\psi }\left(\mathrm{s}\right)\right|\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{|\Delta |{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)|\mathrm{v}\widehat{\left(\mathrm{s}\right)}-\mathrm{v}\left(\mathrm{s}\right)|{\mathrm{d}}_q\mathrm{s}.\hfill \end{array}$$

(22)

By $\left({\mathbf{A}}_{\mathbf{3}}\right)$ and $\left({\mathbf{A}}_{\mathbf{5}}\right)$, we obtain

$$\begin{array}{cc}\hfill |\mathrm{p}\widehat{\left(\mathsf{\xi }\right)}-\mathrm{p}\left(\mathsf{\xi }\right)|⩽& \left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}\right)|\mathrm{x}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|\hfill \\ \hfill & +\frac{{\mathrm{N}}_5{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}|\mathrm{x}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill |\mathrm{v}\widehat{\left(\mathsf{\xi }\right)}-\mathrm{v}\left(\mathsf{\xi }\right)|& ⩽\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)|\mathrm{x}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|.\hfill \end{array}$$

Using (1) of Remark 2 in (22), we obtain

$$\begin{array}{cc}\hfill \left|\mathrm{x}\right(\mathsf{\xi })-\mathrm{z}(\mathsf{\xi }\left)\right|⩽& (\frac{|1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}|{\mathsf{\xi }}^{\alpha }(1-{(1-q)}^{\alpha })}{\mathsf{\lambda }q{\mathsf{\Gamma }}_q(\alpha +1)}+\frac{|(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathsf{\xi }}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}\hfill \\ \hfill & +\sum _{\mathrm{i}=1}^{\mathrm{p}}\frac{|(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_{\mathrm{i}}+2)}\left)\right[\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}\hfill \\ \hfill & +\frac{{\mathrm{N}}_5{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}]|\mathrm{x}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|\hfill \\ \hfill & +(\frac{|1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}|{\mathsf{\xi }}^{\alpha }(1-{(1-q)}^{\alpha })}{\mathsf{\lambda }q{\mathsf{\Gamma }}_q(\alpha +1)}+\frac{|(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathsf{\xi }}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}\hfill \\ \hfill & +\sum _{\mathrm{i}=1}^{\mathrm{p}}\frac{|(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_{\mathrm{i}}+2)})\left|\mathsf{\psi }\left(\mathsf{\xi }\right)\right|\hfill \\ \hfill & +\left[\frac{|1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{|\Delta |q{\mathsf{\Gamma }}_q(\alpha +1)}+\frac{{\mathrm{T}}^{\alpha }(1-{(1-q)}^{\alpha })}{q{\mathsf{\Gamma }}_q\left(\alpha \right)}\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)\right]|\mathrm{x}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|.\hfill \end{array}$$

Since $\mathsf{\xi }\in [0,\mathrm{T}]$ and using Lemma 3, we obtain

$$\begin{array}{cc}\hfill \left|\mathrm{x}\right(\mathsf{\xi })-\mathrm{z}(\mathsf{\xi }\left)\right|⩽& [{\mathfrak{Y}}_{\mathsf{\omega }}\left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}+\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}\right)\hfill \\ \hfill & +{\mathfrak{Y}}_{\chi }\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)]|\mathrm{z}\left(\mathsf{\xi }\right)-\mathrm{x}\left(\mathsf{\xi }\right)|+{\mathfrak{Y}}_{\mathsf{\omega }}\mathsf{ϵ}.\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill \left|\right|\mathrm{x}-{\mathrm{z}\left|\right|}_{\mathbb{X}}⩽\left[\frac{{\mathfrak{Y}}_{\mathsf{\omega }}\mathsf{ϵ}}{1-{\mathfrak{Y}}_{\mathsf{\omega }}\left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}+\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}\right)-{\mathfrak{Y}}_{\chi }\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)}\right].\end{array}$$

This implies

$$\begin{array}{cc}\hfill \left|\right|\mathrm{x}-{\mathrm{z}\left|\right|}_{\mathbb{X}}& ⩽\mathsf{ϵ}{\mathrm{C}}_{\mathrm{f},\mathrm{g}},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\mathrm{C}}_{\mathrm{f},\mathrm{g}}=& \left[\frac{{\mathfrak{Y}}_{\mathsf{\omega }}\mathsf{ϵ}}{1-{\mathfrak{Y}}_{\mathsf{\omega }}\left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}+\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}\right)-{\mathfrak{Y}}_{\chi }\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)}\right].\hfill \end{array}$$

Therefore, (6) is $\mathbb{UHS}$. Furthermore, if we set ${\mathrm{F}}_{\mathrm{f}}\mathsf{ϵ}={\mathrm{C}}_{\mathrm{f}}\mathsf{ϵ},\mathrm{F}\left(0\right)=0$, we see that (6) is $\mathbb{GUHS}$.    ☐

Theorem 7.

If $\left({\mathbf{A}}_{\mathbf{2}}\right)$–$\left({\mathbf{A}}_{\mathbf{9}}\right)$ and (14) hold, then (6) is $\mathbb{UHRS}$, and consequently, it is $\mathbb{GUHRS}$.

Proof. 

Let x be an approximate solution of (5) and let z be the solution of

$$\left\{\begin{array}{cc}& {}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }(\mathcal{D}+\mathsf{\lambda })\mathrm{z}\left(\mathsf{\xi }\right)=\mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathsf{\xi }\right))\hfill \\ \hfill & +{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s},\mathsf{\xi }\in \mathrm{J}=(0,\mathrm{T}),\alpha \in (1,2],\hfill \\ & \mathrm{z}\left(0\right)=-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-q\mathsf{\eta })}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\mathrm{K}(\mathsf{\eta },\mathrm{z}\left(\mathsf{\eta }\right),{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{\alpha }\mathrm{z}\left(\mathsf{\eta }\right)){\mathrm{d}}_q\mathsf{\eta },\hfill \\ \hfill & {}^{\mathrm{C}}{\mathcal{D}}_q\mathrm{z}\left(0\right)=0,{}^{\mathrm{C}}{\mathcal{D}}_q^{{\mathsf{\beta }}_0}\mathrm{z}\left(\mathrm{T}\right)=\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{}^{\mathrm{C}}{\mathcal{D}}_{\mathit{q}}^{{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{z}\left(\mathsf{\xi }\right){\mathrm{d}}_q\mathsf{\xi }.\hfill \end{array}\right.$$

By Lemma 2, we have

$$\begin{array}{cc}\hfill \mathrm{z}\left(\mathsf{\xi }\right)=& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{\Delta }\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\mathrm{p}\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{\Delta {\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)\mathrm{v}\left(\mathrm{s}\right){\mathrm{d}}_q\mathrm{s},\hfill \end{array}$$

where $\mathrm{p},\mathrm{v}\in \mathbb{X}$ are given by

$$\begin{array}{cc}\hfill \mathrm{p}\left(\mathsf{\xi }\right)=& \mathrm{f}(\mathsf{\xi },\mathrm{z}\left(\mathsf{\xi }\right),\mathrm{p}\left(\mathsf{\xi }\right))+{\int }_0^{\mathsf{\xi }}\frac{{(\mathsf{\xi }-q\mathrm{s})}^{\mathsf{\sigma }-1}}{{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\mathrm{g}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right),\mathrm{p}\left(\mathrm{s}\right)){\mathrm{d}}_q\mathrm{s}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathrm{v}\left(\mathsf{\xi }\right)=& \mathrm{K}(\mathsf{\xi },\mathrm{z}(\mathsf{\xi }),\mathrm{v}(\mathsf{\xi }\left)\right).\hfill \end{array}$$

Using the proof procedure of Theorem 6, for each $\mathsf{\xi }\in \mathrm{J}$, we obtain

$$\begin{array}{cc}\hfill \left|\mathrm{x}\right(\mathsf{\xi })-\mathrm{z}(\mathsf{\xi }\left)\right|⩽& {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }|\mathrm{p}\widehat{\left(\mathrm{s}\right)}-\mathrm{p}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}|\mathrm{p}\widehat{\left(\mathrm{s}\right)}-\mathrm{p}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}|\mathrm{p}\widehat{\left(\mathrm{s}\right)}-\mathrm{p}\left(\mathrm{s}\right)|\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & {\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha }\left|\mathsf{\psi }\left(\mathrm{s}\right)\right|\mathrm{d}\mathrm{s}+\frac{(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}{\int }_0^{\mathrm{T}}{\mathrm{e}}^{-\mathsf{\lambda }(\mathrm{T}-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_0}\left|\mathsf{\psi }\left(\mathrm{s}\right)\right|\mathrm{d}\mathrm{s}\hfill \\ \hfill & +\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})}{|\Delta |}\sum _{\mathrm{i}=1}^{\mathrm{p}}{\int }_0^{\mathrm{T}}{\int }_0^{\mathsf{\xi }}{\mathrm{e}}^{-\mathsf{\lambda }(\mathsf{\xi }-\mathrm{s})}{\mathrm{I}}_q^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}}\left|\mathsf{\psi }\left(\mathrm{s}\right)\right|\mathrm{d}\mathrm{s}{\mathrm{d}}_q\mathsf{\xi }\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\left(\frac{(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}){(\mathrm{T}-q\mathrm{s})}^{\alpha -{\mathsf{\beta }}_0-1}}{|\Delta |{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0)}-\frac{{(\mathrm{T}-q\mathrm{s})}^{\alpha -1}}{{\mathsf{\Gamma }}_q\left(\alpha \right)}\right)|\mathrm{v}\widehat{\left(\mathrm{s}\right)}-\mathrm{v}\left(\mathrm{s}\right)|{\mathrm{d}}_q\mathrm{s}.\hfill \end{array}$$

(23)

By $\left({\mathbf{A}}_{\mathbf{3}}\right)$ and $\left({\mathbf{A}}_{\mathbf{5}}\right)$, we obtain

$$\begin{array}{cc}\hfill |\mathrm{p}\widehat{\left(\mathsf{\xi }\right)}-\mathrm{p}\left(\mathsf{\xi }\right)|⩽& \left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}\right)|\mathrm{x}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|\hfill \\ \hfill & +\frac{{\mathrm{N}}_5{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}|\mathrm{x}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill |\mathrm{v}\widehat{\left(\mathsf{\xi }\right)}-\mathrm{v}\left(\mathsf{\xi }\right)|& ⩽\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)|\mathrm{x}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|.\hfill \end{array}$$

Using the last two inequalities and part (1) of Remark 3 in (23), we have

$$\begin{array}{cc}\hfill & \left|\mathrm{x}\right(\mathsf{\xi })-\mathrm{z}(\mathsf{\xi }\left)\right|\hfill \\ \hfill ⩽& (\frac{|1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}|{\mathsf{\xi }}^{\alpha }(1-{(1-q)}^{\alpha })}{\mathsf{\lambda }q{\mathsf{\Gamma }}_q(\alpha +1)}+\frac{|(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathsf{\xi }}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}\hfill \\ \hfill & +\sum _{\mathrm{i}=1}^{\mathrm{p}}\frac{|(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_{\mathrm{i}}+2)}\left)\right[\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}\hfill \\ \hfill & +\frac{{\mathrm{N}}_5{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathsf{\xi }}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}]|\mathrm{x}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|\hfill \\ \hfill & +(\frac{|1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}|{\mathsf{\xi }}^{\alpha }(1-{(1-q)}^{\alpha })}{\mathsf{\lambda }q{\mathsf{\Gamma }}_q(\alpha +1)}+\frac{|(\mathsf{\lambda }\mathsf{\xi }-1+{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{T}})|{\mathsf{\xi }}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_0+1)}\hfill \\ \hfill & +\sum _{\mathrm{i}=1}^{\mathrm{p}}\frac{|(1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})\left|\right|(1-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }})|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_{\mathrm{i}}+1})}{\mathsf{\lambda }|\Delta |q{\mathsf{\Gamma }}_q(\alpha -{\mathsf{\beta }}_{\mathrm{i}}+2)})\left|\mathsf{\psi }\left(\mathsf{\xi }\right)\right|\hfill \\ \hfill & +[\frac{|1-\mathsf{\lambda }\mathsf{\xi }-{\mathrm{e}}^{-\mathsf{\lambda }\mathsf{\xi }}|{\mathrm{T}}^{\alpha -{\mathsf{\beta }}_0}(1-{(1-q)}^{\alpha -{\mathsf{\beta }}_0})}{|\Delta |q{\mathsf{\Gamma }}_q(\alpha +1)}\hfill \\ \hfill & +\frac{{\mathrm{T}}^{\alpha }(1-{(1-q)}^{\alpha })}{q{\mathsf{\Gamma }}_q\left(\alpha \right)}\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)]|\mathrm{x}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|.\hfill \end{array}$$

Since $\mathsf{\xi }\in [0,\mathrm{T}]$, using Lemma 3, we obtain

$$\begin{array}{cc}\hfill \left|\mathrm{x}\right(\mathsf{\xi })-\mathrm{z}(\mathsf{\xi }\left)\right|⩽& [{\mathfrak{Y}}_{\mathsf{\omega }}\left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}+\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}\right)\hfill \\ \hfill & +{\mathfrak{Y}}_{\chi }\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)]|\mathrm{x}\left(\mathsf{\xi }\right)-\mathrm{z}\left(\mathsf{\xi }\right)|+{\mathfrak{Y}}_{\mathsf{\omega }}\mathsf{ϵ}{\mathfrak{L}}_{\mathsf{\psi }}\mathsf{\psi }\left(\mathsf{\xi }\right).\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill \left|\right|\mathrm{x}-{\mathrm{z}\left|\right|}_{\mathbb{X}}⩽\left[\frac{{\mathfrak{Y}}_{\mathsf{\omega }}\mathsf{ϵ}{\mathfrak{L}}_{\mathsf{\psi }}\mathsf{\psi }\left(\mathsf{\xi }\right)}{1-{\mathfrak{Y}}_{\mathsf{\omega }}\left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}+\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}\right)-{\mathfrak{Y}}_{\chi }\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)}\right];\end{array}$$

i.e.,

$$\begin{array}{cc}\hfill \left|\right|\mathrm{x}-{\mathrm{z}\left|\right|}_{\mathbb{X}}& ⩽{\mathrm{C}}_{\mathrm{f},\mathrm{g}}\mathsf{ϵ},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\mathrm{C}}_{\mathrm{f},\mathrm{g}}=& \left[\frac{{\mathfrak{Y}}_{\mathsf{\omega }}{\mathfrak{L}}_{\mathsf{\psi }}\mathsf{\psi }\left(\mathsf{\xi }\right)}{1-{\mathfrak{Y}}_{\mathsf{\omega }}\left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}+\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}\right)-{\mathfrak{Y}}_{\chi }\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)}\right].\hfill \end{array}$$

Therefore, (6) is $\mathbb{UHRS}$. Similarly, we can obtain the $\mathbb{GUHRS}$ of the considered problem.    ☐

5. Examples

In this section, we investigate our theoretical results by two examples.

Example 1.

Take the q-$\mathbb{FDE}$

$$\left\{\begin{array}{cc}\hfill & {}^{\mathrm{C}}{\mathcal{D}}_{\frac{1}{4}}^{\frac{3}{2}}(\mathcal{D}-\frac{9}{10})z\left(\mathsf{\xi }\right)=\frac{5+\left|z\left(\mathsf{\xi }\right)\right|+|{}^{\mathrm{C}}{\mathcal{D}}_{\frac{1}{4}}^{\frac{3}{2}}z\left(\mathsf{\xi }\right)|}{64(4+|z\left(\mathsf{\xi }\right)|+|{}^{\mathrm{C}}{\mathcal{D}}_{\frac{1}{4}}^{\frac{3}{2}}z\left(\mathsf{\xi }\right)\left|\right)}\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\frac{{(\mathsf{\xi }-\frac{1}{4}\mathrm{s})}^{\frac{5}{2}-1}}{{\mathsf{\Gamma }}_{\frac{1}{4}}\left(\frac{5}{2}\right)}\left(\frac{\mathrm{s}sin\left|z\right(\mathrm{s}\left)\right|+\mathrm{s}\mathrm{i}\mathrm{n}|{}^{\mathrm{C}}{\mathcal{D}}_{\frac{1}{4}}^{\frac{3}{2}}z\left(\mathrm{s}\right)|}{50}\right){\mathrm{d}}_{\frac{1}{4}}\mathrm{s},\mathsf{\xi }\in \mathrm{J}=(0,\mathrm{T}),\hfill \\ \hfill & z\left(0\right)=-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-\frac{1}{4}\mathrm{s})}^{\frac{3}{2}-1}}{{\mathsf{\Gamma }}_{\frac{1}{4}}\left(\frac{3}{2}\right)}\left(\frac{\mathrm{s}sin\left|z\right(\mathrm{s}\left)\right|+\mathrm{s}\mathrm{i}\mathrm{n}|{}^{\mathrm{C}}{\mathcal{D}}_{\frac{1}{4}}^{\frac{3}{2}}z\left(\mathrm{s}\right)|}{50}\right){\mathrm{d}}_{\frac{1}{4}}\mathrm{s},\hfill \\ \hfill & {}^{\mathrm{C}}{\mathcal{D}}_{\frac{1}{4}}z\left(0\right)=0,\hfill \\ & {}^{\mathrm{C}}{\mathcal{D}}_{\frac{1}{4}}^{{\mathsf{\beta }}_0}z\left(\mathrm{T}\right)=\sum _{\mathrm{i}=1}^2{\int }_0^{\mathrm{T}}{}^{\mathrm{C}}{\mathcal{D}}_{\frac{1}{4}}^{{\mathsf{\beta }}_{\mathrm{i}}}z\left(\mathsf{\xi }\right){\mathrm{d}}_{\frac{1}{4}}\mathsf{\xi },\hfill \end{array}\right.$$

(24)

where $\mathit{q}=\frac{1}{4}$, $\alpha =\frac{3}{2}$, $\mathsf{\delta }=\mathsf{\sigma }=\frac{5}{2}$, ${\mathsf{\beta }}_1=\frac{7}{6}$, ${\mathsf{\beta }}_2=\frac{1}{6}$, ${\mathsf{\beta }}_0=\frac{1}{3}$, $\mathsf{\lambda }=\frac{-9}{10}$, $\mathrm{T}=\pi$ and $\mathrm{p}=2$. Set

$$\mathrm{f}(\mathsf{\xi },{\mathsf{\zeta }}^{\prime },\mathsf{\theta })=\frac{5+|{\mathsf{\zeta }}^{\prime }|+|\mathsf{\theta }|}{64(4+|{\mathsf{\zeta }}^{\prime }|+|\mathsf{\theta }\left|\right)},{\mathsf{\zeta }}^{\prime }\in \mathfrak{C}(\mathrm{J},\mathbb{R})$$

$$\mathrm{g}(\mathsf{\xi },{\mathsf{\zeta }}^{\prime },\mathsf{\theta })=\frac{\mathsf{\xi }\mathrm{s}\mathrm{i}\mathrm{n}|{\mathsf{\zeta }}^{\prime }|+\mathrm{s}\mathrm{i}\mathrm{n}|\mathsf{\theta }|}{50}$$

and

$$\mathrm{K}(\mathsf{\xi },{\mathsf{\zeta }}^{\prime },\mathsf{\theta })=\frac{\mathsf{\xi }\mathrm{s}\mathrm{i}\mathrm{n}|{\mathsf{\zeta }}^{\prime }|+\mathrm{s}\mathrm{i}\mathrm{n}|\mathsf{\theta }|}{50}.$$

Clearly, the functions $\mathrm{f},\mathrm{g},\mathrm{K}$ are continuous. For each ${\mathsf{\zeta }}^{\prime },\widehat{{\mathsf{\zeta }}^{\prime }}\in \mathbb{X}$, $\mathsf{\theta },\widehat{\mathsf{\theta }}\in \mathbb{R}$, $\mathsf{\xi }\in [0,\pi ]$, we have

$$|\mathrm{f}(\mathsf{\xi },{\mathsf{\zeta }}^{\prime },\mathsf{\theta })-\mathrm{f}(\mathsf{\xi },\widehat{{\mathsf{\zeta }}^{\prime }},\widehat{\mathsf{\theta }})|⩽\frac{1}{64}\left(5+|{\mathsf{\zeta }}^{\prime }|+|\mathsf{\theta }|\right),$$

which satisfies $\left({\mathbf{A}}_{\mathbf{3}}\right)$, ${\mathrm{N}}_1=\frac{1}{64}$, ${\mathrm{N}}_2=\frac{5}{64}$. We can observe that

$$|\mathrm{g}(\mathsf{\xi },{\mathsf{\zeta }}^{\prime },\mathsf{\theta })-\mathrm{g}(\mathsf{\xi },\widehat{{\mathsf{\zeta }}^{\prime }},\widehat{\mathsf{\theta }})|⩽\frac{|{\mathsf{\zeta }}^{\prime }-\widehat{{\mathsf{\zeta }}^{\prime }}|+|\mathsf{\theta }-\widehat{\mathsf{\theta }}|}{50},$$

which satisfies $\left({\mathbf{A}}_{\mathbf{4}}\right)$ and ${\mathrm{N}}_5$=${\mathrm{N}}_6$=$\frac{1}{50}$, and

$$|\mathrm{K}(\mathsf{\xi },{\mathsf{\zeta }}^{\prime },\mathsf{\theta })-\mathrm{K}(\mathsf{\xi },\widehat{{\mathsf{\zeta }}^{\prime }},\widehat{\mathsf{\theta }})|⩽\frac{|{\mathsf{\zeta }}^{\prime }-\widehat{{\mathsf{\zeta }}^{\prime }}|+|\mathsf{\theta }-\widehat{\mathsf{\theta }}|}{50}$$

satisfies $\left({\mathbf{A}}_{\mathbf{5}}\right)$, ${\mathrm{N}}_3$=${\mathrm{N}}_4$=$\frac{1}{50}$. We can see that

$$\begin{array}{cc}& \left[{\mathfrak{Y}}_{\mathsf{\omega }}\left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}+\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}\right)+{\mathfrak{Y}}_{\chi }\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)\right]\hfill \\ \hfill & \approx 0.976974<1.\hfill \end{array}$$

We can see that all the required conditions of Theorem 4 are satisfied; hence, (24) has at least one solution. By letting $\mathsf{\psi }\left(\mathsf{\xi }\right)=\mathsf{\xi }$ for all $\mathsf{\xi }\in \mathrm{J}$, we have

$${\mathrm{I}}_{\frac{1}{4}}^{\frac{3}{2}}\mathsf{\psi }\left(\mathsf{\xi }\right)=\frac{1}{{\mathsf{\Gamma }}_q\left(\frac{3}{2}\right)}{\int }_0^{\mathsf{\xi }}{(\mathsf{\xi }-q\mathrm{s})}^{\frac{3}{2}-1}\mathrm{s}{\mathrm{d}}_q\mathrm{s}=\frac{8{\pi }^{\frac{5}{2}}}{15\sqrt{\pi }}⩽2\mathsf{\xi }.$$

Hence $\left({\mathbf{A}}_{\mathbf{9}}\right)$ is satisfied with ${\mathfrak{L}}_{\mathsf{\psi }}=2$. Therefore, by Theorem 4.2, the given problem is $\mathbb{UHRS}$ and consequently $\mathbb{GUHRS}$.

Example 2.

Assume that q-$\mathbb{FDE}$:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {}^{\mathrm{C}}{\mathcal{D}}_{\frac{2}{3}}^{\frac{3}{2}}(\mathcal{D}-\frac{4}{5})z\left(\mathsf{\xi }\right)=\frac{3+\left|z\left(\mathsf{\xi }\right)\right|+|{}^{\mathrm{C}}{\mathcal{D}}_{\frac{2}{3}}^{\frac{3}{2}}z\left(\mathsf{\xi }\right)|}{21{\mathrm{e}}^{(\mathsf{\xi }+1)}(1+|z\left(\mathsf{\xi }\right)|+|{}^{\mathrm{C}}{\mathcal{D}}_{\frac{2}{3}}^{\frac{3}{2}}z\left(\mathsf{\xi }\right)\left|\right)}\hfill \\ \hfill & +{\int }_0^{\mathrm{T}}\frac{{(\mathsf{\xi }-\frac{2}{3}\mathrm{s})}^{\frac{7}{4}-1}}{{\mathsf{\Gamma }}_{\frac{2}{3}}\left(\frac{7}{4}\right)}\left(\frac{8+\left|z\left(\mathsf{\xi }\right)\right|+|{}^{\mathrm{C}}{\mathcal{D}}_{\frac{2}{3}}^{\frac{3}{2}}z\left(\mathsf{\xi }\right)|}{150{\mathrm{e}}^{\mathsf{\xi }+3}(1+|z\left(\mathsf{\xi }\right)|+|{}^{\mathrm{C}}{\mathcal{D}}_{\frac{2}{3}}^{\frac{3}{2}}z\left(\mathsf{\xi }\right)\left|\right)}\right),\mathsf{\xi }\in \mathrm{J}=(0,\mathrm{T}),\hfill \\ \hfill & z\left(0\right)=-{\int }_0^{\mathrm{T}}\frac{{(\mathrm{T}-\frac{2}{3}\mathrm{s})}^{\frac{3}{2}-1}}{{\mathsf{\Gamma }}_{\frac{2}{3}}\left(\frac{3}{2}\right)}\left(\frac{8+\left|z\left(\mathsf{\xi }\right)\right|+|{}^{\mathrm{C}}{\mathcal{D}}_{\frac{2}{3}}^{\frac{3}{2}}z\left(\mathsf{\xi }\right)|}{150{\mathrm{e}}^{(\mathsf{\xi }+3)}(1+|z\left(\mathsf{\xi }\right)|+|{}^{\mathrm{C}}{\mathcal{D}}_{\frac{2}{3}}^{\frac{3}{2}}z\left(\mathsf{\xi }\right)\left|\right)}\right){\mathrm{d}}_{\frac{2}{3}}\mathrm{s},\hfill \\ \hfill & {}^{\mathrm{C}}{\mathcal{D}}_{\frac{2}{3}}z\left(0\right)=0,\hfill \\ & {}^{\mathrm{C}}{\mathcal{D}}_{\frac{2}{3}}^{{\mathsf{\beta }}_0}z\left(\mathrm{T}\right)=\sum _{\mathrm{i}=1}^3{\int }_0^{\mathrm{T}}{}^{\mathrm{C}}{\mathcal{D}}_{\frac{2}{3}}^{{\mathsf{\beta }}_{\mathrm{i}}}z\left(\mathsf{\xi }\right){\mathrm{d}}_{\frac{2}{3}}\mathsf{\xi },\hfill \end{array}\right.\end{array}$$

(25)

where $\mathit{q}=\frac{2}{3}$, $\alpha =\frac{3}{2}$, $\mathsf{\delta }=\mathsf{\sigma }=\frac{7}{4}$, ${\mathsf{\beta }}_1=\frac{1}{8}$, ${\mathsf{\beta }}_2=\frac{5}{16}$, ${\mathsf{\beta }}_3=\frac{3}{16}$, ${\mathsf{\beta }}_0=\frac{1}{16}$, $\mathsf{\lambda }=\frac{-4}{5}$, $\mathrm{T}=1$ and $\mathrm{p}=3.$ Set

$$\mathrm{f}(\mathsf{\xi },{\mathsf{\zeta }}^{\prime },\mathsf{\theta })=\frac{3+|{\mathsf{\zeta }}^{\prime }|+|\mathsf{\theta }|}{21{\mathrm{e}}^{(\mathsf{\xi }+1)}(1+|{\mathsf{\zeta }}^{\prime }|+|\mathsf{\theta }\left|\right)},{\mathsf{\zeta }}^{\prime }\in \mathfrak{C}(\mathrm{J},\mathbb{R})$$

$$\mathrm{g}(\mathsf{\xi },{\mathsf{\zeta }}^{\prime },\mathsf{\theta })=\frac{8+|{\mathsf{\zeta }}^{\prime }|+|\mathsf{\theta }|}{150{\mathrm{e}}^{(\mathsf{\xi }+3)}(1+|{\mathsf{\zeta }}^{\prime }|+|\mathsf{\theta }\left|\right)}$$

and

$$\mathrm{K}(\mathsf{\xi },{\mathsf{\zeta }}^{\prime },\mathsf{\theta })=\frac{8+|{\mathsf{\zeta }}^{\prime }|+|\mathsf{\theta }|}{150{\mathrm{e}}^{(\mathsf{\xi }+3)}(1+|{\mathsf{\zeta }}^{\prime }|+|\mathsf{\theta }\left|\right)}.$$

Clearly, the functions $\mathrm{f},\mathrm{g},\mathrm{K}$ are continuous. For each ${\mathsf{\zeta }}^{\prime },\widehat{{\mathsf{\zeta }}^{\prime }}\in \mathbb{X}$, $\mathsf{\theta },\widehat{\mathsf{\theta }}\in \mathbb{R}$, $\mathsf{\xi }\in [0,1]$, we have

$$|\mathrm{f}(\mathsf{\xi },{\mathsf{\zeta }}^{\prime },\mathsf{\theta })-\mathrm{f}(\mathsf{\xi },\widehat{{\mathsf{\zeta }}^{\prime }},\widehat{\mathsf{\theta }})|⩽\frac{|{\mathsf{\zeta }}^{\prime }-\widehat{{\mathsf{\zeta }}^{\prime }}|+|\mathsf{\theta }-\widehat{\mathsf{\theta }}|}{21{\mathrm{e}}^1},$$

which satisfies $\left({\mathbf{A}}_{\mathbf{3}}\right)$ and ${\mathrm{N}}_1={\mathrm{N}}_2=\frac{1}{21{\mathrm{e}}^1}$. We can observe that

$$|\mathrm{g}(\mathsf{\xi },{\mathsf{\zeta }}^{\prime },\mathsf{\theta })-\mathrm{g}(\mathsf{\xi },\widehat{{\mathsf{\zeta }}^{\prime }},\widehat{\mathsf{\theta }})|⩽\frac{|{\mathsf{\zeta }}^{\prime }-\widehat{{\mathsf{\zeta }}^{\prime }}|+|\mathsf{\theta }-\widehat{\mathsf{\theta }}|}{150{\mathrm{e}}^3}$$

which satisfies $\left({\mathbf{A}}_{\mathbf{4}}\right)$ and ${\mathrm{N}}_5$=${\mathrm{N}}_6$=$\frac{1}{150{\mathrm{e}}^3}$, and

$$|\mathrm{K}(\mathsf{\xi },{\mathsf{\zeta }}^{\prime },\mathsf{\theta })-\mathrm{K}(\mathsf{\xi },\widehat{{\mathsf{\zeta }}^{\prime }},\widehat{\mathsf{\theta }})|⩽\frac{|{\mathsf{\zeta }}^{\prime }-\widehat{{\mathsf{\zeta }}^{\prime }}|+|\mathsf{\theta }-\widehat{\mathsf{\theta }}|}{150{\mathrm{e}}^3}$$

satisfies $\left({\mathbf{A}}_{\mathbf{5}}\right)$, ${\mathrm{N}}_3$=${\mathrm{N}}_4$=$\frac{1}{150{\mathrm{e}}^3}$. Additionally, we check that

$$\begin{array}{cc}& \left|\left[{\mathfrak{Y}}_{\mathsf{\omega }}\left(\frac{{\mathrm{N}}_1}{1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}}+\frac{{\mathrm{N}}_5{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)\left(1-{\mathrm{N}}_2-{\mathrm{N}}_6\frac{{\mathrm{T}}^{\mathsf{\sigma }}(1-{(1-q)}^{\mathsf{\sigma }})}{{\lceil \mathsf{\sigma }\rceil }_q{\mathsf{\Gamma }}_q\left(\mathsf{\delta }\right)}\right)}\right)+{\mathfrak{Y}}_{\chi }\left(\frac{{\mathrm{N}}_3}{1-{\mathrm{N}}_4}\right)\right]\right|\hfill \\ \hfill & \approx 0.0315193<1.\hfill \end{array}$$

We see that all the required conditions of Theorem 4 are satisfied; hence, (25) has at least one solution. By letting $\mathsf{\psi }\left(\mathsf{\xi }\right)={\mathsf{\xi }}^2$ for all $\mathsf{\xi }\in \mathrm{J}$, we have

$${\mathrm{I}}_{\frac{2}{3}}^{\frac{3}{2}}\mathsf{\psi }\left({\mathsf{\xi }}^2\right)=\frac{1}{{\mathsf{\Gamma }}_q\left(\frac{3}{2}\right)}{\int }_0^{\mathsf{\xi }}{(\mathsf{\xi }-q\mathrm{s})}^{\frac{3}{2}-1}{\mathrm{s}}^2{\mathrm{d}}_q\mathrm{s}=\frac{{32}^{\frac{7}{2}}}{105\sqrt{\pi }}⩽\frac{2}{\sqrt{\pi }}{\mathsf{\xi }}^2.$$

Hence, $\left({\mathbf{A}}_{\mathbf{9}}\right)$ is satisfied with ${\mathfrak{L}}_{\mathsf{\psi }}=\frac{2}{\sqrt{\pi }}$. Therefore, by Theorem 4.2, the given problem is $\mathbb{UHRS}$ and consequently $\mathbb{GUHRS}$.

6. Conclusions

In this article, we present some necessary conditions for obtaining the existence, uniqueness, and Ulam stabilities of a solution to System (1). Schaefer’s $\mathfrak{FP}$ theorem, the Arzela–Ascoli theorem, and the Banach contraction principle are all used to show the desired results of (1). In this way, the Ulam stability results are shown to investigate the solution of our proposed system (1) under specific assumptions and conditions. We conclude by providing an interesting example to implement the theoretical results and demonstrate its validity. From the derived results, we also conclude that such a method is powerful for solving Langevin q-$\mathbb{FDE}$. In our forthcoming work, we consider the inclusion version problem with $\mathfrak{R}\mathcal{L}$ derivatives.