Existence Results for Caputo Tripled Fractional Differential Inclusions with Integral and Multi-Point Boundary Conditions

Abstract

In this study, based on Coitz and Nadler’s fixed point theorem and the non-linear alternative for Kakutani maps, existence results for a tripled system of sequential fractional differential inclusions (SFDIs) with integral and multi-point boundary conditions (BCs) in investigated. A practical examples are given to illustrate the obtained the theoretical results.

1. Introduction

FCs ware first employed in 1695 when L’Hopital summarized his discoveries in a letter to Leibniz. Fractional calculus (FCs) was studied by several twentieth century authors, including Liouville, Grunwald, Letnikov, and Riemann. This field of mathematics, known as fractional differential equations, was invented by mathematicians as a pure branch of mathematics with just a few applications in mathematics. Fractional calculus is a well-established subject with applications in many applied sciences, such as visco-elasticity, medical, and environment, which leads the fractional differential equations to become extremely prevalent. We recommend the monographs [1,2,3,4,5,6,7,8] and the recently mentioned papers [9,10,11,12,13,14,15,16,17]. It is worth noting that most of the works in the field of fractional differentiation focus mostly on R-L and Caputo types. See [18,19,20,21,22].

In 1772, Russian scientists presented a general idea of stability, such as Lyapunov (1758–1817), where the general theme of his doctoral dissertation was movement stability, and his work soon spread all over Russia and later in the West. With the process of research, scientists entered the time delay, and the first to describe these systems with a time delay was the scientist (Boltzman), who studied its effect but did not refer to the time delay in realistic models.

In the early 1900s, a disagreement arose over the necessity of introducing time delays into systems to predict their future development, but this point of view contradicted the Newtonian traditions, which claimed that knowledge of the current values of all relevant variables should suffice for the prediction. Ulam and Hyers, on the other hand, recognized unknown types of stability known as ulam-stability. Hyer’s type of stability study contributes expressively to our understanding of population dynamics and fluid movement, see [23].

In mathematics, differential inclusions relate to one or more functions and their derivatives. In applications, functions generally represent physical quantities, derivatives represent their rates of change, and differential inclusion defines the relationship between the two. Because these relationships are so common, differential equations play a prominent role in many disciplines, including engineering, physics, economics, and biology. The study of differential inclusions mainly consists of studying their solutions (the set of functions that satisfy the equation), and the properties of their solutions. The simplest differential inclusions can be solved by explicit formulas. However, many properties of solutions to particular differential inclusions may be determined without being exactly calculated. If a closed expression is not available for the solutions, the solutions may be numerically approximated using computers. Dynamical systems theory focuses on the qualitative analysis of systems described by differential equations and differential inclusions, while many numerical methods have been developed to determine solutions with a certain degree of precision.

Many of the basic laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations developed first with the sciences in which the equations originated and where the results were put into practice. However, various problems which sometimes arise in quite distinct scientific fields may result in identical differential equations. Whenever this happens, the mathematical theory behind the equations can be seen as a unifying principle behind the various phenomena. For example, consider the propagation of light and sound in the atmosphere, and the waves on the surface of a pond. They can all be described by the same second-order partial differential equation, which is the wave equation, that allows us to think of light and sound as forms of waves, much like the familiar waves in water. Heat conduction, developed by Joseph Fourier, is governed by a second-order partial differential equation, the heat equation. It turns out that many diffusion processes, though apparently different, are described by the same equation; the Black–Scholes equation in finance, for example, is related to the heat equation [24,25]. In [26], the authors were the first who developed the idea of the tripled fixed points. Karakaya et al. [27] introduce tripled fixed points for a class of condensing operators in Banach spaces. In [25], the authors studied the existence results for the following BVP.

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{\mathcal{D}}^{{\psi }_k}{\mathcal{Z}}_k\left(\tau \right)={\mathfrak{f}}_k(\tau ,\mathcal{Z}\left(\tau \right)),1<{\psi }_k\le 2,\hfill \\ {\mathcal{Z}}_k^{\left(j\right)}\left(0\right)=a_{k,j}{\mathcal{Z}}_{\epsilon \left(k\right)}^{\left(j\right)}\left(\mathcal{T}\right),k=1,2,3;j=0,1.\hfill \end{array}\right.\end{array}\end{array}$$

where ${}^c{\mathcal{D}}_0^{{\psi }_k}$ denotes the Caputo fractional derivatives (CFDs) of order ${\psi }_k$, $\tau \in J=[0,\mathcal{T}],{\mathfrak{f}}_k:J\times {\mathcal{R}}_e^3\to {\mathcal{R}}_e$ are continuous functions, $\mathcal{Z}=({\mathcal{Z}}_1,{\mathcal{Z}}_2,{\mathcal{Z}}_3)\in {\mathcal{R}}_e^3,\epsilon =(1,2,3)$ is a cyclic permutation, and $a_{k,j}\in k=1,2,3,j=0,1.$ In this work, motivated by [28], we consider the following system of sequential fractional differential inclusions:

$$\left\{\begin{array}{c}{(}^c{\mathfrak{D}}^{\psi }+{\phi }^c{\mathfrak{D}}^{\psi -1})\mathcal{Z}\left(\varpi \right)\in {\mathcal{F}}_1(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right)),2<\psi \le 3,\hfill \\ {(}^c{\mathfrak{D}}^{\varphi }+{\phi }^c{\mathfrak{D}}^{\varphi -1})\mathcal{Q}\left(\varpi \right)\in {\mathcal{F}}_2(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right)),2<\varphi \le 3,\hfill \\ {(}^c{\mathfrak{D}}^{\omega }+{\phi }^c{\mathfrak{D}}^{\omega -1})\mathcal{Y}\left(\varpi \right)\in {\mathcal{F}}_3(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right)),3<\omega \le 4,\hfill \\ {\displaystyle \mathcal{Z}\left(0\right)=0,{\mathcal{Z}}^{\prime }\left(0\right)=0,\mathcal{Z}\left(\mathcal{T}\right)={\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j\mathcal{Q}\left({\zeta }_j\right)+{\mathrm{\Pi }}_1{\mathcal{I}}^{\varsigma }\mathcal{Q}\left(\vartheta \right),}\hfill \\ {\displaystyle \mathcal{Q}\left(0\right)=0,{\mathcal{Q}}^{\prime }\left(0\right)=0,\mathcal{Q}\left(\mathcal{T}\right)={\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j\mathcal{Y}\left({\zeta }_j\right)+{\mathrm{\Pi }}_2{\mathcal{I}}^{\varrho }\mathcal{Y}\left(\vartheta \right),}\hfill \\ {\displaystyle \mathcal{Y}\left(0\right)=0,{\mathcal{Y}}^{\prime }\left(0\right)=0,{\mathcal{Y}}^{″}\left(0\right)=0,\mathcal{Y}\left(\mathcal{T}\right)={\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j\mathcal{Z}\left({\zeta }_j\right)+{\mathrm{\Pi }}_3{\mathcal{I}}^{\delta }\mathcal{Z}\left(\vartheta \right),}\hfill \end{array}\right.$$

(1)

where ${}^c{D}^{\chi }$ is a CFDs of order $\chi \in \{\psi ,\varphi ,\omega \}$,${\mathcal{F}}_1,{\mathcal{F}}_2,{\mathcal{F}}_3:[0,\mathcal{T}]\times {\mathcal{R}}_e\times {\mathcal{R}}_e\times {\mathcal{R}}_e\to \mathcal{P}\left({\mathcal{R}}_e\right)$ are given continuous functions, $\mathcal{P}\left({\mathcal{R}}_e\right)$ is the family of all non-empty subset of ${\mathcal{R}}_e$, ${\zeta }_j\in {\mathcal{R}}_e,j=1,\dots ,k-2$ and $\vartheta \in [0,\mathcal{T}]$.

The Caputo SFDEs with multi-point and integral boundary conditions discussed in this work are the most widely used Caputo fractional derivatives. The novelty and originality of this work is summarized by using Covitz and Nadler’s fixed point theorem and the non-linear alternative for Kakutani maps in showing the existence results for a tripled system of sequential fractional differential inclusions.

Preliminaries are introduced in the second section, main results are shown in the third section. Finally, in Section 4, we give some numerical examples to show the effectiveness of the obtained theoretical results.

2. Preliminaries

This portion introduces basic fractional calculus concepts, definitions, and tentative results [1,2,3].

Let $\widehat{\mathcal{J}}=\mathcal{C}([0,\mathcal{T}],{\mathcal{R}}_e)$ be a Banach space endowed with the norm $\parallel \mathcal{Z}\parallel =sup\left\{\right|\mathcal{Z}\left(\varpi \right)|,\varpi \in [0,\mathcal{T}\left]\right\}$. Then $(\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}},\parallel (\mathcal{Z},\mathcal{Q},\mathcal{Y}){\parallel }_{\widehat{\mathcal{J}}})$ is also a Banach space equipped with the norm ${\parallel (\mathcal{Z},\mathcal{Q},\mathcal{Y})\parallel }_{\widehat{\mathcal{J}}}=\parallel \mathcal{Z}\parallel +\parallel \mathcal{Q}\parallel +\parallel \mathcal{Y}\parallel ,\mathcal{Z},\mathcal{Q},\mathcal{Y}\in \widehat{\mathcal{J}}.$

Let $({\mathcal{M}}_1,\parallel ·\parallel )$ be a normed space and that ${\mathcal{U}}_{cl}\left({\mathcal{M}}_1\right)=\{\mathcal{A}\in \mathcal{U}\left({\mathcal{M}}_1\right):\mathcal{A}\mathrm{is} \mathrm{closed}\}$, ${\mathcal{U}}_{c,cp}\left({\mathcal{M}}_1\right)=\{\mathcal{A}\in \mathcal{U}\left({\mathcal{M}}_1\right):\mathcal{A}\mathrm{is} \mathrm{convex} \mathrm{and} \mathrm{compact}\}$.

A multi-valued map $\mathcal{Q}:{\mathcal{M}}_1\to \mathcal{U}\left({\mathcal{M}}_1\right)$ is

(a)

Convex valued if $\mathcal{Q}\left(s\right)$ is convex $\forall s\in {\mathcal{M}}_1$;

(b)

Upper semi-continuous (U.S.C.) on ${\mathcal{M}}_1$ if, for each $s_0\in {\mathcal{M}}_1$; the set $\mathcal{Q}\left(s_0\right)$ is a non-empty closed subset of ${\mathcal{M}}_1$ and if, for each open set $\mathcal{V}$ of ${\mathcal{M}}_1$ containing $\mathcal{Q}\left(s_0\right)$, there exists an open neighborhood ${\mathcal{V}}_0$ of $s_0$, such that $\mathcal{Q}\left({\mathcal{V}}_0\right)\subset \mathcal{V}$;

(c)

Lower semi-continuous (L.S.C.) if the set $\{s\in {\mathcal{M}}_1:\mathcal{Q}\left(s\right)\cap \mathcal{E}\ne \oslash \}$ is open for any open set $\mathcal{E}$ in $\mathcal{H}$;

(d)

Completely continuous (C.C) if $\mathcal{Q}\left(\mathcal{E}\right)$ is relatively compact (r.c) for every $\mathcal{E}\in {\mathcal{U}}_b\left({\mathcal{M}}_1\right)=\{\mathcal{A}\in \mathcal{U}\left({\mathcal{M}}_1\right):\mathcal{A}\mathrm{is} \mathrm{bounded}\}$.

A map $\mathcal{Q}:[c,d]\to {\mathcal{U}}_{cl}\left({\mathcal{R}}_e\right)$ of multi-valued is said to be measurable if, for every $s\in {\mathcal{R}}_e$, the function $t⟼d(s,\mathcal{Q}(t\left)\right)=inf\left\{\right|s-l|:l\in \mathcal{Q}(t\left)\right\}$ is measurable.

A multi-valued map $\mathcal{Q}:[c,d]\times {\mathcal{R}}_e\to \mathcal{U}\left({\mathcal{R}}_e\right)$ is said to be Caratheodory if

(i)

$t⟼\mathcal{Q}(t,q,s)$ is measurable for each $q,s\in {\mathcal{R}}_e$;

(ii)

$(q,s)⟼\mathcal{Q}(t,q,s)$ is U.S.C for almost all $t\in [c,d]$.

Further, a Caratheodory function $\mathcal{Q}$ is called ${\mathcal{L}}^1$-Caratheodory if

(i)

For each $\epsilon >0$, ∃${\varphi }_{\epsilon }{\mathcal{L}}^1([c,d],{\mathcal{R}}_e^{+})$∋$\parallel \mathcal{Q}(t,q,s)\parallel =sup\left\{\right|q|:q\in \mathcal{Q}(t,q,s)\le {\varphi }_{\epsilon }\left(t\right)\}$$\forall q,s\in {\mathcal{R}}_e$ with $\parallel q\parallel ,\parallel s\parallel \le \epsilon$ and for a.e. $t\in [c,d]$.

Lemma 1.

Let $\mathcal{M}$ a closed convex subset of a Banach space ${\mathcal{M}}_1$ and $\mathcal{W}$ be an open subset of $\mathcal{K}$ with $0\in \mathcal{W}$. In addition, $\mathcal{H}:\widehat{\mathcal{W}}\to {\mathcal{Z}}_{c,cp}\left(\mathcal{K}\right)$ is an u.s.c compact map. Then either

• $\mathcal{H}$ has fixed point in $\widehat{\mathcal{W}}$ or
• ∃$w\in \partial \mathcal{W}$ and $\lambda \in (0,1)$, such that $w\in \lambda \mathcal{H}\left(w\right)$.

Lemma 2 

([29]). Let $\mathcal{Y}:{\mathcal{M}}_1\to {\mathcal{M}}_1$ be a completely continuous operator in Banach Space ${\mathcal{M}}_1$ and the set $\Psi =\{s\in {\mathcal{M}}_1|s=\delta \mathcal{Y}s,0<\delta <1\}$ is bounded. Then $\mathcal{Y}$ has a fixed point in ${\mathcal{M}}_1$.

Definition 1. 

The fractional integral of order ψ with the lower limit zero for a function k is defined as

$$I^{\psi }k\left(\tau \right)=\frac{1}{\Gamma \left(\psi \right)}{\int }_0^{\tau }\frac{k\left(\rho \right)}{{(\tau -\rho )}^{1-\psi }}ds,\tau >0,\psi >0,$$

(2)

provided the right-hand side is point-wise defined on $[0,\infty )$, where $\Gamma (.)$ is the gamma function, which is defined by $\Gamma \left(\psi \right)={\int }_0^{\infty }{\tau }^{\psi -1}{e}^{-\tau }d\tau .$

Definition 2. 

The R-L fractional derivative of order $\psi >0,n-1<\psi <n,n\in \mathbb{N}$ is defined as

$$\begin{array}{c}\hfill D_{0+}^{\psi }k\left(\tau \right)=\frac{1}{\Gamma (n-\psi )}{\left(\frac{d}{d\tau }\right)}^n{\int }_0^{\tau }{(\tau -\rho )}^{n-\psi -1}k\left(\rho \right)d\rho ,\tau >0,\end{array}$$

(3)

where the function k has absolutely continuous derivative up to order $(n-1)$.

Definition 3.

The Caputo derivative of order $\psi \in [n-1,n)$ for a function $k:[0,\infty )\to \left(\mathbb{R}\right)$ can be written as

$${}^c{D}_{0+}^{\psi }k\left(\tau \right)=D_{0+}^{\psi }\left(k\left(\tau \right)-\sum _{m=0}^{n-1}\frac{{\tau }^m}{m!}{f}^{\left(m\right)}\left(0\right)\right),\tau >0,n-1<r<n.$$

(4)

Note that the CFDs of order $\psi \in [n-1,n)$ almost everywhere on $[0,\infty )$ if $k\in {\mathcal{AC}}^n([0,\infty ),\left(\mathbb{R}\right))$.

Next, we state and prove the auxiliary lemma, which will help us in constructing the existence results for our proposed system.

Lemma 3.

Let ${\mathcal{G}}_1,{\mathcal{G}}_2,{\mathcal{G}}_3\in \mathcal{C}[0,\mathcal{T}]$ and $\mathsf{\Delta }\ne 0$. Then the solution of the linear fractional differential system,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{(}^c{\mathfrak{D}}^{\psi }+{\phi }^c{\mathfrak{D}}^{\psi -1})\mathcal{Z}\left(\varpi \right)={\mathcal{G}}_1,2<\psi \le 3,\hfill \\ {(}^c{\mathfrak{D}}^{\varphi }+{\phi }^c{\mathfrak{D}}^{\varphi -1})\mathcal{Q}\left(\varpi \right)={\mathcal{G}}_2,2<\varphi \le 3,\hfill \\ {(}^c{\mathfrak{D}}^{\omega }+{\phi }^c{\mathfrak{D}}^{\omega -1})\mathcal{Y}\left(\varpi \right)={\mathcal{G}}_3,3<\omega \le 4,\hfill \\ {\displaystyle \mathcal{Z}\left(0\right)=0,{\mathcal{Z}}^{\prime }\left(0\right)=0,\mathcal{Z}\left(\mathcal{T}\right)={\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j\mathcal{Q}\left({\zeta }_j\right)+{\mathrm{\Pi }}_1{\mathcal{I}}^{\varsigma }\mathcal{Q}\left(\vartheta \right),}\hfill \\ {\displaystyle \mathcal{Q}\left(0\right)=0,{\mathcal{Q}}^{\prime }\left(0\right)=0,\mathcal{Q}\left(\mathcal{T}\right)={\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j\mathcal{Y}\left({\zeta }_j\right)+{\mathrm{\Pi }}_2{\mathcal{I}}^{\varrho }\mathcal{Y}\left(\vartheta \right),}\hfill \\ {\displaystyle \mathcal{Y}\left(0\right)=0,{\mathcal{Y}}^{\prime }\left(0\right)=0,{\mathcal{Y}}^{″}\left(0\right)=0,\mathcal{Y}\left(\mathcal{T}\right)={\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j\mathcal{Z}\left({\zeta }_j\right)+{\mathrm{\Pi }}_3{\mathcal{I}}^{\delta }\mathcal{Z}\left(\vartheta \right),}\hfill \end{array}\right.\end{array}$$

(5)

is given by

$$\begin{array}{cc}\hfill {\displaystyle \mathcal{Y}\left(\varpi \right)=}& \frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\mathcal{G}}_2\left(\tau \right)d\tau \right)d\rho \right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\mathcal{G}}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\mathcal{G}}_1\left(\tau \right)d\tau \right)d\rho }\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\mathcal{G}}_3\left(\tau \right)d\tau \right)d\rho \right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\mathcal{G}}_3\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\mathcal{G}}_2\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\mathcal{G}}_2\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\mathcal{G}}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\mathcal{G}}_1\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\mathcal{G}}_1\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\mathcal{G}}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\mathcal{G}}_3\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\mathcal{G}}_1\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

(6)

$$\begin{array}{c}\hfill {\displaystyle \mathcal{Q}\left(\varpi \right)}\\ & =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2\mathsf{\Delta }}\left[\left({\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\mathcal{G}}_3\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\mathcal{G}}_3\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\mathcal{G}}_2\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\mathcal{G}}_2\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\mathcal{G}}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\mathcal{G}}_1\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\mathcal{G}}_1\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\mathcal{G}}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\mathcal{G}}_3\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\mathcal{G}}_2\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

(7)

$$\begin{array}{c}\hfill {\displaystyle \mathcal{Y}\left(\varpi \right)}\\ & =\frac{({\phi }^2{\varpi }^2-2\phi \varpi +2-e^{-\phi \varpi })}{{\phi }^3\mathsf{\Delta }}\left[\left({\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\mathcal{G}}_2\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\mathcal{G}}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\mathcal{G}}_1\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\mathcal{G}}_1\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\mathcal{G}}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\mathcal{G}}_3\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\mathcal{G}}_3\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\mathcal{G}}_3\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\mathcal{G}}_2\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\mathcal{G}}_3\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

(8)

where

$$\begin{array}{cc}\hfill {\mathcal{E}}_1=& {\displaystyle \frac{(\phi \mathcal{T}-1+e^{-\phi \mathcal{T}})}{{\phi }^2},{\mathcal{E}}_3=\frac{(\phi \mathcal{T}-1+e^{-\phi \mathcal{T}})}{{\phi }^2},{\mathcal{E}}_5=\frac{({\phi }^2{\mathcal{T}}^2-2\phi \mathcal{T}+2-2e^{-\phi \mathcal{T}})}{{\phi }^3},}\hfill \\ \hfill {\mathcal{E}}_2=& {\displaystyle \frac{1}{{\phi }^2}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j(\phi {\zeta }_j-1+e^{-\phi {\zeta }_j})+{\mathrm{\Pi }}_1{\int }_0^{\vartheta }\frac{{(\vartheta -\rho )}^{\varsigma -1}}{\Gamma \left(\varsigma \right)}(\phi s-1+e^{-\phi s})d\rho \right],}\hfill \\ \hfill {\mathcal{E}}_4=& {\displaystyle \frac{1}{{\phi }^3}\left[{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j({\phi }^2{{\zeta }_j}^2-2\phi {\zeta }_j+2-e^{-\phi {\zeta }_j})+{\mathrm{\Pi }}_2{\int }_0^{\vartheta }\frac{{(\vartheta -\rho )}^{\varrho -1}}{\Gamma \left(\varrho \right)}({\phi }^2{s}^2-2\phi s+2-2e^{-\phi s})d\rho \right],}\hfill \\ \hfill {\mathcal{E}}_6=& {\displaystyle \frac{1}{{\phi }^2}\left[{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j(\phi {\zeta }_j-1+e^{-\phi {\zeta }_j})+{\mathrm{\Pi }}_3{\int }_0^{\vartheta }\frac{{(\vartheta -\rho )}^{\delta -1}}{\Gamma \left(\delta \right)}(\phi s-1+e^{-\phi s})d\rho \right],}\hfill \\ \hfill \mathsf{\Delta }=& ({\mathcal{E}}_1{\mathcal{E}}_3{\mathcal{E}}_5-{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6).\hfill \end{array}$$

(9)

3. Multi-Valued System

Definition 4.

A function $(\mathcal{Z},\mathcal{Q},\mathcal{Y})\in {\mathcal{C}}^1([0,\mathcal{T}],{\mathcal{R}}_e)\times {\mathcal{C}}^1([0,\mathcal{T}],{\mathcal{R}}_e)\times {\mathcal{C}}^1([0,\mathcal{T}],{\mathcal{R}}_e)$ satisfying the boundary conditions and for which there $\mathfrak{f},\mathfrak{g},\mathfrak{h}={\mathcal{L}}^1([0,\mathcal{T}],{\mathcal{R}}_e)$, such that

$\mathfrak{f}\left(\varpi \right)\in {\mathcal{F}}_1(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right)),\mathfrak{g}\left(\varpi \right)\in {\mathcal{F}}_2(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right)),\mathfrak{h}\left(\varpi \right)\in {\mathcal{F}}_3(\varpi ,$$\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right))$ a.e. on $\varpi \in [0,\mathcal{T}]$ and

With the help of Lemma 3, we define an operator $\mathcal{F}:\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\to \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}$ by

$$\begin{array}{cc}\hfill & {\mathcal{F}}_1\left(\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right)\right)\hfill \\ \hfill =& \frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right.\hfill \\ & +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}(m,\mathcal{Z}\left(m\right),\mathcal{Q}\left(m\right),\mathcal{Y}\left(m\right))dm\right)d\tau \right)d\rho \hfill \\ & -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \hfill \\ & +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right.\right.\hfill \\ & +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}(m,\mathcal{Z}\left(m\right),\mathcal{Q}\left(m\right),\mathcal{Y}\left(m\right))dm\right)d\tau \right)d\rho \hfill \\ & \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right\}\hfill \\ & +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right.\hfill \\ & +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}(m,\mathcal{Z}\left(m\right),\mathcal{Q}\left(m\right),\mathcal{Y}\left(m\right))dm\right)d\tau \right)d\rho \hfill \\ & \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right\}\hfill \\ & +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right.\hfill \\ & +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}(m,\mathcal{Z}\left(m\right),\mathcal{Q}\left(m\right),\mathcal{Y}\left(m\right))dm\right)d\tau \right)d\rho \hfill \\ & \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right\}\right)\right]\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho ,\hfill \end{array}$$

$$\begin{array}{c}\\ \hfill & {\mathcal{F}}_2\left(\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right)\right)\hfill \\ \hfill =& \frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2\mathsf{\Delta }}\left[\left({\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right.\right.\right.\hfill \\ & +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}(m,\mathcal{Z}\left(m\right),\mathcal{Q}\left(m\right),\mathcal{Y}\left(m\right))dm\right)d\tau \right)d\rho \hfill \\ & \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right\}\hfill \\ & +{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right.\hfill \\ & +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}(m,\mathcal{Z}\left(m\right),\mathcal{Q}\left(m\right),\mathcal{Y}\left(m\right))dm\right)d\tau \right)d\rho \hfill \\ & \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right\}\hfill \\ & +{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right.\hfill \\ & +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}(m,\mathcal{Z}\left(m\right),\mathcal{Q}\left(m\right),\mathcal{Y}\left(m\right))dm\right)d\tau \right)d\rho \hfill \\ & \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right\}\right)\right]\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\mathcal{F}}_3\left(\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right)\right)=\frac{({\phi }^2{\varpi }^2-2\phi \varpi +2-e^{-\phi \varpi })}{{\phi }^3\mathsf{\Delta }}\hfill \\ \hfill \times & \left[\left({\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right.\right.\right.\hfill \\ & +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}(m,\mathcal{Z}\left(m\right),\mathcal{Q}\left(m\right),\mathcal{Y}\left(m\right))dm\right)d\tau \right)d\rho \hfill \\ & \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right\}\hfill \\ & +{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right.\hfill \\ & +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}(m,\mathcal{Z}\left(m\right),\mathcal{Q}\left(m\right),\mathcal{Y}\left(m\right))dm\right)d\tau \right)d\rho \hfill \\ & \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right\}\hfill \\ & +{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right.\hfill \\ & +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}(m,\mathcal{Z}\left(m\right),\mathcal{Q}\left(m\right),\mathcal{Y}\left(m\right))dm\right)d\tau \right)d\rho \hfill \\ & \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho \right\}\right)\right]\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}(\tau ,\mathcal{Z}\left(\tau \right),\mathcal{Q}\left(\tau \right),\mathcal{Y}\left(\tau \right))d\tau \right)d\rho .\hfill \end{array}$$

For easy calculations, we set

$$\begin{array}{cc}\hfill {\mathcal{P}}_1=& {\displaystyle \frac{(\phi -1+e^{-\phi })}{{\phi }^2{\mathcal{E}}_1}\left[\frac{{\mathcal{T}}^{\psi -1}(1-e^{-\phi \mathcal{T}})}{\phi \Gamma \left(\psi \right)}+\frac{{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6}{\mathsf{\Delta }}\frac{{\mathcal{T}}^{\psi -1}(1-e^{-\phi \mathcal{T}})}{\phi \Gamma \left(\psi \right)}\right.}\hfill \\ & +\frac{{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4}{\mathsf{\Delta }}\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}\left|{\sigma }_j\right|\left(\frac{{\zeta }_j^{\psi }(1-e^{-\phi {\zeta }_j})}{\phi \Gamma \left(\psi \right)}\right)\right.\hfill \\ & {\displaystyle \left.\left.+{\mathrm{\Pi }}_3\frac{{\delta }^{\psi +\vartheta -1}}{{\phi }^2\Gamma \left(\psi \right)\Gamma \left(\vartheta \right)}(\delta \phi +e^{-\phi \delta }-1)\right\}\right]+\frac{(1-e^{-\phi })}{\phi \Gamma \left(\psi \right)},}\hfill \\ \hfill {\mathcal{Q}}_1=& {\displaystyle \frac{(\phi -1+e^{-\phi })}{{\phi }^2{\mathcal{E}}_1}\left[\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}\left|{\xi }_j\right|\left(\frac{{\zeta }_j^{\varphi }(1-e^{-\phi {\zeta }_j})}{\phi \Gamma \left(\varphi \right)}\right)+{\mathrm{\Pi }}_1\frac{{\varsigma }^{\varphi +\vartheta -1}}{{\phi }^2\Gamma \left(\varphi \right)\Gamma \left(\vartheta \right)}(\varsigma \phi +e^{-\phi \varsigma }-1)\right\}\right.}\hfill \\ & {\displaystyle +\frac{{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6}{\mathsf{\Delta }}\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}\left|{\xi }_j\right|\left(\frac{{\zeta }_j^{\varphi }(1-e^{-\phi {\zeta }_j})}{\phi \Gamma \left(\varphi \right)}\right)+{\mathrm{\Pi }}_1\frac{{\varsigma }^{\varphi +\vartheta -1}}{{\phi }^2\Gamma \left(\varphi \right)\Gamma \left(\vartheta \right)}(\varsigma \phi +e^{-\phi \varsigma }-1)\right\}}\hfill \\ & {\displaystyle \left.+\frac{{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5}{\mathsf{\Delta }}\frac{{\mathcal{T}}^{\varphi -1}(1-e^{-\phi \mathcal{T}})}{\phi \Gamma \left(\varphi \right)}\right],}\hfill \\ \hfill {\mathcal{O}}_1=& {\displaystyle \frac{(\phi -1+e^{-\phi })}{{\phi }^2{\mathcal{E}}_1}\left[\frac{{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4}{\mathsf{\Delta }}\left\{\frac{{\mathcal{T}}^{\omega -1}(1-e^{-\phi \mathcal{T}})}{\phi \Gamma \left(\omega \right)}\right\}+\frac{{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5}{\mathsf{\Delta }}\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}\left|{\nu }_j\right|\left(\frac{{\zeta }_j^{\omega }(1-e^{-\phi {\zeta }_j})}{\phi \Gamma \left(\omega \right)}\right)\right.\right.}\hfill \\ & {\displaystyle \left.\left.+{\mathrm{\Pi }}_2\frac{{\varrho }^{\omega +\vartheta -1}}{{\phi }^2\Gamma \left(\omega \right)\Gamma \left(\vartheta \right)}(\varrho \phi +e^{-\phi \varrho }-1)\right\}\right],}\hfill \\ \hfill {\mathcal{P}}_2=& {\displaystyle \frac{(\phi -1+e^{-\phi })}{{\phi }^2\mathsf{\Delta }}\left[{\mathcal{E}}_4{\mathcal{E}}_6\left\{\frac{{\mathcal{T}}^{\psi -1}(1-e^{-\phi \mathcal{T}})}{\phi \Gamma \left(\psi \right)}\right\}+{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}\left|{\sigma }_j\right|\left(\frac{{\zeta }_j^{\psi }(1-e^{-\phi {\zeta }_j})}{\phi \Gamma \left(\psi \right)}\right)\right.\right.}\hfill \\ & {\displaystyle \left.\left.+{\mathrm{\Pi }}_3\frac{{\delta }^{\psi +\vartheta -1}}{{\phi }^2\Gamma \left(\psi \right)\Gamma \left(\vartheta \right)}(\delta \phi +e^{-\phi \delta }-1)\right\}\right],}\hfill \\ \hfill {\mathcal{Q}}_2=& {\displaystyle \frac{(\phi -1+e^{-\phi })}{{\phi }^2\mathsf{\Delta }}\left[{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}\left|{\xi }_j\right|\left(\frac{{\zeta }_j(1-e^{-\phi {\zeta }_j})}{\phi \Gamma \left(\varphi \right)}\right)+{\mathrm{\Pi }}_1\frac{{\varsigma }^{\varphi +\vartheta -1}}{{\phi }^2\Gamma \left(\varphi \right)\Gamma \left(\vartheta \right)}(\varsigma \phi +e^{-\phi \varsigma }-1)\right\}\right.}\hfill \\ & {\displaystyle \left.+{\mathcal{E}}_1{\mathcal{E}}_5\left\{\frac{{\mathcal{T}}^{\varphi -1}(1-e^{-\phi \mathcal{T}})}{\phi \Gamma \left(\varphi \right)}\right\}\right]+\frac{(1-e^{-\phi })}{\phi \Gamma \left(\varphi \right)},}\hfill \\ \hfill {\mathcal{O}}_2=& {\displaystyle \frac{(\phi -1+e^{-\phi })}{{\phi }^2\mathsf{\Delta }}\left[{\mathcal{E}}_1{\mathcal{E}}_4\left\{\frac{{\mathcal{T}}^{\omega -1}(1-e^{-\phi \mathcal{T}})}{\phi \Gamma \left(\omega \right)}\right\}+{\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}\left|{\nu }_j\right|\left(\frac{{\zeta }_j^{\omega }(1-e^{-\phi {\zeta }_j})}{\phi \Gamma \left(\omega \right)}\right)\right.\right.}\hfill \\ & {\displaystyle \left.\left.+{\mathrm{\Pi }}_2\frac{{\varrho }^{\omega +\vartheta -1}}{{\phi }^2\Gamma \left(\omega \right)\Gamma \left(\vartheta \right)}(\varrho \phi +e^{-\phi \varrho }-1)\right\}\right],}\hfill \\ \hfill {\mathcal{P}}_3=& {\displaystyle \frac{({\phi }^2-2\phi +2-2e^{-\phi })}{{\phi }^3\mathsf{\Delta }}\left[{\mathcal{E}}_3{\mathcal{E}}_6\left\{\frac{{\mathcal{T}}^{\psi -1}(1-e^{-\phi \mathcal{T}})}{\phi \Gamma \left(\psi \right)}\right\}+{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}\left|{\sigma }_j\right|\left(\frac{{\zeta }_j^{\psi }(1-e^{-\phi {\zeta }_j})}{\phi \Gamma \left(\psi \right)}\right)\right.\right.}\hfill \\ & {\displaystyle \left.\left.+{\mathrm{\Pi }}_3\frac{{\delta }^{\psi +\vartheta -1}}{{\phi }^2\Gamma \left(\psi \right)\Gamma \left(\vartheta \right)}(\delta \phi +e^{-\phi \delta }-1)\right\}\right],}\hfill \\ \hfill {\mathcal{Q}}_3=& {\displaystyle \frac{({\phi }^2-2\phi +2-2e^{-\phi })}{{\phi }^3\mathsf{\Delta }}\left[{\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}\left|{\xi }_j\right|\left(\frac{{\zeta }_j^{\varphi }(1-e^{-\phi {\zeta }_j})}{\phi \Gamma \left(\varphi \right)}\right)\right.\right.}\hfill \\ & {\displaystyle \left.\left.+{\mathrm{\Pi }}_1\frac{{\varsigma }^{\varphi +\vartheta -1}}{{\phi }^2\Gamma \left(\varphi \right)\Gamma \left(\vartheta \right)}(\varsigma \phi +e^{-\phi \varsigma }-1)\right\}+{\mathcal{E}}_2{\mathcal{E}}_6\left\{\frac{{\mathcal{T}}^{\varphi -1}(1-e^{-\phi \mathcal{T}})}{\phi \Gamma \left(\varphi \right)}\right\}\right],}\hfill \\ \hfill {\mathcal{O}}_3=& {\displaystyle \frac{({\phi }^2-2\phi +2-2e^{-\phi })}{{\phi }^3\mathsf{\Delta }}\left[{\mathcal{E}}_1{\mathcal{E}}_3\left\{\frac{{\mathcal{T}}^{\omega -1}(1-e^{-\phi \mathcal{T}})}{\phi \Gamma \left(\omega \right)}\right\}+{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}\left|{\nu }_j\right|\left(\frac{{\zeta }_j^{\omega }(1-e^{-\phi {\zeta }_j})}{\phi \Gamma \left(\omega \right)}\right)\right.\right.}\hfill \\ & \left.\left.+{\mathrm{\Pi }}_2\frac{{\varrho }^{\omega +\vartheta -1}}{{\phi }^2\Gamma \left(\omega \right)\Gamma \left(\vartheta \right)}(\varrho \phi +e^{-\phi \varrho }-1)\right\}\right]+\frac{(1-e^{-\phi })}{\phi \Gamma \left(\omega \right)}.\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\mathcal{M}}_1=& ({\mathcal{P}}_1+{\mathcal{P}}_2+{\mathcal{P}}_3)\left|\right|P_1\left|\right|({\mathcal{U}}_1\left(\mathcal{N}\right)+{\mathcal{V}}_1\left(\mathcal{N}\right)+{\mathcal{W}}_1\left(\mathcal{N}\right))+({\mathcal{Q}}_1+{\mathcal{Q}}_2+{\mathcal{Q}}_3)\left|\right|P_2\left|\right|({\mathcal{U}}_2\left(\mathcal{N}\right)+{\mathcal{V}}_2\left(\mathcal{N}\right)+{\mathcal{W}}_2\left(\mathcal{N}\right))\hfill \\ & +({\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3)\left|\right|P_3\left|\right|({\mathcal{U}}_3\left(\mathcal{N}\right)+{\mathcal{V}}_3\left(\mathcal{N}\right)+{\mathcal{W}}_3\left(\mathcal{N}\right))\hfill \end{array}$$

Next, we define the operators ${\mathcal{K}}_1,{\mathcal{K}}_2,{\mathcal{K}}_3:\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\to \mathcal{P}(\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}})$ as follows:

Then, we define an operator $\mathcal{K}:\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\to \mathcal{P}(\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}})$ by

$$\begin{array}{c}\hfill \mathcal{K}(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)=\left(\begin{array}{c}{\mathcal{K}}_1(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)\\ {\mathcal{K}}_2(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)\\ {\mathcal{K}}_3(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)\end{array}\right),\end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{K}}_1=& {\displaystyle \{h_1\in \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}:\mathrm{there} \mathrm{exist}\mathfrak{f}\in {\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},\mathfrak{g}\in {\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},\mathfrak{h}\in {\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},}\hfill \\ & \mathrm{such} \mathrm{that}{h}_1(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)={\mathcal{Q}}_1(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right),\forall \tau \in [0,\mathcal{T}]\},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\mathcal{K}}_2=& {\displaystyle \{h_2\in \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}:\mathrm{there} \mathrm{exist}\mathfrak{f}\in {\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},\mathfrak{g}\in {\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},\mathfrak{h}\in {\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},}\hfill \\ & \mathrm{such} \mathrm{that}{h}_2(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)={\mathcal{Q}}_2(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right),\forall \tau \in [0,\mathcal{T}]\},\hfill \end{array}$$

(12)

and

$$\begin{array}{cc}\hfill {\mathcal{K}}_3=& {\displaystyle \{h_3\in \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}:\mathrm{there} \mathrm{exist}\mathfrak{f}\in {\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},\mathfrak{g}\in {\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},\mathfrak{h}\in {\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},}\hfill \\ & \mathrm{such} \mathrm{that}{h}_3(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)={\mathcal{Q}}_3(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right),\forall \tau \in [0,\mathcal{T}]\},\hfill \end{array}$$

(13)

3.1. The Caratheodory Case

Our first result dealing with convex values $\mathfrak{f},\mathfrak{g}$, and $\mathfrak{h}$ is proved via the Leray–Schauder non-linear alternative for multi-valued maps.

Theorem 1.

Suppose that the following conditions are satisfied:

$\left({\mathcal{B}}_1\right)\mathfrak{f},\mathfrak{g},\mathfrak{h}:[0,\mathcal{T}]\times {\mathcal{R}}^3\to \mathcal{P}\left(\mathcal{R}\right)$ are $L^1$ Caratheodory and have convex values;

$\left({\mathcal{B}}_2\right)$ There exist continuous non-decreasing functions ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3,{\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3,{\mathcal{W}}_1,{\mathcal{W}}_2,{\mathcal{W}}_3:[0,\infty )\to (0,\infty )$ functions $P_1\left(\varpi \right),P_2\left(\varpi \right),P_3\left(\varpi \right)\in (\mathcal{C}[0,\mathcal{T}]\times {\mathcal{R}}_{+})$, such that

$$\begin{array}{cc}\hfill \left|\right|{\mathcal{F}}_1(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y}){\left|\right|}_{\mathcal{P}}:=& sup\left\{\right|\mathfrak{f}|:\mathfrak{f}\in {\mathcal{F}}_1(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y})\}\le P_1[{\mathcal{U}}_1\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_1\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_1\left(\right|\left|\mathcal{Y}\right|\left|\right)],\hfill \\ \hfill \left|\right|{\mathcal{F}}_2(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y}){\left|\right|}_{\mathcal{P}}:=& sup\left\{\right|\mathfrak{g}|:\mathfrak{g}\in {\mathcal{F}}_2(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y})\}\le P_2[{\mathcal{U}}_2\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_2\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_2\left(\right|\left|\mathcal{Y}\right|\left|\right)],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left|\right|{\mathcal{F}}_3(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y}){\left|\right|}_{\mathcal{P}}:=& sup\left\{\right|\mathfrak{h}|:\mathfrak{h}\in {\mathcal{F}}_3(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y})\}\le P_3[{\mathcal{U}}_3\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_3\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_3\left(\right|\left|\mathcal{Y}\right|\left|\right)],\hfill \end{array}$$

for each $(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y})\in [0,\mathcal{T}]\times {\mathcal{R}}_e^3$; $\left({\mathcal{B}}_3\right)$ there exists a number $\mathcal{N}>0$, such that

$$\begin{array}{c}\hfill \frac{\mathcal{N}}{{\mathcal{M}}_1}>1,\end{array}$$

where ${\mathcal{P}}_i,{\mathcal{Q}}_i,$ and ${\mathcal{O}}_i$ are given by (10). The tripled system has at least one solution on $[0,\mathcal{T}]$.

Proof. 

Consider the operator ${\mathcal{K}}_1\times {\mathcal{K}}_2\times {\mathcal{K}}_3:\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\to \mathcal{P}(\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}})$ defined by (11)–(13). From $\left({\mathcal{B}}_1\right)$, it follows that sets ${\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$, ${\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$ and ${\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$ are non-empty for each $(\mathcal{Z},\mathcal{Q},\mathcal{Y})\in \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}$. Then, for $\mathfrak{f}\in {\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},\mathfrak{g}\in {\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},\mathfrak{h}\in {\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$ for $(\mathcal{Z},\mathcal{Q},\mathcal{Y})\in \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}$, we have

$$\begin{array}{cc}\hfill {\mathfrak{H}}_1(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)& =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho }\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\mathfrak{H}}_2(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)& =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2\mathsf{\Delta }}\left[\left({\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\mathfrak{H}}_3(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)\hfill \\ & =\frac{({\phi }^2{\varpi }^2-2\phi \varpi +2-e^{-\phi \varpi })}{{\phi }^3\mathsf{\Delta }}\left[\left({\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

where ${\mathfrak{H}}_1\in {\mathcal{K}}_1(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\tau \right),{\mathfrak{H}}_2\in {\mathcal{K}}_2(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\tau \right),{\mathfrak{H}}_3\in {\mathcal{K}}_3(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\tau \right)$, and $({\mathfrak{H}}_1,{\mathfrak{H}}_2,{\mathfrak{H}}_3)$$\in \mathcal{K}(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)$.

For the applicability of Leray–Schauder non-linear alternative we split our proof into several steps.

Claim 1. The operator $\mathcal{K}(\mathcal{Z},\mathcal{Q},\mathcal{Y})$ is convex. Let $({\mathfrak{H}}_i,{\overline{\mathfrak{H}}}_i,{\widehat{\mathfrak{H}}}_i)\in ({\mathcal{K}}_1,{\mathcal{K}}_2,{\mathcal{K}}_3),i=1,2.$ Then there exist ${\mathfrak{f}}_i\in {\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},{\mathfrak{g}}_i\in {\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},{\mathfrak{h}}_i\in {\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},i=1,2,$ such that, for each $\tau \in [0,\mathcal{T}]$, we have

$$\begin{array}{cc}\hfill {\mathfrak{H}}_i\left(\varpi \right)& =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho }\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \overline{{\mathfrak{H}}_i}\left(\varpi \right)& =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2\mathsf{\Delta }}\left[\left({\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \widehat{{\mathfrak{H}}_i}\left(\varpi \right)& =\frac{({\phi }^2{\varpi }^2-2\phi \varpi +2-e^{-\phi \varpi })}{{\phi }^3\mathsf{\Delta }}\left[\left({\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

Let $0\le \omega \le 1$. Then, for each $\tau \in [0,\mathcal{T}]$, we have $[\omega {\mathcal{F}}_1\left(\mathfrak{a}\right)+(1-\omega ){\mathcal{F}}_2\left(\mathfrak{a}\right)]$

$$\begin{array}{cc}\hfill & [\omega {\mathfrak{H}}_1+(1-\omega ){\mathfrak{H}}_2]\left(\tau \right)\hfill \\ & =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}[\omega {\mathfrak{g}}_1\left(\tau \right)+(1-\omega ){\mathfrak{g}}_2\left(\tau \right)]d\tau \right)d\rho \right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}[\omega {\mathfrak{g}}_1\left(m\right)+(1-\omega ){\mathfrak{g}}_2\left(m\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}[\omega {\mathfrak{f}}_1\left(\tau \right)+(1-\omega ){\mathfrak{f}}_2\left(\tau \right)]d\tau \right)d\rho }\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}[\omega {\mathfrak{h}}_1\left(\tau \right)+(1-\omega ){\mathfrak{h}}_2\left(\tau \right)]d\tau \right)d\rho \right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}[\omega {\mathfrak{h}}_1\left(m\right)+(1-\omega ){\mathfrak{h}}_2\left(m\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}[\omega {\mathfrak{g}}_1\left(\tau \right)+(1-\omega ){\mathfrak{g}}_2\left(\tau \right)]d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}[\omega {\mathfrak{g}}_1\left(\tau \right)+(1-\omega ){\mathfrak{g}}_2\left(\tau \right)]d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}[\omega {\mathfrak{g}}_1\left(m\right)+(1-\omega ){\mathfrak{g}}_2\left(m\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}[\omega {\mathfrak{f}}_1\left(\tau \right)+(1-\omega ){\mathfrak{f}}_2\left(\tau \right)]d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}[\omega {\mathfrak{f}}_1\left(\tau \right)+(1-\omega ){\mathfrak{f}}_2\left(\tau \right)]d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}[\omega {\mathfrak{f}}_1\left(m\right)+(1-\omega ){\mathfrak{f}}_2\left(m\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}[\omega {\mathfrak{h}}_1\left(\tau \right)+(1-\omega ){\mathfrak{h}}_2\left(\tau \right)]d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}[\omega {\mathfrak{f}}_1\left(\tau \right)+(1-\omega ){\mathfrak{f}}_2\left(\tau \right)]d\tau \right)d\rho ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & [\omega {\overline{\mathfrak{H}}}_1+(1-\omega ){\overline{\mathfrak{H}}}_2]\left(\tau \right)\hfill \\ & =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2\mathsf{\Delta }}\left[\left({\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}[\omega {\mathfrak{h}}_1\left(\tau \right)+(1-\omega ){\mathfrak{h}}_2\left(\tau \right)]d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}[\omega {\mathfrak{h}}_1\left(m\right)+(1-\omega ){\mathfrak{h}}_2\left(m\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}[\omega {\mathfrak{g}}_1\left(\tau \right)+(1-\omega ){\mathfrak{g}}_2\left(\tau \right)]d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}[\omega {\mathfrak{g}}_1\left(\tau \right)+(1-\omega ){\mathfrak{g}}_2\left(\tau \right)]d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}[\omega {\mathfrak{g}}_1\left(m\right)+(1-\omega ){\mathfrak{g}}_2\left(m\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}[\omega {\mathfrak{f}}_1\left(\tau \right)+(1-\omega ){\mathfrak{f}}_2\left(\tau \right)]d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}[\omega {\mathfrak{f}}_1\left(\tau \right)+(1-\omega ){\mathfrak{f}}_2\left(\tau \right)]d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}[\omega {\mathfrak{f}}_1\left(m\right)+(1-\omega ){\mathfrak{f}}_2\left(m\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}[\omega {\mathfrak{h}}_1\left(\tau \right)+(1-\omega ){\mathfrak{h}}_2\left(\tau \right)]d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}[\omega {\mathfrak{g}}_1\left(\tau \right)+(1-\omega ){\mathfrak{g}}_2\left(\tau \right)]d\tau \right)d\rho ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & [\omega {\widehat{\mathfrak{H}}}_1+(1-\omega ){\widehat{\mathfrak{H}}}_2]\left(\tau \right)=\frac{({\phi }^2{\varpi }^2-2\phi \varpi +2-e^{-\phi \varpi })}{{\phi }^3\mathsf{\Delta }}\hfill \\ & \times \left[\left({\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}[\omega {\mathfrak{g}}_1\left(\tau \right)+(1-\omega ){\mathfrak{g}}_2\left(\tau \right)]d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}[\omega {\mathfrak{g}}_1\left(m\right)+(1-\omega ){\mathfrak{g}}_2\left(m\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}[\omega {\mathfrak{f}}_1\left(\tau \right)+(1-\omega ){\mathfrak{f}}_2\left(\tau \right)]d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}[\omega {\mathfrak{f}}_1\left(\tau \right)+(1-\omega ){\mathfrak{f}}_2\left(\tau \right)]d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}[\omega {\mathfrak{f}}_1\left(m\right)+(1-\omega ){\mathfrak{f}}_2\left(m\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}[\omega {\mathfrak{h}}_1\left(\tau \right)+(1-\omega ){\mathfrak{h}}_2\left(\tau \right)]d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}[\omega {\mathfrak{h}}_1\left(\tau \right)+(1-\omega ){\mathfrak{h}}_2\left(\tau \right)]d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}[\omega {\mathfrak{h}}_1\left(m\right)+(1-\omega ){\mathfrak{h}}_2\left(m\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}[\omega {\mathfrak{g}}_1\left(\tau \right)+(1-\omega ){\mathfrak{g}}_2\left(\tau \right)]d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}[\omega {\mathfrak{h}}_1\left(\tau \right)+(1-\omega ){\mathfrak{h}}_2\left(\tau \right)]d\tau \right)d\rho ,\hfill \end{array}$$

We deduce that ${\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},{\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$ and ${\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$ are convex valued, since $\mathfrak{f},\mathfrak{g},\mathfrak{h}$ are convex valued. obviously, $[\omega {\mathfrak{H}}_1+(1-\omega ){\mathfrak{H}}_2]\left(\tau \right)\in {\mathcal{K}}_1$,$[\omega {\overline{\mathfrak{H}}}_1+(1-\omega ){\overline{\mathfrak{H}}}_2]\left(\tau \right)\in {\mathcal{K}}_2$, and $[\omega {\overline{\mathfrak{H}}}_1+(1-\omega ){\overline{\mathfrak{H}}}_2]\left(\tau \right)\in {\mathcal{K}}_3$ hence $[\omega ({\mathfrak{H}}_1,{\mathfrak{H}}_2,{\mathfrak{H}}_3)+(1-\omega )({\overline{\mathfrak{h}}}_1,{\overline{\mathfrak{h}}}_2,\widehat{{\mathfrak{H}}_3})]\left(\tau \right)\in \mathcal{K}$.

Claim 2. We show that the operator $\mathcal{K}$ maps bounded sets into bounded sets in $\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}$. Let $\tau >0$, define ${\mathcal{B}}_{\mathfrak{r}}=\{(\mathcal{Z},\mathcal{Q},\mathcal{Y})\in \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}:\left|\right|(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left|\right|\le \mathfrak{r}\}$ be a bounded set in $\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}$. Then, there exist $\mathfrak{f}\in {\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Q},\mathcal{Y})},\mathfrak{g}\in {\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$ and $\mathfrak{h}\in {\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$, such that

$$\begin{array}{cc}\hfill & \left|{\mathfrak{H}}_1(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\tau \right)\right|\hfill \\ & =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\left|\right|P_2\left|\right|[{\mathcal{U}}_2\left(r\right)+{\mathcal{V}}_2\left(r\right)+{\mathcal{W}}_2\left(r\right)]d\tau \right)d\rho \right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\left|\right|P_2\left|\right|[{\mathcal{U}}_2\left(r\right)+{\mathcal{V}}_2\left(r\right)+{\mathcal{W}}_2\left(r\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\left|\right|P_1\left|\right|[{\mathcal{U}}_1\left(r\right)+{\mathcal{V}}_1\left(r\right)+{\mathcal{W}}_1\left(r\right)]d\tau \right)d\rho }\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\left|\right|P_3\left|\right|[{\mathcal{U}}_3\left(r\right)+{\mathcal{V}}_3\left(r\right)+{\mathcal{W}}_3\left(r\right)]d\tau \right)d\rho \right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\left|\right|P_3\left|\right|[{\mathcal{U}}_3\left(r\right)+{\mathcal{V}}_3\left(r\right)+{\mathcal{W}}_3\left(r\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\left|\right|P_2\left|\right|[{\mathcal{U}}_2\left(r\right)+{\mathcal{V}}_2\left(r\right)+{\mathcal{W}}_2\left(r\right)]d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\left|\right|P_2\left|\right|[{\mathcal{U}}_2\left(r\right)+{\mathcal{V}}_2\left(r\right)+{\mathcal{W}}_2\left(r\right)]d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\left|\right|P_2\left|\right|[{\mathcal{U}}_2\left(r\right)+{\mathcal{V}}_2\left(r\right)+{\mathcal{W}}_2\left(r\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\left|\right|P_1\left|\right|[{\mathcal{U}}_1\left(r\right)+{\mathcal{V}}_1\left(r\right)+{\mathcal{W}}_1\left(r\right)]d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\left|\right|P_1\left|\right|[{\mathcal{U}}_1\left(r\right)+{\mathcal{V}}_1\left(r\right)+{\mathcal{W}}_1\left(r\right)]d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\left|\right|P_1\left|\right|[{\mathcal{U}}_1\left(r\right)+{\mathcal{V}}_1\left(r\right)+{\mathcal{W}}_1\left(r\right)]dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\left|\right|P_3\left|\right|[{\mathcal{U}}_3\left(r\right)+{\mathcal{V}}_3\left(r\right)+{\mathcal{W}}_3\left(r\right)]d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\left|\right|P_1\left|\right|[{\mathcal{U}}_1\left(r\right)+{\mathcal{V}}_1\left(r\right)+{\mathcal{W}}_1\left(r\right)]d\tau \right)d\rho ,\hfill \\ & \le {\mathcal{P}}_1\left|\right|P_1\left|\right|{\mathcal{U}}_1\left(r\right)+{\mathcal{V}}_1\left(r\right)+{\mathcal{W}}_1\left(r\right)+{\mathcal{Q}}_1\left|\right|P_2\left|\right|{\mathcal{U}}_2\left(r\right)+{\mathcal{V}}_2\left(r\right)+{\mathcal{W}}_2\left(r\right)+{\mathcal{O}}_1\left|\right|P_3\left|\right|{\mathcal{U}}_3\left(r\right)+{\mathcal{V}}_3\left(r\right)+{\mathcal{W}}_3\left(r\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & |{\mathfrak{H}}_2(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\tau \right)|\hfill \\ & \le {\mathcal{P}}_2\left|\right|P_1\left|\right|{\mathcal{U}}_1\left(r\right)+{\mathcal{V}}_1\left(r\right)+{\mathcal{W}}_1\left(r\right)+{\mathcal{Q}}_2\left|\right|P_2\left|\right|{\mathcal{U}}_2\left(r\right)+{\mathcal{V}}_2\left(r\right)+{\mathcal{W}}_2\left(r\right)+{\mathcal{O}}_2\left|\right|P_3\left|\right|{\mathcal{U}}_3\left(r\right)+{\mathcal{V}}_3\left(r\right)+{\mathcal{W}}_3\left(r\right).\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & |{\mathfrak{H}}_3(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\tau \right)|\hfill \\ & \le {\mathcal{P}}_3\left|\right|P_1\left|\right|{\mathcal{U}}_1\left(r\right)+{\mathcal{V}}_1\left(r\right)+{\mathcal{W}}_1\left(r\right)+{\mathcal{Q}}_3\left|\right|P_2\left|\right|{\mathcal{U}}_2\left(r\right)+{\mathcal{V}}_2\left(r\right)+{\mathcal{W}}_2\left(r\right)+{\mathcal{O}}_3\left|\right|P_3\left|\right|{\mathcal{U}}_3\left(r\right)+{\mathcal{V}}_3\left(r\right)+{\mathcal{W}}_3\left(r\right).\hfill \end{array}$$

Hence we obtain

$$\begin{array}{c}\hfill \left|\right|({\mathfrak{H}}_1,{\mathfrak{H}}_2,{\mathfrak{H}}_3)\left|\right|=\left|\right|{\mathfrak{H}}_1(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left|\right|+\left|\right|{\mathfrak{H}}_2(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left|\right|+\left|\right|{\mathfrak{H}}_3(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left|\right|\end{array}$$

$$\begin{array}{cc}\hfill \le & ({\mathcal{P}}_1+{\mathcal{P}}_2+{\mathcal{P}}_3)\left|\right|P_1\left|\right|({\mathcal{U}}_1\left(\mathcal{N}\right)+{\mathcal{V}}_1\left(\mathcal{N}\right)+{\mathcal{W}}_1\left(\mathcal{N}\right))+({\mathcal{Q}}_1+{\mathcal{Q}}_2+{\mathcal{Q}}_3)\left|\right|P_2\left|\right|({\mathcal{U}}_2\left(\mathcal{N}\right)+{\mathcal{V}}_2\left(\mathcal{N}\right)+{\mathcal{W}}_2\left(\mathcal{N}\right))\hfill \\ & +({\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3)\left|\right|P_3\left|\right|({\mathcal{U}}_3\left(\mathcal{N}\right)+{\mathcal{V}}_3\left(\mathcal{N}\right)+{\mathcal{W}}_3\left(\mathcal{N}\right)).\hfill \end{array}$$

Claim 3. We show the equi-continuity of the operator $\mathcal{K}$. Let ${\varpi }_1,{\varpi }_2\in [0,\mathcal{T}]$ with ${\varpi }_1<{\varpi }_2$. Then there exist $\mathfrak{f}\in {\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$,$\mathfrak{g}\in {\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$, and $\mathfrak{h}\in {\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$, such that

$$\begin{array}{cc}\hfill & \left|{\mathfrak{H}}_1(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left({\varpi }_2\right)-{\mathfrak{H}}_1(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left({\varpi }_1\right)\right|\hfill \\ & \le \left|\frac{\left|\right|P_1\left|\right|{\mathcal{U}}_1\left(r\right)+{\mathcal{V}}_1\left(r\right)+{\mathcal{W}}_1\left(r\right)}{\Gamma \left(\psi \right)}\left({\int }_0^{{\varpi }_1}[{({\varpi }_2-\mathfrak{s})}^{\psi -1}-{({\varpi }_1-\mathfrak{s})}^{\psi -1}]d\mathfrak{s}\right.\right.\hfill \\ & \left.\left.+{\int }_0^{{\varpi }_1}[{({\varpi }_2-\mathfrak{s})}^{\psi -1}-{({\varpi }_1-\mathfrak{s})}^{\psi -1}]d\mathfrak{s}\right)\right|\hfill \\ \hfill \le & \left|\right|P_1\left|\right|{\mathcal{U}}_1\left(r\right)+{\mathcal{V}}_1\left(r\right)+{\mathcal{W}}_1\left(r\right)\left(\frac{2{({\varpi }_2-{\varpi }_1)}^{\psi }+|{\varpi }_2^{\psi }-{\varpi }_1^{\psi }|}{\Gamma (\psi +1)}\right),\hfill \end{array}$$

and,

$$\begin{array}{cc}\hfill \left|{\mathfrak{H}}_2(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left({\varpi }_2\right)-{\mathfrak{H}}_2(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left({\varpi }_1\right)\right|\le & \left|\right|P_2\left|\right|{\mathcal{U}}_2\left(r\right)+{\mathcal{V}}_2\left(r\right)+{\mathcal{W}}_2\left(r\right)\left(\frac{2{({\varpi }_2-{\varpi }_1)}^{\varphi }+|{\varpi }_2^{\varphi }-{\varpi }_1^{\varphi }|}{\Gamma (\varphi +1)}\right).\hfill \end{array}$$

Similar,

$$\begin{array}{cc}\hfill \left|{\mathfrak{H}}_3(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left({\varpi }_2\right)-{\mathfrak{H}}_3(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left({\varpi }_1\right)\right|\le & \left|\right|P_3\left|\right|{\mathcal{U}}_3\left(r\right)+{\mathcal{V}}_3\left(r\right)+{\mathcal{W}}_3\left(r\right)\left(\frac{2{({\varpi }_2-{\varpi }_1)}^{\omega }+|{\varpi }_2^{\omega }-{\varpi }_1^{\omega }|}{\Gamma (\omega +1)}\right).\hfill \end{array}$$

Therefore, the operator $\mathcal{K}(\mathcal{Z},\mathcal{Q},\mathcal{Y})$ is equi-continuous, based on Arzela–Ascoli $\mathcal{K}(\mathcal{Z},\mathcal{Q},\mathcal{Y})$ is completely continuous. We know that a completely continuous operator is upper semi-continuous if it has a closed graph. Thus, we need to prove that $\mathcal{K}$ has a closed graph.

Claim 4. We show that the operator $\mathcal{K}$ has closed graph. As it is known that a completely continuous operator is upper semi-continuous if it has a closed graph. For this we take $({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\to ({\mathcal{Z}}_{*},{\mathcal{Q}}_{*},{\mathcal{Y}}_{*})$, $({\mathfrak{H}}_n,{\overline{\mathfrak{H}}}_n,\widehat{{\mathfrak{H}}_n})\in \mathcal{K}({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)$ and $({\mathfrak{H}}_n,{\overline{\mathfrak{H}}}_n,\widehat{{\mathfrak{H}}_n})\to ({\mathfrak{H}}_{*},{\overline{\mathfrak{H}}}_{*},\widehat{{\mathfrak{H}}_{*}})$, then we need to show $({\mathfrak{H}}_{*},{\overline{\mathfrak{H}}}_{*},\widehat{{\mathfrak{H}}_{*}})\in \mathcal{K}({\mathcal{Z}}_{*},{\mathcal{Q}}_{*},{\mathcal{Y}}_{*})$. Observe that $({\mathfrak{H}}_n,{\overline{\mathfrak{H}}}_n,\widehat{{\mathfrak{H}}_n})\in \mathcal{K}({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)$ implies that exist ${\mathfrak{f}}_n\in {\mathcal{S}}_{{\mathcal{F}}_1,({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)}$, ${\mathfrak{g}}_n\in {\mathcal{S}}_{{\mathcal{F}}_2,({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)}$, and ${\mathfrak{h}}_n\in {\mathcal{S}}_{{\mathcal{F}}_3,({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)}$ such that

$$\begin{array}{cc}\hfill {\mathfrak{H}}_n({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\left(\varpi \right)=& \frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho }\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\overline{\mathfrak{H}}}_n({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\left(\varpi \right)\hfill \\ & =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2\mathsf{\Delta }}\left[\left({\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\widehat{\mathfrak{H}}}_n({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\left(\varpi \right)=& \frac{({\phi }^2{\varpi }^2-2\phi \varpi +2-e^{-\phi \varpi })}{{\phi }^3\mathsf{\Delta }}\hfill \\ & {\displaystyle \times \left[\left({\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.\right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{{(}_{\mathfrak{h}})}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

Let us consider the continuous linear operator ${\mathsf{\Phi }}_1,{\mathsf{\Phi }}_2,{\mathsf{\Phi }}_3:{\mathfrak{L}}^1([0,\mathcal{T}],\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}})\to C([0,\mathcal{T}],\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}})$ given by

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_1(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)& =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho }\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_2(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)\hfill \\ & =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2\mathsf{\Delta }}\left[\left({\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_3(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)\hfill \\ & =\frac{({\phi }^2{\varpi }^2-2\phi \varpi +2-e^{-\phi \varpi })}{{\phi }^3\mathsf{\Delta }}\left[\left({\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\mathfrak{f}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\mathfrak{g}\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\mathfrak{h}\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

From, we know that $({\mathsf{\Phi }}_1,{\mathsf{\Phi }}_2,{\mathsf{\Phi }}_3)\circ ({\mathcal{S}}_{{\mathcal{F}}_1},{\mathcal{S}}_{{\mathcal{F}}_2},{\mathcal{S}}_{{\mathcal{F}}_3})$ is closed graph operator. Further, we have $({\mathfrak{H}}_n,{\overline{\mathfrak{H}}}_n,\widehat{{\mathfrak{H}}_n})\in ({\mathsf{\Phi }}_1,{\mathsf{\Phi }}_2,{\mathsf{\Phi }}_3)\circ ({\mathcal{S}}_{{\mathcal{F}}_1,({\mathcal{Z}}_n,{\mathcal{Q}}_n{\mathcal{Y}}_n)},{\mathcal{S}}_{{\mathcal{F}}_2,({\mathcal{Z}}_n,{\mathcal{Q}}_n{\mathcal{Y}}_n)},{\mathcal{S}}_{{\mathcal{F}}_3,({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)})$ for all n. Since $({\mathcal{Z}}_n,{\mathcal{Q}}_n{\mathcal{Y}}_n)\to ({\mathcal{Z}}_{*},{\mathcal{Q}}_{*},{\mathcal{Y}}_{*})$, $({\mathfrak{H}}_n,{\overline{\mathfrak{H}}}_n,\widehat{{\mathfrak{H}}_n})\to ({\mathfrak{H}}_{*},{\overline{\mathfrak{H}}}_{*},\widehat{{\mathfrak{H}}_n})$ it follows that ${\mathfrak{f}}_{*}\in {\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$ and ${\mathfrak{g}}_{*}\in {\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$, and ${\mathfrak{h}}_{*}\in {\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$, such that

$$\begin{array}{cc}\hfill {\mathfrak{H}}_{*}({\mathcal{Z}}_{*},{\mathcal{Q}}_{*},{\mathcal{Y}}_{*})\left(\varpi \right)=& \frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_{*}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_{*}\left(\tau \right)d\tau \right)d\rho }\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_{*}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_{*}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_{*}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_{*}\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\overline{\mathfrak{H}}}_{*}({\mathcal{Z}}_{*},{\mathcal{Q}}_{*},{\mathcal{Y}}_{*})\left(\varpi \right)\hfill \\ & =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2\mathsf{\Delta }}\left[\left({\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_{*}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_{*}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_{*}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_{*}\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\widehat{\mathfrak{H}}}_{*}({\mathcal{Z}}_{*},{\mathcal{Q}}_{*},{\mathcal{Y}}_{*})\left(\varpi \right)=& \frac{({\phi }^2{\varpi }^2-2\phi \varpi +2-e^{-\phi \varpi })}{{\phi }^3\mathsf{\Delta }}\hfill \\ & {\displaystyle \times \left[\left({\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right.\right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_{*}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_{*}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{{(}_{\mathfrak{h}})}_{*}\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_{*}\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_{*}\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

that is, $({\mathfrak{H}}_n,{\overline{\mathfrak{H}}}_n,\widehat{{\mathfrak{H}}_n})\in \mathcal{K}({\mathcal{Z}}_{*},{\mathcal{Q}}_{*},{\mathcal{Y}}_{*})$.

Exists with: let $(\mathcal{Z},\mathcal{Q},\mathcal{Y})\in \nu \mathcal{K}(\mathcal{Z},\mathcal{Q},\mathcal{Y})$. this implies that the function $\mathfrak{f}\in {\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$, $\mathfrak{g}\in {\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$, and $\mathfrak{h}\in {\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$ exists with:

$$\begin{array}{cc}\hfill \left|\right|\mathcal{Z}\left|\right|\le & {\mathcal{P}}_1\left|\right|P_1\left|\right|[{\mathcal{U}}_1\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_1\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_1\left(\right|\left|\mathcal{Y}\right|\left|\right)]\hfill \\ & +{\mathcal{Q}}_1\left|\right|P_2\left|\right|[{\mathcal{U}}_2\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_2\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_2\left(\right|\left|\mathcal{Y}\right|\left|\right)]\hfill \\ & +{\mathcal{O}}_1\left|\right|P_3\left|\right|[{\mathcal{U}}_3\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_3\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_3\left(\right|\left|\mathcal{Y}\right|\left|\right)],\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left|\right|\mathcal{Q}\left|\right|\le & {\mathcal{P}}_2\left|\right|P_1\left|\right|[{\mathcal{U}}_1\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_1\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_1\left(\right|\left|\mathcal{Y}\right|\left|\right)]\hfill \\ & +{\mathcal{Q}}_2\left|\right|P_2\left|\right|[{\mathcal{U}}_2\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_2\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_2\left(\right|\left|\mathcal{Y}\right|\left|\right)]\hfill \\ & +{\mathcal{O}}_2\left|\right|P_3\left|\right|[{\mathcal{U}}_3\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_3\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_3\left(\right|\left|\mathcal{Y}\right|\left|\right)],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left|\right|\mathcal{Y}\left|\right|\le & {\mathcal{P}}_3\left|\right|P_1\left|\right|[{\mathcal{U}}_1\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_1\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_1\left(\right|\left|\mathcal{Y}\right|\left|\right)]\hfill \\ & +{\mathcal{Q}}_3\left|\right|P_2\left|\right|[{\mathcal{U}}_2\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_2\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_2\left(\right|\left|\mathcal{Y}\right|\left|\right)]\hfill \\ & +{\mathcal{O}}_3\left|\right|P_3\left|\right|[{\mathcal{U}}_3\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_3\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_3\left(\right|\left|\mathcal{Y}\right|\left|\right)],\hfill \end{array}$$

following the same arguments

$$\begin{array}{cc}\hfill \left|\right|(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left|\right|=& \left|\right|\mathcal{Z}\left|\right|+\left|\right|\mathcal{Q}\left|\right|+\left|\right|\mathcal{Y}\left|\right|\hfill \\ \hfill \le & ({\mathcal{P}}_1+{\mathcal{P}}_2+{\mathcal{P}}_3)\left|\right|P_1\left|\right|[{\mathcal{U}}_1\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_1\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_1\left(\right|\left|\mathcal{Y}\right|\left|\right)]\hfill \\ & +({\mathcal{Q}}_1+{\mathcal{Q}}_2+{\mathcal{Q}}_3)\left|\right|P_2\left|\right|[{\mathcal{U}}_2\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_2\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_2\left(\right|\left|\mathcal{Y}\right|\left|\right)]\hfill \\ & +({\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3)\left|\right|P_3\left|\right|[{\mathcal{U}}_3\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_3\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_3\left(\right|\left|\mathcal{Y}\right|\left|\right)].\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \frac{\left|\right|(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left|\right|}{({\mathcal{P}}_1+{\mathcal{P}}_2+{\mathcal{P}}_3)\left|\right|P_1\left|\right|[{\mathcal{U}}_1\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_1\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_1\left(\right|\left|\mathcal{Y}\right|\left|\right)]+({\mathcal{Q}}_1+{\mathcal{Q}}_2+{\mathcal{Q}}_3)\left|\right|P_2\left|\right|[{\mathcal{U}}_2\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_2\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_2\left(\right|\left|\mathcal{Y}\right|\left|\right)]+({\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3)\left|\right|P_3\left|\right|[{\mathcal{U}}_3\left(\right|\left|\mathcal{Z}\right|\left|\right)+{\mathcal{V}}_3\left(\right|\left|\mathcal{Q}\right|\left|\right)+{\mathcal{W}}_3\left(\right|\left|\mathcal{Y}\right|\left|\right)]}\le 1.\end{array}$$

In the light of ${\mathcal{B}}_3$ we can find $\mathcal{N}$ with $\left|\right|(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left|\right|\ne \mathcal{N}$. Consider

$$\begin{array}{c}\hfill \mathsf{\Lambda }=\{(\mathcal{Z},\mathcal{Q},\mathcal{Y})\in \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}:|\left|(\mathcal{Z},\mathcal{Q},\mathcal{Y})\right||<\mathcal{N}\}.\end{array}$$

Here, $\mathcal{K}:\overline{\mathsf{\Lambda }}\to {\mathcal{P}}_{cp,cv}\left(\widehat{\mathcal{J}}\right)\times {\mathcal{P}}_{cp,cv}\left(\widehat{\mathcal{J}}\right)\times {\mathcal{P}}_{cp,cv}\left(\widehat{\mathcal{J}}\right)$ is completely continuous and upper semi-continuous. There is no $(\mathcal{Z},\mathcal{Q},\mathcal{Y})\in \nu \mathcal{K}(\mathcal{Z},\mathcal{Q},\mathcal{Y})$ for some $\nu \in (0,1)$ depending on choosing of $\mathsf{\Lambda }$.

So, by the non-linear alternative of Leray–Schauder type, we conclude that $\mathcal{K}$ has it least one fixed point $(\mathcal{Z},\mathcal{Q},\mathcal{Y})\in \overline{\mathsf{\Lambda }}$, this solution of problem (1). By this, we finalize the proof. □

3.2. The Case of Libschitz

Here, we consider the situation where there are non-convex values in the multi valued maps of system (1).

Consider the metric space $(\mathcal{E},d)$ which is induced from the normed space $(\mathcal{E};||.|\left|\right)$, and consider ${\mathcal{H}}_d:\mathcal{P}\left(\mathcal{E}\right)\times \mathcal{P}\left(\mathcal{E}\right)\to {\mathcal{R}}_e\infty$ be given by ${\mathcal{H}}_d(\mathcal{U},\mathcal{V})=max\{{sup}_{\mathcal{Y}\in \mathcal{U}}d(\mathcal{Y},\mathcal{V}),{sup}_{\nu \in \mathcal{V}}d(\mathcal{U},\nu )\},$ where $d(\mathcal{U},\nu )={inf}_{\mathcal{Y}\in \mathcal{U}}d(\mathcal{Y},\nu )$ and $d(\nu ,\mathcal{V})={inf}_{\nu \in \mathcal{V}}d(\mathcal{Y},\nu ).$

So $({\mathcal{P}}_{b,cl}\left(\mathcal{E}\right),{\mathcal{H}}_d)$ is a metric space and $({\mathcal{P}}_{cl}\left(\mathcal{E}\right),{\mathcal{H}}_d)$ is a generalized one.

In the upcoming result, we take advantage of Covitz and Nadler’s fixed point theorem for multi-valued maps.

Theorem 2. 

$\left({\mathcal{B}}_4\right){\mathcal{F}}_1,{\mathcal{F}}_2,{\mathcal{F}}_3:[0,\mathcal{T}]\times {\mathcal{R}}_e^3\to {\mathcal{P}}_{cp}\left({\mathcal{R}}_e\right)$ are such that ${\mathcal{F}}_1(.,\mathcal{Z},\mathcal{Q},\mathcal{Y}):[0,\mathcal{T}]\to {\mathcal{P}}_{cp}\left({\mathcal{R}}_e\right)$, ${\mathcal{F}}_2(.,\mathcal{Z},\mathcal{Q},\mathcal{Y}):[0,\mathcal{T}]\to {\mathcal{P}}_{cp}\left({\mathcal{R}}_e\right)$, and ${\mathcal{F}}_3(.,\mathcal{Z},\mathcal{Q},\mathcal{Y}):[0,\mathcal{T}]\to {\mathcal{P}}_{cp}\left({\mathcal{R}}_e\right)$ are measurable for each $\mathcal{Z},\mathcal{Q},\mathcal{Y}\in {\mathcal{R}}_e;$

$\left({\mathcal{B}}_5\right)$

$$\begin{array}{c}\hfill {\mathcal{H}}_d({\mathcal{F}}_1(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y}),{\mathcal{F}}_1(\varpi ,\overline{\mathcal{Z}},\overline{\mathcal{X}},\overline{\mathcal{Y}}))\le m_1\left(\varpi \right)\left(\right|\mathcal{Z}-\overline{\mathcal{Z}}|+|\mathcal{Q}-\overline{\mathcal{X}}|+|\mathcal{Y}-\overline{\mathcal{Y}}\left|\right),\end{array}$$

$$\begin{array}{c}\hfill {\mathcal{H}}_d({\mathcal{F}}_2(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y}),{\mathcal{F}}_2(\varpi ,\overline{\mathcal{Z}},\overline{\mathcal{X}},\overline{\mathcal{Y}}))\le m_2\left(\varpi \right)\left(\right|\mathcal{Z}-\overline{\mathcal{Z}}|+|\mathcal{Q}-\overline{\mathcal{X}}|+|\mathcal{Y}-\overline{\mathcal{Y}}\left|\right),\end{array}$$

and

$$\begin{array}{c}\hfill {\mathcal{H}}_d({\mathcal{F}}_3(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y}),{\mathcal{F}}_3(\varpi ,\overline{\mathcal{Z}},\overline{\mathcal{X}},\overline{\mathcal{Y}}))\le m_3\left(\varpi \right)\left(\right|\mathcal{Z}-\overline{\mathcal{Z}}|+|\mathcal{Q}-\overline{\mathcal{X}}|+|\mathcal{Y}-\overline{\mathcal{Y}}\left|\right),\end{array}$$

for the majority of $\varpi \in [0,\mathcal{T}]$ and $\mathcal{Z},\mathcal{Q},\mathcal{Y},\overline{\mathcal{Z}},\overline{\mathcal{X}},\overline{\mathcal{Y}}\in {\mathcal{R}}_e$ with $m_1,m_2,m_3\in C([0,\mathcal{T}],{\mathcal{R}}_e^{+})$ and $d(0,{\mathcal{F}}_1(\varpi ,0,0))\le m_1\left(\varpi \right),d(0,{\mathcal{F}}_2(\varpi ,0,0))\le m_2\left(\varpi \right),d(0,{\mathcal{F}}_3(\varpi ,0,0))\le m_3\left(\varpi \right)$ for almost $\varpi \in [0,\mathcal{T}]$ hold, this implies the existence of solution for system (1) on $[0,\mathcal{T}]$ given that

$$\begin{array}{c}\hfill ({\mathcal{P}}_1+{\mathcal{P}}_2+{\mathcal{P}}_3)\left|\right|m_1\left|\right|+({\mathcal{Q}}_1+{\mathcal{Q}}_2+{\mathcal{Q}}_3)\left|\right|m_2\left|\right|+({\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3)\left|\right|m_3\left|\right|<1.\end{array}$$

(14)

Proof. 

The sets ${\mathcal{S}}_{\mathfrak{f},(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$,${\mathcal{S}}_{\mathfrak{g},(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$ and ${\mathcal{S}}_{\mathfrak{h},(\mathcal{Z},\mathcal{Q},\mathcal{Y})}$ are non-empty for each $(\mathcal{Z},\mathcal{Q},\mathcal{Y})\in \mathcal{E}\times \mathcal{E}\times \mathcal{E}$ by assumption ${\mathcal{B}}_4$, so $\mathfrak{f}$, $\mathfrak{g}$ and $\mathfrak{h}$ have measurable selections. We now demonstrate that the operator $\mathfrak{K}$ meets the criteria of Covitz and Nadler’s fixed point theorem.

We start by $\mathfrak{K}(\mathcal{Z},\mathcal{Q},\mathcal{Y})\in {\mathcal{P}}_{cl}\left(\mathcal{E}\right)\times {\mathcal{P}}_{cl}\left(\mathcal{E}\right)\times {\mathcal{P}}_{cl}\left(\mathcal{E}\right)$ for each $(\mathcal{Z},\mathcal{Q},\mathcal{Y})\in \mathcal{E}\times \mathcal{E}\times \mathcal{E}$. Let $({\mathfrak{H}}_n,{\overline{\mathfrak{H}}}_n,\widehat{\mathfrak{H}})\in \mathfrak{K}({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)$, such that $({\mathfrak{H}}_n,{\overline{\mathfrak{H}}}_n,\widehat{{\mathfrak{H}}_n})\to (\mathfrak{H},\overline{\mathfrak{H}},\widehat{\mathfrak{H}})$ in $\mathcal{E}\times \mathcal{E}\times \mathcal{E}$. Then $(\mathfrak{H},\overline{\mathfrak{H}},\widehat{\mathfrak{H}})\in \mathcal{E}\times \mathcal{E}\times \mathcal{E}$ and there exists ${\mathfrak{f}}_n\in {\mathcal{S}}_{{\mathcal{F}}_1,({\mathcal{Z}}_n,{\mathcal{Y}}_n,{\mathcal{Q}}_n)}$, ${\mathfrak{g}}_n\in {\mathcal{S}}_{{\mathcal{F}}_2,({\mathcal{Z}}_n,{\mathcal{Y}}_n,{\mathcal{Q}}_n)}$, and ${\mathfrak{h}}_n\in {\mathcal{S}}_{{\mathcal{F}}_3,({\mathcal{Z}}_n,{\mathcal{Y}}_n,{\mathcal{Q}}_n)}$, such that

$$\begin{array}{cc}\hfill {\mathfrak{H}}_n({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\left(\varpi \right)=& \frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho }\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\overline{\mathfrak{H}}}_n({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\left(\varpi \right)& =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2\mathsf{\Delta }}\left[\left({\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\widehat{\mathfrak{H}}}_n({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\left(\varpi \right)=& \frac{({\phi }^2{\varpi }^2-2\phi \varpi +2-e^{-\phi \varpi })}{{\phi }^3\mathsf{\Delta }}\hfill \\ & {\displaystyle \times \left[\left({\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.\right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{{(}_{\mathfrak{h}})}_n\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_n\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_n\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

because of the compact values ${\mathcal{F}}_1$, ${\mathcal{F}}_2$, and ${\mathcal{F}}_3$, we take the sub-sequences to show that ${\mathfrak{f}}_n$, ${\mathfrak{g}}_n$ and ${\mathfrak{h}}_n$ tends to $\mathfrak{f}$, $\mathfrak{g}$, and $\mathfrak{h}$ in ${\mathfrak{L}}^1([0,\mathcal{T}],{\mathcal{R}}_e)$, respectively. Thus, $\mathfrak{f}\in {\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Y},\mathcal{Q})}$, $\mathfrak{g}\in {\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Y},\mathcal{Q})}$, and $\mathfrak{h}\in {\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Y},\mathcal{Q})}$ for each $\varpi \in [0,\mathcal{T}]$ and that

$$\begin{array}{cc}\hfill & {\mathfrak{H}}_n({\mathcal{Y}}_n,{\mathcal{Q}}_n)\left(\varpi \right)\to \mathfrak{H}(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)\hfill \\ \hfill =& \frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\left(\mathfrak{g}\right)\left(\tau \right)d\tau \right)d\rho \right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\left(\mathfrak{g}\right)\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\left(\mathfrak{f}\right)\left(\tau \right)d\tau \right)d\rho }\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\left(\mathfrak{h}\right)\left(\tau \right)d\tau \right)d\rho \right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\left(\mathfrak{h}\right)\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\left(\mathfrak{g}\right)\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\left(\mathfrak{g}\right)\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\left(\mathfrak{g}\right)\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\left(\mathfrak{f}\right)\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\left(\mathfrak{f}\right)\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\left(\mathfrak{f}\right)\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\left(\mathfrak{h}\right)\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\left(\mathfrak{f}\right)\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\overline{\mathfrak{H}}}_n({\mathcal{Y}}_n,{\mathcal{Q}}_n)\left(\varpi \right)\to \overline{\mathfrak{H}}(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)\hfill \\ & =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2\mathsf{\Delta }}\left[\left({\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\left(\mathfrak{h}\right)\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\left(\mathfrak{h}\right)\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\left(\mathfrak{g}\right)\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\left(\mathfrak{g}\right)\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\left(\mathfrak{g}\right)\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\left(\mathfrak{f}\right)\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\left(\mathfrak{f}\right)\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\left(\mathfrak{f}\right)\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\left(\mathfrak{h}\right)\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\left(\mathfrak{g}\right)\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \widehat{{\mathfrak{H}}_n}({\mathcal{Y}}_n,{\mathcal{Q}}_n)\left(\varpi \right)\to \widehat{\mathfrak{H}}(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left(\varpi \right)\hfill \\ \hfill =& \frac{({\phi }^2{\varpi }^2-2\phi \varpi +2-e^{-\phi \varpi })}{{\phi }^3\mathsf{\Delta }}\hfill \\ & {\displaystyle \times \left[\left({\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\left(\mathfrak{g}\right)\left(\tau \right)d\tau \right)d\rho \right.\right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}\left(\mathfrak{g}\right)\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\left(\mathfrak{f}\right)\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}\left(\mathfrak{f}\right)\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}\left(\mathfrak{f}\right)\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\left(\mathfrak{h}\right)\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\left(\mathfrak{h}\right)\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}\left(\mathfrak{h}\right)\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}\left(\mathfrak{g}\right)\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}\left(\mathfrak{h}\right)\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

So $(\mathfrak{H},\overline{\mathfrak{H}},\widehat{{\mathfrak{H}}_n})\in \mathcal{K}$, which guarantees that $\mathcal{K}$ is closed. After that, we prove the existence of $\overline{\theta }<1$, such that

$$\begin{array}{c}\hfill {\mathcal{H}}_d\left(\mathfrak{K}(\mathcal{Z},\mathcal{Q},\mathcal{Y}),\mathfrak{K}(\overline{\mathcal{Z}},\overline{\mathcal{X}},\overline{\mathcal{Y}})\right)\le \overline{\theta }\left(\right||\mathcal{Z}-\overline{\mathcal{Z}}\left|\right|+\left|\right|\mathcal{Q}-\overline{\mathcal{X}}\left|\right|+\left|\right|\mathcal{Y}-\overline{\mathcal{Y}}\left|\right|)\mathrm{for} \mathrm{each} \mathcal{Z},\overline{\mathcal{Z}},\mathcal{Q},\overline{\mathcal{X}},\mathcal{Y},\overline{\mathcal{Y}}\in \widehat{\mathcal{J}}.\end{array}$$

Let $(\mathcal{Z},\overline{\mathcal{Z}}),(\mathcal{Q},\overline{\mathcal{X}}),(\mathcal{Y},\overline{\mathcal{Y}})\in \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}\times \widehat{\mathcal{J}}$ and $({\mathfrak{H}}_1,{\overline{\mathfrak{H}}}_{\mathfrak{1}},\widehat{{\mathfrak{H}}_1})\in \mathfrak{K}(\mathcal{Z},\mathcal{Q},\mathcal{Y}).$ then there exist $\mathfrak{f}\in {\mathcal{S}}_{{\mathcal{F}}_1,(\mathcal{Z},\mathcal{Y},\mathcal{Q})}$, $\mathfrak{g}\in {\mathcal{S}}_{{\mathcal{F}}_2,(\mathcal{Z},\mathcal{Y},\mathcal{Q})}$, and $\mathfrak{h}\in {\mathcal{S}}_{{\mathcal{F}}_3,(\mathcal{Z},\mathcal{Y},\mathcal{Q})}$ such that, ∀$\varpi \in [0,\mathcal{T}]$, this gives

$$\begin{array}{cc}\hfill {\mathfrak{H}}_1({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\left(\varpi \right)=& \frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_1\left(\tau \right)d\tau \right)d\rho \right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_1\left(\tau \right)d\tau \right)d\rho }\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_1\left(\tau \right)d\tau \right)d\rho \right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_1\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_1\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_1\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_1\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_1\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_1\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\overline{\mathfrak{H}}}_1({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\left(\varpi \right)& =\frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2\mathsf{\Delta }}\left[\left({\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_1\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_1\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_1\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_1\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_1\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_1\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_1\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\widehat{\mathfrak{H}}}_1({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\left(\varpi \right)=& \frac{({\phi }^2{\varpi }^2-2\phi \varpi +2-e^{-\phi \varpi })}{{\phi }^3\mathsf{\Delta }}\hfill \\ & {\displaystyle \times \left[\left({\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_1\left(\tau \right)d\tau \right)d\rho \right.\right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_1\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_1\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_1\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_1\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_1\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_1\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_1\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

By $\left({\mathcal{B}}_5\right)$, we have

$$\begin{array}{c}\hfill {\mathcal{H}}_d({\mathcal{F}}_1(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y}),{\mathcal{F}}_1(\varpi ,\overline{\mathcal{Z}},\overline{\mathcal{X}},\overline{\mathcal{Y}}))\le m_1\left(\varpi \right)\left(\right|\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)|+|\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)|+|\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)\left|\right)\end{array}$$

$$\begin{array}{c}\hfill {\mathcal{H}}_d({\mathcal{F}}_2(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y}),{\mathcal{F}}_2(\varpi ,\overline{\mathcal{Z}},\overline{\mathcal{X}},\overline{\mathcal{Y}}))\le m_2\left(\varpi \right)\left(\right|\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)|+|\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)|+|\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)\left|\right)\end{array}$$

and

$$\begin{array}{c}\hfill {\mathcal{H}}_d({\mathcal{F}}_3(\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y}),{\mathcal{F}}_3(\varpi ,\overline{\mathcal{Z}},\overline{\mathcal{X}},\overline{\mathcal{Y}}))\le m_3\left(\varpi \right)\left(\right|\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)|+|\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)|+|\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)\left|\right)\end{array}$$

So, there $\mathfrak{f}\in {\mathcal{F}}_1\left((\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y})\right)$, $\mathfrak{g}\in {\mathcal{F}}_2\left((\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y})\right)$ and $\mathfrak{h}\in {\mathcal{F}}_3\left((\varpi ,\mathcal{Z},\mathcal{Q},\mathcal{Y})\right)$ such that

$$\begin{array}{c}\hfill |{\mathfrak{f}}_1\left(\varpi \right)-{\gamma }_1|\le m_1\left(\varpi \right)\left(\right|\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)|+|\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)|+|\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)\left|\right)\end{array}$$

$$\begin{array}{c}\hfill |{\mathfrak{g}}_1\left(\varpi \right)-{\gamma }_2|\le m_2\left(\varpi \right)\left(\right|\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)|+|\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)|+|\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)\left|\right)\end{array}$$

and

$$\begin{array}{c}\hfill |{\mathfrak{h}}_1\left(\varpi \right)-{\gamma }_3|\le m_3\left(\varpi \right)\left(\right|\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)|+|\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)|+|\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)\left|\right)\end{array}$$

Defined ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3:[0,\mathcal{T}]\to \mathcal{P}\left({\mathcal{R}}_e\right)$ by

$$\begin{array}{cc}\hfill {\mathcal{V}}_1\left(\varpi \right)=& \{\mathfrak{f}\in {\mathfrak{L}}^1([0,\mathcal{T}],{\mathcal{R}}_e):|{\mathfrak{f}}_1\left(\varpi \right)-{\gamma }_1|\hfill \\ & \le m_1\left(\varpi \right)\left(\right|\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)|+|\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)|+|\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)\left|\right)\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{V}}_2\left(\varpi \right)=& \{\mathfrak{g}\in {\mathfrak{L}}^1([0,\mathcal{T}],{\mathcal{R}}_e):|{\mathfrak{g}}_1\left(\varpi \right)-{\gamma }_2|\hfill \\ & \le m_2\left(\varpi \right)\left(\right|\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)|+|\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)|+|\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)\left|\right)\},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\mathcal{V}}_3\left(\varpi \right)=& \{\mathfrak{h}\in {\mathfrak{L}}^1([0,\mathcal{T}],{\mathcal{R}}_e):|{\mathfrak{h}}_1\left(\varpi \right)-{\gamma }_3|\hfill \\ & \le m_3\left(\varpi \right)\left(\right|\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)|+|\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)|+|\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)\left|\right)\}.\hfill \end{array}$$

Since the multi-valued operators ${\mathcal{V}}_1\left(\varpi \right)\cap {\mathcal{F}}_1(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right))$, ${\mathcal{V}}_2\left(\varpi \right)\cap {\mathcal{F}}_2(\varpi ,$$\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right))$ and ${\mathcal{V}}_3\left(\varpi \right)\cap {\mathcal{F}}_3(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right))$ are measurable, there exist functions ${\mathfrak{f}}_2\left(\varpi \right),{\mathfrak{g}}_2\left(\varpi \right),{\mathfrak{h}}_2\left(\varpi \right)$ which are a measurable selection for ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3$ and ${\mathfrak{f}}_2\left(\varpi \right)\in {\mathcal{F}}_1(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right))$, ${\mathfrak{g}}_2\left(\varpi \right)\in {\mathcal{F}}_2(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right))$, ${\mathfrak{h}}_2\left(\varpi \right)\in {\mathcal{F}}_3(\varpi ,\mathcal{Z}\left(\varpi \right),$$\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right))$ such that, for a.e. $\varpi \in [0,\mathcal{T}]$, we have

$$\begin{array}{c}\hfill |{\mathfrak{f}}_1\left(\varpi \right)-{\mathfrak{f}}_2\left(\varpi \right)|\le m_1\left(\varpi \right)\left(\right|\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)|+|\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)|+|\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)\left|\right),\end{array}$$

$$\begin{array}{c}\hfill |{\mathfrak{g}}_1\left(\varpi \right)-{\mathfrak{g}}_2\left(\varpi \right)|\le m_2\left(\varpi \right)\left(\right|\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)|+|\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)|+|\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)\left|\right),\end{array}$$

and

$$\begin{array}{c}\hfill |{\mathfrak{h}}_1\left(\varpi \right)-{\mathfrak{h}}_2\left(\varpi \right)|\le m_3\left(\varpi \right)\left(\right|\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)|+|\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)|+|\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)\left|\right).\end{array}$$

Let

$$\begin{array}{cc}\hfill {\mathfrak{H}}_2({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\left(\varpi \right)=& \frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2{\mathcal{E}}_1}\left[{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_2\left(\tau \right)d\tau \right)d\rho \right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle -{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_2\left(\tau \right)d\tau \right)d\rho }\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left({\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_2\left(\tau \right)d\tau \right)d\rho \right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_2\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_2\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_2\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_2{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_2\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_2\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_2\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\overline{\mathfrak{H}}}_2({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\left(\varpi \right)=& \frac{(\phi \varpi -1+e^{-\phi \varpi })}{{\phi }^2\mathsf{\Delta }}\left[\left({\mathcal{E}}_1{\mathcal{E}}_5\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_2\left(\tau \right)d\tau \right)d\rho \right.\right.\right.\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_2\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_4{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_2\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_2\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_4\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_2\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_2\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_2\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\widehat{\mathfrak{H}}}_2({\mathcal{Z}}_n,{\mathcal{Q}}_n,{\mathcal{Y}}_n)\left(\varpi \right)=& \frac{({\phi }^2{\varpi }^2-2\phi \varpi +2-e^{-\phi \varpi })}{{\phi }^3\mathsf{\Delta }}\hfill \\ & {\displaystyle \times \left[\left({\mathcal{E}}_3{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_2\left(\tau \right)d\tau \right)d\rho \right.\right.\right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_1{\int }_0^{\varsigma }\frac{{(\varsigma -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_2\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_1{\mathcal{E}}_3\left\{{\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_2\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_3{\int }_0^{\delta }\frac{{(\delta -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\psi -2}}{\Gamma (\psi -1)}{\left(\mathfrak{f}\right)}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_2\left(\tau \right)d\tau \right)d\rho \right\}}\hfill \\ & {\displaystyle +{\mathcal{E}}_2{\mathcal{E}}_6\left\{{\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j{\int }_0^{{\zeta }_j}{e}^{-\phi ({\zeta }_j-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_2\left(\tau \right)d\tau \right)d\rho \right.}\hfill \\ & {\displaystyle +{\mathrm{\Pi }}_2{\int }_0^{\varrho }\frac{{(\varrho -\rho )}^{\vartheta -1}}{\Gamma \left(\vartheta \right)}\left({\int }_0^{\rho }{e}^{-\phi (\rho -\tau )}\left({\int }_0^{\tau }\frac{{(\tau -m)}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_2\left(m\right)dm\right)d\tau \right)d\rho }\hfill \\ & {\displaystyle \left.\left.\left.-{\int }_0^{\mathcal{T}}{e}^{-\phi (\mathcal{T}-\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\varphi -2}}{\Gamma (\varphi -1)}{\left(\mathfrak{g}\right)}_2\left(\tau \right)d\tau \right)d\rho \right\}\right)\right]}\hfill \\ & +{\int }_0^{\varpi }{e}^{-\phi (\varpi -\rho )}\left({\int }_0^{\rho }\frac{{(\rho -\tau )}^{\omega -2}}{\Gamma (\omega -1)}{\left(\mathfrak{h}\right)}_2\left(\tau \right)d\tau \right)d\rho ,\hfill \end{array}$$

Hence

$$\begin{array}{cc}\hfill \left|\right|{\mathfrak{H}}_1(\mathcal{Z},\mathcal{Q},\mathcal{Y})-{\mathfrak{H}}_2(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left|\right|\le & \left({\mathcal{P}}_1\right|\left|m_1\right||+{\mathcal{Q}}_1|\left|m_2\right||++{\mathcal{O}}_1|\left|m_3\right|\left|\right)\hfill \\ & \times \left(\right||\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)||+||\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)||+||\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)|\left|\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left|\right|{\overline{\mathfrak{H}}}_1(\mathcal{Z},\mathcal{Q},\mathcal{Y})-{\overline{\mathfrak{H}}}_2(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left|\right|\le & \left({\mathcal{P}}_2\right|\left|m_1\right||+{\mathcal{Q}}_2|\left|m_2\right||++{\mathcal{O}}_2|\left|m_3\right|\left|\right)\hfill \\ & \times \left(\right||\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)||+||\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)||+||\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)|\left|\right)\hfill \end{array}$$

Similar, we set

$$\begin{array}{cc}\hfill \left|\right|\widehat{{\mathfrak{H}}_1}(\mathcal{Z},\mathcal{Q},\mathcal{Y})-\widehat{{\mathfrak{H}}_2}(\mathcal{Z},\mathcal{Q},\mathcal{Y})\left|\right|\le & \left({\mathcal{P}}_3\right|\left|m_1\right||+{\mathcal{Q}}_3|\left|m_2\right||++{\mathcal{O}}_3|\left|m_3\right|\left|\right)\hfill \\ & \times \left(\right||\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)||+||\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)||+||\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)|\left|\right)\hfill \end{array}$$

Thus

$$\begin{array}{cc}\hfill \left|\right|({\mathfrak{H}}_1,{\overline{\mathfrak{H}}}_1,\widehat{{\mathfrak{H}}_1}),({\mathfrak{H}}_2,{\overline{\mathfrak{H}}}_{\mathfrak{2}},\widehat{{\mathfrak{H}}_2})\left|\right|\le & ({\mathcal{P}}_1+{\mathcal{P}}_2+{\mathcal{P}}_3)\left|\right|m_1\left|\right|+({\mathcal{Q}}_1+{\mathcal{Q}}_2+{\mathcal{Q}}_3)\left|\right|m_2\left|\right|+({\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3)\left|\right|m_3\left|\right|\hfill \\ & \times \left(\right||\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)||+||\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)||+||\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)|\left|\right)\hfill \end{array}$$

likewise, by reversing the roles of $(\mathcal{Z},\mathcal{Q},\mathcal{Y})$ and $(\overline{\mathcal{Z}},\overline{\mathcal{Y}},\overline{\mathcal{X}})$, it is possible to obtain

$$\begin{array}{cc}\hfill {\mathcal{H}}_d[\mathcal{T}(\mathcal{Z},\mathcal{Q},\mathcal{Y}),\mathcal{T}(\overline{\mathcal{Z}},\overline{\mathcal{X}},\overline{\mathcal{Y}})]\le & ({\mathcal{P}}_1+{\mathcal{P}}_2+{\mathcal{P}}_3)\left|\right|m_1\left|\right|+({\mathcal{Q}}_1+{\mathcal{Q}}_2+{\mathcal{Q}}_3)\left|\right|m_2\left|\right|+({\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3)\left|\right|m_3\left|\right|\hfill \\ & \times \left(\right||\mathcal{Z}\left(\varpi \right)-\overline{\mathcal{Z}}\left(\varpi \right)||+||\mathcal{Q}\left(\varpi \right)-\overline{\mathcal{X}}\left(\varpi \right)||+||\mathcal{Y}\left(\varpi \right)-\overline{\mathcal{Y}}\left(\varpi \right)|\left|\right).\hfill \end{array}$$

In light of the assumption, $\mathfrak{K}$ is a contraction. Therefore, according to Covitz and Nadler’s fixed point theorem, it has a fixed point $(\mathcal{Z},\mathcal{Q},\mathcal{Y})$ that is a solution to problem (1). This concludes the proof. □

4. Example

Example 1.

In consistence with systems (1) the along with the main mentioned theorems, we present an example in this section.

$$\left\{\begin{array}{c}{(}^c{\mathfrak{D}}^{\psi }+{\phi }^c{\mathfrak{D}}^{\psi -1})\mathcal{Z}\left(\varpi \right)\in {\mathcal{F}}_1(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right)),2<\psi \le 3,\hfill \\ {(}^c{\mathfrak{D}}^{\varphi }+{\phi }^c{\mathfrak{D}}^{\varphi -1})\mathcal{Q}\left(\varpi \right)\in {\mathcal{F}}_2(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right)),2<\varphi \le 3,\hfill \\ {(}^c{\mathfrak{D}}^{\omega }+{\phi }^c{\mathfrak{D}}^{\omega -1})\mathcal{Y}\left(\varpi \right)\in {\mathcal{F}}_3(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right)),3<\omega \le 4,\hfill \\ {\displaystyle \mathcal{Z}\left(0\right)=0,{\mathcal{Z}}^{\prime }\left(0\right)=0,\mathcal{Z}\left(\mathcal{T}\right)={\mathsf{{\rm Y}}}_1\sum _{j=1}^{k-2}{\xi }_j\mathcal{Q}\left({\zeta }_j\right)+{\mathrm{\Pi }}_1{\mathcal{I}}^{\varsigma }\mathcal{Q}\left(\vartheta \right),}\hfill \\ {\displaystyle \mathcal{Q}\left(0\right)=0,{\mathcal{Q}}^{\prime }\left(0\right)=0,\mathcal{Q}\left(\mathcal{T}\right)={\mathsf{{\rm Y}}}_2\sum _{j=1}^{k-2}{\nu }_j\mathcal{Y}\left({\zeta }_j\right)+{\mathrm{\Pi }}_2{\mathcal{I}}^{\varrho }\mathcal{Y}\left(\vartheta \right),}\hfill \\ {\displaystyle \mathcal{Y}\left(0\right)=0,{\mathcal{Y}}^{\prime }\left(0\right)=0,{\mathcal{Y}}^{″}\left(0\right)=0,\mathcal{Y}\left(\mathcal{T}\right)={\mathsf{{\rm Y}}}_3\sum _{j=1}^{k-2}{\sigma }_j\mathcal{Z}\left({\zeta }_j\right)+{\mathrm{\Pi }}_3{\mathcal{I}}^{\delta }\mathcal{Z}\left(\vartheta \right),}\hfill \end{array}\right.$$

(15)

Here $\psi =\frac{7}{3},\varphi =\frac{5}{2},\omega =\frac{10}{3};\varsigma =9/20;\varrho =11/20;\delta =13/20;\vartheta =93/50;{\zeta }_j=36/25;{\mathsf{{\rm Y}}}_1=17/400;{\mathsf{{\rm Y}}}_2=15/300;{\mathsf{{\rm Y}}}_3=13/200;{\mathrm{\Pi }}_1=17/200;{\mathrm{\Pi }}_2=8/125;{\mathrm{\Pi }}_3=6/68;\mathcal{T}=1;{\zeta }_1=1/20;{\zeta }_2=2/20;{\nu }_1=1/100;{\nu }_2=1/50;{\sigma }_1=1/1000;{\sigma }_2=1/500;k=4;\mathsf{\Delta }=0.067506818609056$ with the given data, it is found that

$$\begin{array}{cc}\hfill & {\mathcal{P}}_1=1.79143780787545,{\mathcal{P}}_2=0.83611149394660,{\mathcal{P}}_3=0.522203861964576,\hfill \\ & {\mathcal{Q}}_1=0.400530445936702,{\mathcal{Q}}_2=1.56029202354379,{\mathcal{Q}}_3=0.109316638513044,\hfill \\ & {\mathcal{O}}_1=0.190264056004677,{\mathcal{O}}_2=0.748929077567517,{\mathcal{O}}_3=0.745842301275926.\hfill \end{array}$$

To illustrate the use of Theorem 1, we will consider

$$\begin{array}{cc}\hfill & {\mathcal{F}}_1(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right))=\left[0,\frac{1}{12+{\varpi }^2}\left(\frac{\left|\mathcal{Z}\right|}{1+\left|\mathcal{Z}\right|}+\frac{|sin(\mathcal{Q}\left)\right|}{1+|sin(\mathcal{Q}\left)\right|}+{tan}^{-1}\left(\mathcal{Y}\left(\varpi \right)\right)\right),\frac{1}{15+\varpi }\right]\hfill \\ & {\mathcal{F}}_2(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right))=\left[0,\frac{e^{-2\varpi }}{2\sqrt{1600+\varpi }}\left(\frac{1}{270}sin\mathcal{Z}\left(\varpi \right)+\frac{\left|\mathcal{Q}\right|}{1+\left|\mathcal{Q}\right|}+\frac{|cos(\mathcal{Y}\left)\right|}{1+|cos(\mathcal{Y}\left)\right|}\right),\frac{1}{15+\varpi }\right]\hfill \\ & {\mathcal{F}}_3(\varpi ,\mathcal{Z}\left(\varpi \right),\mathcal{Q}\left(\varpi \right),\mathcal{Y}\left(\varpi \right))=\left[0,\frac{1}{2\sqrt{400+{\varpi }^2}}\left(\frac{1}{270}sin\left(\mathcal{Z}\right)+\frac{|tan(\mathcal{Q}\left)\right|}{1+|tan(\mathcal{Q}\left)\right|}+\frac{\left|\mathcal{Y}\right|}{1+\left|\mathcal{Y}\right|}\right),\frac{1}{15+\varpi }\right].\hfill \end{array}$$

Clearly, utilizing the above data, we obtain

$$\begin{array}{c}\hfill ({\mathcal{P}}_1+{\mathcal{P}}_2+{\mathcal{P}}_3)\left|\right|m_1\left|\right|+({\mathcal{Q}}_1+{\mathcal{Q}}_2+{\mathcal{Q}}_3)\left|\right|m_2\left|\right|+({\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3)\left|\right|m_3\left|\right|\approx 0.3304803666<1.\end{array}$$

(16)

All the conditions of Theorem 1, we will consider.

5. Conclusions

In this study, we investigated the existence of solutions for tripled fractional differential inclusions with fractional derivatives of different orders and non-local boundary conditions featuring fractional derivatives and integrals. The existence of both convex and non-convex multi-valued maps was established using the non-linear alternative of Kakutani maps and Covitz and Nadler’s fixed point theorem, see [30].