Existence and H-U stability of a tripled system of sequential fractional differential equations with multipoint boundary conditions

Abstract

In this paper, we introduce a new coupled system of sequential fractional differential equations with coupled boundary conditions. We establish existence and uniqueness results using the Leray–Schauder alternative and Banach contraction principle. We examine the stability of the solutions involved in the Hyers–Ulam type. As an application, we present a few examples to illustrate the main results.

1 Introduction

Over the last few decades, fractional calculus (FC) has evolved as an interesting subject of research. The FC methods greatly improved the study of integer-order mathematical models associated with real-world problems in a variety of scientific and technological disciplines, including finance, control theory [1], ecology [2], signal and image processing [3], blood flow phenomena [4], biophysics [5], and chaotic synchronization [6]. Fractional differential equations (FDEs) are more effective than classic integer-order differential equations (DEs) at representing real-world phenomena such as the knowledge and heredity properties of various materials. As a result, numerous scholars have examined FDEs in the mathematical modeling of a wide range of physical and technical processes [7–13]. Along with Riemann–Liouville, these operators are referred to in the literature as Grünwald–Letnikov, Caputo, Hilfer, and Hadamard.

Coupled systems with fractional differential equations are very important to study since they appear to have a wide range of problems in a variety of real-world scenarios. Scholars have also done numerous investigations of coupled systems of FDEs. Consider the following example: some of the most current results on the problem are contained in a series of papers [11, 14–17] and the references given in [18–22].

Stability analysis is another field of research that has received much attention to fractional differential equations in the last few decades. Various kinds of stability have been investigated in the literature, including Mittag-Leffler, Lyapunov, and others. To our knowledge, the Ulam–Hyers stability of a coupled system of fractional differential equations has been studied very rarely.

Ulam and Hyers discovered a novel type of stability called the Hyers–Ulam stability. This type of research can aid in understanding biochemical processes and fluid motion, as well as semiconductors, population dynamics, heat conduction, and elasticity. This paper summarizes research on integral and nonlocal boundary value problems for coupled FDEs. The papers [16, 23–29] provide more insight into the theoretical approaches to the topic.

Zada, Yar, and Li [26] studied the nonlinear sequential coupled system of Caputo fractional differential equations

$$ \textstyle\begin{cases} (\mathcal{D}^{\eta }+ \varphi \mathcal{D}^{\eta -1}) \mathfrak{p}( \varsigma )= \hat{\mathcal{F}_{1}} (\varsigma ,\mathfrak{p}( \varsigma ),\mathfrak{q}(\varsigma )), \quad 2< \eta \leq 3, \\ (\mathcal{D}^{\xi}+ \varphi \mathcal{D}^{\xi -1})\mathfrak{q}( \varsigma )= \hat{\mathcal{F}_{2}}(\varsigma ,\mathfrak{p}(\varsigma ), \mathfrak{q}(\varsigma )), \quad 2< \xi \leq 3, \\ \mathfrak{p}(0)={0}, \qquad \mathfrak{p}(T)= \sum_{j=1}^{k} \eta _{j} \mathcal{I}^{\rho _{j}} \mathfrak{q}(\varrho _{j}), \\ \mathfrak{q}(0)={0}, \qquad \mathfrak{q}(T)= \sum_{j=1}^{k} \beta _{j} \mathcal{I}^{\gamma _{j}} \mathfrak{q}(\varpi _{j}), \end{cases} $$

where ${}^{c}\mathcal{D}^{\eta}$ and ${}^{c}\mathcal{D}^{\xi}$ denote the Caputo fractional derivatives of orders η and ξ, $\mathcal{I}^{\rho _{j}}$and $\mathcal{I}^{\gamma _{j}}$ are the Riemann–Liouville fractional integrals of orders ${\rho _{j}} ,{\gamma _{j}} > 0$, $\beta _{j},\eta _{j} \in (0, T)$, $k \in \mathbbm{(}\mathbb{R}\mathbbm{)}^{+}$, $\hat{\mathcal{F}_{1}}$, $\hat{\mathcal{F}_{2}} : [0, T] \times \mathbb{R}\in \mathbb{R}^{2} \rightarrow \mathbb{R}$, and ${\rho _{j}} ,{\gamma _{j}} \in \mathbb{R}$, $i = 1, 2, \ldots , n$, $j = 1, 2, \ldots , m$, are real constants. The existence of solutions is established by the Banach contraction principle, and the uniqueness of solutions is established by the Leray–Schauder alternative. The Hyers–Ulam stability was also considered.

In [27] the authors studied a new kind of coupled system of three fractional differential equations with coupled boundary conditions:

$$ \textstyle\begin{cases} {}^{C}{D}_{a^{+}}^{\eta }u(\varsigma )= \rho (\varsigma ,u(\varsigma ), \hat{\mathcal{G}_{1}}(\varsigma ),\hat{\mathcal{G}_{2}}(\varsigma )),\quad 1< \eta \leq 2, \varsigma \in [a,b], \\ {}^{C}{D}_{a^{+}}^{\xi }\hat{\mathcal{G}_{1}}(\varsigma )= \varphi ( \varsigma ,u(\varsigma ),\hat{\mathcal{G}_{1}}(\varsigma ), \hat{\mathcal{G}_{2}}(\varsigma )), \quad 1< \xi \leq 2, \varsigma \in [a,b], \\ {}^{C}{D}_{a^{+}}^{\zeta }\hat{\mathcal{G}_{2}}(\varsigma )= \psi ( \varsigma ,u(\varsigma ),\hat{\mathcal{G}_{1}}(\varsigma ), \hat{\mathcal{G}_{2}}(\varsigma )),\quad 2< \zeta \leq 3, \varsigma \in [a,b], \\ u(a)=u_{0}, \qquad u(b)= \sum_{i=1}^{m} p_{i} \hat{\mathcal{G}_{1}}(\alpha _{i}), \\ \hat{\mathcal{G}_{1}}(a)=0, \qquad \hat{\mathcal{G}_{1}}(b)= \sum_{j=1}^{n} q_{j} \hat{\mathcal{G}_{2}}(\beta _{j}), \\ \hat{\mathcal{G}_{2}}(\xi _{1})=0, \qquad \hat{\mathcal{G}_{2}}(\xi _{2})=0,\qquad \hat{\mathcal{G}_{2}}(b)= \sum_{k=1}^{l}r_{k} u( \gamma _{k}), \\ a< \xi _{1} < \xi _{2} < \alpha _{1}< \cdots < \alpha _{m}< \beta _{1}< \cdots < \beta _{n}< \gamma _{1}< \cdots < \gamma _{l}< b, \end{cases} $$

where ${}^{C}{D}^{\chi }$ is the Caputo fractional derivative of order $\chi \in \{\eta ,\xi ,\zeta \}$, ρ, φ, $\psi :[a,b]\times \mathbb{R}\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ are given functions, and $p_{i},q_{j}, r_{k} \in \mathbb{R}$, $i=1,\ldots ,m$, $j=1,\ldots ,n$, $k=1, \ldots ,l$. The existence is proved via the Leray–Schauder alternative, whereas the existence of a unique solution is established via the Banach contraction mapping principle. We suggest the reader a series of publications on FDE-coupled systems [27, 28, 30–34]. In the last two decades, the fractional-order differential equations appeared and began to study the predator–prey models in the fractional-order form. In [35] the authors studied the predator–prey model of Holling-type II with harvesting and predator in disease

$$ \textstyle\begin{cases} \frac{dx}{dt} = rx(1-\frac{x}{k})-\frac{ayx}{m+x}-azx-h_{1}x, \\ \frac{dy}{dt} = bxy+\alpha yz+\frac{\gamma yx}{m+x}-h_{2}y, \\ \frac{dz}{dt} = bzx-\alpha yz-dz, \end{cases} $$

where x, y, and z are the prey, infected predator, and susceptible predator, respectively, and r, k, a, b, γ, α, $h_{1}$, $h_{2}$, d are assumed to be positive constants. They have studied the existence of a positive biological equilibrium and the uniform boundedness of the system. Local stability conditions are also defined based on Routh–Hurwitz. In [36] the authors discussed the fractional-order model of a two-prey-one-predator system

$$ \textstyle\begin{cases} {{}^{{c}}{D}}^{\alpha}_{*}x_{1}(t) = f_{1}(x_{1},x_{2},x_{3}) = a x_{1}(t)(1-x_{1}(t))-x_{1}(t)x_{3}(t)+x_{1}(t)x_{2}(t)x_{3}(t), \quad t\in [0,T], \\ {{}^{{c}}{D}}^{\alpha}_{*}x_{2}(t) = f_{2}(x_{1},x_{2},x_{3}) = b x_{2}(t)(1-x_{2}(t))-x_{2}(t)x_{3}(t)+x_{1}(t)x_{2}(t)x_{3}(t), \quad t\in [0,T], \\ {{}^{{c}}{D}}^{\alpha}_{*}x_{3}(t) = f_{3}(x_{1},x_{2},x_{3}) = -c x_{3}^{2}(t)+d x_{1}(t)x_{3}(t)+e x_{2}(t)x_{3}(t), \quad t\in [0,T], \end{cases} $$

where c is the death rate of the predator, $0 \leq \alpha \leq 1$, $x_{1}(t)\geq 0$, $x_{2}(t)\geq 0$, $x_{3}(t)\geq 0$, and a, b, c, d, and e are all positive constants. They have studied the local asymptotic stability of the equilibrium solutions of the proposed model. One of the most important disciplines in the study of fractional-order differential equations is the theory of existence, uniqueness, and stability of solutions. In the present paper, inspired by the above-mentioned works, we introduce and investigate the existence and stability of solutions for the following coupled system of sequential fractional differential equations with nonlocal multipoint coupled boundary conditions:

$$ \textstyle\begin{cases} ({}^{c}{\mathcal{D}}^{\eta }+ \varphi {}^{c}{\mathcal{D}}^{\eta -1}) \mathfrak{p}(\varsigma )= \hat{\mathcal{F}_{1}} (\varsigma , \mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma )), \quad 1< \eta \leq 2, \\ ({}^{c}{\mathcal{D}}^{\xi}+ \varphi {}^{c}{\mathcal{D}}^{\xi -1}) \mathfrak{q}(\varsigma )= \hat{\mathcal{F}_{2}}(\varsigma , \mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma )),\quad 1< \xi \leq 2, \\ ({}^{c}{\mathcal{D}}^{\zeta}+ \varphi {}^{c}{\mathcal{D}}^{\zeta -1}) \mathfrak{r}(\varsigma )= \hat{\mathcal{F}_{3}} (\varsigma , \mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma ) ), \quad 2< \zeta \leq 3, \\ \mathfrak{p}(0)={0}, \qquad \mathfrak{p}(1)=\beta _{1} \sum_{j=1}^{k-2} w_{j} \mathfrak{q}(\varrho _{j}), \\ \mathfrak{q}(0)={0}, \qquad \mathfrak{q}(1)= \beta _{2} \sum_{j=1}^{k-2} v_{j} \mathfrak{r}(\varpi _{j}), \\ \mathfrak{r}(0)=0, \qquad \mathfrak{r}'(0)=0, \qquad \mathfrak{r}(1)= \beta _{3} \sum_{j=1}^{k-2}\vartheta _{j} \mathfrak{p}( \rho _{j}), \\ 0 < \varrho _{1} < \varpi _{1} < \rho _{1}< \varrho _{2} < \varpi _{2} < \rho _{2} \ldots < \varrho _{k-2} < \varpi _{k-2} < \rho _{k-2}< 1, \end{cases} $$

(1)

where ${}^{c}{\mathcal{D}}^{\chi}$ is the Caputo fractional derivative (CFD) of order $\chi \in \{\eta ,\xi ,\zeta \}$, f, g, $h :[0,1]\times \mathbb{R}\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ are given functions, φ is a positive real number, and $w_{j},v_{j}, \vartheta _{j} \in \mathbb{R}$, $j=1,\ldots ,k-2$, $\beta _{1}$, $\beta _{2}$, and $\beta _{3}$ are real constants.

The CFD ${}^{c}\mathcal{D}^{\chi}$ of order χ is defined by

$$\begin{aligned} &{}^{C}\mathcal{D}^{\chi}v{(\varsigma )}= \frac{1}{\Gamma (\mathfrak{n}-\chi )} \int ^{\varsigma}_{0} ( \varsigma -\mathfrak{s})^{\mathfrak{n}-\chi -1} \biggl( \frac{d}{d\mathfrak{s}} \biggr)^{\mathfrak{n}}v(\mathfrak{s})\,d\mathfrak{s},\quad \mathfrak{n}-1< \chi < \mathfrak{n}, \\ & \mathfrak{n} = [\chi ]+1, \end{aligned}$$

and the Riemann–Liouville integral of fractional order χ is defined by

$$\begin{aligned} {}^{\mathrm{RL}}\mathcal{I}^{\chi}v{(\varsigma )}=\frac{1}{\Gamma (\chi )} \int ^{\varsigma}_{0} (\varsigma -\mathfrak{s})^{\chi -1}v( \mathfrak{s})\,d\mathfrak{s},\quad \chi > 0, \end{aligned}$$

This investigation is unique in that it also investigates a coupled system of three sequential fractional differential equations (SFDEs) of various orders in an arbitrary domain with multipoint boundary conditions. The multipoint boundary conditions, as we can see, are cyclic in nature and occur in a variety of nonlocal areas. As a result, our findings are more general and have a considerable impact on current research. Existence and uniqueness results can be obtained using fixed point theory. The Hyers–Ulam stability study is also performed.

The rest of the paper is organized as follows. In Sect. 2, we discuss several fundamental definitions and lemmas of fractional calculus. Additionally, we prove an auxiliary lemma involving a linear function of (1), which is necessary for obtaining the main results.

Section 3 summarizes the main results. We obtain the existence of a solution to the problem at hand using the Leray–Schauder alternative and also verify the existence of a unique solution using Banach’s contraction mapping principle.

In Sect. 4, we prove that the proposed problem (1) is Ulam–Hyers stable under certain conditions.

In Sect. 5, we provide examples to illustrate the theoretical results.

2 Preliminaries

Here we recall some notations and definitions of fractional calculus [7–10, 37, 38].

Definition 1

The fractional integral of order α with the lower limit zero for a function $\mathfrak{f}$ is defined as

$$ I^{\alpha} \mathfrak{f}(\varsigma )= \frac{1}{\varGamma (\alpha )} \int _{0}^{\varsigma} \frac{\mathfrak{f}(s)}{(\varsigma -s)^{1-\alpha}}\,ds,\quad \varsigma > 0 , \alpha > 0, $$

(2)

provided that the right-hand side is pointwise defined on $[0,\infty )$, where Γ is the gamma function defined by $\varGamma (\alpha ) = \int _{0}^{\infty} \varsigma ^{\alpha -1} e^{- \varsigma}\,d\varsigma$.

Definition 2

The Riemann–Liouville fractional derivative of order $\alpha > 0$, $n-1 < \alpha < n$, $n \in \mathbb{N}\mathbbm{,}$ is defined as

$$\begin{aligned} D^{\alpha}_{0+} \mathfrak{f}(\varsigma ) = \frac{1}{\varGamma (n-\alpha )} \biggl(\frac{d}{d\varsigma} \biggr)^{n} \int _{0}^{\varsigma} (\varsigma -s)^{n-\alpha -1} \mathfrak{f}(s)\,ds, \quad \varsigma > 0, \end{aligned}$$

(3)

where the function $\mathfrak{f}$ has absolutely continuous derivatives up to order $(n-1)$.

Definition 3

The Caputo derivative of order $r \in {[n-1,n)}$ for a function $\mathfrak{f} : [0,\infty )\rightarrow \mathbb{R}$ can be written as

$$\begin{aligned} {}^{c}\mathcal{D}^{r}_{0+}\mathfrak{f} {( \varsigma )}=\mathcal{D}^{r}_{0+} \Biggl( {\mathfrak{f}}( \varsigma )-\sum_{k=0}^{n-1} \frac{\varsigma ^{k}}{k!}\mathfrak{f}^{(k)}(0) \Biggr),\quad \varsigma >0 , n-1< r< n. \end{aligned}$$

(4)

Note that the Caputo fractional derivative of order $r \in {[n-1,n)}$ exists almost everywhere on $[0,\infty )$ if $\mathfrak{f} \in \mathcal{AC}^{n}([0,\infty ),\mathbbm{(}\mathbb{R}\mathbbm{)})$.

Remark 1

If $\mathfrak{f} \in \mathcal{C}^{n}[0,\infty )$, then

$$\begin{aligned} ^{c}D^{r}_{0+} \hat{\mathfrak{f}}(\varsigma ) = \frac{1}{\varGamma (n-r)} \int _{0}^{\varsigma } \frac{\mathfrak{f}^{(n)}(s)}{(\varsigma -s)^{r+1-n}}\,ds = I^{n-r} \mathfrak{f}^{(n)} (\varsigma ),\quad \varsigma > 0, n-1 < r < n. \end{aligned}$$

Now we are ready to present an essential solution we obtained for (1).

Lemma 1

Let $\hat{\mathcal{G}_{1}}, \hat{\mathcal{G}_{2}},\hat{\mathcal{G}_{3}} \in \mathcal{C}[0,1]$ and $\Upsilon \ne 0$. Then the unique solution of the system

$$ \textstyle\begin{cases} (^{c}\mathcal{D}^{\eta }+ \varphi ^{c}\mathcal{D}^{\eta -1}) \mathfrak{p}(\varsigma )= \hat{\mathcal{G}_{1}}(\varsigma ),\quad 1< \eta \leq 2, \varsigma \in [0,1], \\ (^{c}\mathcal{D}^{\xi}+ \varphi ^{c}\mathcal{D}^{\xi -1})\mathfrak{q}( \varsigma )= \hat{\mathcal{G}_{2}}(\varsigma ), \quad 1< \xi \leq 2, \varsigma \in [0,1], \\ (^{c}\mathcal{D}^{\zeta}+ \varphi ^{c}\mathcal{D}^{\zeta -1}) \mathfrak{r}(\varsigma )= \hat{\mathcal{G}_{3}}(\varsigma ), \quad 2< \zeta \leq 3, \varsigma \in [0,1], \\ \mathfrak{p}(0)={0}, \qquad \mathfrak{p}(1)=\beta _{1} \sum_{j=1}^{k-2} w_{j} \mathfrak{q}(\varrho _{j}), \\ \mathfrak{q}(0)={0}, \qquad \mathfrak{q}(1)= \beta _{2} \sum_{j=1}^{k-2} v_{j} \mathfrak{r}(\varpi _{j}), \\ \mathfrak{r}(0)=0, \qquad \mathfrak{r}'(0)=0,\qquad \mathfrak{r}(1)= \beta _{3} \sum_{j=1}^{k-2}\vartheta _{j} \mathfrak{p}( \rho _{j}), \\ 0 < \varrho _{1} < \varpi _{1} < \rho _{1}< \varrho _{2} < \varpi _{2} < \rho _{2} \ldots < \varrho _{k-2} < \varpi _{k-2} < \rho _{k-2}< 1, \end{cases} $$

(5)

is given by

$$\begin{aligned} &\mathfrak{p}(\varsigma )= \biggl( \frac{1-e^{-\varphi \varsigma}}{\varphi \Upsilon} \biggr) \end{aligned}$$

(6)

$$\begin{aligned} &\hphantom{\mathfrak{p}(\varsigma )=}{} \times \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{1}\sum _{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{- \varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \\ &\hphantom{\mathfrak{p}(\varsigma )=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \Biggr) \\ &\hphantom{\mathfrak{p}(\varsigma )=}{} -\hat{\mathfrak{A}_{5}}\hat{\mathfrak{A}_{2}} \Biggl( \beta _{2} \sum_{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \\ &\hphantom{\mathfrak{p}(\varsigma )=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \Biggr) \\ &\hphantom{\mathfrak{p}(\varsigma )=}{} + \hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{3}} \Biggl( \beta _{3} \sum _{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi ( \rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \\ &\hphantom{\mathfrak{p}(\varsigma )=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \Biggr) \Biggr\rbrace \\ & \hphantom{\mathfrak{p}(\varsigma )=}{}+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds, \end{aligned}$$

(7)

$$\begin{aligned} &\mathfrak{q}(\varsigma )= \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\mathfrak{A}_{2}} \biggr) \Biggl\lbrace \beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{ \varrho _{j}} e^{-\varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \\ &\hphantom{\mathfrak{q}(\varsigma )=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \\ &\hphantom{\mathfrak{q}(\varsigma )=}{} - \frac{1}{\Upsilon} \Biggl[ {\hat{\mathfrak{A}_{1}} \hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}} \Biggl(\beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{-\varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \\ &\hphantom{\mathfrak{q}(\varsigma )=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \Biggr) \\ &\hphantom{\mathfrak{q}(\varsigma )=}{} - \hat{\mathfrak{A}_{1}}\hat{ \mathfrak{A}_{5}} \mathfrak{A}_{2} \Biggl( \beta _{2}\sum_{j=1}^{k-2} v_{j} \int _{0}^{ \varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \\ &\hphantom{\mathfrak{q}(\varsigma )=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \Biggr) \\ &\hphantom{\mathfrak{q}(\varsigma )=}{} + \hat{\mathfrak{A}_{1}} \mathfrak{A}_{2} \hat{\mathfrak{A}_{3}} \Biggl(\beta _{3}\sum_{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi (\rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \\ &\hphantom{\mathfrak{q}(\varsigma )=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \Biggr) \Biggr] \Biggr\rbrace \\ &\hphantom{\mathfrak{q}(\varsigma )=}{}+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds, \end{aligned}$$

(8)

$$\begin{aligned} &\mathfrak{r}(\varsigma )= \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\Upsilon \varphi ^{2}} \\ &\hphantom{\mathfrak{r}(\varsigma )=}{}\times \Biggl\lbrace \hat{\mathfrak{A}_{1}}\hat{ \mathfrak{A}_{4}} \Biggl(\beta _{3}\sum _{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi (\rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \\ &\hphantom{\mathfrak{r}(\varsigma )=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \Biggr) \\ \begin{aligned}&\hphantom{\mathfrak{r}(\varsigma )=}{} - \Biggl[ \hat{\mathfrak{A}_{6}}\hat{\mathfrak{A}_{4}} \Biggl( \beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{- \varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \\ & \hphantom{\mathfrak{r}(\varsigma )=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \Biggr) \end{aligned}\\ & \hphantom{\mathfrak{r}(\varsigma )=}{} - \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{6}} \Biggl( \beta _{2}\sum_{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{- \varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \\ & \hphantom{\mathfrak{r}(\varsigma )=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \Biggr) \Biggr] \Biggr\rbrace \\ &\hphantom{\mathfrak{r}(\varsigma )=}{}+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds, \end{aligned}$$

(9)

where

$$\begin{aligned} &\hat{\mathfrak{A}_{1}}= \frac{(1-e^{-\varphi})}{\varphi}, \qquad \mathfrak{A}_{2}=-\beta _{1}\sum _{j=1}^{k-2}w_{j} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi}, \\ \begin{aligned}&\hat{\mathfrak{A}_{3}}=-\beta _{2}\sum _{j=1}^{k-2}v_{j} \frac{(\varphi \varpi _{j}-1+e^{-\varphi \varpi _{j}})}{\varphi},\qquad \hat{ \mathfrak{A}_{4}}=\frac{(1-e^{-\varphi})}{\varphi}, \\ &\hat{\mathfrak{A}_{5}} = \frac{(\varphi -1+e^{-\varphi})}{\varphi ^{2}},\qquad \hat{ \mathfrak{A}_{6}}=-\beta _{3}\sum _{j=1}^{k-2}\vartheta _{j} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi}, \end{aligned} \end{aligned}$$

(10)

$$\begin{aligned} &\Upsilon = (\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}} \hat{\mathfrak{A}_{5}}+\hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{3}} \hat{\mathfrak{A}_{6}}), \\ &\mathcal{I}_{1}=\beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{ \varrho _{j}} e^{-\varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \\ &\hphantom{\mathcal{I}_{1}=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds, \\ \begin{aligned}&\mathcal{I}_{2}=\beta _{2}\sum _{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \\ &\hphantom{\mathcal{I}_{2}=}{}- \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds, \end{aligned}\\ &\mathcal{I}_{3}=\beta _{3}\sum _{j=1}^{k-2} \vartheta _{j} \int _{0}^{ \rho _{j}} e^{-\varphi (\rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \\ &\hphantom{\mathcal{I}_{3}=}{}- \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds. \end{aligned}$$

(11)

Proof

As argued in [9], the solution of FDEs (5) can be written as

$$\begin{aligned}& \mathfrak{p}(\varsigma )= c_{0}e^{-\varphi \varsigma} + \frac{c_{1}}{\varphi} \bigl(1-e^{-\varphi \varsigma}\bigr)+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds, \end{aligned}$$

(12)

$$\begin{aligned}& \mathfrak{q}(\varsigma )= d_{0}e^{-\varphi \varsigma} + \frac{d_{1}}{\varphi} \bigl(1-e^{-\varphi \varsigma}\bigr)+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds, \end{aligned}$$

(13)

$$\begin{aligned}& \mathfrak{r}(\varsigma )= b_{0}e^{-\varphi \varsigma} + \frac{b_{1}}{\varphi} \bigl(1-e^{-\varphi \varsigma}\bigr)+ \frac{b_{2}}{\varphi ^{2}}\bigl(\varphi \varsigma -1+e^{-\varphi \varsigma}\bigr) \\& \hphantom{\mathfrak{r}(\varsigma )=}{} + \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds. \end{aligned}$$

(14)

Using the condition $\mathfrak{p}(0)=0$ in (12), we get $c_{0}= 0$, and the condition $\mathfrak{q}(0)=0$ in (13) yields $d_{0} = 0$, whereas the conditions $\mathfrak{r}(0)={(0)}$ and $r'{0}=(0)$ in (14) yield $b_{0}=0$ and $b_{1}={0}$. Consequently, we have

$$ \begin{aligned} &\mathfrak{p}(\varsigma )= \frac{c_{1}}{\varphi}\bigl(1-e^{-\varphi \varsigma}\bigr)+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds, \\ &\mathfrak{q}(\varsigma )= \frac{d_{1}}{\varphi}\bigl(1-e^{-\varphi \varsigma}\bigr)+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds, \\ &\mathfrak{r}(\varsigma )=\frac{b_{2}}{\varphi ^{2}}\bigl(\varphi \varsigma -1+e^{-\varphi \varsigma}\bigr)+ \int _{0}^{\varsigma} e^{- \varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds. \end{aligned} $$

(15)

Using the conditions $\mathfrak{p}(1)=\beta _{1} \sum_{j=1}^{k-2} w_{j} \mathfrak{q}(\varrho _{j})$, $\mathfrak{q}(1)= \beta _{2}\sum_{j=1}^{k-2} v_{j} \mathfrak{r}(\varpi _{j})$, and $\mathfrak{r}(1)=\beta _{3} \sum_{j=1}^{k-2}\vartheta _{j} \mathfrak{p}(\rho _{j})$ in (15), we find that

$$\begin{aligned}& c_{1}= \frac{1}{\Upsilon} \Biggl\lbrace \hat{\mathfrak{A}_{4}} \hat{\mathfrak{A}_{5}} \Biggl( \beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{ \varrho _{j}} e^{-\varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \\& \hphantom{ c_{1}= }{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \Biggr) \\& \begin{aligned} & \hphantom{ c_{1}= }{} -\hat{\mathfrak{A}_{5}}\mathfrak{A}_{2} \Biggl( \beta _{2} \sum_{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \\ & \hphantom{ c_{1}= }{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \Biggr) \end{aligned} \end{aligned}$$

(16)

$$\begin{aligned}& \hphantom{ c_{1}= }{} + \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}} \Biggl( \beta _{3} \sum_{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi ( \rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \\& \hphantom{ c_{1}= }{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \Biggr) \Biggr\rbrace , \\& d_{1}= \frac{1}{\mathfrak{A}_{2}} \Biggl\lbrace \beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{-\varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \\& \hphantom{d_{1}=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \\& \hphantom{d_{1}=}{} - \frac{1}{\Upsilon} \Biggl[ {\hat{\mathfrak{A}_{1}} \hat{ \mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}} \Biggl(\beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{-\varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \\& \begin{aligned} &\hphantom{d_{1}=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \Biggr) \\ &\hphantom{d_{1}=}{} - \hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{5}} \mathfrak{A}_{2} \Biggl( \beta _{2}\sum _{j=1}^{k-2} v_{j} \int _{0}^{ \varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \end{aligned} \end{aligned}$$

(17)

$$\begin{aligned}& \hphantom{d_{1}=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \Biggr) \\& \hphantom{d_{1}=}{} + \hat{\mathfrak{A}_{1}}\mathfrak{A}_{2} \hat{\mathfrak{A}_{3}} \Biggl(\beta _{3}\sum _{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi (\rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \\& \hphantom{d_{1}=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \Biggr) \Biggr] \Biggr\rbrace , \\& b_{2}= \frac{1}{\Upsilon} \Biggl\lbrace \hat{\mathfrak{A}_{1}} \hat{\mathfrak{A}_{4}} \Biggl(\beta _{3}\sum_{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi (\rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \\& \hphantom{ b_{2}=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \Biggr) \\& \begin{aligned}& \hphantom{ b_{2}=}{} - \Biggl[ \hat{\mathfrak{A}_{6}}\hat{\mathfrak{A}_{4}} \Biggl( \beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{- \varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \\ &\hphantom{ b_{2}=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{G}_{1}}(u)\,du \biggr)\,ds \Biggr) \end{aligned}\\& \hphantom{ b_{2}=}{} - \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{6}} \Biggl( \beta _{2}\sum_{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{- \varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{G}_{3}}(u)\,du \biggr)\,ds \\& \hphantom{ b_{2}=}{} - \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{G}_{2}}(u)\,du \biggr)\,ds \Biggr) \Biggr] \Biggr\rbrace , \end{aligned}$$

(18)

from which by substituting into (15), we get the solutions (6)–(8)–(9). The converse follows by direct computation. This completes the proof. □

3 Main results

Let $\mathcal{J}=\mathcal{C}([0,1],\mathbb{R})$ be space equipped with the norm $\Vert \mathfrak{q}\Vert =\sup \{\vert \mathfrak{q}(\varsigma )\vert , \varsigma \in [0,1]\}$. Obviously, ($\mathcal{J}$, $\|\cdot \|$) is a Banach space, and, consequently, $(\mathcal{J} \times \mathcal{J} \times \mathcal{J}, \Vert ( \mathfrak{p},\mathfrak{q},\mathfrak{r} )\Vert _{\mathcal{J}})$ is also a Banach space equipped with the norm $\Vert (\mathfrak{p},\mathfrak{q},\mathfrak{r} )\Vert _{\mathcal{J}}= \Vert \mathfrak{p} \Vert + \Vert \mathfrak{q} \Vert +\Vert \mathfrak{r}\Vert , \mathfrak{p},\mathfrak{q}$, $\mathfrak{r} \in \mathcal{J}$.

In view of Lemma 1, we define the operator $\mathcal{S} :\mathcal{J} \times \mathcal{J} \times \mathcal{J} \rightarrow \mathcal{J} \times \mathcal{J} \times \mathcal{J}$ by $\mathcal{S} (\mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ), \mathfrak{r}(\varsigma ) )= (\mathcal{S}_{1} ( \mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma ) ),\mathcal{S}_{2} (\mathfrak{p}(\varsigma ), \mathfrak{q}(\varsigma ),\mathfrak{r}(\varsigma ) ),\mathcal{S}_{3} (\mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma ) ) )$, where

$$\begin{aligned}& \mathcal{S}_{1} \bigl(\mathfrak{p}( \varsigma ),\mathfrak{q}( \varsigma ),\mathfrak{r}(\varsigma ) \bigr) \\& \quad = \biggl( \frac{1-e^{-\varphi \varsigma}}{\varphi \Upsilon} \biggr) \\& \qquad {}\times \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{1}\sum _{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{- \varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}- \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \\& \qquad {}-\hat{\mathfrak{A}_{5}}\mathfrak{A}_{2} \Biggl( \beta _{2} \sum_{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{F}_{3}}\bigl(u,\mathfrak{p}(u),\mathfrak{q}(u),\mathfrak{r}(u) \bigr)\,du \biggr)\,ds \\& \qquad {}- \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \\& \qquad {}+ \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}} \Biggl( \beta _{3} \sum_{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi ( \rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}- \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \Biggr\rbrace \\& \qquad {}+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)}\hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds, \\& \mathcal{S}_{2} \bigl(\mathfrak{p}(\varsigma ),\mathfrak{q}( \varsigma ),\mathfrak{r}(\varsigma ) \bigr) \\& \quad = \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\mathfrak{A}_{2}} \biggr) \\& \qquad {}\times \Biggl\lbrace \beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{ \varrho _{j}} e^{-\varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}- \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}- \frac{1}{\Upsilon} \Biggl[ {\hat{\mathfrak{A}_{1}} \hat{ \mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}} \\& \qquad {}\times \Biggl(\beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{-\varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u,\mathfrak{p}(u),\mathfrak{q}(u),\mathfrak{r}(u) \bigr)\,du \biggr)\,ds \\& \qquad {}- \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \\& \qquad {}- \hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{5}} \mathfrak{A}_{2} \Biggl( \beta _{2}\sum _{j=1}^{k-2} v_{j} \int _{0}^{ \varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}- \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \\& \qquad {}+ \hat{\mathfrak{A}_{1}}\mathfrak{A}_{2} \hat{\mathfrak{A}_{3}} \Biggl(\beta _{3}\sum _{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi (\rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}- \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \Biggr] \Biggr\rbrace \\& \qquad {}+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)}\hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds, \\& \mathcal{S}_{3} \bigl(\mathfrak{p}(\varsigma ),\mathfrak{q}( \varsigma ),\mathfrak{r}(\varsigma ) \bigr) \\& \quad = \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\Upsilon \varphi ^{2}} \\& \qquad {}\times \Biggl\lbrace \hat{\mathfrak{A}_{1}}\hat{ \mathfrak{A}_{4}} \Biggl(\beta _{3}\sum _{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi (\rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}- \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \\& \qquad {}- \Biggl[ \hat{\mathfrak{A}_{6}}\hat{\mathfrak{A}_{4}} \Biggl( \beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{- \varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}- \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \\& \qquad {}- \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{6}} \Biggl( \beta _{2}\sum_{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{- \varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}- \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \Biggr] \Biggr\rbrace \\& \qquad {}+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds. \end{aligned}$$

We further use the following notations:

$$\begin{aligned}& {\mathcal{W}_{1}}= \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\Upsilon} \biggr) \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{ \mathfrak{A}_{5}} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr] + \hat{ \mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}} \Biggl[ \beta _{3}\sum_{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr] \Biggr\rbrace \\& \hphantom{{\mathcal{W}_{1}}=}{}+ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )}, \\& {\mathcal{V}_{1}}= \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\Upsilon} \biggr) \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}} \Biggl( \beta _{1}\sum_{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr) + \biggl(\mathfrak{A}_{2}{\hat{ \mathfrak{A}_{5}}} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr] \biggr) \Biggr\rbrace , \\& {\mathcal{U}_{1}}= \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\Upsilon} \biggr) \Biggl\lbrace \hat{\mathfrak{A}_{3}}\mathfrak{A}_{2} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr] + \hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{2}\sum _{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) \Biggr\rbrace , \\& {\mathcal{W}_{2}}= \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\mathfrak{A}_{2}} \biggr) \Biggl\lbrace \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}}}{\Upsilon} \Biggl( \beta _{3}\sum _{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr) + \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}}{\Upsilon} \biggl(\frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr) \\& \hphantom{{\mathcal{W}_{2}}=}{} + \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \Biggr\rbrace , \\& {\mathcal{V}_{2}}= \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\mathfrak{A}_{2}} \biggr) \Biggl\lbrace \Biggl( \beta _{1}\sum_{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr) \\& \hphantom{{\mathcal{V}_{2}}=}{} + \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}}{\Upsilon} \Biggl( \beta _{1}\sum _{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr) \\& \hphantom{{\mathcal{V}_{2}}=}{} + \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}}{\Upsilon} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr] \Biggr\rbrace + \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr], \\& {\mathcal{U}_{2}}= \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\mathfrak{A}_{2}} \biggr) \Biggl\lbrace \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}}}{\Upsilon} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr) + \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}}{\Upsilon} \Biggl( \beta _{2}\sum_{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) \Biggr\rbrace , \\& {\mathcal{W}_{3}}= \biggl( \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr) \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{1}} \Biggl( \beta _{3}\sum_{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr) + \hat{\mathfrak{A}_{4}}\hat{ \mathfrak{A}_{6}} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr) \Biggr\rbrace , \\& {\mathcal{V}_{3}}= \biggl( \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr) \Biggl\lbrace \hat{\mathfrak{A}_{6}}\hat{\mathfrak{A}_{4}} \Biggl( \beta _{1}\sum_{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr)+\hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{6}} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr) \Biggr\rbrace , \\& {\mathcal{U}_{3}}= \biggl( \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr) \Biggl\lbrace \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{6}} \Biggl( \beta _{2}\sum_{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) + \hat{\mathfrak{A}_{1}}\hat{ \mathfrak{A}_{4}} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr) \Biggr\rbrace \\& \hphantom{{\mathcal{U}_{3}}=}{} +\frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )}. \end{aligned}$$

(19)

Now we provide our first finding, a proof of the existence of a solution to problem (1) using the Leray–Schauder alternative [39].

Lemma 2

Let $\mathfrak{E}: \mathfrak{I}\rightarrow \mathfrak{I}$ be a completely continuous (c.c.) operator. Let $\mathcal{Y} (\mathfrak{E})=\{ \mathfrak{q} \in \mathfrak{I}: \mathfrak{q} = \eta \mathfrak{E}(\mathfrak{q}) \textit{ for some } 0 < \eta <1\}$.

Then either the set $\mathcal{Y}(\mathfrak{E})$ is unbounded, or $\mathfrak{E}$ has at least one fixed point (Leray–Schauder alternative) [39].

Theorem 1

Let $\Upsilon \ne 0$, where ϒ is defined by (10).

Assume that $(\mathscr{M}_{2}) : \hat{\mathcal{F}_{1}},\hat{\mathcal{F}_{2}}, \hat{\mathcal{F}_{3}}:[0,1]\times \mathbb{R} \times \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ are continuous functions and there exist constants $\kappa _{i},\lambda _{i},\varepsilon _{i}\geq 0$ ($i=1,2,3$) and $\kappa _{0}>0$, $\lambda _{0}>0$, $\varepsilon _{0}>0$ such that for all $\mathfrak{p},\mathfrak{q},\mathfrak{r} \in \mathbb{R}$ and $\varsigma \in [0,1]$,

$$ \begin{aligned} & \bigl\vert \hat{\mathcal{F}_{1}}( \varsigma ,\mathfrak{p}, \mathfrak{q},\mathfrak{r}) \bigr\vert \leq \kappa _{0}+ \kappa _{1} \vert \mathfrak{p} \vert + \kappa _{2} \vert \mathfrak{q} \vert + \kappa _{3} \vert \mathfrak{r} \vert , \\ & \bigl\vert \hat{\mathcal{F}_{2}}(\varsigma ,\mathfrak{p}, \mathfrak{q},\mathfrak{r}) \bigr\vert \leq \lambda _{0}+ \lambda _{1} \vert \mathfrak{p} \vert + \lambda _{2} \vert \mathfrak{q} \vert + \lambda _{3} \vert \mathfrak{r} \vert , \\ & \bigl\vert \hat{\mathcal{F}_{3}}(\varsigma ,\mathfrak{p}, \mathfrak{q},\mathfrak{r}) \bigr\vert \leq \varepsilon _{0}+ \varepsilon _{1} \vert \mathfrak{p} \vert + \varepsilon _{2} \vert \mathfrak{q} \vert + \varepsilon _{3} \vert \mathfrak{r} \vert . \end{aligned} $$

Then problem (1) has at least one solution on $[0,1]$, provided that

$$ \begin{aligned} & ( {\mathcal{W}_{1}} + { \mathcal{W}_{2}} + {\mathcal{W}_{3}})\kappa _{1} +({\mathcal{V}_{1}}+ {\mathcal{V}_{2}} + { \mathcal{V}_{3}})\lambda _{1} + ({\mathcal{U}_{1}}+{ \mathcal{U}_{2}}+{\mathcal{U}_{3}}) \varepsilon _{1}< 1, \\ & ( {\mathcal{W}_{1}}+ {\mathcal{W}_{2}} + { \mathcal{W}_{3}})\kappa _{2}+({ \mathcal{V}_{1}} +{\mathcal{V}_{2}} +{\mathcal{V}_{3}}) \lambda _{2}+({ \mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}}) \varepsilon _{2} < 1, \\ & ( {\mathcal{W}_{1}}+ {\mathcal{W}_{2}} + { \mathcal{W}_{3}})\kappa _{3}+({ \mathcal{V}_{1}} +{\mathcal{V}_{2}} +{\mathcal{V}_{3}}) \lambda _{3}+({ \mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}}) \varepsilon _{3} < 1, \end{aligned} $$

(20)

where $\mathcal{W}_{i},\mathcal{V}_{i},\mathcal{U}_{i}$, $i=1,2,3$, are given in (19).

Proof

The operator $\mathcal{S}: \mathcal{J} \times \mathcal{J} \times \mathcal{J} \rightarrow \mathcal{J} \times \mathcal{J}\times \mathcal{J}$ is completely continuous since the functions $\hat{\mathcal{F}_{1}}$, $\hat{\mathcal{F}_{2}}$, and $\hat{\mathcal{F}_{3}}$ are completely continuous Next, let $\hat{\Omega}_{1} \subset \mathcal{J} \times \mathcal{J} \times \mathcal{J}$ be a bounded set to show the uniform boundedness. The operator $\mathcal{S}$ is also continuous such that

$$ \begin{aligned} & \bigl\vert \hat{\mathcal{F}_{1}} \bigl(\varsigma ,\mathfrak{p}( \varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma ) \bigr) \bigr\vert \leq \wp _{1}, \\ & \bigl\vert \hat{\mathcal{F}_{2}} \bigl(\varsigma ,\mathfrak{p}( \varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}(\varsigma ) \bigr) \bigr\vert \leq \wp _{2}, \\ & \bigl\vert \hat{\mathcal{F}_{3}} \bigl(\varsigma ,\mathfrak{p}( \varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}(\varsigma ) \bigr) \bigr\vert \leq \wp _{3},\quad (\mathfrak{p},\mathfrak{q}, \mathfrak{r}) \in \hat{ \Omega}_{1} , \end{aligned} $$

for nonnegative constants $\wp _{1}$, $\wp _{2}$, and $\wp _{3}$. Then, for any $(\mathfrak{p},\mathfrak{q},\mathfrak{r}) \in \hat{\Omega}_{1}$,

$$\begin{aligned}& \bigl\vert \mathcal{S}_{1} \bigl(\mathfrak{p}(\varsigma ), \mathfrak{q}( \varsigma ),\mathfrak{r}(\varsigma ) \bigr) \bigr\vert \\& \quad \leq \biggl( \frac{1-e^{-\varphi \varsigma}}{\varphi \Upsilon} \biggr) \\& \qquad {}\times \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{1}\sum _{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{- \varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \bigl\vert \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \bigl\vert \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \Biggr) \\& \qquad {}+\hat{\mathfrak{A}_{5}}\mathfrak{A}_{2} \Biggl( \beta _{2} \sum_{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \bigl\vert \hat{ \mathcal{F}_{3}}\bigl(u,\mathfrak{p}(u),\mathfrak{q}(u),\mathfrak{r}(u) \bigr) \bigr\vert \,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \bigl\vert \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \Biggr) \\& \qquad {}+ \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}} \Biggl( \beta _{3} \sum_{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi ( \rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \bigl\vert \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \bigl\vert \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \Biggr) \Biggr\rbrace \\& \qquad {}+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \bigl\vert \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \\& \quad \leq \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\Upsilon} \biggr) \Biggl\lbrace \hat{ \mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr] + \hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{3}} \Biggl[ \beta _{3}\sum _{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr] \Biggr\rbrace \\& \qquad {}+ \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}} \Biggl( \beta _{1}\sum_{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr) + \biggl(\mathfrak{A}_{2}{\hat{ \mathfrak{A}_{5}}} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr] \biggr) \Biggr\rbrace \\& \qquad {}+ \Biggl\lbrace \hat{\mathfrak{A}_{3}}\mathfrak{A}_{2} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr] + \hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{2}\sum _{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) \Biggr\rbrace \\& \qquad {}+ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \\& \quad \leq \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\Upsilon} \biggr) ({\mathcal{W}_{1}}\wp _{1}+{\mathcal{V}_{1}}\wp _{2}+{ \mathcal{U}_{1}}\wp _{3}), \end{aligned}$$

which implies that

$$\begin{aligned} \bigl\Vert \mathcal{S}_{1} \bigl(\mathfrak{p}(\varsigma ), \mathfrak{q}( \varsigma ),\mathfrak{r}(\varsigma ) \bigr) \bigr\Vert _{\mathcal{J}}\leq \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\Upsilon} \biggr) ({ \mathcal{W}_{1}} \wp _{1}+{\mathcal{V}_{1}}\wp _{2}+{ \mathcal{U}_{1}} \wp _{3}). \end{aligned}$$

Similarly, we can conclude that

$$\begin{aligned} \bigl\Vert \mathcal{S}_{2} \bigl(\mathfrak{p}(\varsigma ), \mathfrak{q}( \varsigma ),\mathfrak{r}(\varsigma ) \bigr) \bigr\Vert _{\mathcal{J}}\leq \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\mathfrak{A}_{2}\varphi} \biggr) ({\mathcal{W}_{2}} \wp _{1}+{\mathcal{V}_{2}}\wp _{2}+{ \mathcal{U}_{2}}\wp _{3}) \end{aligned}$$

and

$$\begin{aligned}& \bigl\vert \mathcal{S}_{3} \bigl(\mathfrak{p}(\varsigma ), \mathfrak{q}( \varsigma ),\mathfrak{r}(\varsigma ) \bigr) \bigr\vert \\& \quad \leq \sup _{ \varsigma \in [0,1]} \biggl\vert \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr\vert \\& \qquad {}\times \Biggl\lbrace \hat{\mathfrak{A}_{1}}\hat{ \mathfrak{A}_{4}} \Biggl(\beta _{3}\sum _{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi (\rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \bigl\vert \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \bigl\vert \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \Biggr) \\& \qquad {}+ \Biggl[ \hat{\mathfrak{A}_{6}}\hat{\mathfrak{A}_{4}} \Biggl( \beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{- \varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \bigl\vert \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \bigl\vert \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \Biggr) \\& \qquad {}+ \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{6}} \Biggl( \beta _{2}\sum_{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{- \varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \bigl\vert \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \bigl\vert \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \Biggr) \Biggr] \Biggr\rbrace \\& \qquad {}+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \bigl\vert \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \\& \quad \leq \biggl( \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr) \Biggl[ \Biggl\lbrace \hat{ \mathfrak{A}_{4}} \hat{\mathfrak{A}_{1}} \Biggl( \beta _{3}\sum_{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr) + \hat{\mathfrak{A}_{4}}\hat{ \mathfrak{A}_{6}} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr) \Biggr\rbrace \\& \qquad {}+ \Biggl\lbrace \Biggl( \beta _{1}\sum _{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr)+\hat{ \mathfrak{A}_{2}}\hat{\mathfrak{A}_{6}} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr) \Biggr\rbrace \\& \qquad {}+ \Biggl\lbrace \hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{6}} \Biggl( \beta _{2}\sum _{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) + \hat{ \mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr) \Biggr\rbrace \Biggr] +\frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \\& \quad \leq \biggl[ \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr] ({\mathcal{W}_{3}}\wp _{1}+{\mathcal{V}_{3}}\wp _{2}+{ \mathcal{U}_{3}}\wp _{3}), \end{aligned}$$

which accumulates to

$$\begin{aligned} \bigl\Vert \mathcal{S}_{3} \bigl(\mathfrak{p}(\varsigma ), \mathfrak{q}( \varsigma ),\mathfrak{r}(\varsigma ) \bigr) \bigr\Vert _{\mathcal{J}}\leq \biggl[ \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr] ({\mathcal{W}_{3}} \wp _{1}+{\mathcal{V}_{3}}\wp _{2}+{ \mathcal{U}_{3}}\wp _{3}). \end{aligned}$$

As a result, the operator $\mathcal{S}$ is uniformly bounded, that is,

$$\begin{aligned} &\bigl\Vert \mathcal{S} \bigl(\mathfrak{p}(\varsigma ),\mathfrak{q}( \varsigma ),\mathfrak{r}(\varsigma ) \bigr) \bigr\Vert _{\mathcal{J}} \\ &\quad \leq \biggl[ \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} \biggr) + \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\mathfrak{A}_{2}\varphi} \biggr) + \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr] \\ &\qquad {}+({\mathcal{W}_{1}}+{\mathcal{W}_{2}}+{ \mathcal{W}_{3}})\wp _{1}+ ({ \mathcal{V}_{1}}+{ \mathcal{V}_{2}}+{\mathcal{V}_{3}})\wp _{2} + ({ \mathcal{U}_{1}}+{\mathcal{U}_{2}}+{\mathcal{U}_{3}}) \wp _{3}. \end{aligned}$$

Next, we show that $\mathcal{S}$ is equicontinuous.

Let $\varsigma _{1}, \varsigma _{2} \in [0,1]$ with $\varsigma _{1}< \varsigma _{2}$. Then we have

$$\begin{aligned}& \bigl\vert \mathcal{S}_{1} \bigl(\mathfrak{p}(\varsigma _{2}),\mathfrak{q}( \varsigma _{2}),\mathfrak{r}(\varsigma _{2}) \bigr)-\mathcal{S}_{1} \bigl(\mathfrak{p}(\varsigma _{1}),\mathfrak{q}(\varsigma _{1}), \mathfrak{r}(\varsigma _{1}) \bigr) \bigr\vert \\& \quad \leq \frac{(e^{-\varphi{\varsigma _{2}}}-e^{-\varphi{\varsigma _{1}}})}{\Upsilon \varphi} \\& \qquad {}\times \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{1}\sum _{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{- \varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \\& \qquad {}+\hat{\mathfrak{A}_{5}}\mathfrak{A}_{2} \Biggl( \beta _{2} \sum_{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{F}_{3}}\bigl(u,\mathfrak{p}(u),\mathfrak{q}(u),\mathfrak{r}(u) \bigr)\,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \\& \qquad {}+ \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}} \Biggl( \beta _{3} \sum_{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi ( \rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \Biggr\rbrace \\& \qquad {}+ \biggl\vert \int _{0}^{\varsigma _{1}} \bigl(e^{-\varphi (\varsigma _{2}-s)}-e^{- \varphi (\varsigma _{1}-s)} \bigr) \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}+ \int _{\varsigma _{2}}^{\varsigma _{1}} e^{-\varphi ( \varsigma _{2}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \biggr\vert \\& \quad \leq \frac{(e^{-\varphi{\varsigma _{2}}}-e^{-\varphi{\varsigma _{1}}})}{\Upsilon \varphi} \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{ \mathfrak{A}_{5}} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr] + \hat{ \mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}} \Biggl[ \beta _{3}\sum_{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr] \Biggr\rbrace \wp _{1} \\& \qquad {}+ \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}} \Biggl( \beta _{1}\sum_{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr) + \biggl(\mathfrak{A}_{2}{\hat{ \mathfrak{A}_{5}}} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr] \biggr) \Biggr\rbrace \wp _{2} \\& \qquad {}+ \Biggl\lbrace \hat{\mathfrak{A}_{3}}\mathfrak{A}_{2} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr] + \hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{2}\sum _{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) \Biggr\rbrace \wp _{3} \\& \qquad {}+ \biggl\vert \int _{0}^{\varsigma _{1}} \bigl(e^{-\varphi (\varsigma _{2}-s)}-e^{- \varphi (\varsigma _{1}-s)} \bigr) \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)}\,du \biggr)\,ds \\& \qquad {}+ \int _{\varsigma _{2}}^{\varsigma _{1}} e^{-\varphi ( \varsigma _{2}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)}\,du \biggr)\,ds \biggr\vert \wp _{1}. \end{aligned}$$

In a similar way,

$$\begin{aligned}& \bigl\vert \mathcal{S}_{2} \bigl(\mathfrak{p}(\varsigma _{2}),\mathfrak{q}( \varsigma _{2}),\mathfrak{r}(\varsigma _{2}) \bigr)-\mathcal{S}_{2} \bigl(\mathfrak{p}(\varsigma _{1}),\mathfrak{q}(\varsigma _{1}), \mathfrak{r}(\varsigma _{1}) \bigr) \bigr\vert \\& \quad \leq \frac{(e^{-\varphi \varsigma _{2}}-e^{-\varphi \varsigma _{1}})}{\varphi \Upsilon} \\& \qquad {}\times \Biggl\lbrace \beta _{1}\sum _{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{-\varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}+ \frac{1}{\Upsilon} \Biggl[ {\hat{\mathfrak{A}_{1}} \hat{ \mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}} \\& \qquad {}\times \Biggl(\beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{-\varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u,\mathfrak{p}(u),\mathfrak{q}(u),\mathfrak{r}(u) \bigr)\,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \\& \qquad {}+ \hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{5}} \mathfrak{A}_{2} \Biggl( \beta _{2}\sum _{j=1}^{k-2} v_{j} \int _{0}^{ \varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \\& \qquad {}+ \hat{\mathfrak{A}_{1}}\mathfrak{A}_{2} \hat{\mathfrak{A}_{3}} \Biggl(\beta _{3}\sum _{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi (\rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \Biggr) \Biggr] \Biggr\rbrace \\& \qquad {}+ \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)}\hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds, \\& \qquad {}+ \biggl\vert \int _{0}^{\varsigma _{1}} \bigl(e^{-\varphi (\varsigma _{2}-s)}-e^{- \varphi (\varsigma _{1}-s)} \bigr) \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \\& \qquad {}+ \int _{\varsigma _{1}}^{\varsigma _{2}} \bigl(e^{-\varphi ( \varsigma _{2}-s)}\bigr) \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr)\,du \biggr)\,ds \biggr\vert \\& \quad \leq \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\mathfrak{A}_{2}} \biggr) \\& \qquad {}\times \Biggl[ \Biggl\lbrace \frac{\hat{\mathfrak{A}_{1}} \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}}}{\Upsilon} \Biggl( \beta _{3} \sum_{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr) \\& \qquad {}+ \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}}{\Upsilon} \biggl(\frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr)+ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \Biggr\rbrace \wp _{1} \\& \qquad {}+ \Biggl\lbrace \Biggl( \beta _{1}\sum _{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr) + \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}}{\Upsilon} \Biggl( \beta _{1}\sum_{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr) \\& \qquad {}+ \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}}{\Upsilon} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr] \Biggr\rbrace \wp _{2} + \Biggl\lbrace \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}}}{\Upsilon} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr) \\& \qquad {}+ \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}}{\Upsilon} \Biggl( \beta _{2}\sum _{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) \Biggr\rbrace \wp _{3} \Biggr] \\& \qquad {}+ \biggl\vert \int _{0}^{\varsigma _{1}} \bigl(e^{-\varphi (\varsigma _{2}-s)}-e^{- \varphi (\varsigma _{1}-s)} \bigr) \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)}\,du \biggr)\,ds \\& \qquad {}+ \int _{\varsigma _{1}}^{\varsigma _{2}} \bigl(e^{-\varphi ( \varsigma _{2}-s)}\bigr) \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)}\,du \biggr)\,ds \biggr\vert \wp _{2}, \end{aligned}$$

and

$$\begin{aligned}& \bigl\vert \mathcal{S}_{3} \bigl(\mathfrak{p}(\varsigma _{2}),\mathfrak{q}( \varsigma _{2}),\mathfrak{r}(\varsigma _{2}) \bigr)-\mathcal{S}_{3} \bigl(\mathfrak{p}(\varsigma _{1}),\mathfrak{q}(\varsigma _{1}), \mathfrak{r}(\varsigma _{1}) \bigr) \bigr\vert \\& \quad \leq \biggl\vert \frac{(\varphi (\varsigma _{2}-\varsigma _{1}) +e^{-\varphi \varsigma _{2}}-e^{-\varphi \varsigma _{1}} )}{\varphi ^{2}{\Upsilon}} \biggr\vert \\& \qquad {}\times \Biggl\lbrace \hat{\mathfrak{A}_{1}}\hat{ \mathfrak{A}_{4}} \Biggl(\beta _{3}\sum _{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi (\rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \bigl\vert \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \bigl\vert \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \Biggr) \\& \qquad {}+ \Biggl[ \hat{\mathfrak{A}_{6}}\hat{\mathfrak{A}_{4}} \Biggl( \beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{- \varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \bigl\vert \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \bigl\vert \hat{ \mathcal{F}_{1}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \Biggr) \\& \qquad {}+ \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{6}} \Biggl( \beta _{2}\sum_{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{- \varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \bigl\vert \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \\& \qquad {}+ \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \bigl\vert \hat{ \mathcal{F}_{2}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \Biggr) \Biggr] \Biggr\rbrace \\& \qquad {}+ \biggl\vert \int _{0}^{\varsigma _{1}} \bigl(e^{-\varphi (\varsigma _{2}-s)}-e^{- \varphi (\varsigma _{1}-s)} \bigr) \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \bigl\vert \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \\& \qquad {}+ \int _{\varsigma _{1}}^{\varsigma _{2}} e^{-\varphi ( \varsigma _{2}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \bigl\vert \hat{ \mathcal{F}_{3}}\bigl(u, \mathfrak{p}(u),\mathfrak{q}(u), \mathfrak{r}(u)\bigr) \bigr\vert \,du \biggr)\,ds \biggr\vert \\& \quad \leq \biggl\vert \frac{(\varphi (\varsigma _{2}-\varsigma _{1}) +e^{-\varphi \varsigma _{2}}-e^{-\varphi \varsigma _{1}} )}{\varphi ^{2}{\Upsilon}} \biggr\vert \\& \qquad {}\times \Biggl[ \Biggl\lbrace \hat{\mathfrak{A}_{4}} \hat{ \mathfrak{A}_{1}} \Biggl( \beta _{3}\sum _{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr) + \hat{ \mathfrak{A}_{4}}\hat{\mathfrak{A}_{6}} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr) \Biggr\rbrace \wp _{1} \\& \qquad {}+ \Biggl\lbrace \Biggl( \beta _{1}\sum _{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr)+\hat{ \mathfrak{A}_{2}}\hat{\mathfrak{A}_{6}} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr) \Biggr\rbrace \wp _{2} \\& \qquad {}+ \Biggl\lbrace \hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{6}} \Biggl( \beta _{2}\sum _{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) + \hat{ \mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr) \Biggr\rbrace \wp _{3} \Biggr] \\& \qquad {}+ \biggl\vert \int _{0}^{\varsigma _{1}} \bigl(e^{-\varphi (\varsigma _{2}-s)}-e^{- \varphi (\varsigma _{1}-s)} \bigr) \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)}\,du \biggr)\,ds \\& \qquad {}+ \int _{\varsigma _{1}}^{\varsigma _{2}} e^{-\varphi ( \varsigma _{2}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)}\,du \biggr)\,ds \biggr\vert \wp _{3}. \end{aligned}$$

As $\varsigma _{1}\rightarrow \varsigma _{2}$ is independent of $\mathfrak{p}$, $\mathfrak{q}$, $\mathfrak{r}$ with respect to the boundedness of $\hat{\mathcal{F}_{1}}$, $\hat{\mathcal{F}_{2}}$, and $\hat{\mathcal{F}_{3}}$, the operator $\mathcal{S}(\mathfrak{p},\mathfrak{q},\mathfrak{r})$ is equicontinuous. Thus the operator $\mathcal{S}(\mathfrak{p},\mathfrak{q},\mathfrak{r})$ is completely continuous.

Finally, we show that the set $\mathcal{P} = \lbrace (\mathfrak{p},\mathfrak{q},\mathfrak{r}) \in \mathcal{J} \times \mathcal{J} \times \mathcal{J}:(\mathfrak{p}, \mathfrak{q},\mathfrak{r})= \nu \mathcal{S}(\mathfrak{p},\mathfrak{q}, \mathfrak{r}), 0 \leq \nu \leq 1\rbrace$ t is bounded. Let $(\mathfrak{p},\mathfrak{q},\mathfrak{r}) \in \mathcal{P}$ with $(\mathfrak{p},\mathfrak{q},\mathfrak{r})= \nu \mathcal{S}( \mathfrak{p},\mathfrak{q},\mathfrak{r})$. For any $\varsigma \in [0,1]$, we have

$$\begin{aligned} &\mathfrak{p}(\varsigma )=\nu \mathcal{S}_{1}(\mathfrak{p}, \mathfrak{q},\mathfrak{r}) (\varsigma ), \\ & \mathfrak{q}(\varsigma )=\nu \mathcal{S}_{2}(\mathfrak{p}, \mathfrak{q},\mathfrak{r}) (\varsigma ), \\ & \mathfrak{r}(\varsigma )=\nu \mathcal{S}_{3}(\mathfrak{p}, \mathfrak{q},\mathfrak{r}) (\varsigma ). \end{aligned}$$

Then by $(\mathscr{M}_{2})$

$$\begin{aligned}& \bigl\vert \mathfrak{p}(\varsigma ) \bigr\vert \leq \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} \biggr) +{ \mathcal{W}_{1}} \bigl( \kappa _{0}+ \kappa _{1} \vert \mathfrak{p} \vert + \kappa _{2} \vert \mathfrak{q} \vert + \kappa _{3} \vert \mathfrak{r} \vert \bigr) \\& \hphantom{ \bigl\vert \mathfrak{p}(\varsigma ) \bigr\vert \leq }{} + {\mathcal{V}_{1}} \bigl(\lambda _{0}+ \lambda _{1} \vert \mathfrak{p} \vert + \lambda _{2} \vert \mathfrak{q} \vert + \lambda _{3} \vert \mathfrak{r} \vert \bigr) + {\mathcal{U}_{1}} \bigl(\varepsilon _{0} + \varepsilon _{1} \vert \mathfrak{p} \vert + \varepsilon _{2} \vert \mathfrak{q} \vert + \varepsilon _{3} \vert \mathfrak{r} \vert \bigr) \\& \hphantom{ \bigl\vert \mathfrak{p}(\varsigma ) \bigr\vert }\leq \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} \biggr)+ {\mathcal{W}_{1}}\kappa _{0}+ {\mathcal{V}_{1}} \lambda _{0} + { \mathcal{U}_{1}} \varepsilon _{0} \\& \hphantom{ \bigl\vert \mathfrak{p}(\varsigma ) \bigr\vert \leq }{} + ({\mathcal{W}_{1}}\kappa _{1}+ {\mathcal{V}_{1}} \lambda _{1} + { \mathcal{U}_{1}} \varepsilon _{1}) \vert \mathfrak{p} \vert \\& \hphantom{ \bigl\vert \mathfrak{p}(\varsigma ) \bigr\vert \leq }{} + ({\mathcal{W}_{1}}\kappa _{2}+ {\mathcal{V}_{1}} \lambda _{2}+ { \mathcal{U}_{1}} \varepsilon _{2}) \vert \mathfrak{q} \vert \\& \hphantom{ \bigl\vert \mathfrak{p}(\varsigma ) \bigr\vert \leq }{} + ({\mathcal{W}_{1}}\kappa _{3}+ {\mathcal{V}_{1}} \lambda _{3} + { \mathcal{U}_{1}} \varepsilon _{3}) \vert \mathfrak{r} \vert , \\& \bigl\vert \mathfrak{q}(\varsigma ) \bigr\vert \leq \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\varphi \mathfrak{A}_{2}} \biggr) + { \mathcal{W}_{2}}\kappa _{0}+ {\mathcal{V}_{2}} \lambda _{0} + { \mathcal{U}_{2}} \varepsilon _{0} \\& \hphantom{\bigl\vert \mathfrak{q}(\varsigma ) \bigr\vert \leq}{} + ({\mathcal{W}_{2}}\kappa _{1}+ {\mathcal{V}_{2}} \lambda _{1}+ { \mathcal{U}_{2}} \varepsilon _{1}) \vert \mathfrak{p} \vert {} \\& \hphantom{\bigl\vert \mathfrak{q}(\varsigma ) \bigr\vert \leq}{} + ({\mathcal{W}_{2}}\kappa _{2}+ {\mathcal{V}_{2}} \lambda _{2} + { \mathcal{U}_{2}} \varepsilon _{2}) \vert \mathfrak{q} \vert {} \\& \hphantom{\bigl\vert \mathfrak{q}(\varsigma ) \bigr\vert \leq}{} + ({\mathcal{W}_{2}}\kappa _{3}+ {\mathcal{V}_{2}} \lambda _{3} + { \mathcal{U}_{2}} \varepsilon _{3}) \vert \mathfrak{r} \vert , \end{aligned}$$

and

$$ \begin{aligned} \bigl\vert \mathfrak{r}(\varsigma ) \bigr\vert \leq {}& \biggl[ \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr] + {\mathcal{W}_{3}}\kappa _{0}+ {\mathcal{V}_{3}} \lambda _{0} + { \mathcal{U}_{3}} \varepsilon _{0} \\ &{} + ({\mathcal{W}_{3}}\kappa _{1}+ {\mathcal{V}_{3}} \lambda _{1}+ { \mathcal{U}_{3}} \varepsilon _{1}) \vert \mathfrak{p} \vert {} \\ &{}+ ({\mathcal{W}_{3}} \kappa _{2}+ {\mathcal{V}_{3}} \lambda _{2} + { \mathcal{U}_{3}} \varepsilon _{2}) \vert \mathfrak{q} \vert \\ &{}+ ({\mathcal{W}_{3}}\kappa _{3}+ {\mathcal{V}_{3}} \lambda _{3} + { \mathcal{U}_{3}} \varepsilon _{3}) \vert \mathfrak{r} \vert . \end{aligned} $$

As a result, we can conclude that

$$\begin{aligned}& \bigl\Vert \mathfrak{p}(\varsigma ) \bigr\Vert \leq \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} \biggr)+ { \mathcal{W}_{1}}\kappa _{0}+ { \mathcal{V}_{1}} \lambda _{0} + { \mathcal{U}_{1}} \varepsilon _{0} + ({\mathcal{W}_{1}}\kappa _{1}+ { \mathcal{V}_{1}} \lambda _{1} + { \mathcal{U}_{1}} \varepsilon _{1}) \Vert \mathfrak{p} \Vert \\& \hphantom{\bigl\Vert \mathfrak{p}(\varsigma ) \bigr\Vert \leq }{} + ({\mathcal{W}_{1}}\kappa _{2}+ {\mathcal{V}_{1}} \lambda + { \mathcal{U}_{1}} \varepsilon _{2}) \Vert \mathfrak{q} \Vert + ({ \mathcal{W}_{1}}\kappa _{3}+ { \mathcal{V}_{1}} \lambda _{3} + { \mathcal{U}_{1}} \varepsilon _{3}) \Vert \mathfrak{r} \Vert , \\& \bigl\Vert \mathfrak{q}(\varsigma ) \bigr\Vert \leq \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\varphi \mathfrak{A}_{2}} \biggr) + { \mathcal{W}_{2}}\kappa _{0}+ {\mathcal{V}_{2}} \lambda _{0} + { \mathcal{U}_{2}} \varepsilon _{0} + ({\mathcal{W}_{2}}\kappa _{1}+ { \mathcal{V}_{2}} \lambda _{1}+ { \mathcal{U}_{2}} \varepsilon _{1}) \Vert \mathfrak{p} \Vert \\& \hphantom{\bigl\Vert \mathfrak{q}(\varsigma ) \bigr\Vert \leq}{}+ ({\mathcal{W}_{2}}\kappa _{2}+ { \mathcal{V}_{2}}\lambda _{2} + { \mathcal{U}_{2}} \varepsilon _{2}) \Vert \mathfrak{q} \Vert {}+ ({ \mathcal{W}_{2}}\kappa _{3}+ {\mathcal{V}_{2}} \lambda _{3} + { \mathcal{U}_{2}} \varepsilon _{3}) \Vert \mathfrak{r} \Vert , \\& \bigl\Vert \mathfrak{r}(\varsigma ) \bigr\Vert \leq \biggl[ \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr] + {\mathcal{W}_{3}}\kappa _{0}+ {\mathcal{V}_{3}} \lambda _{0} + { \mathcal{U}_{3}} \varepsilon _{0} + ({\mathcal{W}_{3}} \kappa _{1}+ { \mathcal{V}_{3}} \lambda _{1}+ { \mathcal{U}_{3}} \varepsilon _{1}) \Vert \mathfrak{p} \Vert \\& \hphantom{\bigl\Vert \mathfrak{r}(\varsigma ) \bigr\Vert \leq}{}+ ({\mathcal{W}_{3}} \kappa _{2}+ { \mathcal{V}_{3}}\lambda _{2} + { \mathcal{U}_{3}} \varepsilon _{2}) \Vert \mathfrak{q} \Vert + ({ \mathcal{W}_{3}} \kappa _{3}+ {\mathcal{V}_{3}} \lambda _{3} + {\mathcal{U}_{3}} \varepsilon _{3}) \Vert \mathfrak{r} \Vert . \end{aligned}$$

By the previous three inequalities we arrive at

$$\begin{aligned}& \Vert \mathfrak{p} \Vert + \Vert \mathfrak{q} \Vert + \Vert \mathfrak{r} \Vert \\& \quad \leq \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} \biggr) + \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\mathfrak{A}_{2}\varphi} \biggr)+ \biggl[ \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr] \\& \qquad {}+ ({\mathcal{W}_{1}}+ {\mathcal{W}_{2}}+{ \mathcal{W}_{3}})\kappa _{0} +({\mathcal{V}_{1}}+ {\mathcal{V}_{2}}+{\mathcal{V}_{3}})\lambda _{0} {}+ ({\mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}})\varepsilon _{0} \\& \qquad {}+ \bigl[({\mathcal{W}_{1}}+ {\mathcal{W}_{2}}+{ \mathcal{W}_{3}}) \kappa _{1}+({\mathcal{V}_{1}}+ {\mathcal{V}_{2}}+{\mathcal{V}_{3}}) \lambda _{1} + ({\mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}}) \varepsilon _{1} \bigr] \Vert \mathfrak{p} \Vert \\& \qquad {}+ \bigl[({\mathcal{W}_{1}}+ {\mathcal{W}_{2}}+{ \mathcal{W}_{3}}) \kappa _{2} +({\mathcal{V}_{1}}+ {\mathcal{V}_{2}}+{\mathcal{V}_{3}}) \lambda _{2} +({\mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}}) \varepsilon _{2} \bigr] \Vert \mathfrak{q} \Vert \\& \qquad {}+ \bigl[({\mathcal{W}_{1}}+ {\mathcal{W}_{2}}+{ \mathcal{W}_{3}}) \kappa _{3} +({\mathcal{V}_{1}}+ {\mathcal{V}_{2}}+{\mathcal{V}_{3}}) \lambda _{3} +({\mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}}) \varepsilon _{3} \bigr] \Vert \mathfrak{r} \Vert , \end{aligned}$$

implying that

$$ \begin{aligned} \bigl\Vert (\mathfrak{p},\mathfrak{q},\mathfrak{r}) \bigr\Vert _{\mathcal{J}} \leq {}& \frac{1}{\Phi} \biggl[ \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} \biggr) + \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\varphi \mathfrak{A}_{2}} \biggr)+ \biggl[ \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr] \\ & {} +({\mathcal{W}_{1}}+ {\mathcal{W}_{2}}+{ \mathcal{W}_{3}}) \kappa _{0}+({\mathcal{V}_{1}}+ {\mathcal{V}_{2}}+{\mathcal{V}_{3}}) \lambda _{0} {}+({\mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}}) \varepsilon _{0} \biggr] , \end{aligned} $$

where $\Phi =\min \lbrace 1-[({\mathcal{W}_{1}}+{\mathcal{W}_{2}}+{ \mathcal{W}_{3}})\kappa _{i}+({\mathcal{V}_{1}}+{\mathcal{V}_{2}}+{ \mathcal{V}_{3}}) \lambda _{i} +({\mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}})\varepsilon _{i}],i=1,2,3 \rbrace$. which means that $\mathcal{P}$ is bounded. Thus by the Leray–Schauder alternative [39] the operator $\mathcal{S}$ has at least one fixed point, which implies that problem (1) has at least one solution on $[0,1]$. This completes the proof. □

Banach’s principle of contraction mapping provides the basis for our next results on the existence and uniqueness.

Theorem 2

Let $\Upsilon \neq 0$, where ϒ is defined by (10) and (11). In addition, we assume that

$(\mathcal{T}_{1})$ $\hat{\mathcal{F}_{1}},\hat{\mathcal{F}_{2}},\hat{\mathcal{F}_{3}}:[0,1] \times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ are continuous functions, and there exist nonnegative constants $\Theta _{1}$, $\Theta _{2}$, and $\Theta _{3}$ such that for all $\varsigma \in [0,1]$ and $\mathfrak{p}_{i},\mathfrak{q}_{i},\mathfrak{r}_{i} \in \mathbb{R}$, $i = 1,2,3$, we have

$$ \begin{aligned} & \bigl\vert \hat{\mathcal{F}_{1}} ( \varsigma , \mathfrak{q}_{1}, \mathfrak{q}_{2}, \mathfrak{q}_{3}) - \hat{\mathcal{F}_{1}} ( \varsigma , \mathfrak{r}_{1} , \mathfrak{r}_{2}, \mathfrak{r}_{3}) \bigr\vert \leq \Theta _{1} \bigl( \vert \mathfrak{q}_{1}- \mathfrak{r}_{1} \vert + \vert \mathfrak{q}_{2} - \mathfrak{r}_{2} \vert + \vert \mathfrak{q}_{3} - \mathfrak{r}_{3} \vert \bigr), \\ & \bigl\vert \hat{\mathcal{F}_{2}} (\varsigma , \mathfrak{q}_{1}, \mathfrak{q}_{2},\mathfrak{q}_{3}) - \hat{\mathcal{F}_{2}} ( \varsigma , \mathfrak{r}_{1} , \mathfrak{r}_{2},\mathfrak{r}_{3}) \bigr\vert \leq \Theta _{2} \bigl( \vert \mathfrak{q}_{1}- \mathfrak{r}_{1} \vert + \vert \mathfrak{q}_{2} -\mathfrak{r}_{2} \vert + \vert \mathfrak{q}_{3} - \mathfrak{r}_{3} \vert \bigr), \\ & \bigl\vert \hat{\mathcal{F}_{3}} (\varsigma , \mathfrak{q}_{1}, \mathfrak{q}_{2},\mathfrak{q}_{3})- \hat{\mathcal{F}_{3}} (\varsigma , \mathfrak{r}_{1}, \mathfrak{r}_{2},\mathfrak{r}_{3}) \bigr\vert \leq \Theta _{3} \bigl( \vert \mathfrak{q}_{1}- \mathfrak{r}_{1} \vert + \vert \mathfrak{q}_{2}- \mathfrak{r}_{2} \vert + \vert \mathfrak{q}_{3} - \mathfrak{r}_{3} \vert \bigr) \end{aligned} $$

if

$$ ({\mathcal{W}_{1}}+{\mathcal{W}_{2}}+{ \mathcal{W}_{3}})\Theta _{1}+({ \mathcal{V}_{1}}+{ \mathcal{V}_{2}}+{\mathcal{V}_{3}})\Theta _{2}+({ \mathcal{U}_{1}}+{\mathcal{U}_{2}}+{\mathcal{U}_{3}}) \Theta _{3}< 1, $$

(21)

where $\mathcal{W}_{i}$, $\mathcal{V}_{i}$, $\mathcal{U}_{i}$ are given in (19). Then system (1) has a unique solution on $[0,1]$.

Proof

Let $\sup_{\varsigma \in [0,1]} \hat{\mathcal{F}_{1}} (\varsigma ,0,0,0)=Q_{1}< \infty$, $\sup_{\varsigma \in [0,1]} \hat{\mathcal{F}_{2}} (\varsigma ,0,0,0)=Q_{2}< \infty$, and $\sup_{\varsigma \in [0,1]}\hat{\mathcal{F}_{3}} (\varsigma ,0,0,0)=Q_{3}< \infty$, and let $\varPsi > 0$ be such that

$$ \varPsi > \frac{ ( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} ) + ( \frac{1-e^{-\varphi{\varsigma}}}{\mathfrak{A}_{2}\varphi } )+ [ \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} ] + \mathcal{O}_{1}}{1-({\mathcal{W}_{1}}+{\mathcal{W}_{2}}+{\mathcal{W}_{3}})\Theta _{1} -({\mathcal{V}_{1}}+{\mathcal{V}_{2}}+{\mathcal{V}_{3}})\Theta _{2}-({\mathcal{U}_{1}} +{\mathcal{U}_{2}}+{\mathcal{U}_{3}})\Theta _{3}}, $$

where $\mathcal{O}_{1}=({\mathcal{W}_{1}}+{\mathcal{W}_{2}}+{\mathcal{W}_{3}})Q_{1}+({ \mathcal{V}_{1}}+{\mathcal{V}_{2}}+{\mathcal{V}_{3}})Q_{2} +({ \mathcal{U}_{1}}+{\mathcal{U}_{2}}+{\mathcal{U}_{3}})Q_{3}$.

We will show that $\mathcal{S}B_{\varPsi} \subset B_{\varPsi}$, where $B_{\varPsi} = \lbrace (\mathfrak{p},\mathfrak{q},\mathfrak{r})\in X \times X \times X : \Vert (\mathfrak{p},\mathfrak{q},\mathfrak{r}) \Vert \leq \varPsi \rbrace$.

By assumption $(\mathscr{M}_{2})$, for $(\mathfrak{p},\mathfrak{q},\mathfrak{r})\subset B_{\varPsi}$, $\varsigma \in [0,1]$, we have

$$\begin{aligned} \begin{aligned} &\bigl\vert \mathfrak{p} \bigl( \varsigma ,\mathfrak{p}(\varsigma ), \mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma ) \bigr) \bigr\vert \leq \bigl\vert \mathfrak{p} \bigl(\varsigma , \mathfrak{p}( \varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{q}(\varsigma ) \bigr)- \mathfrak{p} (\varsigma ,0,0,0) \bigr\vert \\ &\hphantom{\bigl\vert \mathfrak{p} \bigl( \varsigma ,\mathfrak{p}(\varsigma ), \mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma ) \bigr) \bigr\vert }\leq \Theta _{1} \bigl( \bigl\vert \mathfrak{p}(\varsigma ) \bigr\vert + \bigl\vert \mathfrak{q}(\varsigma ) \bigr\vert + \bigl\vert \mathfrak{r}(\varsigma ) \bigr\vert \bigr)+ Q_{1} \\ &\hphantom{\bigl\vert \mathfrak{p} \bigl( \varsigma ,\mathfrak{p}(\varsigma ), \mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma ) \bigr) \bigr\vert }\leq \Theta _{1} \bigl( \Vert \mathfrak{p} \Vert + \Vert \mathfrak{q} \Vert + \Vert \mathfrak{r} \Vert \bigr)+ Q_{1} \leq \Theta _{1}\varPsi + Q_{1}, \end{aligned} \end{aligned}$$

(22)

$$\begin{aligned} \begin{aligned} &\bigl\vert \mathfrak{q} \bigl( \varsigma ,\mathfrak{p}(\varsigma ), \mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma ) \bigr) \bigr\vert \leq \bigl\vert \mathfrak{q} \bigl(\varsigma , \mathfrak{p}( \varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{q}(\varsigma ) \bigr)- \mathfrak{q} (\varsigma ,0,0,0) \bigr\vert \\ &\hphantom{\bigl\vert \mathfrak{q} \bigl( \varsigma ,\mathfrak{p}(\varsigma ), \mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma ) \bigr) \bigr\vert }\leq \Theta _{2} \bigl( \bigl\vert \mathfrak{p}(\varsigma ) \bigr\vert + \bigl\vert \mathfrak{q}(\varsigma ) \bigr\vert + \bigl\vert \mathfrak{r}(\varsigma ) \bigr\vert \bigr)+ Q_{2} \\ &\hphantom{\bigl\vert \mathfrak{q} \bigl( \varsigma ,\mathfrak{p}(\varsigma ), \mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma ) \bigr) \bigr\vert }\leq \Theta _{2} \bigl( \Vert \mathfrak{p} \Vert + \Vert \mathfrak{q} \Vert + \Vert \mathfrak{r} \Vert \bigr)+ Q_{2} \leq \Theta _{2}\varPsi + Q_{2}, \end{aligned} \end{aligned}$$

(23)

$$\begin{aligned} \begin{aligned} \bigl\vert \mathfrak{r} \bigl( \varsigma ,\mathfrak{p}(\varsigma ), \mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma ) \bigr) \bigr\vert \leq & \bigl\vert \mathfrak{r} \bigl(\varsigma , \mathfrak{p}( \varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{q}(\varsigma ) \bigr)- \mathfrak{r} (\varsigma ,0,0,0) \bigr\vert \\ \leq & \Theta _{3} \bigl( \bigl\vert \mathfrak{p}(\varsigma ) \bigr\vert + \bigl\vert \mathfrak{q}(\varsigma ) \bigr\vert + \bigl\vert \mathfrak{r}(\varsigma ) \bigr\vert \bigr)+ Q_{3} \\ \leq & \Theta _{3} \bigl( \Vert \mathfrak{p} \Vert + \Vert \mathfrak{q} \Vert + \Vert \mathfrak{r} \Vert \bigr)+ Q_{3} \leq \Theta _{3}\varPsi + Q_{3}, \end{aligned} \end{aligned}$$

(24)

using (22), (23), and (24), This leads to

$$\begin{aligned}& \bigl\vert \mathcal{S}_{1} \bigl((\mathfrak{p},\mathfrak{q}, \mathfrak{r}) ( \varsigma ) \bigr) \bigr\vert \\& \quad \leq \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\Upsilon} \biggr) \\& \qquad {}\times \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{ \mathfrak{A}_{5}} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr] + \hat{ \mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}} \Biggl[ \beta _{3}\sum_{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr] \Biggr\rbrace \Vert \hat{\mathcal{F}_{1}} \Vert \\& \qquad {}+ \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}} \Biggl( \beta _{1}\sum_{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr) + \biggl(\mathfrak{A}_{2}{\hat{ \mathfrak{A}_{5}}} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr] \biggr) \Biggr\rbrace \Vert \hat{\mathcal{F}_{2}} \Vert \\& \qquad {}+ \Biggl\lbrace \hat{\mathfrak{A}_{3}}\mathfrak{A}_{2} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr] + \hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{2}\sum _{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) \Biggr\rbrace \Vert \hat{\mathcal{F}_{3}} \Vert \\& \qquad {}+ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \\& \quad \leq \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} \biggr)+ {\mathcal{W}_{1}} (\Theta _{1} \varPsi + Q_{1}) + { \mathcal{V}_{1}}( \Theta _{2} \varPsi +Q_{2}) +{\mathcal{U}_{1}}( \Theta _{3} \varPsi +Q_{3}) \\& \quad \leq \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} \biggr) + ({\mathcal{W}_{1}} \Theta _{1} + {\mathcal{V}_{1}} \Theta _{2} + { \mathcal{U}_{1}} \Theta _{3})\varPsi + { \mathcal{W}_{1}} Q_{1} + { \mathcal{V}_{1}} Q_{2} + {\mathcal{U}_{1}} Q_{3}, \end{aligned}$$

which, on taking the norm on $\varsigma \in [0,1]$, yields

$$\begin{aligned} \bigl\Vert \mathcal{S}_{1}(\mathfrak{p},\mathfrak{q},\mathfrak{r}) \bigr\Vert _{\mathcal{J}} \leq {}& \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} \biggr)+({ \mathcal{W}_{1}}\Theta _{1}+ {\mathcal{V}_{1}} \Theta _{2} + { \mathcal{U}_{1}} \Theta _{3}) \varPsi \\ &{}+ {\mathcal{W}_{1}}Q_{1}+ {\mathcal{V}_{1}}Q_{2}+{ \mathcal{U}_{1}} Q_{3}. \end{aligned}$$

Likewise, we can find that

$$\begin{aligned} \bigl\Vert \mathcal{S}_{2}(\mathfrak{p},\mathfrak{q},\mathfrak{r}) \bigr\Vert _{\mathcal{J}} \leq {}& \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\mathfrak{A}_{2}\varphi} \biggr)+({ \mathcal{W}_{2}}\Theta _{1}+ {\mathcal{V}_{2}} \Theta _{2} + { \mathcal{U}_{2}} \Theta _{3}) \varPsi \\ &{}+ {\mathcal{W}_{2}}Q_{1}+ {\mathcal{V}_{2}}Q_{2}+{ \mathcal{U}_{2}} Q_{3} \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert \mathcal{S}_{3}(\mathfrak{p},\mathfrak{q},\mathfrak{r}) \bigr\Vert _{\mathcal{J}} \leq {}& \biggl[ \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr] +({ \mathcal{W}_{3}}\Theta _{1}+ {\mathcal{V}_{3}} \Theta _{2}+{ \mathcal{U}_{3}}\Theta _{3})\varPsi \\ &{}+ {\mathcal{W}_{3}}Q_{1}+ {\mathcal{V}_{3}}Q_{2}+{ \mathcal{U}_{3}} Q_{3}. \end{aligned}$$

Consequently,

$$ \begin{aligned} \bigl\Vert \mathcal{S}(\mathfrak{p},\mathfrak{q}, \mathfrak{r}) \bigr\Vert _{\mathcal{J}} \leq {}& \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} \biggr) + \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\mathfrak{A}_{2}\varphi} \biggr)+ \biggl[ \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr] \\ &{}+ \bigl[({\mathcal{W}_{1}}+{\mathcal{W}_{2}}+{ \mathcal{W}_{3}}) \Theta _{1}+ ({\mathcal{V}_{1}}+{ \mathcal{V}_{2}}+{\mathcal{V}_{3}}) \Theta _{2} {}+({\mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}}) \Theta _{3} \bigr]\varPsi \\ &{} + ({\mathcal{W}_{1}}+{\mathcal{W}_{2}}+{ \mathcal{W}_{3}})Q_{1}+({ \mathcal{V}_{1}}+{ \mathcal{V}_{2}}+{\mathcal{V}_{3}})Q_{2} {}+ ({ \mathcal{U}_{1}}+{\mathcal{U}_{2}}+{\mathcal{U}_{3}})Q_{3} \\ \leq{}& \varPsi . \end{aligned} $$

Now, for $(\mathfrak{p}_{1},\mathfrak{q}_{1},\mathfrak{r}_{1}),(\mathfrak{p}_{2}, \mathfrak{q}_{2},\mathfrak{r}_{2}) \in \mathcal{J}\times \mathcal{J} \times \mathcal{J}$ and for any $\varsigma \in [0,1]$, we get

$$\begin{aligned}& \bigl\vert \mathcal{S}_{1} \bigl((\mathfrak{p}_{2}, \mathfrak{q}_{2}, \mathfrak{r}_{2}) (\varsigma ) \bigr)- \mathcal{S}_{1} \bigl(( \mathfrak{p}_{1}, \mathfrak{q}_{1},\mathfrak{r}_{1}) (\varsigma ) \bigr) \bigr\vert \\& \quad \leq \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} \biggr) \\& \qquad {}\times \Biggl[ \Biggl\lbrace \hat{\mathfrak{A}_{4}} \hat{ \mathfrak{A}_{5}} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr] + \hat{ \mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}} \Biggl[ \beta _{3}\sum_{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr] \Biggr\rbrace \\& \qquad {}\times \Theta _{1}\bigl( \Vert \mathfrak{p}_{2}- \mathfrak{p}_{1} \Vert + \Vert \mathfrak{q}_{2}- \mathfrak{q}_{1} \Vert + \Vert \mathfrak{r}_{2}- \mathfrak{r}_{1} \Vert \bigr) \\& \qquad {}+ \Biggl\lbrace \hat{\mathfrak{A}_{4}} \hat{ \mathfrak{A}_{5}} \Biggl( \beta _{1}\sum _{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr) + \biggl( \mathfrak{A}_{2}{\hat{\mathfrak{A}_{5}}} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr] \biggr) \Biggr\rbrace \\& \qquad {}\times \Theta _{2}\bigl( \Vert \mathfrak{p}_{2}- \mathfrak{p}_{1} \Vert + \Vert \mathfrak{q}_{2}- \mathfrak{q}_{1} \Vert + \Vert \mathfrak{r}_{2}- \mathfrak{r}_{1} \Vert \bigr) \\& \qquad {}+ \Biggl\lbrace \hat{\mathfrak{A}_{3}}\mathfrak{A}_{2} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr] + \hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{2}\sum _{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) \Biggr\rbrace \Biggr] \\& \qquad {}\times \Theta _{3}\bigl( \Vert \mathfrak{p}_{2}- \mathfrak{p}_{1} \Vert + \Vert \mathfrak{q}_{2}- \mathfrak{q}_{1} \Vert + \Vert \mathfrak{r}_{2}- \mathfrak{r}_{1} \Vert \bigr)+ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \\& \quad \leq (\mathcal{W}_{1}\Theta _{1}+\mathcal{V}_{1} \Theta _{2}+ \mathcal{U}_{1}\Theta _{3}) \bigl( \Vert \mathfrak{p}_{2}-\mathfrak{p}_{1} \Vert + \Vert \mathfrak{q}_{2}-\mathfrak{q}_{1} \Vert + \Vert \mathfrak{r}_{2}-\mathfrak{r}_{1} \Vert \bigr), \end{aligned}$$

from which we obtain

$$\begin{aligned} & \bigl\Vert \mathcal{S}_{1} \bigl((\mathfrak{p}_{2}, \mathfrak{q}_{2}, \mathfrak{r}_{2}) (\varsigma ) \bigr)- \mathcal{S}_{1} \bigl(( \mathfrak{p}_{1}, \mathfrak{q}_{1},\mathfrak{r}_{1}) (\varsigma ) \bigr) \bigr\Vert _{\mathcal{J}} \\ &\quad \leq (\mathcal{W}_{1}\Theta _{1}+\mathcal{V}_{1} \Theta _{2}+ \mathcal{U}_{1}\Theta _{3}) \bigl( \Vert \mathfrak{p}_{2}-\mathfrak{p}_{1} \Vert + \Vert \mathfrak{q}_{2}-\mathfrak{q}_{1} \Vert + \Vert \mathfrak{r}_{2}-\mathfrak{r}_{1} \Vert \bigr). \end{aligned}$$

Similarly,

$$\begin{aligned} & \bigl\Vert \mathcal{S}_{2} \bigl((\mathfrak{p}_{2}, \mathfrak{q}_{2}, \mathfrak{r}_{2}) (\varsigma ) \bigr)- \mathcal{S}_{2} \bigl(( \mathfrak{p}_{1}, \mathfrak{q}_{1},\mathfrak{r}_{1}) (\varsigma ) \bigr) \bigr\Vert _{ \mathcal{J}} \\ &\quad \leq (\mathcal{W}_{2}\Theta _{1}+\mathcal{V}_{2} \Theta _{2}+ \mathcal{U}_{2}\Theta _{3}) \bigl( \Vert \mathfrak{p}_{2}-\mathfrak{p}_{1} \Vert + \Vert \mathfrak{q}_{2}-\mathfrak{q}_{1} \Vert + \Vert \mathfrak{r}_{2}-\mathfrak{r}_{1} \Vert \bigr), \end{aligned}$$

and

$$\begin{aligned}& \bigl\Vert \mathcal{S}_{3} \bigl((\mathfrak{p}_{2}, \mathfrak{q}_{2}, \mathfrak{r}_{2}) (\varsigma ) \bigr)- \mathcal{S}_{3} \bigl(( \mathfrak{p}_{1}, \mathfrak{q}_{1},\mathfrak{r}_{1}) (\varsigma ) \bigr) \bigr\Vert _{ \mathcal{J}} \\& \quad \leq (\mathcal{W}_{3}\Theta _{1}+\mathcal{V}_{3} \Theta _{2}+ \mathcal{U}_{3}\Theta _{3}) \bigl( \Vert \mathfrak{p}_{2}-\mathfrak{p}_{1} \Vert + \Vert \mathfrak{q}_{2}-\mathfrak{q}_{1} \Vert + \Vert \mathfrak{r}_{2}-\mathfrak{r}_{1} \Vert \bigr), \\& \bigl\Vert \mathcal{S}(\mathfrak{p}_{2}, \mathfrak{q}_{2}, \mathfrak{r}_{2})- \mathcal{S}( \mathfrak{p}_{1},\mathfrak{q}_{1}, \mathfrak{r}_{1}) \bigr\Vert _{\mathcal{J}} \\& \quad \leq \bigl[({\mathcal{W}_{1}}+ {\mathcal{W}_{2}}+{ \mathcal{W}_{3}}) \Theta _{1} + ({\mathcal{V}_{1}}+ {\mathcal{V}_{2}}+{\mathcal{V}_{3}}) \Theta _{2}+({\mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}}) \Theta _{3} \bigr] \\& \qquad {}\times \bigl( \Vert \mathfrak{p}_{2}-\mathfrak{p}_{1} \Vert + \Vert \mathfrak{q}_{2}-\mathfrak{q}_{1} \Vert + \Vert \mathfrak{r}_{2}-\mathfrak{r}_{1} \Vert \bigr). \end{aligned}$$

In view of this inequality and (21), $\mathcal{S}$ is a contraction. As a result of Banach’s fixed point theorem, there exists a unique fixed point for the operator $\mathcal{ S}$, which corresponds to a unique solution to problem (1) on $[0,1]$. The proof is complete. □

4 Hyers–Ulam stability

Let us define the nonlinear operators $\mathfrak{Z}_{1},\mathfrak{Z}_{2}, \mathfrak{Z}_{3} \in \mathcal{C}([0,1], \mathbb{R})\times \mathcal{C}([0,1], \mathbb{R})\times \mathcal{C}([0,1], \mathbb{R}) \to \mathcal{C}([0,1],\mathbb{R})$ by

$$ \textstyle\begin{cases} ({}^{c}{\mathcal{D}}^{\eta }+ \varphi {}^{c}{\mathcal{D}}^{\eta -1}) \mathfrak{p}(\varsigma )- \hat{\mathcal{F}_{1}} (\varsigma , \mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma )) ={\mathfrak{Z}}_{1}(\mathfrak{p},\mathfrak{q}, \mathfrak{r}) (\varsigma ), \\ ({}^{c}{\mathcal{D}}^{\xi }+ \varphi {}^{c}{\mathcal{D}}^{\xi -1}) \mathfrak{q}(\varsigma )- \hat{\mathcal{F}_{2}} (\varsigma , \mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma )) = {\mathfrak{Z}}_{2}(\mathfrak{p},\mathfrak{q}, \mathfrak{r}) (\varsigma ), \\ ({}^{c}{\mathcal{D}}^{\zeta }+ \varphi {}^{c}{\mathcal{D}}^{\zeta -1}) \mathfrak{r}(\varsigma )- \hat{\mathcal{F}_{3}}(\varsigma , \mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma )) = {\mathfrak{Z}}_{3}(\mathfrak{p},\mathfrak{q}, \mathfrak{r}) (\varsigma ) \end{cases} $$

(25)

for $\varsigma \in [0,1]$ For some $\pi _{1},\pi _{2},\pi _{3}>0$, we consider the following inequalities:

$$\begin{aligned} &\bigl\Vert {\mathfrak{Z}}_{1}(\mathfrak{p}, \mathfrak{q},\mathfrak{r}) \bigr\Vert \leq \pi _{1}, \qquad \bigl\Vert { \mathfrak{Z}}_{2} ( \mathfrak{p},\mathfrak{q},\mathfrak{r}) \bigr\Vert \leq \pi _{2},\qquad \bigl\Vert {\mathfrak{Z}}_{3} ( \mathfrak{p},\mathfrak{q}, \mathfrak{r}) \bigr\Vert \leq \pi _{3}. \end{aligned}$$

(26)

Definition 4

The coupled system (1) is said to be stable in the Hyers–Ulam sense if there exist $\mathscr{K}_{1}, \mathscr{K}_{2}, \mathscr{K}_{3}>0$ such that there is a unique solution $(\mathfrak{p},\mathfrak{q},\mathfrak{r})\in \mathcal{C}([0,1], \mathbb{R})\times \mathcal{C}([0,1],\mathbb{R})\times \mathcal{C}([0,1], \mathbb{R})$ of problem (1) with

$$\begin{aligned} \bigl\Vert (\mathfrak{p},\mathfrak{q},\mathfrak{r})-( \widehat{\mathfrak{p}}, \widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) \bigr\Vert \leq \mathcal{K}_{1}\pi _{1}+ \mathcal{K}_{2}\pi _{2} + \mathcal{K}_{3}\pi _{3} \end{aligned}$$

for every solution $(\widehat{\mathfrak{p}},\widehat{\mathfrak{q}}, \widehat{\mathfrak{r}})$ belonging to $\mathcal{C}([0,1],\mathbb{R})\times \mathcal{C}([0,1],\mathbb{R}) \times \mathcal{C}([0,1],\mathbb{R})$ of inequality (26).

Theorem 3

Suppose that $(\mathscr{M}_{2})$ holds. Then the BVP (1) is Hyers–Ulam stable.

Proof

Let $(\mathfrak{p},\mathfrak{q},\mathfrak{r})\in \mathcal{C}([0,1], \mathbb{R})\times \mathcal{C}([0,1],\mathbb{R})\times \mathcal{C}([0,1], \mathbb{R})$ be a solution of problem (1) that satisfies the main results. Let $(\widehat{\mathfrak{p}},\widehat{\mathfrak{q}}, \widehat{\mathfrak{r}})$ be any solution satisfying (26):

$$ \textstyle\begin{cases} ({}^{c}{\mathcal{D}}^{\eta }+ \varphi {}^{c}{\mathcal{D}}^{\eta -1}) \mathfrak{p}(\varsigma )= \hat{\mathcal{F}_{1}} (\varsigma , \mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma )) +{\mathfrak{Z}}_{1}(\mathfrak{p},\mathfrak{q}, \mathfrak{r}) (\varsigma ), \\ ({}^{c}{\mathcal{D}}^{\xi }+ \varphi {}^{c}{\mathcal{D}}^{\xi -1}) \mathfrak{q}(\varsigma )= \hat{\mathcal{F}_{2}} (\varsigma , \mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma )) +{\mathfrak{Z}}_{2}(\mathfrak{p},\mathfrak{q}, \mathfrak{r}) (\varsigma ), \\ ({}^{c}{\mathcal{D}}^{\zeta }+ \varphi {}^{c}{\mathcal{D}}^{\zeta -1}) \mathfrak{r}(\varsigma )= \hat{\mathcal{F}_{3}}(\varsigma , \mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ),\mathfrak{r}( \varsigma )) + {\mathfrak{Z}}_{3}(\mathfrak{p},\mathfrak{q}, \mathfrak{r}) (\varsigma ) \end{cases} $$

(27)

for $\varsigma \in [0,1]$. Then

$$\begin{aligned}& \widehat{\mathfrak{p}}(\varsigma )=\mathcal{S}_{1}( \widehat{ \mathfrak{p}},\widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) ( \varsigma )+ \biggl( \frac{1-e^{-\varphi \varsigma}}{\varphi \Upsilon} \biggr) \\& \hphantom{\widehat{\mathfrak{p}}(\varsigma )=}{}\times \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{1}\sum _{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{- \varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \mathfrak{Z}_{2}( \mathfrak{p},\mathfrak{q},\mathfrak{r})\,du \biggr)\,ds \\& \hphantom{\widehat{\mathfrak{p}}(\varsigma )=}{} + \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \mathfrak{Z}_{1}( \mathfrak{p},\mathfrak{q},\mathfrak{r})\,du \biggr)\,ds \Biggr) \\& \hphantom{\widehat{\mathfrak{p}}(\varsigma )=}{} +\hat{\mathfrak{A}_{5}}\mathfrak{A}_{2} \Biggl( \beta _{2} \sum_{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \mathfrak{Z}_{3}(\mathfrak{p},\mathfrak{q},\mathfrak{r})\,du \biggr)\,ds \\& \hphantom{\widehat{\mathfrak{p}}(\varsigma )=}{} + \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \mathfrak{Z}_{2}( \mathfrak{p},\mathfrak{q},\mathfrak{r})\,du \biggr)\,ds \Biggr) \\& \hphantom{\widehat{\mathfrak{p}}(\varsigma )=}{} + \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}} \Biggl( \beta _{3} \sum_{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi ( \rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \mathfrak{Z}_{1}( \mathfrak{p},\mathfrak{q},\mathfrak{r})\,du \biggr)\,ds \\& \hphantom{\widehat{\mathfrak{p}}(\varsigma )=}{} + \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \mathfrak{Z}_{3}( \mathfrak{p},\mathfrak{q},\mathfrak{r})\,du \biggr)\,ds \Biggr) \Biggr\rbrace \\& \hphantom{\widehat{\mathfrak{p}}(\varsigma )=}{} + \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \mathfrak{Z}_{1}( \mathfrak{p},\mathfrak{q},\mathfrak{r})\,du \biggr)\,ds, \\& \bigl\vert \widehat{\mathfrak{p}}(\varsigma )-\mathcal{S}_{1}( \widehat{\mathfrak{p}},\widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) ( \varsigma ) \bigr\vert \\& \quad \leq \biggl( \frac{1-e^{-\varphi \varsigma}}{\varphi \Upsilon} \biggr) \\& \qquad {} \times \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{1}\sum _{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{- \varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \pi _{2}\,du \biggr)\,ds \\& \qquad {} + \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \pi _{1}\,du \biggr)\,ds \Biggr) \\& \qquad {} +\hat{\mathfrak{A}_{5}}\mathfrak{A}_{2} \Biggl( \beta _{2} \sum_{j=1}^{k-2} v_{j} \int _{0}^{\varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \pi _{3}\,du \biggr)\,ds \\& \qquad {} + \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} \pi _{2}\,du \biggr)\,ds \Biggr) \\& \qquad {} + \hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}} \Biggl( \beta _{3} \sum_{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi ( \rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} \pi _{1}\,du \biggr)\,ds \\& \qquad {} + \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} \pi _{3}\,du \biggr)\,ds \Biggr) \Biggr\rbrace \\& \qquad {} + \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)}\pi _{1}\,du \biggr)\,ds \\& \quad \leq \biggl( \frac{1-e^{-\varphi{\varsigma}}}{\Upsilon \varphi} \biggr) \\& \qquad {} \times \Biggl[ \Biggl\lbrace \hat{\mathfrak{A}_{4}} \hat{ \mathfrak{A}_{5}} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr] + \hat{ \mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}} \Biggl[ \beta _{3}\sum_{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr] \pi _{1} \Biggr\rbrace \\& \qquad {} + \Biggl\lbrace \hat{\mathfrak{A}_{4}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{1}\sum _{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr) + \biggl( \mathfrak{A}_{2}{\hat{\mathfrak{A}_{5}}} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr] \biggr) \Biggr\rbrace \pi _{2} \\& \qquad {} + \Biggl\lbrace \hat{\mathfrak{A}_{3}}\mathfrak{A}_{2} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr] + \hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{5}} \Biggl( \beta _{2}\sum _{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) \Biggr\rbrace \pi _{3} \Biggr] \\& \qquad {}+\frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \\& \quad \leq (\mathcal{W}_{1}\pi _{1}+{\mathcal{V}_{1}} \pi _{2}+{ \mathcal{U}_{1}}\pi _{3}). \end{aligned}$$

In the same way,

$$\begin{aligned}& \widehat{\mathfrak{q}}(\varsigma )=\mathcal{S}_{2}( \widehat{ \mathfrak{p}},\widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) ( \varsigma )+ \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\mathfrak{A}_{2}} \biggr) \\& \hphantom{\widehat{\mathfrak{q}}(\varsigma )=}{} \times \Biggl\lbrace \beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{ \varrho _{j}} e^{-\varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} G_{2}({ \mathfrak{p}}, { \mathfrak{q}},{\mathfrak{r}}) (u)\,du \biggr)\,ds \\& \hphantom{\widehat{\mathfrak{q}}(\varsigma )=}{} + \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} G_{1}({ \mathfrak{p}}, { \mathfrak{q}},{\mathfrak{r}}) (u)\,du \biggr)\,ds \\& \hphantom{\widehat{\mathfrak{q}}(\varsigma )=}{} + \frac{1}{\Upsilon} \Biggl[ {\hat{\mathfrak{A}_{1}} \hat{ \mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}} \Biggl(\beta _{1}\sum_{j=1}^{k-2} w_{j} \int _{0}^{\varrho _{j}} e^{-\varphi (\varrho _{j}-s)} \biggl( \int _{0}^{s}\frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} G_{2}({ \mathfrak{p}}, {\mathfrak{q}},{\mathfrak{r}}) (u)\,du \biggr)\,ds \\& \hphantom{\widehat{\mathfrak{q}}(\varsigma )=}{} + \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} G_{1}({ \mathfrak{p}}, { \mathfrak{q}},{\mathfrak{r}}) (u)\,du \biggr)\,ds \Biggr) \\& \hphantom{\widehat{\mathfrak{q}}(\varsigma )=}{} + \hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{5}} \mathfrak{A}_{2} \Biggl( \beta _{2}\sum _{j=1}^{k-2} v_{j} \int _{0}^{ \varpi _{j}} e^{-\varphi (\varpi _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} G_{3}({ \mathfrak{p}}, { \mathfrak{q}},{\mathfrak{r}}) (u)\,du \biggr)\,ds \\& \hphantom{\widehat{\mathfrak{q}}(\varsigma )=}{} + \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)} G_{2}({ \mathfrak{p}}, { \mathfrak{q}},{\mathfrak{r}}) (u)\,du \biggr)\,ds \Biggr) \\& \hphantom{\widehat{\mathfrak{q}}(\varsigma )=}{} + \hat{\mathfrak{A}_{1}}\mathfrak{A}_{2} \hat{\mathfrak{A}_{3}} \Biggl(\beta _{3}\sum _{j=1}^{k-2} \vartheta _{j} \int _{0}^{\rho _{j}} e^{-\varphi (\rho _{j}-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\eta -2}}{\varGamma (\eta -1)} G_{1}({ \mathfrak{p}}, { \mathfrak{q}},{\mathfrak{r}}) (u)\,du \biggr)\,ds \\& \hphantom{\widehat{\mathfrak{q}}(\varsigma )=}{} + \int _{0}^{1} e^{-\varphi (1-s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\zeta -2}}{\varGamma (\zeta -1)} G_{3}({ \mathfrak{p}}, { \mathfrak{q}},{\mathfrak{r}}) (u)\,du \biggr)\,ds \Biggr) \Biggr] \Biggr\rbrace \\& \hphantom{\widehat{\mathfrak{q}}(\varsigma )=}{} + \int _{0}^{\varsigma} e^{-\varphi (\varsigma -s)} \biggl( \int _{0}^{s} \frac{(s-u)^{\xi -2}}{\varGamma (\xi -1)}G_{2}({ \mathfrak{p}}, { \mathfrak{q}},{\mathfrak{r}}) (u)\,du \biggr)\,ds, \\& \bigl\vert \widehat{\mathfrak{q}}(\varsigma )-\mathcal{S}_{2}( \widehat{\mathfrak{p}},\widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) ( \varsigma ) \bigr\vert \\& \quad \leq \biggl( \frac{1-e^{-\varphi \varsigma}}{{\varphi}\mathfrak{A}_{2}} \biggr) \\& \qquad {}\times \Biggl[ \Biggl\lbrace \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}}}{\Upsilon} \Biggl( \beta _{3} \sum_{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr) \\& \qquad {} + \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}}{\Upsilon} \biggl(\frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr)+ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \Biggr\rbrace \pi _{1} \\& \qquad {} + \Biggl\lbrace \Biggl( \beta _{1}\sum _{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr) + \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}}{\Upsilon} \Biggl( \beta _{1}\sum_{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr) \\& \qquad {} + \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}}{\Upsilon} \biggl[ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr] \Biggr\rbrace \pi _{2} + \Biggl\lbrace \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{2}}\hat{\mathfrak{A}_{3}}}{\Upsilon} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr) \\& \qquad {} + \frac{\hat{\mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}}\hat{\mathfrak{A}_{5}}}{\Upsilon} \Biggl( \beta _{2}\sum _{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) \Biggr\rbrace \pi _{3} \Biggr]+ \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \\& \quad \leq ({\mathcal{W}_{2}}\pi _{1}+{\mathcal{V}_{2}} \pi _{2}+{ \mathcal{U}_{2}}\pi _{3}). \end{aligned}$$

Similarly,

$$\begin{aligned}& \bigl\vert \widehat{\mathfrak{r}}(\varsigma )-\mathcal{S}_{3}( \widehat{\mathfrak{p}},\widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) ( \varsigma ) \bigr\vert \\& \quad \leq \biggl( \frac{(\varphi \varsigma -1+e^{-\varphi \varsigma})}{\varphi ^{2}{\Upsilon}} \biggr) \\& \qquad {}\times \Biggl[ \Biggl\lbrace \hat{\mathfrak{A}_{4}} \hat{ \mathfrak{A}_{1}} \Biggl( \beta _{3}\sum _{j=1}^{k-2} \vert \vartheta _{j} \vert \rho _{j}^{\eta -1} \frac{(1-e^{-\varphi \rho _{j}})}{\varphi \varGamma (\eta )} \Biggr) + \hat{ \mathfrak{A}_{4}}\hat{\mathfrak{A}_{6}} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\eta )} \biggr) \Biggr\rbrace \pi _{1} \\& \qquad {}+ \Biggl\lbrace \Biggl( \beta _{1}\sum _{j=1}^{k-2} \vert w_{j} \vert \varrho _{j}^{\xi -1} \frac{(1-e^{-\varphi \varrho _{j}})}{\varphi \varGamma (\xi )} \Biggr)+\hat{ \mathfrak{A}_{2}}\hat{\mathfrak{A}_{6}} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\xi )} \biggr) \Biggr\rbrace \pi _{2} \\& \qquad {}+ \Biggl\lbrace \hat{\mathfrak{A}_{2}}\hat{ \mathfrak{A}_{6}} \Biggl( \beta _{2}\sum _{j=1}^{k-2} \vert v_{j} \vert \varpi _{j}^{\zeta -1} \frac{(1-e^{-\varphi \varpi _{j}})}{\varphi \varGamma (\zeta )} \Biggr) + \hat{ \mathfrak{A}_{1}}\hat{\mathfrak{A}_{4}} \biggl( \frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \biggr) \Biggr\rbrace \pi _{3} \Biggr] \\& \qquad {}+\frac{(1-e^{-\varphi})}{\varphi \varGamma (\zeta )} \\& \quad \leq ({\mathcal{W}_{3}}\pi _{1}+{\mathcal{V}_{3}} \pi _{2}+{ \mathcal{U}_{3}}\pi _{3}), \end{aligned}$$

where $\mathcal{W}_{1}$, $\mathcal{W}_{2}$, $\mathcal{W}_{3}$, $\mathcal{V}_{1}$, $\mathcal{V}_{2}$, $\mathcal{V}_{3}$, $\mathcal{U}_{1}$, $\mathcal{U}_{2}$, and $\mathcal{U}_{3}$ are describes in the main results. Therefore the operator $\mathcal{S}$ defined in the main results can be excluded from the fixed point property as follows. We have

$$ \begin{aligned} \bigl\vert \mathfrak{p}(\varsigma )- \widehat{\mathfrak{p}}(\varsigma ) \bigr\vert ={} & \bigl\vert \mathfrak{p}( \varsigma )-\mathcal{S}_{1}( \widehat{\mathfrak{p}}, \widehat{ \mathfrak{q}},\widehat{\mathfrak{r}}) (\varsigma )+\mathcal{S}_{1}( \widehat{\mathfrak{p}}, \widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) (\varsigma )- \widehat{\mathfrak{p}}(\varsigma ) \bigr\vert \\ \leq {}& \bigl\vert \mathcal{S}_{1}(\mathfrak{p},\mathfrak{q}, \mathfrak{r}) (\varsigma )-\mathcal{S}_{1}(\widehat{\mathfrak{p}}, \widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) (\varsigma ) \bigr\vert + \bigl\vert \mathcal{S} _{1}( \widehat{\mathfrak{\mathfrak{p}}}, \widehat{ \mathfrak{q}}, \widehat{\mathfrak{r}}) (\varsigma )-\widehat{\mathfrak{p}}( \varsigma ) \bigr\vert \\ \leq {}& ({\mathcal{W}_{1}} {k}_{1}+ {\mathcal{V}_{1}} {\lambda}_{1}+ \mathcal{U}_{1}\varepsilon _{1})+ ({\mathcal{W}_{1}} {k}_{2}+ { \mathcal{V}_{1}} { \lambda}_{2}+\mathcal{U}_{1}\varepsilon _{2}) \\ & {}+({\mathcal{W}_{1}} {k}_{3}+ {\mathcal{V}_{1}} {\lambda}_{3}+ \mathcal{U}_{1}\varepsilon _{3}) \bigl\Vert (\mathfrak{p}, \mathfrak{q},\mathfrak{r})-(\widehat{\mathfrak{p}}- \widehat{\mathfrak{q}}-\widehat{\mathfrak{r}}) \bigr\Vert {} \\ & {}+({\mathcal{W}_{1}}\pi _{1}+{\mathcal{V}_{1}} \pi _{2}+{\mathcal{U}_{1}} \pi _{3}), \end{aligned} $$

(28)

so we obtain

$$ \begin{aligned} \bigl\vert \mathfrak{q}(\varsigma )- \widehat{\mathfrak{q}}(\varsigma ) \bigr\vert ={} & \bigl\vert \mathfrak{q}( \varsigma )-\mathcal{S}_{2}( \widehat{\mathfrak{p}}, \widehat{ \mathfrak{q}},\widehat{\mathfrak{r}}) (\varsigma )+\mathcal{S} _{2}( \widehat{\mathfrak{p}}, \widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) (\varsigma )- \widehat{\mathfrak{q}}(\varsigma ) \bigr\vert \\ \leq{} & \bigl\vert \mathcal{S}_{2}(\mathfrak{p},\mathfrak{q}, \mathfrak{r}) (\varsigma )-\mathcal{S}_{2}(\widehat{\mathfrak{p}}, \widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) (\varsigma ) \bigr\vert + \bigl\vert \mathcal{S}_{2}(\widehat{\mathfrak{p}}, \widehat{\mathfrak{q}}, \widehat{\mathfrak{r}}) (\varsigma )- \widehat{\mathfrak{q}}(\varsigma ) \bigr\vert \\ \leq{} & ({\mathcal{W}_{2}} {k}_{1}+ {\mathcal{V}_{2}} {\lambda}_{1}+ \mathfrak {N}_{2}\varepsilon _{1})+ ({\mathcal{W}_{2}} {k}_{2}+ { \mathcal{V}_{2}} {\lambda}_{2}+\mathcal{U}_{2} \varepsilon _{2}) \\ &{} +({\mathcal{W}_{2}} {k}_{3}+ {\mathcal{V}_{2}} {\lambda}_{3}+ \mathcal{U}_{2}\varepsilon _{3}) \bigl\Vert (\mathfrak{p}, \mathfrak{q},\mathfrak{r})-(\widehat{\mathfrak{p}}- \widehat{\mathfrak{q}}-\widehat{\mathfrak{r}}) \bigr\Vert {} \\ &{} +({\mathcal{W}_{2}}\pi _{1}+{\mathcal{V}_{2}} \pi _{2}+{\mathcal{U}_{2}} \pi _{3}) \end{aligned} $$

(29)

and, in the same way,

$$ \begin{aligned} \bigl\vert \mathfrak{r}(\varsigma )-\widehat{ \mathfrak{r}}(\varsigma ) \bigr\vert ={} & \bigl\vert \mathfrak{r}(\varsigma )-\mathcal{S}_{3}( \widehat{\mathfrak{p}}, \widehat{\mathfrak{q}}, \widehat{\mathfrak{r}}) (\varsigma )+\mathcal{S}_{3}(\widehat{ \mathfrak{p}}, \widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) (\varsigma )- \widehat{\mathfrak{r}}(\varsigma ) \bigr\vert \\ \leq{}& \bigl\vert \mathcal{S}_{3}({\mathfrak{p}}, {\mathfrak{q}},{ \mathfrak{r}}) (\varsigma )-\mathcal{S}_{3}(\widehat{\mathfrak{p}}, \widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) (\varsigma ) \bigr\vert + \bigl\vert \mathcal{S} _{3}(\widehat{\mathfrak{p}}, \widehat{ \mathfrak{q}},\widehat{\mathfrak{r}}) (\varsigma )- \widehat{\mathfrak{r}}( \varsigma ) \bigr\vert \\ \leq{}& ({\mathcal{W}_{3}} {k}_{1}+ {\mathcal{V}_{3}} {\lambda}_{1}+ \mathcal{U}_{3}\varepsilon _{1})+ ({\mathcal{W}_{3}} {k}_{2}+ { \mathcal{V}_{3}} { \lambda}_{2}+\mathcal{U}_{3}\varepsilon _{2}) \\ & {}+({\mathcal{W}_{3}} {k}_{3}+ {\mathcal{V}_{3}} {\lambda}_{3}+ \mathcal{U}_{3}\varepsilon _{3}) \big\Vert ({\mathfrak{p}}, { \mathfrak{q}},{\mathfrak{r}})-(\widehat{\mathfrak{p}}- \widehat{\mathfrak{q}}-\widehat{\mathfrak{r}}) \big\Vert \\ &{} +({\mathcal{W}_{3}}\pi _{1}+{\mathcal{V}_{3}} \pi _{2}+{\mathcal{U}_{3}} \pi _{3}). \end{aligned} $$

(30)

From (28), (29), and (4) it follows that

$$\begin{aligned}& \big\Vert ({\mathfrak{p}}, {\mathfrak{q}},{ \mathfrak{r}})-( \widehat{\mathfrak{p}}, \widehat{\mathfrak{q}},\widehat{ \mathfrak{r}}) \big\Vert \\& \quad\leq (\mathcal{W}_{1}+ \mathcal{W}_{2}+ \mathcal{W}_{3}){ \pi }_{1}+(\mathcal{V}_{1}+ \mathcal{V}_{2}+\mathcal{V}_{3}){\pi}_{2}+( \mathcal{U}_{1}+\mathcal{U}_{2}+\mathcal{U}_{3}){ \pi}_{3} \\& \qquad {}+(\mathcal{W}_{1}+ \mathcal{W}_{2}+\mathcal{W}_{3}) (\kappa _{1}+ \lambda _{1}+\varepsilon _{1}) \\& \qquad {}+(\mathcal{V}_{1}+\mathcal{V}_{2}+\mathcal{V}_{3}) (\kappa _{2}+ \lambda _{2}+\varepsilon _{2}) \\& \qquad {}+(\mathcal{U}_{1}+\mathcal{U}_{2}+\mathcal{U}_{3}) (\kappa _{3}+ \lambda _{3}+\varepsilon _{3}) \big\Vert ({\mathfrak{p}}, { \mathfrak{q}},{\mathfrak{r}})- (\widehat{\mathfrak{p}}- \widehat{\mathfrak{q}}-\widehat{\mathfrak{r}}) \big\Vert , \\& \big\| ({\mathfrak{p}}, {\mathfrak{q}},{\mathfrak{r}})-( \widehat{\mathfrak{p}}, \widehat{\mathfrak{q}},\widehat{\mathfrak{r}}) \big\| \\& \quad \leq \frac{(\mathcal{W}_{1} +\mathcal{W}_{2} +\mathcal{W}_{3}){\pi }_{1} +(\mathcal{V}_{1} +\mathcal{V}_{2} +\mathcal{V}_{3}){\pi}_{2} +(\mathcal{U}_{1} +\mathcal{U}_{2} +\mathcal{U}_{3}){\pi}_{3}}{1-((\mathcal{W}_{1} + \mathcal{W}_{2} +\mathcal{W}_{3}) (\kappa _{1} +\lambda _{1} +\varepsilon _{1}) +(\mathcal{V}_{1} +\mathcal{V}_{2} +\mathcal{V}_{3}) (\kappa _{2} +\lambda _{2} +\varepsilon _{2}) +(\mathcal{U}_{1} +\mathcal{U}_{2} +\mathcal{U}_{3}) (\kappa _{3} +\beta _{3} +\varepsilon _{3}))} \\& \quad \leq \mathcal{K}_{1}\pi _{1}+\mathcal{K}_{2} \pi _{2}+\mathcal{K}_{3} \pi _{3} \end{aligned}$$

with

$$\begin{aligned} &\mathcal{K}_{1} \\ &\quad = \frac{ (\mathcal{W}_{1} +\mathcal{W}_{2} +\mathcal{W}_{3})}{1 - ((\mathcal{W}_{1} +\mathcal{W}_{2} +\mathcal{W}_{3}) (\kappa _{1} +\lambda _{1} +\varepsilon _{1}) + (\mathcal{V}_{1} +\mathcal{V}_{2} +\mathcal{V}_{3}) (\kappa _{2} +\lambda _{2} +\varepsilon _{2}) + (\mathcal{U}_{1} +\mathcal{U}_{2} +\mathcal{U}_{3}) (\kappa _{3} +\beta _{3} +\varepsilon _{3}))}, \\ &\mathcal{K}_{2} \\ & \quad =\frac{(\mathcal{V}_{1}+\mathcal{V}_{2}+\mathcal{V}_{3})}{1 - ((\mathcal{W}_{1} +\mathcal{W}_{2} +\mathcal{W}_{3}) (\kappa _{1} +\lambda _{1} +\varepsilon _{1}) + (\mathcal{V}_{1} +\mathcal{V}_{2} +\mathcal{V}_{3}) (\kappa _{2} +\lambda _{2} +\varepsilon _{2}) + (\mathcal{U}_{1} +\mathcal{U}_{2} +\mathcal{U}_{3}) (\kappa _{3} +\beta _{3} +\varepsilon _{3}))}, \\ &\mathcal{K}_{3} \\ & \quad =\frac{(\mathcal{U}_{1}+\mathcal{U}_{2}+\mathcal{U}_{3})}{1 - ((\mathcal{W}_{1} +\mathcal{W}_{2} +\mathcal{W}_{3}) (\kappa _{1} +\lambda _{1} +\varepsilon _{1}) + (\mathcal{V}_{1} +\mathcal{V}_{2} +\mathcal{V}_{3}) (\kappa _{2} +\lambda _{2} +\varepsilon _{2}) + (\mathcal{U}_{1} +\mathcal{U}_{2} +\mathcal{U}_{3}) (\kappa _{3} +\beta _{3} +\varepsilon _{3}))}. \end{aligned}$$

Therefore the BVPs (1) is H-U stable. □

5 Example

Example 1

Consider the following coupled fractional differential system:

$$ \textstyle\begin{cases} ({}^{c}{\mathcal{D}}^{\frac{3}{2}}+ \varphi {}^{c}{\mathcal{D}}^{{ \frac{3}{2}}-1}) \mathfrak{p}(\varsigma )= \hat{\mathcal{F}_{1}} ( \varsigma ,\mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ), \mathfrak{r}(\varsigma )), \quad 1< \eta \leq 2, \\ ({}^{c}{\mathcal{D}}^{\frac{3}{2}}+ \varphi {}^{c}{\mathcal{D}}^{{ \frac{3}{2}}-1})\mathfrak{q}(\varsigma )= \hat{\mathcal{F}_{2}}( \varsigma ,\mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ), \mathfrak{r}(\varsigma )), \quad 1< \xi \leq 2, \\ ({}^{c}{\mathcal{D}}^{\frac{1}{4}}+ \varphi {}^{c}{\mathcal{D}}^{{ \frac{1}{4}}-1})\mathfrak{r}(\varsigma )= \hat{\mathcal{F}_{3}} ( \varsigma ,\mathfrak{p}(\varsigma ),\mathfrak{q}(\varsigma ), \mathfrak{r}(\varsigma ) ), \quad 2< \zeta \leq 3, \\ \mathfrak{p}(0)={0},\qquad \mathfrak{p}(1)=\beta _{1}\sum_{j=1}^{4} w_{j} \mathfrak{q}(\varrho _{j}), \\ \mathfrak{q}(0)={0}, \qquad \mathfrak{q}(1)= \beta _{2}\sum_{j=1}^{4} v_{j} \mathfrak{r}(\varpi _{j}), \\ \mathfrak{r}(0)=0, \qquad \mathfrak{r}'(0)=0, \qquad \mathfrak{r}(1)= \beta _{3}\sum_{j=1}^{4}\vartheta _{j} \mathfrak{p}(\rho _{j}), \\ 0 < \varrho _{1} < \lambda _{1} < \vartheta _{1}< \varrho _{2} < \lambda _{2} < \vartheta _{2} \ldots < \varrho _{k-2} < \lambda _{k-2} < \vartheta _{k-2}< 1, \end{cases} $$

(31)

Here $\eta = {\frac{3}{2}}$, $\xi = {\frac{3}{2}}$, $\zeta ={\frac{1}{4}}$, $\beta _{1}= {\frac{3}{2}}$, $\beta _{2}= {\frac{6}{5}}$, $\beta _{3}={ \frac{1}{3}}$, $w_{1}= {\frac{1}{40}}$, $w_{2}= {\frac{7}{200}}$, $w_{3}= { \frac{9}{200}}$, $w_{4}= {\frac{11}{200}}$, $\varrho _{1}= \frac{5}{4}$, $\varrho _{2}= \frac{7}{5}$, $\varrho _{1}= \frac{33}{20}$, $\varrho _{4}= \frac{9}{5}$, $v_{1}= \frac{1}{50}$, $v_{2}= \frac{9}{200}$, $v_{3}= \frac{3}{50}$, $v_{4}= \frac{17}{200}$, $\varpi _{1}= \frac{6}{5}$, $\varpi _{2}= \frac{29}{20}$, $\varpi _{3}= \frac{8}{5}$, $\varpi _{4}= \frac{37}{20}$, $\vartheta _{1}= \frac{1}{40}$, $\vartheta _{2}= \frac{1}{25}$, $\vartheta _{3}= \frac{13}{200}$, $\vartheta _{4}= \frac{12}{25}$, $\rho _{1}= \frac{21}{40}$, $\rho _{2}= \frac{107}{200}$, $\rho _{3}= \frac{109}{200}$, $\rho _{4}= \frac{111}{200}$, With this data, we find that ${\mathcal{W}_{1}} = 0.8316369829$, ${\mathcal{W}_{2}}= 0.1631960492$, ${ \mathcal{W}_{3}}= 0.0482076794$, ${\mathcal{V}_{1}}= 0.1048341679$, ${ \mathcal{V}_{2}}= 0.8246454202$, ${\mathcal{V}_{3}}= 0.0008716269$, ${ \mathcal{U}_{1}}= 0.0186054209$, ${\mathcal{U}_{2}}= 0.0070833630$, ${\mathcal{U}_{3}}= 0.2236302088$.

(I) To illustrate Theorem 1, we take

$$\begin{aligned} \begin{aligned} &\hat{\mathcal{F}_{1}}( \varsigma ,{\mathfrak{p}}, {\mathfrak{q}},{ \mathfrak{r}}) = \frac{1}{30e} + \frac{7}{50} \mathfrak{p} \cos \mathfrak{q} + \frac{1}{40e} { \mathfrak{q}} \sin \mathfrak{r} + \frac{e^{-\varsigma}}{2} \mathfrak{r} \cos \mathfrak{p}, \\ & \hat{\mathcal{F}_{2}}(\varsigma ,{\mathfrak{p}}, {\mathfrak{q}},{ \mathfrak{r}}) = \varsigma \sqrt{\varsigma +3} + \frac{1}{189} \mathfrak{p} \tan ^{-1}\mathfrak{q}+ \frac{7}{\sqrt{48+\varsigma ^{2}}}\mathfrak{q} + \frac{1}{4} \mathfrak{r} \sin \mathfrak{p}, \\ &\hat{\mathcal{F}_{3}}(\varsigma ,{\mathfrak{p}}, {\mathfrak{q}},{ \mathfrak{r}}) = \frac{e^{-\varsigma}}{4} + \frac{e^{(-\varsigma )}}{3} \mathfrak{p} + \frac{1}{\varsigma +8} \mathfrak{q} + \frac{e^{-\varsigma}}{10} \mathfrak{r} \cos \mathfrak{q}. \end{aligned} \end{aligned}$$

(32)

It is easy to check that condition $(H_{2})$ is satisfied with $\kappa _{0}= \frac{1}{30e}$, $\kappa _{1}= \frac{7}{50}$, $\kappa _{2}= \frac{1}{40e}$, $\kappa _{3} = \frac{e^{-\varsigma}}{2}$, $\lambda _{0}= 2$, $\lambda _{1}= \frac{1}{189}$, $\lambda _{2}= \frac{1}{7}$, $\beta _{3}= \frac{1}{4}$, $\varepsilon _{0}= \frac{e^{-\varsigma}}{4}$, $\varepsilon _{1}= \frac{e^{-\varsigma}}{3}$, $\varepsilon _{2}= \frac{1}{9}$, $\varepsilon _{3}= \frac{e^{-\varsigma}}{10}$. Furthermore,

$$ \begin{aligned} & ( {\mathcal{W}_{1}} + { \mathcal{W}_{2}} + {\mathcal{W}_{3}})\kappa _{1} +({\mathcal{V}_{1}}+ {\mathcal{V}_{2}} + { \mathcal{V}_{3}})\lambda _{1} + ({\mathcal{U}_{1}}+{ \mathcal{U}_{2}}+{\mathcal{U}_{3}}) \varepsilon _{1} \simeq 0.4292961592 < 1, \\ & ( {\mathcal{W}_{1}}+ {\mathcal{W}_{2}} + { \mathcal{W}_{3}})\kappa _{2}+({ \mathcal{V}_{1}} +{\mathcal{V}_{2}} +{\mathcal{V}_{3}}) \lambda _{2}+({ \mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}}) \varepsilon _{2} \simeq 0.1705329068 < 1, \\ & ( {\mathcal{W}_{1}}+ {\mathcal{W}_{2}} + { \mathcal{W}_{3}})\kappa _{3}+({ \mathcal{V}_{1}} +{\mathcal{V}_{2}} +{\mathcal{V}_{3}}) \lambda _{3}+({ \mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}}) \varepsilon _{3} \simeq 0.7699169503 < 1. \end{aligned} $$

Clearly, the hypotheses of Theorem 1 are satisfied, and hence the conclusion of Theorem 1 applies to problem (31) with $\mathfrak{p}$, $\mathfrak{q}$, $\mathfrak{r}$ given by (32).

Example 2

To illustrate Theorem 2, we take

$$\begin{aligned} \begin{aligned} &\hat{\mathcal{F}_{1}}( \varsigma ,{\mathfrak{p}}, {\mathfrak{q}},{ \mathfrak{r}}) = \frac{e^{-\varsigma}}{\sqrt{99+\varsigma ^{2}}} \cos \mathfrak{p} + \cos \varsigma , \\ & \hat{\mathcal{F}_{2}}(\varsigma ,{\mathfrak{p}}, {\mathfrak{q}},{ \mathfrak{r}}) = \frac{1}{ 9+\varsigma ^{2}} \bigl(\sin \mathfrak{p}+ \vert \mathfrak{q} \vert \bigr) +e^{-\varsigma}, \\ &\hat{\mathcal{F}_{3}}(\varsigma ,{\mathfrak{p}}, {\mathfrak{q}},{ \mathfrak{r}}) = \frac{e^{-\varsigma}}{9} \sin \mathfrak{r} +\tan ^{-1} \varsigma . \end{aligned} \end{aligned}$$

(33)

Then condition $(\mathscr{M}_{2})$ is clearly satisfied with ${\Theta _{1}} =\frac{1}{10e}$, ${\Theta _{2}}= \frac{1}{10}$, and ${\Theta _{3}} = \frac{1}{9e}$. Moreover,

$$ ( \mathcal{W}_{1} + \mathcal{W}_{2} + \mathcal{W}_{3})\Theta _{1} +({ \mathcal{V}_{1}}+ {\mathcal{V}_{2}} + {\mathcal{V}_{3}})\Theta _{2 } + ({\mathcal{U}_{1}}+{\mathcal{U}_{2}}+{ \mathcal{U}_{3}}) \Theta _{3} \simeq 0.2080311651 < 1, $$

Thus the hypothesis of Theorem 2 holds, and consequently there exists a unique solution for problem (31) on $[0,1]$ with $\mathfrak{p}$, $\mathfrak{q}$, $\mathfrak{r}$ given by (33).

6 Conclusions

We examined the existence and stability of solutions to a linked system of Caputo sequential fractional differential equations with standard conditions using the Leray–Schauder alternative, Banach–Kranoselskii fixed-point theorem, and Hyer–Ulam stability. We obtain new results for the given system of three sequential fractional differential equations under the specified conditions when we apply the combined solution to all three case values ($w_{j}=0$, $j=1,\ldots,k-2$, $v_{j}=0$, $j=1,\ldots,k-2$, $\vartheta _{j}=0$, $j=1,\ldots,k-2$) to the series of three sequential fractional differential equations. In a fragmented research field, it appears that the first multipoint boundary value problem to be stated in scientific research employs a triple system of sequential fractional differential equations. This paper discusses original research that contributes significantly to the body of expertise on the topic.