Existence and Ulam–Hyers Stability Analysis for Coupled Differential Equations of Fractional-Order with Nonlocal Generalized Conditions via Generalized Liouville–Caputo Derivative

Abstract

In this paper, we investigate the existence and Hyers–Ulam stability of a coupled differential equations of fractional-order with multi-point (discrete) and integral boundary conditions that are related to Katugampola integrals. This manuscript can be categorized into four parts: The Leray–Schauder alternative and Krasnoselskii’s fixed point theorems are used to prove the existence of a solution in the first and third section. The second section emphasizes the analysis of uniqueness, which is based on the Banach fixed point theorem’s concept of contraction mapping, and the fourth section establishes the Hyers–Ulam stability results. We demonstrate Hyers–Ulam stability using the traditional functional analysis technique. Finally, the consequences are validated using examples.

1. Introduction

Differential and integral operators of fractional order are widely used in the mathematical modeling of a variety of scientific and engineering phenomena. The primary advantage of using these operators is their nonlocality, which allows for the description of the processes and material involved in the phenomena’s history. As a result, fractional-order models are more accurate and informative than their integer-order counterparts. Numerous researchers developed this important branch of mathematical analysis as a result of the widespread applications of fractional calculus methods in a variety of real-world phenomena; for example, see the texts [1,2,3,4,5]. Numerous researchers have recently conducted research on fractional differential equations with a variety of boundary conditions. There has been a surge of interest in the study of nonlocal nonlinear fractional-order boundary value problems in particular (BVPs). In the work of Bitsadze and Samarski [6], the concept of nonlocal conditions was first introduced, and these conditions aid in the description of physical phenomena occurring within the boundaries of a given domain. Due to the changing geometry of a blood vessel throughout the vessel, it is difficult to justify the assumption of a circular cross section in computational fluid dynamics studies of blood flow problems. Integral boundary conditions have been introduced to address this problem. Additionally, integral boundary conditions are used to solve ill-posed parabolic backward problems that are not well-posed. Furthermore, integral boundary conditions are crucial in mathematical models of bacterial self-regularization [7]. Existence and uniqueness of solutions, stability and oscillatory properties, analytic and numerical methods are all investigated in the context of nonlocal nonlinear fractional boundary value problems. The literature on the subject has been significantly expanded in recent years to include fractional order differential equations and inclusions involving Riemann–Liouville, Liouville–Caputo (Caputo), and Hadamard-type derivatives, among others. We refer the reader to a series of papers [8,9,10,11,12,13,14] and the references cited therein for some recent works on the subject. Fractional differential systems have received good attention because they are used in mathematical models of physical and engineering processes. For more information on the theoretical development of such systems, see [15,16,17,18,19,20]. In 2018, Ahmad et al. [21] investigated the generalized fractional boundary value problem via generalized fractional integral conditions:

$$\left\{\begin{array}{c}{}^{\rho }{\mathcal{D}}^{\alpha }y\left(t\right)=f(t,y\left(t\right)),t\in [0,T],\hfill \\ y\left(0\right)=0,{\int }_0^{T}y\left(s\right)dH\left(s\right)=\lambda {}^{\rho }{\mathcal{I}}^{\beta }y\left(\xi \right),\xi \in (0,T),\lambda \in \mathbb{R},\hfill \end{array}\right.$$

(1)

where ${}^{\rho }{\mathcal{D}}^{\alpha }$ denotes the generalized fractional derivative of order $1<\alpha \le 2$, $\rho >0$, ${}^{\rho }{\mathcal{I}}^{\beta }$ is the Katugampola type fractional integral of order $\beta >0$, $\lambda \in \mathbb{R}$, and ${\int }_0^{T}y\left(s\right)dH\left(s\right)$ is the Stieltjes integral. In 2019, Ahmad et al. [22] discussed fractional differential equations and inclusions involving generalized Caputo-type derivative with boundary value conditions via generalized fractional integral operator:

$$\left\{\begin{array}{c}{}_C^{\rho }{\mathcal{D}}_{0^{+}}^{\alpha }y\left(t\right)=f(t,y\left(t\right)),t\in J:=[0,T],\hfill \\ \delta y\left(0\right)=0,y\left(T\right)={\sum }_{i=1}^m{\sigma }_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\beta }y\left({\eta }_i\right)+\kappa ,{\eta }_i\in (0,T),\kappa \in \mathbb{R},\hfill \end{array}\right.$$

(2)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}_C^{\rho }{\mathcal{D}}_{0^{+}}^{\alpha }y\left(t\right)\in f(t,y\left(t\right)),t\in J:=[0,T],\hfill \\ \delta y\left(0\right)=0,y\left(T\right)={\sum }_{i=1}^m{\sigma }_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\beta }y\left({\eta }_i\right)+\kappa ,{\eta }_i\in (0,T),\kappa \in \mathbb{R},\hfill \end{array}\right.\end{array}$$

(3)

where ${}_C^{\rho }{\mathcal{D}}_{0^{+}}^{\alpha }$ denotes the generalized Caputo-type fractional derivative of order $1<\alpha \le 2$, and $\rho >0$, ${}^{\rho }{\mathcal{I}}_{0^{+}}^{\beta }$ is the Katugampola type fractional integral of order $\beta >0$ and $\kappa \in \mathbb{R}$. Recently, in [23], the authors derived existence results for a nonlinear coupled system involving both Caputo and Riemann–Liouville generalized fractional derivatives and coupled integral boundary conditions. The robust stability of fractional-order systems represented in a pseudo-state space model with incommensurate fractional orders was recently studied by Tavazoei and Asemani [24]. Using the generalised Nyquist theorem, an existing non-conservative robust stability criterion for integer–order systems is extended to incommensurable-order fractional systems. The author discussed a fractional-order generalisation of the susceptible, infected and recovered (SIR) epidemic model for predicting infectious disease spread [25]. Modelling, researching, evaluating and interpreting biological processes such as species interactions, cohabitation and evolution are all part of mathematical biology. In addition to interactions with the environment, illness and food supply, these interactions can take place between related species, individuals of other species or those of different species. The initial cornerstone in this field was the Lotka–Volterra model, which was created independently by Lotka [26] and Volterra [27]. Later developments included a functional response to the model and density-dependent prey growth [28]. The theory of existence, uniqueness and stability of the solutions is one of the priority components in the study of fractional-order differential equations. Recently, a lot of researchers have become interested in this idea. For some of the recent growth, we can look at [29,30,31] and the references there. As a consequence, Lotka and Volterra published their well-known equations for the prey–predator model in 1920. Here, we state that the concerned model given in the system (4) deals with the relationship of prey and predator in an ecological system as follows:

$$\left\{\begin{array}{c}\dot{u}\left(t\right)=a_1\left(t\right)u\left(t\right)-b_1\left(t\right)v\left(t\right)u\left(\tau \right)={\phi }_2(t,u\left(t\right),v\left(t\right)),\hfill \\ \dot{v}\left(t\right)=a_2\left(t\right)u\left(t\right)v\left(t\right)-b_2\left(t\right)v\left(t\right)={\phi }_2(t,u\left(t\right),v\left(t\right)),\hfill \\ u\left(0\right)=\alpha ,v\left(0\right)=\beta ,\hfill \end{array}\right.$$

(4)

where $\alpha ,\beta \ge 0$. Furthermore, the nonlinear functions ${\phi }_i(i=1,2):\mathcal{J}\times {\mathbb{R}}^2\to \mathbb{R}$ are continuous. Here, we remark that $u\left(t\right)$, $v\left(t\right)$ represent the prey and the predator populations at time t, respectively. Furthermore, $a_1$ is the growth rate of species u, while $b_1$ denotes the impact of predation on $\dot{u}/u$. In addition, $b_2$ is the death rate of v, and $a_2$ is the growth rate (or immigration) of the predator population in response to the size of the prey population. The coefficients are linear continuous and bounded functions. So far, the concerned model has been studied for various purposes and from various directions. In addition, system (4) has been investigated by using the homotopy perturbation method for ordinary Caputo derivative in [32]. Recently, in [33], the authors derived existence theory and an approximate solution to a prey–predator coupled system involving a nonsingular kernel type derivative:

$$\left\{\begin{array}{c}{}^{CF}{\mathcal{D}}_t^{\alpha }u\left(t\right)=a_1\left(t\right)u\left(t\right)-b_1\left(t\right)v\left(t\right)u\left(\tau \right),\hfill \\ {}^{CF}{\mathcal{D}}_t^{\beta }v\left(t\right)=a_2\left(t\right)u\left(t\right)v\left(t\right)-b_2\left(t\right)v\left(t\right),\hfill \end{array}\right.$$

(5)

where $\alpha ,\beta \ge 0$ and $\omega \in (0,1]$. Motivated by the aforesaid works, a new class of BVP of generalized Liouville–Caputo-type coupled differential equations of fractional-order with nonlocal generalized fractional integral (Katugampola type) and multi-point boundary conditions is introduced and studied in this article:

$$\left\{\begin{array}{c}{}_C^{\rho }{\mathcal{D}}_{0^{+}}^{\xi }p\left(\tau \right)=f(\tau ,p\left(\tau \right),q\left(\tau \right)),\tau \in \mathcal{E}:=[0,\mathcal{T}],\hfill \\ {}_C^{\rho }{\mathcal{D}}_{0^{+}}^{\zeta }q\left(\tau \right)=g(\tau ,p\left(\tau \right),q\left(\tau \right)),\tau \in \mathcal{E}:=[0,\mathcal{T}],\hfill \end{array}\right.$$

(6)

enhanced with boundary conditions defined by:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\gamma p\left(0\right)={\sum }_{k=1}^l{v}_{k}q\left({\omega }_k\right),\gamma q\left(0\right)={\sum }_{k=1}^l{\nu }_{k}p\left({\vartheta }_k\right),\hfill \\ p\left(\mathcal{T}\right)={\sum }_{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\varsigma }q\left({\varpi }_i\right),q\left(\mathcal{T}\right)={\sum }_{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\varrho }p\left({\sigma }_i\right),\hfill \\ 0<{\vartheta }_1<{\omega }_1<\cdots <{\vartheta }_l<{\omega }_l<{\sigma }_1<{\varpi }_1<\cdots <{\sigma }_j<{\varpi }_j<\mathcal{T},\hfill \end{array}\right.\end{array}$$

(7)

where ${}_C^{\rho }{\mathcal{D}}_{0^{+}}^{\xi },{}_C^{\rho }{\mathcal{D}}_{0^{+}}^{\zeta }$ are the Liouville–Caputo-type generalized fractional derivative of order $1<\xi ,\zeta \le 2$, ${}_C^{\rho }{\mathcal{I}}_{0^{+}}^{\varsigma },{}_C^{\rho }{\mathcal{I}}_{0^{+}}^{\varrho }$ are the generalized fractional integral of order (Katugampola type) $\varrho ,\varsigma >0,\rho >0,f,g:\mathcal{E}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions, and $u_k,{\nu }_k,{ϵ}_i,{\pi }_i,\in \mathbb{R}$, $i=1,2,\dots ,j,k=1,2,\dots ,l,\gamma ={\tau }^{1-\rho }\frac{d}{d\tau }.$ Furthermore, we are investigating the system (6) under the following conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\gamma p\left(0\right)={\sum }_{k=1}^l{v}_{k}q\left({\omega }_k\right),\gamma q\left(0\right)={\sum }_{k=1}^l{\nu }_{k}p\left({\omega }_k\right),\hfill \\ p\left(\mathcal{T}\right)={\sum }_{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\varsigma }q\left({\varpi }_i\right),q\left(\mathcal{T}\right)={\sum }_{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\varsigma }p\left({\varpi }_i\right),\hfill \\ 0<{\omega }_1<\cdots <{\omega }_l<{\varpi }_1<\cdots <{\varpi }_j<\mathcal{T}.\hfill \end{array}\right.\end{array}$$

(8)

Bear in mind that the the conditions (7) contain strips of varying lengths, whereas the one in (8) contains only one strip of the same length $(0,\varpi ),i=1,2,\dots j$. In contrast to the multi-point boundary conditions described in (8), the multi-point boundary conditions described in (7) contain distinct multi-points. The following is the remainder of the article. Section 2 introduces some fundamental definitions, lemmas and theorems that support our main results. For the existence and uniqueness of solutions to the given system (6) and (7), we use various conditions and some standard fixed-point theorems in Section 3. Example 1 discusses the Ulam–Hyers stability of the given system (6) and (7) under certain conditions. In Example 2, examples are provided to demonstrate the main results. Finally, the consequences of existence, uniqueness and stability for problems (6) and (8) are provided.

2. Essential Background and Representation of Solutions

For our research, we recall some preliminary definitions of generalized Liouville–Caputo fractional derivatives and Katugampola fractional integrals.

The space of all complex-valued Lebesgue measurable functions $\varphi$ on $\mathcal{E}:=(c,d)$ equipped with the norm is denoted by ${\mathcal{Z}}_b^{\rho }(c,d):$

$${\left|\right|\varphi \left|\right|}_{{\mathcal{Z}}_b^{\theta }}={\left({\int }_c^d{\left|z^b\varphi \left(z\right)\right|}^{\theta }\frac{dz}{z}\right)}^{\frac{1}{\theta }}<\infty ,b\in \mathbb{R},1\le \theta \le \infty .$$

Let ${\mathcal{L}}^1(c,d)$ represent the space of all Lebesgue measurable functions $\phi$ on $(c,d)$ endowed with the norm:

$${\parallel \phi \parallel }_{{\mathcal{L}}^1}={\int }_c^d\left|\phi \left(z\right)\right|dz<\infty .$$

We further recall that $\mathcal{A}{\mathcal{C}}^n(\mathcal{E},\mathbb{R})=\{p:\mathcal{E}\to \mathbb{R}:p,p^{{}^{\prime }},\dots ,p^{(n-1)}\in \mathcal{C}(\mathcal{E},\mathbb{R})and{p}^{(n-1)}isabsolutelycontinuous\}$. For $0\le ϵ<1$, we define ${\mathcal{C}}_{ϵ,\rho }(\mathcal{E},\mathbb{R})=\{f:\mathcal{E}\to \mathbb{R}:{\left({\tau }^{\rho -a^{\rho }}\right)}^{ϵ}f\left(\tau \right)\in \mathcal{C}(\mathcal{E},\mathbb{R})$ endowed with the norm ${\parallel f\parallel }_{{\mathcal{C}}_{ϵ,\rho }}={\parallel {\left({\tau }^{\rho -a^{\rho }}\right)}^{ϵ}f\left(\tau \right)\parallel }_{\mathcal{C}}$. Moreover, we define the class of functions f that have absolute continuous ${\delta }^{n-1}$ derivative, denoted by $\mathcal{A}{\mathcal{C}}_{\gamma }^n(\mathcal{E},\mathbb{R})$, as follows: $\mathcal{A}{\mathcal{C}}_{\gamma }^n(\mathcal{E},\mathbb{R})=\{f:\mathcal{E}\to \mathbb{R}:{\gamma }^{n-1}f\in \mathcal{AC}(\mathcal{E},\mathbb{R}),\gamma ={\tau }^{1-\rho }\frac{d}{d\tau }\}$, which is equipped with the norm ${\parallel f\parallel }_{{\mathcal{C}}_{\gamma ,ϵ}^n}={\sum }_{k=0}^{n-1}\parallel {\gamma }^k{f\parallel }_{\mathcal{C}}+{\parallel {\gamma }^{n}f\parallel }_{{\mathcal{C}}_{ϵ,\rho }}$ is defined by

$${\mathcal{C}}_{\gamma ,ϵ}^n(\mathcal{E},\mathbb{R})=\left\{f:\mathcal{E}\to \mathbb{R}:{\gamma }^{n-1}f\in \mathcal{C}(\mathcal{E},\mathbb{R}),{\gamma }^{n}f\in {\mathcal{C}}_{ϵ,\rho }(\mathcal{E},\mathbb{R}),\gamma ={\tau }^{1-\rho }\frac{d}{d\tau }\right\},$$

Notice that ${\mathcal{C}}_{\gamma ,0}^n={\mathcal{C}}_{\gamma }^n$. The space $\mathcal{P}=\mathcal{C}(\mathcal{E},\mathbb{R})$ of all continuous functions is a Banach space under logical norm $\left|\right|p\left|\right|=sup\left\{\right|p\left(\tau \right)|,\tau \in \mathcal{E}\}$, and the product space $\mathcal{P}\times \mathcal{P}$ is also a Banach space under the norm $\left|\right|(p,q)\left|\right|=\left|\right|p\left|\right|+\left|\right|q\left|\right|$ or $\left|\right|(p,q)\left|\right|=max\left\{\right|\left|p\right||,|\left|q\right|\left|\right\}$.

Definition 1

([34]). The left- and right-sided generalized fractional integrals (GFIs) of $f\in {\mathcal{Z}}_b^q(c,d)$ of order $\xi >0$ and $\rho >0$ for $-\infty <c<\tau <d<\infty$, are defined as follows:

$$\begin{array}{c}\hfill {(}^{\rho }{\mathcal{I}}_{c^{+}}^{\xi }f)\left(\tau \right)=\frac{{\rho }^{1-\xi }}{\Gamma \left(\xi \right)}{\int }_c^{\tau }\frac{{\theta }^{\rho -1}}{{({\tau }^{\rho }-{\theta }^{\rho })}^{1-\xi }}f\left(\theta \right)d\theta ,\end{array}$$

(9)

$$\begin{array}{c}\hfill {(}^{\rho }{\mathcal{I}}_{d^{-}}^{\zeta }f)\left(\tau \right)=\frac{{\rho }^{1-\xi }}{\Gamma \left(\xi \right)}{\int }_{\tau }^d\frac{{\theta }^{\rho -1}}{{({\theta }^{\rho }-{\tau }^{\rho })}^{1-\xi }}f\left(\theta \right)d\theta .\end{array}$$

(10)

Definition 2

([35]). The generalized fractional derivatives (GFDs), which are associated with GFIs (9) and (10) for $0\le c<\tau <d<\infty$, are defined as follows:

$$\begin{array}{cc}\hfill {(}^{\rho }{\mathcal{D}}_{c^{+}}^{\xi }f)\left(\tau \right)& ={\left({\tau }^{1-\rho }\frac{d}{d\tau }\right)}^n{(}^{\rho }{\mathcal{I}}_{c^{+}}^{n-\xi }f)\left(\tau \right)\hfill \\ & =\frac{{\rho }^{\xi -n+1}}{\Gamma (n-\xi )}{\left({\tau }^{1-\rho }\frac{d}{d\tau }\right)}^n{\int }_c^{\tau }\frac{{\theta }^{\rho -1}}{{({\tau }^{\rho }-{\theta }^{\rho })}^{\xi -n+1}}f\left(\theta \right)d\theta ,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {(}^{\rho }{\mathcal{D}}_{d^{-}}^{\xi }f)\left(\tau \right)& ={\left(-{\tau }^{1-\rho }\frac{d}{d\tau }\right)}^n{(}^{\rho }{\mathcal{I}}_{d^{-}}^{n-\xi }f)\left(\tau \right)\hfill \\ & =\frac{{\rho }^{\xi -n+1}}{\Gamma (n-\xi )}{\left(-{\tau }^{1-\rho }\frac{d}{d\tau }\right)}^n{\int }_{\tau }^d\frac{{\theta }^{\rho -1}}{{({\theta }^{\rho }-{\tau }^{\rho })}^{\xi -n+1}}f\left(\theta \right)d\theta ,\hfill \end{array}$$

(12)

if the integrals exist.

Definition 3

([36]). The above GFDs define the left- and right-sided generalized Liouville–Caputo type fractional derivatives of $f\in \mathcal{A}{\mathcal{C}}_{\gamma }^n[c,d]$ of order $\xi \ge 0$ if

$$\begin{array}{c}\hfill {}_C^{\rho }{\mathcal{D}}_{c^{+}}^{\xi }f\left(z\right)={}^{\rho }{\mathcal{D}}_{c^{+}}^{\xi }\left[f\left(\tau \right)-\sum _{k=0}^{n-1}\frac{{\gamma }^{k}f\left(c\right)}{k!}{\left(\frac{{\tau }^{\rho }-c^{\rho }}{\rho }\right)}^k\right]\left(z\right),\gamma =z^{1-\rho }\frac{d}{dz},\end{array}$$

(13)

$$\begin{array}{c}\hfill {}_C^{\rho }{\mathcal{D}}_{d^{-}}^{\xi }f\left(z\right){=}^{\rho }{\mathcal{D}}_{d^{-}}^{\xi }\left[f\left(\tau \right)-\sum _{k=0}^{n-1}\frac{{(-1)}^k{\gamma }^{k}f\left(d\right)}{k!}{\left(\frac{d^{\rho }-{\tau }^{\rho }}{\rho }\right)}^k\right]\left(z\right),\gamma =z^{1-\rho }\frac{d}{dz},\end{array}$$

(14)

when $n=\left[\xi \right]+1.$

Lemma 1

([36]). Let $\xi \ge 0,n=\left[\xi \right]+1$ and $f\in \mathcal{A}{\mathcal{C}}_{\gamma }^n[c,d],$ where $0<c<d<\infty$. Then,

1. 

if $\xi \notin \mathbb{N}$

$$\begin{array}{c}\hfill {}_C^{\rho }{\mathcal{D}}_{c^{+}}^{\xi }f\left(\tau \right)=\frac{1}{\Gamma (n-\xi )}{\int }_c^{\tau }{\left(\frac{{\tau }^{\rho }-{\theta }^{\rho }}{\rho }\right)}^{n-\xi -1}\frac{\left({\gamma }^{n}f\right)\left(\theta \right)d\theta }{{\theta }^{1-\rho }}={}^{\rho }{\mathcal{I}}_{c^{+}}^{n-\xi }\left({\gamma }^{n}f\right)\left(\tau \right),\end{array}$$

(15)

$${}_C^{\rho }{\mathcal{D}}_{d^{-}}^{\xi }f\left(\tau \right)=\frac{1}{\Gamma (n-\xi )}{\int }_{\tau }^d{\left(\frac{{\theta }^{\rho }-{\tau }^{\rho }}{\rho }\right)}^{n-\xi -1}\frac{{(-1)}^n\left({\gamma }^{n}f\right)\left(\theta \right)d\theta }{{\theta }^{1-\rho }}={}^{\rho }{\mathcal{I}}_{d^{-}}^{n-\xi }\left({\gamma }^{n}f\right)\left(\tau \right).$$

(16)

2. 

if $\xi \in \mathbb{N}$

$$\begin{array}{c}\hfill {}_C^{\rho }{\mathcal{D}}_{c^{+}}^{\xi }f={\gamma }^{n}f,{}_C^{\rho }{\mathcal{D}}_{d^{-}}^{\xi }f={(-1)}^n{\gamma }^{n}f.\end{array}$$

(17)

Lemma 2

([36]). Let $f\in \mathcal{A}{\mathcal{C}}_{\gamma }^n[c,d]$ or ${\mathcal{C}}_{\gamma }^n[c,d]$ and $\xi \in \mathbb{R}$. Then,

$$\begin{array}{c}\hfill {}^{\rho }{\mathcal{I}}_{c^{+}}^{\xi }{}_C^{\rho }{\mathcal{D}}_{c^{+}}^{\xi }f\left(z\right)=f\left(z\right)-\sum _{k=0}^{n-1}\frac{\left({\gamma }^{k}f\right)\left(c\right)}{k!}{\left(\frac{z^{\rho }-c^{\rho }}{\rho }\right)}^k,\end{array}$$

$$\begin{array}{c}\hfill {}^{\rho }{\mathcal{I}}_{d^{-}}^{\xi }{}_C^{\rho }{\mathcal{D}}_{d^{-}}^{\xi }f\left(z\right)=f\left(z\right)-\sum _{k=0}^{n-1}\frac{{(-1)}^k\left({\gamma }^{k}f\right)\left(d\right)}{k!}{\left(\frac{d^{\rho }-z^{\rho }}{\rho }\right)}^k.\end{array}$$

In particular, for $0<\xi \le 1$, we have

$$\begin{array}{c}\hfill {}^{\rho }{\mathcal{I}}_{c^{+}}^{\xi }{}_C^{\rho }{\mathcal{D}}_{c+}^{\xi }f\left(z\right)=f\left(z\right)-f\left(c\right),{}^{\rho }{\mathcal{I}}_{d^{-}}^{\xi }{}_C^{\rho }{\mathcal{D}}_{d^{-}}^{\xi }f\left(z\right)=f\left(z\right)-f\left(d\right).\end{array}$$

We introduce the following notations for computational ease:

$$\begin{array}{c}\hfill {\mathcal{E}}_1=\sum _{k=1}^l{v}_k,{\mathcal{E}}_2=\sum _{k=1}^l{v}_k\frac{{\omega }_k^{\rho }}{\rho },{\mathcal{E}}_3=\sum _{i=1}^j{ϵ}_i\frac{{\varpi }_i^{\rho \varsigma }}{{\rho }^{\varsigma }\Gamma (\varsigma +1)},{\mathcal{E}}_4=\sum _{i=1}^j{ϵ}_i\frac{{\varpi }_i^{\rho (\varsigma +1)}}{{\rho }^{\varsigma +1}\Gamma (\varsigma +2)},{\mathcal{E}}_5=\frac{{\mathcal{T}}^{\rho }}{\rho },\end{array}$$

(18)

$$\begin{array}{c}\hfill {\mathcal{F}}_1=\sum _{k=1}^l{\nu }_k,{\mathcal{F}}_2=\sum _{k=1}^l{\nu }_k\frac{{\vartheta }_k^{\rho }}{\rho },{\mathcal{F}}_3=\sum _{i=1}^j{\pi }_i\frac{{\sigma }_i^{\rho \varrho }}{{\rho }^{\varrho }\Gamma (\varrho +1)},{\mathcal{F}}_4=\sum _{i=1}^j{\pi }_i\frac{{\sigma }_i^{\rho (\varrho +1)}}{{\rho }^{\varrho +1}\Gamma (\varrho +2)},\end{array}$$

(19)

$$\begin{array}{c}\hfill {\mathcal{G}}_1=1-{\mathcal{E}}_2{\mathcal{F}}_2,{\mathcal{G}}_2={\mathcal{E}}_5-{\mathcal{E}}_4{\mathcal{F}}_3,{\mathcal{G}}_3={\mathcal{F}}_4-{\mathcal{E}}_5{\mathcal{F}}_4,\end{array}$$

(20)

$$\begin{array}{c}\hfill \widehat{{\mathcal{G}}_1}={\mathcal{E}}_4-{\mathcal{E}}_3{\mathcal{E}}_5,\widehat{{\mathcal{G}}_2}={\left({\mathcal{E}}_5\right)}^2-{\mathcal{E}}_4{\mathcal{F}}_3,\widehat{{\mathcal{G}}_3}={\mathcal{F}}_4-{\mathcal{E}}_5{\mathcal{F}}_3,\end{array}$$

(21)

$$\begin{array}{cc}{\mathsf{\Lambda }}_1={\mathcal{G}}_1{\mathcal{G}}_2+{\mathcal{F}}_1{\mathcal{E}}_2\widehat{{\mathcal{G}}_2},{\mathsf{\Lambda }}_2={\mathcal{G}}_1{\mathcal{G}}_3+{\mathcal{F}}_1\widehat{{\mathcal{G}}_2},{\mathsf{\Lambda }}_3={\mathcal{E}}_1\widehat{{\mathcal{G}}_2}+{\mathcal{G}}_1\widehat{{\mathcal{G}}_1},\hfill & \\ {\mathsf{\Lambda }}_4={\mathcal{G}}_1{\mathcal{G}}_3+{\mathcal{E}}_1{\mathcal{F}}_2\widehat{{\mathcal{G}}_2},\mathsf{\Lambda }={\mathsf{\Lambda }}_2{\mathsf{\Lambda }}_3-{\mathsf{\Lambda }}_1{\mathsf{\Lambda }}_4\hfill & \end{array}$$

(22)

$$\begin{array}{c}\hfill {\alpha }_1=({\mathsf{\Lambda }}_4-{\mathsf{\Lambda }}_3{\mathcal{F}}_2)\widehat{{\mathcal{G}}_2},{\alpha }_2=({\mathsf{\Lambda }}_4{\mathcal{E}}_2-{\mathsf{\Lambda }}_3)\widehat{{\mathcal{G}}_2},{\alpha }_3=({\mathsf{\Lambda }}_2-{\mathsf{\Lambda }}_1{\mathcal{F}}_2)\widehat{{\mathcal{G}}_2},{\alpha }_4=({\mathsf{\Lambda }}_2{\mathcal{E}}_2-{\mathsf{\Lambda }}_1)\widehat{{\mathcal{G}}_2},\end{array}$$

(23)

$$\begin{array}{cc}{\beta }_1=({\mathsf{\Lambda }}_4{\mathcal{E}}_5-{\mathsf{\Lambda }}_3{\mathcal{F}}_4){\mathcal{G}}_1,{\beta }_2=({\mathsf{\Lambda }}_4{\mathcal{E}}_4-{\mathsf{\Lambda }}_3{\mathcal{E}}_5){\mathcal{G}}_1,{\beta }_3=({\mathsf{\Lambda }}_2{\mathcal{E}}_5-{\mathsf{\Lambda }}_1{\mathcal{F}}_4){\mathcal{G}}_1,\hfill & \\ {\beta }_4=({\mathsf{\Lambda }}_2{\mathcal{E}}_4-{\mathsf{\Lambda }}_1{\mathcal{E}}_5){\mathcal{G}}_1,\hfill & \end{array}$$

(24)

$$\begin{array}{cc}\hfill \widehat{{\alpha }_1}=1+\frac{{\mathcal{F}}_1{\mathcal{E}}_2{\alpha }_1-{\mathcal{E}}_1{\alpha }_3}{\mathsf{\Lambda }},\widehat{{\alpha }_2}={\mathcal{E}}_2+\frac{{\mathcal{F}}_1{\mathcal{E}}_2{\alpha }_2-{\mathcal{E}}_1{\alpha }_4}{\mathsf{\Lambda }},& \\ \hfill \widehat{{\alpha }_3}={\mathcal{F}}_2+\frac{{\mathcal{F}}_1{\alpha }_1-{\mathcal{E}}_1{\mathcal{F}}_2{\alpha }_3}{\mathsf{\Lambda }},\widehat{{\alpha }_4}=1+\frac{{\mathcal{F}}_1{\alpha }_2-{\mathcal{E}}_1{\mathcal{F}}_2{\alpha }_4}{\mathsf{\Lambda }},\end{array}$$

(25)

$$\begin{array}{cc}\widehat{{\beta }_1}=\frac{{\mathcal{E}}_1{\beta }_1-{\mathcal{F}}_1{\mathcal{E}}_2{\beta }_1}{\mathsf{\Lambda }},\widehat{{\beta }_2}=\frac{{\mathcal{E}}_1{\beta }_4-{\mathcal{F}}_1{\mathcal{E}}_2{\beta }_2}{\mathsf{\Lambda }},\widehat{{\beta }_3}=\frac{{\mathcal{E}}_1{\mathcal{F}}_2{\beta }_3-{\mathcal{F}}_1{\beta }_1}{\mathsf{\Lambda }},\hfill & \\ \widehat{{\beta }_4}=\frac{{\mathcal{E}}_1{\mathcal{F}}_2{\beta }_4-{\mathcal{F}}_1{\beta }_2}{\mathsf{\Lambda }},\hfill & \end{array}$$

(26)

$$\begin{array}{cc}{\delta }_1\left(\tau \right)=\left(\frac{{\alpha }_1}{\mathsf{\Lambda }}+\frac{{\tau }^{\rho }\widehat{{\alpha }_1}}{\rho {\mathcal{G}}_1}\right),{\delta }_2\left(\tau \right)=\left(\frac{{\alpha }_2}{\mathsf{\Lambda }}+\frac{{\tau }^{\rho }\widehat{{\alpha }_2}}{\rho {\mathcal{G}}_1}\right),{\delta }_3\left(\tau \right)=\left(\frac{{\alpha }_3}{\mathsf{\Lambda }}+\frac{{\tau }^{\rho }\widehat{{\alpha }_3}}{\rho {\mathcal{G}}_1}\right),\hfill & \\ {\delta }_4\left(\tau \right)=\left(\frac{{\alpha }_4}{\mathsf{\Lambda }}+\frac{{\tau }^{\rho }\widehat{{\alpha }_4}}{\rho {\mathcal{G}}_1}\right),\hfill & \end{array}$$

(27)

$$\begin{array}{cc}\widehat{{\delta }_1}\left(\tau \right)=\left(\frac{{\tau }^{\rho }\widehat{{\beta }_1}}{\rho {\mathcal{G}}_1}-\frac{{\beta }_1}{\mathsf{\Lambda }}\right),\widehat{{\delta }_2}\left(\tau \right)=\left(\frac{{\tau }^{\rho }\widehat{{\beta }_2}}{\rho {\mathcal{G}}_1}-\frac{{\beta }_2}{\mathsf{\Lambda }}\right),\widehat{{\delta }_3}\left(\tau \right)=\left(\frac{{\tau }^{\rho }\widehat{{\beta }_3}}{\rho {\mathcal{G}}_1}-\frac{{\beta }_3}{\mathsf{\Lambda }}\right),\hfill & \\ \widehat{{\delta }_4}\left(\tau \right)=\left(\frac{{\tau }^{\rho }\widehat{{\beta }_4}}{\rho {\mathcal{G}}_1}-\frac{{\beta }_4}{\mathsf{\Lambda }}\right).\hfill & \end{array}$$

(28)

Next, we prove a lemma, which is vital in converting the given problem to a fixed point problem.

Lemma 3.

Given the functions $\widehat{f},\widehat{g}\in C(0,\mathcal{T})\cap \mathcal{L}(0,\mathcal{T}),p,q\in \mathcal{A}{\mathcal{C}}_{\gamma }^2\left(\mathcal{E}\right)$ and $\mathsf{\Lambda }\ne 0.$ Then, the solution of the coupled BVP:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}_C^{\rho }{\mathcal{D}}_{0+}^{\xi }p\left(\tau \right)=\widehat{f}\left(\tau \right),\tau \in \mathcal{E}:=[0,\mathcal{T}],\hfill \\ {}_C^{\rho }{\mathcal{D}}_{0+}^{\zeta }q\left(\tau \right)=\widehat{g}\left(\tau \right),\tau \in \mathcal{E}:=[0,\mathcal{T}],\hfill \\ \gamma p\left(0\right)={\sum }_{k=1}^l{v}_{k}q\left({\omega }_k\right),\gamma q\left(0\right)={\sum }_{k=1}^l{\nu }_{k}p\left({\vartheta }_k\right),\hfill \\ p\left(\mathcal{T}\right)={\sum }_{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\varsigma }q\left({\varpi }_i\right),q\left(\mathcal{T}\right)={\sum }_{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\varrho }p\left({\sigma }_i\right)\hfill \\ 0<{\vartheta }_1<{\omega }_1<\cdots <{\vartheta }_1<{\omega }_l<{\sigma }_1<{\varpi }_1<\cdots <{\sigma }_i<{\varpi }_j<\mathcal{T},\hfill \end{array}\right.\end{array}$$

(29)

is given by

$$\begin{array}{cc}\hfill p\left(\tau \right)& ={}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\tau \right)+{\delta }_1\left(\tau \right)\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left({\omega }_k\right)+{\delta }_2\left(\tau \right)\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left({\vartheta }_k\right)\hfill \\ & +\widehat{{\delta }_1}\left(\tau \right)\left[\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi }\widehat{f}\left(\mathcal{T}\right)\right]\hfill \\ & +\widehat{{\delta }_2}\left(\tau \right)\left[\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi +\varrho }\widehat{f}\left({\sigma }_i\right)-{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left(\mathcal{T}\right)\right],\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill q\left(\tau \right)& ={}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left(\tau \right)+{\delta }_3\left(\tau \right)\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left({\omega }_k\right)+{\delta }_4\left(\tau \right)\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi }\widehat{f}\left({\vartheta }_k\right)\hfill \\ \hfill & +\widehat{{\delta }_3}\left(\tau \right)\left[\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi }\widehat{f}\left(\mathcal{T}\right)\right]\hfill \\ & +\widehat{{\delta }_4}\left(\tau \right)\left[\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta +\varrho }\widehat{f}\left({\sigma }_i\right)-{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left(\mathcal{T}\right)\right].\hfill \end{array}$$

(31)

Proof. 

When ${}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi }{,}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }$ are applied to the FDEs in (29) and Lemma 2 is used, the solution of the FDEs in (29) for $\tau \in \mathcal{E}$ is

$$\begin{array}{c}\hfill p\left(\tau \right)={}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi }\widehat{f}\left(\tau \right)+a_1+a_2\frac{{\tau }^{\rho }}{\rho }=\frac{{\rho }^{1-\xi }}{\Gamma \left(\xi \right)}{\int }_0^{\tau }{\theta }^{\rho -1}{({\tau }^{\rho }-{\theta }^{\rho })}^{\xi -1}\widehat{f}\left(\theta \right)d\theta +a_1+a_2\frac{{\tau }^{\rho }}{\rho },\end{array}$$

(32)

$$\begin{array}{c}\hfill q\left(\tau \right)={}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left(\tau \right)+b_1+b_2\frac{{\tau }^{\rho }}{\rho }=\frac{{\rho }^{1-\zeta }}{\Gamma \left(\zeta \right)}{\int }_0^{\tau }{\theta }^{\rho -1}{({\tau }^{\rho }-{\theta }^{\rho })}^{\zeta -1}\widehat{g}\left(\theta \right)d\theta +b_1+b_2\frac{{\tau }^{\rho }}{\rho },\end{array}$$

(33)

respectively, for some $a_1,a_2,b_1,b_2\in \mathcal{R}$. Taking γ -derivative of (32) and (33), we obtain

$$\begin{array}{cc}\hfill \gamma p\left(\tau \right)& ={}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi -1}\widehat{f}\left(\tau \right)+a_2=\frac{{\rho }^{2-\xi }}{\Gamma (\xi -1)}{\int }_0^{\tau }{\theta }^{\rho -1}{({\tau }^{\rho }-{\theta }^{\rho })}^{\xi -2}\widehat{f}\left(\theta \right)d\theta +a_2,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \gamma q\left(\tau \right)& ={}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta -1}\widehat{g}\left(\tau \right)+b_2=\frac{{\rho }^{2-\zeta }}{\Gamma (\zeta -1)}{\int }_0^{\tau }{\theta }^{\rho -1}{({\tau }^{\rho }-{\theta }^{\rho })}^{\zeta -2}\widehat{g}\left(\theta \right)d\theta +b_2.\hfill \end{array}$$

(35)

Making use of the boundary conditions $\gamma p\left(0\right)={\sum }_{k=1}^l{v}_{k}q\left({\omega }_k\right),\gamma q\left(0\right)={\sum }_{k=1}^l{\nu }_{k}p\left({\vartheta }_k\right)$ in (34) and (35), respectively, we obtain

$$\begin{array}{c}\hfill a_2=\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left({\omega }_k\right)+b_1\sum _{k=1}^l{v}_k+b_2\sum _{k=1}^l{v}_k\frac{{\omega }_k^{\rho }}{\rho },\end{array}$$

(36)

$$\begin{array}{c}\hfill b_2=\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left({\vartheta }_k\right)+a_1\sum _{k=1}^l{\nu }_k+a_2\sum _{k=1}^l{\nu }_k\frac{{\vartheta }_k^{\rho }}{\rho },\end{array}$$

(37)

which, as a result of (18) and (19), takes the following form:

$$\begin{array}{c}\hfill a_2-b_1{\mathcal{E}}_2-b_2{\mathcal{E}}_2=\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left({\omega }_k\right),\end{array}$$

(38)

$$\begin{array}{c}\hfill b_2-a_1{\mathcal{F}}_1-a_2{\mathcal{F}}_2=\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left({\vartheta }_k\right).\end{array}$$

(39)

Applying the generalised integral operators ${}^{\rho }{\mathcal{I}}_{0+}^{\xi }{,}^{\rho }{\mathcal{I}}_{0+}^{\zeta }$ to (32) and (33), respectively, we obtain

$$\begin{array}{c}\hfill {}^{\rho }{\mathcal{I}}_{0+}^{\varrho }p\left(\tau \right)={}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi +\varrho }\widehat{f}\left(\tau \right)+a_1\frac{{\tau }^{\rho \varrho }}{{\rho }^{\varrho }\Gamma (\varrho +1)}+a_2\frac{{\tau }^{\rho (\varrho +1)}}{{\rho }^{\varrho +1}\Gamma (\varrho +2)},\end{array}$$

(40)

$$\begin{array}{c}\hfill {}^{\rho }{\mathcal{I}}_{0^{+}}^{\varsigma }q\left(\tau \right)={}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta +\varsigma }\widehat{g}\left(\tau \right)+b_1\frac{{\tau }^{\rho \varsigma }}{{\rho }^{\varsigma }\Gamma (\varsigma +1)}+b_2\frac{{\tau }^{\rho (\varsigma +1)}}{{\rho }^{\varsigma +1}\Gamma (\varsigma +2)},\end{array}$$

(41)

which, when combined with the boundary conditions $p\left(\mathcal{T}\right)={\sum }_{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\varsigma }q\left({\varpi }_i\right)$, $q\left(\mathcal{T}\right)={\sum }_{i=1}^j{\pi }_i^{\rho }{\mathcal{I}}_{0+}^{\varrho }p\left({\sigma }_i\right)$, gives the following results:

$${}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi }\widehat{f}\left(\mathcal{T}\right)+a_1+a_2\frac{{\mathcal{T}}^{\rho }}{\rho }=\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)+b_1\sum _{i=1}^j\frac{{ϵ}_i{\varpi }_i^{\rho \varsigma }}{{\rho }^{\varsigma }\Gamma (\varsigma +1)}+b_2\sum _{i=1}^j\frac{{ϵ}_i{\varpi }_i{}^{\rho (\varsigma +1)}}{{\rho }^{\varsigma +1}\Gamma (\varsigma +2)},$$

(42)

$${}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left(\mathcal{T}\right)+b_1+b_2\frac{{\mathcal{T}}^{\rho }}{\rho }=\sum _{i=1}^j{\pi }_i^{\rho }{\mathcal{I}}_{0^{+}}^{\xi +\varrho }\widehat{f}\left({\sigma }_i\right)+a_1\sum _{i=1}^j\frac{{\pi }_i{\sigma }_i^{\rho \varrho }}{{\rho }^{\varrho }\Gamma (\varrho +1)}+a_2\sum _{i=1}^j\frac{{\pi }_i{\sigma }_i^{\rho (\varrho +1)}}{{\rho }^{\varrho +1}\Gamma (\varrho +2)}.$$

(43)

Next, we obtain

$$\begin{array}{c}\hfill a_1+a_2{\mathcal{E}}_5-b_1{\mathcal{E}}_3-b_2{\mathcal{E}}_4=\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\mathcal{T}\right),\end{array}$$

(44)

$$\begin{array}{c}\hfill b_1+b_2{\mathcal{E}}_5-a_1{\mathcal{F}}_3-a_2{\mathcal{F}}_4=\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi +\varrho }\widehat{f}\left({\sigma }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left(\mathcal{T}\right),\end{array}$$

(45)

by employing the notations (18) and (19) in (42) and (43), respectively. We find that when we solve the system of Equations (38), (39), (44) and (45) for $a_1,a_2,b_1$ and $b_2$,

$$\begin{array}{cc}{a}_1=\frac{1}{\mathsf{\Lambda }}\left[{\alpha }_1\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left({\omega }_k\right)+{\alpha }_2\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left({\vartheta }_k\right)-{\beta }_1\left(\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi }\widehat{f}\left(\mathcal{T}\right)\right)\right.\hfill & \\ \left.-{\beta }_2\left(\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi +\varrho }\widehat{f}\left({\sigma }_i\right)-{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left(\mathcal{T}\right)\right)\right],\hfill & \end{array}$$

(46)

$$\begin{array}{cc}{a}_2=\frac{1}{{\mathcal{G}}_1}\left[{\widehat{\alpha }}_1\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left({\omega }_k\right)+{\widehat{\alpha }}_2\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi }\widehat{f}\left({\vartheta }_k\right)+{\widehat{\beta }}_1\left(\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\mathcal{T}\right)\right)\right.\hfill & \\ \left.+{\widehat{\beta }}_2\left(\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi +\varrho }\widehat{f}\left({\sigma }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left(\mathcal{T}\right)\right)\right],\hfill & \end{array}$$

(47)

$$\begin{array}{cc}{b}_1=\frac{-1}{\mathsf{\Lambda }}\left[{\alpha }_3\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left({\omega }_k\right)+{\alpha }_4\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi }\widehat{f}\left({\vartheta }_k\right)-{\beta }_3\left(\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi }\widehat{f}\left(\mathcal{T}\right)\right)\right.\hfill & \\ \left.-{\beta }_4\left(\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi +\varrho }\widehat{f}\left({\sigma }_i\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left(\mathcal{T}\right)\right)\right],\hfill & \end{array}$$

(48)

$$\begin{array}{cc}{b}_2=\frac{1}{{\mathcal{G}}_1}\left[{\widehat{\alpha }}_3\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left({\omega }_k\right)+{\widehat{\alpha }}_4\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi }\widehat{f}\left({\vartheta }_k\right)+{\widehat{\beta }}_3\left(\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right){-}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\mathcal{T}\right)\right)\right.\hfill & \\ \left.+{\widehat{\beta }}_4\left(\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0^{+}}^{\xi +\varrho }\widehat{f}\left({\sigma }_i\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{\zeta }\widehat{g}\left(\mathcal{T}\right)\right)\right].\hfill & \end{array}$$

(49)

Using the notations (18)–(28) and then substituting the values of $a_1,a_2,b_1$ and $b_2$ into (32) and (33), respectively, we obtain the solutions (30) and (31).  □

3. Existence Results for the Problem (6) and (7)

As a result of Lemma 3, we define an operator $\Delta :\mathcal{P}\times \mathcal{P}\to \mathcal{P}\times \mathcal{P}$ by

$$\begin{array}{c}\hfill \Delta (p,q)\left(\tau \right)=({\Delta }_1(p,q)\left(\tau \right),{\Delta }_2(p,q)\left(\tau \right))\end{array}$$

(50)

where

$$\begin{array}{cc}\hfill {\Delta }_1(p,q)\left(\tau \right)=& {}^{\rho }{\mathcal{I}}_{0+}^{\xi }f(\tau ,p\left(\tau \right),q\left(\tau \right))+{\delta }_1\left(\tau \right)\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }g({\omega }_k,p\left({\omega }_k\right),q\left({\omega }_k\right))\hfill \\ & +{\delta }_2\left(\tau \right)\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }f({\vartheta }_k,p\left({\vartheta }_k\right),q\left({\vartheta }_k\right))\hfill \\ & +\widehat{{\delta }_1}\left(\tau \right)\left(\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }g({\varpi }_i,p\left({\varpi }_i\right),q\left({\varpi }_i\right))-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }f(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right)\hfill \\ & +\widehat{{\delta }_2}\left(\tau \right)\left(\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }f({\sigma }_i,p\left({\sigma }_i\right),q\left({\sigma }_i\right))-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }g(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right),\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill {\Delta }_2(p,q)\left(\tau \right)=& {}^{\rho }{\mathcal{I}}_{0+}^{\zeta }g(\tau ,p\left(\tau \right),q\left(\tau \right))+{\delta }_3\left(\tau \right)\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }g({\omega }_k,p\left({\omega }_k\right),q\left({\omega }_k\right))\hfill \\ \hfill & +{\delta }_4\left(\tau \right)\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }f({\vartheta }_k,p\left({\vartheta }_k\right),q\left({\vartheta }_k\right))\hfill \\ & +\widehat{{\delta }_3}\left(\tau \right)\left(\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }g({\varpi }_i,p\left({\varpi }_i\right),q\left({\varpi }_i\right))-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }f(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right)\hfill \\ & +\widehat{{\delta }_4}\left(\tau \right)\left(\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }f({\sigma }_i,p\left({\sigma }_i\right),q\left({\sigma }_i\right))-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }g(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right).\hfill \end{array}$$

(52)

For brevity’s sake, we will use the following notations:

$$\begin{array}{c}\hfill {\mathcal{J}}_1=\left({\mathcal{T}}^{\rho \xi }(1+|\widehat{{\delta }_1}\left|\right)+|{\delta }_2|\sum _{k=1}^l\left|v_k\right|{\vartheta }_k^{\rho \xi }\right)\frac{1}{{\rho }^{\xi }\Gamma (\xi +1)}+|\widehat{{\delta }_2}|\sum _{i=1}^j\left|{\pi }_i\right|\frac{{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)},\end{array}$$

(53)

$$\begin{array}{c}\hfill {\mathcal{K}}_1=\left(|{\delta }_1|\sum _{k=1}^l|v_k|{\omega }_k^{\rho \zeta }+\left|\widehat{{\delta }_2}\right|{\mathcal{T}}^{\rho \zeta }\right)\frac{1}{{\rho }^{\zeta }\Gamma (\zeta +1)}+|\widehat{{\delta }_1}|\sum _{i=1}^j\left|{ϵ}_i\right|\frac{{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)},\end{array}$$

(54)

$$\begin{array}{c}\hfill {\mathcal{J}}_2=\left(|{\delta }_4|\sum _{k=1}^l|{\nu }_k|{\vartheta }_k^{\rho \xi }+{\mathcal{T}}^{\rho \xi }\left|\widehat{{\delta }_3}\right|\right)\frac{1}{{\rho }^{\xi }\Gamma (\xi +1)}+|\widehat{{\delta }_4}|\sum _{i=1}^j\left|{\pi }_i\right|\frac{{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)},\end{array}$$

(55)

$$\begin{array}{c}\hfill {\mathcal{K}}_2=\left({\mathcal{T}}^{\rho \zeta }(1+|\widehat{{\delta }_4}\left|\right)+|{\delta }_3|\sum _{k=1}^l\left|{\nu }_k\right|{\omega }_k^{\rho \zeta }\right)\frac{1}{{\rho }^{\zeta }\Gamma (\zeta +1)}+|\widehat{{\delta }_3}|\sum _{i=1}^j\left|{ϵ}_i\right|\frac{{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}.\end{array}$$

(56)

Our first result is based on the Leray–Schauder alternative [37].

Theorem 1.

Let $\mathsf{\Lambda }\ne 0$, where Λ is defined by (22). Assume that $f,g:\mathcal{E}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions satisfying the condition: $\left({\mathcal{A}}_1\right)$. There exist constants ${\psi }_m,\widehat{{\psi }_m}\ge 0(m=1,2)$ and ${\psi }_0,\widehat{{\psi }_0}>0$ such that, for all $\tau \in \mathcal{E}$ and $o_1,o_2\in \mathbb{R}$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|f\right(\tau ,o_1,o_2\left)\right|& \le {\psi }_0+{\psi }_1|o_1|+{\psi }_2\left|o_2\right|,\hfill \\ \hfill \left|g\right(\tau ,o_1,o_2\left)\right|& \le \widehat{{\psi }_0}+\widehat{{\psi }_1}|o_1|+\widehat{{\psi }_2}\left|o_2\right|,\forall o_m\in \mathbb{R},m=1,2.\hfill \end{array}\end{array}$$

If ${\psi }_1({\mathcal{J}}_1+{\mathcal{J}}_2)+\widehat{{\psi }_1}({\mathcal{K}}_1+{\mathcal{K}}_2)<1,{\psi }_2({\mathcal{J}}_1+{\mathcal{J}}_2)+\widehat{{\psi }_2}({\mathcal{K}}_1+{\mathcal{K}}_2)<1$. Then, ∃ at least one solution for the BVP (6) and (7) on $\mathcal{E}$, where ${\mathcal{J}}_1,{\mathcal{K}}_1,{\mathcal{J}}_2,{\mathcal{K}}_2$ are given by (53)–(56), respectively.

Proof. 

In the first step, we show the operator $\Delta :\mathcal{P}\times \mathcal{P}\to \mathcal{P}\times \mathcal{P}$ as being completely continuous. The continuity of the functions f and g implies that the operators ${\Delta }_1$ and ${\Delta }_2$ are continuous. As a result, the operator $\Delta$ is continuous. To demonstrate that the operator $\Delta$ is uniformly bounded, consider a bounded set $\mathsf{\Psi }\subset \mathcal{P}\times \mathcal{P}$ such that $\left|f(\tau ,p\left(\tau \right),q\left(\tau \right))\right|\le \widehat{{\mathcal{N}}_1},\left|g(\tau ,p\left(\tau \right),q\left(\tau \right))\right|\le \widehat{{\mathcal{N}}_2}$, where $\widehat{{\mathcal{N}}_1}$ and $\widehat{{\mathcal{N}}_2}$ are positive constants $\forall (p,q)\in \mathsf{\Psi }$. Then, we have

$$\begin{array}{cc}\hfill |{\Delta }_1(p,q)\left(\tau \right)|\le & {}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f(\tau ,p\left(\tau \right),q\left(\tau \right))\right|+|{\delta }_1\left(\tau \right)|\sum _{k=1}^l|v_k|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left|g({\omega }_k,p\left({\omega }_k\right),q\left({\omega }_k\right))\right|\hfill \\ & +|{\delta }_2\left(\tau \right)|\sum _{k=1}^l|{\nu }_k|{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f({\vartheta }_k,p\left({\vartheta }_k\right),q\left({\vartheta }_k\right))\right|\hfill \\ & +|\widehat{{\delta }_1}\left(\tau \right)|\left(\sum _{i=1}^j|{ϵ}_i|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }|g({\omega }_i,p\left({\omega }_i\right),q\left({\omega }_i\right))|{+}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right|\right)\hfill \\ & +|\widehat{{\delta }_2}\left(\tau \right)|\left(\sum _{i=1}^j|{\pi }_i|{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }|f({\sigma }_i,p\left({\sigma }_i\right),q\left({\sigma }_i\right))|{+}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left|g(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right|\right)\hfill \\ & \le {\widehat{\mathcal{N}}}_1\left\{\frac{|\widehat{{\delta }_2}|{\sum }_{i=1}^j\left|{\pi }_i\right|{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)}+\frac{\left({\mathcal{T}}^{\rho \xi }(1+|\widehat{{\delta }_1}\left|\right)+|{\delta }_2|{\sum }_{k=1}^l\left|v_k\right|{\vartheta }_k^{\rho \xi }\right)}{{\rho }^{\xi }\Gamma (\xi +1)}\right\}\hfill \\ & +\widehat{{\mathcal{N}}_2}\left\{\frac{|\widehat{{\delta }_1}|{\sum }_{i=1}^j\left|{ϵ}_i\right|{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}+\frac{\left(|{\delta }_1|{\sum }_{k=1}^l|v_k|{\omega }_k^{\rho \zeta }+\left|\widehat{{\delta }_2}\right|{\mathcal{T}}^{\rho \zeta }\right)}{{\rho }^{\zeta }\Gamma (\zeta +1)}\right\},\hfill \end{array}$$

when taking the norm and using (53) and (54), that yields for any $(p,q)\in \mathsf{\Psi }$

$$\begin{array}{c}\hfill \left|\right|{\Delta }_1(p,q)\left|\right|\le {\mathcal{J}}_1{\widehat{\mathcal{N}}}_1+{\mathcal{K}}_1{\widehat{\mathcal{N}}}_2.\end{array}$$

(57)

Likewise, we use (55) and (56) to obtain

$$\begin{array}{cc}\hfill \left|\right|{\Delta }_2(p,q)\left|\right|\le & \widehat{{\mathcal{N}}_2}\left\{\frac{\left({\mathcal{T}}^{\rho \zeta }(1+|\widehat{{\delta }_4}\left|\right)+|\widehat{{\delta }_3}|{\sum }_{k=1}^l\left|{\nu }_k\right|{\omega }_k^{\rho \zeta }\right)}{{\rho }^{\zeta }\Gamma (\zeta +1)}+\frac{|\widehat{{\delta }_3}|{\sum }_{i=1}^j\left|{ϵ}_i\right|{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}\right\}\hfill \\ & +\widehat{{\mathcal{N}}_1}\left\{\frac{\left(|{\delta }_4|{\sum }_{k=1}^l|{\nu }_k|{\vartheta }_k^{\rho \xi }+\left|\widehat{{\delta }_3}\right|{\mathcal{T}}^{\rho \xi }\right)}{{\rho }^{\xi }\Gamma (\xi +1)}+\frac{|\widehat{{\delta }_4}|{\sum }_{i=1}^j\left|{\pi }_i\right|{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)}\right\},\hfill \\ \hfill \le & {\mathcal{J}}_2{\widehat{\mathcal{N}}}_1+{\mathcal{K}}_2{\widehat{\mathcal{N}}}_2.\hfill \end{array}$$

(58)

Based on the inequalities (57) and (58), we can conclude that ${\Delta }_1$ and ${\Delta }_2$ are uniformly bounded, which indicates that the operator $\Delta$ is uniformly bounded. Next, we show that $\Delta$ is equicontinuous. Let ${\tau }_1,{\tau }_2\in \mathcal{E}$ with ${\tau }_1<{\tau }_2$. Then, we have

$$\begin{array}{cc}\hfill & |{\Delta }_1(p,q)\left({\tau }_2\right)-{\Delta }_1(p,q)\left({\tau }_1\right)|\hfill \\ \hfill & {\le |}^{\rho }{\mathcal{I}}_{0+}^{\xi }f({\tau }_2,p\left({\tau }_2\right),q\left({\tau }_2\right))-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }f({\tau }_1,p\left({\tau }_1\right),q\left({\tau }_1\right))|\hfill \\ & +|{\delta }_1\left({\tau }_2\right)-{\delta }_1\left({\tau }_1\right)|\sum _{k=1}^l|v_k|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left|g({\omega }_k,p\left({\omega }_k\right),q\left({\omega }_k\right))\right|\hfill \\ & +|{\delta }_2\left({\tau }_2\right)-{\delta }_2\left({\tau }_1\right)|\sum _{k=1}^l|{\nu }_k|{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f({\vartheta }_k,p\left({\vartheta }_k\right),q\left({\vartheta }_k\right))\right|\hfill \\ & +|\widehat{{\delta }_1}\left({\tau }_2\right)-\widehat{{\delta }_1}\left({\tau }_1\right)|\left(\sum _{i=1}^j|{ϵ}_i|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }|g({\varpi }_i,p\left({\varpi }_i\right),q\left({\varpi }_i\right))|{+}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right|\right)\hfill \\ & +|\widehat{{\delta }_2}\left({\tau }_2\right)-\widehat{{\delta }_2}\left({\tau }_1\right)|\left(\sum _{i=1}^j|{\pi }_i|{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }|f({\sigma }_i,p\left({\sigma }_i\right),q\left({\sigma }_i\right))|{+}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left|g(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right|\right)\hfill \\ \hfill \le & \frac{{\rho }^{1-\xi }\widehat{{\mathcal{N}}_1}}{\Gamma \left(\xi \right)}\left|{\int }_0^{{\tau }_1}\left[\frac{{\theta }^{\rho -1}}{{({\tau }_2^{\rho }-{\theta }^{\rho })}^{1-\xi }}-\frac{{\theta }^{\rho -1}}{{({\tau }_1^{\rho }-{\theta }^{\rho })}^{1-\xi }}\right]d\theta +{\int }_{{\tau }_1}^{{\tau }_2}\frac{{\theta }^{\rho -1}}{{({\tau }_2^{\rho }-{\theta }^{\rho })}^{1-\xi }}d\theta \right|\hfill \\ & +|{\delta }_1\left({\tau }_2\right)-{\delta }_1\left({\tau }_1\right)|\frac{\widehat{{\mathcal{N}}_2}{\sum }_{k=1}^l\left|v_k\right|{\omega }_k^{\rho \zeta }}{{\rho }^{\zeta }\Gamma (\zeta +1)}+|{\delta }_2\left({\tau }_2\right)-{\delta }_2\left({\tau }_1\right)|\frac{\widehat{{\mathcal{N}}_1}{\sum }_{k=1}^l\left|{\nu }_k\right|{\vartheta }_k^{\rho \xi }}{{\rho }^{\xi }\Gamma (\xi +1)}\hfill \\ & +|\widehat{{\delta }_1}\left({\tau }_2\right)-\widehat{{\delta }_1}\left({\tau }_1\right)|+\left[\frac{\widehat{{\mathcal{N}}_2}{\sum }_{i=1}^j\left|{ϵ}_i\right|{\varpi }_i^{\rho \zeta +\varsigma }}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}+\frac{\widehat{{\mathcal{N}}_1}{\mathcal{T}}^{\rho \xi }}{{\rho }^{\xi }\Gamma (\xi +1)}\right]\hfill \\ & +|\widehat{{\delta }_2}\left({\tau }_2\right)-\widehat{{\delta }_2}\left({\tau }_1\right)|+\left[\frac{\widehat{{\mathcal{N}}_1}{\sum }_{i=1}^j\left|{\pi }_i\right|{\sigma }_i^{\rho \xi +\varrho }}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)}+\frac{\widehat{{\mathcal{N}}_2}{\mathcal{T}}^{\rho \zeta }}{{\rho }^{\zeta }\Gamma (\zeta +1)}\right]\hfill \\ & \to 0as{\tau }_2\to {\tau }_1,\hfill \end{array}$$

(59)

independent of $(p,q)$. Similarly, we can express $|{\Delta }_2(p,q)\left({\tau }_2\right)-{\Delta }_2(p,q)\left({\tau }_1\right)|\to 0$ as ${\tau }_2\to {\tau }_1$ independent of $(p,q)$ in terms of the boundedness of the functions f and g. As a result of the equicontinuity of ${\Delta }_1$ and ${\Delta }_2$, the operator $\Delta$ is equicontinuous. As a result of the Arzela–Ascoli theorem, the operator $\Delta$ is compact. Finally, we demonstrate that the set $\Pi (\Delta )=\left\{\right(p,q)\in \mathcal{P}\times \mathcal{P}:(p,q)=\lambda \Delta (p,q);0<\lambda <1\}$ is bounded. Let $(p,q)\in \Pi (\Delta )$. Then $(p,q)=\lambda \Delta (p,q)$. For any $\tau \in \mathcal{E}$, we have $p\left(\tau \right)=\lambda {\Delta }_1(p,q)\left(\tau \right),q\left(\tau \right)=\lambda {\Delta }_2(p,q)\left(\tau \right)$. By utilizing (${\mathcal{A}}_1$) in Theorem 1, we obtain

$$\begin{array}{cc}\hfill & \left|p\right(\tau \left)\right|\hfill \\ & {\le }^{\rho }{\mathcal{I}}_{0+}^{\xi }({\psi }_0,{\psi }_1\left|p\left(\tau \right)\right|,{\psi }_2\left|q\left(\tau \right)\right|)+|{\delta }_1\left(\tau \right)|\sum _{k=1}^l|v_k{|}^{\rho }{\mathcal{I}}_{0+}^{\zeta }(\widehat{{\psi }_0}+\widehat{{\psi }_1}|p\left({\omega }_k\right)|+\widehat{{\psi }_2}|q\left({\omega }_k\right)\left|\right)\hfill \\ & +|{\delta }_2\left(\tau \right)|\sum _{k=1}^l|{\nu }_k{|}^{\rho }{\mathcal{I}}_{0+}^{\xi }({\psi }_0+{\psi }_1|p\left({\vartheta }_k\right)|+{\psi }_2|q\left({\vartheta }_k\right)\left|\right)\hfill \\ & +|\widehat{{\delta }_1}\left(\tau \right)|\left(\sum _{i=1}^j|{ϵ}_i{|}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }(\widehat{{\psi }_0}+\widehat{{\psi }_1}|p\left({\varpi }_i\right)|+\widehat{{\psi }_2}|q\left({\varpi }_i\right)\left|\right)+{}^{\rho }{\mathcal{I}}_{0+}^{\xi }({\psi }_0+{\psi }_0|p\left(\mathcal{T}\right)|+{\psi }_2|q\left(\mathcal{T}\right)\left|\right)\right)\hfill \\ & +\widehat{{\delta }_2}\left(\tau \right)\left(\sum _{i=1}^j|{\pi }_i{|}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }({\psi }_0+{\psi }_1|p\left({\sigma }_i\right)|+{\psi }_2|q\left({\sigma }_i\right)\left|\right){+}^{\rho }{\mathcal{I}}_{0+}^{\zeta }(\widehat{{\psi }_0}+\widehat{{\psi }_1}|p\left(\mathcal{T}\right)|+\widehat{{\psi }_2}|q\left(\mathcal{T}\right)\left|\right)\right),\hfill \end{array}$$

which results when taking the norm for $\tau \in \mathcal{E}$,

$$\begin{array}{c}\hfill \left|\right|p\left|\right|\le ({\psi }_0+{\psi }_1\left|\right|p\left|\right|+{\psi }_2\left|\right|q\left|\right|){\mathcal{J}}_1+(\widehat{{\psi }_0}+\widehat{{\psi }_1}\left|\right|p\left|\right|+\widehat{{\psi }_2}\left|\right|q\left|\right|){\mathcal{K}}_1.\end{array}$$

(60)

Similarly, we are capable of obtaining that

$$\begin{array}{c}\hfill \left|\right|q\left|\right|\le (\widehat{{\psi }_0}+\widehat{{\psi }_1}\left|\right|p\left|\right|+\widehat{{\psi }_2}\left|\right|q\left|\right|){\mathcal{K}}_2+({\psi }_0+{\psi }_1\left|\right|p\left|\right|+{\psi }_2\left|\right|q\left|\right|){\mathcal{J}}_2.\end{array}$$

(61)

From (60) and (61), we obtain

$$\begin{array}{cc}\hfill \left|\right|p\left|\right|+\left|\right|q\left|\right|=& {\psi }_0({\mathcal{J}}_1+{\mathcal{J}}_2)+\widehat{{\psi }_0}({\mathcal{K}}_1+{\mathcal{K}}_2)+\left|\right|p\left|\right|\left[{\psi }_1({\mathcal{J}}_1+{\mathcal{J}}_2)+\widehat{{\psi }_1}({\mathcal{K}}_1+{\mathcal{K}}_2)\right]\hfill \\ & +\left|\right|q\left|\right|\left[{\psi }_1({\mathcal{J}}_1+{\mathcal{J}}_2)+\widehat{{\psi }_1}({\mathcal{K}}_1+{\mathcal{K}}_2)\right],\hfill \end{array}$$

therefore, we have

$$\begin{array}{c}\hfill \left|\right|(p,q)\left|\right|\le \frac{{\psi }_0({\mathcal{J}}_1+{\mathcal{J}}_2)+\widehat{{\psi }_0}({\mathcal{K}}_1+{\mathcal{K}}_2)}{\Phi },\end{array}$$

where

$$\begin{array}{c}\hfill \Phi =min\{1-[{\psi }_1({\mathcal{J}}_1+{\mathcal{J}}_2)+\widehat{{\psi }_1}({\mathcal{K}}_1+{\mathcal{K}}_2)],1-[{\psi }_2({\mathcal{J}}_1+{\mathcal{J}}_2)+\widehat{{\psi }_2}({\mathcal{K}}_1+{\mathcal{K}}_2)]\}.\end{array}$$

(62)

As a result, $\Pi (\Delta )$ is bounded. Thus, the nonlinear alternative of Leray–Schauder is valid, and the operator $\Delta$ has at least one fixed point. It implies that the BVP (6) and (7) contain at least one solution on $\mathcal{E}$. □

Our next result deals with the existence and uniqueness of a solution of the BVP (6) and (7) via the contraction mapping principle.

Theorem 2.

Let $\mathsf{\Lambda }\ne 0$, where Λ is defined by (22). Assume that $f,g:\mathcal{E}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions satisfying the condition:

(${\mathcal{A}}_2$) there exist constants ${\varphi }_m,\widehat{{\varphi }_m}\ge 0(m=1,2)$ such that

$$\begin{array}{cc}\hfill |f(\tau ,o_1,o_2)-f(\tau ,{\widehat{o}}_1,{\widehat{o}}_2)|\le & {\varphi }_1|o_1-{\widehat{o}}_1|+{\varphi }_2|o_2-{\widehat{o}}_2|,\hfill \\ \hfill |g(\tau ,o_1,o_2)-g(\tau ,{\widehat{o}}_1,{\widehat{o}}_2)|\le & \widehat{{\varphi }_1}|o_1-{\widehat{o}}_1|+\widehat{{\varphi }_2}|o_2-{\widehat{o}}_2|,\forall o_m,{\widehat{o}}_m\in \mathbb{R},m=1,2.\hfill \end{array}$$

Then, given that

$$\begin{array}{c}\hfill ({\mathcal{J}}_1+{\mathcal{J}}_2)({\varphi }_1+{\varphi }_2)+({\mathcal{K}}_1+{\mathcal{K}}_2)(\widehat{{\varphi }_1}+\widehat{{\varphi }_2})<1,\end{array}$$

(63)

the BVP (6) and (7) has a unique solution on $\mathcal{E}$, where ${\mathcal{J}}_1,{\mathcal{K}}_1,{\mathcal{J}}_2,{\mathcal{K}}_2$ are given by (53)–(56), respectively.

Proof. 

Define $\left|f(\tau ,0,0)\right|\le {\mathcal{S}}_1<\infty ,\left|g(\tau ,0,0)\right|\le {\mathcal{S}}_2<\infty ,$ and $\phi >0$ such that

$$\phi \le \frac{({\mathcal{J}}_1+{\mathcal{J}}_2){\mathcal{S}}_1+({\mathcal{K}}_1+{\mathcal{K}}_2){\mathcal{S}}_2}{1-(({\mathcal{J}}_1+{\mathcal{J}}_2)({\varphi }_1+{\varphi }_2)+({\mathcal{K}}_1+{\mathcal{K}}_2)(\widehat{{\varphi }_1}+\widehat{{\varphi }_2}))}.$$

Firstly, we demonstrate that $\Delta {\mathcal{B}}_{\phi }\subset {\mathcal{B}}_{\phi }$ when operator $\Delta$ is given by (50) and ${\mathcal{B}}_{\phi }=\{(p,q)\in \mathcal{P}\times \mathcal{P}:|\left|(p,q)\right||\le \phi \}$. For $(p,q)\in {\mathcal{B}}_{\phi },\tau \in \mathcal{E}$, we have

$$\begin{array}{c}\hfill \left|f(\tau ,p\left(\tau \right),q\left(\tau \right))\right|\le {\varphi }_1\left|p\left(\tau \right)\right|+{\varphi }_2\left|q\left(\tau \right)\right|+{\mathcal{S}}_1\\ \hfill \le {\varphi }_1\left|\right|p\left|\right|+{\varphi }_2\left|\right|q\left|\right|+{\mathcal{S}}_1,\end{array}$$

and

$$\begin{array}{c}\hfill \left|g(\tau ,p\left(\tau \right),q\left(\tau \right))\right|\le \widehat{{\varphi }_1}\left|p\left(\tau \right)\right|+\widehat{{\varphi }_2}\left|q\left(\tau \right)\right|+{\mathcal{S}}_2\\ \hfill \le \widehat{{\varphi }_1}\left|\right|p\left|\right|+\widehat{{\varphi }_2}\left|\right|q\left|\right|+{\mathcal{S}}_2.\end{array}$$

(64)

These inequalities guide to

$$\begin{array}{cc}\hfill & |{\Delta }_1(p,q)\left(\tau \right)|\hfill \\ & {\le }^{\rho }{\mathcal{I}}_{0+}^{\xi }\left[\left|f\right(\tau ,p\left(\tau \right),q\left(\tau \right))-f(\tau ,0,0\left)\right|+\left|f\right(\tau ,0,0\left)\right|\right]+\hfill \\ & |{\delta }_1\left(\tau \right)|\sum _{k=1}^l{\left|v_k\right|}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left[|g({\omega }_k,p\left({\omega }_k\right),q\left({\omega }_k\right))-g({\omega }_k,0,0)|+|g({\omega }_k,0,0)|\right]\hfill \\ & +|{\delta }_2\left(\tau \right)|\sum _{k=1}^l|{\nu }_k{|}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f\right[({\vartheta }_k,p\left({\vartheta }_k\right),q\left({\vartheta }_k\right))-f({\vartheta }_k,0,0)|+|f({\vartheta }_k,0,0)\left|\right]\hfill \\ & +|\widehat{{\delta }_1}\left(\tau \right)|\left(\sum _{i=1}^j|{ϵ}_i{|}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }g[({\varpi }_i,p\left({\varpi }_i\right),q\left({\varpi }_i\right))-g({\varpi }_i,0,0)|+|g({\varpi }_i,0,0)\left|\right]\right.\hfill \\ & \left.+{}^{\rho }{\mathcal{I}}_{0+}^{\xi }f[(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))-f(\mathcal{T},0,0)|+\left|f(\mathcal{T},0,0)\right]|\right)\hfill \\ & +|\widehat{{\delta }_2}\left(\tau \right)|\left(\sum _{i=1}^j|{\pi }_i{|}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }f[f({\sigma }_i,p\left({\sigma }_i\right),q\left({\sigma }_i\right))-f({\sigma }_i,0,0)|+|f({\sigma }_i,0,0)\left|\right]\right.\hfill \\ & \left.+{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left[\right|g(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))-g(\mathcal{T},0,0)|+|g(\mathcal{T},0,0)\left|\right]\right),\hfill \\ \hfill \le & ({\varphi }_1\left|\right|p\left|\right|+{\varphi }_2\left|\right|q\left|\right|+{\mathcal{S}}_1)\left\{\frac{|\widehat{{\delta }_2}|{\sum }_{i=1}^j\left|{\pi }_i\right|{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)}+\frac{\left({\mathcal{T}}^{\rho \xi }(1+|\widehat{{\delta }_1}\left|\right)+|{\delta }_2|{\sum }_{k=1}^l\left|{\nu }_k\right|{\vartheta }_k^{\rho \xi }\right)}{{\rho }^{\xi }\Gamma (\xi +1)}\right\}\hfill \\ & +(\widehat{{\varphi }_1}\left|\right|p\left|\right|+\widehat{{\varphi }_2}\left|\right|q\left|\right|+{\mathcal{S}}_2)\left\{\frac{|\widehat{{\delta }_1}|{\sum }_{i=1}^j\left|{ϵ}_i\right|{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}+\frac{\left(|{\delta }_1|{\sum }_{k=1}^l|v_k|{\omega }_k^{\rho \zeta }+\left|\widehat{{\delta }_2}\right|{\mathcal{T}}^{\rho \zeta }\right)}{{\rho }^{\zeta }\Gamma (\zeta +1)}\right\},\hfill \end{array}$$

which, on taking the norm for $\tau \in \mathcal{E}$, gives

$$\begin{array}{c}\hfill \left|\right|{\Delta }_1(p,q)\left|\right|\le ({\varphi }_1\left|\right|p\left|\right|+{\varphi }_2\left|\right|q\left|\right|+{\mathcal{S}}_1){\mathcal{J}}_1+(\widehat{{\varphi }_1}\left|\right|p\left|\right|+\widehat{{\varphi }_2}\left|\right|q\left|\right|+{\mathcal{S}}_2){\mathcal{K}}_1.\end{array}$$

(65)

Similarly, we obtain

$$\begin{array}{cc}\hfill \left|\right|{\Delta }_2(p,q)\left|\right|\le & (\widehat{{\varphi }_1}\left|\right|p\left|\right|+\widehat{{\varphi }_2}\left|\right|q\left|\right|+{\mathcal{S}}_2){\mathcal{K}}_2+({\varphi }_1\left|\right|p\left|\right|+{\varphi }_2\left|\right|q\left|\right|+{\mathcal{S}}_1){\mathcal{J}}_2.\hfill \end{array}$$

(66)

As a result, (65) and (66) follow $\left|\right|\Delta (p,q)\left|\right|\le \phi ,$ and thus $\Delta {\mathcal{B}}_{\phi }\subset {\mathcal{B}}_{\phi }$. Now, for $\left(p_1{q}_2\right),(p_2,q_2)$$\in \mathcal{P}\times \mathcal{P}$ and any $\tau \in \mathcal{E}$, we obtain

$$\begin{array}{cc}& |{\Delta }_1(p_1,q_1)\left(\tau \right)-{\Delta }_1(p_2,q_2)\left(\tau \right)|\hfill \\ \hfill \le & {}^{\rho }{\mathcal{I}}_{0+}^{\xi }|f(\tau ,p_1\left(\tau \right),q_1\left(\tau \right))-f(\tau ,p_2\left(\tau \right),q_2\left(\tau \right))|\hfill \\ & +|{\delta }_1\left(\tau \right)|\sum _{k=1}^l|v_k|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }|g({\omega }_k,p_1\left({\omega }_k\right),q_1\left({\omega }_k\right))-g({\omega }_k,p_2\left({\omega }_k\right),q_2\left({\omega }_k\right))|\hfill \\ & +|{\delta }_2\left(\tau \right)|\sum _{k=1}^l|{\nu }_k|{}^{\rho }{\mathcal{I}}_{0+}^{\xi }|f({\vartheta }_k,p_1\left({\vartheta }_k\right),q_1\left({\vartheta }_k\right))-f({\vartheta }_k,p_2\left({\vartheta }_k\right),q_2\left({\vartheta }_k\right))|\hfill \\ & +|\widehat{{\delta }_1}\left(\tau \right)|\left(\sum _{i=1}^j|{ϵ}_i|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }|g({\varpi }_i,p_1\left({\varpi }_i\right),q_1\left({\varpi }_i\right))-g({\varpi }_i,p_2\left({\varpi }_i\right),q_2\left({\varpi }_i\right))|\right.\hfill \\ & \left.+{}^{\rho }{\mathcal{I}}_{0+}^{\xi }|f(\mathcal{T},p_1\left(\mathcal{T}\right),q_1\left(\mathcal{T}\right))-f(\mathcal{T},p_2\left(\mathcal{T}\right),q_2\left(\mathcal{T}\right))|\right)\hfill \\ & +|\widehat{{\delta }_2}\left(\tau \right)|\left(\sum _{i=1}^j|{\pi }_i|{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }|f({\sigma }_i,p_1\left({\sigma }_i\right),q_1\left({\sigma }_1\right))-f({\sigma }_i,p_2\left({\sigma }_i\right),q_2\left({\sigma }_1\right))|\right.\hfill \\ & \left.+{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }|g(\mathcal{T},p_1\left(\mathcal{T}\right),q_1\left(\mathcal{T}\right))-g(\mathcal{T},p_2\left(\mathcal{T}\right),q_2\left(\mathcal{T}\right))|\right)\hfill \\ \hfill \le & ({\varphi }_1\left|\right|p_1-p_2\left|\right|+{\varphi }_2\left|\right|q_1-q_2\left|\right|)\left\{\frac{\left({\mathcal{T}}^{\rho \xi }(1+|\widehat{{\delta }_1}\left|\right)+|{\delta }_2|{\sum }_{k=1}^l\left|{\nu }_k\right|{\vartheta }_k^{\rho \xi }\right)}{{\rho }^{\xi }\Gamma (\xi +1)}+\frac{|\widehat{{\delta }_2}|{\sum }_{i=1}^j\left|{\pi }_i\right|{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)}\right\}\hfill \\ & +(\widehat{{\varphi }_1}\left|\right|p_1-p_2\left|\right|+\widehat{{\varphi }_2}\left|\right|q_1-q_2\left|\right|)\left\{\frac{\left(|{\delta }_1|{\sum }_{k=1}^l|v_k|{\omega }_k^{\rho \zeta }+\left|\widehat{{\delta }_2}\right|{\mathcal{T}}^{\rho \zeta }\right)}{{\rho }^{\zeta }\Gamma (\zeta +1)}+\frac{|\widehat{{\delta }_1}|{\sum }_{i=1}^j\left|{ϵ}_i\right|{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}\right\}\hfill \\ \hfill \le & ({\mathcal{J}}_1({\varphi }_1+{\varphi }_2)+{\mathcal{K}}_1(\widehat{{\varphi }_1}+\widehat{{\varphi }_2}))\left(\right||p_1-p_2\left|\right|+\left|\right|q_1-q_2\left|\right|).\hfill \end{array}$$

(67)

In a similar manner,

$$\begin{array}{c}\hfill \left|\right|{\Delta }_2(p_1,q_1)\left(\tau \right)-{\Delta }_2(p_2,q_2)\left(\tau \right)\left|\right|\le ({\mathcal{J}}_2({\varphi }_1+{\varphi }_2)+{\mathcal{K}}_2(\widehat{{\varphi }_1}+\widehat{{\varphi }_2}))\left(\right||p_1-p_2\left|\right|+\left|\right|q_1-q_2\left|\right|).\end{array}$$

(68)

Hence, using (67) and (68), we can obtain

$$\begin{array}{cc}\hfill & \left|\right|\Delta (p_1,q_1)\left(\tau \right)-\Delta (p_2,q_2)\left(\tau \right)\left|\right|\hfill \\ & \le (({\mathcal{J}}_1+{\mathcal{J}}_2)({\varphi }_1+{\varphi }_2)+({\mathcal{K}}_1+{\mathcal{K}}_2)(\widehat{{\varphi }_1}+\widehat{{\varphi }_2}))\left(\right||p_1-p_2\left|\right|+\left|\right|q_1-q_2\left|\right|).\hfill \end{array}$$

As a consequence of condition (63), the operator $\Delta$ is a contraction. As an outcome of the Banach fixed point theorem, we can conclude that the operator $\Delta$ has a unique fixed point, which is the unique solution of problems (6) and (7). □

In the following result, we will use Krasnoselskii’s theorem [38] with the aid of the following notations:

$$\begin{array}{c}\hfill {\widehat{\mathsf{\Omega }}}_1={\mathcal{J}}_1-\frac{{\mathcal{T}}^{\rho \xi }}{{\rho }^{\xi }\Gamma (\xi +1)}+{\mathcal{K}}_1,\end{array}$$

(69)

$$\begin{array}{c}\hfill {\widehat{\mathsf{\Omega }}}_2={\mathcal{J}}_2-\frac{{\mathcal{T}}^{\rho \zeta }}{{\rho }^{\zeta }\Gamma (\zeta +1)}+{\mathcal{K}}_2.\end{array}$$

(70)

Theorem 3.

Assume that $f,g:\mathcal{E}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions satisfying the assumption $\left({\mathcal{A}}_2\right)$ in Theorem 2. Furthermore, there exist positive constants ${\mathcal{U}}_1,{\mathcal{U}}_2$ such that $\forall \tau \in \mathcal{E}$ and $r_i\in \mathbb{R},i=1,2$:

$$\begin{array}{c}\hfill |f(\tau ,r_1,r_2)|\le {\mathcal{U}}_1,\left|g(\tau ,r_1,r_2)\right|\le {\mathcal{U}}_2.\end{array}$$

(71)

If

$$\begin{array}{c}\hfill \frac{{\mathcal{T}}^{\rho \xi }({\varphi }_1+{\varphi }_2)}{{\rho }^{\xi }\Gamma (\xi +1)}+\frac{{\mathcal{T}}^{\rho \zeta }(\widehat{{\varphi }_1}+\widehat{{\varphi }_2})}{{\rho }^{\zeta }\Gamma (\zeta +1)}<1,\end{array}$$

(72)

then the BVP (6) and (7) has at least one solution on $\mathcal{E}$.

Proof. 

Let us define a closed ball ${\mathcal{B}}_{\phi }=\{(p,q)\in \mathcal{P}\times \mathcal{P}:|\left|(p,q)\right||\le \phi \}$ and split operators ${\Delta }_1,{\Delta }_2$ as:

$$\begin{array}{cc}\hfill {\Delta }_{1,1}(p,q)\left(\tau \right)=& {\delta }_1\left(\tau \right)\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }f({\omega }_k,p\left({\omega }_k\right),q\left({\omega }_k\right))\hfill \\ & +{\delta }_2\left(\tau \right)\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }f({\vartheta }_k,p\left({\vartheta }_k\right),q\left({\vartheta }_k\right))\hfill \\ & +\widehat{{\delta }_1}\left(\tau \right)\left(\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }g({\varpi }_i,p\left({\varpi }_i\right),q\left({\varpi }_i\right))-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }f(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right)\hfill \\ & +\widehat{{\delta }_2}\left(\tau \right)\left(\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }f({\sigma }_i,p\left({\sigma }_i\right),q\left({\sigma }_i\right))-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }g(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right),\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill {\Delta }_{1,1}(p,q)\left(\tau \right)=& {}^{\rho }{\mathcal{I}}_{0+}^{\xi }f(\tau ,p\left(\tau \right),q\left(\tau \right)),\hfill \end{array}$$

(74)

and

$$\begin{array}{cc}\hfill {\Delta }_{2,1}(p,q)\left(\tau \right)=& {\delta }_3\left(\tau \right)\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }f({\omega }_k,p\left({\omega }_k\right),q\left({\omega }_k\right))\hfill \\ & +\widehat{{\delta }_3}\left(\tau \right)\left(\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }g({\varpi }_i,p\left({\varpi }_i\right),q\left({\varpi }_i\right))-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }f(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right)\hfill \\ \hfill & +{\delta }_4\left(\tau \right)\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }f({\vartheta }_k,p\left({\vartheta }_k\right),q\left({\vartheta }_k\right))\hfill \\ \hfill & +\widehat{{\delta }_4}\left(\tau \right)\left(\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }f({\sigma }_i,p\left({\sigma }_i\right),q\left({\sigma }_i\right))-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }g(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right),\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill {\Delta }_{2,2}(p,q)\left(\tau \right)=& {}^{\rho }{\mathcal{I}}_{0+}^{\xi }g(\tau ,p\left(\tau \right),q\left(\tau \right)).\hfill \end{array}$$

(76)

Clearly, ${\Delta }_1(p,q)\left(\tau \right)={\Delta }_{1,1}(p,q)\left(\tau \right)+{\Delta }_{1,2}(p,q)\left(\tau \right)$, and ${\Delta }_2(p,q)\left(\tau \right)={\Delta }_{2,1}(p,q)\left(\tau \right)+{\Delta }_{2,2}(p,q)\left(\tau \right)$ on ${\mathcal{B}}_{\phi }$ which is a closed, bounded and convex subsets of the Banach space $\mathcal{P}\times \mathcal{P}$. Let us fix $\phi \le max\{{\mathcal{J}}_1{\mathcal{U}}_1+{\mathcal{K}}_1{\mathcal{U}}_2,{\mathcal{J}}_2{\mathcal{U}}_1+{\mathcal{K}}_2{\mathcal{U}}_2\}$ and show that $\Delta {\mathcal{B}}_{\phi }\subset {\mathcal{B}}_{\phi }$ to verify the first condition of Krasnoselskii’s theorem. If we choose $p=(p_1,p_2),q=(q_1,q_2)\in {\mathcal{B}}_{\phi }$, and utilizing condition (71), we obtain

$$\begin{array}{cc}\hfill & |{\Delta }_{1,1}(p,q)\left(\tau \right)+{\Delta }_{1,2}(p,q)\left(\tau \right)|\hfill \\ \hfill \le & {}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f(\tau ,p\left(\tau \right),q\left(\tau \right))\right|+|{\delta }_1\left(\tau \right)|\sum _{k=1}^l|v_k|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left|f({\omega }_k,p\left({\omega }_k\right),q\left({\omega }_k\right))\right|\hfill \\ & +|{\delta }_2\left(\tau \right)|\sum _{k=1}^l|{\nu }_k|{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f({\vartheta }_k,p\left({\vartheta }_k\right),q\left({\vartheta }_k\right))\right|\hfill \\ & +|\widehat{{\delta }_1}\left(\tau \right)|\left(\sum _{i=1}^j|{ϵ}_i|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }|g({\varpi }_i,p\left({\varpi }_i\right),q\left({\varpi }_i\right))|{+}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right|\right)\hfill \\ & +|\widehat{{\delta }_2}\left(\tau \right)|\left(\sum _{i=1}^j|{\pi }_i{|}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }|f({\sigma }_i,p\left({\sigma }_i\right),q\left({\sigma }_i\right))|{+}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left|g(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right|\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {\mathcal{U}}_1\left\{\frac{\left({\mathcal{T}}^{\rho \xi }(1+|\widehat{{\delta }_1}\left|\right)+|{\delta }_2|{\sum }_{k=1}^l\left|{\nu }_k\right|{\vartheta }_k^{\rho \xi }\right)}{{\rho }^{\xi }\Gamma (\xi +1)}+\frac{|\widehat{{\delta }_2}|{\sum }_{i=1}^j\left|{\pi }_i\right|{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)}\right\}\hfill \\ \hfill +& {\mathcal{U}}_2\left\{\frac{\left(|{\delta }_1|{\sum }_{k=1}^l|v_k|{\omega }_k^{\rho \zeta }+\left|\widehat{{\delta }_2}\right|{\mathcal{T}}^{\rho \zeta }\right)}{{\rho }^{\zeta }\Gamma (\zeta +1)}+\frac{|\widehat{{\delta }_1}|{\sum }_{i=1}^j\left|{ϵ}_i\right|{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}\right\}\hfill \\ \hfill \le & {\mathcal{U}}_1{\mathcal{J}}_1+{\mathcal{U}}_2{\mathcal{K}}_1\le \phi .\hfill \end{array}$$

In a similar manner, we can find that

$$\begin{array}{c}\hfill |{\Delta }_{2,1}(p,q)\left(\tau \right)+{\Delta }_{2,2}(p,q)\left(\tau \right)|\le {\mathcal{U}}_1{\mathcal{J}}_2+{\mathcal{U}}_2{\mathcal{K}}_2\le \phi .\end{array}$$

Thus, the above two inequalities show that ${\Delta }_1(p,q)+{\Delta }_2(p,q)\in {\mathcal{B}}_{\phi }$.

Secondly, we prove that the operator $({\Delta }_{1,2},{\Delta }_{2,2})$ is a contraction to satisfy the third condition of Krasnoselskii’s theorem. For $(p_1,q_1),(p_2,q_2)\in {\mathcal{B}}_{\phi }$, we have

$$\begin{array}{cc}\hfill & |{\Delta }_{1,2}(p_1,q_1)\left(\tau \right)-{\Delta }_{1,2}(p_2,q_2)\left(\tau \right)|\hfill \\ \hfill \le & \frac{{\rho }^{1-\xi }}{\Gamma \left(\xi \right)}{\int }_0^{\tau }\frac{{\theta }^{\rho -1}}{{({\tau }^{\rho }-{\theta }^{\rho })}^{1-\xi }}|f(\theta ,p_1\left(\theta \right),q_1\left(\theta \right))-f(\theta ,p_2\left(\theta \right),q_2\left(\theta \right))|d\theta \hfill \\ \hfill \le & \frac{{\mathcal{T}}^{\rho \xi }}{{\rho }^{\xi }\Gamma (\xi +1)}({\varphi }_1\left|\right|p_1-p_2\left|\right|+{\varphi }_2\left|\right|q_1-q_2\left|\right|),\hfill \end{array}$$

(77)

and

$$\begin{array}{cc}\hfill & |{\Delta }_{2,1}(p_1,q_1)\left(\tau \right)-{\Delta }_{2,1}(p_2,q_2)\left(\tau \right)|\hfill \\ \hfill \le & \frac{{\rho }^{1-\zeta }}{\Gamma \left(\zeta \right)}{\int }_0^{\tau }\frac{{\theta }^{\rho -1}}{{({\tau }^{\rho }-{\theta }^{\rho })}^{1-\zeta }}|g(\theta ,p_1\left(\theta \right),q_1\left(\theta \right))-g(\theta ,p_2\left(\theta \right),q_2\left(\theta \right))|d\theta \hfill \\ \hfill \le & \frac{{\mathcal{T}}^{\rho \zeta }}{{\rho }^{\zeta }\Gamma (\zeta +1)}(\widehat{{\varphi }_1}\left|\right|p_1-p_2\left|\right|+\widehat{{\varphi }_2}\left|\right|q_1-q_2\left|\right|).\hfill \end{array}$$

(78)

As a result of the inequalities (77) and (78),

$$\begin{array}{cc}\hfill & \left|\right|({\Delta }_{1,2},{\Delta }_{2,2})(p_1,q_1)-({\Delta }_{1,2},{\Delta }_{2,2})(p_2,q_2)\left|\right|\hfill \\ \hfill \le & \frac{{\mathcal{T}}^{\rho \xi }({\varphi }_1+{\varphi }_2)}{{\rho }^{\xi }\Gamma (\xi +1)}+\frac{{\mathcal{T}}^{\rho \zeta }(\widehat{{\varphi }_1}+\widehat{{\varphi }_2})}{{\rho }^{\zeta }\Gamma (\zeta +1)}\left(\right||p_1-p_2\left|\right|+\left|\right|q_1-q_2\left|\right|),\hfill \end{array}$$

which is a contraction by (72) as required.

Following that, we can establish that the operator (${\Delta }_{1,1},{\Delta }_{2,1}$) satisfies the second condition of Krasnoselskii’s theorem. We can infer the continuous existence of the (${\Delta }_{1,1},{\Delta }_{2,1}$) operator by examining the continuity of the $f,g$ functions. For each $(p,q)\in {\mathcal{B}}_{\phi }$, we have

$$\begin{array}{cc}\hfill & |{\Delta }_{1,1}(p,q)\left(\tau \right)|\hfill \\ \hfill \le & |{\delta }_1\left(\tau \right)|\sum _{k=1}^l|v_k|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left|f({\omega }_k,p\left({\omega }_k\right),q\left({\omega }_k\right))\right|\hfill \\ & +|{\delta }_2\left(\tau \right)|\sum _{k=1}^l|{\nu }_k|{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f({\vartheta }_k,p\left({\vartheta }_k\right),q\left({\vartheta }_k\right))\right|\hfill \\ & +|\widehat{{\delta }_1}\left(\tau \right)|\left(\sum _{i=1}^j|{ϵ}_i|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }|g({\varpi }_i,p\left({\varpi }_i\right),q\left({\varpi }_i\right))|{+}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right|\right)\hfill \\ & +|\widehat{{\delta }_2}\left(\tau \right)|\left(\sum _{i=1}^j|{\pi }_i|{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }|f({\sigma }_i,p\left({\sigma }_i\right),q\left({\sigma }_i\right))|{+}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left|g(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right|\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {\mathcal{U}}_1\left\{\frac{|\widehat{{\delta }_2}|{\sum }_{i=1}^j\left|{\pi }_i\right|{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)}+\frac{\left({\mathcal{T}}^{\rho \xi }|\widehat{{\delta }_1}|+|{\delta }_2|{\sum }_{k=1}^l\left|{\nu }_k\right|{\vartheta }_k^{\rho \xi }\right)}{{\rho }^{\xi }\Gamma (\xi +1)}\right\}\hfill \\ \hfill +& {\mathcal{U}}_2\left\{\frac{|\widehat{{\delta }_1}|{\sum }_{i=1}^j\left|{ϵ}_i\right|{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}+\frac{\left(|{\delta }_1|{\sum }_{k=1}^l|v_k|{\omega }_k^{\rho \zeta }+\left|\widehat{{\delta }_2}\right|{\mathcal{T}}^{\rho \zeta }\right)}{{\rho }^{\zeta }\Gamma (\zeta +1)}\right\}\hfill \\ \hfill =& {\widehat{\mathsf{\Omega }}}_1,\hfill \end{array}$$

and similarly,

$$\begin{array}{cc}\hfill |{\Delta }_{2,1}(p,q)\left(\tau \right)|\le & {\mathcal{U}}_2\left\{\frac{\left({\mathcal{T}}^{\rho \zeta }|\widehat{{\delta }_4}|+|{\delta }_3|{\sum }_{k=1}^l\left|v_k\right|{\omega }_k^{\rho \zeta }\right)}{{\rho }^{\zeta }\Gamma (\zeta +1)}+\frac{|\widehat{{\delta }_3}|{\sum }_{i=1}^j\left|{ϵ}_i\right|{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}\right\}\hfill \\ & +{\mathcal{U}}_1\left\{\frac{\left(|{\delta }_4|{\sum }_{k=1}^l|{\nu }_k|{\vartheta }_k^{\rho \xi }+\left|\widehat{{\delta }_3}\right|{\mathcal{T}}^{\rho \xi }\right)}{{\rho }^{\xi }\Gamma (\xi +1)}+\frac{|\widehat{{\delta }_4}|{\sum }_{i=1}^j\left|{\pi }_i\right|{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)}\right\}\hfill \\ & ={\widehat{\mathsf{\Omega }}}_2.\hfill \end{array}$$

Consequently,

$$\begin{array}{c}\hfill \left|\right|({\Delta }_{1,1},{\Delta }_{2,1})(p,q)\left|\right|\le {\widehat{\mathsf{\Omega }}}_1+\widehat{{\mathsf{\Omega }}_2}.\end{array}$$

Therefore, the set (${\Delta }_{1,1},{\Delta }_{2,1}$)${\mathcal{B}}_{\phi }$ is uniformly bounded. The following step will demonstrate that the set (${\Delta }_{1,1},{\Delta }_{2,1}$)${\mathcal{B}}_{\phi }$ is equicontinuous. For ${\tau }_1,{\tau }_2\in \mathcal{E}$ with ${\tau }_1<{\tau }_2$ and for any $(p,q)\in {\mathcal{B}}_{\phi }$, we obtain

$$\begin{array}{cc}\hfill & |{\Delta }_{1,1}(p,q)\left({\tau }_2\right)-{\Delta }_{1,1}(p,q)\left({\tau }_1\right)|\hfill \\ \hfill \le & |{\delta }_1\left({\tau }_2\right)-{\delta }_1\left({\tau }_1\right)|\sum _{k=1}^l|v_k|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left|f({\omega }_k,p\left({\omega }_k\right),q\left({\omega }_k\right))\right|\hfill \\ & +|{\delta }_2\left({\tau }_2\right)-{\delta }_2\left({\tau }_1\right)|\sum _{k=1}^l|{\nu }_k|{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f({\vartheta }_k,p\left({\vartheta }_k\right),q\left({\vartheta }_k\right))\right|\hfill \\ & +|\widehat{{\delta }_1}\left({\tau }_2\right)-\widehat{{\delta }_1}\left({\tau }_1\right)|\left(\sum _{i=1}^j|{ϵ}_i|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }|g({\omega }_i,p\left({\omega }_i\right),q\left({\omega }_i\right))|{+}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|f(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right|\right)\hfill \\ & +|\widehat{{\delta }_2}\left({\tau }_2\right)-\widehat{{\delta }_2}\left({\tau }_1\right)|\left(\sum _{i=1}^j|{\pi }_i|{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }|f({\sigma }_i,p\left({\sigma }_i\right),q\left({\sigma }_i\right))|{+}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left|g(\mathcal{T},p\left(\mathcal{T}\right),q\left(\mathcal{T}\right))\right|\right)\hfill \\ \hfill \le & |{\delta }_1\left({\tau }_2\right)-{\delta }_1\left({\tau }_1\right)|\frac{{\mathcal{U}}_2{\sum }_{k=1}^l\left|v_k\right|{\omega }_k^{\rho \zeta }}{{\rho }^{\zeta }\Gamma (\zeta +1)}+|{\delta }_2\left({\tau }_2\right)-{\delta }_2\left({\tau }_1\right)|\frac{{\mathcal{U}}_1{\sum }_{k=1}^l\left|{\nu }_k\right|{\vartheta }_k^{\rho \xi }}{{\rho }^{\xi }\Gamma (\xi +1)}\hfill \\ & +|\widehat{{\delta }_1}\left({\tau }_2\right)-\widehat{{\delta }_1}\left({\tau }_1\right)|\left[\frac{{\mathcal{U}}_2{\sum }_{i=1}^j\left|{ϵ}_i\right|{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}+\frac{{\mathcal{U}}_1{\mathcal{T}}^{\rho \xi }}{{\rho }^{\xi }\Gamma (\xi +1)}\right]\hfill \\ & +|\widehat{{\delta }_2}\left({\tau }_2\right)-\widehat{{\delta }_2}\left({\tau }_1\right)|\left[\frac{{\mathcal{U}}_1{\sum }_{i=1}^j\left|{\pi }_i\right|{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)}+\frac{{\mathcal{U}}_2{\mathcal{T}}^{\rho \zeta }}{{\rho }^{\zeta }\Gamma (\zeta +1)}\right].\hfill \end{array}$$

Likewise, we obtain

$$\begin{array}{cc}\hfill & |{\Delta }_{2,1}(p,q)\left({\tau }_2\right)-{\Delta }_{2,1}(p,q)\left({\tau }_1\right)|\hfill \\ \hfill \le & |{\delta }_3\left({\tau }_2\right)-{\delta }_3\left({\tau }_1\right)|\frac{{\mathcal{U}}_2{\sum }_{k=1}^l\left|v_k\right|{\omega }_k^{\rho \zeta }}{{\rho }^{\zeta }\Gamma (\zeta +1)}+|{\delta }_4\left({\tau }_2\right)-{\delta }_4\left({\tau }_1\right)|\frac{{\mathcal{U}}_1{\sum }_{k=1}^l\left|{\nu }_k\right|{\vartheta }_k^{\rho \xi }}{{\rho }^{\xi }\Gamma (\xi +1)}\hfill \end{array}$$

$$\begin{array}{cc}& +|\widehat{{\delta }_3}\left({\tau }_2\right)-\widehat{{\delta }_3}\left({\tau }_1\right)|\left[\frac{{\mathcal{U}}_2{\sum }_{i=1}^j\left|{ϵ}_i\right|{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}+\frac{{\mathcal{U}}_1{\mathcal{T}}^{\rho \xi }}{{\rho }^{\xi }\Gamma (\xi +1)}\right]\hfill \\ & +|\widehat{{\delta }_4}\left({\tau }_2\right)-\widehat{{\delta }_4}\left({\tau }_1\right)|\left[\frac{{\mathcal{U}}_1{\sum }_{i=1}^j\left|{\pi }_i\right|{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)}+\frac{{\mathcal{U}}_2{\mathcal{T}}^{\rho \zeta }}{{\rho }^{\zeta }\Gamma (\zeta +1)}\right].\hfill \end{array}$$

Therefore, $|({\Delta }_{1,1},{\Delta }_{2,1}\left({\tau }_2\right))-({\Delta }_{1,1},{\Delta }_{2,1}\left({\tau }_1\right))|\to 0$ as ${\tau }_2\to {\tau }_1$ independent of $(p,q)\in {\mathcal{B}}_{\phi }$. Thus, the set $({\Delta }_{1,1},{\Delta }_{2,1}){\mathcal{B}}_{\phi }$ is equicontinuous. As an outcome, the Arzela–Ascoli theorem implies that the operator $({\Delta }_{1,1},{\Delta }_{2,1})$ is compact on $\mathcal{B}\phi$. Krasnoselskii’s theorem leads us to problem (6) and (7) having at least one solution on $\mathcal{E}$. □

Example 1.

Consider the following Liouville–Caputo type generalized FDEs coupled system:

$$\left\{\begin{array}{c}{}_C^{\frac{3}{4}}{\mathcal{D}}_{0+}^{\frac{3}{4}}p\left(\tau \right)=f(\tau ,p\left(\tau \right),q\left(\tau \right)),\tau \in \mathcal{E}:=[0,1],\hfill \\ {}_C^{\frac{3}{4}}{\mathcal{D}}_{0+}^{\frac{31}{20}}q\left(\tau \right)=g(\tau ,p\left(\tau \right),q\left(\tau \right)),\tau \in \mathcal{E}:=[0,1],\hfill \end{array}\right.$$

(79)

supplemented with boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\gamma p\left(0\right)=\frac{1}{3}q\left(\frac{3}{10}\right)+\frac{2}{3}q\left(\frac{9}{20}\right),\gamma q\left(0\right)=\frac{1}{4}p\left(\frac{3}{20}\right)+\frac{3}{4}q\left(\frac{1}{4}\right),\hfill \\ p\left(1\right)={\frac{1}{6}}^{\frac{3}{4}}{\mathcal{I}}^{\frac{13}{20}}q\left(\frac{7}{10}\right)+{\frac{5}{6}}^{\frac{3}{4}}{\mathcal{I}}^{\frac{13}{20}}q\left(\frac{9}{20}\right),q\left(1\right)={\frac{1}{7}}^{\frac{3}{4}}{\mathcal{I}}^{\frac{17}{20}}p\left(\frac{1}{2}\right)+{\frac{6}{7}}^{\frac{3}{4}}{\mathcal{I}}^{\frac{17}{20}}p\left(\frac{13}{20}\right),\hfill \end{array}\right.\end{array}$$

(80)

where $\xi =\frac{5}{4},\zeta =\frac{31}{20},\rho =\frac{3}{4},\mathcal{T}=1,v_1=\frac{1}{3},v_2=\frac{2}{3},{\omega }_1=\frac{3}{10},{\omega }_2=\frac{9}{20},{\nu }_1=\frac{1}{4},{\nu }_2=\frac{3}{4},{\vartheta }_1=\frac{3}{20},{\vartheta }_2=\frac{1}{4},{ϵ}_1=\frac{1}{6},{ϵ}_2=\frac{5}{6},{\varpi }_1=\frac{7}{10},{\varpi }_2=\frac{9}{20},{\pi }_1=\frac{1}{7},{\pi }_2=\frac{6}{7},{\sigma }_1=\frac{1}{2},{\sigma }_2=\frac{13}{20},\varsigma =\frac{13}{20},\varrho =\frac{17}{20}$ and

$$\begin{array}{c}\hfill f(\tau ,p\left(\tau \right),q\left(\tau \right))=\frac{(1+\tau )}{30}\left(\frac{\left|p\right(\tau \left)\right|}{1+\left|p\right(\tau \left)\right|}+\frac{1}{3}cos\left(q\left(\tau \right)\right)+3\tau \right),\end{array}$$

(81)

$$\begin{array}{c}\hfill g(\tau ,p\left(\tau \right),q\left(\tau \right))=\frac{e^{-\tau }}{25}\left(\frac{\sqrt{\tau }+1}{5}+\frac{1}{6}cos\left(p\left(\tau \right)\right)+\frac{\left|q\right(\tau \left)\right|}{1+\left|q\right(\tau \left)\right|}\right).\end{array}$$

(82)

With ${\psi }_0=\frac{1}{10},{\psi }_1=\frac{1}{30},{\psi }_2=\frac{1}{90}\widehat{{\psi }_0}=\frac{1}{125},\widehat{{\psi }_1}=\frac{1}{25}$, and $\widehat{{\psi }_2}=\frac{1}{150}$, the functions f and g clearly satisfy the $\left({\mathcal{A}}_1\right)$ condition. Next, we find that $\left({\mathcal{J}}_1\right)=3.576795680900842,\left({\mathcal{K}}_1\right)=2.316467121169555,{\mathcal{J}}_2=2:312227284086664,{\mathcal{K}}_2=3.4501351580508066,{\mathcal{J}}_i,{\mathcal{K}}_i(i=1,2)$ are, respectively, given by (53)–(56), based on the data available. Thus, ${\psi }_1({\mathcal{J}}_1+{\mathcal{J}}_2)+\widehat{{\psi }_1}({\mathcal{K}}_1+{\mathcal{K}}_2)\approxeq 0.426964856668398<1,{\psi }_2({\mathcal{J}}_1+{\mathcal{J}}_2)+\widehat{{\psi }_2}({\mathcal{K}}_1+{\mathcal{K}}_2)\approxeq 0.10387760369466359<1$, all the conditions of Theorem 1 are satisfied, and there is at least one solution for problems (79) and (80) on $[0,1]$ with f and g given by (81) and (82), respectively.

In addition, we will use

$$\begin{array}{c}\hfill f(\tau ,p\left(\tau \right),q\left(\tau \right))=\frac{\tau }{3}+\frac{3}{4(\tau +16)}+\frac{\left|p\right(\tau \left)\right|}{1+\left|p\right(\tau \left)\right|}+\frac{2}{75}cos\left(q\left(\tau \right)\right),\end{array}$$

(83)

$$\begin{array}{c}\hfill g(\tau ,p\left(\tau \right),q\left(\tau \right))=\frac{(1+e^{-\tau })}{4}+\frac{19}{400}cos\left(p\left(\tau \right)\right)+\frac{1}{60}\frac{\left|q\right(\tau \left)\right|}{1+\left|q\right(\tau \left)\right|}.\end{array}$$

(84)

to demonstrate Theorem 2. It is simple to demonstrate that f and g are continuous and satisfy the assumption $\left({\mathcal{A}}_2\right)$ with ${\varphi }_1=\frac{3}{64},{\varphi }_1=\frac{2}{75},\widehat{{\varphi }_1}=\frac{19}{400}$ and $\widehat{{\varphi }_2}=\frac{1}{60}$. All the assumptions of Theorem 2 are also satisfied with $({\mathcal{J}}_1+{\mathcal{J}}_2)({\varphi }_1+{\varphi }_2)+({\mathcal{K}}_1+{\mathcal{K}}_2)(\widehat{{\varphi }_1}+\widehat{{\varphi }_2})\approxeq 0.6771047066230428<1$. As a result, Theorem 2 holds true, and problems (79) and (80) with f and g given by (83) and (84), respectively, have a unique solution on [0,1].

4. Ulam–Hyers Stability Results for the Problem (6) and (7)

The U–H stability of the solutions to BVP (6) and (7) will be discussed in this section using the integral representation of their solutions defined by

$$\begin{array}{c}\hfill p\left(\tau \right)={\Delta }_1(p,q)\left(\tau \right),q\left(\tau \right)={\Delta }_2(p,q)\left(\tau \right),\end{array}$$

(85)

where ${\Delta }_1$ and ${\Delta }_2$ are given by (51) and (52). Consider the following definitions of nonlinear operators

$$\begin{array}{c}\hfill {\mathcal{H}}_1,{\mathcal{H}}_2\in \mathcal{C}(\mathcal{E},\mathbb{R})\times \mathcal{C}(\mathcal{E},\mathbb{R})\to \mathcal{C}(\mathcal{E},\mathbb{R})\end{array}$$

$$\left\{\begin{array}{c}{}_C^{\rho }{\mathcal{D}}_{0+}^{\xi }p\left(\tau \right)-f(\tau ,p\left(\tau \right),q\left(\tau \right))={\mathcal{H}}_1(p,q)\left(\tau \right),\tau \in \mathcal{E},\hfill \\ {}_C^{\rho }{\mathcal{D}}_{0+}^{\zeta }q\left(\tau \right)-g(\tau ,p\left(\tau \right),q\left(\tau \right))={\mathcal{H}}_1(p,q)\left(\tau \right),\tau \in \mathcal{E},\hfill \end{array}\right.$$

It considered the following inequalities for some real constants $\widehat{{\lambda }_1},{\widehat{\lambda }}_2>0:$

$$\begin{array}{c}\hfill \left|\right|{\mathcal{H}}_1(p,q)\left|\right|\le \widehat{{\lambda }_1},\left|\right|{\mathcal{H}}_2(p,q)\left|\right|\le \widehat{{\lambda }_2}.\end{array}$$

(86)

Definition 4.

The coupled system (6) and (7) is said to be U–H stable for real constants ${\mathcal{V}}_1,{\mathcal{V}}_2>0$ if there exists a unique solution $(p,q)\in \mathcal{C}(\mathcal{E},\mathbb{R})$ of problems (6) and (7) with

$$\begin{array}{c}\hfill \left|\right|(p,q)-(p^{*},q^{*})\left|\right|\le {\mathcal{V}}_1\widehat{{\lambda }_1}+{\mathcal{V}}_2\widehat{{\lambda }_2}\end{array}$$

$\forall (p,q)\in \mathcal{C}(\mathcal{E},\mathbb{R})$.

Theorem 4.

Assume that $\left({\mathcal{A}}_2\right)$ holds. Then problems (6) and (7) are U–H stable.

Proof. 

Let $(p,q)\in \mathcal{C}(\mathcal{E},\mathbb{R})\times \mathcal{C}(\mathcal{E},\mathbb{R})$ be the (6) and (7) solution of the problem that satisfies (51) and (52). Let $(p,q)$ be any solution that meets the condition (86):

$$\left\{\begin{array}{c}{}_C^{\rho }{\mathcal{D}}_{0+}^{\xi }p\left(\tau \right)=f(\tau ,p\left(\tau \right),q\left(\tau \right))+{\mathcal{H}}_1(p,q)\left(\tau \right),\tau \in \mathcal{E},\hfill \\ {}_C^{\rho }{\mathcal{D}}_{0+}^{\zeta }q\left(\tau \right)=g(\tau ,p\left(\tau \right),q\left(\tau \right))+{\mathcal{H}}_1(p,q)\left(\tau \right),\tau \in \mathcal{E},\hfill \end{array}\right.$$

so,

$$\begin{array}{cc}\hfill p^{*}\left(\tau \right)& ={\Delta }_1(p^{*},q^{*})\left(\tau \right){+}^{\rho }{\mathcal{I}}_{0+}^{\xi }{\mathcal{H}}_1(p,q)\left(\tau \right)+{\delta }_1\left(\tau \right)\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }{\mathcal{H}}_2(p,q)\left({\omega }_k\right)\hfill \\ & +{\delta }_2\left(\tau \right)\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }{\mathcal{H}}_1(p,q)\left({\vartheta }_k\right)\hfill \\ & +\widehat{{\delta }_1}\left(\tau \right)\left[\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }{\mathcal{H}}_2(p,q)\left({\varpi }_i\right){-}^{\rho }{\mathcal{I}}_{0+}^{\xi }{\mathcal{H}}_1(p,q)\left(\mathcal{T}\right)\right]\hfill \\ & +\widehat{{\delta }_2}\left(\tau \right)\left[\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }{\mathcal{H}}_1(p,q)\left({\sigma }_i\right){-}^{\rho }{\mathcal{I}}_{0+}^{\zeta }{\mathcal{H}}_2(p,q)\left(\mathcal{T}\right)\right].\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill & |{\Delta }_1(p^{*},q^{*})\left(\tau \right)-p^{*}\left(\tau \right)|\hfill \\ \hfill \le & {}^{\rho }{\mathcal{I}}_{0+}^{\xi }|{\mathcal{H}}_1(p,q)\left(\tau \right)|+|{\delta }_1\left(\tau \right)|\sum _{k=1}^l|v_k|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left|{\mathcal{H}}_2(p,q)\left({\omega }_k\right)\right|\hfill \\ & +|{\delta }_2\left(\tau \right)|\sum _{k=1}^l|{\nu }_k|{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|{\mathcal{H}}_1(p,q)\left({\vartheta }_k\right)\right|\hfill \\ & +|\widehat{{\delta }_1}\left(\tau \right)|\left[\sum _{i=1}^j|{ϵ}_i|{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }|{\mathcal{H}}_2(p,q)\left({\varpi }_i\right)|{+}^{\rho }{\mathcal{I}}_{0+}^{\xi }\left|{\mathcal{H}}_1(p,q)\left(\mathcal{T}\right)\right|\right]\hfill \\ & +|\widehat{{\delta }_2}\left(\tau \right)|\left[\sum _{i=1}^j|{\pi }_i|{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }|{\mathcal{H}}_1(p,q)\left({\sigma }_i\right)|{+}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\left|{\mathcal{H}}_2(p,q)\left(\mathcal{T}\right)\right|\right]\hfill \\ & \le {\widehat{\lambda }}_1\left\{\frac{|\widehat{{\delta }_2}|{\sum }_{i=1}^j\left|{\pi }_i\right|{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)}+\frac{\left({\mathcal{T}}^{\rho \xi }(1+|\widehat{{\delta }_1}\left|\right)+|{\delta }_2|{\sum }_{k=1}^l\left|{\nu }_k\right|{\vartheta }_k^{\rho \xi }\right)}{{\rho }^{\xi }\Gamma (\xi +1)}\right\}\hfill \\ & +{\widehat{\lambda }}_2\left\{\frac{|\widehat{{\delta }_1}|{\sum }_{i=1}^j\left|{ϵ}_i\right|{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}+\frac{\left(|{\delta }_1|{\sum }_{k=1}^l|v_k|{\omega }_k^{\rho \zeta }+\left|\widehat{{\delta }_2}\right|{\mathcal{T}}^{\rho \zeta }\right)}{{\rho }^{\zeta }\Gamma (\zeta +1)}\right\}\hfill \\ \hfill \le & {\mathcal{J}}_1\widehat{{\lambda }_1}+{\mathcal{K}}_1\widehat{{\lambda }_2}.\hfill \end{array}$$

Similarly, we obtain

$$\begin{array}{cc}\hfill & |{\Delta }_2(p^{*},q^{*})\left(\tau \right)-q^{*}\left(\tau \right)|\hfill \\ \hfill \le & \widehat{{\lambda }_2}\left\{\frac{\left({\mathcal{T}}^{\rho \zeta }(1+|\widehat{{\delta }_4}\left|\right)+|{\delta }_3|{\sum }_{k=1}^l\left|v_k\right|{\omega }_k^{\rho \zeta }\right)}{{\rho }^{\zeta }\Gamma (\zeta +1)}+\frac{|\widehat{{\delta }_3}|{\sum }_{i=1}^j\left|{ϵ}_i\right|{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}\right\}\hfill \\ & +\widehat{{\lambda }_1}\left\{\frac{\left(|{\delta }_4|{\sum }_{k=1}^l|{\nu }_k|{\vartheta }_k^{\rho \xi }+\left|\widehat{{\delta }_3}\right|{\mathcal{T}}^{\rho \xi }\right)}{{\rho }^{\xi }\Gamma (\xi +1)}+\frac{|\widehat{{\delta }_4}|{\sum }_{i=1}^j\left|{\pi }_i\right|{\sigma }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)}\right\}\hfill \\ \hfill \le & {\mathcal{J}}_2{\widehat{\lambda }}_1+{\mathcal{K}}_2{\widehat{\lambda }}_2,\hfill \end{array}$$

where ${\mathcal{J}}_1,{\mathcal{K}}_1,{\mathcal{J}}_2$ and ${\mathcal{K}}_2$ are defined in (53)–(56), respectively. As an outcome, we deduce from operator $\Delta$’s fixed-point property, which is defined by (51) and (52),

$$\begin{array}{cc}\hfill |p\left(\tau \right)-p^{*}\left(\tau \right)|=& |p\left(\tau \right)-{\Delta }_1(p^{*},q^{*})\left(\tau \right)+{\Delta }_1(p^{*},q^{*})\left(\tau \right)-p^{*}\left(\tau \right)|\hfill \\ \hfill \le & |{\Delta }_1(p,q)\left(\tau \right)-{\Delta }_1(p^{*},q^{*})\left(\tau \right)|+|{\Delta }_1(p^{*},q^{*})\left(\tau \right)-p^{*}\left(\tau \right)|\hfill \\ \hfill \le & (({\mathcal{J}}_1{\varphi }_1+{\mathcal{K}}_1\widehat{{\varphi }_1})+({\mathcal{J}}_1{\varphi }_2+{\mathcal{K}}_1\widehat{{\varphi }_2}))\left|\right|(p,q)-(p^{*},q^{*})\left|\right|\hfill \\ & +{\mathcal{J}}_1\widehat{{\lambda }_1}+{\mathcal{K}}_1\widehat{{\lambda }_2}.\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill |q\left(\tau \right)-q^{*}\left(\tau \right)|=& |q\left(\tau \right)-{\Delta }_2(p^{*},q^{*})\left(\tau \right)+{\Delta }_2(p^{*},q^{*})\left(\tau \right)-q^{*}\left(\tau \right)|\hfill \\ \hfill \le & |{\Delta }_2(p,q)\left(\tau \right)-{\Delta }_2(p^{*},q^{*})\left(\tau \right)|+|{\Delta }_2(p^{*},q^{*})\left(\tau \right)-q^{*}\left(\tau \right)|\hfill \\ \hfill \le & (({\mathcal{J}}_2{\varphi }_1+{\mathcal{K}}_2\widehat{{\varphi }_1})+({\mathcal{J}}_2{\varphi }_2+{\mathcal{K}}_2\widehat{{\varphi }_2}))\left|\right|(p,q)-(p^{*},q^{*})\left|\right|\hfill \\ & +{\mathcal{J}}_2\widehat{{\lambda }_1}+{\mathcal{K}}_2\widehat{{\lambda }_2}.\hfill \end{array}$$

(88)

From the above Equations (87) and (88), it follows that

$$\begin{array}{cc}\hfill \left|\right|(p,q)-(p^{*},q^{*})\left|\right|\le & ({\mathcal{J}}_1+{\mathcal{J}}_2)\widehat{{\lambda }_1}+({\mathcal{K}}_1+{\mathcal{K}}_2)\widehat{{\lambda }_2}\hfill \\ & +(({\mathcal{J}}_1+{\mathcal{J}}_2)({\varphi }_1+{\varphi }_2)+({\mathcal{K}}_1+{\mathcal{K}}_2)(\widehat{{\varphi }_1}+\widehat{{\varphi }_2}))\left|\right|(p,q)-(p^{*},q^{*})\left|\right|.\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left|\right|(p,q)-(p^{*},q^{*})\left|\right|\le & \frac{({\mathcal{J}}_1+{\mathcal{J}}_2)\widehat{{\lambda }_1}+({\mathcal{K}}_1+{\mathcal{K}}_2)\widehat{{\lambda }_2}}{1-(({\mathcal{J}}_1+{\mathcal{J}}_2)({\varphi }_1+{\varphi }_2)+({\mathcal{K}}_1+{\mathcal{K}}_2)(\widehat{{\varphi }_1}+\widehat{{\varphi }_2}))}\hfill \\ \hfill \le & {\mathcal{V}}_1\widehat{{\lambda }_1}+{\mathcal{V}}_2\widehat{{\lambda }_2},\hfill \end{array}$$

with

$$\begin{array}{c}\hfill {\mathcal{V}}_1=\frac{{\mathcal{J}}_1+{\mathcal{J}}_2}{1-(({\mathcal{J}}_1+|{\mathcal{J}}_2)({\varphi }_1+{\varphi }_2)+({\mathcal{K}}_1+|{\mathcal{K}}_2)(\widehat{{\varphi }_1}+\widehat{{\varphi }_2}))},\end{array}$$

$$\begin{array}{c}\hfill {\mathcal{V}}_2=\frac{{\mathcal{K}}_1+{\mathcal{K}}_2}{1-(({\mathcal{J}}_1+{\mathcal{J}}_2)({\varphi }_1+{\varphi }_2)+({\mathcal{K}}_1+{\mathcal{K}}_2)(\widehat{{\varphi }_1}+\widehat{{\varphi }_2}))}.\end{array}$$

Hence, problems (6) and (7) are U–H stable. □

Example 2.

Consider the following Liouville–Caputo type generalized FDEs coupled system:

$$\left\{\begin{array}{c}{}_C^{\frac{19}{20}}{\mathcal{D}}_{0+}^{\frac{5}{4}}p\left(\tau \right)=\frac{\sqrt{\tau }}{2}+\frac{1}{5(\tau +25)}\frac{\left|p\right(\tau \left)\right|}{1+\left|p\right(\tau \left)\right|}+\frac{3}{80}cos\left(q\left(\tau \right)\right),\tau \in [0,1],\hfill \\ {}_C^{\frac{19}{20}}{\mathcal{D}}_{0+}^{\frac{31}{20}}q\left(\tau \right)=\frac{\tau }{5}+\frac{17}{300}cos\left(p\left(\tau \right)\right)+\frac{1}{70}\frac{\left|q\right(\tau \left)\right|}{1+\left|q\right(\tau \left)\right|},\tau \in [0,1],\hfill \end{array}\right.$$

(89)

supplemented with boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\gamma p\left(0\right)=\frac{1}{3}q\left(\frac{3}{10}\right)+\frac{2}{3}q\left(\frac{9}{20}\right),\gamma q\left(0\right)=\frac{1}{4}p\left(\frac{3}{20}\right)+\frac{3}{4}q\left(\frac{1}{4}\right),\hfill \\ p\left(1\right)={\frac{1}{6}}^{\frac{3}{4}}{\mathcal{I}}^{\frac{13}{20}}q\left(\frac{7}{10}\right)+{\frac{5}{6}}^{\frac{3}{4}}{\mathcal{I}}^{\frac{13}{20}}q\left(\frac{9}{20}\right),q\left(1\right)={\frac{1}{7}}^{\frac{3}{4}}{\mathcal{I}}^{\frac{17}{20}}p\left(\frac{1}{2}\right)+{\frac{6}{7}}^{\frac{3}{4}}{\mathcal{I}}^{\frac{17}{20}}p\left(\frac{13}{20}\right),\hfill \end{array}\right.\end{array}$$

(90)

where $\xi =\frac{5}{4},\zeta =\frac{31}{20},\rho =\frac{3}{4},\mathcal{T}=1,v_1=\frac{1}{3},v_2=\frac{2}{3},{\omega }_1=\frac{3}{10},{\omega }_2=\frac{9}{20},{\nu }_1=\frac{1}{4},{\nu }_2=\frac{3}{4},{\vartheta }_1=\frac{3}{20},{\vartheta }_2=\frac{1}{4},{ϵ}_1=\frac{1}{6},{ϵ}_2=\frac{5}{6},{\varpi }_1=\frac{7}{10},{\varpi }_2=\frac{9}{20},{\pi }_1=\frac{1}{7},{\pi }_2=\frac{6}{7},{\sigma }_1=\frac{1}{2},{\sigma }_2=\frac{13}{20},\varsigma =\frac{13}{20},\varrho =\frac{17}{20}$ and

$$\begin{array}{c}\hfill |f(\tau ,p_1\left(\tau \right),q_1\left(\tau \right))-f(\tau ,p_2\left(\tau \right),q_2\left(\tau \right))|=\frac{1}{125}|p_1\left(\tau \right)-p_2\left(\tau \right)|+\frac{3}{80}|q_1\left(\tau \right)-q_2\left(\tau \right)|\end{array}$$

(91)

$$\begin{array}{c}\hfill |g(\tau ,p_1\left(\tau \right),q_1\left(\tau \right))-g(\tau ,p_2\left(\tau \right),q_2\left(\tau \right))|=\frac{17}{300}|p_1\left(\tau \right)-p_2\left(\tau \right)|+\frac{1}{70}|q_1\left(\tau \right)-q_2\left(\tau \right)|.\end{array}$$

(92)

With ${\psi }_1=\frac{1}{125},{\psi }_2=\frac{3}{80},\widehat{{\psi }_1}=\frac{17}{300}$, and $\widehat{{\psi }_2}=\frac{1}{70}$, the functions f and g clearly satisfy the $\left({\mathcal{A}}_2\right)$ condition. Next, we find that $\left({\mathcal{J}}_1\right)=3.576795680900842,\left({\mathcal{K}}_1\right)=2.316467121169555,{\mathcal{J}}_2=2.312227284086664,{\mathcal{K}}_2=3.4501351580508066,{\mathcal{J}}_i,{\mathcal{K}}_i(i=1,2)$ are, respectively, given by (53)–(56), based on the data available. Thus $(({\mathcal{J}}_1+{\mathcal{J}}_2)({\varphi }_1+{\varphi }_2)+({\mathcal{K}}_1+{\mathcal{K}}_2)(\widehat{{\varphi }_1}+\widehat{{\varphi }_2}))\approxeq 0,8313148592<1$, all the conditions of Theorem 5.2 are satisfied, and there is a unique solution for problems (89) and (90) on $[0,1]$, which is stable for Ulam–Hyers, with f and g given by (91) and (92), respectively.

5. Existence Results for the Problem (6) and (8)

We introduce the following notations for computational ease:

$${\mathcal{E}}_1=\sum _{k=1}^l{v}_k,{\mathcal{E}}_2=\sum _{k=1}^l{v}_k\frac{{\omega }_k^{\rho }}{\rho },{\mathcal{E}}_3=\sum _{i=1}^j{ϵ}_i\frac{{\varpi }_i^{\rho \varsigma }}{{\rho }^{\varsigma }\Gamma (\varsigma +1)},{\mathcal{E}}_4=\sum _{i=1}^j{ϵ}_i\frac{{\varpi }_i^{\rho (\varsigma +1)}}{{\rho }^{\varsigma +1}\Gamma (\varsigma +2)},{\mathcal{E}}_5=\frac{{\mathcal{T}}^{\rho }}{\rho },$$

(93)

$$\begin{array}{c}\hfill {\mathcal{F}}_1=\sum _{k=1}^l{\nu }_k,{\mathcal{F}}_2=\sum _{k=1}^l{\nu }_k\frac{{\omega }_k^{\rho }}{\rho },{\mathcal{F}}_3=\sum _{i=1}^j{\pi }_i\frac{{\varpi }_i^{\rho \varrho }}{{\rho }^{\varrho }\Gamma (\varrho +1)},{\mathcal{F}}_4=\sum _{i=1}^j{\pi }_i\frac{{\varpi }_i^{\rho (\varrho +1)}}{{\rho }^{\varrho +1}\Gamma (\varrho +2)},\end{array}$$

(94)

$$\begin{array}{cc}& {\mathcal{G}}_1=1-{\mathcal{E}}_2{\mathcal{F}}_2,{\mathcal{G}}_2={\mathcal{E}}_5-{\mathcal{E}}_4{\mathcal{F}}_3,{\mathcal{G}}_3={\mathcal{F}}_5-{\mathcal{E}}_5{\mathcal{F}}_4,\hfill \end{array}$$

(95)

$$\begin{array}{cc}& \widehat{{\mathcal{G}}_1}={\mathcal{E}}_4-{\mathcal{E}}_3{\mathcal{E}}_5,\widehat{{\mathcal{G}}_2}={\mathcal{E}}_5^2-{\mathcal{E}}_4{\mathcal{F}}_4,\widehat{{\mathcal{G}}_3}={\mathcal{F}}_5-{\mathcal{E}}_5{\mathcal{F}}_4,\hfill \\ & {\mathsf{\Lambda }}_1={\mathcal{G}}_1{\mathcal{G}}_2+{\mathcal{F}}_1{\mathcal{E}}_2\widehat{{\mathcal{G}}_2},{\mathsf{\Lambda }}_2={\mathcal{G}}_1{\mathcal{G}}_3+{\mathcal{F}}_1\widehat{{\mathcal{G}}_2},{\mathsf{\Lambda }}_3={\mathcal{E}}_1\widehat{{\mathcal{G}}_2}+{\mathcal{G}}_1\widehat{{\mathcal{G}}_1},\hfill \end{array}$$

(96)

$$\begin{array}{cc}& {\mathsf{\Lambda }}_4={\mathcal{G}}_1{\mathcal{G}}_3+{\mathcal{E}}_1{\mathcal{F}}_2\widehat{{\mathcal{G}}_2},\mathsf{\Lambda }={\mathsf{\Lambda }}_2{\mathsf{\Lambda }}_3-{\mathsf{\Lambda }}_1{\mathsf{\Lambda }}_4,\hfill \\ & {\alpha }_1=({\mathsf{\Lambda }}_4-{\mathsf{\Lambda }}_3{\mathcal{F}}_2)\widehat{{\mathcal{G}}_2},{\alpha }_2=({\mathsf{\Lambda }}_4{\mathcal{E}}_2-{\mathsf{\Lambda }}_3)\widehat{{\mathcal{G}}_2},{\alpha }_3=({\mathsf{\Lambda }}_2-{\mathsf{\Lambda }}_1{\mathcal{F}}_2)\widehat{{\mathcal{G}}_2},\hfill \end{array}$$

(97)

$$\begin{array}{cc}& {\alpha }_4=({\mathsf{\Lambda }}_2{\mathcal{E}}_2-{\mathsf{\Lambda }}_1)\widehat{{\mathcal{G}}_2},\hfill \\ & {\beta }_1=({\mathsf{\Lambda }}_4{\mathcal{E}}_5-{\mathsf{\Lambda }}_3{\mathcal{F}}_4){\mathcal{G}}_1,{\beta }_2=({\mathsf{\Lambda }}_4{\mathcal{E}}_4-{\mathsf{\Lambda }}_3{\mathcal{E}}_5){\mathcal{G}}_1,{\beta }_3=({\mathsf{\Lambda }}_2{\mathcal{E}}_5-{\mathsf{\Lambda }}_1{\mathcal{F}}_4){\mathcal{G}}_1,\hfill \end{array}$$

(98)

$$\begin{array}{cc}& {\beta }_4=({\mathsf{\Lambda }}_2{\mathcal{E}}_4-{\mathsf{\Lambda }}_1{\mathcal{E}}_5){\mathcal{G}}_1,\hfill \\ & \widehat{{\alpha }_1}=1+\frac{{\mathcal{F}}_1{\mathcal{E}}_2{\alpha }_1-{\mathcal{E}}_1{\alpha }_3}{\mathsf{\Lambda }},\widehat{{\alpha }_2}={\mathcal{E}}_2+\frac{{\mathcal{F}}_1{\mathcal{E}}_2{\alpha }_2-{\mathcal{E}}_1{\alpha }_4}{\mathsf{\Lambda }},\hfill \end{array}$$

(99)

$$\begin{array}{cc}& \widehat{{\alpha }_3}={\mathcal{F}}_2+\frac{{\mathcal{F}}_1{\alpha }_1-{\mathcal{E}}_1{\mathcal{F}}_2{\alpha }_3}{\mathsf{\Lambda }},\widehat{{\alpha }_4}=1+\frac{{\mathcal{F}}_1{\alpha }_2-{\mathcal{E}}_1{\mathcal{F}}_2{\alpha }_4}{\mathsf{\Lambda }},\hfill \\ & \widehat{{\beta }_1}=\frac{{\mathcal{E}}_1{\beta }_1-{\mathcal{F}}_1{\mathcal{E}}_2{\beta }_1}{\mathsf{\Lambda }},\widehat{{\beta }_2}=\frac{{\mathcal{E}}_1{\beta }_4-{\mathcal{F}}_1{\mathcal{E}}_2{\beta }_2}{\mathsf{\Lambda }},\hfill \end{array}$$

(100)

$$\begin{array}{cc}& \widehat{{\beta }_3}=\frac{{\mathcal{E}}_1{\mathcal{F}}_2{\beta }_3-{\mathcal{F}}_1{\beta }_1}{\mathsf{\Lambda }},\widehat{{\beta }_4}=\frac{{\mathcal{E}}_1{\mathcal{F}}_2{\beta }_4-{\mathcal{F}}_1{\beta }_2}{\mathsf{\Lambda }},\hfill \\ & {\delta }_1\left(\tau \right)=\left(\frac{{\alpha }_1}{\mathsf{\Lambda }}+\frac{{\tau }^{\rho }\widehat{{\alpha }_1}}{\rho {\mathcal{G}}_1}\right),{\delta }_2\left(\tau \right)=\left(\frac{{\alpha }_2}{\mathsf{\Lambda }}+\frac{{\tau }^{\rho }\widehat{{\alpha }_2}}{\rho {\mathcal{G}}_1}\right),{\delta }_3\left(\tau \right)=\left(\frac{{\alpha }_3}{\mathsf{\Lambda }}+\frac{{\tau }^{\rho }\widehat{{\alpha }_3}}{\rho {\mathcal{G}}_1}\right),\hfill \end{array}$$

(101)

$$\begin{array}{cc}& {\delta }_4\left(\tau \right)=\left(\frac{{\alpha }_4}{\mathsf{\Lambda }}+\frac{{\tau }^{\rho }\widehat{{\alpha }_4}}{\rho {\mathcal{G}}_1}\right),\hfill \\ & \widehat{{\delta }_1}\left(\tau \right)=\left(\frac{{\tau }^{\rho }\widehat{{\beta }_1}}{\rho {\mathcal{G}}_1}-\frac{{\beta }_1}{\mathsf{\Lambda }}\right),\widehat{{\delta }_2}\left(\tau \right)=\left(\frac{{\tau }^{\rho }\widehat{{\beta }_2}}{\rho {\mathcal{G}}_1}-\frac{{\beta }_2}{\mathsf{\Lambda }}\right),\widehat{{\delta }_3}\left(\tau \right)=\left(\frac{{\tau }^{\rho }\widehat{{\beta }_3}}{\rho {\mathcal{G}}_1}-\frac{{\beta }_3}{\mathsf{\Lambda }}\right),\hfill \end{array}$$

(102)

$$\begin{array}{cc}& \widehat{{\delta }_4}\left(\tau \right)=\left(\frac{{\tau }^{\rho }\widehat{{\beta }_4}}{\rho {\mathcal{G}}_1}-\frac{{\beta }_4}{\mathsf{\Lambda }}\right).\hfill \end{array}$$

(103)

Lemma 4.

Given the functions $\widehat{f},\widehat{g}\in C(0,\mathcal{T})\cap \mathcal{L}(0,\mathcal{T}),p,q\in \mathcal{A}{\mathcal{C}}_{\gamma }^2\left(\mathcal{E}\right)$ and $\mathsf{\Lambda }\ne 0.$ Then, the solution of the coupled BVP:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}_C^{\rho }{\mathcal{D}}_{0+}^{\xi }p\left(\tau \right)=\widehat{f}\left(\tau \right),\tau \in \mathcal{E}:=[0,\mathcal{T}],\hfill \\ {}_C^{\rho }{\mathcal{D}}_{0+}^{\zeta }q\left(\tau \right)=\widehat{g}\left(\tau \right),\tau \in \mathcal{E}:=[0,\mathcal{T}],\hfill \\ \gamma p\left(0\right)={\sum }_{k=1}^l{v}_{k}q\left({\omega }_k\right),\gamma q\left(0\right)={\sum }_{k=1}^l{\nu }_{k}p\left({\omega }_k\right),\hfill \\ p\left(\mathcal{T}\right)={\sum }_{i=1}^j{ϵ}_i^{\rho }{\mathcal{I}}_{0+}^{\varsigma }q\left({\varpi }_i\right),q\left(\mathcal{T}\right)={\sum }_{i=1}^j{\pi }_i^{\rho }{\mathcal{I}}_{0+}^{\varrho }p\left({\varpi }_i\right),\hfill \\ 0<{\omega }_1<\cdots <{\omega }_l<{\varpi }_1<\cdots <{\varpi }_j<\mathcal{T},\hfill \end{array}\right.\end{array}$$

(104)

is given by

$$\begin{array}{cc}\hfill p\left(\tau \right)& ={}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\tau \right)+{\delta }_1\left(\tau \right)\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left({\omega }_k\right)+{\delta }_2\left(\tau \right)\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left({\omega }_k\right)\hfill \\ & +\widehat{{\delta }_1}\left(\tau \right)\left[\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\mathcal{T}\right)\right]\hfill \\ & +\widehat{{\delta }_2}\left(\tau \right)\left[\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varsigma }\widehat{f}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left(\mathcal{T}\right)\right],\hfill \end{array}$$

(105)

and

$$\begin{array}{cc}\hfill q\left(\tau \right)& ={}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left(\tau \right)+{\delta }_3\left(\tau \right)\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left({\omega }_k\right)+{\delta }_4\left(\tau \right)\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left({\omega }_k\right)\hfill \\ & +\widehat{{\delta }_3}\left(\tau \right)\left[\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\mathcal{T}\right)\right]\hfill \\ & +\widehat{{\delta }_4}\left(\tau \right)\left[\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{f}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left(\mathcal{T}\right)\right].\hfill \end{array}$$

(106)

Proof. 

When ${}^{\rho }{\mathcal{I}}_{0+}^{\xi }{,}^{\rho }{\mathcal{I}}_{0+}^{\zeta }$ are applied to the FDEs in (104) and Lemma 4 is used the solution of the FDEs in (104), for $\tau \in \mathcal{E}$ is

$$\begin{array}{c}\hfill p\left(\tau \right)={}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\tau \right)+a_1+a_2\frac{{\tau }^{\rho }}{\rho }=\frac{{\rho }^{1-\xi }}{\Gamma \left(\xi \right)}{\int }_0^{\tau }{\theta }^{\rho -1}{({\tau }^{\rho }-{\theta }^{\rho })}^{\xi -1}\widehat{f}\left(\theta \right)d\theta +a_1+a_2\frac{{\tau }^{\rho }}{\rho },\end{array}$$

(107)

$$\begin{array}{c}\hfill q\left(\tau \right)={}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left(\tau \right)+b_1+b_2\frac{{\tau }^{\rho }}{\rho }=\frac{{\rho }^{1-\zeta }}{\Gamma \left(\zeta \right)}{\int }_0^{\tau }{\theta }^{\rho -1}{({\tau }^{\rho }-{\theta }^{\rho })}^{\zeta -1}\widehat{g}\left(\theta \right)d\theta +b_1+b_2\frac{{\tau }^{\rho }}{\rho },\end{array}$$

(108)

respectively, for some $a_1,a_2,b_1,b_2\in \mathcal{R}$. Taking γ-derivative of (107) and (108), we obtain

$$\begin{array}{cc}\hfill \gamma p\left(\tau \right)& {=}^{\rho }{\mathcal{I}}_{0+}^{\xi -1}\widehat{f}\left(\tau \right)++a_2=\frac{{\rho }^{2-\xi }}{\Gamma (\xi -1)}{\int }_0^{\tau }{\theta }^{\rho -1}{({\tau }^{\rho }-{\theta }^{\rho })}^{\xi -2}\widehat{f}\left(\theta \right)d\theta +a_2,\hfill \end{array}$$

(109)

$$\begin{array}{cc}\hfill \gamma q\left(\tau \right)& {=}^{\rho }{\mathcal{I}}_{0+}^{\zeta -1}\widehat{g}\left(\tau \right)+b_2=\frac{{\rho }^{2-\zeta }}{\Gamma (\zeta -1)}{\int }_0^{\tau }{\theta }^{\rho -1}{({\tau }^{\rho }-{\theta }^{\rho })}^{\zeta -2}\widehat{g}\left(\theta \right)d\theta +b_2.\hfill \end{array}$$

(110)

Making use of the boundary conditions $\gamma p\left(0\right)={\sum }_{k=1}^l{v}_{k}q\left({\omega }_k\right),\gamma q\left(0\right)={\sum }_{k=1}^l{\nu }_{k}p\left({\omega }_k\right)$ in (109) and (110), respectively, we obtain

$$\begin{array}{c}\hfill a_2=\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left({\omega }_k\right)+b_1\sum _{k=1}^l{v}_k+b_2\sum _{k=1}^l{v}_k\frac{{\omega }_k^{\rho }}{\rho },\end{array}$$

(111)

$$\begin{array}{c}\hfill b_2=\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left({\omega }_k\right)+a_1\sum _{k=1}^l{\nu }_k+a_2\sum _{k=1}^l{\nu }_k\frac{{\omega }_k^{\rho }}{\rho },\end{array}$$

(112)

which, as a result of (93) and (94), takes the following form:

$$\begin{array}{c}\hfill a_2-b_1{\mathcal{E}}_2-b_2{\mathcal{E}}_2=\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left({\omega }_k\right)\end{array}$$

(113)

$$\begin{array}{c}\hfill b_2-a_1{\mathcal{F}}_1-a_2{\mathcal{F}}_2=\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left({\omega }_k\right).\end{array}$$

(114)

We obtain by using the generalised integral operators ${}^{\rho }{\mathcal{I}}_{0+}^{\varrho }{,}^{\rho }{\mathcal{I}}_{0+}^{\zeta }$ (107) and (108), respectively,

$$\begin{array}{c}\hfill {}^{\rho }{\mathcal{I}}_{0+}^{\varrho }p\left(\tau \right)={}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }\widehat{f}\left(\tau \right)+a_1\frac{{\tau }^{\rho \varrho }}{{\rho }^{\varrho }\Gamma (\varrho +1)}+a_2\frac{{\tau }^{\rho (\varrho +1)}}{{\rho }^{\varrho +1}\Gamma (\varrho +2)},\end{array}$$

(115)

$$\begin{array}{c}\hfill {}^{\rho }{\mathcal{I}}_{0+}^{\varsigma }q\left(\tau \right)={}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{g}\left(\tau \right)+b_1\frac{{\tau }^{\rho \varsigma }}{{\rho }^{\varsigma }\Gamma (\varsigma +1)}+b_2\frac{{\tau }^{\rho (\varsigma +1)}}{{\rho }^{\varsigma +1}\Gamma (\varsigma +2)},\end{array}$$

(116)

which, when combined with the boundary conditions $p\left(\mathcal{T}\right)={\sum }_{i=1}^j{ϵ}_i^{\rho }{\mathcal{I}}_{0+}^{\varsigma }q\left({\varpi }_i\right)$, $q\left(\mathcal{T}\right)={\sum }_{i=1}^j{\pi }_i^{\rho }{\mathcal{I}}_{0+}^{\varrho }p\left({\varsigma }_i\right)$, gives the following results:

$${}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\mathcal{T}\right)+a_1+a_2\frac{{\mathcal{T}}^{\rho }}{\rho }={\sum }_{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)+b_1{\sum }_{i=1}^j\frac{{ϵ}_i{\varpi }_i^{\rho \varsigma }}{{\rho }^{\varsigma }\Gamma (\varsigma +1)}+b_2{\sum }_{i=1}^j\frac{{ϵ}_i{\varpi }_i{}^{\rho (\varsigma +1)}}{{\rho }^{\varsigma +1}\Gamma (\varsigma +2)},$$

(117)

$${}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left(\mathcal{T}\right)+b_1+b_2\frac{{\mathcal{T}}^{\rho }}{\rho }={\sum }_{i=1}^j{\pi }_i^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }\widehat{f}\left({\varpi }_i\right)+a_1{\sum }_{i=1}^j\frac{{\pi }_i{\varpi }_i^{\rho \varrho }}{{\rho }^{\varrho }\Gamma (\varrho +1)}+a_2{\sum }_{i=1}^j\frac{{\pi }_i{\varpi }_i^{\rho (\varrho +1)}}{{\rho }^{\varrho +1}\Gamma (\varrho +2)}.$$

(118)

Next, we obtain

$$\begin{array}{c}\hfill a_1+a_2{\mathcal{E}}_5-b_1{\mathcal{E}}_3-b_2{\mathcal{E}}_4=\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\mathcal{T}\right),\end{array}$$

(119)

$$\begin{array}{c}\hfill b_1+b_2{\mathcal{E}}_5-a_1{\mathcal{F}}_3-a_2{\mathcal{F}}_4=\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }\widehat{f}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left(\mathcal{T}\right),\end{array}$$

(120)

by employing the notations (93) and (94) in (117) and (118), respectively. We find that when we solve the system of Equations (113), (114), (119) and (120) for $a_1,a_2,b_1$ and $b_2$,

$$\begin{array}{cc}{a}_1=\frac{1}{\mathsf{\Lambda }}\left[{\alpha }_1\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left({\omega }_k\right)+{\alpha }_2\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left({\omega }_k\right)-{\beta }_1\left(\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\mathcal{T}\right)\right)\right.\hfill & \\ \left.-{\beta }_2\left(\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }\widehat{f}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left(\mathcal{T}\right)\right)\right],\hfill & \end{array}$$

(121)

$$\begin{array}{cc}{a}_2=\frac{1}{{\mathcal{G}}_1}\left[{\widehat{\alpha }}_1\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left({\omega }_k\right)+{\widehat{\alpha }}_2\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left({\omega }_k\right)+{\widehat{\beta }}_1\left(\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\mathcal{T}\right)\right)\right.\hfill & \\ \left.+{\widehat{\beta }}_2\left(\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }\widehat{f}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left(\mathcal{T}\right)\right)\right],\hfill & \end{array}$$

(122)

$$\begin{array}{cc}{b}_1=\frac{-1}{\mathsf{\Lambda }}\left[{\alpha }_3\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left({\omega }_k\right)+{\alpha }_4\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left({\omega }_k\right)-{\beta }_3\left(\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\mathcal{T}\right)\right)\right.\hfill & \\ \left.-{\beta }_4\left(\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }\widehat{f}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left(\mathcal{T}\right)\right)\right],\hfill & \end{array}$$

(123)

$$\begin{array}{cc}{b}_2=\frac{1}{{\mathcal{G}}_1}\left[{\widehat{\alpha }}_3\sum _{k=1}^l{v}_k{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left({\omega }_k\right)+{\widehat{\alpha }}_4\sum _{k=1}^l{\nu }_k{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left({\omega }_k\right)+{\widehat{\beta }}_3\left(\sum _{i=1}^j{ϵ}_i{}^{\rho }{\mathcal{I}}_{0+}^{\zeta +\varsigma }\widehat{g}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\xi }\widehat{f}\left(\mathcal{T}\right)\right)\right.\hfill & \\ \left.+{\widehat{\beta }}_4\left(\sum _{i=1}^j{\pi }_i{}^{\rho }{\mathcal{I}}_{0+}^{\xi +\varrho }\widehat{f}\left({\varpi }_i\right)-{}^{\rho }{\mathcal{I}}_{0+}^{\zeta }\widehat{g}\left(\mathcal{T}\right)\right)\right].\hfill & \end{array}$$

(124)

Substituting the values of $a_1,a_2,b_1,b_2$ in (107) and (108), respectively, we obtain the solution for (104). □

For brevity’s sake, we will use the following notations:

$$\widehat{{\mathcal{J}}_1}=\left({\mathcal{T}}^{\rho \xi }(1+|\widehat{{\delta }_1}\left|\right)+|{\delta }_2|{\sum }_{k=1}^l\left|v_k\right|{\omega }_k^{\rho \xi }\right)\frac{1}{{\rho }^{\xi }\Gamma (\xi +1)}+|\widehat{{\delta }_2}|{\sum }_{i=1}^j\left|{\pi }_i\right|\frac{{\varpi }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)},$$

(125)

$$\widehat{{\mathcal{K}}_1}=\left(|{\delta }_1|{\sum }_{k=1}^l|v_k|{\omega }_k^{\rho \zeta }+\left|\widehat{{\delta }_2}\right|{\mathcal{T}}^{\rho \zeta }\right)\frac{1}{{\rho }^{\zeta }\Gamma (\zeta +1)}+|\widehat{{\delta }_1}|{\sum }_{i=1}^j\left|{ϵ}_i\right|\frac{{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)},$$

(126)

$$\widehat{{\mathcal{J}}_2}=\left(|{\delta }_4|{\sum }_{k=1}^l|{\nu }_k|{\omega }_k^{\rho \xi }+\left|\widehat{{\delta }_3}\right|{\mathcal{T}}^{\rho \xi }\right)\frac{1}{{\rho }^{\xi }\Gamma (\xi +1)}+|\widehat{{\delta }_4}|{\sum }_{i=1}^j\left|{\pi }_i\right|\frac{{\varpi }_i^{\rho (\xi +\varrho )}}{{\rho }^{\xi +\varrho }\Gamma (\xi +\varrho +1)},$$

(127)

$$\widehat{{\mathcal{K}}_2}=\left({\mathcal{T}}^{\rho \zeta }(1+|\widehat{{\delta }_4}\left|\right)+|{\delta }_3|{\sum }_{k=1}^l\left|{\nu }_k\right|{\omega }_k^{\rho \zeta }\right)\frac{1}{{\rho }^{\zeta }\Gamma (\zeta +1)}+|\widehat{{\delta }_3}|{\sum }_{i=1}^j\left|{ϵ}_i\right|\frac{{\varpi }_i^{\rho (\zeta +\varsigma )}}{{\rho }^{\zeta +\varsigma }\Gamma (\zeta +\varsigma +1)}.$$

(128)

To finish up, we will go over the results of existence, uniqueness and Ulam–Hyers stability for problems (6) and (8), respectively. For reasons that are similar to those in Section 3, Section 4, Section 5 and Section 6 we are not providing the proof.

Theorem 5.

Assume that $f,g:\mathcal{E}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions satisfying the condition: $\left({\mathcal{A}}_1\right)$. There exist constants ${\psi }_m,\widehat{{\psi }_m}\le 0(m=1,2)$ and ${\psi }_0,\widehat{{\psi }_0}>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|f\right(\tau ,o_1,o_2\left)\right|& \le {\psi }_0+{\psi }_1|o_1|+{\psi }_2\left|o_2\right|,\hfill \\ \hfill \left|g\right(\tau ,o_1,o_2\left)\right|& \le \widehat{{\psi }_0}+\widehat{{\psi }_1}|o_1|+\widehat{{\psi }_2}\left|o_2\right|,\forall o_m\in \mathbb{R},m=1,2.\hfill \end{array}\end{array}$$

If ${\psi }_1(\widehat{{\mathcal{J}}_1}+\widehat{{\mathcal{J}}_2})+\widehat{{\psi }_1}(\widehat{{\mathcal{K}}_1}+\widehat{{\mathcal{K}}_2})<1,{\psi }_2(\widehat{{\mathcal{J}}_1}+\widehat{{\mathcal{J}}_2})+\widehat{{\psi }_2}(\widehat{{\mathcal{K}}_1}+\widehat{{\mathcal{K}}_2})<1$. Then, at least one solution for the BVP (6) and (8) on $\mathcal{E}$, where $\widehat{{\mathcal{J}}_1},\widehat{{\mathcal{K}}_1},\widehat{{\mathcal{J}}_2},\widehat{{\mathcal{K}}_2}$ are given by (125)–(128), respectively.

Theorem 6.

Assume that $f,g:\mathcal{E}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions satisfying the condition: (${\mathcal{A}}_2$). There exist constants ${\varphi }_m,\widehat{{\varphi }_m}\le 0(m=1,2)$ such that

$$\begin{array}{cc}\hfill |f(\tau ,o_1,o_2)-f(\tau ,{\widehat{o}}_1,{\widehat{o}}_2)|\le & {\varphi }_1|o_1-{\widehat{o}}_1|+{\varphi }_2|o_2-{\widehat{o}}_2|,\hfill \\ \hfill |g(\tau ,o_1,o_2)-g(\tau ,{\widehat{o}}_1,{\widehat{o}}_2)|\le & \widehat{{\varphi }_1}|o_1-{\widehat{o}}_1|+\widehat{{\varphi }_2}|o_2-{\widehat{o}}_2|,\forall o_m,{\widehat{o}}_m\in \mathbb{R},m=1,2.\hfill \end{array}$$

If there exist ${\mathcal{S}}_1,{\mathcal{S}}_2>0$ such that $\left|f(\tau ,0,0)\right|\le {\mathcal{S}}_1,\left|f(\tau ,0,0)\right|\le {\mathcal{S}}_2,$ and

$$\begin{array}{c}\hfill ({\mathcal{J}}_1+{\mathcal{J}}_2)({\varphi }_1+{\varphi }_2)+({\mathcal{K}}_1+{\mathcal{K}}_2)(\widehat{{\varphi }_1}+\widehat{{\varphi }_2})<1,\end{array}$$

(129)

the BVP (6) and (8) has a unique solution on $\mathcal{E}$, where $\widehat{{\mathcal{J}}_1},\widehat{{\mathcal{K}}_1},\widehat{{\mathcal{J}}_2},\widehat{{\mathcal{K}}_2}$ are given by (125)–(128), respectively.

Theorem 7.

Assume that $f,g:\mathcal{E}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions satisfying the assumption $\left({\mathcal{A}}_2\right)$ in Theorem 2. Furthermore, there exist positive constants ${\mathcal{U}}_1,{\mathcal{U}}_2$ such that $\forall \tau \in \mathcal{E}$ and $r_i\in \mathbb{R},i=1,2$.

$$\begin{array}{c}\hfill |f(\tau ,r_1,r_2)|\le {\mathcal{U}}_1,\left|g(\tau ,r_1,r_2)\right|\le {\mathcal{U}}_2,\end{array}$$

(130)

If

$$\begin{array}{c}\hfill \frac{{\mathcal{T}}^{\rho \xi }({\varphi }_1+{\varphi }_2)}{{\rho }^{\xi }\Gamma (\xi +1)}+\frac{{\mathcal{T}}^{\rho \zeta }(\widehat{{\varphi }_1}+\widehat{{\varphi }_2})}{{\rho }^{\zeta }\Gamma (\zeta +1)}<1,\end{array}$$

(131)

then the BVP (6), and (8) has at least one solution on $\mathcal{E}$.

Theorem 8.

Assume that (A2) holds. Then problems (6) and (8) are Ulam–Hyers stable.

6. Conclusions

In this study, coupled nonlinear generalized Liouville–Caputo fractional differential equations and Katugampola fractional integral operators are used to solve a new class of boundary value problems. The tools of the fixed-point theory are successfully applied to determine the existence criteria for solutions. The first and third outcomes (Theorems 1 and 3) establish several criteria for the existence of solutions to the given problem, while the second result provides a sufficient criterion to ensure the problem’s unique solution. Then, the Hyers–Ulam stability of the solution was established in the fourth section. It permits us to make the following remarks:

• If $\rho =1$, problem (6) is generalized, and the Liouville–Caputo-type reduces to the classical Caputo form.
• If $\rho =1$, the generalized Riemann–Liouville integral boundary conditions reduce to the Riemann–Liouville integral conditions. Then, the boundary conditions (7) reduce to multi-point and Riemann–Liouville integral conditions.
• If $\rho =0$, the generalized Riemann–Liouville integral boundary conditions reduce to the classical integral conditions. Then the boundary conditions (7) reduces to multi-point and classical integral conditions.

Based on this context, we conclude that our results are novel and can be viewed as an expansion of the qualitative analysis of fractional differential equations. Future research may concentrate on various concepts of stability and existence as they relate to a neutral time-delay system/inclusion and a time-delay system/inclusion with finite delay. In the present configuration, our results are novel and contribute to the literature on nonlinear coupled generalized Liouville–Caputo fractional differential equations with nonlocal multi-point boundary conditions using Katugampolo-type integral operators.