Existence of Solutions to a System of Fractional $\mathfrak{q}$-Difference Boundary Value Problems

Abstract

We are investigating the existence of solutions to a system of two fractional $\mathfrak{q}$-difference equations containing fractional $\mathfrak{q}$-integral terms, subject to multi-point boundary conditions that encompass $\mathfrak{q}$-derivatives and fractional $\mathfrak{q}$-derivatives of different orders. In our main results, we rely on various fixed point theorems, such as the Leray–Schauder nonlinear alternative, the Schaefer fixed point theorem, the Krasnosel’skii fixed point theorem for the sum of two operators, and the Banach contraction mapping principle. Finally, several examples are provided to illustrate our findings.

1. Introduction

We examine the system of fractional $\mathfrak{q}$-difference equations

$$\left\{\begin{array}{c}\left({\mathrm{D}}_{\mathfrak{q}}^{\alpha }\mathrm{u}\right)(\nu )+\mathcal{F}\left(\nu ,\mathrm{u}(\nu ),\mathrm{v}(\nu ),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_1}\mathrm{u}(\nu ),{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_1}\mathrm{v}(\nu )\right)=0,\nu \in (0,1),\hfill \\ \left({\mathrm{D}}_{\mathfrak{q}}^{\beta }\mathrm{v}\right)(\nu )+\mathcal{G}\left(\nu ,\mathrm{u}(\nu ),\mathrm{v}(\nu ),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_2}\mathrm{u}(\nu ),{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_2}\mathrm{v}(\nu )\right)=0,\nu \in (0,1),\hfill \end{array}\right.$$

(1)

subject to the multi-point boundary conditions

$$\left\{\begin{array}{c}{\displaystyle {\mathrm{D}}_{\mathfrak{q}}^i\mathrm{u}(0)=0,i=0,\dots ,n-2,{\mathrm{D}}_{\mathfrak{q}}^{\varsigma }\mathrm{u}(1)=\sum _{i=1}^{\mathfrak{a}}{a}_i{\mathrm{D}}_{\mathfrak{q}}^{{\varrho }_i}\mathrm{u}({\xi }_i)+\sum _{i=1}^{\mathfrak{b}}{b}_i{\mathrm{D}}_{\mathfrak{q}}^{{\sigma }_i}\mathrm{v}({\omega }_i),}\hfill \\ {\displaystyle {\mathrm{D}}_{\mathfrak{q}}^i\mathrm{v}(0)=0,i=0,\dots ,m-2,{\mathrm{D}}_{\mathfrak{q}}^{\vartheta }\mathrm{v}(1)=\sum _{i=1}^{\mathfrak{c}}{c}_i{\mathrm{D}}_{\mathfrak{q}}^{{\eta }_i}\mathrm{u}({\zeta }_i)+\sum _{i=1}^{\mathfrak{d}}{d}_i{\mathrm{D}}_{\mathfrak{q}}^{{\rho }_i}\mathrm{v}({\theta }_i).}\hfill \end{array}\right.$$

(2)

Here, $\mathfrak{q}\in (0,1)$, $\alpha ,\beta \in \mathbb{R}$, $\alpha \in (n-1,n]$, $\beta \in (m-1,m]$, $n,m\in \mathbb{N}$, $n,m\ge 2$; $\mathfrak{a},\mathfrak{b},\mathfrak{c},\mathfrak{d}\in \mathbb{N}$; $\varsigma ,{\varrho }_i,{\eta }_k\in [0,\alpha -1)$; $\vartheta ,{\sigma }_j,{\rho }_{\iota }\in [0,\beta -1)$; $a_i,b_j,c_k,d_{\iota }\in \mathbb{R}$; ${\xi }_i,{\omega }_j,{\zeta }_k,{\theta }_{\iota }\in (0,1)$; ${\mathrm{D}}_{\mathfrak{q}}^{\kappa }$ is the fractional $\mathfrak{q}$-derivative of order $\kappa$, for $\kappa =\alpha ,\beta ,\varsigma ,\vartheta ,{\varrho }_i,{\sigma }_j,{\eta }_k,{\rho }_{\iota }$, for all $i=1,\dots ,\mathfrak{a}$, $j=1,\dots ,\mathfrak{b}$, $k=1,\dots ,\mathfrak{c}$, $\iota =1,\dots ,\mathfrak{d}$; ${\mathrm{D}}_{\mathfrak{q}}^p$ represents the $\mathfrak{q}$-derivative of order p, for $p=0,\dots ,n-2$ and $p=0,\dots ,m-2$; ${\delta }_r,{\gamma }_r>0$ for $r=1,2$; ${\mathrm{I}}_{\mathfrak{q}}^{\kappa }$ is the fractional $\mathfrak{q}$-integral of order $\kappa$, for $\kappa ={\delta }_i,{\gamma }_i,i=1,2$, and $\mathcal{F},\mathcal{G}$ are nonlinear functions satisfying some assumptions.

In this paper, we aim to set forth conditions on the functions $\mathcal{F}$ and $\mathcal{G}$ that guarantee the existence of at least one solution to problem (1), (2). Our proofs will make use of various fixed-point theorems, including the Leray–Schauder nonlinear alternative, the Schaefer fixed-point theorem, the Krasnosel’skii fixed-point theorem for the sum of two operators, and the Banach contraction mapping principle. Furthermore, we will include references to relevant literature closely associated with our investigated problem. In [1], the author studied the existence, uniqueness, and multiplicity of positive solutions for problem (1), (2) under different assumptions than those used in our present paper. The associated Green functions are constructed, and some of their properties are presented. For the proof of the principal findings, the author employed in [1] a range of fixed point theorems, including the Schauder fixed point theorem, the Leggett–Williams fixed point theorem, and the Guo–Krasnosel’skii fixed point theorem. Therefore, the methods used in [1] are distinct from those we will apply in our paper. In [2], the authors investigated the system of nonlinear fractional $\mathfrak{q}$-difference equations

$$\left\{\begin{array}{c}({\mathrm{D}}_{\mathfrak{q}}^{{\alpha }_1}\mathrm{u})(\mathrm{t})+\mathrm{P}(\mathrm{t},\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t}),{\mathrm{I}}_{\mathfrak{q}}^{{\omega }_1}\mathrm{u}(\mathrm{t}),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_1}\mathrm{v}(\mathrm{t}))=0,\mathrm{t}\in (0,1),\hfill \\ ({\mathrm{D}}_q^{{\alpha }_2}\mathrm{v})(\mathrm{t})+\mathrm{Q}(\mathrm{t},\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t}),{\mathrm{I}}_{\mathfrak{q}}^{{\omega }_2}\mathrm{u}(\mathrm{t}),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_2}\mathrm{v}(\mathrm{t}))=0,\mathrm{t}\in (0,1),\hfill \end{array}\right.$$

(3)

with the coupled nonlocal boundary conditions

$$\left\{\begin{array}{c}{\displaystyle {\mathrm{D}}_{\mathfrak{q}}^i\mathrm{u}(0)=0,i=0,\dots ,m-2,{\mathrm{D}}_{\mathfrak{q}}^{{\zeta }_0}\mathrm{u}(1)={\int }_0^1{\mathrm{D}}_{\mathfrak{q}}^{\zeta }\mathrm{v}(\mathrm{t})d_{\mathfrak{q}}\mathrm{H}(\mathrm{t}),}\hfill \\ {\displaystyle {\mathrm{D}}_{\mathfrak{q}}^i\mathrm{v}(0)=0,i=0,\dots ,n-2,{\mathrm{D}}_{\mathfrak{q}}^{{\xi }_0}\mathrm{v}(1)={\int }_0^1{\mathrm{D}}_{\mathfrak{q}}^{\xi }\mathrm{u}(\mathrm{t})d_{\mathfrak{q}}\mathrm{K}(\mathrm{t}),}\hfill \end{array}\right.$$

(4)

where $\mathfrak{q}\in (0,1)$, ${\alpha }_1,{\alpha }_2\in \mathbb{R}$, ${\alpha }_1\in (m-1,m]$, ${\alpha }_2\in (n-1,n]$, $m,n\in \mathbb{N}$, $m\ge 2,n\ge 2$, ${\omega }_i>0,{\delta }_i>0,i=1,2$, $\zeta \in [0,{\alpha }_2-1)$, $\xi \in [0,{\alpha }_1-1)$, ${\zeta }_0\in [0,{\alpha }_1-1)$, ${\xi }_0\in [0,{\alpha }_2-1)$, the integrals from (4) are Riemann–Stieltjes integrals, and $\mathrm{H}$ and $\mathrm{K}$ are bounded variation functions. Utilizing diverse fixed-point theorems, they established results affirming the existence and uniqueness of solutions to problem (3), (4). In [3], the authors analyzed the existence of solutions to the fractional $\mathfrak{q}$-difference equation subject to nonlocal boundary conditions

$$\left\{\begin{array}{c}{(}^C{\mathrm{D}}_{\mathfrak{q}}^{\beta }\mathrm{u})(\mathrm{t})=\mathrm{f}(\mathrm{t},\mathrm{u}(\mathrm{t})),\mathrm{for}\mathrm{a}.\mathrm{e}.\mathrm{t}\in (0,T),\hfill \\ {\displaystyle \mathrm{u}(0)-{\mathrm{u}}^{\prime }(0)={\int }_0^T\mathrm{h}(\mathrm{s},\mathrm{u}(\mathrm{s}))d\mathrm{s},\mathrm{u}(T)+{\mathrm{u}}^{\prime }(T)={\int }_0^T\mathrm{g}(\mathrm{s},\mathrm{u}(\mathrm{s}))d\mathrm{s},}\hfill \end{array}\right.$$

(5)

where $T>0$, $\mathfrak{q}\in (0,1)$, $\beta \in (1,2]$, and ${}^C\mathrm{D}_{\mathfrak{q}}^{\alpha }$ is the Caputo fractional $\mathfrak{q}$-derivative of order $\alpha$. In demonstrating the main result, they employed the Mönch fixed-point theorem, and the method of measures of noncompactness. In [4], the authors examined the existence, uniqueness and multiplicity of positive solutions to the fractional $\mathfrak{q}$-difference equation supplemented with nonlocal boundary conditions

$$\left\{\begin{array}{c}({\mathrm{D}}_{\mathfrak{q}}^{\beta }\mathrm{u})(\mathrm{t})+\mathrm{g}(\mathrm{t},\mathrm{u}(\mathrm{t}))=0,\mathrm{t}\in (0,1),\hfill \\ ({\mathrm{D}}_{\mathfrak{q}}^i\mathrm{u})(0)=0,i=0,\dots ,m-2,({\mathrm{D}}_{\mathfrak{q}}^{\gamma }\mathrm{u})(1)=a({\mathrm{D}}_{\mathfrak{q}}^{\gamma }\mathrm{u})(\eta ),\hfill \end{array}\right.$$

(6)

where $\mathfrak{q}\in (0,1)$, $\beta \in (m-1,m]$, $m>2$, $\gamma \in [1,m-2]$, $\eta \in (0,1)$, $a\in [0,1]$, and $\mathrm{g}:[0,1]\times [0,\infty )\to [0,\infty )$ satisfies Caratheodory type conditions. In proving the main theorems, they utilized multiple fixed-point theorems. In [5], based on the Guo–Krasnosel’skii fixed point theorem, the author explored the existence of positive solutions for the fractional $\mathfrak{q}$-difference equation subject to boundary conditions

$$\left\{\begin{array}{c}({\mathrm{D}}_{\mathfrak{q}}^{\gamma }\mathrm{v})(\mathrm{t})=-\mathrm{g}(\mathrm{t},\mathrm{v}(\mathrm{t})),\mathrm{t}\in (0,1),\hfill \\ \mathrm{v}(0)=({\mathrm{D}}_{\mathfrak{q}}\mathrm{v})(0)=0,({\mathrm{D}}_{\mathfrak{q}}\mathrm{v})(1)=\beta ,\hfill \end{array}\right.$$

(7)

where $\mathfrak{q}\in (0,1)$, $\gamma \in (2,3]$, $\beta \ge 0$, and $\mathrm{g}:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a nonnegative continuous function. In [6], the author studied the existence of nontrivial solutions for the nonlinear $\mathfrak{q}$-fractional boundary value problem

$$\left\{\begin{array}{c}({\mathrm{D}}_{\mathfrak{q}}^{\gamma }\mathrm{v})(\mathrm{t})=-\mathrm{g}(\mathrm{t},\mathrm{v}(\mathrm{t})),\mathrm{t}\in (0,1),\hfill \\ \mathrm{v}(0)=\mathrm{v}(1)=0,\hfill \end{array}\right.$$

(8)

where $\mathfrak{q}\in (0,1)$, $\gamma \in (1,2]$, and $\mathrm{g}:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a nonnegative continuous function. To prove the main results, he also used the Guo–Krasnosel’skii fixed point theorem. For other research works that investigate fractional $\mathfrak{q}$-difference equations and systems of fractional $\mathfrak{q}$-difference equations with either coupled or uncoupled boundary conditions, we refer the reader to the following papers [7,8,9,10,11,12,13,14].

The domain of $\mathfrak{q}$-difference calculus, commonly known as quantum calculus, finds its roots in the seminal contributions of Jackson [15,16]. For a comprehensive exploration of diverse applications within this field, readers are encouraged to delve into the research conducted by Ernst [17]. The inception of fractional $\mathfrak{q}$-difference calculus can be traced back to the works of Al-Salam [18] and Agarwal [19]. To stay updated on advancements in this subfield, covering $\mathfrak{q}$-analogs of integral and differential fractional operators, along with properties such as $\mathfrak{q}$-analogs of Cauchy’s formula, the fractional Leibniz $\mathfrak{q}$-formula, $\mathfrak{q}$-Taylor’s formula, $\mathfrak{q}$-Laplace transform, and $\mathfrak{q}$-analogs of the Mittag–Leffler function, see the papers [19,20,21,22,23,24,25,26,27,28,29,30,31].

The novelty aspects of our problem (1), (2), compared to that examined in [1] are the following. In our paper, we study the existence of solutions for problem (1), (2), in contrast to [1], where the author investigated the existence of positive solutions for (1), (2). For this reason, the assumptions on the orders of the fractional derivatives in [1] are stronger than those used here, and they assure the nonnegativity of the associated Green functions. Indeed, in [1], the orders $\varsigma$ and $\vartheta$ must be greater than or equal to 1, an assumption that does not appear in our present work. In addition, in [1], there are connections between $\varsigma ,{\varrho }_i$ and ${\eta }_k$ for $i=1,\dots ,\mathfrak{a}$ and $k=1,\dots ,\mathfrak{c}$, on the one hand, and $\vartheta ,{\sigma }_j$ and ${\rho }_{\iota }$ for $j=1,\dots ,\mathfrak{b}$ and $\iota =1,\dots ,\mathfrak{d}$, on the other hand. Namely, ${\varrho }_i$ and ${\eta }_k$ are less than or equal to $\varsigma$, for $i=1,\dots ,\mathfrak{a}$ and $k=1,\dots ,\mathfrak{c}$, and ${\sigma }_j$ and ${\rho }_{\iota }$ are less than or equal to $\vartheta$, for $j=1,\dots ,\mathfrak{b}$ and $\iota =1,\dots ,\mathfrak{d}$. These last conditions are not used in our paper. Furthermore, the theorems applied in the present paper are different than those utilized in [1]. Related to paper [2], the differences between [2] and our paper are in the form of boundary conditions, which in our case (boundary conditions (2)) are more general than the conditions (4); our conditions (2) are generalized coupled boundary conditions.

Our paper is structured as follows: Section 2 presents auxiliary results essential for the subsequent sections. In Section 3, we unveil the primary existence results for the problem (1), (2). Moving on, Section 4 offers illustrative examples to showcase the applicability of our theorems. Finally, Section 5 concludes the paper by providing a summary of the findings and presenting comprehensive conclusions.

2. Auxiliary Results

This section provides initial findings that will be utilized in subsequent sections. We begin by examining the linear system associated with our given problem (1), (2), namely

$$\left\{\begin{array}{c}({\mathrm{D}}_{\mathfrak{q}}^{\alpha }\mathrm{u})(\nu )+h(\nu )=0,\nu \in (0,1),\hfill \\ ({\mathrm{D}}_{\mathfrak{q}}^{\beta }\mathrm{v})(\nu )+k(\nu )=0,\nu \in (0,1),\hfill \end{array}\right.$$

(9)

with the boundary conditions (2), where $h,k\in C[0,1]$.

We introduce the constants

$$\begin{array}{c}{\displaystyle {\mathsf{\Lambda }}_1=\frac{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}-\sum _{i=1}^{\mathfrak{a}}{a}_i\frac{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\xi }_i^{\alpha -{\varrho }_i-1},}\hfill \\ {\displaystyle {\mathsf{\Lambda }}_2=\sum _{i=1}^{\mathfrak{b}}{b}_i\frac{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\omega }_i^{\beta -{\sigma }_i-1},{\mathsf{\Lambda }}_3=\sum _{i=1}^{\mathfrak{c}}{c}_i\frac{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\zeta }_i^{\alpha -{\eta }_i-1},}\hfill \\ {\displaystyle {\mathsf{\Lambda }}_4=\frac{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}-\sum _{i=1}^{\mathfrak{d}}{d}_i\frac{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1},}\hfill \\ \Delta ={\mathsf{\Lambda }}_1{\mathsf{\Lambda }}_4-{\mathsf{\Lambda }}_2{\mathsf{\Lambda }}_3.\hfill \end{array}$$

(10)

Lemma 1.

([1]). If $\Delta \ne 0$, then the solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu )),\nu \in [0,1]$ of problem (9), (2) is given by

$$\begin{array}{c}{\displaystyle \mathrm{u}(\nu )=-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}h(\tau )d_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle +\frac{{\nu }^{\alpha -1}}{\Delta }\left[{\displaystyle \frac{{\mathsf{\Lambda }}_4}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}h(\tau )d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle -{\mathsf{\Lambda }}_4\sum _{i=1}^{\mathfrak{a}}\frac{a_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}h(\tau )d_{\mathfrak{q}}\tau }\hfill \\ \left.{\displaystyle -{\mathsf{\Lambda }}_2\sum _{i=1}^{\mathfrak{c}}\frac{c_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}h(\tau )d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle +\frac{{\nu }^{\alpha -1}}{\Delta }\left[{\displaystyle -{\mathsf{\Lambda }}_4\sum _{i=1}^{\mathfrak{b}}\frac{b_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}k(\tau )d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +\frac{{\mathsf{\Lambda }}_2}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}k(\tau )d_{\mathfrak{q}}\tau }\hfill \\ -\left.{\displaystyle {\mathsf{\Lambda }}_2\sum _{i=1}^{\mathfrak{d}}\frac{d_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}k(\tau )d_{\mathfrak{q}}\tau }\right],\nu \in [0,1],\hfill \\ {\displaystyle \mathrm{v}(\nu )=\frac{{\nu }^{\beta -1}}{\Delta }\left[{\displaystyle -{\mathsf{\Lambda }}_1\sum _{i=1}^{\mathfrak{c}}\frac{c_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}h(\tau )d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +\frac{{\mathsf{\Lambda }}_3}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}h(\tau )d_{\mathfrak{q}}\tau }\hfill \\ -\left.{\displaystyle {\mathsf{\Lambda }}_3\sum _{i=1}^{\mathfrak{a}}\frac{a_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}h(\tau )d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle -\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}k(\tau )d_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle +\frac{{\nu }^{\beta -1}}{\Delta }\left[{\displaystyle \frac{{\mathsf{\Lambda }}_1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}k(\tau )d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle -{\mathsf{\Lambda }}_1\sum _{i=1}^{\mathfrak{d}}\frac{d_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}k(\tau )d_{\mathfrak{q}}\tau }\hfill \\ -\left.{\displaystyle {\mathsf{\Lambda }}_3\sum _{i=1}^{\mathfrak{b}}\frac{b_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}k(\tau )d_{\mathfrak{q}}\tau }\right],\nu \in [0,1].\hfill \end{array}$$

(11)

By the definition of fractional $\mathfrak{q}$-integrals, we obtain the next lemma.

Lemma 2.

The following relations are satisfied:

(a) ${\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}{d}_{\mathfrak{q}}\tau =\frac{{\nu }^{\alpha }}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}\left(=({\mathrm{I}}_{\mathfrak{q}}^{\alpha }1)(\nu )\right),\nu \ge 0}$,

(b) ${\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}{d}_{\mathfrak{q}}\tau =\frac{{\nu }^{\beta }}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}\left(=({\mathrm{I}}_{\mathfrak{q}}^{\beta }1)(\nu )\right),\nu \ge 0}$,

(c) ${\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}{d}_{\mathfrak{q}}\tau =\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma +1)}}$,

(d) ${\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}{d}_{\mathfrak{q}}\tau =\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta +1)}}$,

(e) ${\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}{d}_{\mathfrak{q}}\tau =\frac{{\xi }_i^{\alpha -{\varrho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i+1)},i=1,\dots ,\mathfrak{a}}$,

(f) ${\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}{d}_{\mathfrak{q}}\tau =\frac{{\zeta }_i^{\alpha -{\eta }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i+1)},i=1,\dots ,\mathfrak{c}}$,

(g) ${\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}{d}_{\mathfrak{q}}\tau =\frac{{\omega }_i^{\beta -{\sigma }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i+1)},i=1,\dots ,\mathfrak{b}}$,

(h) ${\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}{d}_{\mathfrak{q}}\tau =\frac{{\theta }_i^{\beta -{\rho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i+1)},i=1,\dots ,\mathfrak{d}.}$

Lemma 3.

([1]). If $\mathrm{w}\in C[0,1]$, then for $\kappa >0$, we have

$${\displaystyle |{\mathrm{I}}_{\mathfrak{q}}^{\kappa }\mathrm{w}(\nu )|\le \frac{\parallel \mathrm{w}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}(\kappa +1)},\forall \nu \in [0,1],}$$

(12)

where $\parallel \mathrm{w}\parallel ={sup}_{\nu \in [0,1]}|\mathrm{w}(\nu )|$.

We consider now the Banach space $\mathcal{U}=C([0,1],\mathbb{R})$ with the supremum norm $\parallel \mathrm{u}\parallel ={sup}_{\nu \in [0,1]}|\mathrm{u}(\nu )|$, and the Banach space $\mathcal{V}=\mathcal{U}\times \mathcal{U}$ with the norm ${\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}=\parallel \mathrm{u}\parallel +\parallel \mathrm{v}\parallel$. We define the operator $\mathcal{E}:\mathcal{V}\to \mathcal{V}$, $\mathcal{E}(\mathrm{u},\mathrm{v})=({\mathcal{E}}_1(\mathrm{u},\mathrm{v}),{\mathcal{E}}_2(\mathrm{u},\mathrm{v}))$, with ${\mathcal{E}}_1,{\mathcal{E}}_2:\mathcal{V}\to \mathcal{U}$ given by

$$\begin{array}{c}{\displaystyle {\mathcal{E}}_1(\mathrm{u},\mathrm{v})(\nu )=-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle +\frac{{\nu }^{\alpha -1}}{\Delta }\left[{\displaystyle \frac{{\mathsf{\Lambda }}_4}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle -{\mathsf{\Lambda }}_4\sum _{i=1}^{\mathfrak{a}}\frac{a_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\hfill \\ \left.{\displaystyle -{\mathsf{\Lambda }}_2\sum _{i=1}^{\mathfrak{c}}\frac{c_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle +\frac{{\nu }^{\alpha -1}}{\Delta }\left[{\displaystyle -{\mathsf{\Lambda }}_4\sum _{i=1}^{\mathfrak{b}}\frac{b_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +\frac{{\mathsf{\Lambda }}_2}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\hfill \\ -\left.{\displaystyle {\mathsf{\Lambda }}_2\sum _{i=1}^{\mathfrak{d}}\frac{d_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\right],\nu \in [0,1],\hfill \\ {\displaystyle {\mathcal{E}}_2(\mathrm{u},\mathrm{v})(\nu )=\frac{{\nu }^{\beta -1}}{\Delta }\left[{\displaystyle -{\mathsf{\Lambda }}_1\sum _{i=1}^{\mathfrak{c}}\frac{c_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +\frac{{\mathsf{\Lambda }}_3}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\hfill \\ -\left.{\displaystyle {\mathsf{\Lambda }}_3\sum _{i=1}^{\mathfrak{a}}\frac{a_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle -\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle +\frac{{\nu }^{\beta -1}}{\Delta }\left[{\displaystyle \frac{{\mathsf{\Lambda }}_1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle -{\mathsf{\Lambda }}_1\sum _{i=1}^{\mathfrak{d}}\frac{d_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\hfill \\ -\left.{\displaystyle {\mathsf{\Lambda }}_3\sum _{i=1}^{\mathfrak{b}}\frac{b_i}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\right],\nu \in [0,1],\hfill \end{array}$$

(13)

for $(\mathrm{u},\mathrm{v})\in \mathcal{V}$, where ${\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )=\mathcal{F}(\tau ,\mathrm{u}(\tau ),\mathrm{v}(\tau ),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_1}\mathrm{u}(\tau ),{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_1}\mathrm{v}(\tau ))$, ${\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )=\mathcal{G}(\tau ,\mathrm{u}(\tau ),\mathrm{v}(\tau )$, ${\mathrm{I}}_{\mathfrak{q}}^{{\delta }_2}\mathrm{u}(\tau ),{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_2}\mathrm{v}(\tau )),$ for any $\tau \in [0,1].$

By Lemma 1, we see that $(\mathrm{u},\mathrm{v})$ is a solution of problem (1), (2) if and only if $(\mathrm{u},\mathrm{v})$ is a fixed point of operator $\mathcal{E}$.

3. Existence of Solutions

In this section, we will outline the principal existence results for the problem defined by Equations (1) and (2).

We introduce the fundamental assumptions that form the basis of our theorems.

(J1)

$\mathfrak{q}\in (0,1)$, $\alpha ,\beta \in \mathbb{R}$, $\alpha \in (n-1,n]$, $\beta \in (m-1,m]$, $n,m\in \mathbb{N}$, $n,m\ge 2$; $\mathfrak{a},\mathfrak{b},\mathfrak{c},\mathfrak{d}\in \mathbb{N}$; $\varsigma ,{\varrho }_i,{\eta }_k\in [0,\alpha -1)$; $\vartheta ,{\sigma }_j,{\rho }_{\iota }\in [0,\beta -1)$; $a_i,b_j,c_k,d_{\iota }\in \mathbb{R}$; ${\xi }_i,{\omega }_j,{\zeta }_k,{\theta }_{\iota }\in (0,1)$, for all $i=1,\dots ,\mathfrak{a}$, $j=1,\dots ,\mathfrak{b}$, $k=1,\dots ,\mathfrak{c}$, $\iota =1,\dots ,\mathfrak{d}$; ${\delta }_{\kappa },{\gamma }_{\kappa }>0$ for $\kappa =1,2$; $\Delta \ne 0$ (given by (10)).

We also define the constants

$$\begin{array}{c}{\displaystyle {\mathrm{{\rm Y}}}_1=\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}+\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma +1)}+|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}|a_i|\frac{{\xi }_i^{\alpha -{\varrho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i+1)}}\right.}\hfill \\ \left.{\displaystyle +|{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}|c_i|\frac{{\zeta }_i^{\alpha -{\eta }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i+1)}}\right],\hfill \\ {\displaystyle {\mathrm{{\rm Y}}}_2=\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_2|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta +1)}+|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}|b_i|\frac{{\omega }_i^{\beta -{\sigma }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i+1)}}\right.}\hfill \\ \left.{\displaystyle +|{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}|d_i|\frac{{\theta }_i^{\beta -{\rho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i+1)}}\right]\hfill \\ {\displaystyle {\mathrm{{\rm Y}}}_3=\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_3|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma +1)}+|{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{a}}|a_i|\frac{{\xi }_i^{\alpha -{\varrho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i+1)}}\right.}\hfill \\ \left.{\displaystyle +|{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{c}}|c_i|\frac{{\zeta }_i^{\alpha -{\eta }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i+1)}}\right],\hfill \\ {\displaystyle {\mathrm{{\rm Y}}}_4=\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}+\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_1|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta +1)}+|{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{b}}|b_i|\frac{{\omega }_i^{\beta -{\sigma }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i+1)}}\right.}\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{d}}|d_i|\frac{{\theta }_i^{\beta -{\rho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i+1)}}\right].\hfill \end{array}$$

(14)

Under assumption $(J1)$, we remark that ${\mathrm{{\rm Y}}}_1>0$, ${\mathrm{{\rm Y}}}_2\ge 0$, ${\mathrm{{\rm Y}}}_3\ge 0$, ${\mathrm{{\rm Y}}}_4>0$, and so ${\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3>0$, ${\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4>0$.

The initial existence and uniqueness theorem for problem (1), (2) is as follows, relying on the Banach contraction mapping principle, as detailed in [32].

Theorem 1.

Suppose that $(J1)$ holds. In addition, we assume that the functions $\mathcal{F},\mathcal{G}:[0,1]\times {\mathbb{R}}^4\to \mathbb{R}$ are continuous and satisfy the condition

(J2)

There exist the functions ${\mathcal{H}}_i,{\mathcal{K}}_i\in C([0,1],{\mathbb{R}}_{+})$, $i=1,\dots ,4$, (${\mathbb{R}}_{+}=[0,\infty )$), such that

$$\begin{array}{c}{\displaystyle |\mathcal{F}(\nu ,{\mathrm{u}}_1,{\mathrm{u}}_2,{\mathrm{u}}_3,{\mathrm{u}}_4)-\mathcal{F}(\nu ,{\mathrm{v}}_1,{\mathrm{v}}_2,{\mathrm{v}}_3,{\mathrm{v}}_4)|\le \sum _{i=1}^4{\mathcal{H}}_i(\nu )|{\mathrm{u}}_i-{\mathrm{v}}_i|,}\hfill \\ {\displaystyle |\mathcal{G}(\nu ,{\mathrm{u}}_1,{\mathrm{u}}_2,{\mathrm{u}}_3,{\mathrm{u}}_4)-\mathcal{G}(\nu ,{\mathrm{v}}_1,{\mathrm{v}}_2,{\mathrm{v}}_3,{\mathrm{v}}_4)|\le \sum _{i=1}^4{\mathcal{K}}_i(\nu )|{\mathrm{u}}_i-{\mathrm{v}}_i|,}\hfill \end{array}$$

(15)

for all $\nu \in [0,1]$ and ${\mathrm{u}}_i,{\mathrm{v}}_i\in \mathbb{R}$, $i=1,\dots ,4$.

If

$${\mathsf{\Theta }}_0<1,$$

(16)

where ${\mathsf{\Theta }}_0=max\{{\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2\}$,

$$\begin{array}{c}{\mathsf{\Theta }}_1=\left({\displaystyle {\mathfrak{h}}_1^{\ast }+\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+\left({\displaystyle {\mathfrak{k}}_1^{\ast }+\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4),\hfill \\ {\mathsf{\Theta }}_2=\left({\displaystyle {\mathfrak{h}}_2^{\ast }+\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+\left({\displaystyle {\mathfrak{k}}_2^{\ast }+\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4),\hfill \end{array}$$

(17)

and ${\mathfrak{h}}_i^{\ast }={sup}_{\nu \in [0,1]}{\mathcal{H}}_i(\nu )$, ${\mathfrak{k}}_i^{\ast }={sup}_{\nu \in [0,1]}{\mathcal{K}}_i(\nu )$, $i=1,\dots ,4$, then the boundary value problem (1), (2) has a unique solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu )),\nu \in [0,1]$.

Proof. 

We denote by ${\mathsf{\Xi }}_1={sup}_{\nu \in [0,1]}|\mathcal{F}(\nu ,0,0,0,0)|$ and ${\mathsf{\Xi }}_2={sup}_{\nu \in [0,1]}|\mathcal{G}(\nu ,0,0,0,0)|$. We consider the positive number

$${\displaystyle R\ge \frac{{\mathsf{\Xi }}_1({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+{\mathsf{\Xi }}_2({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)}{1-{\mathsf{\Theta }}_0},}$$

(18)

and let the set $\mathsf{\Omega }=\{(\mathrm{u},\mathrm{v})\in \mathcal{V},\parallel (\mathrm{u},\mathrm{v}){\parallel }_{\mathcal{V}}\le R\}$.

We will show firstly that $\mathcal{E}(\mathsf{\Omega })\subset \mathsf{\Omega }$. For this, let $(\mathrm{u},\mathrm{v})\in \mathsf{\Omega }$, that is $\parallel \mathrm{u}\parallel +\parallel \mathrm{v}\parallel \le R$. Then, by $(J1)$ and Lemma 3, we obtain for all $\nu \in [0,1]$

$$\begin{array}{c}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\nu )|=|\mathcal{F}(\nu ,\mathrm{u}(\nu ),\mathrm{v}(\nu ),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_1}\mathrm{u}(\nu ),{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_1}\mathrm{v}(\nu ))|\hfill \\ \le |\mathcal{F}(\nu ,\mathrm{u}(\nu ),\mathrm{v}(\nu ),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_1}\mathrm{u}(\nu ),{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_1}\mathrm{v}(\nu ))-\mathcal{F}(\nu ,0,0,0,0)|+|\mathcal{F}(\nu ,0,0,0,0)|\hfill \\ \le {\mathcal{H}}_1(\nu )|\mathrm{u}(\nu )|+{\mathcal{H}}_2(\nu )|\mathrm{v}(\nu )|+{\mathcal{H}}_3(\nu )|{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_1}\mathrm{u}(\nu )|+{\mathcal{H}}_4(\nu )|{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_1}\mathrm{v}(\nu )|+{\mathsf{\Xi }}_1\hfill \\ {\displaystyle \le {\mathfrak{h}}_1^{\ast }\parallel \mathrm{u}\parallel +{\mathfrak{h}}_2^{\ast }\parallel \mathrm{v}\parallel +{\mathfrak{h}}_3^{\ast }\frac{\parallel \mathrm{u}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}+{\mathfrak{h}}_4^{\ast }\frac{\parallel \mathrm{v}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}+{\mathsf{\Xi }}_1}\hfill \\ =\left({\displaystyle {\mathfrak{h}}_1^{\ast }+\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_q({\delta }_1+1)}}\right)\parallel \mathrm{u}\parallel +\left({\displaystyle {\mathfrak{h}}_2^{\ast }+\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_q({\gamma }_1+1)}}\right)\parallel \mathrm{v}\parallel +{\mathsf{\Xi }}_1=:A_{\mathrm{u}\mathrm{v}},\hfill \\ |{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\nu )|=|\mathcal{G}(\nu ,\mathrm{u}(\nu ),\mathrm{v}(\nu ),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_2}\mathrm{u}(\nu ),{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_2}\mathrm{v}(\nu ))|\hfill \\ \le |\mathcal{G}(\nu ,\mathrm{u}(\nu ),\mathrm{v}(\nu ),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_2}\mathrm{u}(\nu ),{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_2}\mathrm{v}(\nu ))-\mathcal{G}(\nu ,0,0,0,0)|+|\mathcal{G}(\nu ,0,0,0,0)|\hfill \\ \le {\mathcal{K}}_1(\nu )|\mathrm{u}(\nu )|+{\mathcal{K}}_2(\nu )|\mathrm{v}(\nu )|+{\mathcal{K}}_3(\nu )|{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_2}\mathrm{u}(\nu )|+{\mathcal{K}}_4(\nu )|{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_2}\mathrm{v}(\nu )|+{\mathsf{\Xi }}_2\hfill \\ {\displaystyle \le {\mathfrak{k}}_1^{\ast }\parallel \mathrm{u}\parallel +{\mathfrak{k}}_2^{\ast }\parallel \mathrm{v}\parallel +{\mathfrak{k}}_3^{\ast }\frac{\parallel \mathrm{u}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}+{\mathfrak{k}}_4^{\ast }\frac{\parallel \mathrm{v}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}+{\mathsf{\Xi }}_2}\hfill \\ =\left({\displaystyle {\mathfrak{k}}_1^{\ast }+\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_q({\delta }_2+1)}}\right)\parallel \mathrm{u}\parallel +\left({\displaystyle {\mathfrak{k}}_2^{\ast }+\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_q({\gamma }_2+1)}}\right)\parallel \mathrm{v}\parallel +{\mathsf{\Xi }}_2=:B_{\mathrm{u}\mathrm{v}}.\hfill \end{array}$$

(19)

Therefore, we find

$$\begin{array}{c}{\displaystyle |{\mathcal{E}}_1(\mathrm{u},\mathrm{v})(\nu )|\le \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle +\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\frac{1}{{\mathsf{\Gamma }}_q(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}\frac{|a_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ \left.{\displaystyle +|{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_q(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle +\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}\frac{|b_i|}{{\mathsf{\Gamma }}_q(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_2|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ \left.{\displaystyle +|{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ \le A_{\mathrm{u}\mathrm{v}}\left\{{\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}{d}_{\mathfrak{q}}\tau +\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}{d}_{\mathfrak{q}}\tau }\right.}\right.\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}\frac{|a_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-q\tau )}^{(\alpha -{\varrho }_i-1)}{d}_{\mathfrak{q}}\tau }\hfill \\ \left.\left.{\displaystyle +|{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}{d}_{\mathfrak{q}}\tau }\right]\right\}\hfill \\ {\displaystyle +B_{\mathrm{u}\mathrm{v}}\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}\frac{|b_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}{d}_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_2|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}{d}_{\mathfrak{q}}\tau }\hfill \\ \left.{\displaystyle +|{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}{d}_{\mathfrak{q}}\tau }\right]\hfill \\ =A_{\mathrm{u}\mathrm{v}}\left\{{\displaystyle \frac{{\nu }^{\alpha }}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}+\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma +1)}+|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}|a_i|\frac{{{\xi }_i}^{\alpha -{\varrho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i+1)}}\right.}\right.\hfill \\ \left.\left.{\displaystyle +|{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}|c_i|\frac{{\zeta }_i^{\alpha -{\eta }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i+1)}}\right]\right\}\hfill \\ {\displaystyle +B_{\mathrm{u}\mathrm{v}}\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}|b_i|\frac{{\omega }_i^{\beta -{\sigma }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i+1)}+|{\mathsf{\Lambda }}_2|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta +1)}}\right.}\hfill \\ \left.{\displaystyle +|{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}|d_i|\frac{{\theta }_i^{\beta -{\rho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i+1)}}\right]\le A_{\mathrm{u}\mathrm{v}}{\mathrm{{\rm Y}}}_1+B_{\mathrm{u}\mathrm{v}}{\mathrm{{\rm Y}}}_2,\forall \nu \in [0,1],\hfill \end{array}$$

(20)

and

$$\begin{array}{c}{\displaystyle |{\mathcal{E}}_2(\mathrm{u},\mathrm{v})(\nu )|\le \frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_3|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{a}}\frac{|a_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle +\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle +\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_1|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{b}}\frac{|b_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle \le A_{\mathrm{u}\mathrm{v}}\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}{d}_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_3|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}{d}_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{a}}\frac{|a_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}{d}_{\mathfrak{q}}\tau }\right]\hfill \\ +B_{\mathrm{u}\mathrm{v}}\left\{{\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}{d}_{\mathfrak{q}}\tau +\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_1|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}{d}_{\mathfrak{q}}\tau }\right.}\right.\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}{d}_{\mathfrak{q}}\tau }\hfill \\ +\left.\left.{\displaystyle |{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{b}}\frac{|b_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}{d}_{\mathfrak{q}}\tau }\right]\right\}\hfill \\ {\displaystyle =A_{\mathrm{u}\mathrm{v}}\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{c}}|c_i|\frac{{\zeta }_i^{\alpha -{\eta }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i+1)}+|{\mathsf{\Lambda }}_3|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma +1)}}\right.}\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{a}}|a_i|\frac{{\xi }_i^{\alpha -{\varrho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i+1)}}\right]\hfill \\ +B_{\mathrm{u}\mathrm{v}}\left\{{\displaystyle \frac{{\nu }^{\beta }}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}+\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_1|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta +1)}+|{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{d}}|d_i|\frac{{\theta }_i^{\beta -{\rho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i+1)}}\right.}\right.\hfill \\ +\left.\left.{\displaystyle |{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{b}}|b_i|\frac{{\omega }_i^{\beta -{\sigma }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i+1)}}\right]\right\}\le A_{\mathrm{u}\mathrm{v}}{\mathrm{{\rm Y}}}_3+B_{\mathrm{u}\mathrm{v}}{\mathrm{{\rm Y}}}_4,\forall \nu \in [0,1].\hfill \end{array}$$

(21)

Therefore, by (20), (21) and (18), we deduce

$$\begin{array}{c}{\parallel \mathcal{E}(\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}=\parallel {\mathcal{E}}_1(\mathrm{u},\mathrm{v})\parallel +\parallel {\mathcal{E}}_2(\mathrm{u},\mathrm{v})\parallel \le A_{\mathrm{u}\mathrm{v}}({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+B_{\mathrm{u}\mathrm{v}}({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\hfill \\ =\left[\left({\displaystyle {\mathfrak{h}}_1^{\ast }+\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}}\right)\parallel \mathrm{u}\parallel +\left({\displaystyle {\mathfrak{h}}_2^{\ast }+\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)\parallel \mathrm{v}\parallel +{\mathsf{\Xi }}_1\right]({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)\hfill \\ +\left[\left({\displaystyle {\mathfrak{k}}_1^{\ast }+\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}}\right)\parallel \mathrm{u}\parallel +\left({\displaystyle {\mathfrak{k}}_2^{\ast }+\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)\parallel \mathrm{v}\parallel +{\mathsf{\Xi }}_2\right]({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\hfill \\ =\left[\left({\displaystyle {\mathfrak{h}}_1^{\ast }+\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+\left({\displaystyle {\mathfrak{k}}_1^{\ast }+\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\right]\parallel \mathrm{u}\parallel \hfill \\ +\left[\left({\displaystyle {\mathfrak{h}}_2^{\ast }+\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+\left({\displaystyle {\mathfrak{k}}_2^{\ast }+\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\right]\parallel \mathrm{v}\parallel \hfill \\ +{\mathsf{\Xi }}_1({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+{\mathsf{\Xi }}_2({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\hfill \\ ={\mathsf{\Theta }}_1\parallel \mathrm{u}\parallel +{\mathsf{\Theta }}_2\parallel \mathrm{v}\parallel +{\mathsf{\Xi }}_1({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+{\mathsf{\Xi }}_2({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\hfill \\ \le {\mathsf{\Theta }}_0{\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}+{\mathsf{\Xi }}_1({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+{\mathsf{\Xi }}_2({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\hfill \\ \le {\mathsf{\Theta }}_{0}R+{\mathsf{\Xi }}_1({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+{\mathsf{\Xi }}_2({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\le R.\hfill \end{array}$$

(22)

Therefore, we conclude that $\mathcal{E}(\mathsf{\Omega })\subset \mathsf{\Omega }$.

Subsequently, we will prove that $\mathcal{E}$ is a contraction. For this, let $({\mathrm{u}}_1,{\mathrm{v}}_1),({\mathrm{u}}_2,{\mathrm{v}}_2)\in \mathcal{V}$. By relations (15), we find for any $\tau \in [0,1]$

$$\begin{array}{c}|{\mathcal{F}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )-{\mathcal{F}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|\le {\mathcal{H}}_1(\tau )|{\mathrm{u}}_1(\tau )-{\mathrm{u}}_2(\tau )|+{\mathcal{H}}_2(\tau )|{\mathrm{v}}_1(\tau )-{\mathrm{v}}_2(\tau )|\hfill \\ +{\mathcal{H}}_3(\tau )|{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_1}{\mathrm{u}}_1(\tau )-{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_1}{\mathrm{u}}_2(\tau )|+{\mathcal{H}}_4(\tau )|{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_1}{\mathrm{v}}_1(\tau )-{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_1}{\mathrm{v}}_2(\tau )|\hfill \\ {\displaystyle \le {\mathfrak{h}}_1^{\ast }\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +{\mathfrak{h}}_2^{\ast }\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel +\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel }\hfill \\ =\left({\displaystyle {\mathfrak{h}}_1^{\ast }+\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}}\right)\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +\left({\displaystyle {\mathfrak{h}}_2^{\ast }+\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel =:C_{\mathrm{u}\mathrm{v}},\hfill \\ |{\mathcal{G}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )-{\mathcal{G}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|\le {\mathcal{K}}_1(\tau )|{\mathrm{u}}_1(\tau )-{\mathrm{u}}_2(\tau )|+{\mathcal{K}}_2(\tau )|{\mathrm{v}}_1(\tau )-{\mathrm{v}}_2(\tau )|\hfill \\ +{\mathcal{K}}_3(\tau )|{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_2}{\mathrm{u}}_1(\tau )-{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_2}{\mathrm{u}}_2(\tau )|+{\mathcal{K}}_4(\tau )|{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_2}{\mathrm{v}}_1(\tau )-{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_2}{\mathrm{v}}_2(\tau )|\hfill \\ {\displaystyle \le {\mathfrak{k}}_1^{\ast }\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +{\mathfrak{k}}_2^{\ast }\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel +\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel }\hfill \\ =\left({\displaystyle {\mathfrak{k}}_1^{\ast }+\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}}\right)\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +\left({\displaystyle {\mathfrak{k}}_2^{\ast }+\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel =:D_{\mathrm{u}\mathrm{v}}.\hfill \end{array}$$

(23)

Then, for any $\nu \in [0,1]$, we obtain

$$\begin{array}{c}|{\mathcal{E}}_1({\mathrm{u}}_1,{\mathrm{v}}_1)(\nu )-{\mathcal{E}}_1({\mathrm{u}}_2,{\mathrm{v}}_2)(\nu )|\hfill \\ {\displaystyle \le \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}|{\mathcal{F}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )-{\mathcal{F}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle +\frac{1}{|\Delta |}\left[{\displaystyle \frac{|{\mathsf{\Lambda }}_4|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}|{\mathcal{F}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )-{\mathcal{F}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}\frac{|a_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{\alpha -{\varrho }_i-1)}|{\mathcal{F}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )-{\mathcal{F}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ \left.{\displaystyle +|{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}|{\mathcal{F}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )-{\mathcal{F}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle +\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}\frac{|b_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}|{\mathcal{G}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )-{\mathcal{G}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +\frac{|{\mathsf{\Lambda }}_2|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}|{\mathcal{G}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )-{\mathcal{G}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ \left.{\displaystyle +|{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}|{\mathcal{G}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )-{\mathcal{G}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ \le C_{\mathrm{u}\mathrm{v}}\left\{{\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}{d}_{\mathfrak{q}}\tau +\frac{1}{|\Delta |}\left[{\displaystyle \frac{|{\mathsf{\Lambda }}_4|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}{d}_{\mathfrak{q}}\tau }\right.}\right.\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}\frac{|a_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}{d}_{\mathfrak{q}}\tau }\hfill \\ +\left.\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}{d}_{\mathfrak{q}}\tau }\right]\right\}\hfill \\ {\displaystyle +D_{\mathrm{u}\mathrm{v}}\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}\frac{|b_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}{d}_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +\frac{|{\mathsf{\Lambda }}_2|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}{d}_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}{d}_{\mathfrak{q}}\tau }\right]\le C_{\mathrm{u}\mathrm{v}}{\mathrm{{\rm Y}}}_1+D_{\mathrm{u}\mathrm{v}}{\mathrm{{\rm Y}}}_2.\hfill \end{array}$$

(24)

In a similar manner, for any $\nu \in [0,1]$, we deduce

$$\begin{array}{c}|{\mathcal{E}}_2({\mathrm{u}}_1,{\mathrm{v}}_1)(\nu )-{\mathcal{E}}_2({\mathrm{u}}_2,{\mathrm{v}}_2)(\nu )|\le C_{\mathrm{u}\mathrm{v}}{\mathrm{{\rm Y}}}_3+D_{\mathrm{u}\mathrm{v}}{\mathrm{{\rm Y}}}_4.\hfill \end{array}$$

(25)

Then, by (24), (25) and (17), we conclude that

$$\begin{array}{c}\parallel \mathcal{E}({\mathrm{u}}_1,{\mathrm{v}}_1)-\mathcal{E}({\mathrm{u}}_2,{\mathrm{v}}_2){\parallel }_{\mathcal{V}}=\parallel {\mathcal{E}}_1({\mathrm{u}}_1,{\mathrm{v}}_1)-{\mathcal{E}}_1({\mathrm{u}}_2,{\mathrm{v}}_2)\parallel +\parallel {\mathcal{E}}_2({\mathrm{u}}_1,{\mathrm{v}}_1)-{\mathcal{E}}_2({\mathrm{u}}_2,{\mathrm{v}}_2)\parallel \hfill \\ \le C_{\mathrm{u}\mathrm{v}}({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+D_{\mathrm{u}\mathrm{v}}({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\hfill \\ =\left({\displaystyle {\mathfrak{h}}_1^{\ast }+\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +\left({\displaystyle {\mathfrak{h}}_2^{\ast }+\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel \hfill \\ +\left({\displaystyle {\mathfrak{k}}_1^{\ast }+\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +\left({\displaystyle {\mathfrak{k}}_2^{\ast }+\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel \hfill \\ =\left[\left({\displaystyle {\mathfrak{h}}_1^{\ast }+\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+\left({\displaystyle {\mathfrak{k}}_1^{\ast }+\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\right]\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel \hfill \\ +\left[\left({\displaystyle {\mathfrak{h}}_2^{\ast }+\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+\left({\displaystyle {\mathfrak{k}}_2^{\ast }+\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\right]\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel \hfill \\ ={\mathsf{\Theta }}_1\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +{\mathsf{\Theta }}_2\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel \le {\mathsf{\Theta }}_0(\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel )\hfill \\ ={\mathsf{\Theta }}_0{\parallel ({\mathrm{u}}_1,{\mathrm{v}}_1)-({\mathrm{u}}_2,{\mathrm{v}}_2)\parallel }_{\mathcal{V}}.\hfill \end{array}$$

(26)

By (16) and (26), we deduce that $\mathcal{E}$ is a contraction operator. Therefore, by the Banach contraction mapping principle, the operator $\mathcal{E}$ has a unique fixed point $({\mathrm{u}}^{\ast },{\mathrm{v}}^{\ast })\in \mathsf{\Omega }$. Therefore, problem (1), (2) has a unique solution $({\mathrm{u}}^{\ast }(\nu ),{\mathrm{v}}^{\ast }(\nu )),\nu \in [0,1]$ with $\parallel {\mathrm{u}}^{\ast }\parallel +\parallel {\mathrm{v}}^{\ast }\parallel \le R$. Moreover, for any $({\mathrm{u}}_0,{\mathrm{v}}_0)\in \mathsf{\Omega }$, the sequence ${(({\mathrm{u}}_n,{\mathrm{v}}_n))}_{n\ge 0}$ defined by $({\mathrm{u}}_n,{\mathrm{v}}_n)=\mathcal{E}({\mathrm{u}}_{n-1},{\mathrm{v}}_{n-1})$ for $n\ge 1$ converges to $({\mathrm{u}}^{\ast },{\mathrm{v}}^{\ast })$ as $n\to \infty$. By the proof of Banach theorem, we obtain the error estimate

$${\displaystyle \parallel ({\mathrm{u}}_n,{\mathrm{v}}_n)-({\mathrm{u}}^{\ast },{\mathrm{v}}^{\ast }){\parallel }_{\mathcal{V}}\le \frac{{\mathsf{\Theta }}_0^n}{1-{\mathsf{\Theta }}_0}{\parallel ({\mathrm{u}}_1,{\mathrm{v}}_1)-({\mathrm{u}}_0,{\mathrm{v}}_0)\parallel }_{\mathcal{V}}.}$$

(27)

□

Corollary 1.

Suppose that $(J1)$ holds. In addition, we assume that the functions $\mathcal{F},\mathcal{G}:[0,1]\times {\mathbb{R}}^4\to \mathbb{R}$ are continuous and

(J2)′

There exist $L_i\ge 0,M_i\ge 0$, $i=1,\dots ,4$ such that

$$\begin{array}{c}{\displaystyle |\mathcal{F}(\nu ,{\mathrm{u}}_1,{\mathrm{u}}_2,{\mathrm{u}}_3,{\mathrm{u}}_4)-\mathcal{F}(\nu ,{\mathrm{v}}_1,{\mathrm{v}}_2,{\mathrm{v}}_3,{\mathrm{v}}_4)|\le \sum _{i=1}^4{L}_i|{\mathrm{u}}_i-{\mathrm{v}}_i|,}\hfill \\ {\displaystyle |\mathcal{G}(\nu ,{\mathrm{u}}_1,{\mathrm{u}}_2,{\mathrm{u}}_3,{\mathrm{u}}_4)-\mathcal{G}(\nu ,{\mathrm{v}}_1,{\mathrm{v}}_2,{\mathrm{v}}_3,{\mathrm{v}}_4)|\le \sum _{i=1}^4{M}_i|{\mathrm{u}}_i-{\mathrm{v}}_i|,}\hfill \end{array}$$

(28)

for all $\nu \in [0,1]$ and ${\mathrm{u}}_i,{\mathrm{v}}_i\in \mathbb{R}$, $i=1,\dots ,4$.

If ${\mathsf{\Theta }}_0<1$, where ${\mathsf{\Theta }}_0=max\{{\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2\}$,

$$\begin{array}{c}{\mathsf{\Theta }}_1=\left({\displaystyle L_1+\frac{L_3}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+\left({\displaystyle M_1+\frac{M_3}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4),\hfill \\ {\mathsf{\Theta }}_2=\left({\displaystyle L_2+\frac{L_4}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+\left({\displaystyle M_2+\frac{M_4}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4),\hfill \end{array}$$

(29)

then the boundary value problem (1), (2) has a unique solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu )),\nu \in [0,1]$.

The following two outcomes regarding the existence of solutions to problem (1), (2) rely on the Krasnosel’skii fixed point theorem applied to the combination of two operators (refer to [33] for details).

Theorem 2.

Suppose that assumptions $(J1)$ and $(J2)$ hold. In addition, we assume that the functions $\mathcal{F},\mathcal{G}:[0,1]\times {\mathbb{R}}^4\to \mathbb{R}$ are continuous and

(J3)

There exist the functions $\mathsf{\Phi },\mathsf{\Psi }\in C([0,1],{\mathbb{R}}_{+})$ such that

$$|\mathcal{F}(\nu ,{\mathrm{u}}_1,{\mathrm{u}}_2,{\mathrm{u}}_3,{\mathrm{u}}_4)|\le \mathsf{\Phi }(\nu ),|\mathcal{G}(\nu ,{\mathrm{u}}_1,{\mathrm{u}}_2,{\mathrm{u}}_3,{\mathrm{u}}_4)|\le \mathsf{\Psi }(\nu ),$$

(30)

for all $\nu \in [0,1]$, ${\mathrm{u}}_i\in \mathbb{R}$, $i=1,\dots ,4$.

If

$${\mathfrak{L}}_0<1,$$

(31)

where ${\mathfrak{L}}_0=max\{{\mathfrak{L}}_1,{\mathfrak{L}}_2\}$ with

$$\begin{array}{c}{\mathfrak{L}}_1=\left({\displaystyle {\mathfrak{h}}_1^{\ast }+\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}}\right)\left({\displaystyle {\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}}\right)\hfill \\ +\left({\displaystyle {\mathfrak{k}}_1^{\ast }+\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}}\right)\left({\displaystyle {\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}}\right),\hfill \\ {\mathfrak{L}}_2=\left({\displaystyle {\mathfrak{h}}_2^{\ast }+\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)\left({\displaystyle {\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}}\right)\hfill \\ +\left({\displaystyle {\mathfrak{k}}_2^{\ast }+\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)\left({\displaystyle {\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}}\right),\hfill \end{array}$$

(32)

then problem (1), (2) has at least one solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu )),\nu \in [0,1].$

Proof. 

We define the number $r>0$, which satisfies the condition

$$r\ge ({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)\parallel \mathsf{\Phi }\parallel +({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\parallel \mathsf{\Psi }\parallel ,$$

(33)

and the closed set ${\mathsf{\Omega }}_0={\{(\mathrm{u},\mathrm{v})\in \mathcal{V},\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}\le r\}$. We shall verify the assumptions of the Krasnosel’skii fixed point theorem for the sum of two operators. We split the operator $\mathcal{E}$ defined on ${\mathsf{\Omega }}_0$, as $\mathcal{E}=\mathcal{P}+\mathcal{Q}$, $\mathcal{P}=({\mathcal{P}}_1,{\mathcal{P}}_2)$, $\mathcal{Q}=({\mathcal{Q}}_1,{\mathcal{Q}}_2)$, where ${\mathcal{P}}_i,{\mathcal{Q}}_i,i=1,2$ are defined by

$$\begin{array}{c}{\displaystyle {\mathcal{P}}_1(\mathrm{u},\mathrm{v})(\nu )=-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau ,}\hfill \\ {\mathcal{Q}}_1(\mathrm{u},\mathrm{v})(\nu )={\mathcal{E}}_1(\mathrm{u},\mathrm{v})(\nu )-{\mathcal{P}}_1(\mathrm{u},\mathrm{v})(\nu ),\hfill \\ {\displaystyle {\mathcal{P}}_2(\mathrm{u},\mathrm{v})(\nu )=-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau ,}\hfill \\ {\mathcal{Q}}_2(\mathrm{u},\mathrm{v})(\nu )={\mathcal{E}}_2(\mathrm{u},\mathrm{v})(\nu )-{\mathcal{P}}_2(\mathrm{u},\mathrm{v})(\nu ),\hfill \end{array}$$

(34)

for all $\nu \in [0,1]$ and $(\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0$.

Firstly, we will prove that $\mathcal{P}({\mathrm{u}}_1,{\mathrm{v}}_1)+\mathcal{Q}({\mathrm{u}}_2,{\mathrm{v}}_2)\in {\mathsf{\Omega }}_0$ for all $({\mathrm{u}}_1,{\mathrm{v}}_1),({\mathrm{u}}_2,{\mathrm{v}}_2)\in {\mathsf{\Omega }}_0$. For this, let $({\mathrm{u}}_1,{\mathrm{v}}_1),({\mathrm{u}}_2,{\mathrm{v}}_2)\in {\mathsf{\Omega }}_0$. Then, we find

$$\begin{array}{c}{\displaystyle |{\mathcal{P}}_1({\mathrm{u}}_1,{\mathrm{v}}_1)(\nu )+{\mathcal{Q}}_1({\mathrm{u}}_2,{\mathrm{v}}_2)(\nu )|\le \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}|{\mathcal{F}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\nu )|d_{\mathfrak{q}}\nu }\hfill \\ {\displaystyle +\frac{{\nu }^{\alpha -1}}{|\Delta |}\left[{\displaystyle \frac{|{\mathsf{\Lambda }}_4|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}|{\mathcal{F}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}\frac{|a_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}|{\mathcal{F}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}|{\mathcal{F}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle +\frac{{\nu }^{\alpha -1}}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}\frac{|b_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}|{\mathcal{G}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +\frac{|{\mathsf{\Lambda }}_2|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}|{\mathcal{G}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}|{\mathcal{G}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ \le \parallel \mathsf{\Phi }\parallel \left\{{\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}{d}_{\mathfrak{q}}\tau +\frac{1}{|\Delta |}\left[{\displaystyle \frac{|{\mathsf{\Lambda }}_4|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}{d}_{\mathfrak{q}}\tau }\right.}\right.\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}\frac{|a_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}{d}_{\mathfrak{q}}\tau }\hfill \\ +\left.\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}{d}_{\mathfrak{q}}\tau }\right]\right\}\hfill \\ {\displaystyle +\parallel \mathsf{\Psi }\parallel \frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}\frac{|b_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}{d}_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +\frac{|{\mathsf{\Lambda }}_2|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}{d}_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}{d}_{\mathfrak{q}}\tau }\right]\hfill \\ =\parallel \mathsf{\Phi }\parallel \left\{{\displaystyle \frac{{\nu }^{\alpha }}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}+\frac{1}{|\Delta |}\left[{\displaystyle \frac{|{\mathsf{\Lambda }}_4|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma +1)}+|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}|a_i|\frac{{\xi }_i^{\alpha -{\varrho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i+1)}}\right.}\right.\hfill \\ \left.\left.{\displaystyle +|{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}|c_i|\frac{{\zeta }_i^{\alpha -{\eta }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i+1)}}\right]\right\}\hfill \\ {\displaystyle +\parallel \mathsf{\Psi }\parallel \frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}|b_i|\frac{{\omega }_i^{\beta -{\sigma }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i+1)}+|{\mathsf{\Lambda }}_2|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta +1)}}\right.}\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}|d_i|\frac{{\theta }_i^{\beta -{\rho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i+1)}}\right]\le \parallel \mathsf{\Phi }\parallel {\mathrm{{\rm Y}}}_1+\parallel \mathsf{\Psi }\parallel {\mathrm{{\rm Y}}}_2,\forall \nu \in [0,1].\hfill \end{array}$$

(35)

In a similar manner, we obtain

$$\begin{array}{c}|{\mathcal{P}}_2({\mathrm{u}}_1,{\mathrm{v}}_1)(\nu )+{\mathcal{Q}}_2({\mathrm{u}}_2,{\mathrm{v}}_2)(\nu )|\le \parallel \mathsf{\Phi }\parallel {\mathrm{{\rm Y}}}_3+\parallel \mathsf{\Psi }\parallel {\mathrm{{\rm Y}}}_4,\forall \nu \in [0,1].\hfill \end{array}$$

(36)

Therefore, we deduce

$$\begin{array}{c}\parallel \mathcal{P}({\mathrm{u}}_1,{\mathrm{v}}_1)+\mathcal{Q}({\mathrm{u}}_2,{\mathrm{v}}_2){\parallel }_{\mathcal{V}}=\parallel {\mathcal{P}}_1({\mathrm{u}}_1,{\mathrm{v}}_1)+{\mathcal{Q}}_1({\mathrm{u}}_2,{\mathrm{v}}_2)\parallel +\parallel {\mathcal{P}}_2({\mathrm{u}}_1,{\mathrm{v}}_1)+{\mathcal{Q}}_2({\mathrm{u}}_2,{\mathrm{v}}_2)\parallel \hfill \\ \le ({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)\parallel \mathsf{\Phi }\parallel +({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\parallel \mathsf{\Psi }\parallel \le r,\hfill \end{array}$$

(37)

that is, $\mathcal{P}({\mathrm{u}}_1,{\mathrm{v}}_1)+\mathcal{Q}({\mathrm{u}}_2,{\mathrm{v}}_2)\in {\mathsf{\Omega }}_0$.

Subsequently, we will show that operator $\mathcal{Q}$ is a contraction mapping. Indeed, for all $({\mathrm{u}}_1,{\mathrm{v}}_1),({\mathrm{u}}_2,{\mathrm{v}}_2)\in {\mathsf{\Omega }}_0$, by using assumption $(J2)$, we obtain

$$\begin{array}{c}|{\mathcal{Q}}_1({\mathrm{u}}_1,{\mathrm{v}}_1)(\nu )-{\mathcal{Q}}_1({\mathrm{u}}_2,{\mathrm{v}}_2)(\nu )|\le C_{\mathrm{u}\mathrm{v}}\left({\displaystyle {\mathrm{{\rm Y}}}_1-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}}\right)+D_{\mathrm{u}\mathrm{v}}{\mathrm{{\rm Y}}}_2,\forall \nu \in [0,1],\hfill \\ |{\mathcal{Q}}_2({\mathrm{u}}_1,{\mathrm{v}}_1)(\nu )-{\mathcal{Q}}_2({\mathrm{u}}_2,{\mathrm{v}}_2)(\nu )|\le C_{\mathrm{u}\mathrm{v}}{\mathrm{{\rm Y}}}_3+D_{\mathrm{u}\mathrm{v}}\left({\displaystyle {\mathrm{{\rm Y}}}_4-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}}\right),\forall \nu \in [0,1].\hfill \end{array}$$

(38)

Therefore, we find

$$\begin{array}{c}\parallel \mathcal{Q}({\mathrm{u}}_1,{\mathrm{v}}_1)-\mathcal{Q}({\mathrm{u}}_2,{\mathrm{v}}_2){\parallel }_{\mathcal{V}}\hfill \\ \le C_{\mathrm{u}\mathrm{v}}\left({\displaystyle {\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}}\right)+D_{\mathrm{u}\mathrm{v}}\left({\displaystyle {\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}}\right)\hfill \\ =\left[\left({\displaystyle {\mathfrak{h}}_1^{\ast }+\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}}\right)\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +\left({\displaystyle {\mathfrak{h}}_2^{\ast }+\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel \right]\hfill \\ \times \left({\displaystyle {\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}}\right)\hfill \\ +\left[\left({\displaystyle {\mathfrak{k}}_1^{\ast }+\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}}\right)\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +\left({\displaystyle {\mathfrak{k}}_2^{\ast }+\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel \right]\hfill \\ \times \left({\displaystyle {\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}}\right)\hfill \\ =\left[\left({\displaystyle {\mathfrak{h}}_1^{\ast }+\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}}\right)\left({\displaystyle {\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}}\right)\right.\hfill \\ \left.+\left({\displaystyle {\mathfrak{k}}_1^{\ast }+\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}}\right)\left({\displaystyle {\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}}\right)\right]\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel \hfill \\ +\left[\left({\displaystyle {\mathfrak{h}}_2^{\ast }+\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)\left({\displaystyle {\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}}\right)\right.\hfill \\ \left.+\left({\displaystyle {\mathfrak{k}}_2^{\ast }+\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)\left({\displaystyle {\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}}\right)\right]\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel \hfill \\ ={\mathfrak{L}}_1\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +{\mathfrak{L}}_2\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel \le {\mathfrak{L}}_0{\parallel ({\mathrm{u}}_1,{\mathrm{v}}_1)-({\mathrm{u}}_2,{\mathrm{v}}_2)\parallel }_{\mathcal{V}}.\hfill \end{array}$$

(39)

By condition (31), we conclude that operator $\mathcal{Q}$ is a contraction.

The operators ${\mathcal{P}}_1$, ${\mathcal{P}}_2$ and $\mathcal{P}$ are continuous by the continuity of functions $\mathcal{F}$ and $\mathcal{G}$. Moreover, $\mathcal{P}$ is uniformly bounded on ${\mathsf{\Omega }}_0$, because

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{P}}_1(\mathrm{u},\mathrm{v})\parallel \le \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}\underset{\nu \in [0,1]}{sup}\left({\displaystyle {\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right)}\hfill \\ {\displaystyle \le \underset{\nu \in [0,1]}{sup}\mathsf{\Phi }(\nu )\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}\underset{\nu \in [0,1]}{sup}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}{d}_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle =\parallel \mathsf{\Phi }\parallel \underset{\nu \in [0,1]}{sup}\frac{{\nu }^{\alpha }}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}=\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}\parallel \mathsf{\Phi }\parallel ,\forall (\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0,}\hfill \\ {\displaystyle \parallel {\mathcal{P}}_2(\mathrm{u},\mathrm{v})\parallel \le \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}\underset{\nu \in [0,1]}{sup}\left({\displaystyle {\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right)}\hfill \\ {\displaystyle \le \underset{\nu \in [0,1]}{sup}\mathsf{\Psi }(\nu )\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}\underset{\nu \in [0,1]}{sup}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}{d}_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle =\parallel \mathsf{\Psi }\parallel \underset{\nu \in [0,1]}{sup}\frac{{\nu }^{\beta }}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}=\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}\parallel \mathsf{\Psi }\parallel ,\forall (\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0,}\hfill \end{array}$$

(40)

and then

$${\displaystyle \parallel \mathcal{P}(\mathrm{u},\mathrm{v})\parallel \le \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}\parallel \mathsf{\Phi }\parallel +\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}\parallel \mathsf{\Psi }\parallel ,\forall (\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0.}$$

(41)

In the last part of the proof, we will prove that $\mathcal{P}$ is compact. Let ${\nu }_1,{\nu }_2\in [0,1]$, ${\nu }_1<{\nu }_2$. Then for all $(\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0$, we obtain

$$\begin{array}{c}|{\mathcal{P}}_1(\mathrm{u},\mathrm{v})({\nu }_2)-{\mathcal{P}}_1(\mathrm{u},\mathrm{v})({\nu }_1)|\hfill \\ =\left|{\displaystyle -\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{{\nu }_2}{({\nu }_2-\mathfrak{q}\tau )}^{(\alpha -1)}{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau +\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{{\nu }_1}{({\nu }_1-\mathfrak{q}\tau )}^{(\alpha -1)}{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )d_{\mathfrak{q}}\tau }\right|\hfill \\ {\displaystyle \le \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{{\nu }_1}\left[{({\nu }_2-\mathfrak{q}\tau )}^{(\alpha -1)}-{({\nu }_1-\mathfrak{q}\tau )}^{(\alpha -1)}\right]|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle +\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_{{\nu }_1}^{{\nu }_2}{({\nu }_2-\mathfrak{q}\tau )}^{(\alpha -1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ \le \parallel \mathsf{\Phi }\parallel \left\{{\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{{\nu }_1}\left[{({\nu }_2-\mathfrak{q}\tau )}^{(\alpha -1)}-{({\nu }_1-\mathfrak{q}\tau )}^{(\alpha -1)}\right]d_{\mathfrak{q}}\tau }\right.\hfill \\ +\left.{\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_{{\nu }_1}^{{\nu }_2}{({\nu }_2-\mathfrak{q}\tau )}^{(\alpha -1)}{d}_{\mathfrak{q}}\tau }\right\}\hfill \\ {\displaystyle =\parallel \mathsf{\Phi }\parallel \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}\left({\displaystyle {\int }_0^{{\nu }_2}{({\nu }_2-\mathfrak{q}\tau )}^{(\alpha -1)}{d}_{\mathfrak{q}}\tau -{\int }_0^{{\nu }_1}{({\nu }_1-\mathfrak{q}\tau )}^{(\alpha -1)}{d}_{\mathfrak{q}}\tau }\right)}\hfill \\ {\displaystyle =\parallel \mathsf{\Phi }\parallel \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}\left({\nu }_2^{\alpha }-{\nu }_1^{\alpha }\right),}\hfill \end{array}$$

(42)

which tends to 0 as ${\nu }_2\to {\nu }_1$, independently of $(\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0$.

In a similar manner, we find

$$\begin{array}{c}{\displaystyle |{\mathcal{P}}_2(\mathrm{u},\mathrm{v})({\nu }_2)-{\mathcal{P}}_2(\mathrm{u},\mathrm{v})({\nu }_1)|\le \parallel \mathsf{\Psi }\parallel \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}\left({\nu }_2^{\beta }-{\nu }_1^{\beta }\right),}\hfill \end{array}$$

(43)

which tends to 0 as ${\nu }_2\to {\nu }_1$, independently of $(\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0$.

Therefore, the operators ${\mathcal{P}}_1,{\mathcal{P}}_2$ and $\mathcal{P}$ are equicontinuous. Using the Arzela–Ascoli theorem, we deduce that $\mathcal{P}$ is compact on ${\mathsf{\Omega }}_0$. Then, by the Krasnosel’skii fixed point theorem (see [33]), we conclude that problem (1), (2) has at least one solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu )),\nu \in [0,1]$. □

Theorem 3.

Suppose that $(J1)$ holds and the functions $\mathcal{F},\mathcal{G}:[0,1]\times {\mathbb{R}}^4\to \mathbb{R}$ are continuous and satisfy the assumptions $(J2)$ and $(J3)$. If

$${\mathfrak{M}}_0<1,$$

(44)

where ${\mathfrak{M}}_0=max\{{\mathfrak{M}}_1,{\mathfrak{M}}_2\}$ with

$$\begin{array}{c}{\displaystyle {\mathfrak{M}}_1=\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}\left({\displaystyle {\mathfrak{h}}_1^{\ast }+\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}}\right)+\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}\left({\displaystyle {\mathfrak{k}}_1^{\ast }+\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}}\right),}\hfill \\ {\displaystyle {\mathfrak{M}}_2=\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}\left({\displaystyle {\mathfrak{h}}_2^{\ast }+\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)+\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}\left({\displaystyle {\mathfrak{k}}_2^{\ast }+\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right),}\hfill \end{array}$$

(45)

then problem (1), (2) has at least one solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu )),\nu \in [0,1].$

Proof. 

We consider again, similar to the proof of Theorem 2, the positive number $r\ge ({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)\parallel \mathsf{\Phi }\parallel +({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\parallel \mathsf{\Psi }\parallel$, and the closed set ${\mathsf{\Omega }}_0={\{(\mathrm{u},\mathrm{v})\in \mathcal{V},\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}\le r\}$. We also split the operator $\mathcal{E}$ defined on ${\mathsf{\Omega }}_0$ as $\mathcal{E}=\mathcal{P}+\mathcal{Q}$, $\mathcal{P}=({\mathcal{P}}_1,{\mathcal{P}}_2)$, $\mathcal{Q}=({\mathcal{Q}}_1,{\mathcal{Q}}_2)$, where ${\mathcal{P}}_i,{\mathcal{Q}}_i,i=1,2$ are given by (34). For $({\mathrm{u}}_1,{\mathrm{v}}_1)$, $({\mathrm{u}}_2,{\mathrm{v}}_2)\in {\mathsf{\Omega }}_0$, we deduce, as in the first part of the proof of Theorem 2, that $\parallel \mathcal{P}({\mathrm{u}}_1,{\mathrm{v}}_1)+\mathcal{Q}({\mathrm{u}}_2,{\mathrm{v}}_2){\parallel }_{\mathcal{V}}\le r$.

In what follows we will show that the operator $\mathcal{P}$ is a contraction. Indeed, for $({\mathrm{u}}_1,{\mathrm{v}}_1),$$({\mathrm{u}}_2,{\mathrm{v}}_2)\in {\mathsf{\Omega }}_0$, we obtain

$$\begin{array}{c}|{\mathcal{P}}_1({\mathrm{u}}_1,{\mathrm{v}}_1)(\nu )-{\mathcal{P}}_1({\mathrm{u}}_2,{\mathrm{v}}_2)(\nu )|\hfill \\ =\left|{\displaystyle -\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}{\mathcal{F}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )d_{\mathfrak{q}}\tau +\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}{\mathcal{F}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )d_{\mathfrak{q}}\tau }\right|\hfill \\ {\displaystyle \le \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}|{\mathcal{F}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )-{\mathcal{F}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle \le C_{\mathrm{u}\mathrm{v}}\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}{d}_{\mathfrak{q}}\tau =C_{\mathrm{u}\mathrm{v}}\frac{{\nu }^{\alpha }}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}\le \frac{C_{\mathrm{u}\mathrm{v}}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)},\forall \nu \in [0,1],}\hfill \\ |{\mathcal{P}}_2({\mathrm{u}}_1,{\mathrm{v}}_1)(\nu )-{\mathcal{P}}_2({\mathrm{u}}_2,{\mathrm{v}}_2)(\nu )|\hfill \\ =\left|{\displaystyle -\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}{\mathcal{G}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )d_{\mathfrak{q}}\tau +\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}{\mathcal{G}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )d_{\mathfrak{q}}\tau }\right|\hfill \\ {\displaystyle \le \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}|{\mathcal{G}}_{{\mathrm{u}}_1{\mathrm{v}}_1}(\tau )-{\mathcal{G}}_{{\mathrm{u}}_2{\mathrm{v}}_2}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle \le D_{\mathrm{u}\mathrm{v}}\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}{d}_{\mathfrak{q}}\tau =D_{\mathrm{u}\mathrm{v}}\frac{{\nu }^{\beta }}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}\le \frac{D_{\mathrm{u}\mathrm{v}}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)},\forall \nu \in [0,1],}\hfill \end{array}$$

(46)

where $C_{\mathrm{u}\mathrm{v}},D_{\mathrm{u}\mathrm{v}}$ are given by (23).

So we conclude that

$$\begin{array}{c}\parallel \mathcal{P}({\mathrm{u}}_1,{\mathrm{v}}_1)-\mathcal{P}({\mathrm{u}}_2,{\mathrm{v}}_2){\parallel }_{\mathcal{V}}\hfill \\ \le \left[{\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}\left({\displaystyle {\mathfrak{h}}_1^{\ast }+\frac{{\mathfrak{h}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)}}\right)+\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}\left({\displaystyle {\mathfrak{k}}_1^{\ast }+\frac{{\mathfrak{k}}_3^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)}}\right)}\right]\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel \hfill \\ +\left[{\displaystyle \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}\left({\displaystyle {\mathfrak{h}}_2^{\ast }+\frac{{\mathfrak{h}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)+\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}\left({\displaystyle {\mathfrak{k}}_2^{\ast }+\frac{{\mathfrak{k}}_4^{\ast }}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)}\right]\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel \hfill \\ ={\mathcal{M}}_1\parallel {\mathrm{u}}_1-{\mathrm{u}}_2\parallel +{\mathcal{M}}_2\parallel {\mathrm{v}}_1-{\mathrm{v}}_2\parallel \le {\mathcal{M}}_0{\parallel ({\mathrm{u}}_1,{\mathrm{v}}_1)-({\mathrm{u}}_2,{\mathrm{v}}_2)\parallel }_{\mathcal{V}},\hfill \end{array}$$

(47)

that is, by (44), the operator $\mathcal{P}$ is a contraction.

By the continuity of the functions $\mathcal{F}$ and $\mathcal{G}$, the operators ${\mathcal{Q}}_1,{\mathcal{Q}}_2$ and $\mathcal{Q}$ are continuous. In addition, $\mathcal{Q}$ is uniformly bounded on ${\mathsf{\Omega }}_0$, because we have

$$\begin{array}{c}{\displaystyle |{\mathcal{Q}}_1(\mathrm{u},\mathrm{v})(\nu )|\le \frac{1}{|\Delta |}\left[{\displaystyle \frac{|{\mathsf{\Lambda }}_4|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}\frac{|a_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle +\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}\frac{|b_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +\frac{|{\mathsf{\Lambda }}_2|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle \le \parallel \mathsf{\Phi }\parallel \frac{1}{|\Delta |}\left[{\displaystyle \frac{|{\mathsf{\Lambda }}_4|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma +1)}+|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}|a_i|\frac{{\xi }_i^{\alpha -{\varrho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i+1)}}\right.}\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}|c_i|\frac{{\zeta }_i^{\alpha -{\eta }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i+1)}}\right]\hfill \\ {\displaystyle +\parallel \mathsf{\Psi }\parallel \frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}|b_i|\frac{{\omega }_i^{\beta -{\sigma }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i+1)}+\frac{|{\mathsf{\Lambda }}_2|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta +1)}}\right.}\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}|d_i|\frac{{\theta }_i^{\beta -{\rho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i+1)}}\right]\hfill \\ =\parallel \mathsf{\Phi }\parallel \left({\displaystyle {\mathrm{{\rm Y}}}_1-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}}\right)+\parallel \mathsf{\Psi }\parallel {\mathrm{{\rm Y}}}_2,\forall \nu \in [0,1],(\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0,\hfill \end{array}$$

(48)

and

$$\begin{array}{c}{\displaystyle |{\mathcal{Q}}_2(\mathrm{u},\mathrm{v})(\nu )|\le \frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_3|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ \left.{\displaystyle +|{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{a}}|a_i|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}|{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle +\frac{1}{|\Delta |}\left[{\displaystyle \frac{|{\mathsf{\Lambda }}_1|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{b}}|b_i|\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}|{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle \le \parallel \mathsf{\Phi }\parallel \frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{c}}|c_i|\frac{{{\zeta }_i}^{\alpha -{\eta }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i+1)}+\frac{|{\mathsf{\Lambda }}_3|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma +1)}}\right.}\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{a}}|a_i|\frac{{\xi }_i^{\alpha -{\varrho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i+1)}}\right]\hfill \\ {\displaystyle +\parallel \mathsf{\Psi }\parallel \frac{1}{|\Delta |}\left[{\displaystyle \frac{|{\mathsf{\Lambda }}_1|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta +1)}+|{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{d}}|d_i|\frac{{\theta }_i^{\beta -{\rho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i+1)}}\right.}\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{b}}|b_i|\frac{{\omega }_i^{\beta -{\sigma }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i+1)}}\right]\hfill \\ =\parallel \mathsf{\Phi }\parallel {\mathrm{{\rm Y}}}_3+\parallel \mathsf{\Psi }\parallel \left({\displaystyle {\mathrm{{\rm Y}}}_4-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}}\right),\forall \nu \in [0,1],(\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0.\hfill \end{array}$$

(49)

Therefore we deduce

$$\begin{array}{c}\parallel {\mathcal{Q}}_1(\mathrm{u},\mathrm{v})\parallel \le \parallel \mathsf{\Phi }\parallel \left({\displaystyle {\mathrm{{\rm Y}}}_1-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}}\right)+\parallel \mathsf{\Psi }\parallel {\mathrm{{\rm Y}}}_2,\forall (\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0,\hfill \\ \parallel {\mathcal{Q}}_2(\mathrm{u},\mathrm{v})\parallel \le \parallel \mathsf{\Phi }\parallel {\mathrm{{\rm Y}}}_3+\parallel \mathsf{\Psi }\parallel \left({\displaystyle {\mathrm{{\rm Y}}}_4-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}}\right),\forall (\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0,\hfill \end{array}$$

(50)

and then

$${\parallel \mathcal{Q}(\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}\le \parallel \mathsf{\Phi }\parallel \left({\displaystyle {\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}}\right)+\parallel \mathsf{\Psi }\parallel \left({\displaystyle {\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}}\right),\forall (\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0,$$

(51)

that is, $\mathcal{Q}$ is uniformly bounded on ${\mathsf{\Omega }}_0$.

We finally prove that operator $\mathcal{Q}$ is compact. Let ${\nu }_1,{\nu }_2\in [0,1]$, ${\nu }_1<{\nu }_2$. Then for all $(\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0$, we find

$$\begin{array}{c}|{\mathcal{Q}}_1(\mathrm{u},\mathrm{v})({\nu }_2)-{\mathcal{Q}}_1(\mathrm{u},\mathrm{v})({\nu }_1)|\hfill \\ {\displaystyle \le \frac{{\nu }_2^{\alpha -1}-{\nu }_1^{\alpha -1}}{|\Delta |}\parallel \mathsf{\Phi }\parallel \left[{\displaystyle \frac{|{\mathsf{\Lambda }}_4|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}{d}_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}\frac{|a_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}{d}_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}{d}_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle +\frac{{\nu }_2^{\alpha -1}-{\nu }_1^{\alpha -1}}{|\Delta |}\parallel \mathsf{\Psi }\parallel \left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}\frac{|b_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}{d}_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +\frac{|{\mathsf{\Lambda }}_2|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}{d}_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}{d}_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle =\frac{{\nu }_2^{\alpha -1}-{\nu }_1^{\alpha -1}}{|\Delta |}\parallel \mathsf{\Phi }\parallel \left[{\displaystyle \frac{|{\mathsf{\Lambda }}_4|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma +1)}+|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}|a_i|\frac{{\xi }_i^{\alpha -{\varrho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i+1)}}\right.}\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}|c_i|\frac{{\zeta }_i^{\alpha -{\eta }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i+1)}}\right]\hfill \\ {\displaystyle +\frac{{\nu }_2^{\alpha -1}-{\nu }_1^{\alpha -1}}{|\Delta |}\parallel \mathsf{\Psi }\parallel \left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}|b_i|\frac{{\omega }_i^{\beta -{\sigma }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i+1)}+\frac{|{\mathsf{\Lambda }}_2|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta +1)}}\right.}\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}|d_i|\frac{{\theta }_i^{\beta -{\rho }_i}}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i+1)}}\right]\hfill \\ =({\nu }_2^{\alpha -1}-{\nu }_1^{\alpha -1})\left[\parallel \mathsf{\Phi }\parallel \left({\displaystyle {\mathrm{{\rm Y}}}_1-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}}\right)+\parallel \mathsf{\Psi }\parallel {\mathrm{{\rm Y}}}_2\right],\hfill \end{array}$$

(52)

which tends to zero as ${\nu }_2\to {\nu }_1$, independently of $(\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0$.

In a similar manner, we obtain

$$\begin{array}{c}|{\mathcal{Q}}_2(\mathrm{u},\mathrm{v})({\nu }_2)-{\mathcal{Q}}_2(\mathrm{u},\mathrm{v})({\nu }_1)|\hfill \\ \le ({\nu }_2^{\beta -1}-{\nu }_1^{\beta -1})\left[\parallel \mathsf{\Phi }\parallel {\mathrm{{\rm Y}}}_3+\parallel \mathsf{\Psi }\parallel \left({\displaystyle {\mathrm{{\rm Y}}}_4-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}}\right)\right],\hfill \end{array}$$

(53)

which tends to zero as ${\nu }_2\to {\nu }_1$, independently of $(\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_0$.

Therefore the operators ${\mathcal{Q}}_1$, ${\mathcal{Q}}_2$ and $\mathcal{Q}$ are equicontinuous. Utilizing the Arzela–Ascoli theorem, we ascertain the compactness of $\mathcal{Q}$ on ${\mathsf{\Omega }}_0$. Consequently, employing the Krasnosel’skii fixed-point theorem, we deduce the existence of at least one solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu ))\nu \in [0,1]$ to problem (1), (2) □

The forthcoming result relies on the Schaefer fixed-point theorem (refer to [34]).

Theorem 4.

Suppose that assumption $(J1)$ holds. In addition, we assume that the functions $\mathcal{F},\mathcal{G}:[0,1]\times {\mathbb{R}}^4\to \mathbb{R}$ are continuous and satisfy the condition

(J4)

There exist positive constants $T_1,T_2$ such that

$$\begin{array}{c}|\mathcal{F}(\nu ,{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)|\le T_1,|\mathcal{G}(\nu ,{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)|\le T_2,\hfill \\ \forall \nu \in [0,1],{\mathrm{x}}_i\in \mathbb{R},i=1,\dots ,4.\hfill \end{array}$$

(54)

Then, there exists at least one solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu )),\nu \in [0,1]$ of problem (1), (2).

Proof. 

We prove firstly that operator $\mathcal{E}$ is completely continuous. Operator $\mathcal{E}$ is continuous. Indeed, let $({\mathrm{u}}_n,{\mathrm{v}}_n)\in \mathcal{V},n\in \mathbb{N}$, with $({\mathrm{u}}_n,{\mathrm{v}}_n)\to (\mathrm{u},\mathrm{v})$, as $n\to \infty$ in $\mathcal{V}$. Then, for each $\nu \in [0,1]$, we deduce, as in the proof of Theorem 1, that

$$\begin{array}{c}|{\mathcal{E}}_1({\mathrm{u}}_n,{\mathrm{v}}_n)(\nu )-{\mathcal{E}}_1(\mathrm{u},\mathrm{v})(\nu )|\hfill \\ {\displaystyle \le \frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\alpha -1)}|{\mathcal{F}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle +\frac{1}{|\Delta |}\left[{\displaystyle \frac{|{\mathsf{\Lambda }}_4|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}|{\mathcal{F}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{a}}\frac{|a_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}|{\mathcal{F}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}|{\mathcal{F}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle +\frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_4|\sum _{i=1}^{\mathfrak{b}}\frac{|b_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}|{\mathcal{G}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +\frac{|{\mathsf{\Lambda }}_2|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}|{\mathcal{G}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_2|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}|{\mathcal{G}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right],\hfill \end{array}$$

(55)

and

$$\begin{array}{c}|{\mathcal{E}}_2({\mathrm{u}}_n,{\mathrm{v}}_n)(\nu )-{\mathcal{E}}_2(\mathrm{u},\mathrm{v})(\nu )|\hfill \\ {\displaystyle \le \frac{1}{|\Delta |}\left[{\displaystyle |{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{c}}\frac{|c_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-\mathfrak{q}\tau )}^{(\alpha -{\eta }_i-1)}|{\mathcal{F}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +\frac{|{\mathsf{\Lambda }}_3|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -\varsigma )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\alpha -\varsigma -1)}|{\mathcal{F}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{a}}\frac{|a_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-\mathfrak{q}\tau )}^{(\alpha -{\varrho }_i-1)}|{\mathcal{F}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right]\hfill \\ {\displaystyle +\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta )}{\int }_0^{\nu }{(\nu -\mathfrak{q}\tau )}^{(\beta -1)}|{\mathcal{G}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ {\displaystyle +\frac{1}{|\Delta |}\left[{\displaystyle \frac{|{\mathsf{\Lambda }}_1|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -\vartheta )}{\int }_0^1{(1-\mathfrak{q}\tau )}^{(\beta -\vartheta -1)}|{\mathcal{G}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right.}\hfill \\ {\displaystyle +|{\mathsf{\Lambda }}_1|\sum _{i=1}^{\mathfrak{d}}\frac{|d_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-\mathfrak{q}\tau )}^{(\beta -{\rho }_i-1)}|{\mathcal{G}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\hfill \\ +\left.{\displaystyle |{\mathsf{\Lambda }}_3|\sum _{i=1}^{\mathfrak{b}}\frac{|b_i|}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-\mathfrak{q}\tau )}^{(\beta -{\sigma }_i-1)}|{\mathcal{G}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|d_{\mathfrak{q}}\tau }\right].\hfill \end{array}$$

(56)

Because $\mathcal{F}$ and $\mathcal{G}$ are continuous, we obtain

$$\begin{array}{c}|{\mathcal{F}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{F}}_{\mathrm{u}\mathrm{v}}(\tau )|=|\mathcal{F}(\tau ,{\mathrm{u}}_n(\tau ),{\mathrm{v}}_n(\tau ),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_1}{\mathrm{u}}_n(\tau ),{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_1}{\mathrm{v}}_n(\tau ))\hfill \\ -\mathcal{F}(\tau ,\mathrm{u}(\tau ),\mathrm{v}(\tau ),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_1}\mathrm{u}(\tau ),{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_1}\mathrm{v}(\tau ))|\to 0,\hfill \\ |{\mathcal{G}}_{{\mathrm{u}}_n{\mathrm{v}}_n}(\tau )-{\mathcal{G}}_{\mathrm{u}\mathrm{v}}(\tau )|=|\mathcal{G}(\tau ,{\mathrm{u}}_n(\tau ),{\mathrm{v}}_n(\tau ),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_2}{\mathrm{u}}_n(\tau ),{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_2}{\mathrm{v}}_n(\tau ))\hfill \\ -\mathcal{G}(\tau ,\mathrm{u}(\tau ),\mathrm{v}(\tau ),{\mathrm{I}}_{\mathfrak{q}}^{{\delta }_2}\mathrm{u}(\tau ),{\mathrm{I}}_{\mathfrak{q}}^{{\gamma }_2}\mathrm{v}(\tau ))|\to 0,\hfill \end{array}$$

(57)

as $n\to \infty$, for all $\tau \in [0,1]$. Therefore, by the inequalities (55)–(57), we find

$$\parallel {\mathcal{E}}_1({\mathrm{u}}_n,{\mathrm{v}}_n)-{\mathcal{E}}_1(\mathrm{u},\mathrm{v})\parallel \to 0,\parallel {\mathcal{E}}_2({\mathrm{u}}_n,{\mathrm{v}}_n)-{\mathcal{E}}_2(\mathrm{u},\mathrm{v})\parallel \to 0,\mathrm{as}n\to \infty ,$$

(58)

and then

$$\parallel \mathcal{E}({\mathrm{u}}_n,{\mathrm{v}}_n)-\mathcal{E}(\mathrm{u},\mathrm{v}){\parallel }_{\mathcal{V}}\to 0,\mathrm{as}n\to \infty ,$$

(59)

that is, $\mathcal{E}$ is a continuous operator.

In what follows, we prove that $\mathcal{E}$ maps bounded sets into bounded sets in $\mathcal{V}$. For $R>0$, let ${\mathsf{\Omega }}_1={\{(\mathrm{u},\mathrm{v})\in \mathcal{V},\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}\le R\}$. Then, by using the inequalities (54) and similar computations as those from the first part of the proof of Theorem 1, we obtain

$$|{\mathcal{E}}_1(\mathrm{u},\mathrm{v})(\nu )|\le T_1{\mathrm{{\rm Y}}}_1+T_2{\mathrm{{\rm Y}}}_2,|{\mathcal{E}}_2(\mathrm{u},\mathrm{v})(\nu )|\le T_1{\mathrm{{\rm Y}}}_3+T_2{\mathrm{{\rm Y}}}_4,$$

(60)

for all $\nu \in [0,1]$ and $(\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_1$. Then, we deduce

$${\parallel \mathcal{E}(\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}\le T_1({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+T_2({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4),\forall (\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_1,$$

(61)

that is, $\mathcal{E}({\mathsf{\Omega }}_1)$ is bounded.

Subsequently, we will demonstrate that $\mathcal{E}$ transforms bounded sets into equicontinuous sets. To illustrate this, consider ${\nu }_1,,{\nu }_2\in [0,1]$ with ${\nu }_1<{\nu }_2$ and $(\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_1$. Employing computations akin to those found in the proofs of Theorems 2 and 3, we arrive at the following conclusions

$$\begin{array}{c}|{\mathcal{E}}_1(\mathrm{u},\mathrm{v})({\nu }_2)-{\mathcal{E}}_1(\mathrm{u},\mathrm{v})({\nu }_1)|\hfill \\ \le |{\mathcal{P}}_1(\mathrm{u},\mathrm{v})({\nu }_2)-{\mathcal{P}}_1(\mathrm{u},\mathrm{v})({\nu }_1)|+|{\mathcal{Q}}_1(\mathrm{u},\mathrm{v})({\nu }_2)-{\mathcal{Q}}_1(\mathrm{u},\mathrm{v})({\nu }_1)|\hfill \\ {\displaystyle \le \frac{T_1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}({\nu }_2^{\alpha }-{\nu }_1^{\alpha })+\left[T_1\left({\displaystyle {\mathrm{{\rm Y}}}_1-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\alpha +1)}}\right)+T_2{\mathrm{{\rm Y}}}_2\right]({\nu }_2^{\alpha -1}-{\nu }_1^{\alpha -1})\to 0,}\hfill \\ |{\mathcal{E}}_2(\mathrm{u},\mathrm{v})({\nu }_2)-{\mathcal{E}}_2(\mathrm{u},\mathrm{v})({\nu }_1)|\hfill \\ \le |{\mathcal{P}}_2(\mathrm{u},\mathrm{v})({\nu }_2)-{\mathcal{P}}_2(\mathrm{u},\mathrm{v})({\nu }_1)|+|{\mathcal{Q}}_2(\mathrm{u},\mathrm{v})({\nu }_2)-{\mathcal{Q}}_2(\mathrm{u},\mathrm{v})({\nu }_1)|\hfill \\ {\displaystyle \le \frac{T_2}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}({\nu }_2^{\beta }-{\nu }_1^{\beta })+\left[T_1{\mathrm{{\rm Y}}}_3+T_4\left({\displaystyle {\mathrm{{\rm Y}}}_4-\frac{1}{{\mathsf{\Gamma }}_{\mathfrak{q}}(\beta +1)}}\right)\right]({\nu }_2^{\beta -1}-{\nu }_1^{\beta -1})\to 0,}\hfill \end{array}$$

(62)

as ${\nu }_2\to {\nu }_1$, independently of $(\mathrm{u},\mathrm{v})\in {\mathsf{\Omega }}_2$.

Therefore, the operators ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ are equicontinuous, and so $\mathcal{E}$ is also an equicontinuous operator on ${\mathsf{\Omega }}_2$. Then, the operator $\mathcal{E}:{\mathsf{\Omega }}_2\to \mathcal{V}$ is completely continuous using the Arzela–Ascoli theorem.

In the concluding section of the proof, we establish the boundedness of the set $\mathcal{Z}=\{(\mathrm{u},\mathrm{v})\in \mathcal{V},(\mathrm{u},\mathrm{v})=\lambda \mathcal{E}(\mathrm{u},\mathrm{v}),0\le \lambda \le 1\}$. Take $(\mathrm{u},\mathrm{v})\in \mathcal{V}$, implying there exists $\lambda \in [0,1]$ such that $(\mathrm{u},\mathrm{v})=\lambda \mathcal{E}(\mathrm{u},\mathrm{v})$ or $\mathrm{u}(\nu )=\lambda {\mathcal{E}}_1(\mathrm{u},\mathrm{v})(\nu )$ and $\mathrm{v}(\nu )=\lambda {\mathcal{E}}_2(\mathrm{u},\mathrm{v})(\nu )$ for all $\nu \in [0,1]$. Utilizing $(J4)$, we infer, similarly to the initial part of this proof, that

$$\begin{array}{c}|\mathrm{u}(\nu )|=\lambda |{\mathcal{E}}_1(\mathrm{u},\mathrm{v})(\nu )|\le |{\mathcal{E}}_1(\mathrm{u},\mathrm{v})(\nu )|\le T_1{\mathrm{{\rm Y}}}_1+T_2{\mathrm{{\rm Y}}}_2,\forall \nu \in [0,1],\hfill \\ |\mathrm{v}(\nu )|=\lambda |{\mathcal{E}}_2(\mathrm{u},\mathrm{v})(\nu )|\le |{\mathcal{E}}_2(\mathrm{u},\mathrm{v})(\nu )|\le T_1{\mathrm{{\rm Y}}}_3+T_2{\mathrm{{\rm Y}}}_4,\forall \nu \in [0,1],\hfill \end{array}$$

(63)

and then

$${\parallel (\mathrm{u},\mathrm{v})|}_{\mathcal{V}}=\parallel \mathrm{u}\parallel +\parallel \mathrm{v}\parallel \le T_1({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)+T_2({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4).$$

(64)

This final inequality indicates the boundedness of the set $\mathcal{Z}$. Consequently, employing the Schaefer fixed-point theorem, we establish the existence of at least one fixed point for the operator $\mathcal{E}$. Thus, problem (1), (2) possesses at least one solution. This concludes the proof. □

In the subsequent existence theorem, we will employ the Leray–Schauder nonlinear alternative (refer to [35]).

Theorem 5.

Assume that assumption $(J1)$ holds. In addition, we suppose that the functions $\mathcal{F},\mathcal{G}:[0,1]\times {\mathbb{R}}^4\to \mathbb{R}$ are continuous and the following conditions are satisfied

(J5)

There exist the functions ${\phi }_1,{\phi }_2\in C([0,1],{\mathbb{R}}_{+})$ and the functions ${\psi }_1,{\psi }_2\in C({({\mathbb{R}}_{+})}^4,{\mathbb{R}}_{+})$ nondecreasing in each of variables such that

$$\begin{array}{c}|\mathcal{F}(\nu ,{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)|\le {\phi }_1(\nu ){\psi }_1(|{\mathrm{x}}_1|,|{\mathrm{x}}_2|,|{\mathrm{x}}_3|,|{\mathrm{x}}_4|),\hfill \\ |\mathcal{G}(\nu ,{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)|\le {\phi }_2(\nu ){\psi }_2(|{\mathrm{x}}_1|,|{\mathrm{x}}_2|,|{\mathrm{x}}_3|,|{\mathrm{x}}_4|),\hfill \end{array}$$

(65)

for all $\nu \in [0,1]$, ${\mathrm{x}}_i\in \mathbb{R},i=1,\dots ,4$.

(J6)

There exists a positive constant V such that

$$\begin{array}{c}\parallel {\phi }_1\parallel {\psi }_1\left({\displaystyle V,V,\frac{V}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)},\frac{V}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)\hfill \\ +\parallel {\phi }_2\parallel {\psi }_2\left({\displaystyle V,V,\frac{V}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)},\frac{V}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)<V.\hfill \end{array}$$

(66)

Then, the $\mathfrak{q}$-fractional boundary value problem (1), (2) has at least one solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu )),\nu \in [0,1].$

Proof. 

We define the set $\mathcal{W}=\{(\mathrm{u},\mathrm{v})\in \mathcal{V},\parallel (\mathrm{u},\mathrm{v}){\parallel }_{\mathcal{V}}<V\}$, where V is the constant given by (66). The operator $\mathcal{E}:\overline{\mathcal{W}}\to \mathcal{V}$ is completely continuous. We suppose that there exist $(\mathrm{u},\mathrm{v})\in \partial \mathcal{W}$ such that $(\mathrm{u},\mathrm{v})=\mu \mathcal{E}(\mathrm{u},\mathrm{v})$ for some $\mu \in (0,1)$. Then, we obtain

$$\begin{array}{c}|\mathrm{u}(\nu )|=\mu |{\mathcal{E}}_1(\mathrm{u},\mathrm{v})(\nu )|\le |{\mathcal{E}}_1(\mathrm{u},\mathrm{v})(\nu )|\hfill \\ \le \parallel {\phi }_1\parallel {\psi }_1\left({\displaystyle \parallel \mathrm{u}\parallel ,\parallel \mathrm{v}\parallel ,\frac{\parallel \mathrm{u}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)},\frac{\parallel \mathrm{v}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right){\mathrm{{\rm Y}}}_1\hfill \\ +\parallel {\phi }_2\parallel {\psi }_2\left({\displaystyle \parallel \mathrm{u}\parallel ,\parallel \mathrm{v}\parallel ,\frac{\parallel \mathrm{u}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)},\frac{\parallel \mathrm{v}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right){\mathrm{{\rm Y}}}_2,\hfill \\ |\mathrm{v}(\nu )|=\mu |{\mathcal{E}}_2(\mathrm{u},\mathrm{v})(\nu )|\le |{\mathcal{E}}_2(\mathrm{u},\mathrm{v})(\nu )|\hfill \\ \le \parallel {\phi }_1\parallel {\psi }_1\left({\displaystyle \parallel \mathrm{u}\parallel ,\parallel \mathrm{v}\parallel ,\frac{\parallel \mathrm{u}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)},\frac{\parallel \mathrm{v}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right){\mathrm{{\rm Y}}}_3\hfill \\ +\parallel {\phi }_2\parallel {\psi }_2\left({\displaystyle \parallel \mathrm{u}\parallel ,\parallel \mathrm{v}\parallel ,\frac{\parallel \mathrm{u}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)},\frac{\parallel \mathrm{v}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right){\mathrm{{\rm Y}}}_4,\hfill \end{array}$$

(67)

for all $\nu \in [0,1]$, and so we find

$$\begin{array}{c}{\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}=\parallel \mathrm{u}\parallel +\parallel \mathrm{v}\parallel \hfill \\ \le \parallel {\phi }_1\parallel {\psi }_1\left({\displaystyle \parallel \mathrm{u}\parallel ,\parallel \mathrm{v}\parallel ,\frac{\parallel \mathrm{u}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)},\frac{\parallel \mathrm{v}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)\hfill \\ +\parallel {\phi }_2\parallel {\psi }_2\left({\displaystyle \parallel \mathrm{u}\parallel ,\parallel \mathrm{v}\parallel ,\frac{\parallel \mathrm{u}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)},\frac{\parallel \mathrm{v}\parallel }{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\hfill \\ \le \parallel {\phi }_1\parallel {\psi }_1\left({\displaystyle {\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}},{\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}},\frac{{\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)},\frac{{\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)\hfill \\ +\parallel {\phi }_2\parallel {\psi }_2\left({\displaystyle {\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}},{\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}},\frac{{\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)},\frac{{\parallel (\mathrm{u},\mathrm{v})\parallel }_{\mathcal{V}}}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4).\hfill \end{array}$$

(68)

Therefore, we deduce

$$\begin{array}{c}V/\left[\parallel {\phi }_1\parallel {\psi }_1\left({\displaystyle V,V,\frac{V}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_1+1)},\frac{V}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_1+1)}}\right)({\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_3)\right.\hfill \\ \left.+\parallel {\phi }_2\parallel {\psi }_2\left({\displaystyle V,V,\frac{V}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\delta }_2+1)},\frac{V}{{\mathsf{\Gamma }}_{\mathfrak{q}}({\gamma }_2+1)}}\right)({\mathrm{{\rm Y}}}_2+{\mathrm{{\rm Y}}}_4)\right]\le 1,\hfill \end{array}$$

(69)

which, by (66), is a contradiction.

We conclude that there is no $(\mathrm{u},\mathrm{v})\in \partial \mathcal{W}$ such that $(\mathrm{u},\mathrm{v})=\mu \mathcal{E}(\mathrm{u},\mathrm{v})$ for some $\mu \in (0,1)$. Consequently, by employing the Leray–Schauder nonlinear alternative, we infer that $\mathcal{E}$ possesses a fixed point $(\mathrm{u},\mathrm{v})\in \overline{\mathcal{W}}$, serving as a solution to problem (1), (2). This concludes the proof. □

4. Examples

Let $q=\frac{1}{2}$, $\alpha =\frac{5}{2}$, $n=3$, $\beta =\frac{10}{3}$, $m=4$, ${\delta }_1=\frac{34}{11}$, ${\gamma }_1=\frac{71}{25}$, ${\delta }_2=\frac{48}{13}$, ${\gamma }_2=\frac{95}{32}$, $\varsigma =\frac{6}{5}$, $\vartheta =\frac{9}{7}$, $\mathfrak{a}=\mathfrak{b}=\mathfrak{c}=\mathfrak{d}=1$, ${\varrho }_1=\frac{2}{9}$, ${\sigma }_1=\frac{5}{4}$, ${\eta }_1=\frac{2}{3}$, ${\rho }_1=\frac{21}{10}$, ${\xi }_1=\frac{1}{8}$, ${\omega }_1=\frac{1}{2}$, ${\zeta }_1=\frac{1}{4}$, ${\theta }_1=\frac{1}{16}$, $a_1=3$, $b_1=-\frac{7}{12}$, $c_1=\frac{4}{15}$, $d_1=-2$.

We consider the system of $\mathfrak{q}$-difference equations

$$\left\{\begin{array}{c}({\mathrm{D}}_{1/2}^{5/2}\mathrm{u})(\nu )+\mathcal{F}\left(\nu ,\mathrm{u}(\nu ),\mathrm{v}(\nu ),{\mathrm{I}}_{1/2}^{34/11}\mathrm{u}(\nu ),{\mathrm{I}}_{1/2}^{71/25}\mathrm{v}(\nu )\right)=0,\nu \in (0,1),\hfill \\ ({\mathrm{D}}_{1/2}^{10/3}\mathrm{v})(\nu )+\mathcal{G}\left(\nu ,\mathrm{u}(\nu ),\mathrm{v}(\nu ),{\mathrm{I}}_{1/2}^{48/13}\mathrm{u}(\nu ),{\mathrm{I}}_{1/2}^{95/32}\mathrm{v}(\nu )\right)=0,\nu \in (0,1),\hfill \end{array}\right.$$

(70)

with the boundary conditions

$$\left\{\begin{array}{c}\mathrm{u}(0)={\mathrm{D}}_{1/2}\mathrm{u}(0)=0,{\mathrm{D}}_{1/2}^{6/5}\mathrm{u}(1)=3{\mathrm{D}}_{1/2}^{2/9}\mathrm{u}\left(\frac{1}{8}\right)-\frac{7}{12}{\mathrm{D}}_{1/2}^{5/4}\mathrm{v}\left(\frac{1}{2}\right),\hfill \\ \mathrm{v}(0)={\mathrm{D}}_{1/2}\mathrm{v}(0)={\mathrm{D}}_{1/2}^2\mathrm{v}(0)=0,{\mathrm{D}}_{1/2}^{9/7}\mathrm{v}(1)=\frac{4}{15}{\mathrm{D}}_{1/2}^{2/3}\mathrm{u}\left(\frac{1}{4}\right)-2{\mathrm{D}}_{1/4}^{21/10}\mathrm{v}\left(\frac{1}{16}\right).\hfill \end{array}\right.$$

(71)

By using the Mathematica program, we obtain ${\mathsf{\Lambda }}_1\approx 1.05480868$, ${\mathsf{\Lambda }}_2\approx -0.48038739$, ${\mathsf{\Lambda }}_3\approx 0.10410274$, ${\mathsf{\Lambda }}_4\approx 3.76405767$, and $\Delta \approx 4.02037036\ne 0$. Therefore, assumption $(J1)$ is satisfied.

In addition, after some computations, we find ${\mathrm{{\rm Y}}}_1\approx 1.37638598$, ${\mathrm{{\rm Y}}}_2\approx 0.16737542$, ${\mathrm{{\rm Y}}}_3\approx 0.02789379$, and ${\mathrm{{\rm Y}}}_4\approx 0.49956085$.

Example 1.

We consider the functions

$$\begin{array}{c}{\displaystyle \mathcal{F}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=\frac{sin(3\nu +2)}{\sqrt[4]{{\nu }^3+2}}+\frac{1}{9}{e}^{-{(\nu -1)}^2}\sqrt{{\mathrm{u}}^2+1}-\frac{1}{17}{e}^{-2{\nu }^3+1}arctan\mathrm{v}}\hfill \\ {\displaystyle +\frac{\nu }{4({\nu }^2+1)}sin\mathrm{x}-\frac{3\nu }{25}{cos}^2\mathrm{y},}\hfill \\ {\displaystyle \mathcal{G}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=-{\nu }^2+5+\frac{1}{8(\nu +1)}{sin}^2\mathrm{u}-\frac{2}{31({\nu }^2+4)}\frac{|\mathrm{v}|}{1+|\mathrm{v}|}-\frac{1}{2}\mathrm{x}}\hfill \\ {\displaystyle +\frac{1}{6({\nu }^3+1)}\frac{\mathrm{y}}{{\mathrm{y}}^2+1},}\hfill \end{array}$$

(72)

for all $\nu \in [0,1]$, $\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y}\in \mathbb{R}$.

For these continuous functions, we obtain the following inequalities

$$\begin{array}{c}|\mathcal{F}(\nu ,{\mathrm{u}}_1,{\mathrm{v}}_1,{\mathrm{x}}_1,{\mathrm{y}}_1)-\mathcal{F}(\nu ,{\mathrm{u}}_2,{\mathrm{v}}_2,{\mathrm{x}}_2,{\mathrm{y}}_2)|\hfill \\ \le {\mathcal{H}}_1(\nu )|{\mathrm{u}}_1-{\mathrm{u}}_2|+{\mathcal{H}}_2(\nu )|{\mathrm{v}}_1-{\mathrm{v}}_2|+{\mathcal{H}}_3(\nu )|{\mathrm{x}}_1-{\mathrm{x}}_2|+{\mathcal{H}}_4(\nu )|{\mathrm{y}}_1-{\mathrm{y}}_2|,\hfill \\ |\mathcal{G}(\nu ,{\mathrm{u}}_1,{\mathrm{v}}_1,{\mathrm{x}}_1,{\mathrm{y}}_1)-\mathcal{G}(\nu ,{\mathrm{u}}_2,{\mathrm{v}}_2,{\mathrm{x}}_2,{\mathrm{y}}_2)|\hfill \\ \le {\mathcal{K}}_1(\nu )|{\mathrm{u}}_1-{\mathrm{u}}_2|+{\mathcal{K}}_2(\nu )|{\mathrm{v}}_1-{\mathrm{v}}_2|+{\mathcal{K}}_3(\nu )|{\mathrm{x}}_1-{\mathrm{x}}_2|+{\mathcal{K}}_4(\nu )|{\mathrm{y}}_1-{\mathrm{y}}_2|,\hfill \end{array}$$

(73)

for all $\nu \in [0,1]$, ${\mathrm{u}}_i,{\mathrm{v}}_i,{\mathrm{x}}_i,{\mathrm{y}}_i\in \mathbb{R},i=1,2$, where

$$\begin{array}{c}{\displaystyle {\mathcal{H}}_1(\nu )=\frac{1}{9}{e}^{-{(\nu -1)}^2},{\mathcal{H}}_2(\nu )=\frac{1}{17}{e}^{-2{\nu }^3+1},{\mathcal{H}}_3(\nu )=\frac{\nu }{4({\nu }^2+1)},{\mathcal{H}}_4(\nu )=\frac{6\nu }{25},}\hfill \\ {\displaystyle {\mathcal{K}}_1(\nu )=\frac{1}{4(\nu +1)},{\mathcal{K}}_2(\nu )=\frac{2}{31({\nu }^2+4)},{\mathcal{K}}_3(\nu )=\frac{1}{2},{\mathcal{K}}_4(\nu )=\frac{1}{6({\nu }^3+1)},}\hfill \end{array}$$

(74)

for all $\nu \in [0,1]$. The functions ${\mathcal{H}}_i,{\mathcal{K}}_i,i=1,\dots ,4$ are continuous, and we find ${\mathfrak{h}}_1^{\ast }\approx 0.11111111$, ${\mathfrak{h}}_2^{\ast }\approx 0.15989893$, ${\mathfrak{h}}_3^{\ast }=0.125$, ${\mathfrak{h}}_4^{\ast }=0.24$, ${\mathfrak{k}}_1^{\ast }=0.25$, ${\mathfrak{k}}_2^{\ast }\approx 0.01612903$, ${\mathfrak{k}}_3^{\ast }=0.5$, and ${\mathfrak{k}}_4^{\ast }\approx 0.16666667$. In addition, we obtain ${\mathsf{\Theta }}_1\approx 0.46864611$, ${\mathsf{\Theta }}_2\approx 0.41971398$, and so ${\mathsf{\Theta }}_0={\mathsf{\Theta }}_1<1$. Therefore, assumption $(J2)$ and condition (16) are satisfied. Then, by Theorem 1, we deduce that problem (70), (71) with the functions $\mathcal{F}$ and $\mathcal{G}$ given by (72) has a unique solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu )),\nu \in [0,1]$.

Example 2.

We consider the functions

$$\begin{array}{c}{\displaystyle \mathcal{F}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=\frac{cos({\nu }^3+7)}{2\nu +5}-\frac{1}{{\nu }^2+6}\frac{4{\mathrm{u}}^4+1}{{\mathrm{u}}^4+3}+\frac{({\nu }^5+2)e^{-3\nu +1}}{6({\nu }^4+27)}sin\mathrm{v}}\hfill \\ {\displaystyle -\frac{\nu }{8({\nu }^2+1)}{cos}^2\mathrm{x}+\frac{1}{\nu +4}\frac{|\mathrm{y}|}{1+3|\mathrm{y}|},}\hfill \\ {\displaystyle \mathcal{G}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=-\frac{e^{-\nu +4}}{3{\nu }^2+1}+\frac{1}{{\nu }^2+7}\frac{\mathrm{u}}{{\mathrm{u}}^2+1}-\frac{\nu +1}{9\sqrt{{\nu }^4+5}}{e}^{-{\mathrm{v}}^2}}\hfill \\ {\displaystyle +\frac{2}{11}{sin}^2\mathrm{x}-\frac{\nu |\mathrm{y}|}{6(1+4|\mathrm{y}|)},}\hfill \end{array}$$

(75)

for all $\nu \in [0,1]$, $\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y}\in \mathbb{R}$. For these continuous functions, we obtain the following inequalities

$$\begin{array}{c}{\displaystyle |\mathcal{F}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})\le \frac{1}{2\nu +5}+\frac{4}{{\nu }^2+6}+\frac{({\nu }^5+2)e^{-3\nu +1}}{6({\nu }^4+27)}+\frac{\nu }{8({\nu }^2+1)}+\frac{1}{3(\nu +4)}=:\mathsf{\Phi }(\nu ),}\hfill \\ {\displaystyle |\mathcal{G}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})\le \frac{e^{-\nu +4}}{3{\nu }^2+1}+\frac{1}{2({\nu }^2+7)}+\frac{\nu +1}{9\sqrt{{\nu }^4+5}}+\frac{2}{11}+\frac{\nu }{24}=:\mathsf{\Psi }(\nu ),}\hfill \end{array}$$

(76)

for all $\nu \in [0,1]$, $\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y}\in \mathbb{R}$. In addition, we find

$$\begin{array}{c}|\mathcal{F}(\nu ,{\mathrm{u}}_1,{\mathrm{v}}_1,{\mathrm{x}}_1,{\mathrm{y}}_1)-\mathcal{F}(\nu ,{\mathrm{u}}_2,{\mathrm{v}}_2,{\mathrm{x}}_2,{\mathrm{y}}_2)|\hfill \\ \le {\mathcal{H}}_1(\nu )|{\mathrm{u}}_1-{\mathrm{u}}_2|+{\mathcal{H}}_2(\nu )|{\mathrm{v}}_1-{\mathrm{v}}_2|+{\mathcal{H}}_3(\nu )|{\mathrm{x}}_1-{\mathrm{x}}_2|+{\mathcal{H}}_4(\nu )|{\mathrm{y}}_1-{\mathrm{y}}_2|,\hfill \\ |\mathcal{G}(\nu ,{\mathrm{u}}_1,{\mathrm{v}}_1,{\mathrm{x}}_1,{\mathrm{y}}_1)-\mathcal{G}(\nu ,{\mathrm{u}}_2,{\mathrm{v}}_2,{\mathrm{x}}_2,{\mathrm{y}}_2)|\hfill \\ \le {\mathcal{K}}_1(\nu )|{\mathrm{u}}_1-{\mathrm{u}}_2|+{\mathcal{K}}_2(\nu )|{\mathrm{v}}_1-{\mathrm{v}}_2|+{\mathcal{K}}_3(\nu )|{\mathrm{x}}_1-{\mathrm{x}}_2|+{\mathcal{K}}_4(\nu )|{\mathrm{y}}_1-{\mathrm{y}}_2|,\hfill \end{array}$$

(77)

for all $\nu \in [0,1]$, ${\mathrm{u}}_i,{\mathrm{v}}_i,{\mathrm{x}}_i,{\mathrm{y}}_i\in \mathbb{R},i=1,2$, where

$$\begin{array}{c}{\displaystyle {\mathcal{H}}_1(\nu )=\frac{2.9678}{{\nu }^2+6},{\mathcal{H}}_2(\nu )=\frac{({\nu }^5+2)e^{-3\nu +1}}{6({\nu }^4+27)},{\mathcal{H}}_3(\nu )=\frac{\nu }{4({\nu }^2+1)},{\mathcal{H}}_4(\nu )=\frac{1}{\nu +4},}\hfill \\ {\displaystyle {\mathcal{K}}_1(\nu )=\frac{1}{{\nu }^2+7},{\mathcal{K}}_2(\nu )=\frac{0.8578(\nu +1)}{9\sqrt{{\nu }^4+5}},{\mathcal{K}}_3(\nu )=\frac{4}{11},{\mathcal{K}}_4(\nu )=\frac{\nu }{6},}\hfill \end{array}$$

(78)

for all $\nu \in [0,1]$. Therefore, assumptions $(J2)$ and $(J3)$ are satisfied. In addition, we obtain ${\mathfrak{h}}_1^{\ast }\approx 0.49463333$, ${\mathfrak{h}}_2^{\ast }\approx 0.03355903$, ${\mathfrak{h}}_3^{\ast }=0.125$, ${\mathfrak{h}}_4^{\ast }=0.25$, ${\mathfrak{k}}_1^{\ast }\approx 0.14285714$, ${\mathfrak{k}}_2^{\ast }\approx 0.07782119$, ${\mathfrak{k}}_3^{\ast }\approx 0.36363636$, ${\mathfrak{k}}_4^{\ast }\approx 0.16666667$, and so ${\mathfrak{L}}_1\approx 0.56557327$, ${\mathfrak{L}}_2\approx 0.25625052$, and ${\mathfrak{L}}_0={\mathfrak{L}}_1<1$. Then, condition (31) is also satisfied. Therefore, by Theorem 2, we conclude that problem (70), (71) with functions (75) has at least one solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu )),\nu \in [0,1]$.

Example 3.

We consider the functions

$$\begin{array}{c}{\displaystyle \mathcal{F}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=\frac{e^{-4\nu +3}}{\sqrt{{\nu }^3+1}}cos(\mathrm{u}\mathrm{v}+{\mathrm{x}}^2-\mathrm{y})-\frac{1}{6}\sqrt[3]{{\nu }^2+12},}\hfill \\ {\displaystyle \mathcal{G}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=\frac{1}{2(\nu +1)}{e}^{-{(\mathrm{u}+\mathrm{v})}^2}-arctan\sqrt{{\mathrm{x}}^2+2}+\frac{{\nu }^2}{{\nu }^5+3}\frac{4{\mathrm{y}}^2+1}{{\mathrm{y}}^2+3}-sin(2\nu +1),}\hfill \end{array}$$

(79)

for all $\nu \in [0,1]$, $\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y}\in \mathbb{R}$. For these continuous functions, we obtain the following inequalities

$$|\mathcal{F}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})|<20.4919=T_1,|\mathcal{G}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})|<4.0708=T_2.$$

(80)

Therefore, assumption $(J4)$ is satisfied. Then, by Theorem 4 we infer that problem (70), (71) with functions (79) has at least one solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu )),\nu \in [0,1]$.

Example 4.

We consider the functions

$$\begin{array}{c}{\displaystyle \mathcal{F}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=\frac{2}{{\nu }^3+5}\left({\displaystyle \frac{\mathrm{u}-3\mathrm{x}+1}{34}sin({\mathrm{u}}^2-8\mathrm{v}+{\mathrm{x}}^3)-\frac{\sqrt{{\mathrm{v}}^2+4{\mathrm{y}}^2}}{27}arctan\frac{\mathrm{u}+\mathrm{x}\mathrm{y}}{{\mathrm{y}}^2+3}+\frac{1}{9}}\right),}\hfill \\ {\displaystyle \mathcal{G}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=\frac{3\nu }{{\nu }^4+71}\left({\displaystyle \frac{4\mathrm{u}-5\mathrm{y}}{43}cos(\mathrm{u}\mathrm{x}-3\mathrm{v}\mathrm{y})-\frac{\mathrm{v}+7\mathrm{x}}{11}{e}^{-{(\mathrm{u}+3\mathrm{y})}^2}-\frac{2}{15}}\right),}\hfill \end{array}$$

(81)

for all $\nu \in [0,1]$, $\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y}\in \mathbb{R}$. For these continuous functions, we obtain the inequalities

$$\begin{array}{c}{\displaystyle |\mathcal{F}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})|\le \frac{2}{{\nu }^3+5}\left({\displaystyle \frac{|\mathrm{u}|}{34}+\frac{\pi |\mathrm{v}|}{54}+\frac{3|\mathrm{x}|}{34}+\frac{\pi |\mathrm{y}|}{27}+\frac{43}{306}}\right),}\hfill \\ {\displaystyle |\mathcal{G}(\nu ,\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})|\le \frac{3\nu }{{\nu }^4+71}\left({\displaystyle \frac{4|\mathrm{u}|}{43}+\frac{|\mathrm{v}|}{11}+\frac{7|\mathrm{x}|}{11}+\frac{5|\mathrm{y}|}{43}+\frac{2}{15}}\right),}\hfill \end{array}$$

(82)

for all $\nu \in [0,1]$, $\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y}\in \mathbb{R}$. Therefore, the continuous functions ${\phi }_i,{\psi }_i,i=1,2$ from (65) are given by

$$\begin{array}{c}{\displaystyle {\phi }_1(\nu )=\frac{2}{{\nu }^3+5},{\psi }_1(a,b,c,d)=\frac{a}{34}+\frac{\pi b}{54}+\frac{3c}{34}+\frac{\pi d}{27}+\frac{43}{306},}\hfill \\ {\displaystyle {\phi }_2(\nu )=\frac{3\nu }{{\nu }^4+71},{\psi }_2(a,b,c,d)=\frac{4a}{43}+\frac{b}{11}+\frac{7c}{11}+\frac{5d}{43}+\frac{2}{15},}\hfill \end{array}$$

(83)

for all $\nu \in [0,1]$ and $a,b,c,d\in {\mathbb{R}}_{+}$; that is, assumption $(J5)$ is satisfied. In addition, we find $\parallel {\phi }_1\parallel =\frac{2}{5}$ and $\parallel {\phi }_2\parallel =\frac{3}{72}$. If $V\ge 0.1$, then assumption $(J6)$ is also satisfied. Therefore, by Theorem 5, we conclude the existence of at least one solution $(\mathrm{u}(\nu ),\mathrm{v}(\nu )),\nu \in [0,1]$ for problem (70), (71) with functions (81).

5. Conclusions

In this study, we explored the existence and uniqueness of solutions to a system of fractional $\mathfrak{q}$-difference equations with fractional $\mathfrak{q}$-integral terms (1), subject to the multi-point boundary conditions (2), which encompass $\mathfrak{q}$-derivatives and fractional $\mathfrak{q}$-derivatives of diverse orders. We associated an operator ($\mathcal{E}$) on the space $\mathcal{V}$ with our problem, where the solutions of (1) and (2) correspond to the fixed points of this operator. Consequently, our main results involved the utilization of various fixed-point theorems, including the Banach contraction mapping principle (employed in Theorem 1), the Krasnosel’skii fixed-point theorem for the sum of two operators (applied in Theorems 2 and 3), the Schaefer fixed-point theorem (utilized in Theorem 4), and the Leray–Schauder nonlinear alternative (employed in Theorem 5). To exemplify our findings, we concluded by presenting several illustrative examples.