Positive Solutions for a System of Fractional q-Difference Equations with Multi-Point Boundary Conditions

Abstract

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of fractional q-difference equations that include fractional q-integrals. This investigation is carried out under coupled multi-point boundary conditions featuring q-derivatives and fractional q-derivatives of various orders. The proofs of our principal findings employ a range of fixed-point theorems, including the Guo–Krasnosel’skii fixed-point theorem, the Leggett–Williams fixed-point theorem, the Schauder fixed-point theorem, and the Banach contraction mapping principle.

1. Introduction

In recent decades, there has been a notable surge in the exploration of nonlocal boundary value problems, including those involving multi-point scenarios, within the realm of ordinary differential or difference equations. This area of research is experiencing rapid growth, spurred not only by theoretical interest but also by the practical applications of modeling various phenomena in engineering, physics, and life sciences. To illustrate, consider systems with feedback controls, such as the steady states of a thermostat. In this context, a second-order ordinary differential equation subject to a three-point boundary condition can capture the dynamics, where a controller at one end adjusts the heat based on the temperature recorded at another point. Another instance is found in the vibrations of a guy wire with a uniform cross-section composed of N parts of varying densities. Such scenarios can be effectively modeled as multi-point boundary value problems (refer to [1]).

In this paper, we will investigate the system of fractional q-difference equations

$$\left\{\begin{array}{c}\left(D_q^{\alpha }\mathfrak{u}\right)\left(\mathfrak{t}\right)+\mathfrak{f}\left(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{v}\left(\mathfrak{t}\right),I_q^{{\delta }_1}\mathfrak{u}(\mathfrak{t}),I_q^{{\gamma }_1}\mathfrak{v}\left(\mathfrak{t}\right)\right)=0,\mathfrak{t}\in [0,1],\hfill \\ \left(D_q^{\beta }\mathfrak{v}\right)\left(\mathfrak{t}\right)+\mathfrak{g}\left(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{v}\left(\mathfrak{t}\right),I_q^{{\delta }_2}\mathfrak{u}\left(\mathfrak{t}\right),I_q^{{\gamma }_2}\mathfrak{v}\left(\mathfrak{t}\right)\right)=0,\mathfrak{t}\in [0,1],\hfill \end{array}\right.$$

(1)

supplemented with the multi-point boundary conditions

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

(2)

Here, $q\in (0,1)$, $\alpha ,\beta \in \mathbb{R}$, $\alpha \in (n-1,n]$, $\beta \in (m-1,m]$, $n,m\in \mathbb{N}$, $n,m\ge 3$; $a,b,c,d\in \mathbb{N}$, $0\le {\varrho }_i\le \varsigma <\alpha -1$, $i=1,\dots ,a$, $\varsigma \ge 1$, $0\le {\eta }_i\le \varsigma$, $i=1,\dots ,c$, $0\le {\sigma }_i\le \vartheta <\beta -1$, $i=1,\dots ,b$, $\vartheta \ge 1$, $0\le {\rho }_i\le \vartheta$, $i=1,\dots ,d$; ${\delta }_i,{\gamma }_i>0$, $i=1,2$; ${\xi }_i,{\omega }_j,{\zeta }_k,{\theta }_{\iota }\in (0,1)$, and $a_i,b_j,c_k,d_{\iota }\ge 0$ for $i=1,\dots ,a$, $j=1,\dots ,b$, $k=1,\dots ,c$, $\iota =1,\dots ,d$; $D_q^{\kappa }$ is the fractional q-derivative of order $\kappa$ for $\kappa =\alpha ,\beta ,\varsigma ,\vartheta ,{\varrho }_i,{\sigma }_j,{\eta }_k,{\rho }_{\iota }$, $i=1,\dots ,a$, $j=1,\dots ,b$, $k=1,\dots ,c$, $\iota =1,\dots ,d$; $D_q^i$ is the q-derivative of order i for $i=0,\dots ,n-2$ and $i=0,\dots ,m-2$; and $I_q^{\kappa }$ is the fractional q-integral of order $\kappa$ for $\kappa ={\delta }_i,{\gamma }_i,i=1,2$.

We will establish conditions on the functions f and g under which problem (1),(2) possesses at least one positive solution. Our proofs will leverage several fixed point theorems, including the Guo–Krasnosel’skii fixed point theorem, the Leggett–Williams fixed point theorem, the Schauder fixed point theorem, and the Banach contraction mapping principle. Subsequently, we will provide references to pertinent papers that are closely related to our investigated problem. In [2], the authors studied the solvability for the system of nonlinear fractional q-difference equations

$$\left\{\begin{array}{c}\left(D_q^{\alpha }\mathfrak{u}\right)\left(\mathfrak{t}\right)+P(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{v}\left(\mathfrak{t}\right),I_q^{{\omega }_1}\mathfrak{u}\left(\mathfrak{t}\right),I_q^{{\delta }_1}\mathfrak{v}\left(\mathfrak{t}\right))=0,\mathfrak{t}\in (0,1),\hfill \\ \left(D_q^{\beta }\mathfrak{v}\right)\left(\mathfrak{t}\right)+Q(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{v}\left(\mathfrak{t}\right),I_q^{{\omega }_2}\mathfrak{u}\left(\mathfrak{t}\right),I_q^{{\delta }_2}\mathfrak{v}\left(\mathfrak{t}\right))=0,\mathfrak{t}\in (0,1),\hfill \end{array}\right.$$

(3)

subject to the coupled nonlocal boundary conditions

$$\left\{\begin{array}{c}{\displaystyle D_q^i\mathfrak{u}\left(0\right)=0,i=0,\dots ,n-2,D_q^{{\zeta }_0}\mathfrak{u}\left(1\right)={\int }_0^1{D}_q^{\zeta }\mathfrak{v}\left(\mathfrak{t}\right)d_q\mathfrak{H}\left(\mathfrak{t}\right),\mathfrak{t}\in (0,1),}\hfill \\ {\displaystyle D_q^i\mathfrak{v}\left(0\right)=0,i=0,\dots ,z-2,D_q^{{\xi }_0}\mathfrak{v}\left(1\right)={\int }_0^1{D}_q^{\xi }\mathfrak{u}\left(\mathfrak{t}\right)d_q\mathfrak{K}\left(\mathfrak{t}\right),\mathfrak{t}\in (0,1),}\hfill \end{array}\right.$$

(4)

where $q\in (0,1)$, $\alpha ,\beta \in \mathbb{R}$, $\alpha \in (n-1,n]$, $\beta \in (z-1,z]$, $n,z\in \mathbb{N}$, $n\ge 2,z\ge 2$, ${\omega }_i>0,$${\delta }_i>0,$$i=1,2$, $\zeta \in [0,\beta -1)$, $\xi \in [0,\alpha -1)$, ${\zeta }_0\in [0,\alpha -1)$, ${\xi }_0\in [0,\beta -1)$, and the integrals in (4) are Riemann–Stieltjes integrals with $\mathfrak{H}$- and $\mathfrak{K}$-bounded variation functions. By using varied fixed-point theorems, they obtained the existence and uniqueness results for the solutions of problem (3),(4). In [3], the authors examined the existence of solutions for the fractional q-difference equation with nonlinear integral conditions

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

(5)

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

$$\left\{\begin{array}{c}\left(D_q^{\alpha }\mathfrak{u}\right)\left(\mathfrak{t}\right)+\mathfrak{f}(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right))=0,\mathfrak{t}\in (0,1),\hfill \\ \left(D_q^i\mathfrak{u}\right)\left(0\right)=0,i=0,\dots ,n-2,\left(D_q^{\beta }\mathfrak{u}\right)\left(1\right)=a\left(D_q^{\beta }\mathfrak{u}\right)\left(\eta \right),\hfill \end{array}\right.$$

(6)

where $q\in (0,1)$, $\alpha \in (n-1,n]$, $n>2$, $\beta \in [1,n-2]$, $\eta \in (0,1)$, $a\in [0,1]$, and $\mathfrak{f}:[0,1]\times [0,\infty )\to [0,\infty )$ satisfies Caratheodory-type conditions. In the proof of the main theorems, they applied several fixed-point theorems. In [5], by using the Guo–Krasnosel’skii fixed-point theorem, the author analyzed the existence of positive solutions to the fractional q-difference equation with the boundary conditions

$$\left\{\begin{array}{c}\left(D_q^{\alpha }\mathfrak{y}\right)\left(\mathfrak{t}\right)=-\mathfrak{f}(\mathfrak{t},\mathfrak{y}\left(\mathfrak{t}\right)),\mathfrak{t}\in (0,1),\hfill \\ \mathfrak{y}\left(0\right)=\left(D_q\mathfrak{y}\right)\left(0\right)=0,\left(D_q\mathfrak{y}\right)\left(1\right)=\beta ,\hfill \end{array}\right.$$

(7)

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

$$\left\{\begin{array}{c}\left(D_q^{\alpha }\mathfrak{y}\right)\left(\mathfrak{t}\right)=-\mathfrak{f}(\mathfrak{t},\mathfrak{y}\left(\mathfrak{t}\right)),\mathfrak{t}\in (0,1),\hfill \\ \mathfrak{y}\left(0\right)=\mathfrak{y}\left(1\right)=0,\hfill \end{array}\right.$$

(8)

where $q\in (0,1)$, $\alpha \in (1,2]$, and $\mathfrak{f}:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a nonnegative continuous function. To prove the main finding, he also used the Guo–Krasnosel’skii fixed-point theorem. In [7], the authors proved the existence of solutions to the second-order q-difference equation with the boundary conditions

$$\left\{\begin{array}{c}{D}_q^2\mathfrak{u}\left(\mathfrak{t}\right)=\mathfrak{f}(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right),D_q\mathfrak{u}\left(\mathfrak{t}\right)),\mathfrak{t}\in I,\hfill \\ D_q\mathfrak{u}\left(0\right)=0,D_q\mathfrak{u}\left(1\right)=\alpha \mathfrak{u}\left(1\right),\hfill \end{array}\right.$$

(9)

where $q\in (0,1)$, $I=\{q^n,n\in \mathbb{N}\}\cup \{0,1\}$, $\mathfrak{f}\in C(I\times {\mathbb{R}}^2,\mathbb{R})$, and $\alpha \ne 0$ is a fixed real number. In [8], the authors studied the existence of solutions to the second-order q-difference equation subject to the boundary conditions

$$\left\{\begin{array}{c}{D}_q^2\mathfrak{u}\left(\mathfrak{t}\right)=\mathfrak{f}(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right)),\mathfrak{t}\in I,\hfill \\ \mathfrak{u}\left(0\right)=\eta \mathfrak{u}\left(T\right),D_q\mathfrak{u}\left(0\right)=\eta D_q\mathfrak{u}\left(T\right),\hfill \end{array}\right.$$

(10)

where $I=[0,T]\cap q^{\overline{\mathbb{N}}}$, $q^{\overline{\mathbb{N}}}=\{q^n,n\in \mathbb{N}\}\cup \left\{0\right\}$, $T\in q^{\overline{\mathbb{N}}}$ is a fixed constant, $\eta \ne 1$ is a fixed real number, and $\mathfrak{f}\in C(I\times \mathbb{R},\mathbb{R})$. Regarding additional papers that investigate systems of fractional q-difference equations with either coupled or uncoupled boundary conditions or that focus on fractional q-difference equations specifically, we refer to the following papers: [9,10,11,12,13,14,15,16].

The field of q-difference calculus, also known as quantum calculus, traces its origins back to the pioneering work of Jackson ([17,18]). To explore various applications of this discipline, readers are directed to the research of Ernst ([19]). The fractional q-difference calculus originated in the works of Al-Salam ([20]) and Agarwal ([21]). For advancements in this branch, encompassing q-analogs of integral and differential fractional operators, including properties like the fractional Leibniz q-formula, q-analogs of Cauchy’s formula, q-Laplace transform, q-Taylor’s formula, and q-analogs of the Mittag-Leffler function, refer to the papers [22,23,24,25,26].

The novel aspect of our problem (1),(2), compared to (3),(4) from [2], lies in the inclusion of generalized coupled boundary conditions (2) for the system of q-fractional difference equations in Equation (1). In this formulation, the q-fractional derivative of order $\varsigma$ for the unknown function $\mathfrak{u}$ at the point 1 is contingent upon the q-fractional derivatives of various orders for both functions $\mathfrak{u}$ and $\mathfrak{v}$ at different points within the interval $(0,1)$. Similarly, the q-fractional derivative of order $\vartheta$ for the unknown function $\mathfrak{v}$ at the point 1 is linked to the q-fractional derivatives of distinct orders for functions $\mathfrak{u}$ and $\mathfrak{v}$ at diverse points within the interval $(0,1)$. Furthermore, unlike the approach in the paper [2], we have explored the presence of positive solutions to our specific problem.

Our paper is organized as follows: Section 2 introduces key definitions and properties from q-calculus and fractional q-calculus, along with an existence result for the associated linear problem, the relevant Green functions, and their properties. Section 3 will then present the primary existence results for problem (1),(2), while Section 4 will provide illustrative examples to demonstrate the applicability of our theorems. Finally, Section 5 concludes the paper by summarizing the findings and presenting the overall conclusions.

2. Preliminary Results

In this section, we will introduce certain definitions and properties derived from q-calculus and fractional q-calculus. Additionally, we will outline some auxiliary findings that will play a pivotal role in the subsequent section.

Let $q\in (0,1)$. Define the number

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

(11)

Next, to introduce the q-gamma function, we present the q-analog of the power function ${(\mathfrak{a}-\mathfrak{b})}^n$ with $n\in \mathbb{N}\cup \left\{0\right\}$:

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

(12)

For $\alpha \in \mathbb{R}$, define

$${\displaystyle {(\mathfrak{a}-\mathfrak{b})}^{\left(\alpha \right)}={\mathfrak{a}}^{\alpha }\frac{{\prod }_{n=0}^{\infty }\left(1-\frac{\mathfrak{b}}{\mathfrak{a}}{q}^n\right)}{{\prod }_{n=0}^{\infty }\left(1-\frac{\mathfrak{b}}{\mathfrak{a}}{q}^{\alpha +n}\right)}.}$$

(13)

If $\mathfrak{b}=0$, then ${\mathfrak{a}}^{\left(\alpha \right)}={\mathfrak{a}}^{\alpha }$.

The q-gamma function is defined by

$${\displaystyle {\mathrm{\Gamma }}_q\left(\alpha \right)=\frac{{(1-q)}^{(\alpha -1)}}{{(1-q)}^{\alpha -1}}=\frac{1}{{(1-q)}^{\alpha -1}}\frac{{\prod }_{n=0}^{\infty }(1-q^{n+1})}{{\prod }_{n=0}^{\infty }(1-q^{n+\alpha })},\alpha \in \mathbb{R}\setminus \{0,-1,-2,\dots \}.}$$

(14)

This function satisfies the relation ${\mathrm{\Gamma }}_q(\alpha +1)={\left[\alpha \right]}_q{\mathrm{\Gamma }}_q\left(\alpha \right)$.

Definition 1.

The q-derivative of a real function $\mathfrak{f}$ is defined by

$${\displaystyle \left(D_q\mathfrak{f}\right)\left(\mathfrak{t}\right)=\frac{\mathfrak{f}\left(\mathfrak{t}\right)-\mathfrak{f}\left(q\mathfrak{t}\right)}{(1-q)\mathfrak{t}},\mathfrak{t}\ne 0;\left(D_q\mathfrak{f}\right)\left(0\right)=\underset{\mathfrak{t}\to 0}{lim}\left(D_q\mathfrak{f}\right)\left(\mathfrak{t}\right).}$$

(15)

Definition 2.

The q-derivatives of higher order of a real function $\mathfrak{f}$ are defined by

$$\left(D_q^0\mathfrak{f}\right)\left(\mathfrak{t}\right)=\mathfrak{f}\left(\mathfrak{t}\right),\left(D_q^n\mathfrak{f}\right)\left(\mathfrak{t}\right)=D_q\left(D_q^{n-1}\mathfrak{f}\right)\left(\mathfrak{t}\right),n\in \mathbb{N}.$$

(16)

Definition 3.

The q-integral of a function $\mathfrak{f}$ defined in the interval $[0,b]$ is defined by

$${\displaystyle \left(I_q\mathfrak{f}\right)\left(\mathfrak{t}\right)={\int }_0^{\mathfrak{t}}\mathfrak{f}\left(\mathfrak{s}\right)d_q\mathfrak{s}=\mathfrak{t}(1-q)\sum _{n=0}^{\infty }\mathfrak{f}\left(\mathfrak{t}{q}^n\right)q^n,\mathfrak{t}\in [0,b].}$$

(17)

Definition 4.

If $a\in [0,b]$ and $\mathfrak{f}$ is defined in the interval $[0,b]$, then its q-integral from a to b is given by

$${\displaystyle {\int }_a^b\mathfrak{f}\left(\mathfrak{s}\right)d_q\mathfrak{s}={\int }_0^b\mathfrak{f}\left(s\right)d_q\mathfrak{s}-{\int }_0^a\mathfrak{f}\left(\mathfrak{s}\right)d_q\mathfrak{s}.}$$

(18)

Definition 5.

The q-integrals of a function $\mathfrak{f}$ of higher order are defined by

$$\left(I_q^0\mathfrak{f}\right)\left(\mathfrak{t}\right)=\mathfrak{f}\left(\mathfrak{t}\right),\left(I_q^n\mathfrak{f}\right)\left(\mathfrak{t}\right)=I_q\left(I_q^{n-1}\mathfrak{f}\right)\left(\mathfrak{t}\right),n\in \mathbb{N}.$$

(19)

The fundamental theorem of q-calculus says that $\left(D_q{I}_q\mathfrak{f}\right)\left(\mathfrak{t}\right)=\mathfrak{f}\left(\mathfrak{t}\right)$, and if $\mathfrak{f}$ is continuous at $\mathfrak{t}=0$, then $\left(I_q{D}_q\mathfrak{f}\right)\left(\mathfrak{t}\right)=\mathfrak{f}\left(\mathfrak{t}\right)-\mathfrak{f}\left(0\right)$. The properties of the operators $D_q$ and $I_q$ are presented in [6,27]. Below, we present some properties that will be used later.

Lemma 1.

For $\mathfrak{a},\mathfrak{t},\mathfrak{s},\alpha \in \mathbb{R}$, and $\mathfrak{f}:[0,\mathfrak{b}]\to \mathbb{R}$, we have

(i) ${\left[\mathfrak{a}(\mathfrak{t}-\mathfrak{s})\right]}^{\left(\alpha \right)}={\mathfrak{a}}^{\alpha }{(\mathfrak{t}-\mathfrak{s})}^{\left(\alpha \right)}$;

(ii) ${}_{\mathfrak{t}}{D}_q{(\mathfrak{t}-\mathfrak{s})}^{\left(\alpha \right)}={\left[\alpha \right]}_q{(\mathfrak{t}-\mathfrak{s})}^{(\alpha -1)}$;

(iii) If $\alpha >1$, then $D_q\left({\mathfrak{t}}^{\alpha }\right)={\left[\alpha \right]}_q{\mathfrak{t}}^{\alpha -1}$;

(iv) If $\alpha >0$, then $I_q\left({\mathfrak{t}}^{\alpha }\right)=\frac{1}{{[\alpha +1]}_q}{\mathfrak{t}}^{\alpha +1}$;

(v) If $\alpha >0$ and $\mathfrak{c}\le \mathfrak{d}\le \mathfrak{t}$, then ${(\mathfrak{t}-\mathfrak{c})}^{\left(\alpha \right)}\ge {(\mathfrak{t}-\mathfrak{d})}^{\left(\alpha \right)}$;

(vi) ${\displaystyle {}_{\mathfrak{t}}{D}_q\left({\displaystyle {\int }_0^{\mathfrak{t}}\mathfrak{f}(\mathfrak{t},\mathfrak{s})d_q\mathfrak{s}}\right)\left(\mathfrak{t}\right)={\int }_0^{\mathfrak{t}}{}_{\mathfrak{t}}{D}_q\mathfrak{f}(\mathfrak{t},\mathfrak{s})d_q\mathfrak{s}+\mathfrak{f}(q\mathfrak{t},\mathfrak{t}),\forall \mathfrak{t}\in [0,\mathfrak{b}]}$, where ${}_{\mathfrak{t}}{D}_q$ denotes the q-derivative with respect to variable $\mathfrak{t}$.

Definition 6

([21]). Let $\mathfrak{f}$ be a function defined on $[0,1]$. The fractional q-integral of the Riemann–Liouville type of order $\alpha \ge 0$ is defined by $\left(I_q^0\mathfrak{f}\right)\left(\mathfrak{t}\right)=\mathfrak{f}\left(\mathfrak{t}\right)$ and

$${\displaystyle \left(I_q^{\alpha }\mathfrak{f}\right)\left(\mathfrak{t}\right)=\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\alpha -1)}\mathfrak{f}\left(\mathfrak{s}\right)d_q\mathfrak{s},\mathfrak{t}\in [0,1],\alpha >0.}$$

(20)

Definition 7

([24]). The fractional q-derivative of the Riemann–Liouville type of order $\alpha \ge 0$ is defined by $\left(D_q^0\mathfrak{f}\right)\left(\mathfrak{t}\right)=\mathfrak{f}\left(\mathfrak{t}\right)$ and

$$\left(D_q^{\alpha }\mathfrak{f}\right)\left(\mathfrak{t}\right)=\left(D_q^m{I}_q^{m-\alpha }\mathfrak{f}\right)\left(\mathfrak{t}\right),\alpha >0,$$

(21)

where m is the smallest integer greater than or equal to α.

Lemma 2

([21,24]). Let $\alpha ,\beta \ge 0$, $\gamma >0$, $\zeta \in [0,\gamma )$, and $\lambda \ge 0$, and let $\mathfrak{f}$ be a function defined on $[0,1]$. Then, the following relations are satisfied:

(a) $\left(I_q^{\beta }{I}_q^{\alpha }\mathfrak{f}\right)\left(\mathfrak{t}\right)=\left(I_q^{\alpha +\beta }\mathfrak{f}\right)\left(\mathfrak{t}\right)$;

(b) $\left(D_q^{\alpha }{I}_q^{\alpha }\mathfrak{f}\right)\left(\mathfrak{t}\right)=\mathfrak{f}\left(\mathfrak{t}\right)$;

(c) If $\alpha \ge \beta >0$, then $\left(D_q^{\beta }{I}_q^{\alpha }\mathfrak{f}\right)\left(\mathfrak{t}\right)=\left(D_q^{\beta }{I}_q^{\beta }{I}_q^{\alpha -\beta }\mathfrak{f}\right)\left(\mathfrak{t}\right)=\left(I_q^{\alpha -\beta }\mathfrak{f}\right)\left(\mathfrak{t}\right)$;

(d) $D_q^{\zeta }\left({\mathfrak{t}}^{\gamma }\right)=\frac{{\mathrm{\Gamma }}_q(\gamma +1)}{{\mathrm{\Gamma }}_q(\gamma -\zeta +1)}{\mathfrak{t}}^{\gamma -\zeta }$;

(e) $I_q^{\alpha }\left({\mathfrak{t}}^{\lambda }\right)=\frac{{\mathrm{\Gamma }}_q(\lambda +1)}{{\mathrm{\Gamma }}_q(\alpha +\lambda +1)}{\mathfrak{t}}^{\alpha +\lambda }$.

Lemma 3

([6]). Let $\alpha >0$ and p be a positive integer. Then, the following relation holds:

$${\displaystyle \left(I_q^{\alpha }{D}_q^p\mathfrak{f}\right)\left(\mathfrak{t}\right)=\left(D_q^p{I}_q^{\alpha }\mathfrak{f}\right)\left(\mathfrak{t}\right)-\sum _{k=0}^{p-1}\frac{{\mathfrak{t}}^{\alpha -p+k}}{{\mathrm{\Gamma }}_q(\alpha +k-p+1)}\left(D_q^k\mathfrak{f}\right)\left(0\right).}$$

(22)

Lemma 4.

If $w\in C[0,1]$, then for $\kappa >0$, we have

$${\displaystyle |I_q^{\kappa }w\left(\mathfrak{t}\right)|\le \frac{\parallel w\parallel }{{\mathrm{\Gamma }}_q(\kappa +1)},\forall t\in [0,1],}$$

(23)

where $\parallel w\parallel ={sup}_{\mathfrak{t}\in [0,1]}\left|w\left(\mathfrak{t}\right)\right|$.

Proof. 

By Definition 6 and Lemma 2 (e) (with $\alpha =\kappa$ and $\lambda =0$), we obtain

$$\begin{array}{c}|I_q^{\kappa }w\left(\mathfrak{t}\right)|=\left|{\displaystyle \frac{1}{{\mathrm{\Gamma }}_q\left(\kappa \right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\kappa -1)}w\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right|\hfill \\ =\left|{\displaystyle \frac{1}{{\mathrm{\Gamma }}_q\left(\kappa \right)}\mathfrak{t}(1-q)\sum _{n=0}^{\infty }{(\mathfrak{t}-\mathfrak{t}{q}^{n+1})}^{(\kappa -1)}w\left(\mathfrak{t}{q}^n\right)q^n}\right|\hfill \\ \le \parallel w\parallel \left|{\displaystyle \frac{1}{{\mathrm{\Gamma }}_q\left(\kappa \right)}\mathfrak{t}(1-q)\sum _{n=0}^{\infty }{(\mathfrak{t}-\mathfrak{t}{q}^{n+1})}^{(\kappa -1)}{q}^n}\right|\hfill \\ {\displaystyle =\parallel w\parallel |\left(I_q^{\kappa }1\right)\left(t\right)|=\parallel w\parallel \frac{{\mathfrak{t}}^{\kappa }}{{\mathrm{\Gamma }}_q(\kappa +1)}\le \frac{\parallel w\parallel }{{\mathrm{\Gamma }}_q(\kappa +1)},\forall \mathfrak{t}\in [0,1].}\hfill \end{array}$$

(24)

□

In what follows, we will study the linear problem associated with our problem (1),(2). We consider the system of fractional q-difference equations

$$\left\{\begin{array}{c}\left(D_q^{\alpha }\mathfrak{u}\right)\left(\mathfrak{t}\right)+\mathfrak{h}\left(\mathfrak{t}\right)=0,\mathfrak{t}\in [0,1],\hfill \\ \left(D_q^{\beta }\mathfrak{v}\right)\left(\mathfrak{t}\right)+\mathfrak{k}\left(\mathfrak{t}\right)=0,\mathfrak{t}\in [0,1],\hfill \end{array}\right.$$

(25)

subject to boundary conditions (2), where $\mathfrak{h},\mathfrak{k}\in C[0,1]$.

We define

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

(26)

Lemma 5.

If $\mathrm{\Delta }\ne 0$, then the solution $\left(\mathfrak{u}\right(\mathfrak{t}),\mathfrak{v}(\mathfrak{t}\left)\right),\mathfrak{t}\in [0,1]$, of problem (25),(2) is given by

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)=-\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\alpha -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}+\frac{t^{\alpha -1}}{\mathrm{\Delta }}(A{\mathrm{\Lambda }}_4+B{\mathrm{\Lambda }}_2),\mathfrak{t}\in [0,1],}\hfill \\ {\displaystyle \mathfrak{v}\left(\mathfrak{t}\right)=-\frac{1}{{\mathrm{\Gamma }}_q\left(\beta \right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\beta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}+\frac{t^{\beta -1}}{\mathrm{\Delta }}(B{\mathrm{\Lambda }}_1+A{\mathrm{\Lambda }}_3),\mathfrak{t}\in [0,1],}\hfill \end{array}$$

(27)

where

$$\begin{array}{c}{\displaystyle A=\frac{1}{{\mathrm{\Gamma }}_q(\alpha -\varsigma )}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle -\sum _{i=1}^a\frac{a_i}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-q\mathfrak{s})}^{(\alpha -{\varrho }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle -\sum _{i=1}^b\frac{b_i}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-q\mathfrak{s})}^{(\beta -{\sigma }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s},}\hfill \\ {\displaystyle B=\frac{1}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}{\int }_0^1{(1-q\mathfrak{s})}^{(\beta -\vartheta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle -\sum _{i=1}^c\frac{c_i}{{\mathrm{\Gamma }}_q(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-q\mathfrak{s})}^{(\alpha -{\eta }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle -\sum _{i=1}^d\frac{d_i}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-q\mathfrak{s})}^{(\beta -{\rho }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}.}\hfill \end{array}$$

(28)

Proof. 

We apply $I_q^{\alpha }$ and $I_q^{\beta }$, respectively, to the equations of system (25). Then, by Lemma 2 (a),(b) and Lemma 3, we obtain

$$\begin{array}{c}\mathfrak{u}\left(\mathfrak{t}\right)=-I_q^{\alpha }\mathfrak{h}\left(\mathfrak{t}\right)+{\tilde{a}}_1{\mathfrak{t}}^{\alpha -1}+{\tilde{a}}_2{\mathfrak{t}}^{\alpha -2}+\cdots +{\tilde{a}}_n{\mathfrak{t}}^{\alpha -n}\hfill \\ {\displaystyle =-\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\alpha -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\tilde{a}}_1{\mathfrak{t}}^{\alpha -1}+{\tilde{a}}_2{\mathfrak{t}}^{\alpha -2}+\cdots +{\tilde{a}}_n{\mathfrak{t}}^{\alpha -n},\mathfrak{t}\in [0,1],}\hfill \\ \mathfrak{v}\left(\mathfrak{t}\right)=-I_q^{\beta }\mathfrak{k}\left(\mathfrak{t}\right)+{\tilde{b}}_1{\mathfrak{t}}^{\beta -1}+{\tilde{b}}_2{\mathfrak{t}}^{\beta -2}+\cdots +{\tilde{b}}_m{\mathfrak{t}}^{\beta -m}\hfill \\ {\displaystyle =-\frac{1}{{\mathrm{\Gamma }}_q\left(\beta \right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\beta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\tilde{b}}_1{\mathfrak{t}}^{\beta -1}+{\tilde{b}}_2{\mathfrak{t}}^{\beta -2}+\cdots +{\tilde{b}}_m{\mathfrak{t}}^{\beta -m},\mathfrak{t}\in [0,1],}\hfill \end{array}$$

(29)

for some ${\tilde{a}}_i,{\tilde{b}}_j\in \mathbb{R},i=1,\dots ,n,j=1,\dots ,m$.

From the conditions $D_q^i\mathfrak{u}\left(0\right)=0$ for $i=0,\dots ,n-2$, and $D_q^j\mathfrak{v}\left(0\right)=0$ for $j=0,\dots ,m-2$, we deduce that ${\tilde{a}}_i=0,i=2,\dots ,n$, and ${\tilde{b}}_j=0,j=2,\dots ,m$. So, the solution (29) becomes

$$\left\{\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)={\tilde{a}}_1{\mathfrak{t}}^{\alpha -1}-\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\alpha -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s},\mathfrak{t}\in [0,1],}\hfill \\ {\displaystyle \mathfrak{v}\left(\mathfrak{t}\right)={\tilde{b}}_1{\mathfrak{t}}^{\beta -1}-\frac{1}{{\mathrm{\Gamma }}_q\left(\beta \right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\beta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s},\mathfrak{t}\in [0,1].}\hfill \end{array}\right.$$

(30)

On the other hand, we find

$${\displaystyle D_q^{\kappa }\mathfrak{u}\left(\mathfrak{t}\right)={\tilde{a}}_1\frac{{\mathrm{\Gamma }}_q\left(\alpha \right)}{{\mathrm{\Gamma }}_q(\alpha -\kappa )}{\mathfrak{t}}^{\alpha -\kappa -1}-\frac{1}{{\mathrm{\Gamma }}_q(\alpha -\kappa )}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\alpha -\kappa -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s},}$$

(31)

for $\kappa =\varsigma ,{\varrho }_i,{\eta }_j,i=1,\dots ,a,j=1,\dots ,c$, and

$${\displaystyle D_q^{\kappa }\mathfrak{v}\left(\mathfrak{t}\right)={\tilde{b}}_1\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -\kappa )}{\mathfrak{t}}^{\beta -\kappa -1}-\frac{1}{{\mathrm{\Gamma }}_q(\beta -\kappa )}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\beta -\kappa -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s},}$$

(32)

for $\kappa =\vartheta ,{\sigma }_i,{\rho }_j,i=1,\dots ,b,j=1,\dots ,d$.

By using relations (31) and (32), the boundary conditions $D_q^{\varsigma }\mathfrak{u}\left(1\right)={\sum }_{i=1}^a{a}_i{D}_q^{{\varrho }_i}\mathfrak{u}\left({\xi }_i\right)+{\sum }_{i=1}^b{b}_i{D}_q^{{\sigma }_i}\mathfrak{v}\left({\omega }_i\right)$ and $D_q^{\vartheta }\mathfrak{v}\left(1\right)={\sum }_{i=1}^c{c}_i{D}_q^{{\eta }_i}\mathfrak{u}\left({\zeta }_i\right)+{\sum }_{i=1}^d{d}_i{D}_q^{{\rho }_i}\mathfrak{v}\left({\theta }_i\right)$ become

$$\left\{\begin{array}{c}{\displaystyle {\tilde{a}}_1\frac{{\mathrm{\Gamma }}_q\left(\alpha \right)}{{\mathrm{\Gamma }}_q(\alpha -\varsigma )}-\frac{1}{{\mathrm{\Gamma }}_q(\alpha -\varsigma )}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle =\sum _{i=1}^a{a}_i\left[{\displaystyle {\tilde{a}}_1\frac{{\mathrm{\Gamma }}_q\left(\alpha \right)}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}{\xi }_i^{\alpha -{\varrho }_i-1}-\frac{1}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-q\mathfrak{s})}^{(\alpha -{\varrho }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]}\hfill \\ {\displaystyle +\sum _{i=1}^b{b}_i\left[{\displaystyle {\tilde{b}}_1\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\omega }_i^{\beta -{\sigma }_i-1}-\frac{1}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-q\mathfrak{s})}^{(\beta -{\sigma }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right],}\hfill \\ {\displaystyle {\tilde{b}}_1\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{\mathrm{\Gamma }(\beta -\vartheta )}-\frac{1}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}{\int }_0^1{(1-q\mathfrak{s})}^{(\beta -\vartheta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle =\sum _{i=1}^c{c}_i\left[{\displaystyle {\tilde{a}}_1\frac{{\mathrm{\Gamma }}_q\left(\alpha \right)}{{\mathrm{\Gamma }}_q(\alpha -{\eta }_i)}{\zeta }_i^{\alpha -{\eta }_i-1}-\frac{1}{{\mathrm{\Gamma }}_q(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-q\mathfrak{s})}^{(\alpha -{\eta }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]}\hfill \\ {\displaystyle +\sum _{i=1}^d{d}_i\left[{\displaystyle {\tilde{b}}_1\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1}-\frac{1}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-q\mathfrak{s})}^{(\beta -{\rho }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right].}\hfill \end{array}\right.$$

(33)

or

$$\left\{\begin{array}{c}{\displaystyle {\tilde{a}}_1\left({\displaystyle \frac{{\mathrm{\Gamma }}_q\left(\alpha \right)}{{\mathrm{\Gamma }}_q(\alpha -\varsigma )}-\sum _{i=1}^a{a}_i\frac{{\mathrm{\Gamma }}_q\left(\alpha \right)}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}{\xi }_i^{\alpha -{\varrho }_i-1}}\right)-{\tilde{b}}_1\sum _{i=1}^b{b}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\omega }_i^{\beta -{\sigma }_i-1}}\hfill \\ {\displaystyle =\frac{1}{{\mathrm{\Gamma }}_q(\alpha -\varsigma )}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle -\sum _{i=1}^a\frac{a_i}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-q\mathfrak{s})}^{(\alpha -{\varrho }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle -\sum _{i=1}^b\frac{b_i}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-q\mathfrak{s})}^{(\beta -{\sigma }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s},}\hfill \\ {\displaystyle -{\tilde{a}}_1\sum _{i=1}^c{c}_i\frac{{\mathrm{\Gamma }}_q\left(\alpha \right)}{{\mathrm{\Gamma }}_q(\alpha -{\eta }_i)}{\zeta }_i^{\alpha -{\eta }_i-1}+{\tilde{b}}_1\left({\displaystyle \frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}-\sum _{i=1}^d{d}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1}}\right)}\hfill \\ {\displaystyle =\frac{1}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}{\int }_0^1{(1-q\mathfrak{s})}^{(\beta -\vartheta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle -\sum _{i=1}^c\frac{c_i}{{\mathrm{\Gamma }}_q(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-q\mathfrak{s})}^{(\alpha -{\eta }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle -\sum _{i=1}^d\frac{d_i}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-q\mathfrak{s})}^{(\beta -{\rho }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}.}\hfill \end{array}\right.$$

(34)

The determinant of system (34) in the unknowns ${\tilde{a}}_1$ and ${\tilde{b}}_1$ is $\mathrm{\Delta }$, which, by the assumption of this lemma, is nonzero. So, system (34) has the unique solution

$${\displaystyle {\tilde{a}}_1=\frac{1}{\mathrm{\Delta }}(A{\mathrm{\Lambda }}_4+B{\mathrm{\Lambda }}_2),{\tilde{b}}_1=\frac{1}{\mathrm{\Delta }}(B{\mathrm{\Lambda }}_1+A{\mathrm{\Lambda }}_3).}$$

(35)

By substituting (35) into (30), we obtain the solution $\left(\mathfrak{u}\right(\mathfrak{t}),\mathfrak{v}(\mathfrak{t}\left)\right),\mathfrak{t}\in [0,1]$, of problem (25),(2) given by (27). □

Lemma 6.

If $\mathrm{\Delta }\ne 0$, then the solution $\left(\mathfrak{u}\right(\mathfrak{t}),\mathfrak{v}(\mathfrak{t}\left)\right),\mathfrak{t}\in [0,1]$, of problem (25),(2) can be written as

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)={\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s},\mathfrak{t}\in [0,1],}\hfill \\ {\displaystyle \mathfrak{v}\left(\mathfrak{t}\right)={\int }_0^1{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s},\mathfrak{t}\in [0,1],}\hfill \end{array}$$

(36)

where

$$\begin{array}{c}{\displaystyle {\mathfrak{G}}_1(\mathfrak{t},\mathfrak{s})=g_1(\mathfrak{t},\mathfrak{s})+\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_4\sum _{i=1}^a{a}_i{g}_{1i}({\xi }_i,\mathfrak{s})+{\mathrm{\Lambda }}_2\sum _{i=1}^c{c}_i{g}_{2i}({\zeta }_i,\mathfrak{s})}\right],}\hfill \\ {\displaystyle {\mathfrak{G}}_2(\mathfrak{t},\mathfrak{s})=\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_4\sum _{i=1}^b{b}_i{g}_{3i}({\omega }_i,\mathfrak{s})+{\mathrm{\Lambda }}_2\sum _{i=1}^d{d}_i{g}_{4i}({\theta }_i,\mathfrak{s})}\right],}\hfill \\ {\displaystyle {\mathfrak{G}}_3(\mathfrak{t},\mathfrak{s})=\frac{{\mathfrak{t}}^{\beta -1}}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_3\sum _{i=1}^a{a}_i{g}_{1i}({\xi }_i,\mathfrak{s})+{\mathrm{\Lambda }}_1\sum _{i=1}^c{c}_i{g}_{2i}({\zeta }_i,\mathfrak{s})}\right],}\hfill \\ {\displaystyle {\mathfrak{G}}_4(\mathfrak{t},\mathfrak{s})=g_2(\mathfrak{t},\mathfrak{s})+\frac{{\mathfrak{t}}^{\beta -1}}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_3\sum _{i=1}^b{b}_i{g}_{3i}({\omega }_i,\mathfrak{s})+{\mathrm{\Lambda }}_1\sum _{i=1}^d{d}_i{g}_{4i}({\theta }_i,\mathfrak{s})}\right],}\hfill \end{array}$$

(37)

and

$${\displaystyle \begin{array}{c}{\displaystyle g_1(\mathfrak{t},\mathfrak{s})=\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}\left\{\begin{array}{c}{\mathfrak{t}}^{\alpha -1}{(1-\mathfrak{s})}^{(\alpha -\varsigma -1)}-{(\mathfrak{t}-\mathfrak{s})}^{(\alpha -1)},0\le \mathfrak{s}\le \mathfrak{t}\le 1,\hfill \\ {\mathfrak{t}}^{\alpha -1}{(1-\mathfrak{s})}^{(\alpha -\varsigma -1)},0\le \mathfrak{t}\le \mathfrak{s}\le 1,\hfill \end{array}\right.}\hfill \\ g_2(\mathfrak{t},\mathfrak{s})=\frac{1}{{\mathrm{\Gamma }}_q\left(\beta \right)}\left\{\begin{array}{c}{\mathfrak{t}}^{\beta -1}{(1-\mathfrak{s})}^{(\beta -\vartheta -1)}-{(\mathfrak{t}-\mathfrak{s})}^{(\beta -1)},0\le \mathfrak{s}\le \mathfrak{t}\le 1,\hfill \\ {\mathfrak{t}}^{\beta -1}{(1-\mathfrak{s})}^{(\beta -\vartheta -1)},0\le \mathfrak{t}\le \mathfrak{s}\le 1,\hfill \end{array}\right.\hfill \\ g_{1i}(\mathfrak{t},\mathfrak{s})=\frac{1}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}\left\{\begin{array}{c}{\mathfrak{t}}^{\alpha -{\varrho }_i-1}{(1-\mathfrak{s})}^{(\alpha -\varsigma -1)}-{(\mathfrak{t}-\mathfrak{s})}^{(\alpha -{\varrho }_i-1)},0\le \mathfrak{s}\le \mathfrak{t}\le 1,\hfill \\ {\mathfrak{t}}^{\alpha -{\varrho }_i-1}{(1-\mathfrak{s})}^{(\alpha -\varsigma -1)},0\le \mathfrak{t}\le \mathfrak{s}\le 1,\hfill \end{array}\right.\hfill \\ g_{2j}(\mathfrak{t},\mathfrak{s})=\frac{1}{{\mathrm{\Gamma }}_q(\alpha -{\eta }_j)}\left\{\begin{array}{c}{\mathfrak{t}}^{\alpha -{\eta }_j-1}{(1-\mathfrak{s})}^{(\alpha -\varsigma -1)}-{(\mathfrak{t}-\mathfrak{s})}^{(\alpha -{\eta }_j-1)},0\le \mathfrak{s}\le \mathfrak{t}\le 1,\hfill \\ {\mathfrak{t}}^{\alpha -{\eta }_j-1}{(1-\mathfrak{s})}^{(\alpha -\varsigma -1)},0\le \mathfrak{t}\le \mathfrak{s}\le 1,\hfill \end{array}\right.\hfill \\ g_{3k}(\mathfrak{t},\mathfrak{s})=\frac{1}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_k)}\left\{\begin{array}{c}{\mathfrak{t}}^{\beta -{\sigma }_k-1}{(1-\mathfrak{s})}^{(\beta -\vartheta -1)}-{(\mathfrak{t}-\mathfrak{s})}^{(\beta -{\sigma }_k-1)},0\le \mathfrak{s}\le \mathfrak{t}\le 1,\hfill \\ {\mathfrak{t}}^{\beta -{\sigma }_k-1}{(1-\mathfrak{s})}^{(\beta -\vartheta -1)},0\le \mathfrak{t}\le \mathfrak{s}\le 1,\hfill \end{array}\right.\hfill \\ g_{4\iota }(\mathfrak{t},\mathfrak{s})=\frac{1}{{\mathrm{\Gamma }}_q(\beta -{\rho }_{\iota })}\left\{\begin{array}{c}{\mathfrak{t}}^{\beta -{\rho }_{\iota }-1}{(1-\mathfrak{s})}^{(\beta -\vartheta -1)}-{(\mathfrak{t}-\mathfrak{s})}^{(\beta -{\rho }_{\iota }-1)},0\le \mathfrak{s}\le \mathfrak{t}\le 1,\hfill \\ {\mathfrak{t}}^{\beta -{\rho }_{\iota }-1}{(1-\mathfrak{s})}^{(\beta -\vartheta -1)},0\le \mathfrak{t}\le \mathfrak{s}\le 1,\hfill \end{array}\right.\hfill \end{array}}$$

(38)

for $\mathfrak{t},\mathfrak{s}\in [0,1]$, $i=1,\dots ,a$, $j=1,\dots ,c$, $k=1,\dots ,b$, $\iota =1,\dots ,d$.

Proof. 

For $\mathfrak{u}\left(\mathfrak{t}\right)$, we deduce

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

(39)

Amplifying the fourth term on the right-hand side of the above relation by $\mathrm{\Delta }={\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_4-{\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_3$, we obtain

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)=\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^1{\mathfrak{t}}^{\alpha -1}{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}-\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\alpha -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left\{{\displaystyle -\frac{{\mathrm{\Gamma }}_q\left(\alpha \right)}{{\mathrm{\Gamma }}_q(\alpha -\varsigma )}\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right.}\hfill \\ {\displaystyle +\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}\left({\displaystyle \sum _{i=1}^a{a}_i\frac{{\mathrm{\Gamma }}_q\left(\alpha \right)}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}{\xi }_i^{\alpha -{\varrho }_i-1}}\right)\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\frac{{\mathrm{\Gamma }}_q\left(\alpha \right)}{{\mathrm{\Gamma }}_q(\alpha -\varsigma )}\left({\displaystyle \sum _{i=1}^d{d}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1}}\right)\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle -\left({\displaystyle \sum _{i=1}^a{a}_i\frac{{\mathrm{\Gamma }}_q\left(\alpha \right)}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}{\xi }_i^{\alpha -{\varrho }_i-1}}\right)\left({\displaystyle \sum _{i=1}^d{d}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1}}\right)\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\left({\displaystyle \sum _{i=1}^c{c}_i\frac{{\mathrm{\Gamma }}_q\left(\alpha \right)}{{\mathrm{\Gamma }}_q(\alpha -{\eta }_i)}{\zeta }_i^{\alpha -{\eta }_i-1}}\right)\left({\displaystyle \sum _{i=1}^b{b}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\eta }_i^{\beta -{\eta }_i-1}}\right)\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}\frac{1}{{\mathrm{\Gamma }}_q(\alpha -\varsigma )}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle -\left({\displaystyle \sum _{i=1}^d{d}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1}}\right)\frac{1}{{\mathrm{\Gamma }}_q(\alpha -\varsigma )}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle -\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}\sum _{i=1}^a\frac{a_i}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-q\mathfrak{s})}^{(\alpha -{\varrho }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\left({\displaystyle \sum _{i=1}^d{d}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1}}\right)\sum _{i=1}^a\frac{a_i}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}{\int }_0^{{\xi }_i}{({\xi }_i-q\mathfrak{s})}^{(\alpha -{\varrho }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ -\left.{\displaystyle \left({\displaystyle \sum _{i=1}^b{b}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\eta }_i^{\beta -{\eta }_i-1}}\right)\sum _{i=1}^c\frac{c_i}{{\mathrm{\Gamma }}_q(\alpha -{\eta }_i)}{\int }_0^{{\zeta }_i}{({\zeta }_i-q\mathfrak{s})}^{(\alpha -{\eta }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right\}\hfill \\ {\displaystyle +\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left\{{\displaystyle -\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}\sum _{i=1}^b\frac{b_i}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-q\mathfrak{s})}^{(\beta -{\sigma }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\left(\mathfrak{s}\right)}\right.}\hfill \\ {\displaystyle +\left({\displaystyle \sum _{i=1}^d{d}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1}}\right)\sum _{i=1}^b\frac{b_i}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-q\mathfrak{s})}^{(\beta -{\sigma }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\left({\displaystyle \sum _{i=1}^b{b}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\omega }_i^{\beta -{\sigma }_i-1}}\right)\frac{1}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}{\int }_0^1{(1-q\mathfrak{s})}^{(\beta -\vartheta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ -\left.{\displaystyle \left({\displaystyle \sum _{i=1}^b{b}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\omega }_i^{\beta -{\sigma }_i-1}}\right)\sum _{i=1}^d\frac{d_i}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-q\mathfrak{s})}^{(\beta -{\rho }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right\}\hfill \\ {\displaystyle =\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^1{\mathfrak{t}}^{\alpha -1}{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}-\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\alpha -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left\{{\displaystyle \frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}\sum _{i=1}^a\frac{a_i}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}\left[{\displaystyle {\xi }_i^{\alpha -{\varrho }_i-1}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right.}\right.}\hfill \\ -\left.{\displaystyle {\int }_0^{{\xi }_i}{({\xi }_i-q\mathfrak{s})}^{(\alpha -{\varrho }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]\hfill \\ {\displaystyle -\left({\displaystyle \sum _{i=1}^d{d}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1}}\right)\sum _{i=1}^a\frac{a_i}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}\left[{\displaystyle {\xi }_i^{\alpha -{\varrho }_i-1}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right.}\hfill \\ \left.{\displaystyle -{\int }_0^{{\xi }_i}{({\xi }_i-q\mathfrak{s})}^{(\alpha -{\varrho }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]\hfill \\ {\displaystyle +\left({\displaystyle \sum _{i=1}^b{b}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\eta }_i^{\beta -{\eta }_i-1}}\right)\sum _{i=1}^c\frac{c_i}{{\mathrm{\Gamma }}_q(\alpha -{\eta }_i)}\left[{\displaystyle {\zeta }_i^{\alpha -{\eta }_i-1}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right.}\hfill \\ \left.\left.{\displaystyle -{\int }_0^{{\zeta }_i}{({\zeta }_i-q\mathfrak{s})}^{(\alpha -{\eta }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]\right\}\hfill \\ {\displaystyle +\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left\{{\displaystyle \sum _{i=1}^b{b}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}\frac{1}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}\left[{\displaystyle {\int }_0^1{\omega }_i^{\beta -{\sigma }_i-1}{(1-q\mathfrak{s})}^{({\beta }_i-\vartheta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right.}\right.}\hfill \\ -\left.{\displaystyle {\int }_0^{{\omega }_i}{({\omega }_i-q\mathfrak{s})}^{(\beta -{\sigma }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]\hfill \\ {\displaystyle -\left({\displaystyle \sum _{i=1}^d{d}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1}}\right)\sum _{i=1}^b\frac{b_i}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\int }_0^1{\omega }_i^{\beta -{\sigma }_i-1}{(1-q\mathfrak{s})}^{(\beta -\vartheta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\left({\displaystyle \sum _{i=1}^d{d}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1}}\right)\sum _{i=1}^b\frac{b_i}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\int }_0^{{\omega }_i}{({\omega }_i-q\mathfrak{s})}^{(\beta -{\sigma }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\left({\displaystyle \sum _{i=1}^b{b}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\omega }_i^{\beta -{\sigma }_i-1}}\right)\sum _{i=1}^d\frac{d_i}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\int }_0^1{\theta }_i^{\beta -{\rho }_i-1}{(1-q\mathfrak{s})}^{(\beta -\vartheta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ -\left.{\displaystyle \left({\displaystyle \sum _{i=1}^b{b}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\omega }_i^{\beta -{\sigma }_i-1}}\right)\sum _{i=1}^d\frac{d_i}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\int }_0^{{\theta }_i}{({\theta }_i-q\mathfrak{s})}^{(\beta -{\rho }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right\},\forall \mathfrak{t}\in [0,1].\hfill \end{array}$$

(40)

So, we deduce

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)=\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^1{\mathfrak{t}}^{\alpha -1}{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}-\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\alpha -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left\{\left[{\displaystyle \frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}-\left({\displaystyle \sum _{i=1}^d{d}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1}}\right)}\right]\right.}\hfill \\ {\displaystyle \times \sum _{i=1}^a\frac{a_i}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}\left[{\displaystyle {\xi }_i^{\alpha -{\varrho }_i-1}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}-{\int }_0^{{\xi }_i}{({\xi }_i-q\mathfrak{s})}^{(\alpha -{\varrho }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]}\hfill \\ +\left({\displaystyle \sum _{i=1}^b{b}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\eta }_i^{\beta -{\eta }_i-1}}\right)\hfill \\ \times \left.{\displaystyle \sum _{i=1}^c\frac{c_i}{{\mathrm{\Gamma }}_q(\alpha -{\eta }_i)}\left[{\displaystyle {\zeta }_i^{\alpha -{\eta }_i-1}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\zeta -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}-{\int }_0^{{\zeta }_i}{({\zeta }_i-q\mathfrak{s})}^{(\alpha -{\eta }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]}\right\}\hfill \\ {\displaystyle +\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left\{{\displaystyle \sum _{i=1}^b{b}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -\vartheta )}\frac{1}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}\left[{\displaystyle {\int }_0^1{\omega }_i^{\beta -{\sigma }_i-1}{(1-q\mathfrak{s})}^{(\beta -\vartheta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right.}\right.}\hfill \\ \left.{\displaystyle -{\int }_0^{{\omega }_i}{({\omega }_i-q\mathfrak{s})}^{(\beta -{\sigma }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]-\left({\displaystyle \sum _{i=1}^d{d}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}{\theta }_i^{\beta -{\rho }_i-1}}\right)\hfill \\ {\displaystyle \times \sum _{i=1}^b\frac{b_i}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}\left[{\displaystyle {\int }_0^1{\omega }_i^{\beta -{\sigma }_i-1}{(1-q\mathfrak{s})}^{(\beta -\vartheta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}-{\int }_0^{{\omega }_i}{({\omega }_i-q\mathfrak{s})}^{(\beta -{\sigma }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]}\hfill \\ +\left({\displaystyle \sum _{i=1}^b{b}_i\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}{\omega }_i^{\beta -{\sigma }_i-1}}\right)\hfill \\ {\displaystyle \times \sum _{i=1}^d\frac{d_i}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}\left.\left[{\displaystyle {\int }_0^1{\theta }_i^{\beta -{\rho }_i-1}{(1-q\mathfrak{s})}^{(\beta -\vartheta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}-{\int }_0^{{\theta }_i}{({\theta }_i-q\mathfrak{s})}^{(\beta -{\rho }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]\right\}}\hfill \\ {\displaystyle =\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^1{\mathfrak{t}}^{\alpha -1}{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}-\frac{1}{{\mathrm{\Gamma }}_q\left(\alpha \right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-q\mathfrak{s})}^{(\alpha -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\frac{t^{\alpha -1}}{\mathrm{\Delta }}\left\{{\displaystyle {\mathrm{\Lambda }}_4\sum _{i=1}^a\frac{a_i}{{\mathrm{\Gamma }}_q(\alpha -{\varrho }_i)}\left[{\displaystyle {\xi }_i^{\alpha -{\varrho }_i-1}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}-{\int }_0^{{\xi }_i}{({\xi }_i-q\mathfrak{s})}^{(\alpha -{\varrho }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]}\right.}\hfill \\ +\left.{\displaystyle {\mathrm{\Lambda }}_2\sum _{i=1}^c\frac{c_i}{{\mathrm{\Gamma }}_q(\alpha -{\eta }_i)}\left[{\displaystyle {\zeta }_i^{\alpha -{\eta }_i-1}{\int }_0^1{(1-q\mathfrak{s})}^{(\alpha -\varsigma -1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}-{\int }_0^{{\zeta }_i}{({\zeta }_i-q\mathfrak{s})}^{(\alpha -{\eta }_i-1)}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]}\right\}\hfill \\ {\displaystyle +\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left\{{\displaystyle {\mathrm{\Lambda }}_4\sum _{i=1}^b\frac{b_i}{{\mathrm{\Gamma }}_q(\beta -{\sigma }_i)}\left[{\displaystyle {\int }_0^1{\omega }_i^{\beta -{\sigma }_i-1}{(1-q\mathfrak{s})}^{(\beta -\vartheta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}-{\int }_0^{{\omega }_i}{({\omega }_i-q\mathfrak{s})}^{(\beta -{\sigma }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]}\right.}\hfill \\ +\left.{\displaystyle {\mathrm{\Lambda }}_2\sum _{i=1}^d\frac{d_i}{{\mathrm{\Gamma }}_q(\beta -{\rho }_i)}\left[{\displaystyle {\int }_0^1{\theta }_i^{\beta -{\rho }_i-1}{(1-q\mathfrak{s})}^{(\beta -\vartheta -1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}-{\int }_0^{{\theta }_i}{({\theta }_i-q\mathfrak{s})}^{(\beta -{\rho }_i-1)}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right]}\right\}.\hfill \end{array}$$

(41)

Therefore, we obtain

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)={\int }_0^1\left\{{\displaystyle g_1(\mathfrak{t},q\mathfrak{s})+\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_4\sum _{i=1}^a{a}_i{g}_{1i}({\xi }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_2\sum _{i=1}^c{c}_i{g}_{2i}({\zeta }_i,q\mathfrak{s})}\right]}\right\}\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}{\int }_0^1\left[{\displaystyle {\mathrm{\Lambda }}_4\sum _{i=1}^b{b}_i{g}_{3i}({\omega }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_2\sum _{i=1}^d{d}_i{g}_{4i}({\theta }_i,q\mathfrak{s})}\right]\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle ={\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s},\forall \mathfrak{t}\in [0,1],}\hfill \end{array}$$

(42)

where ${\mathfrak{G}}_1,{\mathfrak{G}}_2,g_1,g_{1i},i=1,\dots ,a,g_{2j},j=1,\dots ,c,g_{3k},k=1,\dots ,b,$ and $g_{4\iota },\iota =1,\dots ,d$, are given by (37) and (38).

In a similar manner, we find

$$\begin{array}{c}{\displaystyle \mathfrak{v}\left(\mathfrak{t}\right)=\frac{{\mathfrak{t}}^{\beta -1}}{\mathrm{\Delta }}{\int }_0^1\left[{\displaystyle {\mathrm{\Lambda }}_3\sum _{i=1}^a{a}_i{g}_{1i}({\xi }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_1\sum _{i=1}^c{c}_i{g}_{2i}({\zeta }_i,q\mathfrak{s})}\right]\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +{\int }_0^1\left\{{\displaystyle g_2(\mathfrak{t},q\mathfrak{s})+\frac{{\mathfrak{t}}^{\beta -1}}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_3\sum _{i=1}^b{b}_i{g}_{3i}({\omega }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_1\sum _{i=1}^d{d}_i{g}_{4i}({\theta }_i,q\mathfrak{s})}\right]}\right\}\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle ={\int }_0^1{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s},\forall \mathfrak{t}\in [0,1],}\hfill \end{array}$$

(43)

where ${\mathfrak{G}}_3,{\mathfrak{G}}_4,$ and $g_2$ are given by (37) and (38). So, we deduce the formulas in (36) for the solution $\left(\mathfrak{u}\right(\mathfrak{t}),\mathfrak{v}(\mathfrak{t}\left)\right),\mathfrak{t}\in [0,1]$, of problem (25),(2). □

With a similar proof to that of Lemma 12 from [4], we obtain the next result.

Lemma 7.

The functions $g_1,g_2,g_{1i},i=1,\dots ,a,g_{2j},j=1,\dots ,c,g_{3k},k=1,\dots ,b,$ and $g_{4\iota },\iota =1,\dots ,d$, have the following properties:

(a) $g_1(\mathfrak{t},q\mathfrak{s})\ge 0,g_2(\mathfrak{t},q\mathfrak{s})\ge 0,\forall \mathfrak{t},\mathfrak{s}\in [0,1]$;

(b) $g_1(\mathfrak{t},q\mathfrak{s})\le g_1(1,q\mathfrak{s}),g_2(\mathfrak{t},q\mathfrak{s})\le g_2(1,q\mathfrak{s}),\forall \mathfrak{t},\mathfrak{s}\in [0,1]$;

(c) $g_1(\mathfrak{t},q\mathfrak{s})\ge {\mathfrak{t}}^{\alpha -1}{g}_1(1,q\mathfrak{s}),g_2(\mathfrak{t},q\mathfrak{s})\ge {\mathfrak{t}}^{\beta -1}{g}_2(1,q\mathfrak{s}),\forall \mathfrak{t},\mathfrak{s}\in [0,1]$;

(d) $g_{1i}(\mathfrak{t},q\mathfrak{s})\ge 0,g_{2j}(\mathfrak{t},q\mathfrak{s})\ge 0,g_{3k}(\mathfrak{t},q\mathfrak{s})\ge 0,g_{4\iota }(\mathfrak{t},q\mathfrak{s})\ge 0,\forall \mathfrak{t},\mathfrak{s}\in [0,1],i=1,\dots ,a,j=1,\dots ,c,k=1,\dots ,b,\iota =1,\dots ,d$.

Remark 1.

If $\varpi =q^n$ with $n\in \mathbb{N}$, then we obtain

$${\displaystyle \underset{\mathfrak{t}\in [\varpi ,1]}{min}{g}_1(\mathfrak{t},q\mathfrak{s})\ge {\varpi }^{\alpha -1}{g}_1(1,q\mathfrak{s}),\underset{\mathfrak{t}\in [\varpi ,1]}{min}{g}_2(\mathfrak{t},q\mathfrak{s})\ge {\varpi }^{\beta -1}{g}_2(1,q\mathfrak{s}),\forall \mathfrak{s}\in [0,1].}$$

(44)

Lemma 8.

If ${\mathrm{\Lambda }}_1>0$, ${\mathrm{\Lambda }}_4>0$, and $\mathrm{\Delta }>0$, then the functions ${\mathfrak{G}}_i,i=1,\dots ,4$ satisfy the following inequalities:

(a) ${\mathfrak{t}}^{\alpha -1}{\mathfrak{G}}_1(1,q\mathfrak{s})\le {\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})\le {\mathfrak{G}}_1(1,q\mathfrak{s}),\forall \mathfrak{t},\mathfrak{s}\in [0,1]$;

(b) ${\mathfrak{t}}^{\alpha -1}{\mathfrak{G}}_2(1,q\mathfrak{s})={\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})\le {\mathfrak{G}}_2(1,q\mathfrak{s}),\forall \mathfrak{t},\mathfrak{s}\in [0,1]$;

(c) ${\mathfrak{t}}^{\beta -1}{\mathfrak{G}}_3(1,q\mathfrak{s})={\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})\le {\mathfrak{G}}_3(1,q\mathfrak{s}),\forall \mathfrak{t},\mathfrak{s}\in [0,1]$;

(d) ${\mathfrak{t}}^{\beta -1}{\mathfrak{G}}_4(1,q\mathfrak{s})\le {\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})\le {\mathfrak{G}}_4(1,q\mathfrak{s}),\forall \mathfrak{t},\mathfrak{s}\in [0,1]$.

Proof. 

$\left(a\right)$–$\left(d\right)$ By using Lemma 7, we deduce the following for all $\mathfrak{t},\mathfrak{s}\in [0,1]$:

$$\begin{array}{c}{\displaystyle {\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})=g_1(\mathfrak{t},q\mathfrak{s})+\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_4\sum _{i=1}^a{a}_i{g}_{1i}({\xi }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_2\sum _{i=1}^c{c}_i{g}_{2i}({\zeta }_i,q\mathfrak{s})}\right]}\hfill \\ {\displaystyle \le g_1(1,q\mathfrak{s})+\frac{1}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_4\sum _{i=1}^a{a}_i{g}_{1i}({\xi }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_2\sum _{i=1}^c{c}_i{g}_{2i}({\zeta }_i,q\mathfrak{s})}\right]={\mathfrak{G}}_1(1,q\mathfrak{s});}\hfill \\ {\displaystyle {\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})\ge {\mathfrak{t}}^{\alpha -1}{g}_1(1,q\mathfrak{s})+\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_4\sum _{i=1}^a{a}_i{g}_{1i}({\xi }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_2\sum _{i=1}^c{c}_i{g}_{2i}({\zeta }_i,q\mathfrak{s})}\right]}\hfill \\ ={\mathfrak{t}}^{\alpha -1}{\mathfrak{G}}_1(1,q\mathfrak{s});\hfill \\ {\displaystyle {\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})=\frac{{\mathfrak{t}}^{\alpha -1}}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_4\sum _{i=1}^b{b}_i{g}_{3i}({\omega }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_2\sum _{i=1}^d{d}_i{g}_{4i}({\theta }_i,q\mathfrak{s})}\right]}\hfill \\ {\displaystyle \le \frac{1}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_4\sum _{i=1}^b{b}_i{g}_{3i}({\omega }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_2\sum _{i=1}^d{d}_i{g}_{4i}({\theta }_i,q\mathfrak{s})}\right]={\mathfrak{G}}_2(1,q\mathfrak{s});}\hfill \\ {\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})={\mathfrak{t}}^{\alpha -1}{\mathfrak{G}}_2(1,q\mathfrak{s});\hfill \\ {\displaystyle {\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})=\frac{{\mathfrak{t}}^{\beta -1}}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_3\sum _{i=1}^a{a}_i{g}_{1i}({\xi }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_1\sum _{i=1}^c{c}_i{g}_{2i}({\zeta }_i,q\mathfrak{s})}\right]}\hfill \\ {\displaystyle \le \frac{1}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_3\sum _{i=1}^a{a}_i{g}_{1i}({\xi }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_1\sum _{i=1}^c{c}_i{g}_{2i}({\zeta }_i,q\mathfrak{s})}\right]={\mathfrak{G}}_3(1,q\mathfrak{s});}\hfill \\ {\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})={\mathfrak{t}}^{\beta -1}{\mathfrak{G}}_3(1,q\mathfrak{s});\hfill \\ {\displaystyle {\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})=g_2(\mathfrak{t},q\mathfrak{s})+\frac{{\mathfrak{t}}^{\beta -1}}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_3\sum _{i=1}^b{b}_i{g}_{3i}({\omega }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_1\sum _{i=1}^d{d}_i{g}_{4i}({\theta }_i,q\mathfrak{s})}\right]}\hfill \\ {\displaystyle \le g_2(1,q\mathfrak{s})+\frac{1}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_3\sum _{i=1}^b{b}_i{g}_{3i}({\omega }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_1\sum _{i=1}^d{d}_i{g}_{4i}({\theta }_i,q\mathfrak{s})}\right]={\mathfrak{G}}_4(1,q\mathfrak{s});}\hfill \\ {\displaystyle {\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})\ge {\mathfrak{t}}^{\beta -1}{g}_2(1,q\mathfrak{s})+\frac{{\mathfrak{t}}^{\beta -1}}{\mathrm{\Delta }}\left[{\displaystyle {\mathrm{\Lambda }}_3\sum _{i=1}^b{b}_i{g}_{3i}({\omega }_i,q\mathfrak{s})+{\mathrm{\Lambda }}_1\sum _{i=1}^d{d}_i{g}_{4i}({\theta }_i,q\mathfrak{s})}\right]}\hfill \\ ={\mathfrak{t}}^{\beta -1}{\mathfrak{G}}_4(1,q\mathfrak{s}).\hfill \end{array}$$

(45)

  □

Lemma 9.

If ${\mathrm{\Lambda }}_1>0$, ${\mathrm{\Lambda }}_4>0$, $\mathrm{\Delta }>0$, $\mathfrak{h}\left(\mathfrak{t}\right)\ge 0$, and $\mathfrak{k}\left(\mathfrak{t}\right)\ge 0$ for all $\mathfrak{t}\in [0,1]$, then the solution $\left(\mathfrak{u}\right(\mathfrak{t}),\mathfrak{v}(\mathfrak{t}\left)\right),\mathfrak{t}\in [0,1]$, of problem (25),(2) satisfies the inequalities $\mathfrak{u}\left(\mathfrak{t}\right)\ge 0$ and $\mathfrak{v}\left(\mathfrak{t}\right)\ge 0$ for all $\mathfrak{t}\in [0,1]$ and $\mathfrak{u}\left(\mathfrak{t}\right)\ge {\mathfrak{t}}^{\alpha -1}\mathfrak{u}\left(\tau \right)$ and $\mathfrak{v}\left(\mathfrak{t}\right)\ge {\mathfrak{t}}^{\beta -1}\mathfrak{v}\left(\tau \right)$ for all $\mathfrak{t},\tau \in [0,1]$.

Proof. 

We can easily verify that $\mathfrak{u}\left(\mathfrak{t}\right)\ge 0$ and $\mathfrak{v}\left(\mathfrak{t}\right)\ge 0$ for all $\mathfrak{t}\in [0,1]$. In addition, by using Lemma 8, we obtain

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)={\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \le {\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})\mathfrak{h}\left(s\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s},\forall \mathfrak{t}\in [0,1],}\hfill \\ {\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)\ge {\int }_0^1{\mathfrak{t}}^{\alpha -1}{\mathfrak{G}}_1(1,q\mathfrak{s})\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{t}}^{\alpha -1}{\mathfrak{G}}_2(1,q\mathfrak{s})\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ ={\mathfrak{t}}^{\alpha -1}\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})\mathfrak{h}\left(s\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right)\ge {\mathfrak{t}}^{\alpha -1}\mathfrak{u}\left(\tau \right),\forall \mathfrak{t},\tau \in [0,1],\hfill \\ {\displaystyle \mathfrak{v}\left(\mathfrak{t}\right)={\int }_0^1{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \le {\int }_0^1{\mathfrak{G}}_3(1,q\mathfrak{s})\mathfrak{h}\left(s\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_4(1,q\mathfrak{s})\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s},\forall \mathfrak{t}\in [0,1],}\hfill \end{array}$$

(46)

$$\begin{array}{c}{\displaystyle \mathfrak{v}\left(\mathfrak{t}\right)\ge {\int }_0^1{\mathfrak{t}}^{\beta -1}{\mathfrak{G}}_3(1,q\mathfrak{s})\mathfrak{h}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{t}}^{\beta -1}{\mathfrak{G}}_4(1,q\mathfrak{s})\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ ={\mathfrak{t}}^{\beta -1}\left({\displaystyle {\int }_0^1{\mathfrak{G}}_3(1,q\mathfrak{s})\mathfrak{h}\left(s\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_4(1,q\mathfrak{s})\mathfrak{k}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right)\ge {\mathfrak{t}}^{\beta -1}\mathfrak{v}\left(\tau \right),\forall \mathfrak{t},\tau \in [0,1].\hfill \end{array}$$

□

Remark 2.

By Lemma 9, we find $\mathfrak{u}\left(\mathfrak{t}\right)\ge {\mathfrak{t}}^{\alpha -1}\parallel u\parallel$ and $\mathfrak{v}\left(\mathfrak{t}\right)\ge {\mathfrak{t}}^{\beta -1}\parallel \mathfrak{v}\parallel$ for all $\mathfrak{t}\in [0,1]$. In addition, by Lemma 8, for $\varpi =q^n$ with $n\in \mathbb{N}$, we have

$$\begin{array}{c}{\displaystyle \underset{\mathfrak{t}\in [\varpi ,1]}{min}{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})\ge {\varpi }^{\alpha -1}{\mathfrak{G}}_1(1,q\mathfrak{s}),\forall \mathfrak{s}\in [0,1];}\hfill \\ {\displaystyle \underset{\mathfrak{t}\in [\varpi ,1]}{min}{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})={\varpi }^{\alpha -1}{\mathfrak{G}}_2(1,q\mathfrak{s}),\forall \mathfrak{s}\in [0,1];}\hfill \\ {\displaystyle \underset{\mathfrak{t}\in [\varpi ,1]}{min}{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})={\varpi }^{\beta -1}{\mathfrak{G}}_3(1,q\mathfrak{s}),\forall \mathfrak{s}\in [0,1];}\hfill \\ {\displaystyle \underset{\mathfrak{t}\in [\varpi ,1]}{min}{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})\ge {\varpi }^{\beta -1}{\mathfrak{G}}_4(1,q\mathfrak{s}),\forall \mathfrak{s}\in [0,1],}\hfill \end{array}$$

(47)

and so

$${\displaystyle \underset{\mathfrak{t}\in [\varpi ,1]}{min}\mathfrak{u}\left(\mathfrak{t}\right)\ge {\varpi }^{\alpha -1}\parallel \mathfrak{u}\parallel ,\underset{\mathfrak{t}\in [\varpi ,1]}{min}\mathfrak{v}\left(\mathfrak{t}\right)\ge {\varpi }^{\beta -1}\parallel \mathfrak{v}\parallel .}$$

(48)

3. Main Results

In this section, we will outline the results concerning the existence of positive solutions for our given problem (1),(2).

Initially, we state our primary assumptions:

(H1)

$q\in (0,1)$, $\alpha ,\beta \in \mathbb{R}$, $\alpha \in (n-1,n]$, $\beta \in (m-1,m]$, $n,m\in \mathbb{N}$, $n,m\ge 3$; $a,b,c,d\in \mathbb{N}$, $0\le {\varrho }_i\le \varsigma <\alpha -1$, $i=1,\dots ,a$, $\varsigma \ge 1$, $0\le {\eta }_i\le \varsigma$, $i=1,\dots ,c$, $0\le {\sigma }_i\le \vartheta <\beta -1$, $i=1,\dots ,b$, $\vartheta \ge 1$, $0\le {\rho }_i\le \vartheta$, $i=1,\dots ,d$; ${\delta }_i,{\gamma }_i>0$, $i=1,2$; ${\xi }_i,{\omega }_j,{\zeta }_k,{\theta }_{\iota }\in (0,1)$, $a_i,b_j,c_k,d_{\iota }\ge 0$ for $i=1,\dots ,a$, $j=1,\dots ,b$, $k=1,\dots ,c$, $\iota =1,\dots ,d$; ${\mathrm{\Lambda }}_1>0$, ${\mathrm{\Lambda }}_4>0$, $\mathrm{\Delta }>0$ (given by (26)).

(H2)

$\mathfrak{f},\mathfrak{g}:[0,1]\times {\mathbb{R}}_{+}^4\to {\mathbb{R}}_{+}$ are continuous functions (${\mathbb{R}}_{+}=[0,\infty )$).

Problem (1),(2) can be expressed equivalently to the following system of fractional q-integral equations:

$$\left\{\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathfrak{t}\right)={\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})\mathfrak{f}(\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right),\mathfrak{v}\left(\mathfrak{s}\right),I_q^{{\delta }_1}\mathfrak{u}\left(\mathfrak{s}\right),I_q^{{\gamma }_1}\mathfrak{v}\left(\mathfrak{s}\right))d_q\mathfrak{s}}\hfill \\ {\displaystyle +{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})\mathfrak{g}(\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right),\mathfrak{v}\left(\mathfrak{s}\right),I_q^{{\delta }_2}\mathfrak{u}\left(\mathfrak{s}\right),I_q^{{\gamma }_2}\mathfrak{v}\left(\mathfrak{s}\right))d_q\mathfrak{s},\mathfrak{t}\in [0,1],}\hfill \\ {\displaystyle \mathfrak{v}\left(\mathfrak{t}\right)={\int }_0^1{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})\mathfrak{f}(\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right),\mathfrak{v}\left(\mathfrak{s}\right),I_q^{{\delta }_1}\mathfrak{u}\left(\mathfrak{s}\right),I_q^{{\gamma }_1}\mathfrak{v}\left(\mathfrak{s}\right))d_q\mathfrak{s}}\hfill \\ {\displaystyle +{\int }_0^1{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})\mathfrak{g}(\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right),\mathfrak{v}\left(\mathfrak{s}\right),I_q^{{\delta }_2}\mathfrak{u}\left(\mathfrak{s}\right),I_q^{{\gamma }_2}\mathfrak{v}\left(\mathfrak{s}\right))d_q\mathfrak{s},\mathfrak{t}\in [0,1].}\hfill \end{array}\right.$$

(49)

Let $\mathcal{X}=C[0,1]$ be the Banach space endowed with the norm $\parallel \mathfrak{u}\parallel ={sup}_{\mathfrak{t}\in [0,1]}\left|\mathfrak{u}\left(\mathfrak{t}\right)\right|$, and let $\mathcal{Y}=\mathcal{X}\times \mathcal{X}$ be the Banach space with the norm ${\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}=\parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel$. We also introduce the cone $\mathcal{P}\subset \mathcal{Y}$ by

$$\mathcal{P}=\{(\mathfrak{u},\mathfrak{v})\in \mathcal{Y},\mathfrak{u}\left(\mathfrak{t}\right)\ge {\mathfrak{t}}^{\alpha -1}\parallel \mathfrak{u}\parallel ,\mathfrak{v}\left(\mathfrak{t}\right)\ge {\mathfrak{t}}^{\beta -1}\parallel \mathfrak{v}\parallel ,\forall \mathfrak{t}\in [0,1]\}.$$

(50)

We now define the operator $\mathcal{A}:\mathcal{P}\to \mathcal{Y}$, $\mathcal{A}(\mathfrak{u},\mathfrak{v})=({\mathcal{A}}_1(\mathfrak{u},\mathfrak{v}),{\mathcal{A}}_2(\mathfrak{u},\mathfrak{v}))$, for $(\mathfrak{u},\mathfrak{v})\in \mathcal{P}$, where ${\mathcal{A}}_1,{\mathcal{A}}_2:\mathcal{P}\to \mathcal{X}$ are given by

$$\begin{array}{c}{\displaystyle {\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)={\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})\mathfrak{f}(\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right),\mathfrak{v}\left(\mathfrak{s}\right),I_q^{{\delta }_1}\mathfrak{u}\left(\mathfrak{s}\right),I_q^{{\gamma }_1}\mathfrak{v}\left(\mathfrak{s}\right))d_q\mathfrak{s}}\hfill \\ {\displaystyle +{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})\mathfrak{g}(\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right),\mathfrak{v}\left(\mathfrak{s}\right),I_q^{{\delta }_2}\mathfrak{u}\left(\mathfrak{s}\right),I_q^{{\gamma }_2}\mathfrak{v}\left(\mathfrak{s}\right))d_q\mathfrak{s},}\hfill \\ {\displaystyle {\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)={\int }_0^1{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})\mathfrak{f}(\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right),\mathfrak{v}\left(\mathfrak{s}\right),I_q^{{\delta }_1}\mathfrak{u}\left(\mathfrak{s}\right),I_q^{{\gamma }_1}\mathfrak{v}\left(\mathfrak{s}\right))d_q\mathfrak{s}}\hfill \\ {\displaystyle +{\int }_0^1{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})\mathfrak{g}(\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right),\mathfrak{v}\left(\mathfrak{s}\right),I_q^{{\delta }_2}\mathfrak{u}\left(\mathfrak{s}\right),I_q^{{\gamma }_2}\mathfrak{v}\left(\mathfrak{s}\right))d_q\mathfrak{s},}\hfill \end{array}$$

(51)

for all $\mathfrak{t}\in [0,1]$ and $(\mathfrak{u},\mathfrak{v})\in \mathcal{P}$.

We observe that $(\mathfrak{u},\mathfrak{v})$ constitutes a positive solution to problem (49) (or equivalently, (1),(2)) if and only if it serves as a fixed point for the operator $\mathcal{A}$. Consequently, our subsequent analysis will focus on examining the existence of fixed points for $\mathcal{A}$.

Under assumptions $\left(H1\right)$ and $\left(H2\right)$, using standard arguments, we deduce that operator $\mathcal{A}$ is completely continuous. In addition, by Lemma 9, we obtain

$${\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)\ge {\mathfrak{t}}^{\alpha -1}\parallel {\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\parallel ,{\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)\ge {\mathfrak{t}}^{\beta -1}\parallel {\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\parallel ,\forall \mathfrak{t}\in [0,1],$$

(52)

that is, $\mathcal{A}\left(\mathcal{P}\right)\subset \mathcal{P}$.

For $\varpi =q^n$, with $n\in \mathbb{N}$, we introduce the constants

$$\begin{array}{c}{\displaystyle L_1={\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})d_q\mathfrak{s},L_2={\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})d_q\mathfrak{s},}\hfill \\ {\displaystyle L_3={\int }_0^1{\mathfrak{G}}_3(1,q\mathfrak{s})d_q\mathfrak{s},L_4={\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})d_q\mathfrak{s},}\hfill \\ M_1=L_1+L_3,M_2=L_2+L_4,\hfill \\ {\displaystyle {\tilde{L}}_1={\int }_{\varpi }^1{\mathfrak{G}}_1(1,q\mathfrak{s})d_q\mathfrak{s},{\tilde{L}}_2={\int }_{\varpi }^1{\mathfrak{G}}_2(1,q\mathfrak{s})d_q\mathfrak{s},}\hfill \\ {\displaystyle {\tilde{L}}_3={\int }_{\varpi }^1{\mathfrak{G}}_3(1,q\mathfrak{s})d_q\mathfrak{s},{\tilde{L}}_4={\int }_{\varpi }^1{\mathfrak{G}}_4(1,q\mathfrak{s})d_q\mathfrak{s},}\hfill \\ {\tilde{M}}_1={\varpi }^{\alpha -1}{\tilde{L}}_1+{\varpi }^{\beta -1}{\tilde{L}}_3,{\tilde{M}}_2={\varpi }^{\alpha -1}{\tilde{L}}_2+{\varpi }^{\beta -1}{\tilde{L}}_4.\hfill \end{array}$$

(53)

We remark that $L_2,L_3,{\tilde{L}}_2,$ and ${\tilde{L}}_3\ge 0$ and $L_1, L_4, {\tilde{L}}_1, {\tilde{L}}_4, M_1, M_2, {\tilde{M}}_1,$ and ${\tilde{M}}_2>0$.

Our initial existence result for positive solutions to problem (1),(2) relies on the Guo–Krasnosel’skii fixed-point theorem (refer to [28]).

Theorem 1.

Let $\varpi =q^n$ with $n\in \mathbb{N}$. Assume that $\left(H1\right)$ and $\left(H2\right)$ are satisfied. In addition, we suppose that there exist two positive constants $r_2>r_1>0$ and the constants ${\tilde{\sigma }}_1\in (0,M_1^{-1}]$, ${\tilde{\sigma }}_2\in (0,M_2^{-1}]$, ${\tilde{\sigma }}_3\in [{\tilde{M}}_1^{-1},\infty )$, and ${\tilde{\sigma }}_4\in [{\tilde{M}}_2^{-1},\infty )$ such that

• (H3)$\left\{\begin{array}{c}\mathfrak{f}(t,u,v,x,y)\ge \frac{{\tilde{\sigma }}_3{r}_1}{2},\forall t\in [\varpi ,1],u,v\ge 0,u+v\le r_1,\hfill \\ x\in \left[0,\frac{r_1}{{\mathrm{\Gamma }}_q({\delta }_1+1)}\right],y\in \left[0,\frac{r_1}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}\right],\hfill \\ \mathfrak{g}(t,u,v,x,y)\ge \frac{{\tilde{\sigma }}_4{r}_1}{2},\forall t\in [\varpi ,1],u,v\ge 0,u+v\le r_1,\hfill \\ x\in \left[0,\frac{r_1}{{\mathrm{\Gamma }}_q({\delta }_2+1)}\right],y\in \left[0,\frac{r_1}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}\right];\hfill \end{array}\right.$
• (H4)$\left\{\begin{array}{c}\mathfrak{f}(t,u,v,x,y)\le \frac{{\tilde{\sigma }}_1{r}_2}{2},\forall t\in [0,1],u,v\ge 0,u+v\le r_2,\hfill \\ x\in \left[0,\frac{r_2}{{\mathrm{\Gamma }}_q({\delta }_1+1)}\right],y\in \left[0,\frac{r_2}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}\right],\hfill \\ \mathfrak{g}(t,u,v,x,y)\le \frac{{\tilde{\sigma }}_2{r}_2}{2},\forall t\in [0,1],u,v\ge 0,u+v\le r_2,\hfill \\ x\in \left[0,\frac{r_2}{{\mathrm{\Gamma }}_q({\delta }_2+1)}\right],y\in \left[0,\frac{r_2}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}\right].\hfill \end{array}\right.$

Then, problem (1),(2) has at least one positive solution$\left(\mathfrak{u}\right(\mathfrak{t}),\mathfrak{v}(\mathfrak{t}\left)\right),\mathfrak{t}\in [0,1]$, such that $(\mathfrak{u},\mathfrak{v})\in \mathcal{P}$ and $r_1\le {\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}\le r_2$.

Proof. 

We introduce the set ${\mathrm{\Omega }}_2={\{(\mathfrak{u},\mathfrak{v})\in \mathcal{Y},\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}<r_2\}$. Then, for $(\mathfrak{u},\mathfrak{v})\in \mathcal{P}\cap \partial {\mathrm{\Omega }}_2$, we have $\parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel =r_2$, so $\mathfrak{u}\left(\mathfrak{t}\right)+\mathfrak{v}\left(\mathfrak{t}\right)\le r_2$ for all $\mathfrak{t}\in [0,1]$. Because $\parallel \mathfrak{u}\parallel \le r_2$ and $\parallel \mathfrak{v}\parallel \le r_2$, by Lemma 4, we find $|I_q^{{\delta }_i}\mathfrak{u}\left(\mathfrak{s}\right)|\le \frac{r_2}{{\mathrm{\Gamma }}_q({\delta }_i+1)}$, and $|I_q^{{\gamma }_j}\mathfrak{v}\left(\mathfrak{s}\right)|\le \frac{r_2}{{\mathrm{\Gamma }}_q({\gamma }_j+1)}$ for all $\mathfrak{s}\in [0,1]$, $i,j=1,2$. We define $F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)=\mathfrak{f}(\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right),\mathfrak{v}\left(\mathfrak{s}\right),I_q^{{\delta }_1}\mathfrak{u}\left(\mathfrak{s}\right),I_q^{{\gamma }_1}\mathfrak{v}\left(\mathfrak{s}\right))$ and $G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)=\mathfrak{g}(\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right),\mathfrak{v}\left(\mathfrak{s}\right),I_q^{{\delta }_2}\mathfrak{u}\left(\mathfrak{s}\right),I_q^{{\gamma }_2}\mathfrak{v}\left(\mathfrak{s}\right))$ for $\mathfrak{s}\in [0,1]$.

Then, by Lemma 8 and $\left(H4\right)$, we obtain

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\parallel =\underset{\mathfrak{t}\in [0,1]}{sup}\left|{\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)\right|}\hfill \\ {\displaystyle \le \underset{\mathfrak{t}\in [0,1]}{sup}{\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+\underset{\mathfrak{t}\in [0,1]}{sup}{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \le {\int }_0^1\underset{\mathfrak{t}\in [0,1]}{sup}{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1\underset{\mathfrak{t}\in [0,1]}{sup}{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \le {\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \le \frac{{\tilde{\sigma }}_1{r}_2}{2}{\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})d_q\mathfrak{s}+\frac{{\tilde{\sigma }}_2{r}_2}{2}{\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle =\frac{{\tilde{\sigma }}_1{r}_2}{2}{L}_1+\frac{{\tilde{\sigma }}_2{r}_2}{2}{L}_2=\left({\displaystyle \frac{{\tilde{\sigma }}_1{L}_1}{2}+\frac{{\tilde{\sigma }}_2{L}_2}{2}}\right)r_2,}\hfill \end{array}$$

(54)

and

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\parallel =\underset{\mathfrak{t}\in [0,1]}{sup}\left|{\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)\right|}\hfill \\ {\displaystyle \le \underset{\mathfrak{t}\in [0,1]}{sup}{\int }_0^1{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+\underset{\mathfrak{t}\in [0,1]}{sup}{\int }_0^1{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \le {\int }_0^1\underset{\mathfrak{t}\in [0,1]}{sup}{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1\underset{\mathfrak{t}\in [0,1]}{sup}{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \le {\int }_0^1{\mathfrak{G}}_3(1,q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_4(1,q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \le \frac{{\tilde{\sigma }}_1{r}_2}{2}{\int }_0^1{\mathfrak{G}}_3(1,q\mathfrak{s})d_q\mathfrak{s}+\frac{{\tilde{\sigma }}_2{r}_2}{2}{\int }_0^1{\mathfrak{G}}_4(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle =\frac{{\tilde{\sigma }}_1{r}_2}{2}{L}_3+\frac{{\tilde{\sigma }}_2{r}_2}{2}{L}_4=\left({\displaystyle \frac{{\tilde{\sigma }}_1{L}_3}{2}+\frac{{\tilde{\sigma }}_2{L}_4}{2}}\right)r_2.}\hfill \end{array}$$

(55)

Then, we deduce

$$\begin{array}{c}{\parallel \mathcal{A}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}=\parallel {\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\parallel +\parallel {\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\parallel \le \left[{\displaystyle \frac{{\tilde{\sigma }}_1(L_1+L_3)}{2}+\frac{{\tilde{\sigma }}_2(L_2+L_4)}{2}}\right]r_2\hfill \\ =\left({\displaystyle \frac{{\tilde{\sigma }}_1{M}_1}{2}+\frac{{\tilde{\sigma }}_2{M}_2}{2}}\right)r_2\le r_2,\hfill \end{array}$$

(56)

that is,

$${\parallel \mathcal{A}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}\le {\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}},\forall (\mathfrak{u},\mathfrak{v})\in \mathcal{P}\cap \partial {\mathrm{\Omega }}_2.$$

(57)

Now, we define the set ${\mathrm{\Omega }}_1={\{(\mathfrak{u},\mathfrak{v})\in \mathcal{Y},\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}<r_1\}$. Then, for $(\mathfrak{u},\mathfrak{v})\in \mathcal{P}\cap \partial {\mathrm{\Omega }}_1$, we have $\parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel =r_1$, so $\mathfrak{u}\left(\mathfrak{t}\right)+\mathfrak{v}\left(\mathfrak{t}\right)\le r_1$ for all $\mathfrak{t}\in [0,1]$.

Therefore, by Lemma 8 and $\left(H3\right)$, we obtain

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\parallel =\underset{\mathfrak{t}\in [0,1]}{sup}\left|{\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)\right|}\hfill \\ {\displaystyle =\underset{\mathfrak{t}\in [0,1]}{sup}\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right)}\hfill \\ {\displaystyle \ge \underset{\mathfrak{t}\in [\varpi ,1]}{inf}\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right)}\hfill \\ {\displaystyle \ge \underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \ge {\int }_0^1\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \ge {\int }_{\varpi }^1{\varpi }^{\alpha -1}{\mathfrak{G}}_1(1,q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_{\varpi }^1{\varpi }^{\alpha -1}{\mathfrak{G}}_2(1,q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \ge \frac{{\tilde{\sigma }}_3{r}_1{\varpi }^{\alpha -1}}{2}{\int }_{\varpi }^1{\mathfrak{G}}_1(1,q\mathfrak{s})d_q\mathfrak{s}+\frac{{\tilde{\sigma }}_4{r}_1{\varpi }^{\alpha -1}}{2}{\int }_{\varpi }^1{\mathfrak{G}}_2(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle =\frac{{\tilde{\sigma }}_3{r}_1{\varpi }^{\alpha -1}{\tilde{L}}_1}{2}+\frac{{\tilde{\sigma }}_4{r}_1{\varpi }^{\alpha -1}{\tilde{L}}_2}{2}=\left({\displaystyle \frac{{\tilde{\sigma }}_3{\varpi }^{\alpha -1}{\tilde{L}}_1}{2}+\frac{{\tilde{\sigma }}_4{\varpi }^{\alpha -1}{\tilde{L}}_2}{2}}\right)r_1,}\hfill \end{array}$$

(58)

and

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\parallel =\underset{\mathfrak{t}\in [0,1]}{sup}\left|{\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)\right|}\hfill \\ {\displaystyle =\underset{\mathfrak{t}\in [0,1]}{sup}\left({\displaystyle {\int }_0^1{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right)}\hfill \\ {\displaystyle \ge \underset{\mathfrak{t}\in [\varpi ,1]}{inf}\left({\displaystyle {\int }_0^1{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right)}\hfill \\ {\displaystyle \ge \underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\int }_0^1{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\int }_0^1{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \ge {\int }_0^1\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \ge {\int }_{\varpi }^1{\varpi }^{\beta -1}{\mathfrak{G}}_3(1,q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_{\varpi }^1{\varpi }^{\beta -1}{\mathfrak{G}}_4(1,q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \ge \frac{{\tilde{\sigma }}_3{r}_1{\varpi }^{\beta -1}}{2}{\int }_{\varpi }^1{\mathfrak{G}}_3(1,q\mathfrak{s})d_q\mathfrak{s}+\frac{{\tilde{\sigma }}_4{r}_1{\varpi }^{\beta -1}}{2}{\int }_{\varpi }^1{\mathfrak{G}}_4(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle =\frac{{\tilde{\sigma }}_3{r}_1{\varpi }^{\beta -1}{\tilde{L}}_3}{2}+\frac{{\tilde{\sigma }}_4{r}_1{\varpi }^{\beta -1}{\tilde{L}}_4}{2}=\left({\displaystyle \frac{{\tilde{\sigma }}_3{\varpi }^{\beta -1}{\tilde{L}}_3}{2}+\frac{{\tilde{\sigma }}_4{\varpi }^{\beta -1}{\tilde{L}}_4}{2}}\right)r_1.}\hfill \end{array}$$

(59)

Then, we conclude

$$\begin{array}{c}{\parallel \mathcal{A}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}=\parallel {\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\parallel +\parallel {\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\parallel \hfill \\ \ge \left[{\displaystyle \frac{{\tilde{\sigma }}_3({\varpi }^{\alpha -1}{\tilde{L}}_1+{\varpi }^{\beta -1}{\tilde{L}}_3)}{2}+\frac{{\tilde{\sigma }}_4({\varpi }^{\alpha -1}{\tilde{L}}_2+{\varpi }^{\beta -1}{\tilde{L}}_4)}{2}}\right]r_1\hfill \\ =\left({\displaystyle \frac{{\tilde{\sigma }}_3{\tilde{M}}_1}{2}+\frac{{\tilde{\sigma }}_4{\tilde{M}}_2}{2}}\right)r_1\ge r_1,\hfill \end{array}$$

(60)

and so

$${\parallel \mathcal{A}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}\ge {\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}},\forall (\mathfrak{u},\mathfrak{v})\in \mathcal{P}\cap \partial {\mathrm{\Omega }}_1.$$

(61)

As we mentioned before, the operator $\mathcal{A}$ is completely continuous. Then, by (57), (61), and the Guo–Krasnosel’skii fixed-point theorem, we deduce that the operator $\mathcal{A}$ has a fixed point $(\mathfrak{u},\mathfrak{v})\in \mathcal{P}\cap ({\overline{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1)$, which is a solution of problem (1),(2). This solution satisfies $r_1\le {\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}\le r_2$, $\mathfrak{u}\left(\mathfrak{t}\right)\ge 0$, $\mathfrak{v}\left(\mathfrak{t}\right)\ge 0$ for all $\mathfrak{t}\in [0,1]$, $\mathfrak{u}\left(\mathfrak{t}\right)\ge {\mathfrak{t}}^{\alpha -1}\parallel u\parallel$, $\mathfrak{v}\left(\mathfrak{t}\right)\ge {\mathfrak{t}}^{\beta -1}\parallel v\parallel$ for all $\mathfrak{t}\in [0,1]$; because $\parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel \ge r_1$, we obtain $\parallel \mathfrak{u}\parallel >0$ or $\parallel \mathfrak{v}\parallel >0$, that is, $\mathfrak{u}\left(\mathfrak{t}\right)>0$ for all $\mathfrak{t}\in (0,1]$ or $\mathfrak{v}\left(\mathfrak{t}\right)>0$ for all $\mathfrak{t}\in (0,1]$. □

Subsequently, we will establish the existence of at least three positive solutions to problem (1),(2) using the Leggett–Williams fixed-point theorem (refer to Theorem 3.3 in [29] or Theorem 2.3 in [30]).

Theorem 2.

Let $\varpi =q^n$ with $n\in \mathbb{N}$. Assume that $\left(H1\right)$ and $\left(H2\right)$ are satisfied. In addition, we suppose that there exist positive constants $0<r_1<r_2<r_3$ such that 

• (H5)$\left\{\begin{array}{c}\mathfrak{f}(t,u,v,x,y)<\frac{r_1}{2M_1},\forall t\in [0,1],u,v\ge 0,u+v\le r_1,\hfill \\ x\in \left[0,\frac{r_1}{{\mathrm{\Gamma }}_q({\delta }_1+1)}\right],y\in \left[0,\frac{r_1}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}\right],\hfill \\ \mathfrak{g}(t,u,v,x,y)<\frac{r_1}{2M_2},\forall t\in [0,1],u,v\ge 0,u+v\le r_1,\hfill \\ x\in \left[0,\frac{r_1}{{\mathrm{\Gamma }}_q({\delta }_2+1)}\right],y\in \left[0,\frac{r_1}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}\right];\hfill \end{array}\right.$
• (H6)$\left\{\begin{array}{c}\mathfrak{f}(t,u,v,x,y)>\frac{r_2}{2{\tilde{M}}_1},\forall t\in [\varpi ,1],u,v\ge 0,r_2\le u+v\le r_3,\hfill \\ x\in \left[0,\frac{r_3}{{\mathrm{\Gamma }}_q({\delta }_1+1)}\right],y\in \left[0,\frac{r_3}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}\right],\hfill \\ \mathfrak{g}(t,u,v,x,y)>\frac{r_2}{2{\tilde{M}}_2},\forall t\in [\varpi ,1],u,v\ge 0,r_2\le u+v\le r_3,\hfill \\ x\in \left[0,\frac{r_3}{{\mathrm{\Gamma }}_q({\delta }_2+1)}\right],y\in \left[0,\frac{r_3}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}\right];\hfill \end{array}\right.$
• (H7)$\left\{\begin{array}{c}\mathfrak{f}(t,u,v,x,y)\le \frac{r_3}{2M_1},\forall t\in [0,1],u,v\ge 0,u+v\le r_3,\hfill \\ x\in \left[0,\frac{r_3}{{\mathrm{\Gamma }}_q({\delta }_1+1)}\right],y\in \left[0,\frac{r_3}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}\right],\hfill \\ \mathfrak{g}(t,u,v,x,y)\le \frac{r_3}{2M_2},\forall t\in [0,1],u,v\ge 0,u+v\le r_3,\hfill \\ x\in \left[0,\frac{r_3}{{\mathrm{\Gamma }}_q({\delta }_2+1)}\right],y\in \left[0,\frac{r_3}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}\right].\hfill \end{array}\right.$

Then, problem (1),(2) has at least three positive solutions $({\mathfrak{u}}_i\left(\mathfrak{t}\right),{\mathfrak{v}}_i\left(\mathfrak{t}\right)),\mathfrak{t}\in [0,1],i=1,\dots ,3$, such that $({\mathfrak{u}}_i,{\mathfrak{v}}_i)\in \mathcal{P}$, $i=1,\dots ,3$, $\parallel ({\mathfrak{u}}_1,{\mathfrak{v}}_1)\parallel <r_1$, $\parallel ({\mathfrak{u}}_3,{\mathfrak{v}}_3)\parallel >r_1$, ${inf}_{\mathfrak{t}\in [\varpi ,1]}({\mathfrak{u}}_2\left(\mathfrak{t}\right)+{\mathfrak{v}}_2\left(\mathfrak{t}\right))>r_2$, and ${inf}_{\mathfrak{t}\in [\varpi ,1]}({\mathfrak{u}}_3\left(\mathfrak{t}\right)+{\mathfrak{v}}_3\left(\mathfrak{t}\right))<r_2$.

Proof. 

Firstly, we prove that $\mathcal{A}:{\overline{\mathcal{P}}}_{r_3}\to {\overline{\mathcal{P}}}_{r_3}$, where ${\mathcal{P}}_{r_3}=\{(\mathfrak{u},\mathfrak{v})\in \mathcal{P},\parallel (\mathfrak{u},\mathfrak{v})\parallel <r_3\}$. For any $(\mathfrak{u},\mathfrak{v})\in {\overline{\mathcal{P}}}_{r_3}$, we have $\parallel (\mathfrak{u},\mathfrak{v})\parallel \le r_3$, and so $\parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel \le r_3$ and $0\le \mathfrak{u}\left(\mathfrak{t}\right)+\mathfrak{v}\left(\mathfrak{t}\right)\le r_3$ for all $\mathfrak{t}\in [0,1]$. By using $\left(H7\right)$ and Lemma 8, we find

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\parallel =\underset{\mathfrak{t}\in [0,1]}{sup}\left|{\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)\right|}\hfill \\ {\displaystyle \le {\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \le \frac{r_3}{2M_1}{\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})d_q\mathfrak{s}+\frac{r_3}{2M_2}{\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle =\frac{r_3}{2M_1}{L}_1+\frac{r_3}{2M_2}{L}_2=\left({\displaystyle \frac{L_1}{2M_1}+\frac{L_2}{2M_2}}\right)r_3,}\hfill \end{array}$$

(62)

and

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\parallel =\underset{\mathfrak{t}\in [0,1]}{sup}\left|{\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)\right|}\hfill \\ {\displaystyle \le {\int }_0^1{\mathfrak{G}}_3(1,q\mathfrak{s})F_{\mathfrak{u}\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_4(1,q\mathfrak{s})G_{\mathfrak{u}\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \le \frac{r_3}{2M_1}{\int }_0^1{\mathfrak{G}}_3(1,q\mathfrak{s})d_q\mathfrak{s}+\frac{r_3}{2M_2}{\int }_0^1{\mathfrak{G}}_4(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle =\frac{r_3}{2M_1}{L}_3+\frac{r_3}{2M_2}{L}_4=\left({\displaystyle \frac{L_3}{2M_1}+\frac{L_4}{2M_2}}\right)r_3.}\hfill \end{array}$$

(63)

Then, we obtain

$$\parallel \mathcal{A}(\mathfrak{u},\mathfrak{v})\parallel \le \left({\displaystyle \frac{L_1+L_3}{2M_1}+\frac{L_2+L_4)}{2M_2}}\right)r_3=\left({\displaystyle \frac{M_1}{2M_1}+\frac{M_2}{2M_2}}\right)r_3=r_3,\forall (\mathfrak{u},\mathfrak{v})\in {\overline{\mathcal{P}}}_{r_3}.$$

(64)

So, $\mathcal{A}\left({\overline{\mathcal{P}}}_{r_3}\right)\subset {\overline{\mathcal{P}}}_{r_3}$.

We consider ${\tilde{r}}_3\in (r_2,r_3)$, and we define the concave nonnegative continuous functional $\mathrm{\Theta }$ on $\mathcal{P}$ by $\mathrm{\Theta }(\mathfrak{u},\mathfrak{v})={inf}_{\mathfrak{t}\in [\varpi ,1]}(\mathfrak{u}\left(\mathfrak{t}\right)+\mathfrak{v}\left(\mathfrak{t}\right)),(\mathfrak{u},\mathfrak{v})\in \mathcal{P}$. We see that $\mathrm{\Theta }(\mathfrak{u},\mathfrak{v})\le \parallel (\mathfrak{u},\mathfrak{v})\parallel$ for all $(\mathfrak{u},\mathfrak{v})\in {\overline{\mathcal{P}}}_{r_3}$.

Next, we will verify conditions (i)–(iii) of Theorem 2.3 from [30], with $E=\mathcal{Y}$, $K=\mathcal{P}$, $A=\mathcal{A}$, $m=r_1$, $c=r_2$, $l=r_3$, and $d={\tilde{r}}_3$.

Firstly, we verify condition (ii). For $(\mathfrak{u},\mathfrak{v})\in {\overline{\mathcal{P}}}_{r_1}$, we will show that $\parallel \mathcal{A}(\mathfrak{u},\mathfrak{v})\parallel <r_1$. For this, let $(\mathfrak{u},\mathfrak{v})\in {\overline{\mathcal{P}}}_{r_1}$. In a similar manner to that in (62) and (63), by using $\left(H5\right)$, we obtain

$$\parallel {\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\parallel <\left({\displaystyle \frac{L_1}{2M_1}+\frac{L_2}{2M_2}}\right)r_1,\parallel {\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\parallel <\left({\displaystyle \frac{L_3}{2M_1}+\frac{L_4}{2M_2}}\right)r_1,$$

(65)

and so $\parallel \mathcal{A}(\mathfrak{u},\mathfrak{v})\parallel <r_1$. So, we have assumption (ii) of Theorem 2.3 from [30].

We now verify condition (i). We set the element $({\mathfrak{u}}_0\left(\mathfrak{t}\right),{\mathfrak{v}}_0\left(\mathfrak{t}\right))=(\frac{r_2+{\tilde{r}}_3}{4},\frac{r_2+{\tilde{r}}_3}{4}),$$\mathfrak{t}\in [0,1]$. Because ${\mathfrak{u}}_0\left(\mathfrak{t}\right)\ge 0$, ${\mathfrak{v}}_0\left(\mathfrak{t}\right)\ge 0$ for all $\mathfrak{t}\in [0,1]$, $\parallel ({\mathfrak{u}}_0,{\mathfrak{v}}_0)\parallel =\frac{r_2+{\tilde{r}}_3}{2}<{\tilde{r}}_3$, and $\mathrm{\Theta }({\mathfrak{u}}_0,{\mathfrak{v}}_0)=\frac{r_2+{\tilde{r}}_3}{2}>r_2$, we deduce that $({\mathfrak{u}}_0,{\mathfrak{v}}_0)\in \{(\mathfrak{u},\mathfrak{v}),(\mathfrak{u},\mathfrak{v})\in \mathcal{P}(\mathrm{\Theta },r_2,{\tilde{r}}_3),\mathrm{\Theta }(\mathfrak{u},\mathfrak{v})>r_2\}$. Now, let $(\mathfrak{u},\mathfrak{v})\in \mathcal{P}(\mathrm{\Theta },r_2,{\tilde{r}}_3)$, that is, $(\mathfrak{u},\mathfrak{v})\in \mathcal{P},\mathrm{\Theta }(\mathfrak{u},\mathfrak{v})\ge r_2$ and $\parallel (\mathfrak{u},\mathfrak{v})\parallel \le {\tilde{r}}_3$. So, we have $\mathfrak{u}\left(\mathfrak{t}\right)+\mathfrak{v}\left(\mathfrak{t}\right)\le {\tilde{r}}_3$ for all $\mathfrak{t}\in [0,1]$ and ${inf}_{\mathfrak{t}\in [\varpi ,1]}(\mathfrak{u}\left(\mathfrak{t}\right)+\mathfrak{v}\left(\mathfrak{t}\right))\ge r_2$. Then, by $\left(H6\right)$ and Lemma 8, we find

$$\begin{array}{c}{\displaystyle \mathrm{\Theta }\left(\mathcal{A}(\mathfrak{u},\mathfrak{v})\right)=\underset{\mathfrak{t}\in [\varpi ,1]}{inf}({\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)+{\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right))}\hfill \\ {\displaystyle \ge \underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)+\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)}\hfill \\ {\displaystyle \ge \underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u},\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u},\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\int }_0^1{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u},\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\int }_0^1{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u},\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \ge {\int }_0^1\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u},\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u},\mathfrak{v}}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle +{\int }_0^1\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\mathfrak{G}}_3(\mathfrak{t},q\mathfrak{s})F_{\mathfrak{u},\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1\underset{\mathfrak{t}\in [\varpi ,1]}{inf}{\mathfrak{G}}_4(\mathfrak{t},q\mathfrak{s})G_{\mathfrak{u},\mathfrak{v}}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\hfill \\ {\displaystyle \ge {\int }_{\varpi }^1{\varpi }^{\alpha -1}{\mathfrak{G}}_1(1,q\mathfrak{s})F_{\mathfrak{u},\mathfrak{v}}^{{\delta }_1{\gamma }_1}{d}_q\mathfrak{s}+{\int }_{\varpi }^1{\varpi }^{\alpha -1}{\mathfrak{G}}_2(1,q\mathfrak{s})G_{\mathfrak{u},\mathfrak{v}}^{{\delta }_2{\gamma }_2}{d}_q\mathfrak{s}}\hfill \\ {\displaystyle +{\int }_{\varpi }^1{\varpi }^{\beta -1}{\mathfrak{G}}_3(1,q\mathfrak{s})F_{\mathfrak{u},\mathfrak{v}}^{{\delta }_1{\gamma }_1}{d}_q\mathfrak{s}+{\int }_{\varpi }^1{\varpi }^{\beta -1}{\mathfrak{G}}_4(1,q\mathfrak{s})G_{\mathfrak{u},\mathfrak{v}}^{{\delta }_2{\gamma }_2}{d}_q\mathfrak{s}}\hfill \\ {\displaystyle >\frac{r_2{\varpi }^{\alpha -1}}{2{\tilde{M}}_1}{\int }_{\varpi }^1{\mathfrak{G}}_1(1,q\mathfrak{s})d_q\mathfrak{s}+\frac{r_2{\varpi }^{\alpha -1}}{2{\tilde{M}}_2}{\int }_{\varpi }^1{\mathfrak{G}}_2(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle +\frac{r_2{\varpi }^{\beta -1}}{2{\tilde{M}}_1}{\int }_{\varpi }^1{\mathfrak{G}}_3(1,q\mathfrak{s})d_q\mathfrak{s}+\frac{r_2{\varpi }^{\beta -1}}{2{\tilde{M}}_2}{\int }_{\varpi }^1{\mathfrak{G}}_4(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle =\frac{r_2{\varpi }^{\alpha -1}}{2{\tilde{M}}_1}{\tilde{L}}_1+\frac{r_2{\varpi }^{\alpha -1}}{2{\tilde{M}}_2}{\tilde{L}}_2+\frac{r_2{\varpi }^{\beta -1}}{2{\tilde{M}}_1}{\tilde{L}}_3+\frac{r_2{\varpi }^{\beta -1}}{2{\tilde{M}}_2}{\tilde{L}}_4}\hfill \\ =\left[{\displaystyle \frac{{\varpi }^{\alpha -1}{\tilde{L}}_1+{\varpi }^{\beta -1}{\tilde{L}}_3}{2{\tilde{M}}_1}+\frac{{\varpi }^{\alpha -1}{\tilde{L}}_2+{\varpi }^{\beta -1}{\tilde{L}}_4}{2{\tilde{M}}_2}}\right]r_2\hfill \\ =\left({\displaystyle \frac{{\tilde{M}}_1}{2{\tilde{M}}_1}+\frac{{\tilde{M}}_2}{2{\tilde{M}}_2}}\right)r_2=r_2.\hfill \end{array}$$

(66)

Therefore, $\mathrm{\Theta }\left(\mathcal{A}(\mathfrak{u},\mathfrak{v})\right)>r_2$, and we have assumption (i) of Theorem 2.3 from [30].

We now verify condition (iii), namely, $\mathrm{\Theta }\left(\mathcal{A}(\mathfrak{u},\mathfrak{v})\right)>r_2$ for all $(\mathfrak{u},\mathfrak{v})\in \mathcal{P}(\mathrm{\Theta },r_2,r_3)$ and $\parallel \mathcal{A}(\mathfrak{u},\mathfrak{v})\parallel >{\tilde{r}}_3$. So, let $(\mathfrak{u},\mathfrak{v})\in \mathcal{P}(\mathrm{\Theta },r_2,r_3)$ and $\parallel \mathcal{A}(\mathfrak{u},\mathfrak{v})\parallel >{\tilde{r}}_3$. By using similar arguments to those used before, we deduce that $\mathrm{\Theta }\left(\mathcal{A}(\mathfrak{u},\mathfrak{v})\right)>r_2$; that is, assumption (iii) of Theorem 2.3 from [30] is satisfied.

We know that $\mathcal{A}$ is a completely continuous operator. Then, by the Leggett–Williams fixed-point theorem (see Theorem 2.3 from [30]), we conclude that problem (1),(2) has at least three positive solutions $({\mathfrak{u}}_i\left(\mathfrak{t}\right),{\mathfrak{v}}_i\left(\mathfrak{t}\right))\mathfrak{t}\in [0,1],i=1,\dots ,3$, with $({\mathfrak{u}}_i,{\mathfrak{v}}_i)\in \mathcal{P}$, $i=1,\dots ,3$ and $\parallel ({\mathfrak{u}}_1,{\mathfrak{v}}_1)\parallel <r_1$, $\parallel ({\mathfrak{u}}_3,{\mathfrak{v}}_3)\parallel >r_1$, $\mathrm{\Theta }({\mathfrak{u}}_2,{\mathfrak{v}}_2)={inf}_{\mathfrak{t}\in [\varpi ,1]}({\mathfrak{u}}_2\left(\mathfrak{t}\right)+{\mathfrak{v}}_2\left(\mathfrak{t}\right))>r_2$ and $\mathrm{\Theta }({\mathfrak{u}}_3,{\mathfrak{v}}_3)={inf}_{\mathfrak{t}\in [\varpi ,1]}({\mathfrak{u}}_3\left(\mathfrak{t}\right)+{\mathfrak{v}}_3\left(\mathfrak{t}\right))<r_2$. □

The following result is derived from the Schauder fixed-point theorem (refer to [31]).

Theorem 3.

Assume that $\left(H1\right)$ and $\left(H2\right)$ are satisfied. In addition, we suppose that there exist continuous functions $P_i,Q_i:[0,1]\to {\mathbb{R}}_{+}$, $i=1,\dots ,5$, such that

$$\begin{array}{c}{\displaystyle \mathfrak{f}(\mathfrak{t},u_1,u_2,u_3,u_4)\le \sum _{i=1}^4{P}_i\left(\mathfrak{t}\right)u_i+P_5\left(\mathfrak{t}\right),}\hfill \\ {\displaystyle \mathfrak{g}(\mathfrak{t},u_1,u_2,u_3,u_4)\le \sum _{i=1}^4{Q}_i\left(\mathfrak{t}\right)u_i+Q_5\left(\mathfrak{t}\right),}\hfill \end{array}$$

(67)

for all $\mathfrak{t}\in [0,1]$ and $u_i\in {\mathbb{R}}_{+}$, $i=1,\dots ,4$. If

$${\mathrm{\Psi }}_0=max\{{\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2\}<1,$$

(68)

where

$$\begin{array}{c}{\mathrm{\Psi }}_1=(L_1+L_3)\left({\displaystyle p_1^{*}+\frac{p_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_1+1)}}\right)+(L_2+L_4)\left({\displaystyle q_1^{*}+\frac{q_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_2+1)}}\right),\hfill \\ {\mathrm{\Psi }}_2=(L_1+L_3)\left({\displaystyle p_2^{*}+\frac{p_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}}\right)+(L_2+L_4)\left({\displaystyle q_2^{*}+\frac{q_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}}\right),\hfill \\ {\displaystyle p_i^{*}=\underset{\mathfrak{t}\in [0,1]}{sup}{P}_i\left(\mathfrak{t}\right),q_i^{*}=\underset{\mathfrak{t}\in [0,1]}{sup}{Q}_i\left(\mathfrak{t}\right),i=1,\dots ,4,}\hfill \end{array}$$

(69)

then the boundary value problem (1),(2) has at least one positive solution$\left(\mathfrak{u}\right(\mathfrak{t}),\mathfrak{v}(\mathfrak{t}\left)\right),\mathfrak{t}\in [0,1]$.

Proof. 

We consider a positive number $R_0$ satisfying the condition

$${\displaystyle R_0\ge \frac{(L_1+L_3)p_5^{*}+(L_2+L_4)q_5^{*}}{1-{\mathrm{\Psi }}_0},}$$

(70)

where $p_5^{*}={sup}_{\mathfrak{t}\in [0,1]}{P}_5\left(\mathfrak{t}\right)$, $q_5^{*}={sup}_{\mathfrak{t}\in [0,1]}{Q}_5\left(\mathfrak{t}\right).$ We define the set ${\mathrm{\Omega }}_1=\{(\mathfrak{u},\mathfrak{v})\in \mathcal{P},$$\parallel (\mathfrak{u},\mathfrak{v})\parallel \le R_0\}$.

Firstly, we show that $\mathcal{A}\left({\mathrm{\Omega }}_1\right)\subset {\mathrm{\Omega }}_1$. For this, let $(\mathfrak{u},\mathfrak{v})\in {\mathrm{\Omega }}_1$, that is, ${\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}\le R_0$, so $\parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel \le R_0$, which implies $\parallel \mathfrak{u}\parallel \le R_0$ and $\parallel \mathfrak{v}\parallel \le R_0$. Then, by using (67), Lemma 4 and Lemma 8, we obtain

$$\begin{array}{c}{\mathcal{A}}_1(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)\hfill \\ {\displaystyle \le {\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})(P_1\left(\mathfrak{s}\right)\mathfrak{u}\left(\mathfrak{s}\right)+P_2\left(\mathfrak{s}\right)\mathfrak{v}\left(\mathfrak{s}\right)+P_3\left(\mathfrak{s}\right)I_q^{{\delta }_1}\mathfrak{u}\left(\mathfrak{s}\right)+P_4\left(\mathfrak{s}\right)I_q^{{\gamma }_1}\mathfrak{v}\left(\mathfrak{s}\right)+P_5\left(\mathfrak{s}\right))d_q\mathfrak{s}}\hfill \\ {\displaystyle +{\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})(Q_1\left(\mathfrak{s}\right)\mathfrak{u}\left(\mathfrak{s}\right)+Q_2\left(\mathfrak{s}\right)\mathfrak{v}\left(\mathfrak{s}\right)+Q_3\left(\mathfrak{s}\right)I_q^{{\delta }_2}\mathfrak{u}\left(\mathfrak{s}\right)+Q_4\left(\mathfrak{s}\right)I_q^{{\gamma }_2}\mathfrak{v}\left(\mathfrak{s}\right)+Q_5\left(\mathfrak{s}\right))d_q\mathfrak{s}}\hfill \\ {\displaystyle \le \left({\displaystyle p_1^{*}\parallel \mathfrak{u}\parallel +p_2^{*}\parallel \mathfrak{v}\parallel +p_3^{*}\frac{\parallel \mathfrak{u}\parallel }{{\mathrm{\Gamma }}_q({\delta }_1+1)}+p_4^{*}\frac{\parallel \mathfrak{v}\parallel }{{\mathrm{\Gamma }}_q({\gamma }_1+1)}+p_5^{*}}\right){\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle +\left({\displaystyle q_1^{*}\parallel \mathfrak{u}\parallel +q_2^{*}\parallel \mathfrak{v}\parallel +q_3^{*}\frac{\parallel \mathfrak{u}\parallel }{{\mathrm{\Gamma }}_q({\delta }_2+1)}+q_4^{*}\frac{\parallel \mathfrak{v}\parallel }{{\mathrm{\Gamma }}_q({\gamma }_2+1)}+q_5^{*}}\right){\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ =L_1\left[\left({\displaystyle p_1^{*}+\frac{p_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_1+1)}}\right)\parallel \mathfrak{u}\parallel +\left({\displaystyle p_2^{*}+\frac{p_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}}\right)\parallel \mathfrak{v}\parallel +p_5^{*}\right]\hfill \\ +L_2\left[\left({\displaystyle q_1^{*}+\frac{q_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_2+1)}}\right)\parallel \mathfrak{u}\parallel +\left({\displaystyle q_2^{*}+\frac{q_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}}\right)\parallel \mathfrak{v}\parallel +q_5^{*}\right],\forall \mathfrak{t}\in [0,1],\hfill \end{array}$$

(71)

and

$$\begin{array}{c}{\mathcal{A}}_2(\mathfrak{u},\mathfrak{v})\left(\mathfrak{t}\right)\hfill \\ {\displaystyle \le {\int }_0^1{\mathfrak{G}}_3(1,q\mathfrak{s})(P_1\left(\mathfrak{s}\right)\mathfrak{u}\left(\mathfrak{s}\right)+P_2\left(\mathfrak{s}\right)\mathfrak{v}\left(\mathfrak{s}\right)+P_3\left(\mathfrak{s}\right)I_q^{{\delta }_1}\mathfrak{u}\left(\mathfrak{s}\right)+P_4\left(\mathfrak{s}\right)I_q^{{\gamma }_1}\mathfrak{v}\left(\mathfrak{s}\right)+P_5\left(\mathfrak{s}\right))d_q\mathfrak{s}}\hfill \\ {\displaystyle +{\int }_0^1{\mathfrak{G}}_4(1,q\mathfrak{s})(Q_1\left(\mathfrak{s}\right)\mathfrak{u}\left(\mathfrak{s}\right)+Q_2\left(\mathfrak{s}\right)\mathfrak{v}\left(\mathfrak{s}\right)+Q_3\left(\mathfrak{s}\right)I_q^{{\delta }_2}\mathfrak{u}\left(\mathfrak{s}\right)+Q_4\left(\mathfrak{s}\right)I_q^{{\gamma }_2}\mathfrak{v}\left(\mathfrak{s}\right)+Q_5\left(\mathfrak{s}\right))d_q\mathfrak{s}}\hfill \\ {\displaystyle \le \left({\displaystyle p_1^{*}\parallel \mathfrak{u}\parallel +p_2^{*}\parallel \mathfrak{v}\parallel +p_3^{*}\frac{\parallel \mathfrak{u}\parallel }{{\mathrm{\Gamma }}_q({\delta }_1+1)}+p_4^{*}\frac{\parallel \mathfrak{v}\parallel }{{\mathrm{\Gamma }}_q({\gamma }_1+1)}+p_5^{*}}\right){\int }_0^1{\mathfrak{G}}_3(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle +\left({\displaystyle q_1^{*}\parallel \mathfrak{u}\parallel +q_2^{*}\parallel \mathfrak{v}\parallel +q_3^{*}\frac{\parallel \mathfrak{u}\parallel }{{\mathrm{\Gamma }}_q({\delta }_2+1)}+q_4^{*}\frac{\parallel \mathfrak{v}\parallel }{{\mathrm{\Gamma }}_q({\gamma }_2+1)}+q_5^{*}}\right){\int }_0^1{\mathfrak{G}}_4(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ =L_3\left[\left({\displaystyle p_1^{*}+\frac{p_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_1+1)}}\right)\parallel \mathfrak{u}\parallel +\left({\displaystyle p_2^{*}+\frac{p_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}}\right)\parallel \mathfrak{v}\parallel +p_5^{*}\right]\hfill \\ +L_4\left[\left({\displaystyle q_1^{*}+\frac{q_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_2+1)}}\right)\parallel \mathfrak{u}\parallel +\left({\displaystyle q_2^{*}+\frac{q_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}}\right)\parallel \mathfrak{v}\parallel +q_5^{*}\right],\forall \mathfrak{t}\in [0,1].\hfill \end{array}$$

(72)

Then, by (70)–(72), we deduce

$$\begin{array}{c}{\parallel \mathcal{A}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}\le (L_1+L_3)\left[\left({\displaystyle p_1^{*}+\frac{p_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_1+1)}}\right)\parallel \mathfrak{u}\parallel +\left({\displaystyle p_2^{*}+\frac{p_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}}\right)\parallel \mathfrak{v}\parallel +p_5^{*}\right]\hfill \\ +(L_2+L_4)\left[\left({\displaystyle q_1^{*}+\frac{q_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_2+1)}}\right)\parallel \mathfrak{u}\parallel +\left({\displaystyle q_2^{*}+\frac{q_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}}\right)\parallel \mathfrak{v}\parallel +q_5^{*}\right]\hfill \\ =\left[(L_1+L_3)\left({\displaystyle p_1^{*}+\frac{p_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_1+1)}}\right)+(L_2+L_4)\left({\displaystyle q_1^{*}+\frac{q_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_2+1)}}\right)\right]\parallel \mathfrak{u}\parallel \hfill \\ +\left[(L_1+L_3)\left({\displaystyle p_2^{*}+\frac{p_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}}\right)+(L_2+L_4)\left({\displaystyle q_2^{*}+\frac{q_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}}\right)\right]\parallel \mathfrak{v}\parallel \hfill \\ +(L_1+L_3)p_5^{*}+(L_2+L_4)q_5^{*}\hfill \\ ={\mathrm{\Psi }}_1\parallel \mathfrak{u}\parallel +{\mathrm{\Psi }}_2\parallel \mathfrak{v}\parallel +(L_1+L_3)p_5^{*}+(L_2+L_4)q_5^{*}\hfill \\ \le {\mathrm{\Psi }}_0{\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}+(L_1+L_3)p_5^{*}+(L_2+L_4)q_5^{*}\hfill \\ \le {\mathrm{\Psi }}_0{R}_0+(L_1+L_3)p_5^{*}+(L_2+L_4)q_5^{*}\le R_0.\hfill \end{array}$$

(73)

Therefore, $\mathcal{A}\left({\mathrm{\Omega }}_1\right)\subset {\mathrm{\Omega }}_1$. Because the operator $\mathcal{A}$ is completely continuous, then by the Schauder fixed-point theorem, we conclude that $\mathcal{A}$ has a fixed point $(\mathfrak{u},\mathfrak{v})\in \mathcal{P}$ with ${\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Y}}\le R_0$, which is a positive solution of problem (1),(2). □

In the final theorem, we will employ the Banach contraction mapping principle.

Theorem 4.

Assume that $\left(H1\right)$ and $\left(H2\right)$ are satisfied. In addition, we suppose that there exist continuous functions $H_i,K_i:[0,1]\to {\mathbb{R}}_{+}$, $i=1,\dots ,4$, such that

$$\begin{array}{c}{\displaystyle |\mathfrak{f}(\mathfrak{t},u_1,u_2,u_3,u_4)-\mathfrak{f}(\mathfrak{t},v_1,v_2,v_3,v_4)| \le \sum _{i=1}^4{H}_i\left(\mathfrak{t}\right)|u_i-v_i|,}\hfill \\ {\displaystyle |\mathfrak{g}(\mathfrak{t},u_1,u_2,u_3,u_4)-\mathfrak{g}(\mathfrak{t},v_1,v_2,v_3,v_4)| \le \sum _{i=1}^4{K}_i\left(\mathfrak{t}\right)|u_i-v_i|,}\hfill \end{array}$$

(74)

for all $\mathfrak{t}\in [0,1]$, $u_i,v_i\in {\mathbb{R}}_{+}$, $i=1,\dots ,4$. If

$${\mathrm{\Phi }}_0=max\{{\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2\}<1,$$

(75)

where

$$\begin{array}{c}{\mathrm{\Phi }}_1=(L_1+L_3)\left({\displaystyle h_1^{*}+\frac{h_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_1+1)}}\right)+(L_2+L_4)\left({\displaystyle k_1^{*}+\frac{k_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_2+1)}}\right),\hfill \\ {\mathrm{\Phi }}_2=(L_1+L_3)\left({\displaystyle h_2^{*}+\frac{h_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}}\right)+(L_2+L_4)\left({\displaystyle k_2^{*}+\frac{k_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}}\right),\hfill \\ {\displaystyle h_i^{*}=\underset{\mathfrak{t}\in [0,1]}{sup}{H}_i\left(\mathfrak{t}\right),k_i^{*}=\underset{\mathfrak{t}\in [0,1]}{sup}{K}_i\left(\mathfrak{t}\right),i=1,\dots ,4,}\hfill \end{array}$$

(76)

then the boundary value problem given by (1),(2) possesses a unique positive solution $({\mathfrak{u}}^{*}\left(\mathfrak{t}\right),{\mathfrak{v}}^{*}\left(\mathfrak{t}\right)),$$\mathfrak{t}\in [0,1]$. Moreover, for any initial point $({\mathfrak{u}}_0,{\mathfrak{v}}_0)\in \mathcal{P}$, the sequence ${\left(({\mathfrak{u}}_n,{\mathfrak{v}}_n)\right)}_{n\ge 0}$ defined as $({\mathfrak{u}}_n,{\mathfrak{v}}_n)=\mathcal{A}({\mathfrak{u}}_{n-1},{\mathfrak{v}}_{n-1}),n\ge 1$, converges to $({\mathfrak{u}}^{*},{\mathfrak{v}}^{*})$ as $n\to \infty$. Additionally, an error estimate is provided by the following inequality:

$${\displaystyle \parallel ({\mathfrak{u}}_n,{\mathfrak{v}}_n)-({\mathfrak{u}}^{*},{\mathfrak{v}}^{*}){\parallel }_{\mathcal{Y}}\le \frac{{\mathrm{\Phi }}_0^n}{1-{\mathrm{\Phi }}_0}{\parallel ({\mathfrak{u}}_1,{\mathfrak{v}}_1)-({\mathfrak{u}}_0,{\mathfrak{v}}_0)\parallel }_{\mathcal{Y}}.}$$

(77)

Proof. 

By using (74), Lemma 4, and Lemma 8, for any $({\mathfrak{u}}_1,{\mathfrak{v}}_1),({\mathfrak{u}}_2,{\mathfrak{v}}_2)\in \mathcal{P}$, we obtain

$$\begin{array}{c}|{\mathcal{A}}_1({\mathfrak{u}}_1,{\mathfrak{v}}_1)\left(\mathfrak{t}\right)-{\mathcal{A}}_1({\mathfrak{u}}_2,{\mathfrak{v}}_2)\left(\mathfrak{t}\right)|\hfill \\ =\left|{\displaystyle {\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})F_{{\mathfrak{u}}_1{\mathfrak{v}}_1}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})G_{{\mathfrak{u}}_1{\mathfrak{v}}_1}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right.\hfill \\ -\left.{\displaystyle {\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})F_{{\mathfrak{u}}_2{\mathfrak{v}}_2}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)d_q\mathfrak{s}-{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})G_{{\mathfrak{u}}_2{\mathfrak{v}}_2}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)d_q\mathfrak{s}}\right|\hfill \\ {\displaystyle \le {\int }_0^1{\mathfrak{G}}_1(\mathfrak{t},q\mathfrak{s})\left|F_{{\mathfrak{u}}_1{\mathfrak{v}}_1}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)-F_{{\mathfrak{u}}_2{\mathfrak{v}}_2}^{{\delta }_1{\gamma }_1}\left(\mathfrak{s}\right)\right|d_q\mathfrak{s}+{\int }_0^1{\mathfrak{G}}_2(\mathfrak{t},q\mathfrak{s})\left|G_{{\mathfrak{u}}_1{\mathfrak{v}}_1}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)-G_{{\mathfrak{u}}_2{\mathfrak{v}}_2}^{{\delta }_2{\gamma }_2}\left(\mathfrak{s}\right)\right|d_q\mathfrak{s}}\hfill \\ {\displaystyle \le {\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})\left[H_1\left(\mathfrak{s}\right)|{\mathfrak{u}}_1\left(\mathfrak{s}\right)-{\mathfrak{u}}_2\left(\mathfrak{s}\right)|+H_2\left(\mathfrak{s}\right)|{\mathfrak{v}}_1\left(\mathfrak{s}\right)-{\mathfrak{v}}_2\left(\mathfrak{s}\right)|\right.}\hfill \\ +\left.H_3\left(\mathfrak{s}\right)|I_q^{{\delta }_1}{\mathfrak{u}}_1\left(\mathfrak{s}\right)-I_q^{{\delta }_1}{\mathfrak{u}}_2\left(\mathfrak{s}\right)|+H_4\left(\mathfrak{s}\right)|I_q^{{\gamma }_1}{\mathfrak{v}}_1\left(\mathfrak{s}\right)-I_q^{{\gamma }_1}{\mathfrak{v}}_2\left(\mathfrak{s}\right)|\right]d_q\mathfrak{s}\hfill \\ {\displaystyle +{\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})\left[K_1\left(\mathfrak{s}\right)|{\mathfrak{u}}_1\left(\mathfrak{s}\right)-{\mathfrak{u}}_2\left(\mathfrak{s}\right)|+K_2\left(\mathfrak{s}\right)|{\mathfrak{v}}_1\left(\mathfrak{s}\right)-{\mathfrak{v}}_2\left(\mathfrak{s}\right)|\right.}\hfill \\ +\left.K_3\left(\mathfrak{s}\right)|I_q^{{\delta }_2}{\mathfrak{u}}_1\left(\mathfrak{s}\right)-I_q^{{\delta }_2}{\mathfrak{u}}_2\left(\mathfrak{s}\right)|+K_4\left(\mathfrak{s}\right)|I_q^{{\gamma }_2}{\mathfrak{v}}_1\left(\mathfrak{s}\right)-I_q^{{\gamma }_2}{\mathfrak{v}}_2\left(\mathfrak{s}\right)|\right]d_q\mathfrak{s}\hfill \\ {\displaystyle \le {\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})\left[H_1\left(\mathfrak{s}\right)\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel +H_2\left(\mathfrak{s}\right)\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel \right.}\hfill \\ +\left.{\displaystyle H_3\left(\mathfrak{s}\right)\frac{1}{{\mathrm{\Gamma }}_q({\delta }_1+1)}\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel +H_4\left(\mathfrak{s}\right)\frac{1}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel }\right]d_q\mathfrak{s}\hfill \\ {\displaystyle +{\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})\left[K_1\left(\mathfrak{s}\right)\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel +K_2\left(\mathfrak{s}\right)\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel \right.}\hfill \\ +\left.{\displaystyle K_3\left(\mathfrak{s}\right)\frac{1}{{\mathrm{\Gamma }}_q({\delta }_2+1)}\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel +K_4\left(\mathfrak{s}\right)\frac{1}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel }\right]d_q\mathfrak{s}\hfill \\ {\displaystyle \le \left({\displaystyle h_1^{*}+\frac{h_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_1+1)}}\right)\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel {\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle +\left({\displaystyle h_2^{*}+\frac{h_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}}\right)\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel {\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle +\left({\displaystyle k_1^{*}+\frac{k_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_2+1)}}\right)\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel {\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ {\displaystyle +\left({\displaystyle k_2^{*}+\frac{k_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}}\right)\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel {\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})d_q\mathfrak{s}}\hfill \\ =\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel \left[\left({\displaystyle h_1^{*}+\frac{h_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_1+1)}}\right)L_1+\left({\displaystyle k_1^{*}+\frac{k_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_2+1)}}\right)L_2\right]\hfill \\ +\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel \left[\left({\displaystyle h_2^{*}+\frac{h_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}}\right)L_1+\left({\displaystyle k_2^{*}+\frac{k_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}}\right)L_2\right],\forall \mathfrak{t}\in [0,1].\hfill \end{array}$$

(78)

In a similar manner, we deduce

$$\begin{array}{c}|{\mathcal{A}}_2({\mathfrak{u}}_1,{\mathfrak{v}}_1)\left(\mathfrak{t}\right)-{\mathcal{A}}_2({\mathfrak{u}}_2,{\mathfrak{v}}_2)\left(\mathfrak{t}\right)|\hfill \\ \le \parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel \left[\left({\displaystyle h_1^{*}+\frac{h_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_1+1)}}\right)L_3+\left({\displaystyle k_1^{*}+\frac{k_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_2+1)}}\right)L_4\right]\hfill \\ +\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel \left[\left({\displaystyle h_2^{*}+\frac{h_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}}\right)L_3+\left({\displaystyle k_2^{*}+\frac{k_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}}\right)L_4\right],\forall \mathfrak{t}\in [0,1].\hfill \end{array}$$

(79)

Therefore, by (78) and (79), we conclude

$$\begin{array}{c}\parallel \mathcal{A}({\mathfrak{u}}_1,{\mathfrak{v}}_1)-\mathcal{A}({\mathfrak{u}}_2,{\mathfrak{v}}_2){\parallel }_{\mathcal{Y}}\hfill \\ \le \parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel \left[\left({\displaystyle h_1^{*}+\frac{h_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_1+1)}}\right)(L_1+L_3)+\left({\displaystyle k_1^{*}+\frac{k_3^{*}}{{\mathrm{\Gamma }}_q({\delta }_2+1)}}\right)(L_2+L_4)\right]\hfill \\ +\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel \left[\left({\displaystyle h_2^{*}+\frac{h_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_1+1)}}\right)(L_1+L_3)+\left({\displaystyle k_2^{*}+\frac{k_4^{*}}{{\mathrm{\Gamma }}_q({\gamma }_2+1)}}\right)(L_2+L_4)\right]\hfill \\ \le max\{{\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2\}\parallel ({\mathfrak{u}}_1,{\mathfrak{v}}_1)-({\mathfrak{u}}_2,{\mathfrak{v}}_2){\parallel }_{\mathcal{Y}}={\mathrm{\Phi }}_0{\parallel ({\mathfrak{u}}_1,{\mathfrak{v}}_1)-({\mathfrak{u}}_2,{\mathfrak{v}}_2)\parallel }_{\mathcal{Y}}.\hfill \end{array}$$

(80)

By satisfying condition (75), we establish that the operator $\mathcal{A}$ is a contraction mapping. Consequently, according to the Banach fixed-point theorem, it follows that $\mathcal{A}$ possesses a unique fixed point $({\mathfrak{u}}^{*},{\mathfrak{v}}^{*})\in \mathcal{P}$. This fixed point corresponds to the unique positive solution of problem (1),(2). Furthermore, for any $({\mathfrak{u}}_0,{\mathfrak{v}}_0)\in \mathcal{P}$, the sequence ${\left(({\mathfrak{u}}_n,{\mathfrak{v}}_n)\right)}_{n\ge 0}$ defined by $({\mathfrak{u}}_n,{\mathfrak{v}}_n)=\mathcal{A}({\mathfrak{u}}_{n-1},{\mathfrak{v}}_{n-1})$ for $n\ge 1$ converges to $({\mathfrak{u}}^{*},{\mathfrak{v}}^{*})$ as $n\to \infty$. The proof of the Banach theorem yields the error estimate (77). □

4. Examples

In this section, we will provide various examples to demonstrate and illustrate our key findings.

Let $q=\frac{1}{2}$, $\alpha =\frac{7}{2}$, $n=4$, $\beta =\frac{19}{4}$, $m=5$, $\varsigma =\frac{5}{3}$, $\vartheta =\frac{12}{5}$, ${\delta }_1=\frac{46}{5}$, ${\gamma }_1=\frac{31}{6}$, ${\delta }_2=\frac{59}{8}$, ${\gamma }_2=\frac{21}{13}$, $a=1$, $b=1$, $c=1$, $d=1$, ${\varrho }_1=\frac{14}{9}$, ${\sigma }_1=\frac{2}{11}$, ${\eta }_1=1$, ${\rho }_1=\frac{13}{7}$, ${\xi }_1=\frac{1}{2}$, ${\omega }_1=\frac{1}{16}$, ${\zeta }_1=\frac{1}{8}$, ${\theta }_1=\frac{1}{4}$, $a_1=\frac{1}{6}$, $b_1=31$, $c_1=25$, $d_1=\frac{1}{2}$, and $\varpi =\frac{1}{4}$.

We consider the system of fractional q-difference equations

$$\left\{\begin{array}{c}\left(D_{1/2}^{7/2}\mathfrak{u}\right)\left(\mathfrak{t}\right)+\mathfrak{f}\left(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{v}\left(\mathfrak{t}\right),I_{1/2}^{46/5}\mathfrak{u}\left(\mathfrak{t}\right),I_{1/2}^{31/6}\mathfrak{v}\left(\mathfrak{t}\right)\right)=0,\mathfrak{t}\in [0,1],\hfill \\ \left(D_{1/2}^{19/4}\mathfrak{v}\right)\left(\mathfrak{t}\right)+\mathfrak{g}\left(\mathfrak{t},\mathfrak{u}\left(\mathfrak{t}\right),\mathfrak{v}\left(\mathfrak{t}\right),I_{1/2}^{59/8}\mathfrak{u}\left(\mathfrak{t}\right),I_{1/2}^{21/13}\mathfrak{v}\left(\mathfrak{t}\right)\right)=0,\mathfrak{t}\in [0,1],\hfill \end{array}\right.$$

(81)

with the boundary conditions

$$\left\{\begin{array}{c}\mathfrak{u}\left(0\right)=D_{1/2}\mathfrak{u}\left(0\right)=D_{1/2}^2\mathfrak{u}\left(0\right)=0,D_{1/2}^{5/3}\mathfrak{u}\left(1\right)=\frac{1}{6}{D}_{1/2}^{14/9}\mathfrak{u}\left(\frac{1}{2}\right)+31D_{1/2}^{2/11}\mathfrak{v}\left(\frac{1}{16}\right),\hfill \\ \mathfrak{v}\left(0\right)=D_{1/2}\mathfrak{v}\left(0\right)=D_{1/2}^2\mathfrak{v}\left(0\right)=D_{1/2}^3\mathfrak{v}\left(0\right)=0,D_{1/2}^{12/5}\mathfrak{v}\left(1\right)=25D_{1/2}\mathfrak{u}\left(\frac{1}{8}\right)+\frac{1}{2}{D}_{1/2}^{13/7}\mathfrak{v}\left(\frac{1}{4}\right).\hfill \end{array}\right.$$

(82)

We obtain ${\mathrm{\Lambda }}_1\approx 1.86833829$, ${\mathrm{\Lambda }}_2\approx 0.00175841$, ${\mathrm{\Lambda }}_3\approx 1.8190837$, ${\mathrm{\Lambda }}_4\approx 3.62744165$, and $\mathrm{\Delta }={\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_4-{\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_3\approx 6.77408943>0$. So, assumption $\left(H1\right)$ is satisfied. We also have

$${\displaystyle \begin{array}{c}{g}_1(\mathfrak{t},\mathfrak{s})=\frac{1}{{\mathrm{\Gamma }}_{1/2}(7/2)}\left\{\begin{array}{c}{\mathfrak{t}}^{5/2}{(1-\mathfrak{s})}^{(5/6)}-{(\mathfrak{t}-\mathfrak{s})}^{(5/2)},0\le \mathfrak{s}\le \mathfrak{t}\le 1,\hfill \\ {\mathfrak{t}}^{5/2}{(1-\mathfrak{s})}^{(5/6)},0\le \mathfrak{t}\le \mathfrak{s}\le 1,\hfill \end{array}\right.\hfill \\ g_2(\mathfrak{t},\mathfrak{s})=\frac{1}{{\mathrm{\Gamma }}_{1/2}(19/4)}\left\{\begin{array}{c}{\mathfrak{t}}^{15/4}{(1-\mathfrak{s})}^{(27/20)}-{(\mathfrak{t}-\mathfrak{s})}^{(15/4)},0\le \mathfrak{s}\le \mathfrak{t}\le 1,\hfill \\ {\mathfrak{t}}^{15/4}{(1-\mathfrak{s})}^{(27/20)},0\le \mathfrak{t}\le \mathfrak{s}\le 1,\hfill \end{array}\right.\hfill \\ g_{11}(\mathfrak{t},\mathfrak{s})=\frac{1}{{\mathrm{\Gamma }}_{1/2}(35/18)}\left\{\begin{array}{c}{\mathfrak{t}}^{17/18}{(1-\mathfrak{s})}^{(5/6)}-{(\mathfrak{t}-\mathfrak{s})}^{(17/18)},0\le \mathfrak{s}\le \mathfrak{t}\le 1,\hfill \\ {\mathfrak{t}}^{17/18}{(1-\mathfrak{s})}^{(5/6)},0\le \mathfrak{t}\le \mathfrak{s}\le 1,\hfill \end{array}\right.\hfill \\ g_{21}(\mathfrak{t},\mathfrak{s})=\frac{1}{{\mathrm{\Gamma }}_{1/2}(5/2)}\left\{\begin{array}{c}{\mathfrak{t}}^{3/2}{(1-\mathfrak{s})}^{(5/6)}-{(\mathfrak{t}-\mathfrak{s})}^{(3/2)},0\le \mathfrak{s}\le \mathfrak{t}\le 1,\hfill \\ {\mathfrak{t}}^{3/2}{(1-\mathfrak{s})}^{(5/6)},0\le \mathfrak{t}\le \mathfrak{s}\le 1,\hfill \end{array}\right.\hfill \\ g_{31}(\mathfrak{t},\mathfrak{s})=\frac{1}{{\mathrm{\Gamma }}_{1/2}(201/44)}\left\{\begin{array}{c}{\mathfrak{t}}^{157/44}{(1-\mathfrak{s})}^{(27/20)}-{(\mathfrak{t}-\mathfrak{s})}^{(157/44)},0\le \mathfrak{s}\le \mathfrak{t}\le 1,\hfill \\ {\mathfrak{t}}^{157/44}{(1-\mathfrak{s})}^{(27/20)},0\le \mathfrak{t}\le \mathfrak{s}\le 1,\hfill \end{array}\right.\hfill \\ g_{41}(\mathfrak{t},\mathfrak{s})=\frac{1}{{\mathrm{\Gamma }}_{1/2}(81/28)}\left\{\begin{array}{c}{\mathfrak{t}}^{53/28}{(1-\mathfrak{s})}^{(27/20)}-{(\mathfrak{t}-\mathfrak{s})}^{(53/28)},0\le \mathfrak{s}\le \mathfrak{t}\le 1,\hfill \\ {\mathfrak{t}}^{53/28}{(1-\mathfrak{s})}^{(27/20)},0\le \mathfrak{t}\le \mathfrak{s}\le 1.\hfill \end{array}\right.\hfill \end{array}}$$

(83)

In addition, we find

$$\begin{array}{c}{\displaystyle {\mathfrak{G}}_1(\mathfrak{t},\mathfrak{s})=g_1(\mathfrak{t},\mathfrak{s})+\frac{{\mathfrak{t}}^{5/2}}{\mathrm{\Delta }}\left[{\displaystyle \frac{1}{6}{\mathrm{\Lambda }}_4{g}_{11}\left({\displaystyle \frac{1}{2},\mathfrak{s}}\right)+25{\mathrm{\Lambda }}_2{g}_{21}\left({\displaystyle \frac{1}{8},\mathfrak{s}}\right)}\right],}\hfill \\ {\displaystyle {\mathfrak{G}}_2(\mathfrak{t},\mathfrak{s})=\frac{{\mathfrak{t}}^{5/2}}{\mathrm{\Delta }}\left[{\displaystyle 31{\mathrm{\Lambda }}_4{g}_{31}\left({\displaystyle \frac{1}{16},\mathfrak{s}}\right)+\frac{1}{2}{\mathrm{\Lambda }}_2{g}_{41}\left({\displaystyle \frac{1}{4},\mathfrak{s}}\right)}\right],}\hfill \\ {\displaystyle {\mathfrak{G}}_3(\mathfrak{t},\mathfrak{s})=\frac{{\mathfrak{t}}^{15/4}}{\mathrm{\Delta }}\left[{\displaystyle \frac{1}{6}{\mathrm{\Lambda }}_3{g}_{11}\left({\displaystyle \frac{1}{2},\mathfrak{s}}\right)+25{\mathrm{\Lambda }}_1{g}_{21}\left({\displaystyle \frac{1}{8},\mathfrak{s}}\right)}\right],}\hfill \\ {\displaystyle {\mathfrak{G}}_4(\mathfrak{t},\mathfrak{s})=g_2(\mathfrak{t},\mathfrak{s})+\frac{{\mathfrak{t}}^{15/4}}{\mathrm{\Delta }}\left[{\displaystyle 31{\mathrm{\Lambda }}_3{g}_{31}\left({\displaystyle \frac{1}{16},\mathfrak{s}}\right)+\frac{1}{2}{\mathrm{\Lambda }}_1{g}_{41}\left({\displaystyle \frac{1}{4},\mathfrak{s}}\right)}\right],}\hfill \end{array}$$

(84)

for all $\mathfrak{t},\mathfrak{s}\in [0,1]$.

After complex computations using the Mathematica program, we obtain

$$\begin{array}{c}{L}_1={\int }_0^1{\mathfrak{G}}_1(1,q\mathfrak{s})d_q\mathfrak{s}\approx 0.09173103,L_2={\int }_0^1{\mathfrak{G}}_2(1,q\mathfrak{s})d_q\mathfrak{s}\approx 0.00012801,\hfill \\ L_3={\int }_0^1{\mathfrak{G}}_3(1,q\mathfrak{s})d_q\mathfrak{s}\approx 0.17043259,L_4={\int }_0^1{\mathfrak{G}}_4(1,q\mathfrak{s})d_q\mathfrak{s}\approx 0.02776846,\hfill \\ {\tilde{L}}_1={\int }_{1/4}^1{\mathfrak{G}}_1(1,q\mathfrak{s})d_q\mathfrak{s}\approx 0.08337987,{\tilde{L}}_2={\int }_{1/4}^1{\mathfrak{G}}_2(1,q\mathfrak{s})d_q\mathfrak{s}\approx 0.00008466,\hfill \\ {\tilde{L}}_3={\int }_{1/4}^1{\mathfrak{G}}_3(1,q\mathfrak{s})d_q\mathfrak{s}\approx 0.13008493,{\tilde{L}}_4={\int }_{1/4}^1{\mathfrak{G}}_4(1,q\mathfrak{s})d_q\mathfrak{s}\approx 0.02446965,\hfill \\ M_1\approx 0.26216363,M_2\approx 0.02789647,{\tilde{M}}_1\approx 0.00332425,{\tilde{M}}_2\approx 0.00013782.\hfill \end{array}$$

(85)

Example 1.

We consider the functions

$$\begin{array}{c}{\displaystyle \mathfrak{f}(\mathfrak{t},u,v,x,y)=\frac{{(u+v)}^2}{25}+\frac{1}{8}arctanx+\frac{1}{7}{cos}^{2}y+\frac{3({\mathfrak{t}}^2+1)}{\mathfrak{t}+4},}\hfill \\ {\displaystyle \mathfrak{g}(\mathfrak{t},u,v,x,y)=e^{-(u+v)}+\frac{x^2}{3}+y+\frac{9(\mathfrak{t}+1)}{{\mathfrak{t}}^3+2},}\hfill \end{array}$$

(86)

for all $\mathfrak{t}\in [0,1]$, $u,v,x,y\in {\mathbb{R}}_{+}$. Assumption $\left(H2\right)$ is satisfied.

We choose $r_1=\frac{1}{1000}$, $r_2=10$, ${\tilde{\sigma }}_1=3<\frac{1}{M_1}\approx 3.8144$, ${\tilde{\sigma }}_2=35<\frac{1}{M_2}\approx 35.8468$, ${\tilde{\sigma }}_3=301>\frac{1}{{\tilde{M}}_1}\approx 300.8201$, and ${\tilde{\sigma }}_4=7256>\frac{1}{{\tilde{M}}_2}\approx 7255.6974$.

Then, we obtain

$$\begin{array}{c}\mathfrak{f}(\mathfrak{t},u,v,x,y)\ge \underset{\mathfrak{t}\in [1/4,1]}{min}\frac{3({\mathfrak{t}}^2+1)}{\mathfrak{t}+4}=0.75\ge \frac{{\tilde{\sigma }}_3{r}_1}{2}=0.1505,\hfill \\ \forall \mathfrak{t}\in \left[\frac{1}{4},1\right],u,v\ge 0,u+v\le \frac{1}{1000},x\in \left[0,\frac{1}{1000{\mathrm{\Gamma }}_{1/2}(51/5)}\right],y\in \left[0,\frac{1}{1000{\mathrm{\Gamma }}_{1/2}(37/6)}\right],\hfill \\ \mathfrak{g}(\mathfrak{t},u,v,x,y)\ge e^{-1/1000}+\underset{\mathfrak{t}\in [1/4,1]}{min}\frac{9(\mathfrak{t}+1)}{{\mathfrak{t}}^3+2}\approx 6.5804\ge \frac{{\tilde{\sigma }}_4{r}_1}{2}\approx 3.628,\hfill \\ \forall \mathfrak{t}\in \left[\frac{1}{4},1\right],u,v\ge 0,u+v\le \frac{1}{1000},x\in \left[0,\frac{1}{1000{\mathrm{\Gamma }}_{1/2}(67/8)}\right],y\in \left[0,\frac{1}{1000{\mathrm{\Gamma }}_{1/2}(34/13)}\right],\hfill \end{array}$$

(87)

that is, assumption $\left(H3\right)$ is satisfied. In addition, we have

$$\begin{array}{c}\mathfrak{f}(\mathfrak{t},u,v,x,y)\le 4+\frac{1}{8}arctan\frac{10}{{\mathrm{\Gamma }}_{1/2}(51/5)}+\frac{1}{7}+\underset{\mathfrak{t}\in [0,1]}{max}\frac{3({\mathfrak{t}}^2+1)}{\mathfrak{t}+4}\approx 5.3502\le \frac{{\tilde{\sigma }}_1{r}_2}{2}=15,\hfill \\ \forall \mathfrak{t}\in [0,1],u,v\ge 0,u+v\le 10,x\in \left[0,\frac{10}{{\mathrm{\Gamma }}_{1/2}(51/5)}\right],y\in \left[0,\frac{10}{{\mathrm{\Gamma }}_{1/2}(37/6)}\right],\hfill \\ \mathfrak{g}(\mathfrak{t},u,v,x,y)\le 1+{\left(\frac{10}{{\mathrm{\Gamma }}_{1/2}(67/8)}\right)}^2+\frac{10}{{\mathrm{\Gamma }}_{1/2}(34/13)}+\underset{\mathfrak{t}\in [0,1]}{max}\frac{9(\mathfrak{t}+1)}{{\mathfrak{t}}^3+2}\approx 15.5719\le \frac{{\tilde{\sigma }}_2{r}_2}{2}=175,\hfill \\ \forall \mathfrak{t}\in [0,1],u,v\ge 0,u+v\le 10,x\in \left[0,\frac{10}{{\mathrm{\Gamma }}_{1/2}(67/8)}\right],y\in \left[0,\frac{10}{{\mathrm{\Gamma }}_{1/2}(34/13)}\right],\hfill \end{array}$$

(88)

so assumption $\left(H4\right)$ is verified.

By Theorem 1, we deduce that problem (81),(82) with the functions in (86) has at least one positive solution $\left(\mathfrak{u}\right(\mathfrak{t}),\mathfrak{v}(\mathfrak{t}\left)\right),\mathfrak{t}\in [0,1]$, such that $\frac{1}{1000}\le \parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel \le 10$, and ${min}_{\mathfrak{t}\in [1/4,1]}\mathfrak{u}\left(\mathfrak{t}\right)\ge {\left(\frac{1}{4}\right)}^{5/2}\parallel \mathfrak{u}\parallel$, ${min}_{\mathfrak{t}\in [1/4,1]}\mathfrak{v}\left(\mathfrak{t}\right)\ge {\left(\frac{1}{4}\right)}^{15/4}\parallel \mathfrak{v}\parallel$.

Example 2.

Let the functions

$$\begin{array}{c}\mathfrak{f}(\mathfrak{t},u,v,x,y)=\left\{\begin{array}{c}\frac{{\mathfrak{t}}^2+2}{4}+\frac{u}{3(1+u)}+\frac{v}{5(1+v)}+x^{1/3}+\frac{1}{6}{e}^{-y},\hfill \\ \forall \mathfrak{t}\in [0,1],u,v\ge 0,u+v\le 1,\hfill \\ \frac{{\mathfrak{t}}^2+2}{4}+\frac{1}{3}\left(1+4175\left(arctan\frac{\sqrt{3}(u+v)}{3}-\frac{\pi }{6}\right)\right)\frac{u}{1+u}\hfill \\ +\frac{1}{5}\left(1+3985\left(1-sin\frac{\pi }{2}(u+v)\right)\right)\frac{v}{1+v}+x^{1/3}+\frac{1}{6}{e}^{-y},\hfill \\ \forall \mathfrak{t}\in [0,1],u,v\ge 0,1<u+v\le 4,\hfill \\ \frac{{\mathfrak{t}}^2+2}{4}+\frac{1}{3}\left(1+4175\left(arctan\frac{4\sqrt{3}}{3}-\frac{\pi }{6}\right)\right)\frac{u}{1+u}\hfill \\ +\frac{3986}{5}\frac{v}{1+v}+x^{1/3}+\frac{1}{6}{e}^{-y},\hfill \\ \forall \mathfrak{t}\in [0,1],u,v\ge 0,u+v>4,\hfill \end{array}\right.\hfill \\ \mathfrak{g}(\mathfrak{t},u,v,x,y)=\left\{\begin{array}{c}\frac{14{\mathfrak{t}}^5+3}{7}+\frac{4u}{1+u}+\frac{v}{1+v}+\frac{x}{x^2+1}+y^{1/4},\hfill \\ \forall \mathfrak{t}\in [0,1],u,v\ge 0,u+v\le 1,\hfill \\ \frac{14{\mathfrak{t}}^5+3}{7}+4\left(1+38547\left(1-\frac{1}{\sqrt{u+v}}\right)\right)\frac{u}{1+u}\hfill \\ +\left(1+18452{cos}^2\frac{3\pi }{2}(u+v)\right)\frac{v}{1+v}+\frac{x}{x^2+1}+y^{1/4},\hfill \\ \forall \mathfrak{t}\in [0,1],u,v\ge 0,1<u+v\le 4,\hfill \\ \frac{14{\mathfrak{t}}^5+3}{7}+77098\frac{u}{1+u}+18453\frac{v}{1+v}+\frac{x}{x^2+1}+y^{1/4},\hfill \\ \forall \mathfrak{t}\in [0,1],u,v\ge 0,u+v>4.\hfill \end{array}\right.\hfill \end{array}$$

(89)

Assumption $\left(H2\right)$ is satisfied.

We choose $r_1=1$, $r_2=4$, and $r_3=6000$. Then, we obtain

$$\begin{array}{c}\mathfrak{f}(\mathfrak{t},u,v,x,y)\le \frac{3}{4}+\frac{1}{3}+\frac{1}{5}+{\left(\frac{1}{{\mathrm{\Gamma }}_{1/2}(51/5)}\right)}^{1/3}+\frac{1}{6}\approx 1.63046876<\frac{r_1}{2M_1}\approx 1.90720585,\hfill \\ \forall \mathfrak{t}\in [0,1],u,v\ge 0,u+v\le 1,x\in \left[0,\frac{1}{{\mathrm{\Gamma }}_{1/2}(51/5)}\right],y\in \left[0,\frac{1}{{\mathrm{\Gamma }}_{1/2}(37/6)}\right],\hfill \\ \mathfrak{g}(\mathfrak{t},u,v,x,y)\le \frac{17}{7}+5+\frac{1/{\mathrm{\Gamma }}_{1/2}(67/8)}{{(1/{\mathrm{\Gamma }}_{1/2}(67/8))}^2+1}+{\left(\frac{1}{{\mathrm{\Gamma }}_{1/2}(34/13)}\right)}^{1/4}\approx 8.39492155\hfill \\ <\frac{r_1}{2M_2}\approx 17.92341375,\forall \mathfrak{t}\in [0,1],u,v\ge 0,u+v\le 1,\hfill \\ x\in \left[0,\frac{1}{{\mathrm{\Gamma }}_{1/2}(67/8)}\right],y\in \left[0,\frac{1}{{\mathrm{\Gamma }}_{1/2}(34/13)}\right],\hfill \end{array}$$

(90)

that is, assumption $\left(H5\right)$ is satisfied.

Next, we find

$$\begin{array}{c}\mathfrak{f}(\mathfrak{t},u,v,x,y)\ge \frac{1}{64}+\frac{1}{2}+\frac{1}{3}\left(1+4175\left(arctan\frac{4\sqrt{3}}{3}-\frac{\pi }{6}\right)\right)\frac{u}{1+u}+\frac{3986}{5}\frac{v}{1+v}+\frac{1}{6}{e}^{-6000/{\mathrm{\Gamma }}_{1/2}(37/6)}\hfill \\ \ge \frac{33}{64}+min\left\{\frac{1}{3}\left(1+4175\left(arctan\frac{4\sqrt{3}}{3}-\frac{\pi }{6}\right)\right),\frac{3986}{5}\right\}\left(\frac{u}{1+u}+\frac{v}{1+v}\right)+\frac{1}{6}{e}^{-6000/{\mathrm{\Gamma }}_{1/2}(37/6)}\hfill \\ \ge \frac{33}{64}+min\left\{\frac{1}{3}\left(1+4175\left(arctan\frac{4\sqrt{3}}{3}-\frac{\pi }{6}\right)\right),\frac{3986}{5}\right\}\frac{u+v}{1+u+v}+\frac{1}{6}{e}^{-6000/{\mathrm{\Gamma }}_{1/2}(37/6)}\hfill \\ \ge \frac{33}{64}+min\left\{\frac{1}{3}\left(1+4175\left(arctan\frac{4\sqrt{3}}{3}-\frac{\pi }{6}\right)\right),\frac{3986}{5}\right\}\frac{4}{5}+\frac{1}{6}{e}^{-6000/{\mathrm{\Gamma }}_{1/2}(37/6)}\hfill \\ \approx 638.275625>\frac{r_2}{2{\tilde{M}}_1}\approx 601.64029336,\hfill \\ \forall \mathfrak{t}\in \left[\frac{1}{4},1\right],u,v\ge 0,4\le u+v\le 6000,x\in \left[0,\frac{6000}{{\mathrm{\Gamma }}_{1/2}(51/5)}\right],y\in \left[0,\frac{6000}{{\mathrm{\Gamma }}_{1/2}(37/6)}\right],\hfill \\ \mathfrak{g}(\mathfrak{t},u,v,x,y)\ge \frac{14{(1/4)}^5+3}{7}+min\{77098,18453\}\left(\frac{u}{1+u}+\frac{v}{1+v}\right)\hfill \\ \ge \frac{14{(1/4)}^5+3}{7}+18453\frac{u+v}{1+u+v}\ge \frac{14{(1/4)}^5+3}{7}+18453·\frac{4}{5}\hfill \\ \approx 14762.83052455>\frac{r_2}{2{\tilde{M}}_2}\approx 14511.39486712,\hfill \\ \forall \mathfrak{t}\in \left[\frac{1}{4},1\right],u,v\ge 0,4\le u+v\le 6000,x\in \left[0,\frac{6000}{{\mathrm{\Gamma }}_{1/2}(67/8)}\right],y\in \left[0,\frac{6000}{{\mathrm{\Gamma }}_{1/2}(34/13)}\right],\hfill \end{array}$$

(91)

and so assumption $\left(H6\right)$ is verified.

Lastly, we deduce

$$\begin{array}{c}\mathfrak{f}(\mathfrak{t},u,v,x,y)\le \frac{3}{4}+\frac{1}{3}\left(1+4175\left(arctan\frac{4\sqrt{3}}{3}-\frac{\pi }{6}\right)\right)\frac{6000}{6001}+\frac{3986}{5}·\frac{6000}{6001}\hfill \\ +{\left(\frac{6000}{{\mathrm{\Gamma }}_{1/2}(51/5)}\right)}^{1/3}+\frac{1}{6}\approx 1690.110593\le \frac{r_3}{2M_1}\approx 11443.23507,\hfill \\ \forall \mathfrak{t}\in [0,1],u,v\ge 0,u+v\le 6000,x\in \left[0,\frac{6000}{{\mathrm{\Gamma }}_{1/2}(51/5)}\right],y\in \left[0,\frac{6000}{{\mathrm{\Gamma }}_{1/2}(37/6)}\right],\hfill \\ \mathfrak{g}(\mathfrak{t},u,v,x,y)\le \frac{17}{7}+77098·\frac{6000}{6001}+18453·\frac{6000}{6001}+\frac{1}{2}+{\left(\frac{6000}{{\mathrm{\Gamma }}_{1/2}(34/13)}\right)}^{1/4}\hfill \\ \approx 95546.32860601\le \frac{r_3}{2M_2}\approx 107540.48247,\hfill \\ \forall \mathfrak{t}\in [0,1],u,v\ge 0,u+v\le 6000,x\in \left[0,\frac{6000}{{\mathrm{\Gamma }}_{1/2}(67/8)}\right],y\in \left[0,\frac{6000}{{\mathrm{\Gamma }}_{1/2}(34/13)}\right],\hfill \end{array}$$

(92)

that is, assumption $\left(H7\right)$ is also satisfied.

Therefore, by Theorem 2, we conclude that problem (81),(82) with the nonlinearities in (89) has at least three positive solutions $({\mathfrak{u}}_i,{\mathfrak{v}}_i)\in \mathcal{P},i=1,\dots ,3$, such that $\parallel ({\mathfrak{u}}_1,{\mathfrak{v}}_1){\parallel }_{\mathcal{Y}} <1$, $\parallel ({\mathfrak{u}}_3,{\mathfrak{v}}_3){\parallel }_{\mathcal{Y}} >1$, ${inf}_{\mathfrak{t}\in [1/4,1]}({\mathfrak{u}}_2\left(\mathfrak{t}\right)+{\mathfrak{v}}_2\left(\mathfrak{t}\right))>4$, and ${inf}_{\mathfrak{t}\in [1/4,1]}({\mathfrak{u}}_3\left(\mathfrak{t}\right)+{\mathfrak{v}}_3\left(\mathfrak{t}\right))<4$.

Example 3.

We consider the functions

$$\begin{array}{c}{\displaystyle \mathfrak{f}(\mathfrak{t},u,v,x,y)=\frac{{(\mathfrak{t}+1)}^{1/3}{e}^{-\mathfrak{t}+2}u}{4(1+u^2)}+\frac{{(\mathfrak{t}-1)}^{2}v}{2(\mathfrak{t}+5)}+\frac{e^{\mathfrak{t}}x}{3}+\frac{\mathfrak{t}y}{1+3y}+\frac{{\mathfrak{t}}^{11/3}}{{\mathfrak{t}}^2+6},}\hfill \\ {\displaystyle \mathfrak{g}(\mathfrak{t},u,v,x,y)=\frac{{\mathfrak{t}}^{2/7}u}{3(\mathfrak{t}+4)}+\frac{(\mathfrak{t}+3)e^{\mathfrak{t}+1}v}{2({\mathfrak{t}}^2+3)(1+2v^3)}+\frac{x}{{(\mathfrak{t}+1)}^2(1+2x)}+\frac{\mathfrak{t}y}{{\mathfrak{t}}^3+1}+\frac{{\mathfrak{t}}^{6/5}}{\mathfrak{t}+2},}\hfill \end{array}$$

(93)

for all $\mathfrak{t}\in [0,1],u,v,x,y\ge 0$. We obtain the following inequalities:

$$\begin{array}{c}\mathfrak{f}(\mathfrak{t},u,v,x,y)\le P_1\left(\mathfrak{t}\right)u+P_2\left(\mathfrak{t}\right)v+P_3\left(\mathfrak{t}\right)x+P_4\left(\mathfrak{t}\right)y+P_5\left(\mathfrak{t}\right),\hfill \\ \mathfrak{g}(\mathfrak{t},u,v,x,y)\le Q_1\left(\mathfrak{t}\right)u+Q_2\left(\mathfrak{t}\right)v+Q_3\left(\mathfrak{t}\right)x+Q_4\left(\mathfrak{t}\right)y+Q_5\left(\mathfrak{t}\right),\hfill \end{array}$$

(94)

for all $\mathfrak{t}\in [0,1]$, $u,v,x,y\ge 0$, where

$$\begin{array}{c}{P}_1\left(\mathfrak{t}\right)=\frac{{(\mathfrak{t}+1)}^{1/3}{e}^{-\mathfrak{t}+2}}{4},P_2\left(\mathfrak{t}\right)=\frac{{(\mathfrak{t}-1)}^2}{2(\mathfrak{t}+5)},P_3\left(\mathfrak{t}\right)=\frac{e^t}{3},P_4\left(\mathfrak{t}\right)=t,P_5\left(\mathfrak{t}\right)=\frac{{\mathfrak{t}}^{11/3}}{{\mathfrak{t}}^2+6},\hfill \\ Q_1\left(\mathfrak{t}\right)=\frac{{\mathfrak{t}}^{2/7}}{3(\mathfrak{t}+4)},Q_2\left(\mathfrak{t}\right)=\frac{(\mathfrak{t}+3)e^{\mathfrak{t}+1}}{2({\mathfrak{t}}^2+3)},Q_3\left(\mathfrak{t}\right)=\frac{1}{{(\mathfrak{t}+1)}^2},Q_4\left(\mathfrak{t}\right)=\frac{\mathfrak{t}}{{\mathfrak{t}}^3+1},Q_5\left(\mathfrak{t}\right)=\frac{{\mathfrak{t}}^{6/5}}{\mathfrak{t}+2},\hfill \end{array}$$

(95)

for all $\mathfrak{t}\in [0,1]$.

The functions $P_i$ and $Q_i,i=1,\dots ,5$, are continuous and satisfy condition (67). In addition, we find $p_1^{*}\approx 1.84726402$, $p_2^{*}=0.1$, $p_3^{*}\approx 0.90609394$, $p_4^{*}=1$, $q_1^{*}\approx 0.06666667$, $q_2^{*}\approx 3.69452805$, $q_3^{*}=1$, $q_4^{*}\approx 0.52913368$, ${\mathrm{\Psi }}_1\approx 0.48811984$, ${\mathrm{\Psi }}_2\approx 0.16566027$, and ${\mathrm{\Psi }}_0\approx 0.4881<1$; that is, assumption (68) is satisfied. Therefore, by Theorem 3, we conclude that problem (81),(82) with the nonlinearities in (93) has at least one positive solution $\left(\mathfrak{u}\right(\mathfrak{t}),\mathfrak{v}(\mathfrak{t}\left)\right),\mathfrak{t}\in [0,1]$.

Example 4.

Let the functions

$$\begin{array}{c}{\displaystyle \mathfrak{f}(\mathfrak{t},u,v,x,y)=\frac{2{(\mathfrak{t}+1)}^3{e}^{-5\mathfrak{t}+1}}{3}\sqrt{u^2+4}+\frac{{(\mathfrak{t}-2)}^2}{2}{e}^{3\mathfrak{t}-2}{cos}^2(v+1)}\hfill \\ {\displaystyle +\frac{{\mathfrak{t}}^{15/4}}{{\mathfrak{t}}^2+2}+\frac{\mathfrak{t}x}{4(\mathfrak{t}+1)}+\frac{\mathfrak{t}+1}{{\mathfrak{t}}^2+3}arctany,}\hfill \\ {\displaystyle \mathfrak{g}(\mathfrak{t},u,v,x,y)=\frac{{\mathfrak{t}}^3{e}^{-4\mathfrak{t}+1}}{2}arctanu+\frac{e^{2\mathfrak{t}}}{3{(\mathfrak{t}+2)}^4}\sqrt{v^2+1}+\frac{{\mathfrak{t}}^{9/8}}{\mathfrak{t}+3}}\hfill \\ {\displaystyle +\frac{\mathfrak{t}x}{10(x^2+1)}+\frac{e^t}{9}{sin}^{2}y,}\hfill \end{array}$$

(96)

for all $\mathfrak{t}\in [0,1],u,v,x,y\ge 0$. We obtain the following inequalities:

$$\begin{array}{c}|\mathfrak{f}(\mathfrak{t},u_1,v_1,x_1,y_1)-\mathfrak{f}(\mathfrak{t},u_2,v_2,x_2,y_2)|\hfill \\ \le H_1\left(\mathfrak{t}\right)|u_1-u_2|+H_2\left(\mathfrak{t}\right)|v_1-v_2|+H_3\left(\mathfrak{t}\right)|x_1-x_2|+H_4\left(\mathfrak{t}\right)|y_1-y_2|,\hfill \\ |\mathfrak{g}(\mathfrak{t},u_1,v_1,x_1,y_1)-\mathfrak{g}(\mathfrak{t},u_2,v_2,x_2,y_2)|\hfill \\ \le K_1\left(\mathfrak{t}\right)|u_1-u_2|+K_2\left(\mathfrak{t}\right)|v_1-v_2|+K_3\left(\mathfrak{t}\right)|x_1-x_2|+K_4\left(\mathfrak{t}\right)|y_1-y_2|,\hfill \end{array}$$

(97)

for all $\mathfrak{t}\in [0,1],u_i,v_i,x_i,y_i\ge 0,i=1,2$, where

$$\begin{array}{c}{H}_1\left(\mathfrak{t}\right)=\frac{2{(\mathfrak{t}+1)}^3{e}^{-5\mathfrak{t}+1}}{3},H_2\left(\mathfrak{t}\right)={(\mathfrak{t}-2)}^2{e}^{3\mathfrak{t}-2},H_3\left(\mathfrak{t}\right)=\frac{\mathfrak{t}}{4(\mathfrak{t}+1)},H_4\left(\mathfrak{t}\right)=\frac{\mathfrak{t}+1}{{\mathfrak{t}}^2+3},\hfill \\ K_1\left(\mathfrak{t}\right)=\frac{{\mathfrak{t}}^3{e}^{-4\mathfrak{t}+1}}{2},K_2\left(\mathfrak{t}\right)=\frac{e^{2\mathfrak{t}}}{3{(\mathfrak{t}+2)}^4},K_3\left(\mathfrak{t}\right)=\frac{\mathfrak{t}}{10},K_4\left(\mathfrak{t}\right)=\frac{2e^{\mathfrak{t}}}{9},\hfill \end{array}$$

(98)

for all $\mathfrak{t}\in [0,1]$.

The functions $H_i$ and $K_i,i=1,\dots ,4$ are continuous and satisfy condition (74). In addition, we find $h_1^{*}\approx 1.81218788$, $h_2^{*}\approx 2.71828183$, $h_3^{*}=0.125$, $h_4^{*}=0.5$, $k_1^{*}\approx 0.02854728$, $k_2^{*}\approx 0.03040764$, $k_3^{*}=0.1$, $k_4^{*}\approx 0.60406263$, ${\mathrm{\Phi }}_1\approx 0.47613656$, ${\mathrm{\Phi }}_2\approx 0.73924549$, and ${\mathrm{\Phi }}_0\approx 0.7392<1$; that is, assumption (75) is verified. Then, by Theorem 4, we deduce that problem (81),(82) with the functions in (96) has a unique positive solution $({\mathfrak{u}}^{*}\left(\mathfrak{t}\right),{\mathfrak{v}}^{*}\left(\mathfrak{t}\right)),$$\mathfrak{t}\in [0,1]$.

5. Conclusions

In this paper, we explore the existence, uniqueness, and multiplicity of positive solutions for a system of fractional q-difference equations (Equation (1)). These equations involve fractional q-integrals and are subject to coupled multi-point boundary conditions (2). These boundary conditions encompass q-derivatives and fractional q-derivatives of unknown functions with varying orders. Initially, our focus was on investigating the linear boundary value problem related to (1),(2), along with studying the associated Green functions and their properties. Subsequently, we reformulated our problem equivalently to a system of fractional q-integral equations (Equation (49)). We associated these equations with an operator, and the fixed points of this operator correspond to the positive solutions of (49). Our primary results hinge on the application of the Guo–Krasnosel’skii fixed-point theorem (in Theorem 1), the Leggett–Williams fixed-point theorem (for Theorem 2), the Schauder fixed-point theorem (in the case of Theorem 3), and the Banach contraction mapping principle (in the context of Theorem 4). In the second-to-last section of the paper, we provide several examples to elucidate and illustrate the implications of our results.